On the potential role of lateral connectivity in retinal anticipation

Selma Souihel; Bruno Cessac

doi:10.1186/s13408-020-00101-z

On the potential role of lateral connectivity in retinal anticipation

Souihel, Selma; Cessac, Bruno 2021-01-09 00:00:00 Biovision Team and Neuromod We analyse the potential eﬀects of lateral connectivity (amacrine cells and gap Institute, Inria, Université Côte d’Azur, Nice, France junctions) on motion anticipation in the retina. Our main result is that lateral connectivity can—under conditions analysed in the paper—trigger a wave of activity enhancing the anticipation mechanism provided by local gain control (Berry et al. in Nature 398(6725):334–338, 1999; Chen et al. in J. Neurosci. 33(1):120–132, 2013). We illustrate these predictions by two examples studied in the experimental literature: diﬀerential motion sensitive cells (Baccus and Meister in Neuron 36(5):909–919, 2002) and direction sensitive cells where direction sensitivity is inherited from asymmetry in gap junctions connectivity (Trenholm et al. in Nat. Neurosci. 16:154–156, 2013). We ﬁnally present reconstructions of retinal responses to 2D visual inputs to assess the ability of our model to anticipate motion in the case of three diﬀerent 2D stimuli. Keywords: Retina; Motion anticipation; Lateral connectivity; 2D 1 Introduction Our visual system has to constantly handle moving objects. Static images do not exist for it, as the environment, our body, our head, our eyes are constantly moving. A “computa- tional”, contemporary view, likens the retina to an “encoder”, converting the light photons coming from a visual scene into spike trains sent—via the axons of ganglion cells (GCells) that constitute the optic nerve—to the thalamus, and then to the visual cortex acting as a “decoder”. In this view, comparing the size and the number of neurons in the retina, i.e. about 1 million GCells (humans), to the size, structure and number of neurons in the visual cortex (around 538 million per hemisphere in the human visual cortex [22]), the “en- coder” has to be quite smart to eﬃciently compress the visual information coming from a world made of moving objects. Although it has long been thought that the retina was not more than a simple camera, there is more and more evidence that this organ is “smarter than neuroscientists believed” [41]. It is indeed able to perform complex tasks and general motion feature extractions, such as approaching motion, diﬀerential motion and motion anticipation, allowing the visual cortex to process visual stimuli with more eﬃciency. Computations occurring in neuronal networks downstream photoreceptors are crucial to make sense out of the “motion-agnostic” signal delivered by these retinal sensors [11]. There exists a wide range of theoretical and biological approaches to studying retinal pro- cessing of motion. Moving stimuli are generally considered as a spatiotemporal pattern of © The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 2 of 60 light intensity projected on the retina, from which it extracts relevant information, such as the direction of image motion. Detecting motion requires then neural networks to be able to process, in a nonlinear fashion, moving stimuli, asymmetrically in time. [6, 10][37, 88] The ﬁrst motion detector model was proposed by Hassenstein and Reichardt [42], based on the optomotor behaviour of insects. The model relies on changes in contrast of two spatially distinct locations inside the receptive ﬁeld of a motion sensitive neuron. The neuron will only produce a response if contrast changes are temporally delayed, and is thus not only selective to direction, but also to motion velocity. Several models have then been developed based on Reichardt detectors, attempting to explain motion processing in vertebrate species as well, but they all share the common feature of integrating spatio- temporal variations of the contrast. To perceive motion as coherent and uninterrupted, an additional integration over motion detectors is hypothesised to take place. This inte- gration usually takes the form of a pooling mechanism over visual space and time. However, before performing these high-level motion detection computations, the pho- tons received by the retina need ﬁrst to be converted into electrical signals that will be transmitted to the visual cortex, a process known as phototransduction. This process takes about 30–100 milliseconds. Though this might look fast, it is actually too slow. A tennis ball moving at 30 m/s–108 km/h (the maximum measured speed is about 250 km/h) cov- ers between 0.9 and 3 m during this time. So, without a mechanism compensating this delay, it would not be possible to play tennis (not to speak of survival, a necessary con- dition for a species to reach the level where playing tennis becomes possible). The visual system is indeed able to extrapolate the trajectory of a moving object to perceive it at its actual location. This corresponds to anticipation mechanisms taking place in the visual cortex and in the retina with diﬀerent modalities [4, 56, 57, 84]. In the early visual cortex an object moving across the visual ﬁeld triggers a wave of ac- tivity ahead of motion thanks to the cortical lateral connectivity [8, 46, 77]. Jancke et al. [46] ﬁrst demonstrated the existence of anticipatory mechanisms in the cat primary vi- sual cortex. They recorded cells in the central visual ﬁeld of area 17 (corresponding to the primary visual cortex) of anaesthetised cats responding to small squares of light either ﬂashed or moving in diﬀerent directions and with diﬀerent speeds. When presented with the moving stimulus, cells show a reduction of neural latencies, as compared to the ﬂashed stimulus. Subramaniyan et al. [77] reported the existence of similar anticipatory eﬀects in the macaque primary visual cortex, showing that a moving bar is processed faster than a ﬂashed bar. They give two possible explanations to this phenomenon: either a shift in the cells receptive ﬁelds induced by motion, or a faster propagation of motion signals as compared to the ﬂash signal. In the retina, anticipation takes a diﬀerent form. One observes a peak in the ﬁring rate response of GCells to a moving object occurring before thepeakresponsetothe same object when ﬂashed. This eﬀect can be explained by purely local mechanisms at individual cells level [9, 20]. To our best knowledge, collective eﬀects similar to the cortical ones, that is, a rise in the cell’s activity before the object enters in its receptive ﬁeld due to a wave of activity ahead of the moving object, have not been reported yet. In a classical Hubel–Wiezel–Barlow [5, 44, 60] view of vision, each retinal ganglion cell carries a ﬂow of information with an eﬃcient coding strategy maximising the available channel capacity by minimising the redundancy between GCells. From this point of view, the most eﬃcient coding is provided when GCells are independent encoders (parallel Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 3 of 60 Figure 1 Synthetic view of the retina model. A stimulus is perceived by the retina, triggering diﬀerent pathways. Pathway I (blue) corresponds to a feed-forward response where, from top to bottom: The stimulus is ﬁrst convolved with a spatio-temporal receptive ﬁeld that mimics the outer plexiform layer (OPL) (“Bipolar receptive ﬁeld response”). This response is rectiﬁed by low voltage threshold (blue squares). Bipolar cell responses are then pooled (blue circles with blue arrows) and input ganglion cells. The ﬁring rate response of a ganglion cell is a sigmoidal function of the voltage (blue square). Gain control can be applied at the bipolar and ganglion cell level (pink circles) triggering anticipation. This corresponds to the label II (pink)inthe ﬁgure. Lateral connectivity is featured by pathway III (brown) through ACells and pathway IV (green)through gap-junctions at the level of GCells streaming identiﬁed by a “I” in Fig. 1). In this setting one can propose a simple and satis- factory mechanism explaining anticipation in the retina based on gain control at the level of bipolar cells (BCells) and GCells (label “II” in Fig. 1)[9, 20]. Yet, some GCells are connected. Either directly, by electric synapses–gap junctions (pathway IV in Fig. 1), or indirectly via speciﬁc amacrine cells (ACells, pathway III in Fig. 1). It is known that these pathways are involved in motion processing by the retina. AII ACells play a fundamental role in the interaction between the ON and OFF cone path- way [54]. There are GCells able to detect the diﬀerential motion of an object onto a mov- ing background [1] thanks to ACell lateral connectivity. Some GCells are direction sensi- tive because they are connected via a speciﬁc asymmetric gap junctions connectivity [80]. Could lateral connectivity play a role in motion anticipation, inducing a wave of activity ahead of the motion similar to the cortical anticipation mechanism? Whilesomestud- ies hypothesise that local gain control mechanisms can be explained by the prevalence of inhibition in the retinal connectome [47], the mechanistic aspects of the role of lateral connectivity on motion anticipation has not, to the best of our knowledge, been addressed yet on either experimental or computational grounds. In this paper, we address this question from a modeler, computational neuroscientist, point of view. We propose here a simpliﬁed description of pathways I, II, III, IV of Fig. 1, grounded on biology, but not sticking at it, to numerically study the potential eﬀects of gain Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 4 of 60 control combined with lateral connectivity—gap junctions or ACells—on motion antici- pation. The goal here is not to be biologically realistic but, instead, to propose from bi- ological observations potential mechanisms enhancing the retina’s capacity to anticipate motion and compensate the delay introduced by photo-transduction and feed-forward processing in the cortical response. We want the mechanisms to be as generic as possible, so that the detailed biological implementation is not essential. This has the advantage of making the model more prone to mathematical analysis. The ﬁrst contribution of our work lies in the development of a model of retinal antici- pation where GCells have gain control, orientation selectivity and are laterally connected. It is based on a model introduced by Chen et al. in [20] (itself based on [9]) reproduc- ing several motion processing features: anticipation, alert response to motion onset and motion reversal. The original model handles one-dimensional motions and its cells are not laterally connected (only pathways I and II were considered). The extension proposed here features cells with oriented receptive ﬁeld, although our numerical simulations do not consider this case (see discussion). Lateral connectivity is based on biophysical mod- elling and existing literature [1, 28, 43, 78, 80]. In this framework, we study diﬀerent types of motion. We start with a bar moving with constant speed and study the eﬀect of con- trast, bar size and speed on anticipation, generalising previous studies by Berry et al. [9] and Chen et al. [20]. We then extend the analysis to two-dimensional motions, investigat- ing, e.g. angular motion and curved trajectories. Far from making an exhaustive study of anticipation in complex stimuli, the goal here is to calibrate anticipation without lateral connectivity so as to compare the eﬀect when connectivity is switched on. The second contribution emphasises a potential role of lateral connectivity (gap junc- tions and ACells) on anticipation. For this, we ﬁrst make a general mathematical analysis concluding that lateral connectivity can induce a wave triggered by the stimulus which un- der speciﬁc conditions can improve anticipation. The eﬀect depends on the connectivity graph and is nonlinearly tuned by gain control. In the case of gap junctions, the wave prop- agation depends on whether connectivity is symmetric (the standard case) or asymmetric, as proposed by Trenholm et al. in [80] for a speciﬁc type of direction sensitive GCells. In the case of ACells, the connectivity graph is involved in the spectrum of a propagation operator controlling the time evolution of the network response to a moving stimulus. We instantiate this general analysis by studying diﬀerential motion sensitive cells [1]with two types of connectivity: nearest neighbours, and a random connectivity, inspired from biology [78], where only numerical results are shown. In general, the anticipation eﬀect depends on the connectivity graph structure and the intensity of coupling between cells as well as on the respective characteristic times of response of cells in a way that we analyse mathematically and illustrate numerically. We actually observe two forms of anticipation. The ﬁrst one, discussed in the beginning of this introduction and already observed in [9, 20], is a shift in the peak of a retinal GCell response occurring before the object reaches the centre of its receptive ﬁeld. In our case, lateral connectivity can enhance the shift improving the mere eﬀect of gain control. The second anticipation eﬀect we observe is a raise in GCells activity before the bar reaches the receptive ﬁeld of the cell, similarly to what is observed in the cortex [8]. To the best of our knowledge, this eﬀect has not been studied in the retina and constitutes therefore a prediction of our model. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 5 of 60 The paper is organised as follows. Section 2 introduces the model of retinal organisa- tion and cell types dynamics, ending up with a system of nonlinear diﬀerential equations driven by a time-dependent stimulus. Section 3 is divided into four parts. The ﬁrst part analyses mathematically the potential anticipation eﬀects in a general setting before con- sidering the role of ACells and lateral inhibition on anticipation (Sect. 3.2)and gapjunc- tions (Sect. 3.3). Both sections contain general mathematical results as well as numerical simulations for one-dimensional motion. The fourth part investigates examples of two- dimensional motions. The last section is devoted to discussion and conclusion. In Ap- pendix A, we have added the values of parameters used in simulations and in Appendix B the receptive ﬁelds mathematical form used in the paper as well as the numerical method to compute eﬃciently the response of oriented two-dimensional receptive ﬁelds to spatio- temporal stimuli. Appendix C presents a model of random connectivity from amacrine to bipolar cells inspired from biological data [78]. Finally, Appendix D contains mathemati- cal results which constitute the skeleton of the work, but whose proof would be too long to integrate in the core of the paper. This work is based on Selma Souihel’s PhD thesis where more extensive results can be found [74]. In particular, there is an analysis of the conjugated eﬀects of retinal and cortical anticipation, subject of a forthcoming paper and brieﬂy discussed in the conclusion. In all the following simulations, we use the CImg Library, an open-source C++ tool kit for image processing in order to load the stimuli and reconstruct the retina activity. The source code is available on demand. 2 Material and methods 2.1 Retinal organisation In the retinal processing light photons coming from a visual scene are converted into volt- age variations by photoreceptors (cones and rods). The complex hierarchical and layered structure of the retina allows to convert these variations into spike trains, produced by ganglion cells (GCells) and conveyed to the thalamus via their axons. We considerably sim- plify this process here. Light response induces a voltage variations of bipolar cells (BCells), laterally connected via amacrine cells (ACells), and feeding GCells, as depicted in Fig. 1. We describe this structure in details here. Note that neither BCells nor ACells are spiking. They act synaptically on each other by graded variations of their potential. We assimilate the retina to a ﬂat, two-dimensional square of edge length L mm. There- fore, we do not integrate the three-dimensional structure of the retina in the model, merely for mathematical convenience. Spatial coordinates are noted x, y (see Fig. 2 for the whole structure). In the model, each cell population tiles the retina with a regular square lattice. The den- sity of cells is therefore uniform for convenience but the extension to nonuniform density canbeaﬀorded.For thepopulation p,wedenoteby δ the lattice spacing in mm and by N p p the total number of cells. Without loss of generality we assume that L, the retina’s edge size, is a multiple of δ .Wedenoteby L = thenumberofcells p per row or column so that p p N = L . Each cell in the population p has thus Cartesian coordinates (x, y)=(i δ , i δ ), p x p y p (i , i ) ∈{1,..., L } . To avoid multiples indices, we associate with each pair (i , i ) a unique x y p x y index i = i +(i –1)L . The cell of population p located at coordinates (i δ , i δ )is then x y p x p y p denoted by p .Wedenoteby d[p , p ] the Euclidean distance between p and p . i i i j j Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 6 of 60 Figure 2 Example of a retina grid tiling and indexing. The green and blue ellipses denote respectively the positive centre and the negative surround of the BCell receptive ﬁeld K . The centre of RF coincides with the position of the cell (blue and green arrows). The red ellipse denotes the ganglion cell pooling over bipolar cells (Eq. (17)) We use the notation V for the membrane potential of cell p . Cells are coupled. The p i synaptic weight from cell p to cell q reads W . Thus, the pre-synaptic neuron is ex- j i pressed in the upper index; the post-synaptic, in the lower index. Dynamics of cells is voltage-based. This is because our model is constructed from Chen et al. model [20], itself derived from Berry et al. [9], where a voltage-based description is used. Implicitly, volt- age is measured with respect to the rest state of the cell (V = 0 when the cell receives no input). 2.2 Bipolar cell layer The model consists ﬁrst of a set of N BCells, regularly spaced by a distance δ ,withspatial B B coordinates x , y , i = 1,..., N . Their voltage, a function of the stimulus, is computed as i i B follows. 2.2.1 Stimulus response and receptive ﬁeld The projection of the visual scene on the retina (“stimulus”) is a function S(x, y, t)where t is the time coordinate. As we do not consider colour sensitivity here, S characterises a black and white scene with a control on the level of contrast ∈ [0, 1]. A receptive ﬁeld (RF) is a region of the visual ﬁeld (the physical space) in which stimulation alters the voltage of a cell. Thus, BCell i has a spatio-temporal receptive ﬁeld K , featuring the biophysical processes occurring at the level of the outer plexiform layer (OPL), that is, photo-receptors (rod-cones) response modulated by horizontal cells (HCells). As a consequence, in our model, the voltage of BCell i is stimulus-driven by the term +∞ +∞ t x,y,t V (t)=[K ∗ S](t)= K(x–x , y–y , t –s)S(x, y, s) dx dy ds,(1) i B i i drive i x=–∞ y=–∞ s=–∞ Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 7 of 60 x,y,t where ∗ means space-time convolution. We consider only one family of BCells so that the kernel K is the same for all BCells. What changes is the centre of the RF, located at x , y , which also corresponds to the coordinates of the BCell i. We consider in the paper sep- arable kernel K(x, y, t)= K (x, y)K (t)where K is the spatial part and K is the temporal S T S T part. The detailed form of K is given in Appendix B. We have dV x,y,t dS drive = K ∗ (t), (2) dt dt resulting from the condition K (x, y,0) = 0 (see Appendix B). Note that the exponential decay of the spatial and temporal part at inﬁnity ensures the existence of the space-time integral. The spatial integral K (x, y)S(x, y, u) dx dy is numerically computed using er- 2 S ror function in the case of circular RF and a computer vision method from Geusenroek et al. [39] in the case of anisotropic RF, allowing to integrate generalised Gaussians with an eﬃcient computational time. This method is described in Appendix B. For explanations purposes, we will often use the approximation of V by a Gaussian drive pulse, with width σ , propagating at constant speed v along the direction e : 2 2 (x–vt) (x–vt) 1 1 A V 0 – 0 – 2 2 2 2 σ σ V (t)= √ e ≡ √ e,(3) drive 2πσ 2π where x = iδ is the horizontal coordinate of BCell i and where σ is in mm, A is in mV.mm B 0 (and is proportional to stimulus contrast), V is in mV. 2.2.2 BCells voltage and gain control In our model, the BCell voltage is the sum of external drive (1)receivedbythe BCelland of a post-synaptic potential P induced by connected ACells: V (t)= V (t)+ P (t). (4) B i B i drive i The form of P is given by Eq. (11)inSect. 2.3.1. P (t) = 0 when no ACells are considered. B B i i BCells have voltage threshold [9]: 0, if V ≤ θ ; B B N (V )= (5) B B V – θ ,else. B B Values of parameters are given in Appendix A. BCells have gain control, a desensitisation when activated by a steady illumination [92]. 2+ This desensitisation is mediated by a rise in intracellular calcium Ca at the origin of a feedback inhibition preventing thus prolonged signalling of the ON BCell [20, 73]. Fol- lowing Chen et al., we introduce the dimensionless activity variable A obeying the dif- ferential equation dA A B B i i =– + h N V (t).(6) B B dt τ a Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 8 of 60 Assuming an initial condition A (t ) = 0 at initial time t ,the solution is B 0 0 t–s A (t)= h e N V (s) ds.(7) B B B i i The bipolar output to ACells and GCells is then characterised by a nonlinear response to its voltage variation given by R (V , A )= N (V )G (A ), (8) B B B B B B B i i i i i where 0, if A ≤ 0; G (A )= (9) B B i 1 ,else. 1+A Note that R has the physical dimension of a voltage, whereas, from Eq. (9), the activity A is dimensionless. As a consequence, the parameter h in Eq. (6)mustbeexpressed in B B –1 –1 ms mV .Form(9) and its 6th power are based on experimental ﬁts made by Chen et al. Itsformisshown in Fig. 3. In the course of the paper we will use the following piecewise linear approximation also represented in Fig. 3: 0, if A ∈]– ∞,0[ ∪ [ ,+∞[, silent region; G (A)= (10) 1, if A ∈ [0, ], maximal gain; 3 2 4 – A +2, if A ∈ [ , ], fast decay. 2 3 3 Thanks to this approximation, we roughly distinguish three regions for the gain function G (A). This shape is useful to understand the mechanism of anticipation (Sect. 3.1). Figure 3 Gain control (9) as a function of activity A. The function l(A), in dashed line, is a piecewise linear approximation of G (A) from which three regions are roughly deﬁned. In the region “Silent” the gain vanishes so the cell does not respond to stimuli; in the region “Max”, the gain is maximal so that cell behaviour does not show any diﬀerence with a not gain-controlled cell; the region “Fast decay” is the one which contributes to anticipation by shifting the peak in the cell’s activity (see Sect. 3.1). The value A = corresponds to the value of activity where gain control, in the piecewise linear approximation, becomes eﬀective Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 9 of 60 2.3 Amacrine cell layer There is a wide variety of ACells (about 30–40 diﬀerent types for humans) [64]. Some spe- ciﬁc types are well studied such as starburst amacrine cells, which are involved in direction sensitivity [33, 34, 81], as well as contrast impression and suppression of GCells response [58], or AII, a central element of the vertebrate rod-cone pathway [54]. Here, we do not want to consider speciﬁc types of ACells with a detailed biophysical description. Instead, we want to point out the potential role they can play in motion an- ticipation thanks to the inhibitory lateral connectivity they induce. We focus on a speciﬁc circuitry involved in diﬀerential motion: an object with a diﬀerent motion from its back- ground induces more salient activity. The mechanism, observed in mice and rabbit retinas [41, 61], is featured in Fig. 1, pathway III. When the left pathway receives a diﬀerent illumi- nation from the right pathway (corresponding, e.g. to a moving object), this asymmetry is ampliﬁed by the ACells’ mutual inhibition, enhancing the response of the left pathway in a “push–pull” eﬀect. We want to propose that such a mutual inhibition circuit, deployed in a lattice through the whole retina, can generate—under speciﬁc conditions mathematically analysed—a wave of activity propagation triggered by the moving object. In the model, ACells tile the retina with a lattice spacing δ . We index them with j = 1,..., N . 2.3.1 Synaptic connections between ACells and BCells We consider here a simple model of ACells. We assimilate them to passive cells (no active ionic channels) acting as a simple relay between BCells. This aspect is further discussed later in the paper. The ACell A , connected to the BCell B , induces on the latter the post- j i synaptic potential: A A j j – P (t)= W (t) γ (t – s)V (s) ds; γ (t)= e H(t), B A B B B j i i –∞ where the Heaviside function H ensures causality. Thus, the post synaptic potential is the mere convolution of the pre synaptic ACell voltage, with an exponential α-proﬁle [28]. In addition, we assume the propagation to be instantaneous. Here, the synaptic weight W < 0 mimics the inhibitory connection from ACell to BCell (glycine or GABA) with the convention that W = 0 if there is no connection from A to B . In general, several ACells input the BCell B giving a total PSP: B t P (t)= W γ (t – s)V (s) ds. (11) B B A i B j –∞ j=1 Conversely, the BCell B connected to A induces, on this cell, a synaptic response char- i j acterised by a post-synaptic potential (PSP) P (t). As ACells are passive elements, their voltage V (t) is equal to this PSP. We have thus V (t)= W γ (t – s)R (s) ds (12) A A B j A i –∞ i=1 Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 10 of 60 with γ (t)= e H(t). Here, W > 0 corresponding to the excitatory eﬀect of BCells on ACells through a glutamate release. Note that the voltage of the BCell is rectiﬁed and gain- controlled. 2.3.2 Dynamics The coupled dynamics of bipolar and amacrine cells can be described by a dynamical system that we derive now. Bipolar voltage By diﬀerentiating (11), (4) and introducing x,y,t V dV S dS i i drive drive F (t)= K ∗ + (t)= + , (13) B B i i τ dt τ dt B B we end up with the following equation for the bipolar voltage: dV 1 B j =– V + W V + F (t), (14) B A B i B j i dt τ j=1 wherewehaveused(2). This is a diﬀerential equation driven by the time-dependent term F containing the stimulus and its time derivative. To illustrate the role of F , let us consider an object moving with a speed v depending dv on time, thus with a nonzero acceleration γ = .Thisstimulushas theform S(x, y, t)= dt dS g(X – v(t)t)with X = ,sothat =–∇g(X – v(t)t).(v + γ t), where ∇ denotes the gra- dt dient. Therefore, thanks to Eq. (14), BCells are sensitive to changes in directions, thereby justifying a study of two-dimensional stimuli (Sect. 3.4). Note that this property is in- dV drive herited from the simple, diﬀerential structure of the dynamics, the term resulting dt from the diﬀerentiation of V . This term does not appear in the classical formulation (1) of the bipolar response without amacrine connectivity. It appears here because synaptic response involves an implicit time derivative via convolution (12). Coupled dynamics Likewise, diﬀerentiating (12)gives dV A 1 j B =– V + W R . (15) A B j A i dt τ i=1 Equation (6) (activity), (14)and(15)deﬁneasetof 2N +N diﬀerential equations, ruling B A the behaviour of coupled BCells and ACells, under the drive of the stimulus, appearing in the term F (t). We summarise the diﬀerential system here: dV A B N j 1 A =– V + W V + F (t), B A B dt τ i j=1 B j i ⎪ B dV j N B 1 B i (16) =– V + W R , A B j i=1 i dt τ A ⎪ A ⎩ dA A B B i i =– + h N (V ). B B dt τ We have used the classical dynamical systems convention where time appears explicitly only in the driving term F (t)to emphasise that (16) is non-autonomous. Note that BCells act on ACells via a rectiﬁed voltage (gain control and piecewise linear rectiﬁcation), in agreement with Fig. 1,pathway III.Weanalyse this dynamics in Sect. 3.2.1. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 11 of 60 2.3.3 Connectivity graph The way ACells connect to BCells and reciprocally have a deep impact on dynamics (16). In this paper, we want to point out the role of relative excitation (from BCells to ACells) and inhibition (from ACells to BCells) as well as the role of the network topology. For math- ematical convenience when dealing with square matrices, we assume from now on that there are as many BCell as ACells, and we set N ≡ N = N . At the core of our mathemati- A B calstudiesisamatrix L,deﬁnedinSect. 3.2.1, whose spectrum conditions the evolution of the BCells–ACells network under the inﬂuence of a stimulus. It is interesting and relevant to relate the spectrum of L to the spectrum of the connectivity matrices ACells to BCells and BCells to ACells. There is not such a general relation for arbitrary matrices of connec- tivity. A simple case holds when the two connectivity matrices commute. Here, we choose an even simpler situation based on the fact that we compare the role of the direct feed- forward pathway on anticipation in the presence of ACell lateral connectivity. We feature the direct pathway by assuming that a BCell connects only one ACell with a weight w uni- B + + form for all BCell, so that W = w I , w >0, where I is the N-dimensional identity N,N N,N matrix. In contrast, we assume that ACells connect to BCells with a connectivity matrix – – A – W, not necessarily symmetric, with a uniform weight –w , w >0, so that W =–w W. We consider then two types of network topology for W: 1. Nearest neighbours. An ACell connects its 2d nearest BCell neighbours where d =1,2 is the lattice dimension. 2. Random ACell connectivity. This model is inspired from the paper [78]onthe shape and arrangement of starburst ACells in the rabbit retina. Each cell (ACell and BCell) has a random number of branches (dendritic tree), each of which has a random length and a random angle with respect to the horizontal axis. The length of branches L follows an exponential distribution with spatial scale ξ.The number of branches n is also a random variable, Gaussian with mean n ¯ and variance σ .The angle distribution is taken to be isotropic in the plane, i.e. uniform on [0, 2π[.Whena branch of an ACell A intersects a branch of a BCell B, there is a chemical synapse from A to B. The probability that two branches intersect follows a nearly exponential probability distribution that can be analytically computed (see Appendix C). 2.4 Ganglion cells There are many diﬀerent types of GCells in the retina, with diﬀerent physiologies and func- tions [3, 70]. In the present computational study, we focus on speciﬁc subtypes associated with pathways I–II (fast OFF cells with gain control), III (diﬀerential motion sensitive cells) and IV (direction selective cells) in Fig. 1.All thesehavecommon features: BCells pooling and gain control. 2.4.1 BCells pooling In theretina, GCells of thesametypecover thesurface of theretina, formingamosaic. The degree of overlap between GCells indicates the extent to which their dendritic arbours are entangled in one another. This overlap remains, however, very limited between cells of the same type [68]. We denote by k the index of the GCells, k = 1,..., N ,and by δ the G G spacing between two consecutive GCells lying on the grid (Fig. 2). Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 12 of 60 In the model, GCell k pools over the output of BCells in its neighbourhood [20]. Its voltage reads as follows: (P) B V = W R , (17) G G i k k where the superscript “P” stands for “pool”. We use this notation to diﬀerentiate this volt- age from the total GCell voltage V when they are diﬀerent. This happens in the case when GCells are directly coupled by gap junctions (Sects. 2.4.4, 3.3). When there is no ambiguity, we will drop the superscript “P”. In Eq. (17), the weights W are Gaussian: d [B ,G ] i k B 2σ W = a e , (18) where σ has the dimension of a distance and a is dimensionless. p p 2.4.2 Ganglion cell response The voltage V is processed through a gain control loop similar to the BCell layer [20]. As GCells are spiking cells, a nonlinearity is ﬁxed so as to impose an upper limit over the ﬁringrate. Here,itismodelledbyasigmoidfunction, e.g. 0, if V ≤ θ ; ⎪ G max N (V)= (19) α (V – θ ), if θ ≤ V ≤ N /α + θ ; G G G G G max N ,else. This function corresponds to a probability of ﬁring in a time interval. Thus, it is expressed –1 max in Hz. Consequently, α is expressed in Hz mV and N in Hz. Parameter values can be found in Appendix A. Gain control is implemented with an activation function A , solving the following dif- ferential equation: dA A G G k k =– + h N (V ), (20) G G G dt τ and a gain function 0, if A <0; G (A)= (21) ,else. 1+A Note that the origin of this gain control is diﬀerent from the BCell gain control (9). Indeed, Chen et al. hypothesise that the biophysical mechanisms that could lie behind ganglion gain control are spike-dependent inactivation of Na and K channels, while the study by Jacoby et al. [45] hypothesises that GCells gain control is mediated by feed- forward inhibition that they receive from ACells. The speciﬁc forms of the nonlinear- ity and the gain control function used in this paper match, however, the ﬁrst hypothesis, namely the suppression of the Na current [20]. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 13 of 60 Finally, the response function of this GCell type is R (V , A )= N (V )G (A ). (22) G G G G G G G k k k k In contrast to BCell response R (8), which is a voltage, here R is a ﬁring rate. B G Gain control has been reported for OFF GCells only [9, 20]. Therefore, we restrict our study to OFF cells, i.e with a negative centre of the spatial RF kernel. However, on mathe- matical grounds, it is easier to carry our explanation when the RF centre is positive. Thus, forconvenience,wehaveadopted achangeinconventioninterms of contrast measure- ment. We take the reference value 0 of the stimulus to be white rather than black, black corresponding then to 1. The spatial RF kernel is also inverted, with a positive centre and a negative surround. The problem is therefore mathematically equivalent to an ON cell submitted to positive stimulus. 2.4.3 Diﬀerential motion sensitive cells We consider here a class of GCells connected to ACells according to pathways III in Fig. 1, acting as diﬀerential motion detectors. They are able to respond saliently to an object moving over a stationary surround while being strongly inhibited by global motion. Here, stationary is meant in a general, probabilistic sense. This can be a uniform background or a noisy background where the probability distribution of the noise is time-translation invariant. These cells are hence able to ﬁlter head and eye movements. Baccus et al. [1] emphasised a pathway accountable for this type of response involving polyaxonal ACells which selectively suppress GCells response to global motion and enhance their response to diﬀerential motion, as shown in Fig. 1,pathway III.The GCellreceivesanexcitatoryinput from the BCells lying in its receptive ﬁeld which respond to the central object motion and an indirect inhibitory input from ACells that are connected to BCells which respond to the background motion. When motion is global, the excitatory signal is equivalent to the inhibitory one, resulting in an overall suppression. However, when the object in the cen- tre moves distinctively from the surrounding background, the cell in the centre responds strongly. There are here two concomitant eﬀects. When a moving object (say, from left to right) enters the BCell pool connected to a central GCell k , the BCells in the periphery of the pool respond ﬁrst, with no signiﬁcant change on the GCell response, because of the Gaus- sian shape (18) of the pooling: weights are small in the periphery. Those BCells excite, however, the ACells they are connected to, with the eﬀect of inhibiting the BCells of neigh- bouring GCells pools. This has the eﬀect of decreasing the voltage of these BCells which in turn excite less ACells which, in turn, inhibit less the BCells of the pool k .Thus, the response of the GCell k is enhanced, while the cells on the background are inhibited. We call this eﬀect “push–pull” eﬀect. Note that propagation delays ought to play an important role here, although we are not going to consider them in this paper. 2.4.4 Direction selective GCells and gap junction connectivity These cells correspond to pathway IV in Fig. 1. They are only coupled via electric synapses (gap junctions). In several animals, like the mouse, this enables the corresponding GCells to be direction sensitive. Note that other mechanisms, involving lateral inhibition via star- burst amacrine cells have also been widely reported [33, 34, 71, 72, 81, 85, 88]. Here we Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 14 of 60 focus on gap junctions direction sensitive cells (DSGCs). There exist four major types of these DSGCs, each responding to edges moving in one of the four cardinal directions. Trenholm et al. [80] emphasised the role of these cells coupling in lag normalisation: un- coupled cells begin responding when a bar enters their receptive ﬁeld, i.e. their dendritic ﬁeld extension, whereas coupled cells start responding before the bar reaches their den- dritic ﬁeld. This anticipated response is due to the eﬀective propagation of activity from neighbouring cells through gap junctions and is particularly interesting when comparing the responses for diﬀerent velocities of the bar. Trenholm et al. showed that the uncou- pled DSGCs detect the bar at a position which is further shifted as the velocity grows, while coupled cells respond at an almost constant position regardless of the velocity. In our work, we analyse this eﬀect in terms of a propagating wave driven by the stimulus and show that temporally this spatial lag normalisation induces a motion extrapolation that confers to the retina more than just the ability to compensate for processing delays, but to anticipate motion. Classical, symmetric bidirectional gap junctions coupling between neighbouring cells would involve a current of the form –g(V – V )– g(V – V ), where g is the gap G G G G k k–1 k k+1 junction conductance. In contrast, here, the current takes the form –g(V – V ). This G G k k–1 is due to the speciﬁc asymmetric structure of the direction selective GCell dendritic tree [80]. The experimental results of these authors suggest that the eﬀect of the possible gap junction input from downstream cells, in the direction of motion, can be neglected due to oﬀset inhibition and gain control suppression. This, along with the asymmetry of the dendritic arbour, justiﬁes the approximation whereby the cell k+1 does not inﬂuence the current in the cell k. This induces a strong diﬀerence in the propagation of a perturbation. Indeed, consider the case V – V = V – V = δ. In the symmetric form the total G G G G k k–1 k k+1 current vanishes, whereas in the asymmetric form the current is –gδ. Still, the current can have both directions depending on the sign of δ. This has a strong consequence on the way GCells connected by gap junctions respond to a propagating stimulus, as shown in Sect. 3.3. (P) The total GCell voltage is the sum of the pooled BCell voltage V and of the eﬀect of neighbours GCells connected to k by gap junctions: (P) V (t)= V – V (s)– V (s) ds, G G G k k k–1 –∞ where C is the membrane capacitance. Deriving the previous equation with respect to time, we obtain the following diﬀerential equation governing the GCell voltage: (P) dV dV G G k k = – w V (t)– V (t) , (23) gap G G k k–1 dt dt where w = . (24) gap Gain control is then applied on V as in (22). An alternative is to consider that gain control occurs before gap junctions eﬀect. We investigated this eﬀect as well (not shown, see [74]). Mainly, the anticipatory eﬀect is enhanced when the gain control is applied after the gap junction coupling; thus, from now, we focus on the formulation (23)inthe paper. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 15 of 60 Note that our voltage-based model of gap junctions takes a diﬀerent from as Trenholm et al. (expressed in terms of currents), because we had to adapt it so as to deal with the pooling voltage form (17). Still, our model reproduces the lag normalisation as in the orig- inal model as we checked (not shown, see [74]). 3Results 3.1 The mechanism of motion anticipation and the role of gain control The (smooth) trajectory of a moving object can be extrapolated from its past position and velocity to obtain an estimate of its current location [4, 56, 57]. When human subjects are shown a moving bar travelling at constant velocity, while a second bar is brieﬂy ﬂashed in alignment with the moving bar, the subjects report seeing the ﬂashed bar trailing behind the moving bar. This led Berry et al. [9] to investigate the potential role of the retina in anticipation mechanisms. Under constraints on the bar speed and contrast they were able to exhibit a positive anticipation time, deﬁned as the time lag between the peak in the retinal GCell response to a ﬂashed bar and the corresponding peak when the stimulus is a moving bar. In this paper we adopt a slightly diﬀerent deﬁnition although inspired by it. Indeed, the goal of this modelling paper is to dissect the various potential stages of retinal anticipation as developed in the next subsections. Several layers and mechanisms are involved in the model, each one deﬁning a response time and potentially contributing to anticipation under conditions that we now analyse. 3.1.1 Anticipation at the level of a single isolated BCell; the local eﬀect of gain control We consider ﬁrst a single BCell without lateral connectivity so that V = V .The very B i i drive mechanism of anticipation at this stage is illustrated in Fig. 4. The peak response time of the convolution of the stimulus with the RF of one BCell occurs at a time t (dashed line in Fig. 4(a)). The increase in V leads to an increase in activity (Fig. 4(c)) and an increase drive of R (Fig. 4(e)). When activity becomes large enough, gain control switches on (Fig. 4(d)) leading to a sharp decrease of the response R (Fig. 4(e)) and a peak in R occurring at B B time t (dashed line in Fig. 4(e)) before t .The bipolar anticipation time, = t – t ,is B B B B B A A therefore positive. dV drive Mathematically, > 0 results from the intermediate value theorem using that ≥ dt 0on[0, t ]and that t is deﬁned by B B dV G (A ) dA B B drive B i i =–V (t ) , i B drive A dt G (A ) dt B B t=t i t=t B B A A where the right-hand side is positive provided that the parameters h , τ are tuned such B a dA that ≥ 0on[0, t ]. An important consequence is that the amplitude of the response at dt the peak is smaller in the presence of gain control (compare the amplitude of the voltage in Fig. 4(a) to 4(e)). The anticipation time at the BCell level depends on parameters such as h , τ .Itdepends B a as well on characteristics of the stimulus such as contrast, size and speed. An easy way to ﬁgure this out is to consider that the peak in BCell response (Fig. 4(d), (e)) arises when the gain control function G (A )starts to drop oﬀ (Fig. 4(e)), which from the piecewise B B linear approximation (10)ofBCell arises when A = .When V has the form (3). this drive 3 Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 16 of 60 Figure 4 The mechanism of motion anticipation and the role of gain control. The ﬁgure illustrates the bipolar anticipation time without lateral connectivity. We see the response of OFF BCells with gain control to a dark moving bar. The curves correspond to three cells spaced by 450 μm. The ﬁrst line (a)shows thelinear ﬁltering of the stimulus corresponding to V (t)(Eq. (1)). Line (b) corresponds to the threshold nonlinearity drive N applied to the linear response; (c) represents the adaptation variable (16), and (d) shows the gain control time curse. Finally, the last line (e) corresponds to the response R of the BCell. The two dashed lines correspond respectively to t and t , the peak in the response of the (purple) BCell without pooling B B gives, using N (V )= V (7) and letting the initial time t → –∞ (which corresponds i i 0 drive drive to assuming that the initial state was taken in a distant past, quite longer than the time scales in the model): 1 σ h 1 x – vt σ 2 B 2 2 B 2 (x–vt ) A τ v B τ v a a A A (t )= A e e 1– + = , (25) B B 0 i A v σ τ v 3 where (x) is the cumulative distribution function of the standard Gaussian probability (see deﬁnition, Eq. (60)inAppendix B). This establishes an explicit equation for the time t as a function of contrast (A ), size (σ ) and speed (v) as well as the parameters h and τ . B 0 B a We do not show the corresponding curves here (see [74]for adetailedstudy)preferring to illustrate the global anticipation at the level of GCells, illustrated in Fig. 5 below. 3.1.2 Anticipation time of the BCells pooled voltage The main eﬀects we want to illustrate in the paper (impact of lateral connectivity on GCell anticipation) are evidenced by the shift of the peak in activity of the BCells pooled voltage occurring at time t .Wefocus on this time here,postponingtoSect. 3.1.3 the subsequent eﬀect of GCells gain control. We assume therefore here that h =0 so that A =0 and G G G (A )=1 in (19). Thus, the ﬁring rate of GCell k is N (V ). For mathematical sim- G G G G k k Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 17 of 60 plicity, we will consider that the ﬁring rate function (5)of G is a smooth, monotonously increasing sigmoid function such that N (V )>0. We deﬁne t as thetimewhen V is G G G G k k dV d V G G k k maximum, after the stimulus is switched on. This corresponds to =0 and <0. dt dt Equivalently, from Eqs. (17), (23), we have dR dV dA B B B B B i i i i i W = W G (A )N (V ) + N (V )G (A ) B B B B B B B B G G i i i i k k dt dt dt i i (26) = w [V – V ], gap G G k k–1 where this equation holds at time t = t (we have not written explicitly t to alleviate G G notation). This is the most general equation for the anticipation time at the level of BCells pooling. In the sum , there are two types of BCells. The inactive ones where V ≤ , B B i i dR N (V )=0 and = 0, so they do not contribute to the activity. The active BCells B B dt V > obey N (V )= V . For the moment we assume that, at time t , thereisnoBCell B B B B B G i i i switching from one state (active/inactive) to the other, postponing this case to the end of the section. Then, Eq. (26)reduces to 1 A B j W G (A ) – V + W V + F (t) B B B A B G i i B j i k i i j=1 (II) (I) (V) (III) (27) dA B i =– W G (A ) V (t) + w [V – V ] . B B gap G G G B i i k k–1 dt (II) (IV) (V) This general equation emphasises the respective role of (I), stimulus (term F (t)); (II), gain control (terms G (A ), G (A )); (III), ACell lateral connectivity (term W ); (IV), gap B B B i B i B junctions (term w [V (t )–V (t )]); (V), pooling (terms W ). Note that we could gap G G G G G k k–1 A A k as well consider a symmetric gap junctions connectivity where we would have a term w [–V +2V – V ] in IV. The equation terms have been arranged this way for gap G G G k+1 k k–1 reasons that become clear in the next lines. It is not possible to solve this equation in full generality, but it can be used to understand the respective role of each component. In the absence of gain control and lateral connectivity (W =0, w =0), the peak in gap GCell G voltage at time t is given by dV B i drive W = 0. (28) dt This generalises the deﬁnition of t , time of peak of a single BCell, to a set of pooled BCells, and we will proceed along the same lines as in Sect. 3.1.1. We ﬁx as reference time 0 the time when the pooled voltage becomes positive. It increases then until the time t when dV dV B i B i i i drive drive W =0. Thus, W is positive on [0, t [ and vanishes at t . i G i G G G k dt k dt We now show that, in the presence of gain control, the peak occurs at time t < t leading to anticipation induced by gain control and generalising the eﬀect observed for one BCell Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 18 of 60 in Sect. 3.1.1. Indeed, Eq. (27) reads now as follows: dV dA i B B B i drive i i W G (A ) =– W G (A )V (t) . (29) B B B i G i G B i drive k k dt dt i i dV dV B i B i i drive i drive Because 0 ≤ G (A ) ≤ 1, W G (A ) ≤ W so that the left-hand B B B B i i G i i G dt dt k k side in (29)reaches 0 atatime t ≤ t . The right-hand side is positive for the same reasons as in Sect. 3.1.1. The same mathematical argument holds as well, using the intermediate value theorem to show that t < t . We now investigate Eq. (27) with the two terms of lateral connectivity: (III) ACells and (IV) gap junctions. The eﬀect of gap junctions is straightforward. A positive term w [V – V ] increases the right-hand side of Eq. (27). As developed in Sect. 3.3,this gap G G k k–1 arises when the stimulus propagates in the preferred direction of the cell inducing a wave of activity propagating ahead of the stimulus. In view of the qualitative argument devel- oped above using the intermediate value theorem, this can enhance the anticipation time. This deserves, however, a deeper study developed in Sect. 3.3. N j Theeﬀect of ACells cellsislessevident, asthe term (– V + W V + F (t)) can B A B i j=1 B j i τ i have any sign, so that network eﬀect can either anticipate or delay the ganglion response, as illustrated in several examples in the next section. As we show, this term is in general related to a wave of activity, enhancing or weakening the anticipation eﬀect as shown in Sect. 3.2. Let us ﬁnally discuss what happens when some BCell switches from one state (ac- tive/inactive) to the other (i.e. V = ). In this case, taking into account the deﬁnition B B (5), the derivative N (V )= . Thus, when a BCell reaches the lower threshold, there is B i a big variation in N (V ) thereby leading to a positive contribution in (26) and an addi- B i B N j 1 1 i A tional term W G (A )(– V + W V + F (t)) in the left-hand side of (27), B B B A B i G i i j=1 B j i 2 τ i k B where the sum holds on switching state cells. As we see in section (3.2), this can have an important impact on the anticipation time. 3.1.3 Anticipation time at the GCell level We now show that the ﬁring rate of the GCell k,given by (22), reaches its maximum at a time t < t .From(22), at time t : G G G A A dV V dA G G G k k k = . (30) dt 1+ A dt dV V starts from 0 and increases on the time interval [0, t ], thus is positive on [0, t ] G G G dt dV and vanishes at t .Thus, thereisatime t < t such that increases on [0, t ]and de- G d G d dt creases on [t , t ]. The right-hand side of (30)startsfrom 0 at t =0 and stays strictly pos- d G dA itive until either V vanishes, which occurs for t > t ,oruntil vanishes. We choose G G k dt dA the characteristic time τ and the intensity h in (20)sothat >0 on [0, t ]. Thus, G G G dt V dA dV G G G k k k >0 on [0, t ]. Therefore, in the time interval [t , t ], decreases to 0, while G d G 1+A dt dt V dA G G k k increases from 0. From the intermediate value theorem, these two curves have 1+A dt to intersect at a time t < t . G G A Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 19 of 60 Figure 5 Maximum ﬁring rate and anticipation time variability with stimulus parameters in the gain control layer of the model. Left: contrast (with v = 1 mm/s et size = 90 μm); middle:size(with v = 2 mm/s et contrast = 1); right: speed (with contrast 1 and size = 162 μm) We ﬁnally deﬁne the total anticipation time of a GCell as follows: = t – t , (31) B G c A where t is the peak of the BCell at the centre of the BCells pooling to that GCell. 3.1.4 Anticipation variability: stimulus characteristics In general, depends on gain control, lateral connectivity as well as characteristics of the stimulus such as speed and contrast. This has been shown mathematically in Eq. (25)for a single BCell. Here, we investigate numerically the dependence of the total anticipation time of a GCell when the stimulus is a bar of inﬁnite height, width σ mm, travelling in one dimension at speed v mm/s with contrast C ∈ [0,1]. Resultsare showninFig. 5.This ﬁgure is a calibration later used to compare to the eﬀects induced by lateral connectivity. We ﬁrst observe that anticipation increases with contrast, as it has experimentally been observed [9]. Indeed, increasing the contrast increases V (t) thereby accelerating the drive growth of A so that gain control takes place earlier (Fig. 5(a)). We also notice that an- ticipation increases with the width of the object until a maximum (Fig. 5(b)). Finally, the model shows a decrease in anticipation as a function of velocity, as it was evidenced exper- imentally [9, 47](Fig. 4(c)). Indeed, when the velocity increases, V varies faster than drive the characteristic activation time τ , and the adaptation peak value is lower. Consequently, gain control has a weaker eﬀect and the peak activity is less shifted than when the bar is slow. A large part of these eﬀects can be understood from Eq. (25). Note, however, here that simulation of Fig. 5 takes into account the convolution of a moving bar with the receptive ﬁeld, the pooling eﬀect and gain control at the stage of GCells. In Fig. 5 we also show the evolution of GCells maximum ﬁring rate as a function of the moving bar velocity, contrast and size. We observe that it increases with these parameters, an expected result. 3.2 The potential role of ACell lateral inhibition on anticipation In this section we study the potential eﬀect of ACells (pathway III of Fig. 1)onmotionan- ticipation. We restrict to the case where there are as many BCells as ACells (N = N ≡ N) B A A B so that the matrices W and W are square matrices. We ﬁrst derive general mathemat- B A ical results (for the full derivation, see Appendix D) before considering the two types of connectivity described in Sect. 2.3.3. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 20 of 60 3.2.1 Mathematical study Dynamical system We study mathematically dynamical system (16)thatwewrite in a more convenient form. We use Greek indices α, β, γ = 1,...,3N and deﬁne the state vector X as follows: V , α = i, i = 1,..., N; ⎪ B X = V , α = N + i, i = 1,..., N; A , α =2N + i, i = 1,..., N. Likewise,wedeﬁne thestimulusvector F = F if α = 1,..., N and F = 0 otherwise. Then α B α dynamical system (16) has the general form dX = H(X)+ F(t), (32) dt where H(X ) is a nonlinear function, via the function R (V , A )of Eq. (8), featuring B B B i i i gain control and low voltage threshold. The nonlinear problem can be simpliﬁed using 3N the piecewise linear approximation (10). Indeed, there is a domain of R = V ≥ θ , A ∈ 0, , i = 1,..., N , (33) B B B i i where R (V , A )= V so that (16) is linear and can be written in the form B B B B i i i i dX = L.X + F(t), (34) dt with ⎛ ⎞ N,N A – W 0 N,N ⎜ I ⎟ B N,N L = W – 0 , (35) ⎝ ⎠ N,N N,N h I 0 – B N,N N,N where I is the N × N identity matrix and 0 is the N × N zero matrix. This corre- N,N N,N sponds to intermediate activity, where neither BCells gain control (9) nor low threshold (5) are active. We ﬁrst study this case and describe then what happens when trajectories of (32) get out of this domain, activating low voltage threshold or gain control. The idea of using such a phase space decomposition with piecewise linear approxima- tions has been used in a diﬀerent context by Coombes et al. [23]and in [14, 15, 18]. We consider the evolution of the state vector X (t) from an initial time t . Typically, t is 0 0 a reference time where the network is at rest before the stimulus is applied. So, the initial condition X (t ) will be set to 0 without loss of generality. Linear analysis The general solution of (34)is L(t–s) X (t)= e .F(s) ds. (36) 0 Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 21 of 60 The behaviour of solution (36) depends on the eigenvalues λ , β = 1,...,3N,of L and its eigenvectors P with entries P .The matrix P transforms L in Jordan form (L is not β αβ diagonalizable when h =0, see Appendix D.1). Whatever the form of the connectivity B A matrices W , W ,the N last eigenvalues are always λ =– , β =2N + 1,...,3N. A B In Appendix D.1 we show the following general result (not depending on the speciﬁc B A form of W , W , they just need to be square matrices and to be diagonalizable): A B B B B X (t)= V (t)+ E (t)+ E (t)+ E (t), α = 1,...,3N, (37) α α drive B,α A,α a,α where drive term (1) is extended here to 3N-dimensions with V (t)=0 if α > N.The drive other terms have the following deﬁnition and meaning: N N B –1 λ (t–s) E (t)= + λ P P e V (s) ds, α = 1,..., N, (38) β αβ γ B,α βγ drive B t β=1 γ =1 corresponds to the indirect eﬀect, via the ACell connectivity, of the BCells drive on BCells voltages (i.e. the drive excites BCell i, which acts on BCell j via the ACells network); 2N N B –1 λ (t–s) E (t)= + λ P P e V (s) ds, α = N + 1,...,2N, (39) β αβ γ A,α βγ drive B t β=N+1 γ =1 corresponds to the eﬀect of BCell drive on ACell voltages, and 2N N λ + B –1 B λ (t–s) E (t)= h P P e V (s) ds B α–2N β γ a,α βγ drive λ + β t β=1 γ =1 a (40) 1 1 – + τ τ B a 0 + A (t) , α =2N + 1,...,3N, α–2N λ + corresponds to the eﬀect of the BCells drive on the dynamics of BCell activity variables. The ﬁrst term of (40) corresponds to the action of BCells and ACells on the activity of BCells via lateral connectivity. In the second term t–s A (t)= e V (s) ds (41) α–2N α–2N drive corresponds to the direct eﬀect of the BCell voltage with index α –2N on its activity (see Eq. (7)). To sumup, Eq.(37) describes the direct eﬀect of a time-dependent stimulus (ﬁrst term) and the indirect lateral network eﬀects it induces. The term E (t)is what activates the a,α gain control. In the piecewise linear approximation (10), the BCell i triggers its gain control when its activity E (t)> , α =2N + i. (42) a,α This relation extends the computation made in Sect. 3.1.1 for isolated BCells to the case of a BCell under the inﬂuence of ACells. On this basis, let us now discuss how the network eﬀect inﬂuences the activation of gain control and, thereby, anticipation. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 22 of 60 Figure 6 Front (43) for diﬀerent values of λ (purple) as a function of time for the cell γ =0. All ﬁgures are –1 –1 drawn with v = 2 mm/s; σ =0.2 mm. Top. λ =–0.5 ms ; Bottom. λ =0.5 ms β β The structure of terms (38), (39)(40) is interpreted as follows. The drive (index γ = –1 1,..., N) excites the eigenmodes β = 1,...,3N of L with a weight proportional to P . βγ The mode β in turn excites the variable α = 1,...,3N with a weight proportional to P . The time dependence and the eﬀect of the drive are controlled by the integral αβ λ (t–s) e V (s) ds. For example, when the stimulus has the Gaussian form (3)and t drive cells are spaced with a distance δ so that cell γ is located at x = γδ,wehave, taking t → –∞: 2 2 σ λ β λ 1 β A λ σ 1 0 β λ (t–s) – (γδ–vt) β 2 2 v v e V (s) ds = e e – (γδ – vt) , (43) drive v v σ –∞ where (x) is the cumulative distribution function of the standard Gaussian probabil- ity (see deﬁnition, Eq. (60)inAppendix B). This is actually the same computation as (25)with λ =– .Equation(43) corresponds to a front, separating a region where [...] = 0 from a region where [...] = 1, propagating at speed v with an interface of 1 – (γδ–vt) width multiplied by an exponential factor e . Here, the sign of the real part of λ , λ is important. If λ < 0, the front has the shape depicted in Fig. 6(top). It β β,r β,r decays exponentially fast as t → +∞ with a time scale .Onthe opposite,itin- |λ | β,r creases exponentially fast with a time scale as t → +∞ when λ >0, thereby β,r β,r enhancing the network eﬀect and accelerating the activation of nonlinear eﬀect (low threshold or gain control) leading the trajectory out of . Remark that the peak of the drive occurs at γδ – vt = 0. The inﬂexion point of the function (x)is at x =0. Thus, when λ < 0, the front is a bit behind the drive, whereas it is a bit ahead when λ >0. Having unstable eigenvalues is not the only way to get out of . Indeed, even if all eigen- values are stable, the drive itself can lead some cells to get out of this set. When the tra- jectory of dynamical system (34)getsout of , two cases are then possible: Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 23 of 60 (i) Either a BCell i is such that V < θ .Inthiscase, R (V , A )=0.Then, in the B B B B B i i i i matrix L, there is a line of zeros replacing the line i in the matrix W ,i.e.atthe line i + N of L. This corresponds to a stable eigenvalue – for L, controlling the exponential instability observed in Fig. 6(bottom). Thus, too low BCell voltages trigger a re-stabilisation of the dynamical system. (ii) There are BCells such that condition (42) holds, then gain control is activated and system (32) becomes nonlinear. Here, we get out of the linear analysis, and we have not been able to solve the problem mathematically. There is, however, a simple qualitative argument. If the cell i enters the gain control region, then the B i corresponding line i + N in the matrix W of L is replaced by W G (A ),which B B A A i rapidly decays to 0 (see,e.g.Fig. 4(e)). From thesameargumentasin(i),this generates a stable eigenvalue ∼– controlling as well the exponential instability. Equation (37) features therefore the direct eﬀect of the stimulus as well as the indirect eﬀect via the amacrine network, corresponding to a weighted sum of propagating fronts, generated by the stimulus and inﬂuencing a given cell through the connectivity pathways. These fronts interfere either constructively, inducing a wave of activity enhancing the ef- fect of the stimulus and, thereby, anticipation, or destructively somewhat lowering the stimulus eﬀect. The ﬁne tuning between “constructive” and “destructive” interferences depends on the connectivity matrix via the spectrum of L and its projection vectors P . For example, complex eigenvalues introduce time oscillations which are likely to gener- ate destructive interferences, unless some speciﬁc resonance conditions exist between the imaginary parts of the eigenvalues λ . Such resonances are known to exist, e.g. in neural network models exhibiting a Ruelle–Takens transition to chaos [65], and they are closely related to the spectrum of the connectivity matrix [16]. Although we are not in this sit- uation here, our linear analysis clearly shows the inﬂuence of the spectrum of L,itself constrained by W, on the network response to stimuli and anticipation. Spectrum of L This argumentation invites us to consider diﬀerent situations where one can ﬁgure out how connectivity impacts the spectrum of L and thereby anticipation. We therefore provide some general results about the spectrum of L and potential linear insta- bilities before considering speciﬁc examples. These results are proved in Appendix D.2. As stated in Sect. 2.3.3, to go further in the analysis, we now assume that a BCell connects + B + + only one ACell, with a weight w uniform for all BCells, so that W = w I , w >0. We N,N also assume that ACells connect to BCells with a connectivity matrix W, not necessarily – – A – symmetric, with a uniform weight –w , w >0, so that W =–w W. We denote by κ , n = 1,..., N, the eigenvalues of W ordered as |κ |≤|κ | ≤ ··· ≤ |κ |, n 1 2 n and ψ is the corresponding eigenvector. We normalise ψ so that ψ .ψ =1 where † is n n n the adjoint. (Note that, as W is not symmetric in general, eigenvectors are complex). From the eigenvalues and eigenvectors of W, one can compute the eigenvalues and eigenvectors of L (see Appendix D.2),and infer stability conditions for the linear system. The main conclusions are the following: 1. The stability of the linear system is controlled by the reduced, a-dimensional parameter: – + 2 μ = w w τ ≥ 0, (44) Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 24 of 60 where: 1 1 1 = – , (45) τ τ τ A B with a degenerate case when τ = τ considered in the Appendix. A B 2. If W is symmetric, its eigenvalues κ are real, but the eigenvalues of L can be real or complex. Each κ corresponds actually to eigenvalues λ of L (see Eq. (71)). (a) If κ <0, the two corresponding eigenvalues of L are real and one of the two corresponding eigenmodes of L becomes unstable when 1 1 – + w w >– . (46) τ τ κ A B n (b) If κ >0 and if =0, the corresponding eigenvalues of L are complex conjugate if μ > ≡ μ . (47) n,c 4κ The corresponding eigenmodes are always stable. 3. If W is asymmetric, eigenvalues κ are complex, κ = κ + iκ . The eigenvalues of n n n,r n,i L have the form λ = λ + iλ , β = 1,...,2N,with β β,r β,i 1 1 1 λ =– ± a + u ; β,r n n 2τ 2τ AB 2 (48) ⎩ 1 1 λ = ± u – a , β,i n n 2τ ! ! 2 2 2 2 2 where a =1 – 4μκ and u = (1 – 4μκ ) +16μ κ = 1–8μκ +16μ |κ | . n n,r n n,r n n,i n,r Note that we recover the real case when κ =0 by setting u = a . n,i n n Instability occurs if λ >0 for some β.Thisgives β,r a + u >2 , (49) n n AB a condition on μ depending on κ and κ . n,r n,i Remarks The introduction of a dimensional parameter μ allows us to simplify the study – + of the joint inﬂuence of w , w , τ on dynamics because stability is controlled by μ only. In other words, a bifurcation condition of the form μ = μ signiﬁes that this bifurcation – + – + 2 holds when the parameters w , w , τ lay on the manifold deﬁned by w w τ = μ . We now show this in two examples of connectivity and aﬀerent instabilities. 3.2.2 Nearest neighbours connectivity Eigenmodes of the linear regime We consider the case where the matrix W, connecting ACells to BCells, is a matrix of nearest neighbours symmetric connections. In this case, W can be written in terms of the discrete Laplacian on a d dimensional regular lattice, d = 1, 2, with lattice spacing δ = δ set here equal to 1 without loss of generality: A B W =2dI + . (50) Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 25 of 60 Because of this relation, we will often use the terminology Laplacian connectivity for the nearest-neighbours connectivity. We also assume that dynamics holds on a square lattice with null boundary conditions. That is, ACell and BCells are located on d-dimensional grid with indices i , i = 0,..., L + 1 where the voltage and activity of cells with indices x y i =0, i = L +1, i =0 or i = L +1 vanish. x x y y The eigenvalues and eigenvectors are explicitly known in this case. They are parame- trized by a quantum number n = n ∈{1,..., L = N } in one dimension and by two quan- tum numbers (n , n ) ∈{1,..., L = N } in two dimensions. They deﬁne a wave vector x y n π n π L+1 L+1 k =( , ) corresponding to wave lengths ( , ). Hence, the ﬁrst eigenmode (n = n x L+1 L+1 n n x y 1, n = 1) corresponds to the largest space scale (scale of the whole retina) with the small- π π est eigenvalue (in absolute value) s =2(cos( )+ cos( ) – 2). To each of these eigen- (1,1) L+1 L+1 modes is related a characteristic time τ = . The slowest mode is the mode (1, 1). In contrast, the fastest mode is the mode (n = L, n = L) corresponding to the smallest space x y scale, the scale of the lattice spacing δ =1. Eigenvalues κ can be positive or negative. Consider for example the one-dimensional nπ case, where κ =2 cos( ). We choose L even to avoid having a zero eigenvalue κ L . Eigen- L+1 values κ , n = 1,..., , are positive, thus the corresponding eigenvalues λ of L are com- plex and stable. The modes with the largest space scale are therefore stable for the linear dynamical system with oscillations. Eigenvalues κ , n = + 1,..., L, are negative, thus the corresponding eigenvalues λ of L are real. From (46)the mode n becomes unstable when 1 1 – + w w >– . (51) nπ τ τ 2 cos( ) A B L+1 Therefore, the ﬁrst mode to become unstable is the mode L with the smallest space scale 1 1 – + 1 (lattice spacing). For large L,thishappens for w w ∼ . This instability induces 2 τ τ A B – + spatial oscillations at the scale of the lattice spacing. When w w further increases, the next modes become unstable. This instability results in a wave packet following the drive (as shown in Fig. 6). The width of this wave packet is controlled by the unstable modes and by nonlinear eﬀects. We now illustrate the relations of these spectral properties with the mechanism of anticipation. Numerical results In all the following 1D simulations, we consider a bar with a width 150 μm, moving in one dimension at constant speed v = 3 mm/s. We simulate 100 BCells, 100 ACells and 100 GCells placed on a 1D horizontal grid with a uniform spacing of δ = δ = δ =30 μm between to consecutive cells. At time t = 0, the ﬁrst cell lies at b a g 100 μm to the right of the leading edge of the moving bar. We set τ = 300 ms, τ =50 ms, B a + – τ = 100 ms, corresponding to τ = 150 ms (Eq. (45)). We vary the value of weights w , w . + – For the sake of simplicity, we also choose w =–w = w to have only one control parameter. We investigate how the bipolar anticipation time and the maximum in the response R depend on w.ThisissummarisedinFig. 7(top), where we have shown the eﬀect of gain control alone (blue horizontal line, independent of w), the eﬀect of ACell lateral connec- tivity alone (red triangles) and the compound eﬀect (white squares). Anticipation time is averaged over all cells. On the same ﬁgure (bottom) we see the responses of two neighbour cells lying at the centre of the lattice. As w increases, we observe three areas of interest: the ﬁrst (A) corresponds to a regime where ACell connectivity has a negative eﬀect on anticipation, competing with gain con- Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 26 of 60 Figure 7 Anticipation in the Laplacian (nearest-neighbours) case. Top. Anticipation time and maximum bipolar response as a function of the connectivity weight w. The blue line corresponds to gain control alone (it does not depend on w). Red triangles correspond to the eﬀect of lateral ACell connectivity without gain control. White squares correspond to the compound eﬀect of ACell connectivity and gain control. The three regimes A, B, C are commented in the text. Bottom. Response curves of ACells and BCells corresponding to the three –1 –1 regimes: (A) w =0.05 ms with a small cross-inhibition, (B) w =0.3 ms with an opposition in activity –1 between the blue (cell 50) and red cell (51), (C) w =0.6 ms , where the red cell (51) is completely inhibited by cell 50 trol. As w is small, the anticipation is controlled by the direct pathway I, II of Fig. 1,from BCells to GCells, with a small inhibition coming from ACells, thereby decreasing the volt- age of BCells and impairing the eﬀect of gain control. This explains why the anticipation time in the case of lateral connectivity + gain control is smaller than the anticipation time of gain control alone. The network eﬀect (red triangles) on anticipation time increases with w though. This corresponds to the “push–pull” eﬀect already evoked in Sect. 2.3. When a BCell B feels the stimulus, its activity increases favoured by the stimulus, it in- creases the voltage of the connected ACell, inhibiting the next BCell B , thereby inducing i+1 a feedback loop, the push–pull eﬀect, enhancing the voltage of B . In zone (B) the push–pull eﬀect becomes more eﬃcient than gain control alone. In this region, the voltage of the BCell feeling the bar increases fast, while the voltage of its neigh- bours becomes more and more negative, enhancing the feedback loop. This holds until the voltage rectiﬁcation (5) takes place. This is the time when the dynamical system gets out of . The push–pull eﬀect then saturates and V reaches a maximum, corresponding to a peak in activity. This peak is reached faster than the peak in the function G (A). Thus, thepeakof R (t) occurs at the same time as the peak of N (V (t)) and, thus, before the B B B i i reference peak (time t for isolated BCells deﬁned in Sect. 3.1.1). In other words, the ACell lateral connectivity allows the BCell to outperform the gain control mechanism for antic- ipation. As w increases in zone B the push–pull eﬀect (averaged over BCells) reaches a maximum, then decreases. This is because the increase in w makes the inhibitory eﬀect of ACells stronger and stronger on silent BCells which then remain silent longer and longer Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 27 of 60 because the ACell voltage increases with w, and it takes longer for it to decrease and de- inhibit the neighbours. The silent cells are less and less sensitive to the stimulus, being strongly and durably inhibited. In region C, the anticipation is again dominated by gain control. In this case, the eﬀect on cells depends on the parity of their index. The response of BCells is either completely suppressed or identical to the response of the reference case (with gain control alone). This is why the average anticipation time with gain control is about half of the gain control without network eﬀect. Cells that are inhibited do no participate to anticipation, and the others anticipate in the same way than with gain control alone. Note that this “parity” eﬀect is due to the nearest neighbours connectivity and the symmetry of interactions. We now interpret and complete these results from the point of view of the spectrum of L and associated dynamics. The fastest mode to destabilise corresponds to the smallest space scale, i.e. the lattice spacing. This is a mode with alternate sign at the scale of the lattice. We call it the “push–pull” mode, as it is precisely what makes the push–pull eﬀect. When the push–pull mode becomes unstable, the excited BCell becomes more and more excited and the next BCell more and more inhibited. However, the time it takes τ has to be compared to the time where the bar stays in the RF, τ (and more generally the time it bar takes to RF kernel to respond to the bar). In the case of the simulation σ =90 μm(see center Appendix A,Table 1)and v =3 μm/ms giving a characteristic time τ = 270ms, whereas, bar as we observed, τ < 100 ms. The push–pull mode is therefore quite faster than τ ,so L bar the push–pull eﬀect takes place fast and leads to a fast exponential increase of the front depicted in Fig. 6(right). This explains the rapid increase of network anticipation eﬀect observed in regions A, B of Fig. 7. 3.2.3 Random connectivity In this section, we study the behaviour of the model using the more realistic, probabilistic type of connectivity presented in Sect. 2.3.3 and more thoroughly studied in Appendix C. Within this framework, a given ACell A receives the upstream activity from the BCell ly- ing at the same position B with a constant weight w. The same ACell inhibits BCells with which it is coupled through the random adjacency matrix W, generated by the probabilis- tic model of connectivity, and the weight matrix W =–wW. We recall that the connec- tivity depends on a scale parameter ξ for the branch length) and the mean and variance n ¯, σ for the distribution of the number of branches. These parameters can be found in Table 1 in Appendix A. Eigenmodes of the linear regime Similarly to Sect. 3.2.2 we now analyse the spectrum of L when W is a random connectivity matrix. Although a couple of results can be es- tablished (using the Perron–Frobenius theorem), we have not been able to ﬁnd general mathematical results on the spectrum or eigenvectors of this family of random matrices. We thus performed numerical simulations. The spectrum of L is deduced from the spectrum of W as exposed above. The spec- trum of W depends on n ¯, σ and ξ.InFig. 8 we have plotted, on the left, an example of such a spectrum. This is the spectral density (distributions of eigenvalues in the complex plane) obtained from the diagonalization of 10,000 matrices 100 × 100 (so the statistics is made over 10 eigenvalues). We note that the largest eigenvalues are always real positive, a straightforward consequence of the Perron–Frobenius theorem [38, 69]. More generally, Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 28 of 60 Figure 8 Spectral density of eigenvalues for ξ =2, n ¯ =4, σ =1.Top left. For the matrix W (density estimated over 10,000 samples), density is represented in colour plots, in log scale. The colour bar refers to powers of 10. Top right.Spectraldensity of L for w =0.05. Bottom left.Spectraldensity of L for w =0.1. Bottom right.Spectral density of L for w = 0.15. Unstable eigenvalues are on the right to the vertical dashed line x =0 Figure 9 Heat map for the largest real part eigenvalue in the plane w, ξ for diﬀerent values of n ¯. Left. n ¯ =1. Right. n ¯ = 4. Colour lines are level lines. The level line 0 is the frontier of instability of the linear dynamical system we observe an over-density of real eigenvalues. The same holds for random Gaussian ma- trices with independent entries N (0, )[32] whose asymptotic density converges to the circular law [40]. The shape of the spectral density in our model diﬀers from the circular law though, and it depends on the parameters n ¯, σ and ξ. On the same ﬁgure we show the corresponding spectral density of L obtained from Eq. (48)for w = 0.05, 0.1, 015. We have taken here τ =30, τ =10 ms to see better the A B transitions with w (level lines in Fig. 9). There is an evident symmetry with respect to = –0.066 expected from the mathematical analysis. We see that the largest eigenvalue AB is real (although it is not necessarily related to the largest eigenvalue of W). We also see that, as w increases, a large number of (complex) eigenvalues become unstable. There is actually a frontier of instability that we have plotted in the plane w, ξ for diﬀerent values Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 29 of 60 of n ¯. This is shown in Fig. 9 (dashed line). The level line 0 is the frontier of instability of the linear dynamical system. This frontier has the (empirical) form (ξ – ξ ).(w – w0) = c, where c has the dimension of a characteristic speed. What matters here is that there are complex unstable eigenvalues with no speciﬁc reso- nance relations between them. They are therefore prone to generate destructive interfer- ences in (37). Numerical results In Fig. 10 we consider, similarly to Fig. 7 for Laplacian connectivity, the eﬀect of random connectivity on anticipation, compared to pure gain control mechanism. In contrast to the Laplacian case, we have here more parameters to handle: ξ,which con- trols the characteristic length of branches and n, σ which control the number of branches distribution. We present here a few results where ξ varies, whereas the average number of branches n ¯ =2(σ = 1). A more systematic study is done in [74]. The interest of varying ξ is to start from a situation which is close to the Laplacian case (characteristic distance ξ =1) and to increase ξ to see how the size of the dendritic tree of ACells may impact anticipa- tion. This is a preliminary step toward considering diﬀerent physiological ACells type (e.g. narrow-, medium-, or wide-ﬁeld [29]). Note, however, that the probability of connection given the distance of cells (ﬁxed by n ¯, σ ) implicitly impacts w and the anticipation eﬀects. The main diﬀerence with the Laplacian case is the asymmetry of connections. Here, symmetry means that if ACell j connects the BCell i, then the ACell i connects the BCell j too. This does not necessarily hold for random connectivity, and this has a strong impact on the push–pull eﬀect and anticipation. So, even if the connectivity is short-range when ξ is small, mainly connecting nearest neighbours, we observe already a big diﬀerence with theLaplacian case.Thisisshown in Fig. 10,where ξ = 1. Similarly to the Laplacian case, Figure 10 Average anticipation in the random connectivity case. Top. Bipolar anticipation time and maximum in the response R as a function of the connectivity weight w in the case of a random connectivity graph with ξ =1, n ¯ =2 and σ =1. Bottom. Response curves of ACells and BCells 51 – 52 corresponding to the three –1 –1 –1 regimes: (A) w =0.05 ms ,(B) w =0.3 ms ,(C) w =0.6 ms Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 30 of 60 we observe three main regions depending on w.Tohavethe same representation as Fig. 7, we present V , N , R for two connected cells (here, ACell 51 and BCell 52). However, in A B B this case, connection is not symmetric: ACell 51 inhibits BCell 52, but ACell 52 does not inhibit BCell 51. We observe three regimes, as in the Laplacian case. In the ﬁrst region (A) ACell ran- dom connectivity has a negative eﬀect on anticipation, as compared to gain control alone. However, since in this case the “push–pull” eﬀect is not evoked, this decay simply comes from the fact that BCell 52 receives an inhibition for ACell 51 that reduces the eﬀect of gain control. This inhibition, however, is not strong enough to signiﬁcantly shift the peak response as in region (B). Indeed, in region (B), the inhibition of BCell 52 is strong enough to outperform the eﬀect of gain control. In this case, and similarly to the Laplacian case, the peak of R (t)occurs at the same time as the peak of N (V (t)) and before the reference peak. However, this eﬀect B B is not consistent over all cells and only occurs for BCells that receive active inhibition. This explains why the performance of the Laplacian connectivity is better, on average, in this region. Finally, as w grows higher, the inhibition grows stronger, completely inhibiting BCell 51. Cells that do not receive any inhibition, as BCell 49 in this example, keep a response that is identical to the response without ACell connectivity. The fraction of cells receiving inhibition in this case being quite small (about 15), this explains why the stationary value of anticipation is fairly close to the value with gain control alone. The role of the characteristic distance In Fig. 11,weanalyse theeﬀect of thecharacter- istic length ξ on anticipation. On the top of the ﬁgure we represent the joint eﬀect of the random ACell connectivity and gain control on anticipation for three values of ξ.Atthe bottom we represent the only eﬀect of the random ACell connectivity for the same values of ξ. We observe that performance in anticipation decreases with ξ.Moreprecisely,we observe an anticipatory eﬀect in this case, as shown in Fig. 11(bottom), but this eﬀect is not able to compete with gain control alone. Even worse, the compound eﬀect shown in 11(top) is disastrous since increasing w renders the anticipation time smaller and smaller. This spurious eﬀect can be interpreted through the analysis made in Sect. 3.2.1,Eq. (37). From the spectrum of L, we see that there are unstable complex eigenvalues whose num- ber increases with w. These eigenvalues are prone to generate destructive interferences, especially when their number becomes large as w increases, explaining the small peak in region B. The consequence on cells activity and gain control can be dramatic as seen in the red trace of Fig. 10(B bottom), line R . This depends on the precise connectivity pat- tern when long range connections from ACells to BCells induce a desensitisation of BCells, which is not counterbalanced by the push–pull eﬀect as in the Laplacian connectivity case. 3.2.4 Conclusion The two numerical examples considered in this section emphasise the role of symmetry in the synapses and, more generally, the role of complex versus real eigenvalues in the spec- trum of L. Recall that, from Sect. 3.2.1,if W is symmetric, complex eigenvalues are always stable, so, for the type of architecture considered here, unstable destructive interferences only occur when W is asymmetric. This leads to several questions, potential subjects for further studies. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 31 of 60 Figure 11 Role of the characteristic branch length ξ on anticipation. Top. The joint eﬀect of the random ACell connectivity and gain control on anticipation for ξ = 1,2,3. Left. Average bipolar anticipation time. Right. Maximum value of the bipolar response R . Bottom. The single eﬀect of the random ACell connectivity on anticipation with the same representation 1. How much does anticipation depend on the degree of asymmetry in the matrix W? The way we generate the random connectivity in the model does not allow us to tune thedegreeofasymmetry(i.e. theprobability that aconnection A → B exists j i simultaneously with a connection A → B ). Therefore, one has to ﬁnd a diﬀerent way i j to generate the connectivity. From the mathematical analysis made in Appendix C,a distribution depending exponentially on the distance, with a tunable probability to have a symmetric connection, could be appropriate. We do not know about any experimental results characterising this degree of symmetry of the connections in the retina. On mathematical grounds, and from the analogy of the spectrum of W with a circular law, one could expect the spectrum of W to become more and more elongated on the real axis as the degree of symmetry increases in an elliptic like law [55]. 2. Nonlinear eﬀects. The destructive interference eﬀect in our model is partly due to the linear nature of the ACell dynamics. In nonlinear dynamics, eigenvalues of the evolution operator can display resonance conditions favouring constructive interferences. On biological grounds, it is for example known that starburst amacrine cells display periodic bursting activity during development, disappearing a few days after birth [94]. Bursting and its disappearance can be understood in the framework of bifurcation theory of a nonlinear dynamical system featuring these cells [50]. In this setting, even if they are not bursting in the mature stage, SACs remain sensitive Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 32 of 60 to speciﬁc stimulation that can temporally synchronise them, thereby enhancing the network eﬀect with a potential eﬀect on anticipation. 3.3 The potential role of gap junctions on anticipation In this section, we study the network ability to improve anticipation in the presence of gap junctions coupling, as in Eq. (23), and gain control at the level of GCells. We start ﬁrst with mathematical results and show then simulation results. 3.3.1 Mathematical study We use a continuous space limit for a one-dimensional lattice. The extension to two di- mensions is straightforward. Here, x corresponds to the preferred direction of the direc- tion sensitive cells. We consider a continuous spatio-temporal ﬁeld V(x, t), x ∈ R,such (P) (P) that V ≡ V(kδ , t). We assume likewise that V ≡ V (kδ , t)for some continuous G G G k G (P) function V (x, t) corresponding to the GCells bipolar pooling input (17), and we take the limit δ → 0. In this limit Eq. (23)becomes ∂V ∂V G G = f (x, t)– v + O δ , (52) gap ∂t ∂x (P) ∂V (x,t) where v ≡ w δ has the dimension of a speed and ≡ f (x, t). Finally, we denote gap gap G ∂t by C(x) the initial proﬁle so that V(x, t )= C(x). Solution Neglecting terms of order δ , the general solution of (52)is V (x, t)= C x – v (t – t ) + f x – v (t – u), u du. G gap 0 gap Equation (52) is a transport equation of ballistic type [74]. For example, if we consider (P) a stimulation of the form V (x, t)= h(x – vt), where h is a Gaussian pulse of the form (3), propagating with speed v, and an initial proﬁle C(x)= h(x – vt ), the voltage of GCells obeys v v gap V (x, t)= h(x – vt) – h x – v t –(v – v )t . (53) G gap gap 0 v – v v – v gap gap π π gap stim When v = 0, the GCell voltage follows the stimulation, i.e. V (x, t)= h(x – vt). In the gap G presence of gap junctions, there are two pulses: the ﬁrst one π with amplitude stim v–v gap propagating at speed v and following the stimulation; the second one π with amplitude gap gap – propagating at speed v . gap v–v gap We have the following cases (we take t = 0 for simplicity). An illustration is given in Fig. 12. 1. If v and v have thesamesign: gap (A) If v < v, the front π is ampliﬁed by a factor ,whereas thereisa gap stim v–v gap refractory front π ,proportionalto v , behind the excitatory pulse. gap gap (B) If v = v , V (x, t)= h(x – v t)+ v (t – t )h (x – v t) which diverges like t gap G gap gap 0 gap when t →∞ and x → +∞. This divergence is a consequence of the limit δ → 0 in (52). Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 33 of 60 Figure 12 Anticipation for non symmetric gap junctions. Top. GCell anticipation time and maximum ﬁring rate as a function of the gap junction velocity v . Bottom: response curves of GCells corresponding to the three gap regimes: (A) v = 0.6 mm/s, (B) v =3 mm/s,(C) v = 12 mm/s. The curves display 3 main regimes (see gap gap gap text): In (A) v < v and the positive front propagates at the same speed as the pooling voltage triggered by gap the stimulus; In (B), v = v, the positive front and the negative fronts both propagate at the speed v and gap gap the amplitude of the positive front (V (t)) increases with t;In(C), v > v and the positive front propagates G gap faster than the stimulus so that the peak of activity arises earlier. The negative front propagates at the stimulus speed (C) If v > v, the amplitude of π follows the stimulation with a negative sign gap stim (hyper polarization), whereas π is ahead of the stimulation, with a positive gap sign, travelling at speed v . gap 2. If v and v have theoppositesign, we set v =–αv with α >0. Then the front π gap gap stim follows the stimulus but is attenuated by a factor . The front π propagates in gap 1+α the opposite direction with an attenuated amplitude . 1+α This shows that these gap junctions favour the response to motion in the preferred direc- tion and attenuate the motion in the opposite direction although the attenuation is weak. The eﬀect is reinforced by gain control [74]. The most interesting case is 1 c where these gap junctions can induce a wave of activation ahead of the stimulation. Eﬀect of gain control When the low voltage threshold N (19) and the gain control G (A) G G (21) are applied to V (x, t), there are two eﬀects: (i) the hyperpolarised front is cut by N ; G G (ii) the positive pulse induces a raise in activity, which in turn triggers the ganglion gain control G (A) inducing an anticipated peak in the response of the GCell, similar to what happens with BCells, with a diﬀerent form for the GCell gain control though. Moreover, in contrast to pathway II of Fig. 1, where only gain control generates anticipation, in pathway IV the wave of activity generated by gap junctions increases anticipation by two distinct eﬀects. If v < v, the cell’s response propagates at the same speed as the stimulus, but gap its amplitude is larger than the case with no gap junction (term π ). From Eq. (53)this stim results in an increase of h to an eﬀective value h inducing an improvement in the B B v–v gap Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 34 of 60 anticipation time (with a saturation of the eﬀect, though, as v → v). If v > v,the cell’s gap gap response propagates at a larger speed than the stimulus (term π ), so that the cell re- gap sponds before the time of response without gaps. This induces as well an increase in the anticipation time. 3.3.2 Numerical illustrations We consider a bar with a width 200 μm moving in one dimension at constant speed v = 3 mm/s. We simulate here 100 GCells, placed on a 1D horizontal grid, with a spacing of 30 μm between two consecutive cells. At time t = 0, the ﬁrst cell lies at 100 μmfrom the leading edge of the moving bar. We investigate how the GCell anticipation time and GCell ﬁring rate depend on v in gap Fig. 12. The top shows the eﬀect of gain control alone (blue horizontal line independent of v ), the eﬀect of the asymmetric gap junction connectivity alone (red triangles) and gap the compound eﬀect (white squares). Anticipation time is averaged over all GCells. In the bottom part of the ﬁgure, we show the responses of two GCells of indices 30 and 60, spaced by 900 μm. As explained in Sect. 3.3.1,weobserve thethree regimesA,B,Cmathematically antic- ipated above. Note that, for these parameter values, the negative trailing front predicted in A is not visible. 3.3.3 Symmetric gap junctions The asymmetry observed by Trenholm et al. is due to the speciﬁc structure of the direc- tion selective GCell dendritic tree [80]. However, in general gap junctions connectivity is expected to be symmetric. So, to be complete, we consider here the eﬀect of symmetric gap junctions on anticipation. It is not diﬃcult to derive the equivalent of Eq. (52)inthis case too. This is a diﬀusion equation of the form ∂V = f (x, t)+ D V + O δ , gap G ∂t where D = w δG is the diﬀusion coeﬃcient and is the Laplacian operator. gap gap The response to a Gaussian stimulus of the form (3) reads as follows: x,y,t V (x, y, t)=[H ∗ f ], (54) where 2 2 x +y 4Dgapt H(x, y, t)= (55) 4πD t gap which is the heat equation diﬀusion kernel. (P) ∂V (x,t) (P) Recall that f (x, t) ≡ .So, if V (x, t)= h(x – vt), where h is a Gaussian pulse of the ∂t form (3) propagating with speed v, f is a bimodal function of the form h(x–vt) ×(x–vt), the shape of which can be seen in Fig. 13(bottom), second row. The convolution with the heat kernel leads to a front propagating at the same rate as the stimulus, with a diﬀusive spreading whose rate is controlled by D . In particular, there is positive bump ahead of gap the motion, which can induce anticipation, as shown in Fig. 13(top). The eﬀect is weak Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 35 of 60 Figure 13 Anticipation for symmetric gap junctions. Top. GCell anticipation time and maximum ﬁring rate as a function of the gap junction velocity v . Bottom. Response curves of GCells corresponding for three values gap of v :(A) v = 0.6 mm/s, (B) v = 3 mm/s, (C) v = 12 mm/s. For consistency, we have kept the same gap gap gap gap values as in the asymmetric case. Here, anticipation time grows continuously until saturation, while the maximum ﬁring decreases like a power law as a function of v gap though, essentially because the diﬀusive spreading makes the amplitude of the response decrease fast as a function of D . gap Although this positive front, for small D , increases a bit the anticipation time by accel- gap erating the gain control triggering, rapidly the peak in the response R is led by the voltage peak corresponding to the positive bump with a low voltage. The position of this peak is, 2 2 roughly, at a distance σ = σ + σ from the peak of the Gaussian pool, where σ is center center B the width of the centre RF and σ the width of the bar. This corresponds to a time ahead of the peak in the drive, ﬁxing a maximal value to the anticipation time (see the saturation of the anticipation time curve in Fig. 13(top, left)). In our case, σ ∼ 134 given a satura- 134 μm tion peak at = 44.84 ms. A consequence of the voltage decay is the corresponding 3 μm/ms power law ( for large D ) decay of the ﬁring rate (Fig. 13(top, right)). gap gap To conclude, the situation with symmetric gap junctions is in high contrast with di- rection selective gap junctions where the response to stimuli was ballistic and was not decreasing with time. On this basis we consider that, for symmetric gap junctions, the an- ticipation eﬀect is irrelevant, especially taking into account the smallness of the voltage response in case C. 3.3.4 Numerical results We investigate in this section how the GCell anticipation time and GCell ﬁring rate depend on the gap junction conductance in the case of symmetric gap junctions. In Fig. 13(top), we use the same representation as in Fig. 12. For consistency with the direction sensitive case, gap we choose v = as a control parameter. We also take (A) v = 0.6 mm/s, (B) v = gap gap gap δG Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 36 of 60 3 mm/s, (C) v = 12 mm/s in Fig. 13(bottom). This corresponds to a diﬀusion coeﬃcient gap –3 2 –3 2 –3 2 (A) D =18 × 10 mm /s, (B) D =90 × 10 mm /s, (C) D = 360 × 10 mm /s. gap gap gap 3.3.5 Conclusion In this section we have shown how gap junctions direction sensitive cells can display an- ticipation due to the propagation of a wave of activity ahead of the stimulus. This eﬀect is negligible for symmetric gap junctions. Note that symmetric gap junctions are known to favour waves propagation, for example in the early development (stage I, see [49]and the reference therein for a recent numerical investigation). Here, gap junctions are considered in a diﬀerent context due to the presence of a nonstationary stimulus triggering the wave. Let us now comment our computational result. How does it ﬁt to biological reality? Depending on the gap-junction conductance value, the propagation patterns we predict are quite diﬀerent. What is the typical value of v in biology? It is diﬃcult to make an estimate from the gap g δ gap G expression v = . The membrane capacity C and gap junctions conductance can be gap obtained from the literature (for connexins Cx36, g ∼ 10–15 pS [75]), but the distance gap δ is more diﬃcult to evaluate. In the model, this is the average distance between GCells’ soma which corresponds to ∼200–300 μm. But in the computation with gap junctions what matters is the length of a connexin channel which is quite smaller. Taking δ as the distance between GCells assumes a propagation speed between somas at the speed of a connexin, which is wrong because most of the speed is constrained by the propagation of action potential along the dendritic tree. So we used a phenomenological argument (we thank O. Marre for pointing it out to us). The correlation of spiking activity between GCellneighboursisabout 2–5ms for cellsseparated by ∼200–300 μm[86]. This gives a speed v in the interval [40, 150] mm/s, which is quite fast compared to the bar speed in gap experiments. So we are in case 1 b, and should one observe an experimental eﬀect? To the best of our knowledge, an eﬀect of DSGC gap junctions on motion anticipation has not been ob- served. But we do not know about experiments targeting precisely this eﬀect. It would be interesting to block gap junctions and address Berry et al. [9]orChenetal. [20]ex- perimentsinthiscase. Thediﬃcultyisthatblockinggap junctionsblocksmanyessential retinal pathways. We do not pursue this discussion further concluding that our model proposes a computational prediction that could be interesting to be experimentally inves- tigated. 3.4 Response to two-dimensional stimuli In this section, we present some examples of retinal responses and anticipation to trajec- tories more complex than a bar moving in one dimension with a uniform speed. The aim here is not to do an exhaustive study but, instead, to assess qualitatively some anticipatory eﬀects not considered in the previous sections. 3.4.1 Flash lag eﬀect The ﬂash lag eﬀect is an optical illusion where a bar moving along a smooth trajectory and a ﬂashed bar are presented to the subject and are perceived with a spatial displacement, while they are actually aligned. A variation of this illusion consists of a bar moving in rotation, a bar ﬂashed in angular alignment, giving rise to a perceived angular discrepancy. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 37 of 60 Figure 14 Flash lag eﬀect with diﬀerent anticipatory mechanisms.(A) Response to a ﬂash lag stimulus: a bar moving in smooth motion with a second bar ﬂashed in alignment with the ﬁrst bar for one time frame. The ﬁrst line shows the stimulus, the second line shows the GCells response with gain control, the third line –1 presents the eﬀect of lateral ACell Laplacian connectivity with w =0.3 ms ,and thelastlineshows theeﬀect of asymmetric gap junctions with v = 9 mm/s. (B) Time course response of (top) a cell responding to the gap ﬂashed bar and (bottom) a cell responding to the moving bar. Dashed lines indicate the peak of each curve We have investigated this eﬀect in our model in the presence of the diﬀerent anticipatory eﬀects considered in the paper. Figure 14 shows the response to a bar moving with a smooth motion, while a second bar is ﬂashed in alignment with the ﬁrst bar at one time frame. The ﬁrst line shows the stimuli, consisting of 130 frames, of a bar moving at 2.7 mm/s, with a refreshment rate of 100 Hz. The second line shows the GCell response with gain control, and the third line presents the eﬀect of lateral amacrine connectivity in the case of a Laplacian graph. Keeping the –1 same values of parameters as in the 1D case, we set w =0.3 ms corresponding to case BinFig. 7. Finally, the last line shows the eﬀect of asymmetric gap junctions, having a preferred orientation in the direction of motion, with v = 9 mm/s. gap In thecaseofthe gain controlresponse, thepeakofresponsetothe moving barisshifted by about 10 ms in the direction of motion, as compared to the static bar. The ﬂashed bar elicits a lower response, given its very short appearance in comparison with the character- istic time of adaptation. We choose this time short enough to avoid gain control triggering, explaining the diﬀerence with the strong response observed by Chen et al. [20]inthe pres- ence of a still bar. In the case of amacrine connectivity, the moving bar representation is shrunk as com- pared to the gain control case given the prevalence of inhibition, while the level of activity for the ﬂashed bar remains roughly the same. In this case, cells responding to the moving bar reach their peak activity slightly earlier (about 19 ms for these parameters value) than in the gain control case (Fig. 14(B, top)). Finally, asymmetric gap junction connectivity displays a wave propagating ahead of the bar, increasing the central blob, which is much larger than the size of the bar in the stim- ulus, while the ﬂashed bar activity remains similar to the previous cases. 3.4.2 Parabolic trajectory In this subsection, we assess the eﬀect of the three anticipatory mechanisms on a parabolic trajectory. The interest is to have a trajectory with a change in direction and speed, thus an acceleration. The stimulus consists of 20 frames displayed at 10 Hz. The simulations parameters and connectivity weights are the same as the ones used in the previous section. Figure 15 shows the response to a dot moving along a parabolic trajectory. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 38 of 60 Figure 15 Eﬀect of anticipatory mechanisms on a parabolic trajectory.(A) Response to a dot moving along a parabolic trajectory. The ﬁrst line shows the stimulus, the second line shows the GCell response with gain –1 control, the third line presents the eﬀect of lateral ACell connectivity with w =0.3 ms ,and thelastline shows the eﬀect of asymmetric gap junctions with v = 9 mm/s. (B) Time course response of a cell gap responding to the dot near the trajectory turning point. Linear response corresponds to the response to the stimulus without gain control In the case of gain control, GCell response is more elongated, which has a distortion eﬀect on the dot representation near the turning point of the trajectory (1400–1600 ms). Cells responding near the trajectory turning point are still anticipating motion, as the peak response of the gain control curve is slightly shifted to the left, compared to the RF re- sponse (Fig. 14(B)). In the case of amacrine connectivity, the elicited response is also more localised, as com- pared to the gain control response, and the ﬂow of activity follows more accurately the stimulus. This is a direct consequence of the sensitivity of the ACell connectivity model to the stimulus acceleration. In this case, the peak response is also more shifted as compared to the gain control case. Finally, the gap junction connectivity model performs worse in this case, giving rise to a propagating wave that does not follow the trajectory, since the latter is not parallel to the direction to which GCells are sensitive. Cells responding near the trajectory turning point have a higher level of activity and an increased latency, while the peak response roughly corresponds to the gain control case. 3.4.3 Angular anticipation We investigate in this subsection a two-dimensional example of motion where angular anticipation takes place. The stimulus consists here of 72 frames, displayed at 100 Hz, of a bar moving at a constant angular speed of 4.25 rad/ms. Figure 16 shows the retina response to a rotating bar with the angular orientation of activity as a function of time for the diﬀerent models. We used Matlab to estimate the bar orientation from the displayed activity, ﬁtting the set of activated points by an ellipse whose principal axis determines the response orientation. In the three cases, one can see that the response around the centre of the bar is sup- pressed due to gain control adaptation. While the gain control activity orientation roughly follows thelinearresponse(Fig. 16(B)), the ACell response shows a slight angular shift (frames: 250–300 ms), which is also visible on the response orientation time course. The ACell angular anticipation is, however, only observed during the ﬁrst period of the bar. Interestingly, this eﬀect vanishes during the second rotation due to a persistent eﬀect of Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 39 of 60 Figure 16 Anticipation for a rotating bar.(A) Response to a bar rotating at 4.25 rad/ms. The ﬁrst line shows the stimulus, the second line shows the GCell response with gain control, the third line presents the eﬀect of –1 lateral ACell connectivity with w =0.2 ms , and the last line shows the eﬀect of asymmetric gap junctions with v = 9 mm/s. (B) Time course response of the bar orientation in the reconstructed retinal gap representations the activation function generating a sort of a suppressive eﬀect erasing the second occur- rence of the bar (frame: 450 ms). We shall point out that the ACell connectivity weight in –1 –1 this simulation has been reduced to w =0.2 ms , since with a value of w =0.3 ms used in the previous simulations, the response to the second rotation of the bar is completely suppressed. Finally, similarly to the parabolic trajectory, the gap junction connectivity model per- forms worse due to the wave propagating from left to right, distorting once more the bar shape. Consequently, the bar activity orientation in this case has been discarded in Fig. 16(B), the orientation estimate giving poor results. 3.5 Conclusion This section shows how lateral connectivity can play a role in motion anticipation of 2D stimuli, both in the case of the classical ﬂash lag eﬀect and more complex trajectories. Indeed, for a given network setting, ACell connectivity can noticeably improve anticipa- tion with respect to gain control in all three stimuli and has also the advantage of being sensitive to trajectory shifts (Sect. 3.4.2). While gap junction connectivity improves anticipation when the trajectory of the bar is parallel to the preferred GCells direction, it also induces more blur around the bar and shape distortion in the case of parabolic motion and rotation, suggesting a trade-oﬀ be- tween anticipation and object recognition for this speciﬁc model. 4 Discussion Using a simpliﬁed model, mathematically analysed with numerical simulations exam- ples, we have been able to give strong evidences that lateral connectivity—inhibition with ACells, gap junctions—could participate to motion anticipation in the retina. The main argument is that a moving stimulus can, under speciﬁc conditions mathematically con- trolled, induce a wave of activity which propagates ahead of the stimulus thanks to lateral connectivity. This suggests that, in addition to local gain control mechanism inducing an anticipated peak of GCells activity, lateral connectivity could induce a mechanism of neu- ral latencies reduction similar to what is observed in the cortex [8, 46, 77]. This is visible in particular in Fig. 14, where the gap junction coupling induces a wave which increases the GCell level activity before the bar reaches its RF. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 40 of 60 Yet, these studies raise several questions and remarks. The ﬁrst one is, of course, the biological plausibility. At the core of the model, what makes the mathematical analysis tractable is the fact that we can reduce dynamics, in some region of the phase space, to a linear dynamical system. This structure is aﬀorded by two facts: (i) Synapses are char- acterised by a simple convolution; (ii) Cells, especially ACells, have a simple passive dy- namics, where nonlinear eﬀects induced e.g. by ionic channels are neglected, as well as propagation delays. As stated in the introduction, the goal here is not to be biologically realistic, but instead to illustrate potential general spatio-temporal response mechanisms taking into account speciﬁcities of the retina, as compared to e.g. the cortex. Essentially, most neurons are not spiking (except GCells and some type of BCells or ACells, not con- sidered here [2]). Yet, synapses follow the same biophysics as their cortical counterpart. As it is standard to model the whole chain of biophysical machinery triggering a post-synaptic potential upon a sharp increase of the pre-synaptic voltage by a convolution kernel [27], we adopted here the same approach. Note that it is absolutely not required, in this convo- lutional approach, for the pre-synaptic increase in voltage to be a spike; it can be a smooth variation of the voltage. Note also that higher order convolution kernels can be consid- ered, integrating more details of the biological machinery. These higher order kernels are represented by higher order linear diﬀerential equations [35]. Concerning point (ii), non- linear eﬀects are neglected, especially for ACells (BCells have gain control), there are not so many available models of ACells. A linear model for predictive coding using linear ACells has been used by Hosoya et al. [43]. We discuss it in more detail below. The nonlinear models of ACells we know have been developed to study the retina in its early stage (reti- nal waves) and feature either AII ACells [21]orstarburst ACells [50]. In Sect. 3.2.4 we have brieﬂy commented how nonlinear mechanisms could enhance resonance eﬀects in the network and, thereby, favour the propagation of a lateral wave of activity induced by a moving stimulus. This would of course deserve more detailed study. Another potentially interesting nonlinear mechanism is short term plasticity discussed below. The second question one may ask about the model is about the robustness of this mech- anism with respect to parameters. The model contains many parameters, some of them (BCells, GCells and gain control) coming from the previous paper from Berry et al. [9] and Chen et al. [20]. Although they did not perform a structural stability analysis of their model (i.e. stability of the model with respect to small variations of parameters), we believe that they are tuned away from bifurcation points so that slight changes in their (isolated cells) model parameters would not induce big changes. As we have shown, the situation changes dramatically when cells are connected via lateral connectivity. Here, many types of dynamical behaviour can be expected simply by changing the connectivity patterns in the case of ACells. A more detailed analysis would require a closer investigation of ACell to BCell connectivity and an estimation of synaptic coupling, implying to deﬁne more speciﬁcally the type of ACell (AII, starburst, A17, wide ﬁeld, medium ﬁeld, narrow ﬁeld, etc.) and the type of functional circuit one wants to consider. Note that ACells are diﬃcult to access experimentally due to their location inside the retina. Even more diﬃcult is a measurement of ACell connectivity, especially the degree of symmetry discussed in our paper. Such studies can be performed at the computational level, though, where the math- ematical framework proposed here can be applied and extended. Computational results do not tell us what is the reality, but they shed light on what it could be. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 41 of 60 We would like now to address several possible extensions of this work. The retino-thalamico-cortical pathway Theretinaisonlythe earlystage of thevisual system. Visual responses are then processed via the thalamus and the cortex. As exposed in the introduction, anticipation is also observed in V1 with a diﬀerent modality than in the retina. In this paper, our main focus was on the shift of the peak response, while when studying anticipation in the cortex, the main focus lies in the increase of the response latency, i.e. the delay between the time the bar reaches the receptive ﬁeld of a cortical column and the eﬀective time its activity starts rising. How do these two eﬀects combine? How does retinal anticipation impact cortical anticipation? To answer these questions at a computational level, one would need to propose a model of the retino-thalamico-cortical pathway which, to the best of our knowledge, has never been done. Yet, we have developed a retino-cortical (V1) model—thus, short-cutting the thalamus—based on a mean-ﬁeld model of the V1 cortex, developed earlier by the groups of F. Chavane and A. Destexhe [19, 93], able to reproduce V1 anticipation as observed in VSDI imaging. The aim of this work is to understand, computationally, the eﬀect of retinal anticipation on the cortical one and more generally the combined eﬀects of motion extrapolation in the retina and V1. This is the object of a forthcoming paper (S. Souihel, M. di Volo, S. Chemla, A. Destexhe, F. Chavane and B. Cessac., in preparation). See [74] for preliminary results. Retinal circuits BCells, ACells and GCells are organised into multiple, local, functional circuits with speciﬁc connectivity patterns and dynamics in response to stimuli. Each cir- cuit is related to a speciﬁc task, such as light intensity or contrast adaptation, motion de- tection, orientation, motion direction and so on. Here, we have considered a circuit allow- ing the retina to detect a moving object on a moving background, where motion sensitive retinal cells remain silent under global motion of the visual scene, but ﬁre when the image patch in their receptive ﬁeld moves diﬀerently from the background. From our study we have emitted the hypothesis that this circuit, spread over the retina, could improve mo- tion anticipation thanks to what we have called the “push–pull” eﬀect. Yet, other circuits could be studied in their role to process motion and anticipation. We especially think of the ON-OFF cone and rod-cone pathways responsible for the separation of highlights and shadows, allowing to provide information to the GCells concerning brighter than back- ground stimuli (ON-centre) or darker than background stimuli (OFF-centre) [54]. This circuit involves both gap junction and ACell (AII) connectivity, and our model could al- low to study its dynamics in the presence of a moving object. Adaptation eﬀects In a paper from 2005, Hosoya et al. [43] studied dynamic predictive coding in the retina and showed how spatio-temporal receptive ﬁelds of retinal GCells change after a few seconds in a new environment, allowing the retina to adjust its pro- cessing dynamically when encountering changes in its visual environment. They showed that an amacrine network model with plastic synapses can account for the large variety of observed adaptations. They feature a linear network model of ACells, similar to ours, with, in addition, anti-Hebbian plasticity. Their mathematical analysis, based on linear algebra, allows to determine the behaviour of the model in terms of eigenvalues and eigenvectors. However, their analysis does not carry out to the gain control introduced by Berry et al., which, as we show, renders the spectral analysis quite more complex. It would therefore be Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 42 of 60 interesting to explore how plasticity in the ACell synaptic network, conjugated with local gain control, contributes to anticipation. Correlations The trajectory of a moving object—which is, in general, quite more com- plex than a moving bar with constant speed—involves long-range correlations in space and in time. Local information about this motion is encoded by retinal GCells. De- coders based on the ﬁring rates of these cells can extract some of the motion features [25, 43, 51, 62, 66, 67, 76]. Yet lateral connectivity plays a central role in motion processing (see, e.g. [41]). One may expect it to induce spatial and temporal correlations in spiking activity, as an echo, a trace, of the object’s trajectory. These correlations cannot be read in the variations of ﬁring rate; they also cannot be read in synchronous pairwise corre- lations, as the propagation of information due to lateral connectivity necessarily involves delays. This example raises the question about what information can be extracted from spatio-temporal correlations in a network of connected neurons submitted to a transient stimulus. What is the eﬀect of the stimulus on these correlations? How can one handle this information from data where one has to measure transient correlations? This question has been addressed in [17]. The potential impact of these spatio-temporal correlations on de- coding and anticipating a trajectory will be the object of further studies. Orientation selective cells Our model aﬀords the possibility to consider BCells with ori- entation sensitive RF. The potential role of such BCells for predictive coding has been outlined by Johnston et al. [48]. In their model individual GCells receive excitatory BCell inputs tuned to diﬀerent orientations, generating a dynamic predictive code, while feed- forward inhibition generates a high-pass ﬁlter that only transmits the initial activation of these inputs, removing redundancy. Should such circuits play a role in motion anticipa- tion? We did not elaborate on this in the present paper, leaving it to a potential forthcoming work. Another important question is “how to model a retinal network with cells having diﬀerent orientation selectivity?” A V1 cortical model has been proposed by Baspinar et al. [7] for the generation of orientation preference maps, considering both orientation and scale features. Each point (cortical column) is characterised by intrinsic variables, orien- tation and scale, and the corresponding RF is a rotated Gabor function. The visual stim- ulus is lifted in a four-dimensional space, characterised by coordinate variables, position, orientation and scale. The authors infer from the V1 connectivity a “natural” geometry from which they can apply methods from diﬀerential geometry. This type of mathemati- cal construction could be interesting to investigate in the case of the retina with families of orientation selective cells, although the retinal connectivity between these cells is not the same as the V1 orientation preference map structure [12, 13, 82, 91]. Biologically inspired vision systems When the retina receives a visual stimulus, it deter- mines which component of it is signiﬁcant and needs to be further transmitted to the brain. This eﬃcient coding heuristic has inspired many recent studies in developing biologically inspired systems, both for static image and motion representations [59, 87, 90]. Two ma- jor applications of biologically inspired vision systems are retinal prostheses [53, 63, 79] and navigational robotics [24, 52]. Focusing on the second ﬁeld of application, the abil- ity of a mobile device to navigate in its environment is of utmost interest, especially in order to avoid dangerous situations such as collisions. To be able to move, the robot re- quires a mapped representation of its environment, but also the ability to interpret and Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 43 of 60 process this representation. Motion processing mechanisms such as anticipation can be thus implemented to assess the eﬃciency of bio-inspired vision in obstacle avoidance. Psychophysical study of motion anticipation We brieﬂy come back to the ﬂash lag eﬀect introduced in 3.4.1. Neuroscience has explored several explanations for this illusion. The ﬁrst explanation is that the visual system processes moving objects with a smaller latency than ﬂashed objects [89]. The second explanation suggests that the ﬂash lag eﬀect is due to postdiction: the perception of the ﬂash is conditioned by events happening after its ap- pearance [30, 31]. This hypothesis is inspired by the colour phi illusion, where two dots of diﬀerent colours appearing at two discrete yet close positions, with a small latency, will be perceived as a single moving dot whose colour has changed. Another explanation in- cludes motion extrapolation. The visual system being predictive, it processes diﬀerently a bar in smooth motion, whose motion can be extrapolated, and a ﬂashed bar, which can- not be predicted by the system. The study conducted in this article falls under this third explanation, suggesting that the retina could possibly be assisting the cortex in the motion extrapolation task. Appendix A: Parameters of the model Table 1 Model parameters values used in simulations, unless stated otherwise Function Parameter Value Unit K (Eq. (56)) σ (centre) 90 μm B,S 1 σ (surround) 290 μm A (centre) 1.2 mV A (surround) 0.2 mV K (t)(Eq.(57)) μ 60 ms T 1 μ 180 ms σ 20 ms σ 44 ms K 0.22 unitless K 0.1 unitless BCell dynamics τ 100 ms –1 –3 –1 h 6.11e mV .ms θ 5.32 mV τ 200 ms ACell dynamics τ 200 ms –1 w [0, 1] ms ACell connectivity ξ {1, 2, 3, 4} mm n ¯ {1, 2, 3, 4} unitless σ 1unitless GCell dynamics a 0.5 unitless σ 90 μm τ 189.5 ms –4 h 3.59e unitless α 1110 Hz/mV θ 0mV max N 212 Hz G Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 44 of 60 Appendix B: Spatio-temporal ﬁltering B.1 Receptive ﬁelds The spatial kernel of the BCell i is modelled with a diﬀerence of Gaussians (DOG): A 1 A 1 –1 –1 1 ˜ 2 ˜ – X .C .X – X .C .X i i i i 2 1 2 2 K (x, y)= √ e – √ e , (56) B ,S 2π det C 2π det C 1 2 x–x where X = ," denotes the transpose, x and y are the coordinates of the receptive i i i y–y ﬁeld centre which coincide with the coordinates of the cell, C , C are positive deﬁnite ma- 1 2 trix whose main principal axis represents the preferred orientation. For circular DOGs(no 2 2 preferred orientation) C ≡ σ I, C ≡ σ I where I is the identity matrix in two dimen- 1 2 1 2 sions. The two Gaussians of the DOG are thus concentric. They have the same principal axes. X has the physical dimension of a length (mm), thus the entries of C , a =1,2, are i a expressed in mm . A , a = 1,...,2, have the dimension of mV so that convolution (1)has the dimension of a voltage. We model the temporal part of the RF with a diﬀerence of non-concentric Gaussians whose integral on the time domain is zero. This kernel well ﬁts the shape of the temporal projection of the bipolar RF observed in experiments [74]. 2 2 (t–μ ) (t–μ ) 1 2 – – K K 1 2 2 2 2σ 2σ 1 2 K (t)= √ e – √ e H(t), (57) 2πσ 2πσ 1 2 where H(t) is the Heaviside function. The parameters μ , σ , b = 1, 2, have the dimension b b of a time (s), whereas K are dimensionless. The following condition must hold to ensure the continuity of K (t)at zero: 2 2 μ μ 1 2 – – K K 2 2 1 2 2σ 2σ 1 2 e = e . (58) σ σ 1 2 Thus, K (x, y, 0) = 0. In addition, we require that the integral of a constant stimulus con- verges to zero, so that the cell is only reactive to changes. This reads as follows: μ μ 1 2 K = K , (59) 1 2 σ σ 1 2 where (x)= √ e dy (60) 2π –∞ is the cumulative distribution function of the standard Gaussian probability. B.2 Numerical convolution Here we describe the method used to numerically integrate convolution (1)ofthe recep- tive ﬁeld K (56) in the spatial domain with a stimulus S. For the sake of clarity, we B ,S restrict the computation to one Gaussian in the DOG. The extension to a diﬀerence of Gaussian is straightforward. In the following, we consider a spatially discretized stimulus. When dealing with a 2D stimulus, we have to integrate over two axes. In the case where the Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 45 of 60 eigenvectors of the 2D of Gaussians are the axes of integration, the spatial ﬁlter is separable in the stimulus coordinate system. Considering the stimulus as a grid of pixels, we can in- tegrate using the following discretization: let L be the size of the stimulus along the x axis in pixels, L be its size along the y axis, and δ be the pixel length. We set S (t) ≡ S(iδ, jδ, u), y ij L y with i = 0,..., and j = 0,..., . The spatial convolution becomes then δ δ x,y [K ∗ S](t) 2 2 (x–x ) (y–y ) 0 0 – – 1 2 2 2σ 2σ x y = S(x, y, t)e dx dy 2πσ σ 2 x y R i + δ – x i – x j + δ – y j – y 0 0 0 0 = S (t) erf √ – erf √ erf √ – erf √ . ij 2σ 2σ 2σ 2σ x x y y i,j In the case where the eigenvectors of the 2D of Gaussians are not the axes of integration, the spatial ﬁlter is not separable in the stimulus coordinates system. There exist methods that perform the computation by making a linear combination of basis ﬁlters [36], others that use Fourier based deconvolution techniques [83] and others using recursive ﬁlter- ing techniques [26]. However, these methods are of high computational complexity. We choose instead to use a computer vision method from Geusenroek et al. [39]. It is based on a projection in a non-orthogonal basis, where the ﬁrst axis is x and the second is parametrized by an angle φ (see Fig. 17). The new standard deviations read as follows: σ σ x y σ = ! , 2 2 2 2 σ cos θ + σ sin θ x y 2 2 2 2 σ cos θ + σ sin θ y x σ = sin φ Figure 17 Filter transformation description [39]. Theoriginalsystemofaxesisrepresented by x and y,and the ellipse system of axes by u and v. φ represents the angle of the second axis of the non-orthogonal basis. The integration domain of a pixel is limited by four lines of equations: x = iδ, x =(i +1)δ, y = jδ and y =(j +1)δ. Rewriting these fours equations in the new system of axes through a coordinate change enables us to write Eq. (61) Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 46 of 60 with 2 2 2 2 σ cos θ + σ sin θ y x tan(φ)= 2 2 (σ – σ ) cos θ sin θ x y and σ = σ (in the orientation sensitive case). x y We adapt the implementation to the spatially discretized stimulus using an integration scheme similar to the one introduced in the separable case. The spatial convolution reads now as follows: x,y [K ∗ S](x , y , t) B 0 0 δ (y –y ) (y+1) sin(φ) 2 2σ = σ C e x ij (61) sin(φ) (i;j)∈[0,s ]×[0,s ] x y (– cos(φ)y + x +1)δ – x (– cos(φ)y + x)δ – x 0 0 × erf √ – erf √ dy , 2σ 2σ x x where s (resp. s ) denotes the size of the stimulus in pixels along the x axis (resp. y axis), x y and C is thecontrastatthe pixellocated at (i, j). ij The integral is then computed numerically. The advantage of this formulation is to re- place a two-dimensional integration by a one-dimensional one. Appendix C: Random connectivity Here, we deﬁne the random connectivity matrix from ACell to BCells considered in Sect. 2.3.3. Each cell (ACell and BCell) has a random number of branches (dendritic tree), each of which has a random length and a random angle with respect to the horizontal axis. The length of branches L follows an exponential distribution f (l)= e , l ≥ 0, (62) with spatial scale ξ. The number of branches n is also a random variable, Gaussian with mean n ¯ and variance σ . The angle distribution is taken to be isotropic in the plane, i.e. uniform on [0, 2π[. When a branch of an ACell A intersects a branch of a BCell B, there is achemicalsynapse from AtoB. Here, we assume that both cell types have the same probability distributions for branches, thus neglecting the actual shape of ACell and BCell dendritic trees. On biolog- ical grounds, this assumption is relevant if we consider the shape of BCell dendritic tree in the inner plexiform layer (IPL)(seee.g https://webvision.med.utah.edu/book/part-iii- retinal-circuits/roles-of-ACell-cells/ Figs. 5, 16, 17). While out of the IPL, BCells have the form of a dipole, in the IPL their dendrites have a form well approximated by our two- dimensional model. A potential reﬁnement would consist of considering diﬀerent set of parameters in the probability laws respectively deﬁning BCell and ACell dendritic tree. We show in Fig. 18 an example of connectivity matrix produced this way, as well as the probability that two branches intersect as a function of the distance of the two cells. We now compute this probability. We use the standard notation in probability theory where the random variable is written in capitals and its realisation in a small letter. Thus, Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 47 of 60 Figure 18 Random connectivity. Left. Example of a random connectivity matrix from ACells A to BCells cells B . j i White points correspond to connection from A to B . Right. Probability P(d) that two branches intersect as a j i function of the distance between two cells. ‘Exp’ corresponds to numerical estimation and ‘Th’ corresponds to the theoretical prediction. Here, ξ =2 F (x)= P[X < x] is the cumulative distribution function of the random variable X and dF f (x)= is its density. dx We consider the oriented connection between the cell A (ACell) of coordinates (x , y ) A A to a cell B (BCell) of coordinates (x , y ), so that the distance between the two cells is B B 2 2 d = (x – x ) +(y – y ) . AB B A B A The vector AB makes an oriented angle η = (AB, Ax) with the positive horizontal axis, where y –y ⎨ B A arctan ,if x > x ; B A x –x B A η = (63) y –y ⎩ B A π + arctan ,if x < x . B A x –x B A Here, we neglect the eﬀects of boundaries, taking e.g. an inﬁnite lattice or periodic bound- ary conditions, so that the probability to connect A to B is invariant by rotation. Thus, we y –y B A compute this probability in the ﬁrst quadrant x > x , y > y .Inthiscase η = arctan . B A B A x –x B A Each cell has a random number of branches (dendritic tree), each of which has a random length and a random angle with respect to the horizontal axis (Fig. 19). The length of branches L follows the exponential distribution (62): f (l)= e , l ≥ 0, with repartition function F (l)=1 – e . (64) The spatial scale ξ favours short range connections. The number of branches N distribu- tion follows a normal distribution with mean n ¯ and variance σ . The angle distribution is taken to be isotropic in the plane, i.e. uniform on [0, 2π[. We compute the probability that a branch of ACell A of length L intersects at point C abranchofBCell B of length L .Wedenoteby α the oriented angle (Ax, AC); by β the oriented angle (Bx, BC); by θ the oriented angle (AB, AC). In the ﬁrst quadrant, α = θ + η. Note that the condition to be in the ﬁrst quadrant constraints η but not α. sin(β–α+θ) sin θ sin(β–α) From the sin rule we have = = .Thisholds,however,if A, B, C is a d d d AC BC AB triangle, that is, if the two branches are long enough to intersect at C,which reads: 0 ≤ sin(β–α+θ) sin θ d = d ≤ L and 0 ≤ d = d ≤ L . These are necessary and suﬃcient AC AB A BC AB B sin(β–α) sin(β–α) conditions for the branches to intersect. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 48 of 60 Figure 19 Geometry of connection between two neurons. α (β) is the angle of the neuron A’s branch with length L (neuron B’s branch with length L ) with respect to the horizontal axis. θ is the angle between the A B segment connecting AB and the branch A. C represents the virtual point that lies at the intersection of the branches of length L and L . d (resp. d , d ) denotes the distance between A and B (resp. A, C and B, C). A B AB AC BC Note that d ≤ L , d ≤ L AC A BC B Note that the positivity of these quantities imposes conditions linking the angles α, β, η with θ = α – η. 1. If sin(β – α)>0 ⇔ 0< β – α < π ⇔ α < β < π + α,wemusthave sin θ >0 ⇔ 0< θ = α – η < π so that η < α < π + η and sin(β – α + θ)>0 ⇔ 0< β – α + θ = β – η < π so that η < β < π + η (because η ≥ 0). All these constraints are satisﬁed if η < α < β < min(π + α, π + η)= π + η. 2. If sin(β – α)<0 ⇔ –π < β – α <0 ⇔ –π + α < β < α,wemusthave sin θ <0 ⇔ –π < θ = α – η <0 so that –π + η < α < η and sin(β – α + θ)<0 ⇔ –π < β – α + θ = β – η <0 so that –π + η < β < η.All these constraints are satisﬁed if max(–π + α,–π + η)=–π + η < β < α < η. Modulo these conditions, the conditional probability ρ to have intersection given c|(α,β) the angles α, β is ρ = P[Connection | α, β] c|(α,β) sin(β – α + θ) sin θ = P 0 ≤ d ≤ L ,0 ≤ d ≤ L α, β . AB A AB B sin(β – α) sin(β – α) Using the cumulative distribution function (64) of the exponential distribution and the independence of L , L gives A B α+β sin( –η) AB 2 sin(β – α + θ) sin θ ξ β–α sin( ) ρ = 1– F d 1– F d = e . c|(α,β) L AB L AB sin(β – α) sin(β – α) Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 49 of 60 The probability to connect the two branches in the ﬁrst quadrant is then α+β sin( –η) AB 2 π+η π+η 1 ξ β–α sin( ) ρ (d , η)= e dα dβ c AB 4π α=η β=α (65) α+β d sin( –η) AB 2 η α ξ β–α sin( ) + e dα dβ , α=–π+η β=–π+η which depends on the distance between the two cells and their angle η, depending para- metrically on the characteristic length ξ. Note that the condition of positivity of the sine ratio ensures an exponential decay of the probability as d increases. AB Remark The positivity of arguments in the exponential implies that π+η π+η η α 2 2 4π ρ (d , η) ≤ dα dβ + dα dβ = π , c AB α=η β=α α=–π+η β=–π+η so that ρ (d , η) ≤ . c AB Appendix D: Linear analysis D.1 General solution of the linear dynamical system dX Here, we consider dynamical system (34), = L.X + F(t), whose general solution is dt L(t–s) X (t)= e .F(s) ds. The behaviour of this integral depends on the spectrum of L.The diﬃcultyisthat L is not diagonalisable (because of the activity term h I ). We write it in the following form: B N,N ⎛ ⎞ ⎛ ⎞ 0 0 0 I N,N N,N N,N ⎜ ⎟ % N,N A & – W ⎜ τ B ⎟ ⎜ ⎟ L = + . ⎜ 0 0 ⎟ ⎝ 0 0 0 ⎠ N,N N,N N,N N,N N,N B N,N ⎝ ⎠ W – A τ h I 0 0 B N,N N,N N,N N,N 0 0 – N,N N,N We assume that the matrix M is diagonalisable. Even in this case, L is not diagonalisable because of the Jordan matrix J .Wedenoteby φ the normalised eigenvectors of M and by λ the corresponding eigenvalues with β = 1,...,2N. The eigenvalues of D are then the 1 1 2N eigenvalues of M plus N eigenvalues – .Wedenotethemby λ too, with λ =– , β β τ τ a a β =2N + 1,...,3N. The eigenvectors of D have the form , β = 1,...,2N; P = e , β =2N + 1,...,3N, where 0 is the N-dimensional vector with entries 0 and e is the canonical basis vector N β in direction β.The matrix P made by the columns P is the matrix which diagonalises D. β Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 50 of 60 –1 We denote by = P DP the diagonal form, where = Diag{λ , β = 1,...,3N }. P and –1 P have the following block form: –1 0 0 2N,N 2N,N –1 P = ; yP = , (66) 0 I 0 I N,2N N,N N,2N N,N where is the matrix whose columns are the eigenvectors φ of M.Thisformimplies –1 that P = P = δ for β =2N + 1,...,3N. αβ αβ αβ +∞ Lt Lt t n We now compute e using the series expansion e = (D + J ) .Using therela- n=0 n! tions 2 n J =0 ; D .J = – J , 3N,3N one proves that n–1 k n n n–1–k (D + J ) = D + J . – D . k=0 Therefore +∞ n–1 t 1 Lt Dt n–1–k e = e + J . – D . n! τ n=1 k=0 –1 We use the matrices P, P to write it in the form +∞ n–1 t 1 Lt t –1 n–1–k –1 e = P.e .P + J . – P. .P . n! τ n=1 k=0 From relation (36) we obtain, for the entries of X (t), 3N –1 λ (t–s) X (t)= P P e F (s) ds α αβ γ βγ β,γ =1 (67) 3N +∞ n–1 t n (t – s) 1 –1 n–1–k + J P P – λ F (s) ds. α,δ δβ γ βγ β n! τ t a δ,β,γ =1 n=1 k=0 We consider the ﬁrst term of this equation. We use F = F , γ = i = 1,..., N (BCells). γ B V dV i i drive drive We recall that, from (13), F (t)= + ,sothat i τ dt t t λ (t–s) λ (t–s) β β e F (s) ds = V (t)+ + λ e V (s) ds. (68) γ γ β γ drive drive t B t 0 0 Moreover, 3N 3N 3N 3N 3N –1 –1 P P V (t)= V (t) P P = V (t)δ = V (t). αβ γ γ αβ γ αγ α βγ drive drive βγ drive drive β=1 γ =1 γ =1 β=1 γ =1 Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 51 of 60 We extend the deﬁnition of drive term (1)to 3N dimensions such that V (t)=0 if drive α > N.Thus 3N –1 λ (t–s) P P e F (s) ds αβ γ βγ β,γ =1 3N N –1 λ (t–s) = V (t)+ + λ P P e V (s) ds. α β αβ γ drive βγ drive B –∞ β=1 γ =1 We decompose the sum over β in three sums: β = 1,..., N corresponding to BCells; β = N + 1,...,2N corresponding to ACells; β =2N + 1,...,3N corresponding to activities of BCells. We deﬁne (Eq. (38)inthe text): N N B –1 λ (t–s) E (t)= + λ P P e V (s) ds, α = 1,..., N, β αβ γ B,α βγ drive β=1 γ =1 corresponding to the indirect eﬀect, via the ACell connectivity, of the drive on BCell volt- ages. The term 2N N B –1 λ (t–s) E (t)= + λ P P e V (s) ds, α = N + 1,...,2N β αβ γ A,α βγ drive γ =1 β=N+1 (Eq. (39) in the text) corresponds to the eﬀect of BCell drive on ACell voltages. The third term 3N N –1 λ (t–s) + λ P P e V (s) ds =0, α =2N + 1,...,3N, β αβ γ βγ drive B t β=2N+1 γ =1 –1 because P = δ . βγ βγ To compute the second term in Eq. (67), we ﬁrst ﬁrst remark that J =0 if α = 1,...,2N α,δ and J = h δ if α =2N + 1,...,3N, so that this term is nonzero only if α =2N + α,δ B α–2N,δ –1 1,...,3N (BCell activities). Also F =0 for γ = 1,..., N,while P = δ for β =2N + γ βγ βγ 1,...,3N. Therefore, for α =2N + 1,...,3N,the second term in X (t)is 2N N +∞ n–1 t k (t – s) 1 –1 n–1–k h P P – λ F (s) ds. B α–2N β γ βγ β n! τ β=1 γ =1 n=1 k=0 We now simplify the series: +∞ n–1 +∞ n–1 k k n n (t – s) 1 (t – s) 1 n–1–k n–1 – λ = λ – β β n! τ n! τ λ a a β n=1 k=0 n=1 k=0 1 n +∞ 1–(– ) (t – s) τ λ n–1 = λ n! 1+ n=1 τ λ a β Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 52 of 60 ' ( +∞ +∞ t–s (– ) 1 1 (λ (t – s)) = – λ 1+ n! n! τ λ n=1 n=1 a β 1 t–s λ (t–s) – β τ = e – e . λ + Thetimeintegraliscomputedthe same wayasEq. (68): +∞ n–1 t n (t – s) 1 n–1–k – λ F (s) ds n! τ t a n=1 k=0 t t 1 t–s λ (t–s) – β τ = e F (s) ds – e F (s) ds γ γ λ + β t t 0 0 t t 1 1 1 1 t–s λ (t–s) = + λ e V (s) ds – – e V (s) ds . β γ γ 1 drive drive λ + τ τ τ B B a β t t 0 0 Similar to Eqs. (38), (39)inthe text,weintroduce ' ( 2N N 1 λ (t–s) 1 ( + λ ) e V (s) ds β γ B –1 τ t drive B 0 E (t)= h P P , t–s B α–2N β a,α βγ t 1 1 1 – λ + –( – ) e V (s) ds τ t drive β=1 γ =1 a τ τ B a 0 α =2N + 1,...,3N, corresponding to the action of BCells and ACells on the activity of BCells via the net- 2N –1 work eﬀect. Let us consider in more detail the second term. From (66), P P = α–2N β β=1 βγ δ ,thus α–2N γ 2N N N 2N t t t–s t–s – – –1 –1 τ τ a a P P e V (s) ds = e V (s) ds P P α–2N β γ γ α–2N β βγ drive drive βγ t t 0 0 γ =1 γ =1 β=1 β=1 α–2N γ t–s – 0 = e V (s) ds ≡ A (t) α–2N α–2N drive (Eq. (41)inthe text). This ﬁnally leads to ((40)inthe text) 2N N 1 1 1 λ + – + τ τ τ B –1 B λ (t–s) B a 0 E (t)= h P P e V (s) ds + A (t) , B α–2N β γ a,α βγ drive α–2N 1 1 λ + λ + β t β τ τ β=1 γ =1 a a α =2N + 1,...,3N. D.2 Spectrum of L and stability of dynamical system (34) Here, we assume that a BCell connects only one ACell, with a weight w uniform for all B + + BCells, so that W = w I , w > 0.WealsoassumethatACellsconnect to BCells with N,N – – a connectivity matrix W, not necessarily symmetric, with a uniform weight –w , w >0, A – so that W =–w W. We have shown in the previous section that the 2N ﬁrst eigenvalues and eigenvectors of L are given by the 2N eigenvalues and eigenvectors of M,which reads Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 53 of 60 now as follows: N,N – – –w W M = . (69) + N,N w I – N,N We now show that this speciﬁc structure allows us to compute the spectrum of M in terms of the spectrum of W. D.2.1 Eigenvalues and eigenvectors of M We denote by κ , n = 1,..., N, the eigenvalues of W ordered as |κ |≤|κ | ≤ ··· ≤ |κ |,and n 1 2 n ψ is the corresponding eigenvector. We normalise ψ so that ψ .ψ =1, where † is the n n n adjoint. (Note that, as W is not symmetric in general, eigenvectors are complex.) We shall neglect the case where, simultaneously, =0 and κ =0 for some n. ± ± ± n Proposition For each n, there is a pair of eigenvalues λ and eigenvectors φ = c n n n ± ρ ψ n n ± 1 of M with c = √ (normalisation factor) and 1+(ρ ) 1 1 ⎪ (1 ± 1–4μκ ), κ =0, =0; – n n 2τw κ τ ± 1 ρ = w τ, κ =0, =0; (70) ⎪ ! w 1 1 ± – , =0, w κ τ where 1 1 1 = – τ τ τ A B and 1 1 1 = + . τ τ τ AB A B Eigenvalues are given by ⎨ 1 1 1 – ∓ 1–4μκ , =0; ± 2τ 2τ τ AB λ = (71) ⎩ 1 1 – + – ∓ –w w κ , =0, τ τ with – + 2 μ = w w τ ≥ 0. As a consequence, in addition to the N last eigenvalues – , L admits 2N eigenvalues given by (71), while the 2N ﬁrst columns of the matrix P (eigenvectors of L) are as follows: ⎛ ⎞ ⎛ ⎞ ψ ψ n n 1 1 ⎜ ⎟ ⎜ ⎟ – + P = $ ρ ψ ; P = $ ρ ψ , β = n = 1,..., N. (72) β ⎝ ⎠ β+N ⎝ ⎠ n n n n – 2 + 2 1+(ρ ) 1+(ρ ) n n 0 0 N N For the N last eigenvectors P = e , β =2N + 1,...,3N. β β Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 54 of 60 Remark The structure of these eigenvectors is quite instructive. Indeed, the factors ρ control the projection of the eigenvectors P on the space of ACells, thereby tuning the inﬂuence of ACells via lateral connectivity. Proof We use here the generic notation λ , φ , β = 1,...,2N, for the eigenvalues and as- β β sociated eigenvectors of M. If we assume that φ is of the form φ = for some n, β β ρψ then we have I 1 – – – –w W (– – w ρκ ).ψ ψ ψ n n n n τ τ B B M.φ = . = = λ , β β I ρ + + w I – ρψ (– + w ).ψ ρψ n n n τ τ A A which gives ⎨ 1 – (– – w ρκ )= λ , n β (– + w )= λ ρ, leading to – 2 + w κ ρ – ρ + w =0, where 1 1 1 = – . τ τ τ A B This gives, if κ =0 and =0, ρ = (1 ± 1–4μκ ), 2τw κ where – + 2 μ = w w τ ≥ 0. Thus, for each n, there are two eigenvalues: 1 1 λ =– ∓ 1–4μκ , 2τ 2τ AB with 1 1 1 = + . τ τ τ AB A B 1 1 Note that ≥ . τ τ AB 1 ± + If κ =0, =0, ρ = w τ ,then λ =– – w ρκ . β n B Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 55 of 60 1 ± w 1 ± 1 1 – + Finally, if κ =0, =0, (τ = τ ), ρ =– and λ =– ± –w w κ .If κ =0, =0, n A B – n n n n τ w κ τ τ n B there is no solution for ρ. Remark When μ =0, M is diagonal: the N ﬁrst eigenvalues are – ,the N next eigen- 1 1 1 + – values are – .Wehaveinthiscase λ =– and λ =– . Therefore, in order to be n n τ τ τ A B A coherent with this diagonal form of L when μ = 0, we order eigenvalues and eigenvectors + – of M such that the N ﬁrst eigenvalues are λ = λ , β = 1,..., N,and the N next are λ = λ , β β n n β = N + 1,...,2N. D.2.2 Stability of eigenmodes Stability of eigenmodes when W is symmetric If W is symmetric, its eigenvalues κ are real, but λ , β = 1,...,2N, can be real or complex, depending on κ ,as μ is positive. β n We have four cases: • κ <0. Then, from (71), λ s are real, and there are two cases. If > 0, the eigenvalues n β λ , β = 1,..., N, can have a positive real part (unstable), while λ , β = N + 1,...,2N,has β β always a negative real part (stable); for < 0, the situation is inverted. In both cases, the eigenvalue λ has a positive real part if 1 τ τ A B μ >– ≡ μ , n,u κ (τ – τ ) n B A which reads, using the deﬁnition of μ, as follows: 1 1 – + w w >– . (73) τ τ κ A B n Thus, τ , τ play a symmetric role. If =0 (τ = τ ), all eigenvalues are real. Eigenvalues A B A B – + – + 1 1 λ are all stable. The eigenvalue λ becomes unstable if w w >– , corresponding to n n 2 τ n (73). Proof There are two cases. – >0 ⇔ τ < τ . A B 1 1 λ =– ± 1–4μκ >0 β n 2τ 2τ AB ⇔± 1–4μκ > . AB Only + is possible (because τ and τ are positive). This gives AB 1–4μκ > , AB τ τ τ A B 1– =–4 >4μκ , 2 2 τ (τ – τ ) B A AB which is possible because κ <0. Thus, λ is unstable if n β 1 τ τ A B μ >– ≡ μ . n,u κ (τ – τ ) n B A Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 56 of 60 – <0 ⇔ τ > τ . A B 1 1 – ± 1–4μκ >0 2τ 2τ AB ⇔± 1–4μκ < . AB Only – is possible (because τ <0). This gives 1–4μκ > , AB the same condition as in the previous item. 1 1 – + –If =0, λ =– ∓ –w w κ so that eigenvalues are real. The eigenvalue with the β n τ τ – + minus sign (λ ) are all stable. The eigenvalue λ becomes unstable if n n 1 1 – + w w >– . τ κ • κ >0. Then λ , β = 1,...,2N, are real or complex. If =0, they are complex if n β μ > ≡ μ . n,c 4κ 1 1 In this case the real part is – , the imaginary part is ± 1–4μκ , and all eigenvalues 2τ 2τ AB are stable. If μ ≤ μ , eigenvalues λ are real and all modes are stable as well. Indeed: n,c β 1 1 –If =0, all eigenvalues are equal to – , hence are stable. τ 2τ AB –If >0 ⇔ τ < τ . A B 1 1 – ± 1–4μκ >0 2τ 2τ AB ⇔± 1–4μκ > , AB which is not possible because >1,whereas 1–4μκ <1. AB –If <0 ⇔ τ > τ . A B 1 1 – ± 1–4μκ >0 2τ 2τ AB ⇔± 1–4μκ < . AB Only – is possible because τ <0. 1–4μκ > , AB which is not possible because >1,whereas 1–4μκ <1. AB Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 57 of 60 Stability of eigenmodes when W is asymmetric If W is asymmetric, eigenvalues κ are complex, κ = κ + iκ .Wewrite λ = λ + iλ , β = 1,...,2N,with n n,r n,i β β,r β,i 1 1 1 λ =– ± a + u ; β,r n n 2τ 2τ AB 2 1 1 λ = ± u – a , β,i n n 2τ 2 2 2 2 2 2 where a =1 – 4μκ and u = (1 – 4μκ ) +16μ κ = 1–8μκ +16μ |κ | .Note n n,r n n,r n n,i n,r that we recover the real case when κ =0 by setting u = a . n,i n n Instability occurs if a + u >2 , n n AB a condition depending on κ and κ . n,r n,i Acknowledgements We warmly acknowledge Olivier Marre and Frédéric Chavane for their insightful comments, as well as Michael Berry, Matthias Hennig, Benoit Miramond, Stephanie Palmer and Laurent Perrinet for their thorough feedback as Jury members of Selma Souihel’s PhD. Funding This work was supported by the National Research Agency (ANR), in the project “Trajectory”, https://anr.fr/Project-ANR-15- CE37-0011, funding Selma Souihel’s PhD, and by the interdisciplinary Institute for Modelling in Neuroscience and Cognition (NeuroMod http://univ-cotedazur.fr/en/idex/projet-structurant/neuromod) of the Université Côte d’Azur. Availability of data and materials The model source code is available upon request. Ethics approval and consent to participate Not applicable. Competing interests The authors declare that they have no competing interests. Consent for publication The authors transfer to Springer the non-exclusive publication rights, and they warrant that their contribution is original and that they have full power to make this grant. Authors’ contributions Both authors contributed to the ﬁnal version of the manuscript. BC supervised the project. Both authors read and approved the ﬁnal manuscript. Endnote dA a B From (6) >0 if A (t)< h τ V (t). 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Publisher: Springer Journals
Copyright: Copyright © The Author(s) 2021
eISSN: 2190-8567
DOI: 10.1186/s13408-020-00101-z
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Abstract

Biovision Team and Neuromod We analyse the potential eﬀects of lateral connectivity (amacrine cells and gap Institute, Inria, Université Côte d’Azur, Nice, France junctions) on motion anticipation in the retina. Our main result is that lateral connectivity can—under conditions analysed in the paper—trigger a wave of activity enhancing the anticipation mechanism provided by local gain control (Berry et al. in Nature 398(6725):334–338, 1999; Chen et al. in J. Neurosci. 33(1):120–132, 2013). We illustrate these predictions by two examples studied in the experimental literature: diﬀerential motion sensitive cells (Baccus and Meister in Neuron 36(5):909–919, 2002) and direction sensitive cells where direction sensitivity is inherited from asymmetry in gap junctions connectivity (Trenholm et al. in Nat. Neurosci. 16:154–156, 2013). We ﬁnally present reconstructions of retinal responses to 2D visual inputs to assess the ability of our model to anticipate motion in the case of three diﬀerent 2D stimuli. Keywords: Retina; Motion anticipation; Lateral connectivity; 2D 1 Introduction Our visual system has to constantly handle moving objects. Static images do not exist for it, as the environment, our body, our head, our eyes are constantly moving. A “computa- tional”, contemporary view, likens the retina to an “encoder”, converting the light photons coming from a visual scene into spike trains sent—via the axons of ganglion cells (GCells) that constitute the optic nerve—to the thalamus, and then to the visual cortex acting as a “decoder”. In this view, comparing the size and the number of neurons in the retina, i.e. about 1 million GCells (humans), to the size, structure and number of neurons in the visual cortex (around 538 million per hemisphere in the human visual cortex [22]), the “en- coder” has to be quite smart to eﬃciently compress the visual information coming from a world made of moving objects. Although it has long been thought that the retina was not more than a simple camera, there is more and more evidence that this organ is “smarter than neuroscientists believed” [41]. It is indeed able to perform complex tasks and general motion feature extractions, such as approaching motion, diﬀerential motion and motion anticipation, allowing the visual cortex to process visual stimuli with more eﬃciency. Computations occurring in neuronal networks downstream photoreceptors are crucial to make sense out of the “motion-agnostic” signal delivered by these retinal sensors [11]. There exists a wide range of theoretical and biological approaches to studying retinal pro- cessing of motion. Moving stimuli are generally considered as a spatiotemporal pattern of © The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 2 of 60 light intensity projected on the retina, from which it extracts relevant information, such as the direction of image motion. Detecting motion requires then neural networks to be able to process, in a nonlinear fashion, moving stimuli, asymmetrically in time. [6, 10][37, 88] The ﬁrst motion detector model was proposed by Hassenstein and Reichardt [42], based on the optomotor behaviour of insects. The model relies on changes in contrast of two spatially distinct locations inside the receptive ﬁeld of a motion sensitive neuron. The neuron will only produce a response if contrast changes are temporally delayed, and is thus not only selective to direction, but also to motion velocity. Several models have then been developed based on Reichardt detectors, attempting to explain motion processing in vertebrate species as well, but they all share the common feature of integrating spatio- temporal variations of the contrast. To perceive motion as coherent and uninterrupted, an additional integration over motion detectors is hypothesised to take place. This inte- gration usually takes the form of a pooling mechanism over visual space and time. However, before performing these high-level motion detection computations, the pho- tons received by the retina need ﬁrst to be converted into electrical signals that will be transmitted to the visual cortex, a process known as phototransduction. This process takes about 30–100 milliseconds. Though this might look fast, it is actually too slow. A tennis ball moving at 30 m/s–108 km/h (the maximum measured speed is about 250 km/h) cov- ers between 0.9 and 3 m during this time. So, without a mechanism compensating this delay, it would not be possible to play tennis (not to speak of survival, a necessary con- dition for a species to reach the level where playing tennis becomes possible). The visual system is indeed able to extrapolate the trajectory of a moving object to perceive it at its actual location. This corresponds to anticipation mechanisms taking place in the visual cortex and in the retina with diﬀerent modalities [4, 56, 57, 84]. In the early visual cortex an object moving across the visual ﬁeld triggers a wave of ac- tivity ahead of motion thanks to the cortical lateral connectivity [8, 46, 77]. Jancke et al. [46] ﬁrst demonstrated the existence of anticipatory mechanisms in the cat primary vi- sual cortex. They recorded cells in the central visual ﬁeld of area 17 (corresponding to the primary visual cortex) of anaesthetised cats responding to small squares of light either ﬂashed or moving in diﬀerent directions and with diﬀerent speeds. When presented with the moving stimulus, cells show a reduction of neural latencies, as compared to the ﬂashed stimulus. Subramaniyan et al. [77] reported the existence of similar anticipatory eﬀects in the macaque primary visual cortex, showing that a moving bar is processed faster than a ﬂashed bar. They give two possible explanations to this phenomenon: either a shift in the cells receptive ﬁelds induced by motion, or a faster propagation of motion signals as compared to the ﬂash signal. In the retina, anticipation takes a diﬀerent form. One observes a peak in the ﬁring rate response of GCells to a moving object occurring before thepeakresponsetothe same object when ﬂashed. This eﬀect can be explained by purely local mechanisms at individual cells level [9, 20]. To our best knowledge, collective eﬀects similar to the cortical ones, that is, a rise in the cell’s activity before the object enters in its receptive ﬁeld due to a wave of activity ahead of the moving object, have not been reported yet. In a classical Hubel–Wiezel–Barlow [5, 44, 60] view of vision, each retinal ganglion cell carries a ﬂow of information with an eﬃcient coding strategy maximising the available channel capacity by minimising the redundancy between GCells. From this point of view, the most eﬃcient coding is provided when GCells are independent encoders (parallel Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 3 of 60 Figure 1 Synthetic view of the retina model. A stimulus is perceived by the retina, triggering diﬀerent pathways. Pathway I (blue) corresponds to a feed-forward response where, from top to bottom: The stimulus is ﬁrst convolved with a spatio-temporal receptive ﬁeld that mimics the outer plexiform layer (OPL) (“Bipolar receptive ﬁeld response”). This response is rectiﬁed by low voltage threshold (blue squares). Bipolar cell responses are then pooled (blue circles with blue arrows) and input ganglion cells. The ﬁring rate response of a ganglion cell is a sigmoidal function of the voltage (blue square). Gain control can be applied at the bipolar and ganglion cell level (pink circles) triggering anticipation. This corresponds to the label II (pink)inthe ﬁgure. Lateral connectivity is featured by pathway III (brown) through ACells and pathway IV (green)through gap-junctions at the level of GCells streaming identiﬁed by a “I” in Fig. 1). In this setting one can propose a simple and satis- factory mechanism explaining anticipation in the retina based on gain control at the level of bipolar cells (BCells) and GCells (label “II” in Fig. 1)[9, 20]. Yet, some GCells are connected. Either directly, by electric synapses–gap junctions (pathway IV in Fig. 1), or indirectly via speciﬁc amacrine cells (ACells, pathway III in Fig. 1). It is known that these pathways are involved in motion processing by the retina. AII ACells play a fundamental role in the interaction between the ON and OFF cone path- way [54]. There are GCells able to detect the diﬀerential motion of an object onto a mov- ing background [1] thanks to ACell lateral connectivity. Some GCells are direction sensi- tive because they are connected via a speciﬁc asymmetric gap junctions connectivity [80]. Could lateral connectivity play a role in motion anticipation, inducing a wave of activity ahead of the motion similar to the cortical anticipation mechanism? Whilesomestud- ies hypothesise that local gain control mechanisms can be explained by the prevalence of inhibition in the retinal connectome [47], the mechanistic aspects of the role of lateral connectivity on motion anticipation has not, to the best of our knowledge, been addressed yet on either experimental or computational grounds. In this paper, we address this question from a modeler, computational neuroscientist, point of view. We propose here a simpliﬁed description of pathways I, II, III, IV of Fig. 1, grounded on biology, but not sticking at it, to numerically study the potential eﬀects of gain Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 4 of 60 control combined with lateral connectivity—gap junctions or ACells—on motion antici- pation. The goal here is not to be biologically realistic but, instead, to propose from bi- ological observations potential mechanisms enhancing the retina’s capacity to anticipate motion and compensate the delay introduced by photo-transduction and feed-forward processing in the cortical response. We want the mechanisms to be as generic as possible, so that the detailed biological implementation is not essential. This has the advantage of making the model more prone to mathematical analysis. The ﬁrst contribution of our work lies in the development of a model of retinal antici- pation where GCells have gain control, orientation selectivity and are laterally connected. It is based on a model introduced by Chen et al. in [20] (itself based on [9]) reproduc- ing several motion processing features: anticipation, alert response to motion onset and motion reversal. The original model handles one-dimensional motions and its cells are not laterally connected (only pathways I and II were considered). The extension proposed here features cells with oriented receptive ﬁeld, although our numerical simulations do not consider this case (see discussion). Lateral connectivity is based on biophysical mod- elling and existing literature [1, 28, 43, 78, 80]. In this framework, we study diﬀerent types of motion. We start with a bar moving with constant speed and study the eﬀect of con- trast, bar size and speed on anticipation, generalising previous studies by Berry et al. [9] and Chen et al. [20]. We then extend the analysis to two-dimensional motions, investigat- ing, e.g. angular motion and curved trajectories. Far from making an exhaustive study of anticipation in complex stimuli, the goal here is to calibrate anticipation without lateral connectivity so as to compare the eﬀect when connectivity is switched on. The second contribution emphasises a potential role of lateral connectivity (gap junc- tions and ACells) on anticipation. For this, we ﬁrst make a general mathematical analysis concluding that lateral connectivity can induce a wave triggered by the stimulus which un- der speciﬁc conditions can improve anticipation. The eﬀect depends on the connectivity graph and is nonlinearly tuned by gain control. In the case of gap junctions, the wave prop- agation depends on whether connectivity is symmetric (the standard case) or asymmetric, as proposed by Trenholm et al. in [80] for a speciﬁc type of direction sensitive GCells. In the case of ACells, the connectivity graph is involved in the spectrum of a propagation operator controlling the time evolution of the network response to a moving stimulus. We instantiate this general analysis by studying diﬀerential motion sensitive cells [1]with two types of connectivity: nearest neighbours, and a random connectivity, inspired from biology [78], where only numerical results are shown. In general, the anticipation eﬀect depends on the connectivity graph structure and the intensity of coupling between cells as well as on the respective characteristic times of response of cells in a way that we analyse mathematically and illustrate numerically. We actually observe two forms of anticipation. The ﬁrst one, discussed in the beginning of this introduction and already observed in [9, 20], is a shift in the peak of a retinal GCell response occurring before the object reaches the centre of its receptive ﬁeld. In our case, lateral connectivity can enhance the shift improving the mere eﬀect of gain control. The second anticipation eﬀect we observe is a raise in GCells activity before the bar reaches the receptive ﬁeld of the cell, similarly to what is observed in the cortex [8]. To the best of our knowledge, this eﬀect has not been studied in the retina and constitutes therefore a prediction of our model. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 5 of 60 The paper is organised as follows. Section 2 introduces the model of retinal organisa- tion and cell types dynamics, ending up with a system of nonlinear diﬀerential equations driven by a time-dependent stimulus. Section 3 is divided into four parts. The ﬁrst part analyses mathematically the potential anticipation eﬀects in a general setting before con- sidering the role of ACells and lateral inhibition on anticipation (Sect. 3.2)and gapjunc- tions (Sect. 3.3). Both sections contain general mathematical results as well as numerical simulations for one-dimensional motion. The fourth part investigates examples of two- dimensional motions. The last section is devoted to discussion and conclusion. In Ap- pendix A, we have added the values of parameters used in simulations and in Appendix B the receptive ﬁelds mathematical form used in the paper as well as the numerical method to compute eﬃciently the response of oriented two-dimensional receptive ﬁelds to spatio- temporal stimuli. Appendix C presents a model of random connectivity from amacrine to bipolar cells inspired from biological data [78]. Finally, Appendix D contains mathemati- cal results which constitute the skeleton of the work, but whose proof would be too long to integrate in the core of the paper. This work is based on Selma Souihel’s PhD thesis where more extensive results can be found [74]. In particular, there is an analysis of the conjugated eﬀects of retinal and cortical anticipation, subject of a forthcoming paper and brieﬂy discussed in the conclusion. In all the following simulations, we use the CImg Library, an open-source C++ tool kit for image processing in order to load the stimuli and reconstruct the retina activity. The source code is available on demand. 2 Material and methods 2.1 Retinal organisation In the retinal processing light photons coming from a visual scene are converted into volt- age variations by photoreceptors (cones and rods). The complex hierarchical and layered structure of the retina allows to convert these variations into spike trains, produced by ganglion cells (GCells) and conveyed to the thalamus via their axons. We considerably sim- plify this process here. Light response induces a voltage variations of bipolar cells (BCells), laterally connected via amacrine cells (ACells), and feeding GCells, as depicted in Fig. 1. We describe this structure in details here. Note that neither BCells nor ACells are spiking. They act synaptically on each other by graded variations of their potential. We assimilate the retina to a ﬂat, two-dimensional square of edge length L mm. There- fore, we do not integrate the three-dimensional structure of the retina in the model, merely for mathematical convenience. Spatial coordinates are noted x, y (see Fig. 2 for the whole structure). In the model, each cell population tiles the retina with a regular square lattice. The den- sity of cells is therefore uniform for convenience but the extension to nonuniform density canbeaﬀorded.For thepopulation p,wedenoteby δ the lattice spacing in mm and by N p p the total number of cells. Without loss of generality we assume that L, the retina’s edge size, is a multiple of δ .Wedenoteby L = thenumberofcells p per row or column so that p p N = L . Each cell in the population p has thus Cartesian coordinates (x, y)=(i δ , i δ ), p x p y p (i , i ) ∈{1,..., L } . To avoid multiples indices, we associate with each pair (i , i ) a unique x y p x y index i = i +(i –1)L . The cell of population p located at coordinates (i δ , i δ )is then x y p x p y p denoted by p .Wedenoteby d[p , p ] the Euclidean distance between p and p . i i i j j Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 6 of 60 Figure 2 Example of a retina grid tiling and indexing. The green and blue ellipses denote respectively the positive centre and the negative surround of the BCell receptive ﬁeld K . The centre of RF coincides with the position of the cell (blue and green arrows). The red ellipse denotes the ganglion cell pooling over bipolar cells (Eq. (17)) We use the notation V for the membrane potential of cell p . Cells are coupled. The p i synaptic weight from cell p to cell q reads W . Thus, the pre-synaptic neuron is ex- j i pressed in the upper index; the post-synaptic, in the lower index. Dynamics of cells is voltage-based. This is because our model is constructed from Chen et al. model [20], itself derived from Berry et al. [9], where a voltage-based description is used. Implicitly, volt- age is measured with respect to the rest state of the cell (V = 0 when the cell receives no input). 2.2 Bipolar cell layer The model consists ﬁrst of a set of N BCells, regularly spaced by a distance δ ,withspatial B B coordinates x , y , i = 1,..., N . Their voltage, a function of the stimulus, is computed as i i B follows. 2.2.1 Stimulus response and receptive ﬁeld The projection of the visual scene on the retina (“stimulus”) is a function S(x, y, t)where t is the time coordinate. As we do not consider colour sensitivity here, S characterises a black and white scene with a control on the level of contrast ∈ [0, 1]. A receptive ﬁeld (RF) is a region of the visual ﬁeld (the physical space) in which stimulation alters the voltage of a cell. Thus, BCell i has a spatio-temporal receptive ﬁeld K , featuring the biophysical processes occurring at the level of the outer plexiform layer (OPL), that is, photo-receptors (rod-cones) response modulated by horizontal cells (HCells). As a consequence, in our model, the voltage of BCell i is stimulus-driven by the term +∞ +∞ t x,y,t V (t)=[K ∗ S](t)= K(x–x , y–y , t –s)S(x, y, s) dx dy ds,(1) i B i i drive i x=–∞ y=–∞ s=–∞ Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 7 of 60 x,y,t where ∗ means space-time convolution. We consider only one family of BCells so that the kernel K is the same for all BCells. What changes is the centre of the RF, located at x , y , which also corresponds to the coordinates of the BCell i. We consider in the paper sep- arable kernel K(x, y, t)= K (x, y)K (t)where K is the spatial part and K is the temporal S T S T part. The detailed form of K is given in Appendix B. We have dV x,y,t dS drive = K ∗ (t), (2) dt dt resulting from the condition K (x, y,0) = 0 (see Appendix B). Note that the exponential decay of the spatial and temporal part at inﬁnity ensures the existence of the space-time integral. The spatial integral K (x, y)S(x, y, u) dx dy is numerically computed using er- 2 S ror function in the case of circular RF and a computer vision method from Geusenroek et al. [39] in the case of anisotropic RF, allowing to integrate generalised Gaussians with an eﬃcient computational time. This method is described in Appendix B. For explanations purposes, we will often use the approximation of V by a Gaussian drive pulse, with width σ , propagating at constant speed v along the direction e : 2 2 (x–vt) (x–vt) 1 1 A V 0 – 0 – 2 2 2 2 σ σ V (t)= √ e ≡ √ e,(3) drive 2πσ 2π where x = iδ is the horizontal coordinate of BCell i and where σ is in mm, A is in mV.mm B 0 (and is proportional to stimulus contrast), V is in mV. 2.2.2 BCells voltage and gain control In our model, the BCell voltage is the sum of external drive (1)receivedbythe BCelland of a post-synaptic potential P induced by connected ACells: V (t)= V (t)+ P (t). (4) B i B i drive i The form of P is given by Eq. (11)inSect. 2.3.1. P (t) = 0 when no ACells are considered. B B i i BCells have voltage threshold [9]: 0, if V ≤ θ ; B B N (V )= (5) B B V – θ ,else. B B Values of parameters are given in Appendix A. BCells have gain control, a desensitisation when activated by a steady illumination [92]. 2+ This desensitisation is mediated by a rise in intracellular calcium Ca at the origin of a feedback inhibition preventing thus prolonged signalling of the ON BCell [20, 73]. Fol- lowing Chen et al., we introduce the dimensionless activity variable A obeying the dif- ferential equation dA A B B i i =– + h N V (t).(6) B B dt τ a Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 8 of 60 Assuming an initial condition A (t ) = 0 at initial time t ,the solution is B 0 0 t–s A (t)= h e N V (s) ds.(7) B B B i i The bipolar output to ACells and GCells is then characterised by a nonlinear response to its voltage variation given by R (V , A )= N (V )G (A ), (8) B B B B B B B i i i i i where 0, if A ≤ 0; G (A )= (9) B B i 1 ,else. 1+A Note that R has the physical dimension of a voltage, whereas, from Eq. (9), the activity A is dimensionless. As a consequence, the parameter h in Eq. (6)mustbeexpressed in B B –1 –1 ms mV .Form(9) and its 6th power are based on experimental ﬁts made by Chen et al. Itsformisshown in Fig. 3. In the course of the paper we will use the following piecewise linear approximation also represented in Fig. 3: 0, if A ∈]– ∞,0[ ∪ [ ,+∞[, silent region; G (A)= (10) 1, if A ∈ [0, ], maximal gain; 3 2 4 – A +2, if A ∈ [ , ], fast decay. 2 3 3 Thanks to this approximation, we roughly distinguish three regions for the gain function G (A). This shape is useful to understand the mechanism of anticipation (Sect. 3.1). Figure 3 Gain control (9) as a function of activity A. The function l(A), in dashed line, is a piecewise linear approximation of G (A) from which three regions are roughly deﬁned. In the region “Silent” the gain vanishes so the cell does not respond to stimuli; in the region “Max”, the gain is maximal so that cell behaviour does not show any diﬀerence with a not gain-controlled cell; the region “Fast decay” is the one which contributes to anticipation by shifting the peak in the cell’s activity (see Sect. 3.1). The value A = corresponds to the value of activity where gain control, in the piecewise linear approximation, becomes eﬀective Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 9 of 60 2.3 Amacrine cell layer There is a wide variety of ACells (about 30–40 diﬀerent types for humans) [64]. Some spe- ciﬁc types are well studied such as starburst amacrine cells, which are involved in direction sensitivity [33, 34, 81], as well as contrast impression and suppression of GCells response [58], or AII, a central element of the vertebrate rod-cone pathway [54]. Here, we do not want to consider speciﬁc types of ACells with a detailed biophysical description. Instead, we want to point out the potential role they can play in motion an- ticipation thanks to the inhibitory lateral connectivity they induce. We focus on a speciﬁc circuitry involved in diﬀerential motion: an object with a diﬀerent motion from its back- ground induces more salient activity. The mechanism, observed in mice and rabbit retinas [41, 61], is featured in Fig. 1, pathway III. When the left pathway receives a diﬀerent illumi- nation from the right pathway (corresponding, e.g. to a moving object), this asymmetry is ampliﬁed by the ACells’ mutual inhibition, enhancing the response of the left pathway in a “push–pull” eﬀect. We want to propose that such a mutual inhibition circuit, deployed in a lattice through the whole retina, can generate—under speciﬁc conditions mathematically analysed—a wave of activity propagation triggered by the moving object. In the model, ACells tile the retina with a lattice spacing δ . We index them with j = 1,..., N . 2.3.1 Synaptic connections between ACells and BCells We consider here a simple model of ACells. We assimilate them to passive cells (no active ionic channels) acting as a simple relay between BCells. This aspect is further discussed later in the paper. The ACell A , connected to the BCell B , induces on the latter the post- j i synaptic potential: A A j j – P (t)= W (t) γ (t – s)V (s) ds; γ (t)= e H(t), B A B B B j i i –∞ where the Heaviside function H ensures causality. Thus, the post synaptic potential is the mere convolution of the pre synaptic ACell voltage, with an exponential α-proﬁle [28]. In addition, we assume the propagation to be instantaneous. Here, the synaptic weight W < 0 mimics the inhibitory connection from ACell to BCell (glycine or GABA) with the convention that W = 0 if there is no connection from A to B . In general, several ACells input the BCell B giving a total PSP: B t P (t)= W γ (t – s)V (s) ds. (11) B B A i B j –∞ j=1 Conversely, the BCell B connected to A induces, on this cell, a synaptic response char- i j acterised by a post-synaptic potential (PSP) P (t). As ACells are passive elements, their voltage V (t) is equal to this PSP. We have thus V (t)= W γ (t – s)R (s) ds (12) A A B j A i –∞ i=1 Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 10 of 60 with γ (t)= e H(t). Here, W > 0 corresponding to the excitatory eﬀect of BCells on ACells through a glutamate release. Note that the voltage of the BCell is rectiﬁed and gain- controlled. 2.3.2 Dynamics The coupled dynamics of bipolar and amacrine cells can be described by a dynamical system that we derive now. Bipolar voltage By diﬀerentiating (11), (4) and introducing x,y,t V dV S dS i i drive drive F (t)= K ∗ + (t)= + , (13) B B i i τ dt τ dt B B we end up with the following equation for the bipolar voltage: dV 1 B j =– V + W V + F (t), (14) B A B i B j i dt τ j=1 wherewehaveused(2). This is a diﬀerential equation driven by the time-dependent term F containing the stimulus and its time derivative. To illustrate the role of F , let us consider an object moving with a speed v depending dv on time, thus with a nonzero acceleration γ = .Thisstimulushas theform S(x, y, t)= dt dS g(X – v(t)t)with X = ,sothat =–∇g(X – v(t)t).(v + γ t), where ∇ denotes the gra- dt dient. Therefore, thanks to Eq. (14), BCells are sensitive to changes in directions, thereby justifying a study of two-dimensional stimuli (Sect. 3.4). Note that this property is in- dV drive herited from the simple, diﬀerential structure of the dynamics, the term resulting dt from the diﬀerentiation of V . This term does not appear in the classical formulation (1) of the bipolar response without amacrine connectivity. It appears here because synaptic response involves an implicit time derivative via convolution (12). Coupled dynamics Likewise, diﬀerentiating (12)gives dV A 1 j B =– V + W R . (15) A B j A i dt τ i=1 Equation (6) (activity), (14)and(15)deﬁneasetof 2N +N diﬀerential equations, ruling B A the behaviour of coupled BCells and ACells, under the drive of the stimulus, appearing in the term F (t). We summarise the diﬀerential system here: dV A B N j 1 A =– V + W V + F (t), B A B dt τ i j=1 B j i ⎪ B dV j N B 1 B i (16) =– V + W R , A B j i=1 i dt τ A ⎪ A ⎩ dA A B B i i =– + h N (V ). B B dt τ We have used the classical dynamical systems convention where time appears explicitly only in the driving term F (t)to emphasise that (16) is non-autonomous. Note that BCells act on ACells via a rectiﬁed voltage (gain control and piecewise linear rectiﬁcation), in agreement with Fig. 1,pathway III.Weanalyse this dynamics in Sect. 3.2.1. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 11 of 60 2.3.3 Connectivity graph The way ACells connect to BCells and reciprocally have a deep impact on dynamics (16). In this paper, we want to point out the role of relative excitation (from BCells to ACells) and inhibition (from ACells to BCells) as well as the role of the network topology. For math- ematical convenience when dealing with square matrices, we assume from now on that there are as many BCell as ACells, and we set N ≡ N = N . At the core of our mathemati- A B calstudiesisamatrix L,deﬁnedinSect. 3.2.1, whose spectrum conditions the evolution of the BCells–ACells network under the inﬂuence of a stimulus. It is interesting and relevant to relate the spectrum of L to the spectrum of the connectivity matrices ACells to BCells and BCells to ACells. There is not such a general relation for arbitrary matrices of connec- tivity. A simple case holds when the two connectivity matrices commute. Here, we choose an even simpler situation based on the fact that we compare the role of the direct feed- forward pathway on anticipation in the presence of ACell lateral connectivity. We feature the direct pathway by assuming that a BCell connects only one ACell with a weight w uni- B + + form for all BCell, so that W = w I , w >0, where I is the N-dimensional identity N,N N,N matrix. In contrast, we assume that ACells connect to BCells with a connectivity matrix – – A – W, not necessarily symmetric, with a uniform weight –w , w >0, so that W =–w W. We consider then two types of network topology for W: 1. Nearest neighbours. An ACell connects its 2d nearest BCell neighbours where d =1,2 is the lattice dimension. 2. Random ACell connectivity. This model is inspired from the paper [78]onthe shape and arrangement of starburst ACells in the rabbit retina. Each cell (ACell and BCell) has a random number of branches (dendritic tree), each of which has a random length and a random angle with respect to the horizontal axis. The length of branches L follows an exponential distribution with spatial scale ξ.The number of branches n is also a random variable, Gaussian with mean n ¯ and variance σ .The angle distribution is taken to be isotropic in the plane, i.e. uniform on [0, 2π[.Whena branch of an ACell A intersects a branch of a BCell B, there is a chemical synapse from A to B. The probability that two branches intersect follows a nearly exponential probability distribution that can be analytically computed (see Appendix C). 2.4 Ganglion cells There are many diﬀerent types of GCells in the retina, with diﬀerent physiologies and func- tions [3, 70]. In the present computational study, we focus on speciﬁc subtypes associated with pathways I–II (fast OFF cells with gain control), III (diﬀerential motion sensitive cells) and IV (direction selective cells) in Fig. 1.All thesehavecommon features: BCells pooling and gain control. 2.4.1 BCells pooling In theretina, GCells of thesametypecover thesurface of theretina, formingamosaic. The degree of overlap between GCells indicates the extent to which their dendritic arbours are entangled in one another. This overlap remains, however, very limited between cells of the same type [68]. We denote by k the index of the GCells, k = 1,..., N ,and by δ the G G spacing between two consecutive GCells lying on the grid (Fig. 2). Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 12 of 60 In the model, GCell k pools over the output of BCells in its neighbourhood [20]. Its voltage reads as follows: (P) B V = W R , (17) G G i k k where the superscript “P” stands for “pool”. We use this notation to diﬀerentiate this volt- age from the total GCell voltage V when they are diﬀerent. This happens in the case when GCells are directly coupled by gap junctions (Sects. 2.4.4, 3.3). When there is no ambiguity, we will drop the superscript “P”. In Eq. (17), the weights W are Gaussian: d [B ,G ] i k B 2σ W = a e , (18) where σ has the dimension of a distance and a is dimensionless. p p 2.4.2 Ganglion cell response The voltage V is processed through a gain control loop similar to the BCell layer [20]. As GCells are spiking cells, a nonlinearity is ﬁxed so as to impose an upper limit over the ﬁringrate. Here,itismodelledbyasigmoidfunction, e.g. 0, if V ≤ θ ; ⎪ G max N (V)= (19) α (V – θ ), if θ ≤ V ≤ N /α + θ ; G G G G G max N ,else. This function corresponds to a probability of ﬁring in a time interval. Thus, it is expressed –1 max in Hz. Consequently, α is expressed in Hz mV and N in Hz. Parameter values can be found in Appendix A. Gain control is implemented with an activation function A , solving the following dif- ferential equation: dA A G G k k =– + h N (V ), (20) G G G dt τ and a gain function 0, if A <0; G (A)= (21) ,else. 1+A Note that the origin of this gain control is diﬀerent from the BCell gain control (9). Indeed, Chen et al. hypothesise that the biophysical mechanisms that could lie behind ganglion gain control are spike-dependent inactivation of Na and K channels, while the study by Jacoby et al. [45] hypothesises that GCells gain control is mediated by feed- forward inhibition that they receive from ACells. The speciﬁc forms of the nonlinear- ity and the gain control function used in this paper match, however, the ﬁrst hypothesis, namely the suppression of the Na current [20]. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 13 of 60 Finally, the response function of this GCell type is R (V , A )= N (V )G (A ). (22) G G G G G G G k k k k In contrast to BCell response R (8), which is a voltage, here R is a ﬁring rate. B G Gain control has been reported for OFF GCells only [9, 20]. Therefore, we restrict our study to OFF cells, i.e with a negative centre of the spatial RF kernel. However, on mathe- matical grounds, it is easier to carry our explanation when the RF centre is positive. Thus, forconvenience,wehaveadopted achangeinconventioninterms of contrast measure- ment. We take the reference value 0 of the stimulus to be white rather than black, black corresponding then to 1. The spatial RF kernel is also inverted, with a positive centre and a negative surround. The problem is therefore mathematically equivalent to an ON cell submitted to positive stimulus. 2.4.3 Diﬀerential motion sensitive cells We consider here a class of GCells connected to ACells according to pathways III in Fig. 1, acting as diﬀerential motion detectors. They are able to respond saliently to an object moving over a stationary surround while being strongly inhibited by global motion. Here, stationary is meant in a general, probabilistic sense. This can be a uniform background or a noisy background where the probability distribution of the noise is time-translation invariant. These cells are hence able to ﬁlter head and eye movements. Baccus et al. [1] emphasised a pathway accountable for this type of response involving polyaxonal ACells which selectively suppress GCells response to global motion and enhance their response to diﬀerential motion, as shown in Fig. 1,pathway III.The GCellreceivesanexcitatoryinput from the BCells lying in its receptive ﬁeld which respond to the central object motion and an indirect inhibitory input from ACells that are connected to BCells which respond to the background motion. When motion is global, the excitatory signal is equivalent to the inhibitory one, resulting in an overall suppression. However, when the object in the cen- tre moves distinctively from the surrounding background, the cell in the centre responds strongly. There are here two concomitant eﬀects. When a moving object (say, from left to right) enters the BCell pool connected to a central GCell k , the BCells in the periphery of the pool respond ﬁrst, with no signiﬁcant change on the GCell response, because of the Gaus- sian shape (18) of the pooling: weights are small in the periphery. Those BCells excite, however, the ACells they are connected to, with the eﬀect of inhibiting the BCells of neigh- bouring GCells pools. This has the eﬀect of decreasing the voltage of these BCells which in turn excite less ACells which, in turn, inhibit less the BCells of the pool k .Thus, the response of the GCell k is enhanced, while the cells on the background are inhibited. We call this eﬀect “push–pull” eﬀect. Note that propagation delays ought to play an important role here, although we are not going to consider them in this paper. 2.4.4 Direction selective GCells and gap junction connectivity These cells correspond to pathway IV in Fig. 1. They are only coupled via electric synapses (gap junctions). In several animals, like the mouse, this enables the corresponding GCells to be direction sensitive. Note that other mechanisms, involving lateral inhibition via star- burst amacrine cells have also been widely reported [33, 34, 71, 72, 81, 85, 88]. Here we Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 14 of 60 focus on gap junctions direction sensitive cells (DSGCs). There exist four major types of these DSGCs, each responding to edges moving in one of the four cardinal directions. Trenholm et al. [80] emphasised the role of these cells coupling in lag normalisation: un- coupled cells begin responding when a bar enters their receptive ﬁeld, i.e. their dendritic ﬁeld extension, whereas coupled cells start responding before the bar reaches their den- dritic ﬁeld. This anticipated response is due to the eﬀective propagation of activity from neighbouring cells through gap junctions and is particularly interesting when comparing the responses for diﬀerent velocities of the bar. Trenholm et al. showed that the uncou- pled DSGCs detect the bar at a position which is further shifted as the velocity grows, while coupled cells respond at an almost constant position regardless of the velocity. In our work, we analyse this eﬀect in terms of a propagating wave driven by the stimulus and show that temporally this spatial lag normalisation induces a motion extrapolation that confers to the retina more than just the ability to compensate for processing delays, but to anticipate motion. Classical, symmetric bidirectional gap junctions coupling between neighbouring cells would involve a current of the form –g(V – V )– g(V – V ), where g is the gap G G G G k k–1 k k+1 junction conductance. In contrast, here, the current takes the form –g(V – V ). This G G k k–1 is due to the speciﬁc asymmetric structure of the direction selective GCell dendritic tree [80]. The experimental results of these authors suggest that the eﬀect of the possible gap junction input from downstream cells, in the direction of motion, can be neglected due to oﬀset inhibition and gain control suppression. This, along with the asymmetry of the dendritic arbour, justiﬁes the approximation whereby the cell k+1 does not inﬂuence the current in the cell k. This induces a strong diﬀerence in the propagation of a perturbation. Indeed, consider the case V – V = V – V = δ. In the symmetric form the total G G G G k k–1 k k+1 current vanishes, whereas in the asymmetric form the current is –gδ. Still, the current can have both directions depending on the sign of δ. This has a strong consequence on the way GCells connected by gap junctions respond to a propagating stimulus, as shown in Sect. 3.3. (P) The total GCell voltage is the sum of the pooled BCell voltage V and of the eﬀect of neighbours GCells connected to k by gap junctions: (P) V (t)= V – V (s)– V (s) ds, G G G k k k–1 –∞ where C is the membrane capacitance. Deriving the previous equation with respect to time, we obtain the following diﬀerential equation governing the GCell voltage: (P) dV dV G G k k = – w V (t)– V (t) , (23) gap G G k k–1 dt dt where w = . (24) gap Gain control is then applied on V as in (22). An alternative is to consider that gain control occurs before gap junctions eﬀect. We investigated this eﬀect as well (not shown, see [74]). Mainly, the anticipatory eﬀect is enhanced when the gain control is applied after the gap junction coupling; thus, from now, we focus on the formulation (23)inthe paper. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 15 of 60 Note that our voltage-based model of gap junctions takes a diﬀerent from as Trenholm et al. (expressed in terms of currents), because we had to adapt it so as to deal with the pooling voltage form (17). Still, our model reproduces the lag normalisation as in the orig- inal model as we checked (not shown, see [74]). 3Results 3.1 The mechanism of motion anticipation and the role of gain control The (smooth) trajectory of a moving object can be extrapolated from its past position and velocity to obtain an estimate of its current location [4, 56, 57]. When human subjects are shown a moving bar travelling at constant velocity, while a second bar is brieﬂy ﬂashed in alignment with the moving bar, the subjects report seeing the ﬂashed bar trailing behind the moving bar. This led Berry et al. [9] to investigate the potential role of the retina in anticipation mechanisms. Under constraints on the bar speed and contrast they were able to exhibit a positive anticipation time, deﬁned as the time lag between the peak in the retinal GCell response to a ﬂashed bar and the corresponding peak when the stimulus is a moving bar. In this paper we adopt a slightly diﬀerent deﬁnition although inspired by it. Indeed, the goal of this modelling paper is to dissect the various potential stages of retinal anticipation as developed in the next subsections. Several layers and mechanisms are involved in the model, each one deﬁning a response time and potentially contributing to anticipation under conditions that we now analyse. 3.1.1 Anticipation at the level of a single isolated BCell; the local eﬀect of gain control We consider ﬁrst a single BCell without lateral connectivity so that V = V .The very B i i drive mechanism of anticipation at this stage is illustrated in Fig. 4. The peak response time of the convolution of the stimulus with the RF of one BCell occurs at a time t (dashed line in Fig. 4(a)). The increase in V leads to an increase in activity (Fig. 4(c)) and an increase drive of R (Fig. 4(e)). When activity becomes large enough, gain control switches on (Fig. 4(d)) leading to a sharp decrease of the response R (Fig. 4(e)) and a peak in R occurring at B B time t (dashed line in Fig. 4(e)) before t .The bipolar anticipation time, = t – t ,is B B B B B A A therefore positive. dV drive Mathematically, > 0 results from the intermediate value theorem using that ≥ dt 0on[0, t ]and that t is deﬁned by B B dV G (A ) dA B B drive B i i =–V (t ) , i B drive A dt G (A ) dt B B t=t i t=t B B A A where the right-hand side is positive provided that the parameters h , τ are tuned such B a dA that ≥ 0on[0, t ]. An important consequence is that the amplitude of the response at dt the peak is smaller in the presence of gain control (compare the amplitude of the voltage in Fig. 4(a) to 4(e)). The anticipation time at the BCell level depends on parameters such as h , τ .Itdepends B a as well on characteristics of the stimulus such as contrast, size and speed. An easy way to ﬁgure this out is to consider that the peak in BCell response (Fig. 4(d), (e)) arises when the gain control function G (A )starts to drop oﬀ (Fig. 4(e)), which from the piecewise B B linear approximation (10)ofBCell arises when A = .When V has the form (3). this drive 3 Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 16 of 60 Figure 4 The mechanism of motion anticipation and the role of gain control. The ﬁgure illustrates the bipolar anticipation time without lateral connectivity. We see the response of OFF BCells with gain control to a dark moving bar. The curves correspond to three cells spaced by 450 μm. The ﬁrst line (a)shows thelinear ﬁltering of the stimulus corresponding to V (t)(Eq. (1)). Line (b) corresponds to the threshold nonlinearity drive N applied to the linear response; (c) represents the adaptation variable (16), and (d) shows the gain control time curse. Finally, the last line (e) corresponds to the response R of the BCell. The two dashed lines correspond respectively to t and t , the peak in the response of the (purple) BCell without pooling B B gives, using N (V )= V (7) and letting the initial time t → –∞ (which corresponds i i 0 drive drive to assuming that the initial state was taken in a distant past, quite longer than the time scales in the model): 1 σ h 1 x – vt σ 2 B 2 2 B 2 (x–vt ) A τ v B τ v a a A A (t )= A e e 1– + = , (25) B B 0 i A v σ τ v 3 where (x) is the cumulative distribution function of the standard Gaussian probability (see deﬁnition, Eq. (60)inAppendix B). This establishes an explicit equation for the time t as a function of contrast (A ), size (σ ) and speed (v) as well as the parameters h and τ . B 0 B a We do not show the corresponding curves here (see [74]for adetailedstudy)preferring to illustrate the global anticipation at the level of GCells, illustrated in Fig. 5 below. 3.1.2 Anticipation time of the BCells pooled voltage The main eﬀects we want to illustrate in the paper (impact of lateral connectivity on GCell anticipation) are evidenced by the shift of the peak in activity of the BCells pooled voltage occurring at time t .Wefocus on this time here,postponingtoSect. 3.1.3 the subsequent eﬀect of GCells gain control. We assume therefore here that h =0 so that A =0 and G G G (A )=1 in (19). Thus, the ﬁring rate of GCell k is N (V ). For mathematical sim- G G G G k k Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 17 of 60 plicity, we will consider that the ﬁring rate function (5)of G is a smooth, monotonously increasing sigmoid function such that N (V )>0. We deﬁne t as thetimewhen V is G G G G k k dV d V G G k k maximum, after the stimulus is switched on. This corresponds to =0 and <0. dt dt Equivalently, from Eqs. (17), (23), we have dR dV dA B B B B B i i i i i W = W G (A )N (V ) + N (V )G (A ) B B B B B B B B G G i i i i k k dt dt dt i i (26) = w [V – V ], gap G G k k–1 where this equation holds at time t = t (we have not written explicitly t to alleviate G G notation). This is the most general equation for the anticipation time at the level of BCells pooling. In the sum , there are two types of BCells. The inactive ones where V ≤ , B B i i dR N (V )=0 and = 0, so they do not contribute to the activity. The active BCells B B dt V > obey N (V )= V . For the moment we assume that, at time t , thereisnoBCell B B B B B G i i i switching from one state (active/inactive) to the other, postponing this case to the end of the section. Then, Eq. (26)reduces to 1 A B j W G (A ) – V + W V + F (t) B B B A B G i i B j i k i i j=1 (II) (I) (V) (III) (27) dA B i =– W G (A ) V (t) + w [V – V ] . B B gap G G G B i i k k–1 dt (II) (IV) (V) This general equation emphasises the respective role of (I), stimulus (term F (t)); (II), gain control (terms G (A ), G (A )); (III), ACell lateral connectivity (term W ); (IV), gap B B B i B i B junctions (term w [V (t )–V (t )]); (V), pooling (terms W ). Note that we could gap G G G G G k k–1 A A k as well consider a symmetric gap junctions connectivity where we would have a term w [–V +2V – V ] in IV. The equation terms have been arranged this way for gap G G G k+1 k k–1 reasons that become clear in the next lines. It is not possible to solve this equation in full generality, but it can be used to understand the respective role of each component. In the absence of gain control and lateral connectivity (W =0, w =0), the peak in gap GCell G voltage at time t is given by dV B i drive W = 0. (28) dt This generalises the deﬁnition of t , time of peak of a single BCell, to a set of pooled BCells, and we will proceed along the same lines as in Sect. 3.1.1. We ﬁx as reference time 0 the time when the pooled voltage becomes positive. It increases then until the time t when dV dV B i B i i i drive drive W =0. Thus, W is positive on [0, t [ and vanishes at t . i G i G G G k dt k dt We now show that, in the presence of gain control, the peak occurs at time t < t leading to anticipation induced by gain control and generalising the eﬀect observed for one BCell Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 18 of 60 in Sect. 3.1.1. Indeed, Eq. (27) reads now as follows: dV dA i B B B i drive i i W G (A ) =– W G (A )V (t) . (29) B B B i G i G B i drive k k dt dt i i dV dV B i B i i drive i drive Because 0 ≤ G (A ) ≤ 1, W G (A ) ≤ W so that the left-hand B B B B i i G i i G dt dt k k side in (29)reaches 0 atatime t ≤ t . The right-hand side is positive for the same reasons as in Sect. 3.1.1. The same mathematical argument holds as well, using the intermediate value theorem to show that t < t . We now investigate Eq. (27) with the two terms of lateral connectivity: (III) ACells and (IV) gap junctions. The eﬀect of gap junctions is straightforward. A positive term w [V – V ] increases the right-hand side of Eq. (27). As developed in Sect. 3.3,this gap G G k k–1 arises when the stimulus propagates in the preferred direction of the cell inducing a wave of activity propagating ahead of the stimulus. In view of the qualitative argument devel- oped above using the intermediate value theorem, this can enhance the anticipation time. This deserves, however, a deeper study developed in Sect. 3.3. N j Theeﬀect of ACells cellsislessevident, asthe term (– V + W V + F (t)) can B A B i j=1 B j i τ i have any sign, so that network eﬀect can either anticipate or delay the ganglion response, as illustrated in several examples in the next section. As we show, this term is in general related to a wave of activity, enhancing or weakening the anticipation eﬀect as shown in Sect. 3.2. Let us ﬁnally discuss what happens when some BCell switches from one state (ac- tive/inactive) to the other (i.e. V = ). In this case, taking into account the deﬁnition B B (5), the derivative N (V )= . Thus, when a BCell reaches the lower threshold, there is B i a big variation in N (V ) thereby leading to a positive contribution in (26) and an addi- B i B N j 1 1 i A tional term W G (A )(– V + W V + F (t)) in the left-hand side of (27), B B B A B i G i i j=1 B j i 2 τ i k B where the sum holds on switching state cells. As we see in section (3.2), this can have an important impact on the anticipation time. 3.1.3 Anticipation time at the GCell level We now show that the ﬁring rate of the GCell k,given by (22), reaches its maximum at a time t < t .From(22), at time t : G G G A A dV V dA G G G k k k = . (30) dt 1+ A dt dV V starts from 0 and increases on the time interval [0, t ], thus is positive on [0, t ] G G G dt dV and vanishes at t .Thus, thereisatime t < t such that increases on [0, t ]and de- G d G d dt creases on [t , t ]. The right-hand side of (30)startsfrom 0 at t =0 and stays strictly pos- d G dA itive until either V vanishes, which occurs for t > t ,oruntil vanishes. We choose G G k dt dA the characteristic time τ and the intensity h in (20)sothat >0 on [0, t ]. Thus, G G G dt V dA dV G G G k k k >0 on [0, t ]. Therefore, in the time interval [t , t ], decreases to 0, while G d G 1+A dt dt V dA G G k k increases from 0. From the intermediate value theorem, these two curves have 1+A dt to intersect at a time t < t . G G A Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 19 of 60 Figure 5 Maximum ﬁring rate and anticipation time variability with stimulus parameters in the gain control layer of the model. Left: contrast (with v = 1 mm/s et size = 90 μm); middle:size(with v = 2 mm/s et contrast = 1); right: speed (with contrast 1 and size = 162 μm) We ﬁnally deﬁne the total anticipation time of a GCell as follows: = t – t , (31) B G c A where t is the peak of the BCell at the centre of the BCells pooling to that GCell. 3.1.4 Anticipation variability: stimulus characteristics In general, depends on gain control, lateral connectivity as well as characteristics of the stimulus such as speed and contrast. This has been shown mathematically in Eq. (25)for a single BCell. Here, we investigate numerically the dependence of the total anticipation time of a GCell when the stimulus is a bar of inﬁnite height, width σ mm, travelling in one dimension at speed v mm/s with contrast C ∈ [0,1]. Resultsare showninFig. 5.This ﬁgure is a calibration later used to compare to the eﬀects induced by lateral connectivity. We ﬁrst observe that anticipation increases with contrast, as it has experimentally been observed [9]. Indeed, increasing the contrast increases V (t) thereby accelerating the drive growth of A so that gain control takes place earlier (Fig. 5(a)). We also notice that an- ticipation increases with the width of the object until a maximum (Fig. 5(b)). Finally, the model shows a decrease in anticipation as a function of velocity, as it was evidenced exper- imentally [9, 47](Fig. 4(c)). Indeed, when the velocity increases, V varies faster than drive the characteristic activation time τ , and the adaptation peak value is lower. Consequently, gain control has a weaker eﬀect and the peak activity is less shifted than when the bar is slow. A large part of these eﬀects can be understood from Eq. (25). Note, however, here that simulation of Fig. 5 takes into account the convolution of a moving bar with the receptive ﬁeld, the pooling eﬀect and gain control at the stage of GCells. In Fig. 5 we also show the evolution of GCells maximum ﬁring rate as a function of the moving bar velocity, contrast and size. We observe that it increases with these parameters, an expected result. 3.2 The potential role of ACell lateral inhibition on anticipation In this section we study the potential eﬀect of ACells (pathway III of Fig. 1)onmotionan- ticipation. We restrict to the case where there are as many BCells as ACells (N = N ≡ N) B A A B so that the matrices W and W are square matrices. We ﬁrst derive general mathemat- B A ical results (for the full derivation, see Appendix D) before considering the two types of connectivity described in Sect. 2.3.3. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 20 of 60 3.2.1 Mathematical study Dynamical system We study mathematically dynamical system (16)thatwewrite in a more convenient form. We use Greek indices α, β, γ = 1,...,3N and deﬁne the state vector X as follows: V , α = i, i = 1,..., N; ⎪ B X = V , α = N + i, i = 1,..., N; A , α =2N + i, i = 1,..., N. Likewise,wedeﬁne thestimulusvector F = F if α = 1,..., N and F = 0 otherwise. Then α B α dynamical system (16) has the general form dX = H(X)+ F(t), (32) dt where H(X ) is a nonlinear function, via the function R (V , A )of Eq. (8), featuring B B B i i i gain control and low voltage threshold. The nonlinear problem can be simpliﬁed using 3N the piecewise linear approximation (10). Indeed, there is a domain of R = V ≥ θ , A ∈ 0, , i = 1,..., N , (33) B B B i i where R (V , A )= V so that (16) is linear and can be written in the form B B B B i i i i dX = L.X + F(t), (34) dt with ⎛ ⎞ N,N A – W 0 N,N ⎜ I ⎟ B N,N L = W – 0 , (35) ⎝ ⎠ N,N N,N h I 0 – B N,N N,N where I is the N × N identity matrix and 0 is the N × N zero matrix. This corre- N,N N,N sponds to intermediate activity, where neither BCells gain control (9) nor low threshold (5) are active. We ﬁrst study this case and describe then what happens when trajectories of (32) get out of this domain, activating low voltage threshold or gain control. The idea of using such a phase space decomposition with piecewise linear approxima- tions has been used in a diﬀerent context by Coombes et al. [23]and in [14, 15, 18]. We consider the evolution of the state vector X (t) from an initial time t . Typically, t is 0 0 a reference time where the network is at rest before the stimulus is applied. So, the initial condition X (t ) will be set to 0 without loss of generality. Linear analysis The general solution of (34)is L(t–s) X (t)= e .F(s) ds. (36) 0 Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 21 of 60 The behaviour of solution (36) depends on the eigenvalues λ , β = 1,...,3N,of L and its eigenvectors P with entries P .The matrix P transforms L in Jordan form (L is not β αβ diagonalizable when h =0, see Appendix D.1). Whatever the form of the connectivity B A matrices W , W ,the N last eigenvalues are always λ =– , β =2N + 1,...,3N. A B In Appendix D.1 we show the following general result (not depending on the speciﬁc B A form of W , W , they just need to be square matrices and to be diagonalizable): A B B B B X (t)= V (t)+ E (t)+ E (t)+ E (t), α = 1,...,3N, (37) α α drive B,α A,α a,α where drive term (1) is extended here to 3N-dimensions with V (t)=0 if α > N.The drive other terms have the following deﬁnition and meaning: N N B –1 λ (t–s) E (t)= + λ P P e V (s) ds, α = 1,..., N, (38) β αβ γ B,α βγ drive B t β=1 γ =1 corresponds to the indirect eﬀect, via the ACell connectivity, of the BCells drive on BCells voltages (i.e. the drive excites BCell i, which acts on BCell j via the ACells network); 2N N B –1 λ (t–s) E (t)= + λ P P e V (s) ds, α = N + 1,...,2N, (39) β αβ γ A,α βγ drive B t β=N+1 γ =1 corresponds to the eﬀect of BCell drive on ACell voltages, and 2N N λ + B –1 B λ (t–s) E (t)= h P P e V (s) ds B α–2N β γ a,α βγ drive λ + β t β=1 γ =1 a (40) 1 1 – + τ τ B a 0 + A (t) , α =2N + 1,...,3N, α–2N λ + corresponds to the eﬀect of the BCells drive on the dynamics of BCell activity variables. The ﬁrst term of (40) corresponds to the action of BCells and ACells on the activity of BCells via lateral connectivity. In the second term t–s A (t)= e V (s) ds (41) α–2N α–2N drive corresponds to the direct eﬀect of the BCell voltage with index α –2N on its activity (see Eq. (7)). To sumup, Eq.(37) describes the direct eﬀect of a time-dependent stimulus (ﬁrst term) and the indirect lateral network eﬀects it induces. The term E (t)is what activates the a,α gain control. In the piecewise linear approximation (10), the BCell i triggers its gain control when its activity E (t)> , α =2N + i. (42) a,α This relation extends the computation made in Sect. 3.1.1 for isolated BCells to the case of a BCell under the inﬂuence of ACells. On this basis, let us now discuss how the network eﬀect inﬂuences the activation of gain control and, thereby, anticipation. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 22 of 60 Figure 6 Front (43) for diﬀerent values of λ (purple) as a function of time for the cell γ =0. All ﬁgures are –1 –1 drawn with v = 2 mm/s; σ =0.2 mm. Top. λ =–0.5 ms ; Bottom. λ =0.5 ms β β The structure of terms (38), (39)(40) is interpreted as follows. The drive (index γ = –1 1,..., N) excites the eigenmodes β = 1,...,3N of L with a weight proportional to P . βγ The mode β in turn excites the variable α = 1,...,3N with a weight proportional to P . The time dependence and the eﬀect of the drive are controlled by the integral αβ λ (t–s) e V (s) ds. For example, when the stimulus has the Gaussian form (3)and t drive cells are spaced with a distance δ so that cell γ is located at x = γδ,wehave, taking t → –∞: 2 2 σ λ β λ 1 β A λ σ 1 0 β λ (t–s) – (γδ–vt) β 2 2 v v e V (s) ds = e e – (γδ – vt) , (43) drive v v σ –∞ where (x) is the cumulative distribution function of the standard Gaussian probabil- ity (see deﬁnition, Eq. (60)inAppendix B). This is actually the same computation as (25)with λ =– .Equation(43) corresponds to a front, separating a region where [...] = 0 from a region where [...] = 1, propagating at speed v with an interface of 1 – (γδ–vt) width multiplied by an exponential factor e . Here, the sign of the real part of λ , λ is important. If λ < 0, the front has the shape depicted in Fig. 6(top). It β β,r β,r decays exponentially fast as t → +∞ with a time scale .Onthe opposite,itin- |λ | β,r creases exponentially fast with a time scale as t → +∞ when λ >0, thereby β,r β,r enhancing the network eﬀect and accelerating the activation of nonlinear eﬀect (low threshold or gain control) leading the trajectory out of . Remark that the peak of the drive occurs at γδ – vt = 0. The inﬂexion point of the function (x)is at x =0. Thus, when λ < 0, the front is a bit behind the drive, whereas it is a bit ahead when λ >0. Having unstable eigenvalues is not the only way to get out of . Indeed, even if all eigen- values are stable, the drive itself can lead some cells to get out of this set. When the tra- jectory of dynamical system (34)getsout of , two cases are then possible: Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 23 of 60 (i) Either a BCell i is such that V < θ .Inthiscase, R (V , A )=0.Then, in the B B B B B i i i i matrix L, there is a line of zeros replacing the line i in the matrix W ,i.e.atthe line i + N of L. This corresponds to a stable eigenvalue – for L, controlling the exponential instability observed in Fig. 6(bottom). Thus, too low BCell voltages trigger a re-stabilisation of the dynamical system. (ii) There are BCells such that condition (42) holds, then gain control is activated and system (32) becomes nonlinear. Here, we get out of the linear analysis, and we have not been able to solve the problem mathematically. There is, however, a simple qualitative argument. If the cell i enters the gain control region, then the B i corresponding line i + N in the matrix W of L is replaced by W G (A ),which B B A A i rapidly decays to 0 (see,e.g.Fig. 4(e)). From thesameargumentasin(i),this generates a stable eigenvalue ∼– controlling as well the exponential instability. Equation (37) features therefore the direct eﬀect of the stimulus as well as the indirect eﬀect via the amacrine network, corresponding to a weighted sum of propagating fronts, generated by the stimulus and inﬂuencing a given cell through the connectivity pathways. These fronts interfere either constructively, inducing a wave of activity enhancing the ef- fect of the stimulus and, thereby, anticipation, or destructively somewhat lowering the stimulus eﬀect. The ﬁne tuning between “constructive” and “destructive” interferences depends on the connectivity matrix via the spectrum of L and its projection vectors P . For example, complex eigenvalues introduce time oscillations which are likely to gener- ate destructive interferences, unless some speciﬁc resonance conditions exist between the imaginary parts of the eigenvalues λ . Such resonances are known to exist, e.g. in neural network models exhibiting a Ruelle–Takens transition to chaos [65], and they are closely related to the spectrum of the connectivity matrix [16]. Although we are not in this sit- uation here, our linear analysis clearly shows the inﬂuence of the spectrum of L,itself constrained by W, on the network response to stimuli and anticipation. Spectrum of L This argumentation invites us to consider diﬀerent situations where one can ﬁgure out how connectivity impacts the spectrum of L and thereby anticipation. We therefore provide some general results about the spectrum of L and potential linear insta- bilities before considering speciﬁc examples. These results are proved in Appendix D.2. As stated in Sect. 2.3.3, to go further in the analysis, we now assume that a BCell connects + B + + only one ACell, with a weight w uniform for all BCells, so that W = w I , w >0. We N,N also assume that ACells connect to BCells with a connectivity matrix W, not necessarily – – A – symmetric, with a uniform weight –w , w >0, so that W =–w W. We denote by κ , n = 1,..., N, the eigenvalues of W ordered as |κ |≤|κ | ≤ ··· ≤ |κ |, n 1 2 n and ψ is the corresponding eigenvector. We normalise ψ so that ψ .ψ =1 where † is n n n the adjoint. (Note that, as W is not symmetric in general, eigenvectors are complex). From the eigenvalues and eigenvectors of W, one can compute the eigenvalues and eigenvectors of L (see Appendix D.2),and infer stability conditions for the linear system. The main conclusions are the following: 1. The stability of the linear system is controlled by the reduced, a-dimensional parameter: – + 2 μ = w w τ ≥ 0, (44) Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 24 of 60 where: 1 1 1 = – , (45) τ τ τ A B with a degenerate case when τ = τ considered in the Appendix. A B 2. If W is symmetric, its eigenvalues κ are real, but the eigenvalues of L can be real or complex. Each κ corresponds actually to eigenvalues λ of L (see Eq. (71)). (a) If κ <0, the two corresponding eigenvalues of L are real and one of the two corresponding eigenmodes of L becomes unstable when 1 1 – + w w >– . (46) τ τ κ A B n (b) If κ >0 and if =0, the corresponding eigenvalues of L are complex conjugate if μ > ≡ μ . (47) n,c 4κ The corresponding eigenmodes are always stable. 3. If W is asymmetric, eigenvalues κ are complex, κ = κ + iκ . The eigenvalues of n n n,r n,i L have the form λ = λ + iλ , β = 1,...,2N,with β β,r β,i 1 1 1 λ =– ± a + u ; β,r n n 2τ 2τ AB 2 (48) ⎩ 1 1 λ = ± u – a , β,i n n 2τ ! ! 2 2 2 2 2 where a =1 – 4μκ and u = (1 – 4μκ ) +16μ κ = 1–8μκ +16μ |κ | . n n,r n n,r n n,i n,r Note that we recover the real case when κ =0 by setting u = a . n,i n n Instability occurs if λ >0 for some β.Thisgives β,r a + u >2 , (49) n n AB a condition on μ depending on κ and κ . n,r n,i Remarks The introduction of a dimensional parameter μ allows us to simplify the study – + of the joint inﬂuence of w , w , τ on dynamics because stability is controlled by μ only. In other words, a bifurcation condition of the form μ = μ signiﬁes that this bifurcation – + – + 2 holds when the parameters w , w , τ lay on the manifold deﬁned by w w τ = μ . We now show this in two examples of connectivity and aﬀerent instabilities. 3.2.2 Nearest neighbours connectivity Eigenmodes of the linear regime We consider the case where the matrix W, connecting ACells to BCells, is a matrix of nearest neighbours symmetric connections. In this case, W can be written in terms of the discrete Laplacian on a d dimensional regular lattice, d = 1, 2, with lattice spacing δ = δ set here equal to 1 without loss of generality: A B W =2dI + . (50) Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 25 of 60 Because of this relation, we will often use the terminology Laplacian connectivity for the nearest-neighbours connectivity. We also assume that dynamics holds on a square lattice with null boundary conditions. That is, ACell and BCells are located on d-dimensional grid with indices i , i = 0,..., L + 1 where the voltage and activity of cells with indices x y i =0, i = L +1, i =0 or i = L +1 vanish. x x y y The eigenvalues and eigenvectors are explicitly known in this case. They are parame- trized by a quantum number n = n ∈{1,..., L = N } in one dimension and by two quan- tum numbers (n , n ) ∈{1,..., L = N } in two dimensions. They deﬁne a wave vector x y n π n π L+1 L+1 k =( , ) corresponding to wave lengths ( , ). Hence, the ﬁrst eigenmode (n = n x L+1 L+1 n n x y 1, n = 1) corresponds to the largest space scale (scale of the whole retina) with the small- π π est eigenvalue (in absolute value) s =2(cos( )+ cos( ) – 2). To each of these eigen- (1,1) L+1 L+1 modes is related a characteristic time τ = . The slowest mode is the mode (1, 1). In contrast, the fastest mode is the mode (n = L, n = L) corresponding to the smallest space x y scale, the scale of the lattice spacing δ =1. Eigenvalues κ can be positive or negative. Consider for example the one-dimensional nπ case, where κ =2 cos( ). We choose L even to avoid having a zero eigenvalue κ L . Eigen- L+1 values κ , n = 1,..., , are positive, thus the corresponding eigenvalues λ of L are com- plex and stable. The modes with the largest space scale are therefore stable for the linear dynamical system with oscillations. Eigenvalues κ , n = + 1,..., L, are negative, thus the corresponding eigenvalues λ of L are real. From (46)the mode n becomes unstable when 1 1 – + w w >– . (51) nπ τ τ 2 cos( ) A B L+1 Therefore, the ﬁrst mode to become unstable is the mode L with the smallest space scale 1 1 – + 1 (lattice spacing). For large L,thishappens for w w ∼ . This instability induces 2 τ τ A B – + spatial oscillations at the scale of the lattice spacing. When w w further increases, the next modes become unstable. This instability results in a wave packet following the drive (as shown in Fig. 6). The width of this wave packet is controlled by the unstable modes and by nonlinear eﬀects. We now illustrate the relations of these spectral properties with the mechanism of anticipation. Numerical results In all the following 1D simulations, we consider a bar with a width 150 μm, moving in one dimension at constant speed v = 3 mm/s. We simulate 100 BCells, 100 ACells and 100 GCells placed on a 1D horizontal grid with a uniform spacing of δ = δ = δ =30 μm between to consecutive cells. At time t = 0, the ﬁrst cell lies at b a g 100 μm to the right of the leading edge of the moving bar. We set τ = 300 ms, τ =50 ms, B a + – τ = 100 ms, corresponding to τ = 150 ms (Eq. (45)). We vary the value of weights w , w . + – For the sake of simplicity, we also choose w =–w = w to have only one control parameter. We investigate how the bipolar anticipation time and the maximum in the response R depend on w.ThisissummarisedinFig. 7(top), where we have shown the eﬀect of gain control alone (blue horizontal line, independent of w), the eﬀect of ACell lateral connec- tivity alone (red triangles) and the compound eﬀect (white squares). Anticipation time is averaged over all cells. On the same ﬁgure (bottom) we see the responses of two neighbour cells lying at the centre of the lattice. As w increases, we observe three areas of interest: the ﬁrst (A) corresponds to a regime where ACell connectivity has a negative eﬀect on anticipation, competing with gain con- Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 26 of 60 Figure 7 Anticipation in the Laplacian (nearest-neighbours) case. Top. Anticipation time and maximum bipolar response as a function of the connectivity weight w. The blue line corresponds to gain control alone (it does not depend on w). Red triangles correspond to the eﬀect of lateral ACell connectivity without gain control. White squares correspond to the compound eﬀect of ACell connectivity and gain control. The three regimes A, B, C are commented in the text. Bottom. Response curves of ACells and BCells corresponding to the three –1 –1 regimes: (A) w =0.05 ms with a small cross-inhibition, (B) w =0.3 ms with an opposition in activity –1 between the blue (cell 50) and red cell (51), (C) w =0.6 ms , where the red cell (51) is completely inhibited by cell 50 trol. As w is small, the anticipation is controlled by the direct pathway I, II of Fig. 1,from BCells to GCells, with a small inhibition coming from ACells, thereby decreasing the volt- age of BCells and impairing the eﬀect of gain control. This explains why the anticipation time in the case of lateral connectivity + gain control is smaller than the anticipation time of gain control alone. The network eﬀect (red triangles) on anticipation time increases with w though. This corresponds to the “push–pull” eﬀect already evoked in Sect. 2.3. When a BCell B feels the stimulus, its activity increases favoured by the stimulus, it in- creases the voltage of the connected ACell, inhibiting the next BCell B , thereby inducing i+1 a feedback loop, the push–pull eﬀect, enhancing the voltage of B . In zone (B) the push–pull eﬀect becomes more eﬃcient than gain control alone. In this region, the voltage of the BCell feeling the bar increases fast, while the voltage of its neigh- bours becomes more and more negative, enhancing the feedback loop. This holds until the voltage rectiﬁcation (5) takes place. This is the time when the dynamical system gets out of . The push–pull eﬀect then saturates and V reaches a maximum, corresponding to a peak in activity. This peak is reached faster than the peak in the function G (A). Thus, thepeakof R (t) occurs at the same time as the peak of N (V (t)) and, thus, before the B B B i i reference peak (time t for isolated BCells deﬁned in Sect. 3.1.1). In other words, the ACell lateral connectivity allows the BCell to outperform the gain control mechanism for antic- ipation. As w increases in zone B the push–pull eﬀect (averaged over BCells) reaches a maximum, then decreases. This is because the increase in w makes the inhibitory eﬀect of ACells stronger and stronger on silent BCells which then remain silent longer and longer Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 27 of 60 because the ACell voltage increases with w, and it takes longer for it to decrease and de- inhibit the neighbours. The silent cells are less and less sensitive to the stimulus, being strongly and durably inhibited. In region C, the anticipation is again dominated by gain control. In this case, the eﬀect on cells depends on the parity of their index. The response of BCells is either completely suppressed or identical to the response of the reference case (with gain control alone). This is why the average anticipation time with gain control is about half of the gain control without network eﬀect. Cells that are inhibited do no participate to anticipation, and the others anticipate in the same way than with gain control alone. Note that this “parity” eﬀect is due to the nearest neighbours connectivity and the symmetry of interactions. We now interpret and complete these results from the point of view of the spectrum of L and associated dynamics. The fastest mode to destabilise corresponds to the smallest space scale, i.e. the lattice spacing. This is a mode with alternate sign at the scale of the lattice. We call it the “push–pull” mode, as it is precisely what makes the push–pull eﬀect. When the push–pull mode becomes unstable, the excited BCell becomes more and more excited and the next BCell more and more inhibited. However, the time it takes τ has to be compared to the time where the bar stays in the RF, τ (and more generally the time it bar takes to RF kernel to respond to the bar). In the case of the simulation σ =90 μm(see center Appendix A,Table 1)and v =3 μm/ms giving a characteristic time τ = 270ms, whereas, bar as we observed, τ < 100 ms. The push–pull mode is therefore quite faster than τ ,so L bar the push–pull eﬀect takes place fast and leads to a fast exponential increase of the front depicted in Fig. 6(right). This explains the rapid increase of network anticipation eﬀect observed in regions A, B of Fig. 7. 3.2.3 Random connectivity In this section, we study the behaviour of the model using the more realistic, probabilistic type of connectivity presented in Sect. 2.3.3 and more thoroughly studied in Appendix C. Within this framework, a given ACell A receives the upstream activity from the BCell ly- ing at the same position B with a constant weight w. The same ACell inhibits BCells with which it is coupled through the random adjacency matrix W, generated by the probabilis- tic model of connectivity, and the weight matrix W =–wW. We recall that the connec- tivity depends on a scale parameter ξ for the branch length) and the mean and variance n ¯, σ for the distribution of the number of branches. These parameters can be found in Table 1 in Appendix A. Eigenmodes of the linear regime Similarly to Sect. 3.2.2 we now analyse the spectrum of L when W is a random connectivity matrix. Although a couple of results can be es- tablished (using the Perron–Frobenius theorem), we have not been able to ﬁnd general mathematical results on the spectrum or eigenvectors of this family of random matrices. We thus performed numerical simulations. The spectrum of L is deduced from the spectrum of W as exposed above. The spec- trum of W depends on n ¯, σ and ξ.InFig. 8 we have plotted, on the left, an example of such a spectrum. This is the spectral density (distributions of eigenvalues in the complex plane) obtained from the diagonalization of 10,000 matrices 100 × 100 (so the statistics is made over 10 eigenvalues). We note that the largest eigenvalues are always real positive, a straightforward consequence of the Perron–Frobenius theorem [38, 69]. More generally, Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 28 of 60 Figure 8 Spectral density of eigenvalues for ξ =2, n ¯ =4, σ =1.Top left. For the matrix W (density estimated over 10,000 samples), density is represented in colour plots, in log scale. The colour bar refers to powers of 10. Top right.Spectraldensity of L for w =0.05. Bottom left.Spectraldensity of L for w =0.1. Bottom right.Spectral density of L for w = 0.15. Unstable eigenvalues are on the right to the vertical dashed line x =0 Figure 9 Heat map for the largest real part eigenvalue in the plane w, ξ for diﬀerent values of n ¯. Left. n ¯ =1. Right. n ¯ = 4. Colour lines are level lines. The level line 0 is the frontier of instability of the linear dynamical system we observe an over-density of real eigenvalues. The same holds for random Gaussian ma- trices with independent entries N (0, )[32] whose asymptotic density converges to the circular law [40]. The shape of the spectral density in our model diﬀers from the circular law though, and it depends on the parameters n ¯, σ and ξ. On the same ﬁgure we show the corresponding spectral density of L obtained from Eq. (48)for w = 0.05, 0.1, 015. We have taken here τ =30, τ =10 ms to see better the A B transitions with w (level lines in Fig. 9). There is an evident symmetry with respect to = –0.066 expected from the mathematical analysis. We see that the largest eigenvalue AB is real (although it is not necessarily related to the largest eigenvalue of W). We also see that, as w increases, a large number of (complex) eigenvalues become unstable. There is actually a frontier of instability that we have plotted in the plane w, ξ for diﬀerent values Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 29 of 60 of n ¯. This is shown in Fig. 9 (dashed line). The level line 0 is the frontier of instability of the linear dynamical system. This frontier has the (empirical) form (ξ – ξ ).(w – w0) = c, where c has the dimension of a characteristic speed. What matters here is that there are complex unstable eigenvalues with no speciﬁc reso- nance relations between them. They are therefore prone to generate destructive interfer- ences in (37). Numerical results In Fig. 10 we consider, similarly to Fig. 7 for Laplacian connectivity, the eﬀect of random connectivity on anticipation, compared to pure gain control mechanism. In contrast to the Laplacian case, we have here more parameters to handle: ξ,which con- trols the characteristic length of branches and n, σ which control the number of branches distribution. We present here a few results where ξ varies, whereas the average number of branches n ¯ =2(σ = 1). A more systematic study is done in [74]. The interest of varying ξ is to start from a situation which is close to the Laplacian case (characteristic distance ξ =1) and to increase ξ to see how the size of the dendritic tree of ACells may impact anticipa- tion. This is a preliminary step toward considering diﬀerent physiological ACells type (e.g. narrow-, medium-, or wide-ﬁeld [29]). Note, however, that the probability of connection given the distance of cells (ﬁxed by n ¯, σ ) implicitly impacts w and the anticipation eﬀects. The main diﬀerence with the Laplacian case is the asymmetry of connections. Here, symmetry means that if ACell j connects the BCell i, then the ACell i connects the BCell j too. This does not necessarily hold for random connectivity, and this has a strong impact on the push–pull eﬀect and anticipation. So, even if the connectivity is short-range when ξ is small, mainly connecting nearest neighbours, we observe already a big diﬀerence with theLaplacian case.Thisisshown in Fig. 10,where ξ = 1. Similarly to the Laplacian case, Figure 10 Average anticipation in the random connectivity case. Top. Bipolar anticipation time and maximum in the response R as a function of the connectivity weight w in the case of a random connectivity graph with ξ =1, n ¯ =2 and σ =1. Bottom. Response curves of ACells and BCells 51 – 52 corresponding to the three –1 –1 –1 regimes: (A) w =0.05 ms ,(B) w =0.3 ms ,(C) w =0.6 ms Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 30 of 60 we observe three main regions depending on w.Tohavethe same representation as Fig. 7, we present V , N , R for two connected cells (here, ACell 51 and BCell 52). However, in A B B this case, connection is not symmetric: ACell 51 inhibits BCell 52, but ACell 52 does not inhibit BCell 51. We observe three regimes, as in the Laplacian case. In the ﬁrst region (A) ACell ran- dom connectivity has a negative eﬀect on anticipation, as compared to gain control alone. However, since in this case the “push–pull” eﬀect is not evoked, this decay simply comes from the fact that BCell 52 receives an inhibition for ACell 51 that reduces the eﬀect of gain control. This inhibition, however, is not strong enough to signiﬁcantly shift the peak response as in region (B). Indeed, in region (B), the inhibition of BCell 52 is strong enough to outperform the eﬀect of gain control. In this case, and similarly to the Laplacian case, the peak of R (t)occurs at the same time as the peak of N (V (t)) and before the reference peak. However, this eﬀect B B is not consistent over all cells and only occurs for BCells that receive active inhibition. This explains why the performance of the Laplacian connectivity is better, on average, in this region. Finally, as w grows higher, the inhibition grows stronger, completely inhibiting BCell 51. Cells that do not receive any inhibition, as BCell 49 in this example, keep a response that is identical to the response without ACell connectivity. The fraction of cells receiving inhibition in this case being quite small (about 15), this explains why the stationary value of anticipation is fairly close to the value with gain control alone. The role of the characteristic distance In Fig. 11,weanalyse theeﬀect of thecharacter- istic length ξ on anticipation. On the top of the ﬁgure we represent the joint eﬀect of the random ACell connectivity and gain control on anticipation for three values of ξ.Atthe bottom we represent the only eﬀect of the random ACell connectivity for the same values of ξ. We observe that performance in anticipation decreases with ξ.Moreprecisely,we observe an anticipatory eﬀect in this case, as shown in Fig. 11(bottom), but this eﬀect is not able to compete with gain control alone. Even worse, the compound eﬀect shown in 11(top) is disastrous since increasing w renders the anticipation time smaller and smaller. This spurious eﬀect can be interpreted through the analysis made in Sect. 3.2.1,Eq. (37). From the spectrum of L, we see that there are unstable complex eigenvalues whose num- ber increases with w. These eigenvalues are prone to generate destructive interferences, especially when their number becomes large as w increases, explaining the small peak in region B. The consequence on cells activity and gain control can be dramatic as seen in the red trace of Fig. 10(B bottom), line R . This depends on the precise connectivity pat- tern when long range connections from ACells to BCells induce a desensitisation of BCells, which is not counterbalanced by the push–pull eﬀect as in the Laplacian connectivity case. 3.2.4 Conclusion The two numerical examples considered in this section emphasise the role of symmetry in the synapses and, more generally, the role of complex versus real eigenvalues in the spec- trum of L. Recall that, from Sect. 3.2.1,if W is symmetric, complex eigenvalues are always stable, so, for the type of architecture considered here, unstable destructive interferences only occur when W is asymmetric. This leads to several questions, potential subjects for further studies. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 31 of 60 Figure 11 Role of the characteristic branch length ξ on anticipation. Top. The joint eﬀect of the random ACell connectivity and gain control on anticipation for ξ = 1,2,3. Left. Average bipolar anticipation time. Right. Maximum value of the bipolar response R . Bottom. The single eﬀect of the random ACell connectivity on anticipation with the same representation 1. How much does anticipation depend on the degree of asymmetry in the matrix W? The way we generate the random connectivity in the model does not allow us to tune thedegreeofasymmetry(i.e. theprobability that aconnection A → B exists j i simultaneously with a connection A → B ). Therefore, one has to ﬁnd a diﬀerent way i j to generate the connectivity. From the mathematical analysis made in Appendix C,a distribution depending exponentially on the distance, with a tunable probability to have a symmetric connection, could be appropriate. We do not know about any experimental results characterising this degree of symmetry of the connections in the retina. On mathematical grounds, and from the analogy of the spectrum of W with a circular law, one could expect the spectrum of W to become more and more elongated on the real axis as the degree of symmetry increases in an elliptic like law [55]. 2. Nonlinear eﬀects. The destructive interference eﬀect in our model is partly due to the linear nature of the ACell dynamics. In nonlinear dynamics, eigenvalues of the evolution operator can display resonance conditions favouring constructive interferences. On biological grounds, it is for example known that starburst amacrine cells display periodic bursting activity during development, disappearing a few days after birth [94]. Bursting and its disappearance can be understood in the framework of bifurcation theory of a nonlinear dynamical system featuring these cells [50]. In this setting, even if they are not bursting in the mature stage, SACs remain sensitive Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 32 of 60 to speciﬁc stimulation that can temporally synchronise them, thereby enhancing the network eﬀect with a potential eﬀect on anticipation. 3.3 The potential role of gap junctions on anticipation In this section, we study the network ability to improve anticipation in the presence of gap junctions coupling, as in Eq. (23), and gain control at the level of GCells. We start ﬁrst with mathematical results and show then simulation results. 3.3.1 Mathematical study We use a continuous space limit for a one-dimensional lattice. The extension to two di- mensions is straightforward. Here, x corresponds to the preferred direction of the direc- tion sensitive cells. We consider a continuous spatio-temporal ﬁeld V(x, t), x ∈ R,such (P) (P) that V ≡ V(kδ , t). We assume likewise that V ≡ V (kδ , t)for some continuous G G G k G (P) function V (x, t) corresponding to the GCells bipolar pooling input (17), and we take the limit δ → 0. In this limit Eq. (23)becomes ∂V ∂V G G = f (x, t)– v + O δ , (52) gap ∂t ∂x (P) ∂V (x,t) where v ≡ w δ has the dimension of a speed and ≡ f (x, t). Finally, we denote gap gap G ∂t by C(x) the initial proﬁle so that V(x, t )= C(x). Solution Neglecting terms of order δ , the general solution of (52)is V (x, t)= C x – v (t – t ) + f x – v (t – u), u du. G gap 0 gap Equation (52) is a transport equation of ballistic type [74]. For example, if we consider (P) a stimulation of the form V (x, t)= h(x – vt), where h is a Gaussian pulse of the form (3), propagating with speed v, and an initial proﬁle C(x)= h(x – vt ), the voltage of GCells obeys v v gap V (x, t)= h(x – vt) – h x – v t –(v – v )t . (53) G gap gap 0 v – v v – v gap gap π π gap stim When v = 0, the GCell voltage follows the stimulation, i.e. V (x, t)= h(x – vt). In the gap G presence of gap junctions, there are two pulses: the ﬁrst one π with amplitude stim v–v gap propagating at speed v and following the stimulation; the second one π with amplitude gap gap – propagating at speed v . gap v–v gap We have the following cases (we take t = 0 for simplicity). An illustration is given in Fig. 12. 1. If v and v have thesamesign: gap (A) If v < v, the front π is ampliﬁed by a factor ,whereas thereisa gap stim v–v gap refractory front π ,proportionalto v , behind the excitatory pulse. gap gap (B) If v = v , V (x, t)= h(x – v t)+ v (t – t )h (x – v t) which diverges like t gap G gap gap 0 gap when t →∞ and x → +∞. This divergence is a consequence of the limit δ → 0 in (52). Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 33 of 60 Figure 12 Anticipation for non symmetric gap junctions. Top. GCell anticipation time and maximum ﬁring rate as a function of the gap junction velocity v . Bottom: response curves of GCells corresponding to the three gap regimes: (A) v = 0.6 mm/s, (B) v =3 mm/s,(C) v = 12 mm/s. The curves display 3 main regimes (see gap gap gap text): In (A) v < v and the positive front propagates at the same speed as the pooling voltage triggered by gap the stimulus; In (B), v = v, the positive front and the negative fronts both propagate at the speed v and gap gap the amplitude of the positive front (V (t)) increases with t;In(C), v > v and the positive front propagates G gap faster than the stimulus so that the peak of activity arises earlier. The negative front propagates at the stimulus speed (C) If v > v, the amplitude of π follows the stimulation with a negative sign gap stim (hyper polarization), whereas π is ahead of the stimulation, with a positive gap sign, travelling at speed v . gap 2. If v and v have theoppositesign, we set v =–αv with α >0. Then the front π gap gap stim follows the stimulus but is attenuated by a factor . The front π propagates in gap 1+α the opposite direction with an attenuated amplitude . 1+α This shows that these gap junctions favour the response to motion in the preferred direc- tion and attenuate the motion in the opposite direction although the attenuation is weak. The eﬀect is reinforced by gain control [74]. The most interesting case is 1 c where these gap junctions can induce a wave of activation ahead of the stimulation. Eﬀect of gain control When the low voltage threshold N (19) and the gain control G (A) G G (21) are applied to V (x, t), there are two eﬀects: (i) the hyperpolarised front is cut by N ; G G (ii) the positive pulse induces a raise in activity, which in turn triggers the ganglion gain control G (A) inducing an anticipated peak in the response of the GCell, similar to what happens with BCells, with a diﬀerent form for the GCell gain control though. Moreover, in contrast to pathway II of Fig. 1, where only gain control generates anticipation, in pathway IV the wave of activity generated by gap junctions increases anticipation by two distinct eﬀects. If v < v, the cell’s response propagates at the same speed as the stimulus, but gap its amplitude is larger than the case with no gap junction (term π ). From Eq. (53)this stim results in an increase of h to an eﬀective value h inducing an improvement in the B B v–v gap Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 34 of 60 anticipation time (with a saturation of the eﬀect, though, as v → v). If v > v,the cell’s gap gap response propagates at a larger speed than the stimulus (term π ), so that the cell re- gap sponds before the time of response without gaps. This induces as well an increase in the anticipation time. 3.3.2 Numerical illustrations We consider a bar with a width 200 μm moving in one dimension at constant speed v = 3 mm/s. We simulate here 100 GCells, placed on a 1D horizontal grid, with a spacing of 30 μm between two consecutive cells. At time t = 0, the ﬁrst cell lies at 100 μmfrom the leading edge of the moving bar. We investigate how the GCell anticipation time and GCell ﬁring rate depend on v in gap Fig. 12. The top shows the eﬀect of gain control alone (blue horizontal line independent of v ), the eﬀect of the asymmetric gap junction connectivity alone (red triangles) and gap the compound eﬀect (white squares). Anticipation time is averaged over all GCells. In the bottom part of the ﬁgure, we show the responses of two GCells of indices 30 and 60, spaced by 900 μm. As explained in Sect. 3.3.1,weobserve thethree regimesA,B,Cmathematically antic- ipated above. Note that, for these parameter values, the negative trailing front predicted in A is not visible. 3.3.3 Symmetric gap junctions The asymmetry observed by Trenholm et al. is due to the speciﬁc structure of the direc- tion selective GCell dendritic tree [80]. However, in general gap junctions connectivity is expected to be symmetric. So, to be complete, we consider here the eﬀect of symmetric gap junctions on anticipation. It is not diﬃcult to derive the equivalent of Eq. (52)inthis case too. This is a diﬀusion equation of the form ∂V = f (x, t)+ D V + O δ , gap G ∂t where D = w δG is the diﬀusion coeﬃcient and is the Laplacian operator. gap gap The response to a Gaussian stimulus of the form (3) reads as follows: x,y,t V (x, y, t)=[H ∗ f ], (54) where 2 2 x +y 4Dgapt H(x, y, t)= (55) 4πD t gap which is the heat equation diﬀusion kernel. (P) ∂V (x,t) (P) Recall that f (x, t) ≡ .So, if V (x, t)= h(x – vt), where h is a Gaussian pulse of the ∂t form (3) propagating with speed v, f is a bimodal function of the form h(x–vt) ×(x–vt), the shape of which can be seen in Fig. 13(bottom), second row. The convolution with the heat kernel leads to a front propagating at the same rate as the stimulus, with a diﬀusive spreading whose rate is controlled by D . In particular, there is positive bump ahead of gap the motion, which can induce anticipation, as shown in Fig. 13(top). The eﬀect is weak Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 35 of 60 Figure 13 Anticipation for symmetric gap junctions. Top. GCell anticipation time and maximum ﬁring rate as a function of the gap junction velocity v . Bottom. Response curves of GCells corresponding for three values gap of v :(A) v = 0.6 mm/s, (B) v = 3 mm/s, (C) v = 12 mm/s. For consistency, we have kept the same gap gap gap gap values as in the asymmetric case. Here, anticipation time grows continuously until saturation, while the maximum ﬁring decreases like a power law as a function of v gap though, essentially because the diﬀusive spreading makes the amplitude of the response decrease fast as a function of D . gap Although this positive front, for small D , increases a bit the anticipation time by accel- gap erating the gain control triggering, rapidly the peak in the response R is led by the voltage peak corresponding to the positive bump with a low voltage. The position of this peak is, 2 2 roughly, at a distance σ = σ + σ from the peak of the Gaussian pool, where σ is center center B the width of the centre RF and σ the width of the bar. This corresponds to a time ahead of the peak in the drive, ﬁxing a maximal value to the anticipation time (see the saturation of the anticipation time curve in Fig. 13(top, left)). In our case, σ ∼ 134 given a satura- 134 μm tion peak at = 44.84 ms. A consequence of the voltage decay is the corresponding 3 μm/ms power law ( for large D ) decay of the ﬁring rate (Fig. 13(top, right)). gap gap To conclude, the situation with symmetric gap junctions is in high contrast with di- rection selective gap junctions where the response to stimuli was ballistic and was not decreasing with time. On this basis we consider that, for symmetric gap junctions, the an- ticipation eﬀect is irrelevant, especially taking into account the smallness of the voltage response in case C. 3.3.4 Numerical results We investigate in this section how the GCell anticipation time and GCell ﬁring rate depend on the gap junction conductance in the case of symmetric gap junctions. In Fig. 13(top), we use the same representation as in Fig. 12. For consistency with the direction sensitive case, gap we choose v = as a control parameter. We also take (A) v = 0.6 mm/s, (B) v = gap gap gap δG Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 36 of 60 3 mm/s, (C) v = 12 mm/s in Fig. 13(bottom). This corresponds to a diﬀusion coeﬃcient gap –3 2 –3 2 –3 2 (A) D =18 × 10 mm /s, (B) D =90 × 10 mm /s, (C) D = 360 × 10 mm /s. gap gap gap 3.3.5 Conclusion In this section we have shown how gap junctions direction sensitive cells can display an- ticipation due to the propagation of a wave of activity ahead of the stimulus. This eﬀect is negligible for symmetric gap junctions. Note that symmetric gap junctions are known to favour waves propagation, for example in the early development (stage I, see [49]and the reference therein for a recent numerical investigation). Here, gap junctions are considered in a diﬀerent context due to the presence of a nonstationary stimulus triggering the wave. Let us now comment our computational result. How does it ﬁt to biological reality? Depending on the gap-junction conductance value, the propagation patterns we predict are quite diﬀerent. What is the typical value of v in biology? It is diﬃcult to make an estimate from the gap g δ gap G expression v = . The membrane capacity C and gap junctions conductance can be gap obtained from the literature (for connexins Cx36, g ∼ 10–15 pS [75]), but the distance gap δ is more diﬃcult to evaluate. In the model, this is the average distance between GCells’ soma which corresponds to ∼200–300 μm. But in the computation with gap junctions what matters is the length of a connexin channel which is quite smaller. Taking δ as the distance between GCells assumes a propagation speed between somas at the speed of a connexin, which is wrong because most of the speed is constrained by the propagation of action potential along the dendritic tree. So we used a phenomenological argument (we thank O. Marre for pointing it out to us). The correlation of spiking activity between GCellneighboursisabout 2–5ms for cellsseparated by ∼200–300 μm[86]. This gives a speed v in the interval [40, 150] mm/s, which is quite fast compared to the bar speed in gap experiments. So we are in case 1 b, and should one observe an experimental eﬀect? To the best of our knowledge, an eﬀect of DSGC gap junctions on motion anticipation has not been ob- served. But we do not know about experiments targeting precisely this eﬀect. It would be interesting to block gap junctions and address Berry et al. [9]orChenetal. [20]ex- perimentsinthiscase. Thediﬃcultyisthatblockinggap junctionsblocksmanyessential retinal pathways. We do not pursue this discussion further concluding that our model proposes a computational prediction that could be interesting to be experimentally inves- tigated. 3.4 Response to two-dimensional stimuli In this section, we present some examples of retinal responses and anticipation to trajec- tories more complex than a bar moving in one dimension with a uniform speed. The aim here is not to do an exhaustive study but, instead, to assess qualitatively some anticipatory eﬀects not considered in the previous sections. 3.4.1 Flash lag eﬀect The ﬂash lag eﬀect is an optical illusion where a bar moving along a smooth trajectory and a ﬂashed bar are presented to the subject and are perceived with a spatial displacement, while they are actually aligned. A variation of this illusion consists of a bar moving in rotation, a bar ﬂashed in angular alignment, giving rise to a perceived angular discrepancy. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 37 of 60 Figure 14 Flash lag eﬀect with diﬀerent anticipatory mechanisms.(A) Response to a ﬂash lag stimulus: a bar moving in smooth motion with a second bar ﬂashed in alignment with the ﬁrst bar for one time frame. The ﬁrst line shows the stimulus, the second line shows the GCells response with gain control, the third line –1 presents the eﬀect of lateral ACell Laplacian connectivity with w =0.3 ms ,and thelastlineshows theeﬀect of asymmetric gap junctions with v = 9 mm/s. (B) Time course response of (top) a cell responding to the gap ﬂashed bar and (bottom) a cell responding to the moving bar. Dashed lines indicate the peak of each curve We have investigated this eﬀect in our model in the presence of the diﬀerent anticipatory eﬀects considered in the paper. Figure 14 shows the response to a bar moving with a smooth motion, while a second bar is ﬂashed in alignment with the ﬁrst bar at one time frame. The ﬁrst line shows the stimuli, consisting of 130 frames, of a bar moving at 2.7 mm/s, with a refreshment rate of 100 Hz. The second line shows the GCell response with gain control, and the third line presents the eﬀect of lateral amacrine connectivity in the case of a Laplacian graph. Keeping the –1 same values of parameters as in the 1D case, we set w =0.3 ms corresponding to case BinFig. 7. Finally, the last line shows the eﬀect of asymmetric gap junctions, having a preferred orientation in the direction of motion, with v = 9 mm/s. gap In thecaseofthe gain controlresponse, thepeakofresponsetothe moving barisshifted by about 10 ms in the direction of motion, as compared to the static bar. The ﬂashed bar elicits a lower response, given its very short appearance in comparison with the character- istic time of adaptation. We choose this time short enough to avoid gain control triggering, explaining the diﬀerence with the strong response observed by Chen et al. [20]inthe pres- ence of a still bar. In the case of amacrine connectivity, the moving bar representation is shrunk as com- pared to the gain control case given the prevalence of inhibition, while the level of activity for the ﬂashed bar remains roughly the same. In this case, cells responding to the moving bar reach their peak activity slightly earlier (about 19 ms for these parameters value) than in the gain control case (Fig. 14(B, top)). Finally, asymmetric gap junction connectivity displays a wave propagating ahead of the bar, increasing the central blob, which is much larger than the size of the bar in the stim- ulus, while the ﬂashed bar activity remains similar to the previous cases. 3.4.2 Parabolic trajectory In this subsection, we assess the eﬀect of the three anticipatory mechanisms on a parabolic trajectory. The interest is to have a trajectory with a change in direction and speed, thus an acceleration. The stimulus consists of 20 frames displayed at 10 Hz. The simulations parameters and connectivity weights are the same as the ones used in the previous section. Figure 15 shows the response to a dot moving along a parabolic trajectory. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 38 of 60 Figure 15 Eﬀect of anticipatory mechanisms on a parabolic trajectory.(A) Response to a dot moving along a parabolic trajectory. The ﬁrst line shows the stimulus, the second line shows the GCell response with gain –1 control, the third line presents the eﬀect of lateral ACell connectivity with w =0.3 ms ,and thelastline shows the eﬀect of asymmetric gap junctions with v = 9 mm/s. (B) Time course response of a cell gap responding to the dot near the trajectory turning point. Linear response corresponds to the response to the stimulus without gain control In the case of gain control, GCell response is more elongated, which has a distortion eﬀect on the dot representation near the turning point of the trajectory (1400–1600 ms). Cells responding near the trajectory turning point are still anticipating motion, as the peak response of the gain control curve is slightly shifted to the left, compared to the RF re- sponse (Fig. 14(B)). In the case of amacrine connectivity, the elicited response is also more localised, as com- pared to the gain control response, and the ﬂow of activity follows more accurately the stimulus. This is a direct consequence of the sensitivity of the ACell connectivity model to the stimulus acceleration. In this case, the peak response is also more shifted as compared to the gain control case. Finally, the gap junction connectivity model performs worse in this case, giving rise to a propagating wave that does not follow the trajectory, since the latter is not parallel to the direction to which GCells are sensitive. Cells responding near the trajectory turning point have a higher level of activity and an increased latency, while the peak response roughly corresponds to the gain control case. 3.4.3 Angular anticipation We investigate in this subsection a two-dimensional example of motion where angular anticipation takes place. The stimulus consists here of 72 frames, displayed at 100 Hz, of a bar moving at a constant angular speed of 4.25 rad/ms. Figure 16 shows the retina response to a rotating bar with the angular orientation of activity as a function of time for the diﬀerent models. We used Matlab to estimate the bar orientation from the displayed activity, ﬁtting the set of activated points by an ellipse whose principal axis determines the response orientation. In the three cases, one can see that the response around the centre of the bar is sup- pressed due to gain control adaptation. While the gain control activity orientation roughly follows thelinearresponse(Fig. 16(B)), the ACell response shows a slight angular shift (frames: 250–300 ms), which is also visible on the response orientation time course. The ACell angular anticipation is, however, only observed during the ﬁrst period of the bar. Interestingly, this eﬀect vanishes during the second rotation due to a persistent eﬀect of Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 39 of 60 Figure 16 Anticipation for a rotating bar.(A) Response to a bar rotating at 4.25 rad/ms. The ﬁrst line shows the stimulus, the second line shows the GCell response with gain control, the third line presents the eﬀect of –1 lateral ACell connectivity with w =0.2 ms , and the last line shows the eﬀect of asymmetric gap junctions with v = 9 mm/s. (B) Time course response of the bar orientation in the reconstructed retinal gap representations the activation function generating a sort of a suppressive eﬀect erasing the second occur- rence of the bar (frame: 450 ms). We shall point out that the ACell connectivity weight in –1 –1 this simulation has been reduced to w =0.2 ms , since with a value of w =0.3 ms used in the previous simulations, the response to the second rotation of the bar is completely suppressed. Finally, similarly to the parabolic trajectory, the gap junction connectivity model per- forms worse due to the wave propagating from left to right, distorting once more the bar shape. Consequently, the bar activity orientation in this case has been discarded in Fig. 16(B), the orientation estimate giving poor results. 3.5 Conclusion This section shows how lateral connectivity can play a role in motion anticipation of 2D stimuli, both in the case of the classical ﬂash lag eﬀect and more complex trajectories. Indeed, for a given network setting, ACell connectivity can noticeably improve anticipa- tion with respect to gain control in all three stimuli and has also the advantage of being sensitive to trajectory shifts (Sect. 3.4.2). While gap junction connectivity improves anticipation when the trajectory of the bar is parallel to the preferred GCells direction, it also induces more blur around the bar and shape distortion in the case of parabolic motion and rotation, suggesting a trade-oﬀ be- tween anticipation and object recognition for this speciﬁc model. 4 Discussion Using a simpliﬁed model, mathematically analysed with numerical simulations exam- ples, we have been able to give strong evidences that lateral connectivity—inhibition with ACells, gap junctions—could participate to motion anticipation in the retina. The main argument is that a moving stimulus can, under speciﬁc conditions mathematically con- trolled, induce a wave of activity which propagates ahead of the stimulus thanks to lateral connectivity. This suggests that, in addition to local gain control mechanism inducing an anticipated peak of GCells activity, lateral connectivity could induce a mechanism of neu- ral latencies reduction similar to what is observed in the cortex [8, 46, 77]. This is visible in particular in Fig. 14, where the gap junction coupling induces a wave which increases the GCell level activity before the bar reaches its RF. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 40 of 60 Yet, these studies raise several questions and remarks. The ﬁrst one is, of course, the biological plausibility. At the core of the model, what makes the mathematical analysis tractable is the fact that we can reduce dynamics, in some region of the phase space, to a linear dynamical system. This structure is aﬀorded by two facts: (i) Synapses are char- acterised by a simple convolution; (ii) Cells, especially ACells, have a simple passive dy- namics, where nonlinear eﬀects induced e.g. by ionic channels are neglected, as well as propagation delays. As stated in the introduction, the goal here is not to be biologically realistic, but instead to illustrate potential general spatio-temporal response mechanisms taking into account speciﬁcities of the retina, as compared to e.g. the cortex. Essentially, most neurons are not spiking (except GCells and some type of BCells or ACells, not con- sidered here [2]). Yet, synapses follow the same biophysics as their cortical counterpart. As it is standard to model the whole chain of biophysical machinery triggering a post-synaptic potential upon a sharp increase of the pre-synaptic voltage by a convolution kernel [27], we adopted here the same approach. Note that it is absolutely not required, in this convo- lutional approach, for the pre-synaptic increase in voltage to be a spike; it can be a smooth variation of the voltage. Note also that higher order convolution kernels can be consid- ered, integrating more details of the biological machinery. These higher order kernels are represented by higher order linear diﬀerential equations [35]. Concerning point (ii), non- linear eﬀects are neglected, especially for ACells (BCells have gain control), there are not so many available models of ACells. A linear model for predictive coding using linear ACells has been used by Hosoya et al. [43]. We discuss it in more detail below. The nonlinear models of ACells we know have been developed to study the retina in its early stage (reti- nal waves) and feature either AII ACells [21]orstarburst ACells [50]. In Sect. 3.2.4 we have brieﬂy commented how nonlinear mechanisms could enhance resonance eﬀects in the network and, thereby, favour the propagation of a lateral wave of activity induced by a moving stimulus. This would of course deserve more detailed study. Another potentially interesting nonlinear mechanism is short term plasticity discussed below. The second question one may ask about the model is about the robustness of this mech- anism with respect to parameters. The model contains many parameters, some of them (BCells, GCells and gain control) coming from the previous paper from Berry et al. [9] and Chen et al. [20]. Although they did not perform a structural stability analysis of their model (i.e. stability of the model with respect to small variations of parameters), we believe that they are tuned away from bifurcation points so that slight changes in their (isolated cells) model parameters would not induce big changes. As we have shown, the situation changes dramatically when cells are connected via lateral connectivity. Here, many types of dynamical behaviour can be expected simply by changing the connectivity patterns in the case of ACells. A more detailed analysis would require a closer investigation of ACell to BCell connectivity and an estimation of synaptic coupling, implying to deﬁne more speciﬁcally the type of ACell (AII, starburst, A17, wide ﬁeld, medium ﬁeld, narrow ﬁeld, etc.) and the type of functional circuit one wants to consider. Note that ACells are diﬃcult to access experimentally due to their location inside the retina. Even more diﬃcult is a measurement of ACell connectivity, especially the degree of symmetry discussed in our paper. Such studies can be performed at the computational level, though, where the math- ematical framework proposed here can be applied and extended. Computational results do not tell us what is the reality, but they shed light on what it could be. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 41 of 60 We would like now to address several possible extensions of this work. The retino-thalamico-cortical pathway Theretinaisonlythe earlystage of thevisual system. Visual responses are then processed via the thalamus and the cortex. As exposed in the introduction, anticipation is also observed in V1 with a diﬀerent modality than in the retina. In this paper, our main focus was on the shift of the peak response, while when studying anticipation in the cortex, the main focus lies in the increase of the response latency, i.e. the delay between the time the bar reaches the receptive ﬁeld of a cortical column and the eﬀective time its activity starts rising. How do these two eﬀects combine? How does retinal anticipation impact cortical anticipation? To answer these questions at a computational level, one would need to propose a model of the retino-thalamico-cortical pathway which, to the best of our knowledge, has never been done. Yet, we have developed a retino-cortical (V1) model—thus, short-cutting the thalamus—based on a mean-ﬁeld model of the V1 cortex, developed earlier by the groups of F. Chavane and A. Destexhe [19, 93], able to reproduce V1 anticipation as observed in VSDI imaging. The aim of this work is to understand, computationally, the eﬀect of retinal anticipation on the cortical one and more generally the combined eﬀects of motion extrapolation in the retina and V1. This is the object of a forthcoming paper (S. Souihel, M. di Volo, S. Chemla, A. Destexhe, F. Chavane and B. Cessac., in preparation). See [74] for preliminary results. Retinal circuits BCells, ACells and GCells are organised into multiple, local, functional circuits with speciﬁc connectivity patterns and dynamics in response to stimuli. Each cir- cuit is related to a speciﬁc task, such as light intensity or contrast adaptation, motion de- tection, orientation, motion direction and so on. Here, we have considered a circuit allow- ing the retina to detect a moving object on a moving background, where motion sensitive retinal cells remain silent under global motion of the visual scene, but ﬁre when the image patch in their receptive ﬁeld moves diﬀerently from the background. From our study we have emitted the hypothesis that this circuit, spread over the retina, could improve mo- tion anticipation thanks to what we have called the “push–pull” eﬀect. Yet, other circuits could be studied in their role to process motion and anticipation. We especially think of the ON-OFF cone and rod-cone pathways responsible for the separation of highlights and shadows, allowing to provide information to the GCells concerning brighter than back- ground stimuli (ON-centre) or darker than background stimuli (OFF-centre) [54]. This circuit involves both gap junction and ACell (AII) connectivity, and our model could al- low to study its dynamics in the presence of a moving object. Adaptation eﬀects In a paper from 2005, Hosoya et al. [43] studied dynamic predictive coding in the retina and showed how spatio-temporal receptive ﬁelds of retinal GCells change after a few seconds in a new environment, allowing the retina to adjust its pro- cessing dynamically when encountering changes in its visual environment. They showed that an amacrine network model with plastic synapses can account for the large variety of observed adaptations. They feature a linear network model of ACells, similar to ours, with, in addition, anti-Hebbian plasticity. Their mathematical analysis, based on linear algebra, allows to determine the behaviour of the model in terms of eigenvalues and eigenvectors. However, their analysis does not carry out to the gain control introduced by Berry et al., which, as we show, renders the spectral analysis quite more complex. It would therefore be Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 42 of 60 interesting to explore how plasticity in the ACell synaptic network, conjugated with local gain control, contributes to anticipation. Correlations The trajectory of a moving object—which is, in general, quite more com- plex than a moving bar with constant speed—involves long-range correlations in space and in time. Local information about this motion is encoded by retinal GCells. De- coders based on the ﬁring rates of these cells can extract some of the motion features [25, 43, 51, 62, 66, 67, 76]. Yet lateral connectivity plays a central role in motion processing (see, e.g. [41]). One may expect it to induce spatial and temporal correlations in spiking activity, as an echo, a trace, of the object’s trajectory. These correlations cannot be read in the variations of ﬁring rate; they also cannot be read in synchronous pairwise corre- lations, as the propagation of information due to lateral connectivity necessarily involves delays. This example raises the question about what information can be extracted from spatio-temporal correlations in a network of connected neurons submitted to a transient stimulus. What is the eﬀect of the stimulus on these correlations? How can one handle this information from data where one has to measure transient correlations? This question has been addressed in [17]. The potential impact of these spatio-temporal correlations on de- coding and anticipating a trajectory will be the object of further studies. Orientation selective cells Our model aﬀords the possibility to consider BCells with ori- entation sensitive RF. The potential role of such BCells for predictive coding has been outlined by Johnston et al. [48]. In their model individual GCells receive excitatory BCell inputs tuned to diﬀerent orientations, generating a dynamic predictive code, while feed- forward inhibition generates a high-pass ﬁlter that only transmits the initial activation of these inputs, removing redundancy. Should such circuits play a role in motion anticipa- tion? We did not elaborate on this in the present paper, leaving it to a potential forthcoming work. Another important question is “how to model a retinal network with cells having diﬀerent orientation selectivity?” A V1 cortical model has been proposed by Baspinar et al. [7] for the generation of orientation preference maps, considering both orientation and scale features. Each point (cortical column) is characterised by intrinsic variables, orien- tation and scale, and the corresponding RF is a rotated Gabor function. The visual stim- ulus is lifted in a four-dimensional space, characterised by coordinate variables, position, orientation and scale. The authors infer from the V1 connectivity a “natural” geometry from which they can apply methods from diﬀerential geometry. This type of mathemati- cal construction could be interesting to investigate in the case of the retina with families of orientation selective cells, although the retinal connectivity between these cells is not the same as the V1 orientation preference map structure [12, 13, 82, 91]. Biologically inspired vision systems When the retina receives a visual stimulus, it deter- mines which component of it is signiﬁcant and needs to be further transmitted to the brain. This eﬃcient coding heuristic has inspired many recent studies in developing biologically inspired systems, both for static image and motion representations [59, 87, 90]. Two ma- jor applications of biologically inspired vision systems are retinal prostheses [53, 63, 79] and navigational robotics [24, 52]. Focusing on the second ﬁeld of application, the abil- ity of a mobile device to navigate in its environment is of utmost interest, especially in order to avoid dangerous situations such as collisions. To be able to move, the robot re- quires a mapped representation of its environment, but also the ability to interpret and Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 43 of 60 process this representation. Motion processing mechanisms such as anticipation can be thus implemented to assess the eﬃciency of bio-inspired vision in obstacle avoidance. Psychophysical study of motion anticipation We brieﬂy come back to the ﬂash lag eﬀect introduced in 3.4.1. Neuroscience has explored several explanations for this illusion. The ﬁrst explanation is that the visual system processes moving objects with a smaller latency than ﬂashed objects [89]. The second explanation suggests that the ﬂash lag eﬀect is due to postdiction: the perception of the ﬂash is conditioned by events happening after its ap- pearance [30, 31]. This hypothesis is inspired by the colour phi illusion, where two dots of diﬀerent colours appearing at two discrete yet close positions, with a small latency, will be perceived as a single moving dot whose colour has changed. Another explanation in- cludes motion extrapolation. The visual system being predictive, it processes diﬀerently a bar in smooth motion, whose motion can be extrapolated, and a ﬂashed bar, which can- not be predicted by the system. The study conducted in this article falls under this third explanation, suggesting that the retina could possibly be assisting the cortex in the motion extrapolation task. Appendix A: Parameters of the model Table 1 Model parameters values used in simulations, unless stated otherwise Function Parameter Value Unit K (Eq. (56)) σ (centre) 90 μm B,S 1 σ (surround) 290 μm A (centre) 1.2 mV A (surround) 0.2 mV K (t)(Eq.(57)) μ 60 ms T 1 μ 180 ms σ 20 ms σ 44 ms K 0.22 unitless K 0.1 unitless BCell dynamics τ 100 ms –1 –3 –1 h 6.11e mV .ms θ 5.32 mV τ 200 ms ACell dynamics τ 200 ms –1 w [0, 1] ms ACell connectivity ξ {1, 2, 3, 4} mm n ¯ {1, 2, 3, 4} unitless σ 1unitless GCell dynamics a 0.5 unitless σ 90 μm τ 189.5 ms –4 h 3.59e unitless α 1110 Hz/mV θ 0mV max N 212 Hz G Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 44 of 60 Appendix B: Spatio-temporal ﬁltering B.1 Receptive ﬁelds The spatial kernel of the BCell i is modelled with a diﬀerence of Gaussians (DOG): A 1 A 1 –1 –1 1 ˜ 2 ˜ – X .C .X – X .C .X i i i i 2 1 2 2 K (x, y)= √ e – √ e , (56) B ,S 2π det C 2π det C 1 2 x–x where X = ," denotes the transpose, x and y are the coordinates of the receptive i i i y–y ﬁeld centre which coincide with the coordinates of the cell, C , C are positive deﬁnite ma- 1 2 trix whose main principal axis represents the preferred orientation. For circular DOGs(no 2 2 preferred orientation) C ≡ σ I, C ≡ σ I where I is the identity matrix in two dimen- 1 2 1 2 sions. The two Gaussians of the DOG are thus concentric. They have the same principal axes. X has the physical dimension of a length (mm), thus the entries of C , a =1,2, are i a expressed in mm . A , a = 1,...,2, have the dimension of mV so that convolution (1)has the dimension of a voltage. We model the temporal part of the RF with a diﬀerence of non-concentric Gaussians whose integral on the time domain is zero. This kernel well ﬁts the shape of the temporal projection of the bipolar RF observed in experiments [74]. 2 2 (t–μ ) (t–μ ) 1 2 – – K K 1 2 2 2 2σ 2σ 1 2 K (t)= √ e – √ e H(t), (57) 2πσ 2πσ 1 2 where H(t) is the Heaviside function. The parameters μ , σ , b = 1, 2, have the dimension b b of a time (s), whereas K are dimensionless. The following condition must hold to ensure the continuity of K (t)at zero: 2 2 μ μ 1 2 – – K K 2 2 1 2 2σ 2σ 1 2 e = e . (58) σ σ 1 2 Thus, K (x, y, 0) = 0. In addition, we require that the integral of a constant stimulus con- verges to zero, so that the cell is only reactive to changes. This reads as follows: μ μ 1 2 K = K , (59) 1 2 σ σ 1 2 where (x)= √ e dy (60) 2π –∞ is the cumulative distribution function of the standard Gaussian probability. B.2 Numerical convolution Here we describe the method used to numerically integrate convolution (1)ofthe recep- tive ﬁeld K (56) in the spatial domain with a stimulus S. For the sake of clarity, we B ,S restrict the computation to one Gaussian in the DOG. The extension to a diﬀerence of Gaussian is straightforward. In the following, we consider a spatially discretized stimulus. When dealing with a 2D stimulus, we have to integrate over two axes. In the case where the Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 45 of 60 eigenvectors of the 2D of Gaussians are the axes of integration, the spatial ﬁlter is separable in the stimulus coordinate system. Considering the stimulus as a grid of pixels, we can in- tegrate using the following discretization: let L be the size of the stimulus along the x axis in pixels, L be its size along the y axis, and δ be the pixel length. We set S (t) ≡ S(iδ, jδ, u), y ij L y with i = 0,..., and j = 0,..., . The spatial convolution becomes then δ δ x,y [K ∗ S](t) 2 2 (x–x ) (y–y ) 0 0 – – 1 2 2 2σ 2σ x y = S(x, y, t)e dx dy 2πσ σ 2 x y R i + δ – x i – x j + δ – y j – y 0 0 0 0 = S (t) erf √ – erf √ erf √ – erf √ . ij 2σ 2σ 2σ 2σ x x y y i,j In the case where the eigenvectors of the 2D of Gaussians are not the axes of integration, the spatial ﬁlter is not separable in the stimulus coordinates system. There exist methods that perform the computation by making a linear combination of basis ﬁlters [36], others that use Fourier based deconvolution techniques [83] and others using recursive ﬁlter- ing techniques [26]. However, these methods are of high computational complexity. We choose instead to use a computer vision method from Geusenroek et al. [39]. It is based on a projection in a non-orthogonal basis, where the ﬁrst axis is x and the second is parametrized by an angle φ (see Fig. 17). The new standard deviations read as follows: σ σ x y σ = ! , 2 2 2 2 σ cos θ + σ sin θ x y 2 2 2 2 σ cos θ + σ sin θ y x σ = sin φ Figure 17 Filter transformation description [39]. Theoriginalsystemofaxesisrepresented by x and y,and the ellipse system of axes by u and v. φ represents the angle of the second axis of the non-orthogonal basis. The integration domain of a pixel is limited by four lines of equations: x = iδ, x =(i +1)δ, y = jδ and y =(j +1)δ. Rewriting these fours equations in the new system of axes through a coordinate change enables us to write Eq. (61) Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 46 of 60 with 2 2 2 2 σ cos θ + σ sin θ y x tan(φ)= 2 2 (σ – σ ) cos θ sin θ x y and σ = σ (in the orientation sensitive case). x y We adapt the implementation to the spatially discretized stimulus using an integration scheme similar to the one introduced in the separable case. The spatial convolution reads now as follows: x,y [K ∗ S](x , y , t) B 0 0 δ (y –y ) (y+1) sin(φ) 2 2σ = σ C e x ij (61) sin(φ) (i;j)∈[0,s ]×[0,s ] x y (– cos(φ)y + x +1)δ – x (– cos(φ)y + x)δ – x 0 0 × erf √ – erf √ dy , 2σ 2σ x x where s (resp. s ) denotes the size of the stimulus in pixels along the x axis (resp. y axis), x y and C is thecontrastatthe pixellocated at (i, j). ij The integral is then computed numerically. The advantage of this formulation is to re- place a two-dimensional integration by a one-dimensional one. Appendix C: Random connectivity Here, we deﬁne the random connectivity matrix from ACell to BCells considered in Sect. 2.3.3. Each cell (ACell and BCell) has a random number of branches (dendritic tree), each of which has a random length and a random angle with respect to the horizontal axis. The length of branches L follows an exponential distribution f (l)= e , l ≥ 0, (62) with spatial scale ξ. The number of branches n is also a random variable, Gaussian with mean n ¯ and variance σ . The angle distribution is taken to be isotropic in the plane, i.e. uniform on [0, 2π[. When a branch of an ACell A intersects a branch of a BCell B, there is achemicalsynapse from AtoB. Here, we assume that both cell types have the same probability distributions for branches, thus neglecting the actual shape of ACell and BCell dendritic trees. On biolog- ical grounds, this assumption is relevant if we consider the shape of BCell dendritic tree in the inner plexiform layer (IPL)(seee.g https://webvision.med.utah.edu/book/part-iii- retinal-circuits/roles-of-ACell-cells/ Figs. 5, 16, 17). While out of the IPL, BCells have the form of a dipole, in the IPL their dendrites have a form well approximated by our two- dimensional model. A potential reﬁnement would consist of considering diﬀerent set of parameters in the probability laws respectively deﬁning BCell and ACell dendritic tree. We show in Fig. 18 an example of connectivity matrix produced this way, as well as the probability that two branches intersect as a function of the distance of the two cells. We now compute this probability. We use the standard notation in probability theory where the random variable is written in capitals and its realisation in a small letter. Thus, Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 47 of 60 Figure 18 Random connectivity. Left. Example of a random connectivity matrix from ACells A to BCells cells B . j i White points correspond to connection from A to B . Right. Probability P(d) that two branches intersect as a j i function of the distance between two cells. ‘Exp’ corresponds to numerical estimation and ‘Th’ corresponds to the theoretical prediction. Here, ξ =2 F (x)= P[X < x] is the cumulative distribution function of the random variable X and dF f (x)= is its density. dx We consider the oriented connection between the cell A (ACell) of coordinates (x , y ) A A to a cell B (BCell) of coordinates (x , y ), so that the distance between the two cells is B B 2 2 d = (x – x ) +(y – y ) . AB B A B A The vector AB makes an oriented angle η = (AB, Ax) with the positive horizontal axis, where y –y ⎨ B A arctan ,if x > x ; B A x –x B A η = (63) y –y ⎩ B A π + arctan ,if x < x . B A x –x B A Here, we neglect the eﬀects of boundaries, taking e.g. an inﬁnite lattice or periodic bound- ary conditions, so that the probability to connect A to B is invariant by rotation. Thus, we y –y B A compute this probability in the ﬁrst quadrant x > x , y > y .Inthiscase η = arctan . B A B A x –x B A Each cell has a random number of branches (dendritic tree), each of which has a random length and a random angle with respect to the horizontal axis (Fig. 19). The length of branches L follows the exponential distribution (62): f (l)= e , l ≥ 0, with repartition function F (l)=1 – e . (64) The spatial scale ξ favours short range connections. The number of branches N distribu- tion follows a normal distribution with mean n ¯ and variance σ . The angle distribution is taken to be isotropic in the plane, i.e. uniform on [0, 2π[. We compute the probability that a branch of ACell A of length L intersects at point C abranchofBCell B of length L .Wedenoteby α the oriented angle (Ax, AC); by β the oriented angle (Bx, BC); by θ the oriented angle (AB, AC). In the ﬁrst quadrant, α = θ + η. Note that the condition to be in the ﬁrst quadrant constraints η but not α. sin(β–α+θ) sin θ sin(β–α) From the sin rule we have = = .Thisholds,however,if A, B, C is a d d d AC BC AB triangle, that is, if the two branches are long enough to intersect at C,which reads: 0 ≤ sin(β–α+θ) sin θ d = d ≤ L and 0 ≤ d = d ≤ L . These are necessary and suﬃcient AC AB A BC AB B sin(β–α) sin(β–α) conditions for the branches to intersect. Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 48 of 60 Figure 19 Geometry of connection between two neurons. α (β) is the angle of the neuron A’s branch with length L (neuron B’s branch with length L ) with respect to the horizontal axis. θ is the angle between the A B segment connecting AB and the branch A. C represents the virtual point that lies at the intersection of the branches of length L and L . d (resp. d , d ) denotes the distance between A and B (resp. A, C and B, C). A B AB AC BC Note that d ≤ L , d ≤ L AC A BC B Note that the positivity of these quantities imposes conditions linking the angles α, β, η with θ = α – η. 1. If sin(β – α)>0 ⇔ 0< β – α < π ⇔ α < β < π + α,wemusthave sin θ >0 ⇔ 0< θ = α – η < π so that η < α < π + η and sin(β – α + θ)>0 ⇔ 0< β – α + θ = β – η < π so that η < β < π + η (because η ≥ 0). All these constraints are satisﬁed if η < α < β < min(π + α, π + η)= π + η. 2. If sin(β – α)<0 ⇔ –π < β – α <0 ⇔ –π + α < β < α,wemusthave sin θ <0 ⇔ –π < θ = α – η <0 so that –π + η < α < η and sin(β – α + θ)<0 ⇔ –π < β – α + θ = β – η <0 so that –π + η < β < η.All these constraints are satisﬁed if max(–π + α,–π + η)=–π + η < β < α < η. Modulo these conditions, the conditional probability ρ to have intersection given c|(α,β) the angles α, β is ρ = P[Connection | α, β] c|(α,β) sin(β – α + θ) sin θ = P 0 ≤ d ≤ L ,0 ≤ d ≤ L α, β . AB A AB B sin(β – α) sin(β – α) Using the cumulative distribution function (64) of the exponential distribution and the independence of L , L gives A B α+β sin( –η) AB 2 sin(β – α + θ) sin θ ξ β–α sin( ) ρ = 1– F d 1– F d = e . c|(α,β) L AB L AB sin(β – α) sin(β – α) Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 49 of 60 The probability to connect the two branches in the ﬁrst quadrant is then α+β sin( –η) AB 2 π+η π+η 1 ξ β–α sin( ) ρ (d , η)= e dα dβ c AB 4π α=η β=α (65) α+β d sin( –η) AB 2 η α ξ β–α sin( ) + e dα dβ , α=–π+η β=–π+η which depends on the distance between the two cells and their angle η, depending para- metrically on the characteristic length ξ. Note that the condition of positivity of the sine ratio ensures an exponential decay of the probability as d increases. AB Remark The positivity of arguments in the exponential implies that π+η π+η η α 2 2 4π ρ (d , η) ≤ dα dβ + dα dβ = π , c AB α=η β=α α=–π+η β=–π+η so that ρ (d , η) ≤ . c AB Appendix D: Linear analysis D.1 General solution of the linear dynamical system dX Here, we consider dynamical system (34), = L.X + F(t), whose general solution is dt L(t–s) X (t)= e .F(s) ds. The behaviour of this integral depends on the spectrum of L.The diﬃcultyisthat L is not diagonalisable (because of the activity term h I ). We write it in the following form: B N,N ⎛ ⎞ ⎛ ⎞ 0 0 0 I N,N N,N N,N ⎜ ⎟ % N,N A & – W ⎜ τ B ⎟ ⎜ ⎟ L = + . ⎜ 0 0 ⎟ ⎝ 0 0 0 ⎠ N,N N,N N,N N,N N,N B N,N ⎝ ⎠ W – A τ h I 0 0 B N,N N,N N,N N,N 0 0 – N,N N,N We assume that the matrix M is diagonalisable. Even in this case, L is not diagonalisable because of the Jordan matrix J .Wedenoteby φ the normalised eigenvectors of M and by λ the corresponding eigenvalues with β = 1,...,2N. The eigenvalues of D are then the 1 1 2N eigenvalues of M plus N eigenvalues – .Wedenotethemby λ too, with λ =– , β β τ τ a a β =2N + 1,...,3N. The eigenvectors of D have the form , β = 1,...,2N; P = e , β =2N + 1,...,3N, where 0 is the N-dimensional vector with entries 0 and e is the canonical basis vector N β in direction β.The matrix P made by the columns P is the matrix which diagonalises D. β Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 50 of 60 –1 We denote by = P DP the diagonal form, where = Diag{λ , β = 1,...,3N }. P and –1 P have the following block form: –1 0 0 2N,N 2N,N –1 P = ; yP = , (66) 0 I 0 I N,2N N,N N,2N N,N where is the matrix whose columns are the eigenvectors φ of M.Thisformimplies –1 that P = P = δ for β =2N + 1,...,3N. αβ αβ αβ +∞ Lt Lt t n We now compute e using the series expansion e = (D + J ) .Using therela- n=0 n! tions 2 n J =0 ; D .J = – J , 3N,3N one proves that n–1 k n n n–1–k (D + J ) = D + J . – D . k=0 Therefore +∞ n–1 t 1 Lt Dt n–1–k e = e + J . – D . n! τ n=1 k=0 –1 We use the matrices P, P to write it in the form +∞ n–1 t 1 Lt t –1 n–1–k –1 e = P.e .P + J . – P. .P . n! τ n=1 k=0 From relation (36) we obtain, for the entries of X (t), 3N –1 λ (t–s) X (t)= P P e F (s) ds α αβ γ βγ β,γ =1 (67) 3N +∞ n–1 t n (t – s) 1 –1 n–1–k + J P P – λ F (s) ds. α,δ δβ γ βγ β n! τ t a δ,β,γ =1 n=1 k=0 We consider the ﬁrst term of this equation. We use F = F , γ = i = 1,..., N (BCells). γ B V dV i i drive drive We recall that, from (13), F (t)= + ,sothat i τ dt t t λ (t–s) λ (t–s) β β e F (s) ds = V (t)+ + λ e V (s) ds. (68) γ γ β γ drive drive t B t 0 0 Moreover, 3N 3N 3N 3N 3N –1 –1 P P V (t)= V (t) P P = V (t)δ = V (t). αβ γ γ αβ γ αγ α βγ drive drive βγ drive drive β=1 γ =1 γ =1 β=1 γ =1 Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 51 of 60 We extend the deﬁnition of drive term (1)to 3N dimensions such that V (t)=0 if drive α > N.Thus 3N –1 λ (t–s) P P e F (s) ds αβ γ βγ β,γ =1 3N N –1 λ (t–s) = V (t)+ + λ P P e V (s) ds. α β αβ γ drive βγ drive B –∞ β=1 γ =1 We decompose the sum over β in three sums: β = 1,..., N corresponding to BCells; β = N + 1,...,2N corresponding to ACells; β =2N + 1,...,3N corresponding to activities of BCells. We deﬁne (Eq. (38)inthe text): N N B –1 λ (t–s) E (t)= + λ P P e V (s) ds, α = 1,..., N, β αβ γ B,α βγ drive β=1 γ =1 corresponding to the indirect eﬀect, via the ACell connectivity, of the drive on BCell volt- ages. The term 2N N B –1 λ (t–s) E (t)= + λ P P e V (s) ds, α = N + 1,...,2N β αβ γ A,α βγ drive γ =1 β=N+1 (Eq. (39) in the text) corresponds to the eﬀect of BCell drive on ACell voltages. The third term 3N N –1 λ (t–s) + λ P P e V (s) ds =0, α =2N + 1,...,3N, β αβ γ βγ drive B t β=2N+1 γ =1 –1 because P = δ . βγ βγ To compute the second term in Eq. (67), we ﬁrst ﬁrst remark that J =0 if α = 1,...,2N α,δ and J = h δ if α =2N + 1,...,3N, so that this term is nonzero only if α =2N + α,δ B α–2N,δ –1 1,...,3N (BCell activities). Also F =0 for γ = 1,..., N,while P = δ for β =2N + γ βγ βγ 1,...,3N. Therefore, for α =2N + 1,...,3N,the second term in X (t)is 2N N +∞ n–1 t k (t – s) 1 –1 n–1–k h P P – λ F (s) ds. B α–2N β γ βγ β n! τ β=1 γ =1 n=1 k=0 We now simplify the series: +∞ n–1 +∞ n–1 k k n n (t – s) 1 (t – s) 1 n–1–k n–1 – λ = λ – β β n! τ n! τ λ a a β n=1 k=0 n=1 k=0 1 n +∞ 1–(– ) (t – s) τ λ n–1 = λ n! 1+ n=1 τ λ a β Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 52 of 60 ' ( +∞ +∞ t–s (– ) 1 1 (λ (t – s)) = – λ 1+ n! n! τ λ n=1 n=1 a β 1 t–s λ (t–s) – β τ = e – e . λ + Thetimeintegraliscomputedthe same wayasEq. (68): +∞ n–1 t n (t – s) 1 n–1–k – λ F (s) ds n! τ t a n=1 k=0 t t 1 t–s λ (t–s) – β τ = e F (s) ds – e F (s) ds γ γ λ + β t t 0 0 t t 1 1 1 1 t–s λ (t–s) = + λ e V (s) ds – – e V (s) ds . β γ γ 1 drive drive λ + τ τ τ B B a β t t 0 0 Similar to Eqs. (38), (39)inthe text,weintroduce ' ( 2N N 1 λ (t–s) 1 ( + λ ) e V (s) ds β γ B –1 τ t drive B 0 E (t)= h P P , t–s B α–2N β a,α βγ t 1 1 1 – λ + –( – ) e V (s) ds τ t drive β=1 γ =1 a τ τ B a 0 α =2N + 1,...,3N, corresponding to the action of BCells and ACells on the activity of BCells via the net- 2N –1 work eﬀect. Let us consider in more detail the second term. From (66), P P = α–2N β β=1 βγ δ ,thus α–2N γ 2N N N 2N t t t–s t–s – – –1 –1 τ τ a a P P e V (s) ds = e V (s) ds P P α–2N β γ γ α–2N β βγ drive drive βγ t t 0 0 γ =1 γ =1 β=1 β=1 α–2N γ t–s – 0 = e V (s) ds ≡ A (t) α–2N α–2N drive (Eq. (41)inthe text). This ﬁnally leads to ((40)inthe text) 2N N 1 1 1 λ + – + τ τ τ B –1 B λ (t–s) B a 0 E (t)= h P P e V (s) ds + A (t) , B α–2N β γ a,α βγ drive α–2N 1 1 λ + λ + β t β τ τ β=1 γ =1 a a α =2N + 1,...,3N. D.2 Spectrum of L and stability of dynamical system (34) Here, we assume that a BCell connects only one ACell, with a weight w uniform for all B + + BCells, so that W = w I , w > 0.WealsoassumethatACellsconnect to BCells with N,N – – a connectivity matrix W, not necessarily symmetric, with a uniform weight –w , w >0, A – so that W =–w W. We have shown in the previous section that the 2N ﬁrst eigenvalues and eigenvectors of L are given by the 2N eigenvalues and eigenvectors of M,which reads Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 53 of 60 now as follows: N,N – – –w W M = . (69) + N,N w I – N,N We now show that this speciﬁc structure allows us to compute the spectrum of M in terms of the spectrum of W. D.2.1 Eigenvalues and eigenvectors of M We denote by κ , n = 1,..., N, the eigenvalues of W ordered as |κ |≤|κ | ≤ ··· ≤ |κ |,and n 1 2 n ψ is the corresponding eigenvector. We normalise ψ so that ψ .ψ =1, where † is the n n n adjoint. (Note that, as W is not symmetric in general, eigenvectors are complex.) We shall neglect the case where, simultaneously, =0 and κ =0 for some n. ± ± ± n Proposition For each n, there is a pair of eigenvalues λ and eigenvectors φ = c n n n ± ρ ψ n n ± 1 of M with c = √ (normalisation factor) and 1+(ρ ) 1 1 ⎪ (1 ± 1–4μκ ), κ =0, =0; – n n 2τw κ τ ± 1 ρ = w τ, κ =0, =0; (70) ⎪ ! w 1 1 ± – , =0, w κ τ where 1 1 1 = – τ τ τ A B and 1 1 1 = + . τ τ τ AB A B Eigenvalues are given by ⎨ 1 1 1 – ∓ 1–4μκ , =0; ± 2τ 2τ τ AB λ = (71) ⎩ 1 1 – + – ∓ –w w κ , =0, τ τ with – + 2 μ = w w τ ≥ 0. As a consequence, in addition to the N last eigenvalues – , L admits 2N eigenvalues given by (71), while the 2N ﬁrst columns of the matrix P (eigenvectors of L) are as follows: ⎛ ⎞ ⎛ ⎞ ψ ψ n n 1 1 ⎜ ⎟ ⎜ ⎟ – + P = $ ρ ψ ; P = $ ρ ψ , β = n = 1,..., N. (72) β ⎝ ⎠ β+N ⎝ ⎠ n n n n – 2 + 2 1+(ρ ) 1+(ρ ) n n 0 0 N N For the N last eigenvectors P = e , β =2N + 1,...,3N. β β Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 54 of 60 Remark The structure of these eigenvectors is quite instructive. Indeed, the factors ρ control the projection of the eigenvectors P on the space of ACells, thereby tuning the inﬂuence of ACells via lateral connectivity. Proof We use here the generic notation λ , φ , β = 1,...,2N, for the eigenvalues and as- β β sociated eigenvectors of M. If we assume that φ is of the form φ = for some n, β β ρψ then we have I 1 – – – –w W (– – w ρκ ).ψ ψ ψ n n n n τ τ B B M.φ = . = = λ , β β I ρ + + w I – ρψ (– + w ).ψ ρψ n n n τ τ A A which gives ⎨ 1 – (– – w ρκ )= λ , n β (– + w )= λ ρ, leading to – 2 + w κ ρ – ρ + w =0, where 1 1 1 = – . τ τ τ A B This gives, if κ =0 and =0, ρ = (1 ± 1–4μκ ), 2τw κ where – + 2 μ = w w τ ≥ 0. Thus, for each n, there are two eigenvalues: 1 1 λ =– ∓ 1–4μκ , 2τ 2τ AB with 1 1 1 = + . τ τ τ AB A B 1 1 Note that ≥ . τ τ AB 1 ± + If κ =0, =0, ρ = w τ ,then λ =– – w ρκ . β n B Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 55 of 60 1 ± w 1 ± 1 1 – + Finally, if κ =0, =0, (τ = τ ), ρ =– and λ =– ± –w w κ .If κ =0, =0, n A B – n n n n τ w κ τ τ n B there is no solution for ρ. Remark When μ =0, M is diagonal: the N ﬁrst eigenvalues are – ,the N next eigen- 1 1 1 + – values are – .Wehaveinthiscase λ =– and λ =– . Therefore, in order to be n n τ τ τ A B A coherent with this diagonal form of L when μ = 0, we order eigenvalues and eigenvectors + – of M such that the N ﬁrst eigenvalues are λ = λ , β = 1,..., N,and the N next are λ = λ , β β n n β = N + 1,...,2N. D.2.2 Stability of eigenmodes Stability of eigenmodes when W is symmetric If W is symmetric, its eigenvalues κ are real, but λ , β = 1,...,2N, can be real or complex, depending on κ ,as μ is positive. β n We have four cases: • κ <0. Then, from (71), λ s are real, and there are two cases. If > 0, the eigenvalues n β λ , β = 1,..., N, can have a positive real part (unstable), while λ , β = N + 1,...,2N,has β β always a negative real part (stable); for < 0, the situation is inverted. In both cases, the eigenvalue λ has a positive real part if 1 τ τ A B μ >– ≡ μ , n,u κ (τ – τ ) n B A which reads, using the deﬁnition of μ, as follows: 1 1 – + w w >– . (73) τ τ κ A B n Thus, τ , τ play a symmetric role. If =0 (τ = τ ), all eigenvalues are real. Eigenvalues A B A B – + – + 1 1 λ are all stable. The eigenvalue λ becomes unstable if w w >– , corresponding to n n 2 τ n (73). Proof There are two cases. – >0 ⇔ τ < τ . A B 1 1 λ =– ± 1–4μκ >0 β n 2τ 2τ AB ⇔± 1–4μκ > . AB Only + is possible (because τ and τ are positive). This gives AB 1–4μκ > , AB τ τ τ A B 1– =–4 >4μκ , 2 2 τ (τ – τ ) B A AB which is possible because κ <0. Thus, λ is unstable if n β 1 τ τ A B μ >– ≡ μ . n,u κ (τ – τ ) n B A Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 56 of 60 – <0 ⇔ τ > τ . A B 1 1 – ± 1–4μκ >0 2τ 2τ AB ⇔± 1–4μκ < . AB Only – is possible (because τ <0). This gives 1–4μκ > , AB the same condition as in the previous item. 1 1 – + –If =0, λ =– ∓ –w w κ so that eigenvalues are real. The eigenvalue with the β n τ τ – + minus sign (λ ) are all stable. The eigenvalue λ becomes unstable if n n 1 1 – + w w >– . τ κ • κ >0. Then λ , β = 1,...,2N, are real or complex. If =0, they are complex if n β μ > ≡ μ . n,c 4κ 1 1 In this case the real part is – , the imaginary part is ± 1–4μκ , and all eigenvalues 2τ 2τ AB are stable. If μ ≤ μ , eigenvalues λ are real and all modes are stable as well. Indeed: n,c β 1 1 –If =0, all eigenvalues are equal to – , hence are stable. τ 2τ AB –If >0 ⇔ τ < τ . A B 1 1 – ± 1–4μκ >0 2τ 2τ AB ⇔± 1–4μκ > , AB which is not possible because >1,whereas 1–4μκ <1. AB –If <0 ⇔ τ > τ . A B 1 1 – ± 1–4μκ >0 2τ 2τ AB ⇔± 1–4μκ < . AB Only – is possible because τ <0. 1–4μκ > , AB which is not possible because >1,whereas 1–4μκ <1. AB Souihel and Cessac Journal of Mathematical Neuroscience (2021) 11:3 Page 57 of 60 Stability of eigenmodes when W is asymmetric If W is asymmetric, eigenvalues κ are complex, κ = κ + iκ .Wewrite λ = λ + iλ , β = 1,...,2N,with n n,r n,i β β,r β,i 1 1 1 λ =– ± a + u ; β,r n n 2τ 2τ AB 2 1 1 λ = ± u – a , β,i n n 2τ 2 2 2 2 2 2 where a =1 – 4μκ and u = (1 – 4μκ ) +16μ κ = 1–8μκ +16μ |κ | .Note n n,r n n,r n n,i n,r that we recover the real case when κ =0 by setting u = a . n,i n n Instability occurs if a + u >2 , n n AB a condition depending on κ and κ . n,r n,i Acknowledgements We warmly acknowledge Olivier Marre and Frédéric Chavane for their insightful comments, as well as Michael Berry, Matthias Hennig, Benoit Miramond, Stephanie Palmer and Laurent Perrinet for their thorough feedback as Jury members of Selma Souihel’s PhD. Funding This work was supported by the National Research Agency (ANR), in the project “Trajectory”, https://anr.fr/Project-ANR-15- CE37-0011, funding Selma Souihel’s PhD, and by the interdisciplinary Institute for Modelling in Neuroscience and Cognition (NeuroMod http://univ-cotedazur.fr/en/idex/projet-structurant/neuromod) of the Université Côte d’Azur. Availability of data and materials The model source code is available upon request. Ethics approval and consent to participate Not applicable. Competing interests The authors declare that they have no competing interests. Consent for publication The authors transfer to Springer the non-exclusive publication rights, and they warrant that their contribution is original and that they have full power to make this grant. Authors’ contributions Both authors contributed to the ﬁnal version of the manuscript. BC supervised the project. Both authors read and approved the ﬁnal manuscript. Endnote dA a B From (6) >0 if A (t)< h τ V (t). 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The Journal of Mathematical Neuroscience – Springer Journals

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On the potential role of lateral connectivity in retinal anticipation

On the potential role of lateral connectivity in retinal anticipation

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On the potential role of lateral connectivity in retinal anticipation

On the potential role of lateral connectivity in retinal anticipation

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