Canard solutions in neural mass models: consequences on critical regimes

Elif Köksal Ersöz; Fabrice Wendling

doi:10.1186/s13408-021-00109-z

Canard solutions in neural mass models: consequences on critical regimes

Köksal Ersöz, Elif; Wendling, Fabrice 2021-09-16 00:00:00 Univ Rennes, INSERM, LTSI-U1099, Mathematical models at multiple temporal and spatial scales can unveil the Campus de Beaulieu, F - 35000 Rennes, France fundamental mechanisms of critical transitions in brain activities. Neural mass models (NMMs) consider the average temporal dynamics of interconnected neuronal subpopulations without explicitly representing the underlying cellular activity. The mesoscopic level oﬀered by the neural mass formulation has been used to model electroencephalographic (EEG) recordings and to investigate various cerebral mechanisms, such as the generation of physiological and pathological brain activities. In this work, we consider a NMM widely accepted in the context of epilepsy, which includes four interacting neuronal subpopulations with diﬀerent synaptic kinetics. Due to the resulting three-time-scale structure, the model yields complex oscillations of relaxation and bursting types. By applying the principles of geometric singular perturbation theory, we unveil the existence of the canard solutions and detail how they organize the complex oscillations and excitability properties of the model. In particular, we show that boundaries between pathological epileptic discharges and physiological background activity are determined by the canard solutions. Finally we report the existence of canard-mediated small-amplitude frequency-speciﬁc oscillations in simulated local ﬁeld potentials for decreased inhibition conditions. Interestingly, such oscillations are actually observed in intracerebral EEG signals recorded in epileptic patients during pre-ictal periods, close to seizure onsets. MSC: 34E15; 34E17; 37N25; 92B25 Keywords: Multiple time-scale systems; Canards; Bursting; Excitability; Epilepsy; Neural mass model 1 Introduction Brain dynamics emerges from neural entities interacting at diﬀerent levels, from sin- gle neurons to large-scale neural networks. At each level, transitions between diﬀerent regimes, such as ﬁring/resting states in single neurons and up/down states in neural net- works, are associated with both physiological functions and pathological activity [1–3]. One of the features of the system that determines how these transitions would occur is excitability. The concept of neural excitability for single neurons was introduced initially by Louis Lapique in 1907 [4, 5]. Alan Hodgkin, who then re-introduced the concept [6], classiﬁed the neural excitability with respect the ﬁring rate of neurons in response to in- jected steps of currents. Excitability properties of neural systems can vary with internal © The Author(s) 2021. 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Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 2 of 27 dynamics, leading to diﬀerent physiological and pathological behavior [7–10]. At the cor- tical scale, for instance, variations in excitability [11] and loss of network resilience [12] are associated with epileptic seizures. Yet, what may be as important as a transition itself is the dynamics preceding the transition. In the context of epilepsy, for example, identi- ﬁcation of the dynamic features along the path to a transition is crucial for intervention and prevention of seizures. Mathematical models of brain activity range from microscopic level of single cell dy- namics to macroscopic level of interactions between large scale neural systems. Neural mass models (NMMs) consider the average temporal dynamics of interconnected neural subpopulations without explicitly representing the underlying mechanisms at the level of single cells. The mesoscopic level oﬀered by the neural mass formulation has been used to model brain signals, from local ﬁeld potentials (LFPs) to global electroencephalo- graphic (EEG) recordings, and to investigate various cerebral rhythms [13–15]. NMMs have also been used extensively to study pathological dynamics such as in epilepsy [16– 19], Alzheimer’s disease [20] and Parkinson’s disease [21, 22]. Interactions between slow and fast components of neural systems, hence, of their math- ematical models, result in multiple time-scale complex oscillations, such as relaxation, bursting and mixed-mode oscillations. Geometric singular perturbation theory (GSPT) is a key tool for understanding the interaction between the geometry of the system and the emerging multiple time-scale dynamics. In particular, canard solutions, which can exist in multiple time-scale systems with a folded geometry, appear as building blocks of complex oscillations in both phenomenological and neurophysiologically plausible models ranging from single cell [23–26] to neural networks [27, 28]. The canard phenomenon in such sys- tems has been related to neural excitability [29], excitability thresholds [23, 30–34], and boundaries between diﬀerent type of solutions, such as subthreshold oscillations and large amplitude spiking/bursting oscillations [24, 28, 35–43]. While such canard-organized ﬁne structures have been shown in a wide range of two-time-scale models, recent studies started to explore canard-mediated processes in systems with three or more time-scales [44–46]. In this study we investigate critical regimes in the NMM initially presented in [16]. This physiologically-grounded model has been extensively used for modeling structural and functional changes leading to epileptic activity observed in intracranial (stereoelec- troencephalography, SEEG) signals. The model includes four interacting neuronal sub- populations: two interconnected subpopulations of glutamatergic pyramidal neurons and GABAergic inhibitory interneurons (somatostatin positive (SOM+), and parparvalbunim positive (PV+), also called dendrite-projecting slow and soma-projecting fast interneu- rons, respectively). Although the model was introduced for the CA1 region of the hip- pocampus, implementation of these four subpopulations mediating glutamatergic and GABAergic signaling makes it generic enough to be considered for many other cortical regions [47]. Activity of each subpopulation is given by the corresponding average post- synaptic potential (PSP) that is determined by two functions: 1) a “pulse to wave” function, S(v)=5/(1 + exp(0.56(6 – v))), transforming the incoming synaptic potentials into a ﬁring rate; and 2) a “wave to pulse” converting the input average ﬁring rate into a mean PSP at the input of each subpopulation, that is, h(t)= Wt/τ exp(–t/τ ), where W represents the w w average synaptic gain and τ is the average synaptic time constant mimicking the rise and w Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 3 of 27 decay of actual PSPs. The system schematized in Fig. 1areads y ˙ = y,(1a) 0 5 A 2 1 y ˙ = S[y – y – y ]– y – y,(1b) 5 1 2 3 5 0 τ τ τ a a y ˙ = y (1c) 1 6 A 2 1 y ˙ = p(t)+ C S[C y ] – y – y,(1d) 6 2 1 1 6 1 τ τ τ a a y ˙ = y , (1e) 2 7 B 2 1 y ˙ = C S[C y ]– y – y,(1f) 7 4 3 0 7 2 τ τ τ b b y ˙ = y,(1g) 3 8 G C 2 1 y ˙ = C S C y – y – y – y.(1h) 8 7 5 0 4 8 3 τ C τ τ g 4 g The variables y stand for the excitatory PSPs mediated by two pyramidal neuron sub- 0,1 populations, y and y are inhibitory PSPs mediated by the SOM+ and PV+ interneuron 2 3 subpopulations, respectively. Variables y (j ∈{5, 6, 7, 8}) are the auxiliary variables that are introduced to convert the second-order diﬀerential equations describing the wave to pulse functions to ﬁrst-order diﬀerential equations [13]. The parameters A, B, G are the synaptic gains, the C are the connectivity constants representing the average number of synaptic contacts, p(t) is the external (noisy) cortical input (p(t)= p + ξ,where p is the mean of external input ξ is a random variable following a normal distribution with N (0, σ )). The synaptic time constants are given by τ , τ , τ . The major contribution to LFPs (as recorded a b g by intracranial electrodes in patients candidate to surgery) corresponds to the PSPs sum- mated at the level of pyramidal neurons, which are geometrically aligned “in palisades”, i.e. one relative to the other and perpendicular to the plane of the cortical layers. In the model, the LFP is given by the sum of excitatory PSP (EPSP) and inhibitory PSPs (IPSPs) received by the glutamatergic pyramidal cells, hence LFP = y – y – y . 1 2 3 As introduced in [48], under the following variable conversion: t y y y y 0 1 2 3 , , , , , y , y , y , y ⇒ (t, v , v , v , v , y , y , y , y ), 5 6 7 8 0 1 2 3 5 6 7 8 τ τ τ τ τ g a a b g with δ = τ /τ and ε = τ /τ ,system(1a)–(1h) can be written in the following deterministic g a a b (σ =0) slow–fast form: dv = y := F (y ), (2a) 8 3 8 dt dy = GS[C τ v – C τ v ]– v –2y := F (v , v , v , y ), (2b) 5 a 0 6 b 2 3 8 8 0 2 3 8 dt dv = δy := δF (y ), (2c) 5 0 5 dt Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 4 of 27 dy = δ AS[Aτ p + C τ v – C τ v – C τ v ]– v –2y (2d) a 2 a 1 4 b 2 7 g 3 0 5 dt := δF (v , v , v , v , y ), 5 0 1 2 3 5 dv = δy := δF (y ), (2e) 6 1 6 dt dy = δ AS[C τ v ]– v –2y := δF (v , v , y ), (2f) 1 a 0 1 6 6 0 1 6 dt dv = δεy := δεF (y ), (2g) 7 2 7 dt dy = δε BS[C τ v ]– v –2y := δεF (v , v , y ). (2h) 3 a 0 2 7 7 0 2 7 dt In this manuscript, we will consider system (2a)–(2h) for the slow–fast analysis and (1a)–(1h) for simulations under the stochastic input. We will be using the parameter set given in Table 1,unlessotherwise stated,for which δ =0.3 and ε =0.2. Numerical bi- furcation analysis is performed in AUTO-07p software [49]. The stochastic diﬀerential equations were integrated using the Euler–Murayama method with a step size dt = 10e–4 second in XPPAUT software [50]. As noticed in [48], system (2a)–(2h) is a three-time-scale system written in fast form with (v , y )being fast,(v , y , v , y )slow and (v , y ) super-slow variables. Köksal Ersöz et 3 8 0 5 1 6 2 7 al. [48] have focused on electrophysiological pre-ictal bursting patterns recorded in hu- man patients just before the onset of seizure. Pre-ictal bursting patterns are characterized by fast oscillatory discharges (which will be referred as spikes) followed by a slower oscil- lation (a simulated pattern with the parameter set in Table 1 is exempliﬁed in Fig. 1b). The authors have reproduced pre-ictal bursting and unveiled the mechanism yielding these so- lutions by decorticating the three-time-scale structure of the model. They have discussed appropriate stimulation strategies for aborting of the pre-ictal bursting, hence, for pre- venting a subsequent epileptic seizure. However, they did not focus on possible slow–fast transitions. Here we extend the slow–fast analysis initiated in [48] by investigating the role of slow-manifolds in transitions to relaxation and bursting type of solutions. We will focus on how canard trajectories shape the diﬀerent routes from physiological to patho- logical brain activity. In what follows, we will go brieﬂy through the multiple-time-scale analysis presented in [48], and then show diﬀerent canard structures present in the model and how they take a part in critical transitions. Finally, we will see the system’s response to stochastic inputs near critical regimes, and make a remark on the slow oscillations ob- served along the path to seizure in SEEG signals recorded during pre-surgical evaluation of two patients with drug-resistant epilepsy. Table 1 Parameter values for the bursting-type discharges A (mV) B (mV) G (mV) p (Hz) C C C C C C C τ (s) τ (s) τ (s) 1 2 3 4 5 6 7 a b g 5 5 35 90 135 108 80 25 450 121 121 0.01 0.05 0.003 Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 5 of 27 Figure 1 Model diagram and time series of a bursting solution. (a) Model diagram showing excitatory (red connections) and inhibitory (blue connections) interactions between subpopulations of pyramidal neurons (PYR and PYR’) and inhibitory interneurons (PV+ and SOM+). The post-synaptic potential of a subpopulation, which is the output of h(t), is multiplied by a synaptic coeﬃcient C before being received by another subpopulation. (b) Time series of a bursting solution for the parameter set in Table 1. The panels from the top to the bottom show the time courses of post-synaptic potentials of PV+ (v ), PYR (v ), SOM+ (v )and thelocal 3 0 2 ﬁeld potential (LFP), i.e. Aτ p + C τ v – C τ v – C τ v a 2 a 1 4 b 2 7 g 3 2Preliminaries System (2a)–(2h) expressed in the fast time t is called a fast system.The slow system is ˜ ˜ obtained by deﬁning t = δt, dv δ = F (y ), (3a) 3 8 dt dy δ = F (v , v , v , y ), (3b) 8 0 2 3 8 dt dv = F (y ), (3c) 0 5 dt dy = F (v , v , v , v , y ), (3d) 5 0 1 2 3 5 dt dv = F (y ), (3e) 1 6 dt dy = F (v , v , y ), (3f) 6 0 1 6 dt dv = εF (y ), (3g) 2 7 dt dy = εF (v , v , y ), (3h) 7 0 2 7 dt where the functions F (·) are as deﬁned in (2a)–(2h). The super-slow system is obtained by ˜ ˜ ˜ deﬁning t = εt = εδt: ss s dv δε = F (y ), (4a) 3 8 dt ss dy δε = F (v , v , v , y ), (4b) 8 0 2 3 8 dt ss Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 6 of 27 dv ε = F (y ), (4c) 0 5 dt ss dy ε = F (v , v , v , v , y ), (4d) 5 0 1 2 3 5 dt ss dv ε = F (y ), (4e) 1 6 dt ss dy ε = F (v , v , y ), (4f) 6 0 1 6 dt ss dv = F (y ), (4g) 2 7 dt ss dy = F (v , v , y ). (4h) 7 0 2 7 dt ss Systems (2a)–(2h), (3a)–(3h)and (4a)–(4h) describe diﬀerent dynamics in the singular limits ε → 0 and/or δ → 0, although they are equivalent for ε =0 and δ = 0. Letting δ → 0 in (2a)–(2h)yieldsthe fast layer problem (2a)–(2b) which describes the dynamics of the fast variables (v , y ) for ﬁxed values of the slow (v )and super-slow (v )variables.The 3 8 0 2 critical manifold is deﬁned by the equilibrium points of the fast layer problem, that is, 0 8 S = (v , y , v , y , v , y v , y ) ∈ R | GS[C τ v – C τ v ]– v =0,(5) 3 8 0 5 1 6 2 7 5 a 0 6 b 2 3 which is eventually in the (y = 0)-space. Since the eigenvalues of the Jacobian matrix of the fast layer problem deﬁned by (2a)–(2b)withrespect to (v , y )are λ =–1, the 6- 3 8 1,2 dimensional critical manifold S is normally hyperbolic and stable, thus, it is perturbed to local slow manifolds for suﬃciently small δ > 0. Therefore, the fast dynamics can be approximated by slow dynamics as suggested by the Fenichel theorem [51]. Setting δ → 0in (3a)–(3h) gives an algebraic-diﬀerential slow reduced problem, 0= F (y ), (6a) 3 8 0= F (v , v , v , y ), (6b) 8 0 2 3 8 dv = F (y ), (6c) 0 5 dt dy = F (v , v , v , v , y ), (6d) 5 0 1 2 3 5 dt dv = F (y ), (6e) 1 6 dt dy = F (v , v , y ), (6f) 6 0 1 6 dt dv = εF (y ), (6g) 2 7 dt dy = εF (v , v , y ), (6h) 7 0 2 7 dt which describes the slow dynamics restricted to S .System(6a)–(6h) is a two-time-scale system of 4slow/2super-slowvariables with ε being the time-scaling parameter. The equi- libria of the slow layer problem in the ε → 0 limit deﬁnes the super-slow manifold L ,which Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 7 of 27 is reduced to 0 0 L = (v , y , v , y , v , y v , y ) ∈ S | (7) 3 8 0 5 1 6 2 7 AS Aτ p + C τ AS[C τ v ]– C τ v – C τ v – v =0 , a 2 a 1 a 0 4 b 2 7 g 3 0 and restricted to S by the algebraic condition v = GS[C τ v –C τ v ]= K(v , v )in (6a)– 3 5 a 0 6 b 2 0 2 0 0 (6b). The super-slow dynamics restricted to the 2-dimensional manifold L ,hence to S , are given by the super-slow reduced system in the ε → 0 limit of (4a)–(4h). In order to investigate the super-slow ﬂow on L , we consider the two-time-scale system (6a)–(6h)withthe fast variable v on S ,i.e. v = K(v , v ), and rewrite the slow reduced 3 3 0 2 system (6a)–(6h)as dv = F (y ), (8a) 0 5 dt dy = F v , v , v , K(v , v ), y,(8b) 5 0 1 2 0 2 5 dt dv = F (y ), (8c) 1 6 dt dy = F (v , v , y ), (8d) 6 0 1 6 dt dv = εF (y ), (8e) 2 7 dt dy = εF (v , v , y ). (8f) 7 0 2 7 dt Applying the time-scaling τ˜ = εt and taking the singular limit ε → 0 give the algebraic- s s diﬀerential system 0= F (y ), (9a) 0 5 0= F v , v , v , K(v , v ), y,(9b) 5 0 1 2 0 2 5 0= F (y ), (9c) 1 6 0= F (v , v , y ), (9d) 6 0 1 6 dv = F (y ), (9e) 2 7 dτ˜ dy = F (v , v , y ). (9f) 7 0 2 7 dτ˜ The algebraic conditions (9a)–(9d) deﬁne the ‘critical manifold’ of (8a)–(8f)which is 0 0 equivalent to L given by (7). Notice that L is restricted in the zero plane of the (y , y )- 5 6 0 0 space. Assuming that v is some function of v on L ,i.e. v = M(v ), the fold points on L 2 0 2 0 are deﬁned by ∂M(v ) F = (v , v , v , v , y , y , y , y ) ∈ L v = M(v ), =0 . (10) 0 1 2 3 5 6 7 8 2 0 ∂v 0 Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 8 of 27 Figure 2 Critical manifold, slow manifold and folded singular points. (a) Critical manifold S (green surface), 0 0 super-slow manifold L (red curve) and a bursting orbit in the (v , v , v )-space. The curve L is divided into 0 2 3 0 0 three branches at F and F (red dots) where it changes stability. The middle branch of the L (L )curve 1 2 0 0 between F and F is unstable (dashed). The stable left-hand side branch (L ), F , L and F are entirely on 1 2 1 2 l m 0 0 the almost horizontal part of S (approximately on the (v ≈ 0)-plane). The stable right-hand side branch of L 0 0 (L ) on expands both on the horizontal and vertical parts of S . Arrows indicate the corresponding time-scale (single-headed for super-slow, double-headed for slow dynamics). (b) Super-slow manifold L (red surface), fold curves F (black lines) and folded singular points p (red dots)inthe (y , v , v )-space. Arrows indicate 1,2 1,2 7 2 0 the corresponding time-scale 0 0 0 Figure 2ashows S and L in the (v , v , v )-space, and Fig. 2b L in the (y , v , v )-space. 0 2 3 7 2 0 The super-slow manifold L expands between the lower horizontal and vertical planes of 0 0 0 S .The part of curve L on the lower horizontal plane of S is folded with respect to v at along the fold curves F and F deﬁned by (10), i.e. F = F ∪ F .Inthisprojection, 1 2 1 2 0 0 0 the 1-D fold curves divide L into two stable (left-hand side L and right-hand side L ) l r and one unstable (middle L ) branches on the (v , v , v )-space. We also verify that four 0 2 3 eigenvalues of L (two real and two complex conjugate) have negative real parts along the stable parts of L . One of the real eigenvalues changes sign along F ,hence theunstable 1,2 middle branch is of saddle type. Along the stable and unstable branches L is normally hyperbolic, so L is perturbed to local super-slow manifolds for small values of ε >0 within (6a)–(6h); see the extension of Fenichel theory for systems with more than two time-scales [52]. On the other hand, the dynamics near the non-hyperbolic fold curves F should be 1,2 investigated by using the elements of GSPT. As being the usual strategy, we consider the desingularized version of the super-slow dynamics on L is given by the desingularized slow reduced system (DSRS), reading dv =–y , (11a) dτˆ dy ∂M(v ) 7 0 =– F v , M(v ), y , (11b) 7 0 0 7 dτˆ ∂v s 0 ∂M(v ) where τ˜ = τˆ .The equilibria of (11a)–(11b)onthe fold set F are located at s s ∂v p1 p1 p2 p2 (v , y ) = (1.2343, 0) and (v , y ) = (9.9976, 0) for the parameter set given in Table 1. 0 7 0 7 These equilibrium points, which are not generally the true equilibria of (8a)–(8f), are re- lated to the folded singularities of (8a)–(8f), hence of (2a)–(2h). On the other hand, equi- F F F F 7 7 7 7 librium points (v , y ), i.e. F (v , y ) = 0, are ordinary singularities since they are also 0 7 0 7 equilibria of (8a)–(8f), hence of (2a)–(2h). Figure 2bshows L ,foldcurves F and folded 1,2 singular points p in the (y , v , v )-space. 1,2 7 2 0 Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 9 of 27 Stability of the equilibrium points of the desingularized (slow) reduced system on the fold set determines the type of the folded singularities of the original system. Classiﬁca- tion of these equilibrium points is based on the linear stability analysis. When the desin- gularized (slow) reduced system is planar, this analysis can be done using the trace and the determinant of the Jacobian matrix at the fold equilibrium. If both are diﬀerent from zero, the fold equilibrium can be a folded saddle, a folded node or a folded focus. If the determinant equals zero but not the trace, then the desingularized ﬂow has a degenerate equilibrium point, which is a folded saddle-node. A folded saddle-node is either related to a saddle-node bifurcation of the folded equilibria or a transcritical bifurcation of a folded equilibrium with an ordinary equilibrium. The latter case refers to the folded saddle-node type II (FSN II) singularity [53, 54], where a folded node becomes a folded saddle and a regular saddle becomes a regular node. The original system exhibits a singular Hopf bi- furcation close to a FSN II singularity [55, 56]. The Jacobian matrix of (11a)–(11b) has the following general form: ⎡ ⎤ 0–1 ⎣ ⎦ 2 ∗ ∗ ∗ ∗ ∗ J = , (12) ∂ M(v ) ∂M(v ) ∂F (v ,y ) ∂M(v ) 0 ∗ ∗ 0 0 7 0 – F (v , y )– 2 2 7 0 7 ∂v ∂v ∂v ∂v 0 0 0 ∗ ∗ where (v , y ) stands for the equilibrium point of interest. Since on the folded equilibria 0 7 p1,p2 ∂M(v ) 2 = 0, the trace and determinant of (12) on the folded equilibria read ∂v p1,p2 ∂ M(v ) p1,p2 p1,p2 p1,p2 p1,p2 0 tr J =0, det J =– F v , y . (13) 2 0 7 ∂v The trace and determinant of (12) on the regular equilibria read F F F F 7 7 7 7 ∂M(v ) ∂M(v ) ∂F (v , y ) F 0 F 0 0 7 7 7 tr J =2 , det J =– . (14) ∂v ∂v ∂v 0 0 0 Notice that the generic folded singularity condition is violated due to the fact that ∂(F v ˙ +F y ˙ ) v 2 y 7 2 7 p1,p2 =0 in (12), and tr(J )=0 in (13). Therefore, the folded singularities de- ∂v termined by (11a)–(11b) are not generic and a folded equilibrium is one of the following p1,p2 p1,p2 p1,p2 types: a saddle for det(J )<0, a center for det(J )>0, a nilpotent for det(J )=0. The latter degenerate type corresponds to a point in the parameter space at which a F F 7 7 folded singularity and a regular singularity meet, i.e. tr(J )=0 and det(J )=0 in (14). Consequently, the equilibrium points of (11a)–(11b) related to the folded and regu- lar singularities undergo degenerate transcritical bifurcations where (12)has twozero- eigenvalues. Figure 3 shows the bifurcation diagram of (11a)–(11b)withrespect to B in the region of p1 p1 p2 p2 interest. Two straight lines of the equilibria (v , y )and (v , y ) intersect with the regu- 0 7 0 7 lar equilibria curve F (v , y )at two bifurcation points, BP at B ≈ 16.7817 and BP at 7 0 7 1 BP1 2 B ≈ 5.4817, which are degenerate transcritical bifurcations. For B < B ,the equilib- BP2 BP1 p1 p1 rium (v , y ) is a center with two complex conjugate eigenvalues. After the bifurcation at 0 7 p1 p1 BP ,(v , y ) becomes a saddle. Consequently, the system (8a)–(8f)(and(2a)–(2h)) has a 0 7 p2 p2 folded-saddle singularity near p for B > B .The equilibrium (v , y ) is of a saddle type 1 BP1 0 7 for B < B and becomes a center with two complex conjugate eigenvalues at B = B . BP2 BP2 Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 10 of 27 Figure 3 Bifurcation diagram of (11a)–(11b)withrespect to B. p1 p1 p1 Equilibrium points (v , y ) lie on the lower red horizontal line (v , 0 7 0 p2 p2 p2 p2 y ), and (v , y ) on the upper red horizontal line (v , y ). Dashed 0 7 0 7 parts of the red lines represent saddle type, solid parts represent center type solutions. The true equilibrium points line on the black curve, F . Dashed part of the black curve represents saddle type, solid parts stable focus type solutions. The saddle type equilibrium points p1 p1 p2 p2 along (v , y )and (v , y ) change to center at the intersections 0 7 0 7 with F at BP and BP , respectively 7 1 2 Hence, the system (8a)–(8f)(and(2a)–(2h)) has a folded-saddle singularity near p for B B . Similarly, in a neigh- BP1 BP1 borhood of BP , the equilibrium points along the F (v , y )curve are of stable focus type 2 7 0 7 for B B . BP2 BP2 As mentioned above, a generic transcritical bifurcation of regular and folded singular- ities is related to a FSN II singularity. In our case, a folded saddle becomes a folded cen- terand astablefocus becomesasaddleatthe degenerate transcritical bifurcation points BP and BP . Furthermore, system (2a)–(2h) can undergo the (singular) Hopf bifurcations 1 2 close to BP and BP in the parameter space (see Fig. 4d), as it will be detailed in the follow- 2 1 ing sections. Hence, the interaction of regular and non-generic folded singularities can be referred as a degenerate FSN II singularity. A degenerate FSN II singularity in (2a)–(2h) stems from the structure of the NMM, which is deﬁned as a second-order system that violates the generic folded singularity condition tr(J) =0. Folded singularities can lead to canard solutions in the original system. In a planar slow– fast system, a singular Hopf bifurcation can occur near a folded singularity, which is then called a canard point. In such case, the amplitude of the periodic orbits bifurcated at the singular Hopf point increases stiﬄy in a narrow interval of the parameter (scaled by the time-scale separation parameter) that controls the transition from small amplitude to relaxation oscillations [57]. This phenomenon is known as canard explosion [26, 58]. A canard-explosive branch hosts small canards following the unstable branch of the criti- cal manifold and one stable branch (so-called canard-without-head solutions), large ca- nards following the unstable branch of the critical manifold and two stable branches (so-called canard-with-head solutions), and a maximal canard solution that follows the longest the repelling branch. In planar multiple time-scale systems, canard solutions are tightly connected to excitability and ﬁring thresholds [30, 31]. In higher dimensional multiple time-scale systems with at least two slow variables, the folded-singularities are generic, hence they are robust to small parameter perturbations, and canard solutions as- sociated with folded singularities connect stable and unstable branches of a folded critical manifold [36, 53, 59–61]. Canards of folded node and FSN II singularities support mixed mode oscillations [27, 36, 44]. FSN II singularities have been identiﬁed in neuronal models where the exchange from an excitable to a relaxation oscillatory state is accompanied by subthreshold oscillations [24, 28, 42, 62]. Folded-saddle canards have been shown to be sculpting ﬁring threshold manifolds, as well [33, 34, 63–65]. In our problem, the critical manifold S (5) is hyperbolic, whereas the super-slow man- ifold L (8a)–(8f) has a folded structure. Thus, the critical transitions occur mainly in the 6-dimensional reduced system given by (8a)–(8f). As the analysis above have shown, (8a)– Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 11 of 27 Figure 4 Bifurcation diagrams of (2a)–(2h) with as a function of (B, C ). (a) Bifurcation diagram of (2a)–(2h)on the (B, C ) plane. Curves are named, respectively, after the limit point (LP, black curves), Hopf (H, red curves) and homoclinic (HOM, blue curves) bifurcations in panels (b–f). Only the LP bifurcations interacting with canard solutions are plotted. Black squares indicate cusp (CP), red circles indicate Bogdanov–Takens (BT) and red squares indicate generalized Hopf (GF) bifurcations. The regions marked by black, green and purple boxes are zoomed in black, green and purple framed insets. The region where the homoclinic curve tips to the LP is zoomed inside the green inset. (b–f) Bifurcation diagrams of (2a)–(2h) as a function of B for diﬀerent values of C . The limit point bifurcations of interest are marked by black squares, Hopf bifurcations by red circles, and homoclinic connections by blue stars. Stable and unstable solutions are represented by continuous and dashed curves, respectively. Along the curves of equilibrium points, (2a)–(2h) undergoes four Hopf bifurcations (H )for C = {50, 80, 145} (c, d, e) and three Hopf bifurcations (H )for C =15 (b)and 1,2,3,4 3 1,2,3 3 (H )for C = 200 (f) 1,2,4 3 (8f) has degenerate folded singularities along the fold curve at p and p .Noticethat, since 1 2 the system has neither a folded node nor a FSN II, small amplitude oscillations do not exist near p or p . But the folded saddle, degenerate FSN II and singular Hopf bifurcations can 1 2 lead to canard solutions governing the critical transitions in (8a)–(8f)(hencein(2a)–(2h)). On the other hand, the bursting behavior cannot be captured by (8a)–(8f)because (8a)– (8f) is restricted in the critical manifold S , whereas the fast oscillations of the bursting orbits leave oﬀ S . So the bursting solutions exist in the full system (2a)–(2h)(see[48]for a detailed analysis of the bursting solutions). As a result, our problem yields both three and two time-scaled behaviors. In the next section, we investigate canard dynamics near p and p . 1 2 3 Multiple time-scale oscillations and canard transitions Transitions near the folded singularities of (2a)–(2h) which lead to canard solutions de- pend on the system parameters. The reader may refer to Table 1 forthe parametervalues, Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 12 of 27 unless otherwise stated. The connectivity strength from the pyramidal cell population on the subpopulation of the SOM+ interneurons, C , and their synaptic gain, B,appearas two crucial parameters controlling the transitions by aﬀecting the curve F in (11a)–(11b) (see Fig. 3). Figure 4a shows a 2-parameter bifurcation diagram in the plane (B, C ). De- pending on C ,system(2a)–(2h) undergoes several Hopf bifurcations as a function of B. The ﬁrst two Hopf bifurcations, H and H , yield harmonic oscillations, whereas the pe- 1 2 riodic branches appearing at H and/or H connect to multiple time-scale oscillations. 3 4 Under the variations in (B, C ), H and H persist; and the emerging periodic orbits do 3 1 2 not change qualitatively. On the other hand, H and H undergo Bogdanov–Takens (BT) 3 4 bifurcations BT and the corresponding periodic branches vary qualitatively. The peri- 1,2 odic orbits emerging at H and H can end on homoclinic connections, namely HOM . 3 4 1,2,3,4 Figures 4b–f exemplify qualitative variations in (2a)–(2h) as a function of B for diﬀerent values of C .For C < C ≈ 18.9, the system undergoes three Hopf bifurcations, for in- 3 3 3,BT1 stance in Fig. 4bfor C = 15. The branch of periodic solutions starting at H terminates at 3 3 a homoclinic connection, HOM .As C increases, HOM and LP get closer while the am- 1 3 1 1 plitude and the number of spikes of the periodic orbits increase. The spike adding occurs as the HOM curve folds back and forth in the (B, C )-space(seethe blackframedinset 1 3 in Fig. 4a for an example folding). At C = C another Hopf bifurcation, H ,appears 3 3,BT1 4 yielding a new branch of periodic orbits making a second homoclinic connection, HOM (green framed inset in Fig. 4a). Consequently, HOM and HOM points coexist in a nar- 1 2 row range of (B, C ). The HOM curve touches the LP curve at (B, C ) ≈ (23.98, 22.43), 3 1 1 3 foldsbackand continuesinthe parameterspace,which then we name as HOM curve (dashed zone in the green framed inset in Fig. 4a). The curves HOM and HOM stay 1 3 very close to each other in (23.98 < B < 24.46, 21.94 < C < 22.43), before HOM bends 3 3 in the C direction at (B, C ) ≈ (24.46, 22.43). As it happens, the branch of periodic orbits 3 3 curls below LP in the B-space and eventually connects to HOM . With increasing C ,this 1 3 3 branch of periodic orbits advances further towards the stable equilibrium points while in- troducing a region of multi-attractors of nodes, saddles, unstable small oscillations and stable large amplitude bursting oscillations (see Fig. 4cfor C = 50, dynamics will be de- tailed in Sect. 3.2). Concurrently, H moves away from LP and HOM approach LP . 4 1 2,3 2 In (20.09 54.43, 2 3 the branch of periodic orbits initiated at H connects to the branch of large amplitude multiple time-scale oscillations (Fig. 4d). System (2a)–(2h) does not have any LPs between H and H for C ≤ C ≤ C .At 3 4 3,CP1 3 3,CP2 C = C ≈ 141.4, as the lower branch of equilibrium points curls below H ,the con- 3 3,CP2 3 nection between the large amplitude orbits and H is broken up on a saddle-saddle homo- clinic bifurcation (the equilibrium points in a neighborhood of LP for B ≥ B are saddles 3 LP3 [66, 67]). As a consequence, the branch of periodic orbits starting from H terminates on a homoclinic connection, HOM (for C = 145 in Fig. 4e). This homoclinic connection 4 3 remains until H disappears at C = C ≈ 157. Beyond C > C ,the largeamplitude 3 3 3,BT2 3 3,BT2 bursting orbits introduced by H terminate on a saddle-node homoclinic connection (for instance at C = 200 in Fig. 4f). The Hopf bifurcations H and H occur close to the folded singularities p and p ,re- 3 4 2 1 spectively. System (2a)–(2h) can yield canard solutions close to these points in the param- eter space of B,suchas B ≈ B and B ≈ B . In the following section, we will show the H3 H4 Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 13 of 27 canard-mediated transition from sinusoidal oscillations initiated by H to large amplitude bursting/relaxation type solutions. Subsequently, Sect. 3.2 will detail the canard dynamics and related excitability near H , in particular, the type-I excitability for C = 50 and type-II 4 3 excitability for C =80. 3.1 Canard-mediated transitions between sinusoidal and multiple time-scale oscillations Köksal Ersöz et al. [48] have realized that the number of spikes of a bursting solution of (2a)–(2h) depends on the amount of the PSP received by the PV+ interneuron subpop- ulation, hence on the EPSP coming from the pyramidal cell subpopulation and the IPSP from the SOM+ interneurons. For instance, increasing the IPSP on the PV+ interneurons by increasing B decreases the number of spikes while driving the oscillations one peak to the next one in the parameter space (see Fig. 4c–f and 5). The connectivity constant from the pyramidal cell subpopulation to the PV+ interneuron subpopulation, C , directly scales the EPSP on this subpopulation, therefore determines the maximum number of fast spikes of the bursting oscillations, or more generally, the type of the multiple time-scale oscillations. Figure 5 exempliﬁes how C modulates the large amplitude oscillations between H and 5 3 H on the bifurcation diagram of (2a)–(2h)for C =80. Increasing C decreases the ampli- 4 3 5 tude of the oscillations, moves H and H slightly to the right, but does not aﬀect consider- 1 2 ably the locations H and H with respect to B (Fig. 5a). The supercritical Hopf bifurcation 3 4 at H yields a branch of sinusoidal periodic oscillations (Fig. 5b) that folds back and forth as B varies and enters in a regime of multiple time-scale periodic oscillations. These os- cillations are of relaxation type for small values of C , and of bursting type for large values of C . Furthermore, the stable sinusoidal and multiple time-scale periodic oscillations can coexist depending on the values of C (see Fig. 5bat B ≈ 4.8). The form of the branch of periodic solutions between H and H in Fig. 5a indicates 3 4 the type of the multiple time-scale oscillations for a certain parameter combination. For C = 80 (black diagram in Fig. 5) the smoothly decreasing amplitude of v with B in- 5 0 dicates that the corresponding orbits are of relaxation type (exempliﬁed in Fig. 6). The Figure 5 Variation of large amplitude solutions with respect to (B, C ). (a) Bifurcation diagrams of (2a)–(2h) with respect to B for C = 80 and diﬀerent values of C . Curves and Hopf bifurcations (H , dots) are colored 3 5 1,2,3,4 with respect to the color codes of C values. Stable and unstable solutions are represented by continuous and dashed curves, respectively. For the sake of simplicity, the periodic solutions between H and H are not 1 2 shown. (b) Zoom into the region of transitions between sinusoidal and large amplitude multiple-time-scale solutions in B ∈ [4.7, 4.9] Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 14 of 27 Figure 6 Example canard orbits along the transition from sinusoidal oscillations to relaxation oscillations. Zoom near the bifurcation diagram for C =80, C =80 and B ∈ [4.7, 4.9] (see Fig. 5a for the whole diagram). 3 5 Continuous and dash curves represent stable and unstable solutions, respectively. Hopf (H ,red dot) is marked on the diagram. Numbered orbits from 1-7 are given in panels (b–d). The orange curve traces the frequency of the oscillations emerging at H The orange curve traces the frequency of the oscillations emerging at H .(b) Periodic orbits marked in panel (a), L (red curve), fold curves F (red points) and the 3 1,2 critical surface S (green surface) are projected on the (v , v , v )-space. Arrows indicate the corresponding 0 2 3 time-scale (single-headed for super-slow, double-headed for slow dynamics). (c) Periodic orbits marked in panel (a), L (red surface), fold curves F (black curves) and folded singular points p (red dots) are 1,2 1,2 projected on the (y , v , v )-space. Arrows indicate the corresponding time-scale. (d) Time series of the 7 2 0 ˜ ˜ ˜ periodic orbits on panels (b, c) with respective color codes. Period is normalized to 1 (t/T =1, where T represents period of a cycle) horizontal zigzags along the upper part of the periodic branches obtained for greater values of C indicate the presence of bursting solutions along these periodic branches and the number of their fast spikes. For instance, the 5 peaks that we count between H and H for C = 350 (blue diagram in Fig. 5) signify that the maximum number of fast 4 5 spikes for C = 350 is 4. Such a bursting orbit is obtained for suﬃciently small values of B (B = 5, for instance). Then as B increases, the bursting orbits lose their fast spikes one by one through the peaks of the horizontal branch. They become relaxation cycles (B = 16, for instance), before shrinking and disappearing via a subcritical Hopf bifurcation at H . 4 Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 15 of 27 The periodic solutions connected to H and H do notinteractwiththe singular fold 1 2 points of L , p and p ,but theonesnear H and H do because H and H take place 1 2 3 4 3 4 close to the degenerate FSN II singularities on the fold curve F. As a consequence, the multiple time-scale orbits emanating at singular Hopf bifurcations H and H can un- 3 4 dergo canard explosion along which canard trajectories sculpt the periodic oscillations. Figure 6 shows example orbits along the periodic branch that follow H for C =80 and 3 3 C = 80. As the periodic branch folds with respect to B and becomes unstable (Fig. 6a), the sinusoidal orbits of 4.5–6 Hz start to interact with p . In particular, they move along 0 0 0 theunstablebranchof L , L , before jumping back to the stable branch L .Hence,the m r periodic orbits become canard orbits (the 1st orbit). As B varies along the periodic branch in the parameter space, the canard orbits grow in amplitude along L (the second, third and fourth orbits) until they stretch out between F and F (the ﬁfth orbit). The canard 1 2 0 0 orbits that oscillate between L and L can be interpreted as canard-without-head orbits, m r and the ﬁfth orbit as the maximal canard since it has the largest period of the canard fam- ily of the periodic branch under consideration. Soon after the ﬁfth orbit, the trajectories jump to the attracting branch L , get a shape of canard-with-head solutions, and become stable (the sixth orbit). As B increases, the relaxation cycles appear with parts exclusively following the attracting branches of L and jumping close to the fold points. As mentioned in the introduction, canard solutions play a fundamental role in sepa- rating diﬀerent dynamical regimes. The unstable canard orbits in Fig. 6 (from the ﬁrst to the ﬁfth) appear as an other example of this phenomenon by accompanying the transi- tion from sinusoidal oscillations to relaxation oscillations. For instance, sinusoidal oscil- lations and large amplitude canard-with-head cycles coexist for B ∈ (4.75, 4.81) and the canard-without-head cycles form the boundary between them, as seen clearly in Fig. 6a. While increasing C introduces bursting type of solutions, it can preserve the bistability between the bursting and sinusoidal oscillations, for example for C = {250, 350, 425} in Fig. 5. Notice that with increasing C , the initially smooth branch of periodic orbits be- comes steeper, gains vertical zigzags that move to the right along the B-axis, and the region of bistability decreases. Figure 7 zooms into the region of canard orbits following the sinusoidal solutions of 4.8–6 Hz started at H for C = 450. As the stable sinusoidal oscillations grow in amplitude 3 5 with increasing B, they start to interact with p and to follow the bits of L (the ﬁrst orbit). Soon after, the orbits undergo a LP bifurcation (where the branch of the periodic orbits folds back at B ≈ 4.821) and become unstable. As B varies in the parameter space along the periodic branch, the orbits moving along L in the super-slow time-scale grow in 0 0 amplitude and they start to interact with the vertical panel of S as they jump to L in the slow time-scale. So, the orbits become canard orbits. As the part of the trajectory along L grows in amplitude, the trajectory gets attracted 0 0 0 by L along the vertical panel of S and it spirals around L before landing on the horizon- r r 0 0 tal plane of S . This interaction with the vertical panel of S occurs in the fast time-scale, and eventually yields fast spikes, i.e., bursting-type canard oscillations. For instance, the second orbit in Fig. 7cand 7d has one fast spike. The number of spikes increases as the trajectory stays longer and longer along L while B varies. More precisely, the number of spikes changes by one as we pass from one fold to another on the same side of along the snaking periodic branch with respect to B (Fig. 7b). For instance, solution-2 has 1 spike, solution-3, which is two fold below, has 3 spikes and so on. The spike adding continues Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 16 of 27 Figure 7 Example canard orbits along the transition from sinusoidal oscillations to bursting oscillations (a) Bifurcation diagram for C =80, C = 450 and B ∈ [4.7, 4.9] (see Fig. 5a for the whole diagram). Stable and 3 5 unstable solutions are represented by continuous and dashed curves, respectively. Hopf (H ,red dot) is marked on the diagram. The rectangular region is zoomed in panel (b). Numbered orbits from 1-7 are given in panels (c–d). The orange curve traces the frequency of the oscillations emerging at H .(c) Periodic orbits 0 0 marked in panels (a, b), L (red curve), fold curves F (red points) and the critical surface S (green surface) 1,2 are projected on the (v , v , v )-space. Arrows indicate the ﬂow direction and its time-scale (single-head for 0 2 3 super-slow, double-head for slow, triple-head for fast). (d) Periodic orbits marked in panel (a), L (red surface), (black curves) and folded singular points p (red dots) are projected on the (y , v , v )-space. fold curves F 1,2 1,2 7 2 0 Arrows indicate the corresponding time-scale. (e) Time series of the periodic orbits on panels are (b, c)shown ˜ ˜ ˜ with respective color codes. Period is normalized to 1 (t/T =1, where T represents period of a cycle) until the canard orbits (analogous to canard-without-head orbits) expands between two folded singularities, hence till the occurrence of the maximal canard of the family (ap- proximated by the sixth orbit). After the maximal canard, the canard cycles start to follow L in the super-slow time-scale (analogous to canard-with-head orbits) and they become stable (the seventh orbit). The part of the trajectory along L decreases as B increases further. Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 17 of 27 For C = 80, the system undergoes a complete canard explosion along the periodic branch following H since it visits the whole canard family from small to large (Figs. 6 and 7). However, what we may observe for small values of C is an incomplete canard ex- plosion terminating at a homoclinic connection. For instance, for C =15 (see Fig. 4b), the sinusoidal oscillations along the periodic branch initiated at H change qualitatively by interacting with L as B varies and we observe homoclinic canard-without-head orbits at HOM . The orbits terminating on HOM are homoclinic canard-with-head orbits sur- 1 3 rounded by stable large amplitude oscillations. Increasing C completes the canard explo- sion and the system enters into an excitable regime which will be detailed in the following section. 3.2 Canard-mediated transitions and excitability According to Hodgkin’s [6] classiﬁcation of neural excitability, type-I excitable neurons have continuous frequency-injected current curves, whereas type-II excitable neurons have discontinuous frequency-injected current curves. Rinzel and Ermentrout [68, 69] linked the type-I excitability to a SNIC bifurcation and the type-II excitability to a Hopf bifurcation. De Maesschalck and Wechselberger [29] explained the transition between the two excitability types via an intermediate regime of type-I excitability associated with a codimension-2 Bogdanov–Takens (BT) bifurcation in a planar system. They showed the existence of incomplete canard transitions in this transitory regime. Later on, transitions between the neuronal excitability types was shown to be induced by the inhibitory and excitatory autapse in the Morris-Lecar model [70]. Folded singularities and correspond- ing canard solutions in higher dimensional systems also have been shown to be shaping systems’ excitability properties [24, 28, 33, 34, 63–65]. System (2a)–(2h) can yield large amplitude oscillations in response to certain forms of stimulation (due to stochastic inputs, for instance) after being initiated from an equilib- rium point for a B value close to H , LP and LP in Figs. 4c–4f. Hence, system (2a)– 4 1 2 (2h) is excitable in these regions and the excitability properties of (2a)–(2h)determined by the parameter C (see Fig. 4). Indeed, the local pictures in these regions are similar to the ones investigated in [29, 70]. In particular, system (2a)–(2h)istype-Iexcitable for C ∈ (22.43, 54.43), basically between the homoclinic/saddle-saddle interactions near LP 3 1 and LP . In this parameter region, the large amplitude oscillations terminate on a homo- clinic orbit for which the ﬁring frequency is zero. System (2a)–(2h)istype-II excitablefor C > 54.43 for which the termination is issued via a Hopf bifurcation. In both cases, canard solutions shape the resulting dynamics. Figure 8 zoomsinnearthe excitableregionfor C =50 (see Fig. 4c for the whole dia- gram). For a particular value of B for B < B , the only attractor is the large amplitude LP1 bursting oscillation (the 1st orbit). In B < B < B theunstableattractorsofthe equi- LP1 H4 librium points appear. The subcritical Hopf bifurcation at B = B initiates a branch of H4 periodic orbits that terminates on the homoclinic point HOM , which bounds the canard explosion near H .For B , a homoclinic canard-without-head orbit (the second or- 4 HOM2 bit) coexists with a large stable bursting orbit of canard type (the third orbit). At B a HOM3 homoclinic canard-with-head orbit (the ﬁfth orbit) appears together with an outer large amplitude canard cycle (the fourth orbit). The large amplitude canard cycle grows in am- plitude and disappears on a saddle-node of periodic orbits (SNPO) at B = B (the sixth SNPO orbit). We also notice that, as HOM gets closer to LP for C ≈ 54.4 and B ≈ B ,the 3 2 3 LP2 canard orbits on the HOM become of without-head type. 3 Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 18 of 27 Figure 8 Example canard orbits near the type-I excitable regime. (a) Bifurcation diagram for C =50, C = 450 3 5 and B ∈ [18, 22.5] (see Fig. 4c for the whole diagram). Stable and unstable solutions are represented by continuous and dashed curves, respectively. Limit point (LP ,black squares),Hopf(H , red dot), homoclinic 1,2 4 (HOM , blue stars) bifurcations and saddle-node bifurcation of periodic orbits (SNOP, orange purple square) 2,3 are marked on the diagram. Numbered solutions are presented in panels (b–d). The orange curves trace the frequency of the oscillations. (b) Periodic orbits marked in panel (a), L (red curve), fold curves F (red points) 1,2 and the critical surface S (green surface) are projected on the (v , v , v )-space. Arrows indicate the ﬂow 0 2 3 direction and its time-scale (single-headed for super-slow, double-headed for slow dynamics. The homoclinic points HOM and HOM are marked by cyan and dark blue stars. (c) Periodic orbits marked in panel (a), L (red 2 3 surface), fold curves F (black curves) and folded singular points p (red dots) are projected on the 1,2 1,2 (y , v , v )-space. Arrows indicate the corresponding time-scale. (d) Time series of the periodic orbits on panels 7 2 0 ˜ ˜ ˜ (b, c) with respective color codes. Period is normalized to 1 (t/T =1, where T represents period of a cycle) Foraparametersetensuringthetype-IIexcitability(C = 80, for instance), the fast spikes of the bursting oscillations disappear and the ﬁnal oscillation turns out to be a relaxation type running in slow and super-slow timescales. These relaxation oscillations terminate via a complete canard-explosion near the singular Hopf bifurcation point H .Thishap- pens in a similar manner for all C values under consideration. Figure 9 provides an ex- ample for C = 450. As the large amplitude periodic solutions decrease in amplitude, they start to follow L and take the shape of canard-with-head solutions (the second and third m Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 19 of 27 Figure 9 Example canard orbits near the type-II excitable regime. (a) Bifurcation diagram for C =80, C = 450 and B ∈ [16, 18] (see Fig. 5a for the whole diagram). Stable and unstable solutions are represented by continuous and dashed curves, respectively. Hopf bifurcation (H , red dot) is marked on the diagram. Numbered orbits on the lower branch of periodic solutions are presented in panels (c–d). The orange curve traces the frequency of the oscillations emerging at H .(b) Periodic orbits marked in panel (a), L (red curve), fold curves F (red points) and the critical surface S (green surface) are projected on the (v , v , v )-space. 1,2 0 2 3 Arrows indicate the ﬂow direction and its time-scale (single-headed for super-slow, double-headed for slow dynamics). (c) Periodic orbits marked in panel (a), L (red surface), fold curves F (black curves) and folded 1,2 singular points p (red dots) are projected on the (y , v , v )-space. Arrows indicate the corresponding 1,2 7 2 0 time-scale. (d) Time series of the periodic orbits on panels (b, c) are shown with respective color codes. Period ˜ ˜ ˜ is normalized to 1 (t/T =1, where T represents period of a cycle) orbits). The maximal canard of this canard family is the fourth orbit that stays along to the super-slow manifolds as long as possible. After the fourth orbit, we observe canard- without-head orbits (the 5th and the 6th orbits) that shrink to p . The frequency of the oscillations along the canard explosion ranges in 1.8–3.5 Hz. We also notice a region of bistability between large amplitude bursting oscillations and equilibrium points. Once again the canard solutions construct the boundary between them. For a parameter set giving relaxation oscillations in this region (e.g. C = {80, 250, 350} in Fig. 5), the relax- ation oscillations shrink H via a ‘classical’ canard explosion, similar to the one in the 2D van der Pol system, without having any fast component in v . 3 Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 20 of 27 4 Local ﬁeld potential in critical regimes In the previous section we have shown two diﬀerent regions in parameter space where canard solutions determine boundaries and organize transitions between diﬀerent dy- namical regimes. The narrow-band sinusoidal activity of 4.5–6 Hz emerging near H and of 1.8–3.5 Hz emerging near H are connected to large amplitude periodic multiple- timescale solutions through canard orbits. System (1a)–(1h) emits aperiodic large ampli- tude epileptic discharges under stochastic input (p(t)= p + ξ,with ξ = N (0, 2 )) when it is initialized near the critical regions of H and H (Fig. 10–11). A parameter setting ensur- 3 4 ing the type-I excitability without any canard solutions near H gives a board band activity between the large amplitude spikes (Fig. 10a1–a3). On the other hand, taking the system to type-II excitability near H introduces transient small amplitude oscillations of ≈ 3.5 Hz due to the presence of the canard cycles in this region (Fig. 10b1–b3). We observe transi- tions between large amplitude discharges and harmonic oscillations of ≈ 6 Hz when the system is initialized close to the Hopf bifurcation H (Fig. 10c1–c3). Simulated PSPs at the levelofthe pyramidalcellsubpopulation aregiven in Fig. 11. Figures 12 and 13 show LFPs recorded by the SEEG electrodes in two diﬀerent patients with drug-resistant focal epilepsy during presurgical evaluation (see Table 2 for the de- tails). Multiple-contact depth electrodes were implanted according to the SEEG technique as a standard clinical procedure in the care of patients who consented the possible use of data for research purpose. The positioning of the electrodes is determined in each patient from hypotheses about the localization of the epileptogenic areas. Implantation accuracy Figure 10 LFP traces of system (1a)–(1h) near critical transitions under stochastic input. (a1) Transitions between multiple time-scale oscillations and background regime for a type-I setting at B =23, C =35, C = 200. Panel (a2) zooms between two large amplitude discharges (blue) and panel (a3)shows the normalized power spectral density of the signal. (b1) Transitions between multiple time-scale oscillations and background regime with slow oscillations of ≈ 3.5 Hz for a type-II B = 17.8, C =80, C = 200. Panel (b2) 3 5 zooms between two large amplitude discharges (red) and panel (b3) shows the normalized power spectral density of the signal. (c1) Transitions between multiple time-scale oscillations and sinusoidal oscillations for B =4.7, C =80, C = 200. Panel (c2) zooms between two large amplitude discharges (cyan) and panel (c3) 3 5 shows the normalized power spectral density of the signal Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 21 of 27 Figure 11 Corresponding PSPs of the LFPs given in Fig. 10.(a1)–(a3) The EPSP, slow IPSP (IPSP ) and fast IPSP (IPSP ) for the type-I setting given in Fig. 10a. (b1)–(b3) The EPSP, slow IPSP (IPSP ) and fast IPSP (IPSP )for the f s f type-IIsetting giveninFig. 10b. (c1)–(c3) The EPSP, slow IPSP (IPSP ) and fast IPSP (IPSP ) for the setting given s f in Fig. 10c Figure 12 SEEG signals recorded in a patient with epilepsy during the inter-ictal and ictal transition. (a) Transition from inter-ictal to ictal period in the ﬁrst patient. Background activity (BKG) observed further away from seizure (panel b1) has a broad-band frequency distribution (normalized power density spectrum in panel b2). A sporadic spike is preceded by narrow band low amplitude resembling canard-mediated oscillations (CMO, marked in red, zoomed in panel (c1)) at ≈ 3.5 Hz (normalized power density spectrum in panel (c2)) peri-operatively is veriﬁed by an X-ray CT scan. A post-operative CT scan without con- trast product is then used to verify the precise 3D location of each electrode contact. After SEEG exploration, intracerebral electrodes are removed. An MRI is performed on which Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 22 of 27 Figure 13 SEEG signals recorded in a patient with epilepsy during the inter-ictal and ictal transition. (a) Transition from inter-ictal to ictal period in the second patient. Background activity (BKG) observed further away from seizure (panel b1) has a broad-band frequency distribution (normalized power density spectrum in panel b2). A sporadic spike is preceded by narrow band low amplitude resembling canard-mediated oscillations (CMO, marked in red, zoomed in panel (c1)) at ≈ 7 Hz (normalized power density spectrum in panel (c2)) Table 2 Summary of patients’ features Feature Patient 1 (Fig. 12)Patient2(Fig. 13) Age at SEEG 16 y 14 y Gender Female Female MRI Right occipito-temporal focal cortical Left hippocampal sclerosis dysplasia Syndrome Temporal lobe epilepsy (temporal Mesial temporal lobe epilepsy plus) Recorded Cerebral Medial part of the middle temporal Internal temporal pole (electrode Region gyrus (electrode contact: B4) contact: Pt’1) Surgical outcome II (cortectomy) IA (anterior temporal lobectomy) (Engel Class) the trajectory of each electrode remains visible. Finally, a CT-scan/MRI data fusion is per- formed to anatomically locate each contact along each electrode trajectory. The patient had electrodes implanted in the temporal region. For this study, signals were selected as they exhibited clear transitions in electrophysiological patterns. In particular, we selected pre-ictal events followed by a fast discharge typical of the seizure onset, which is one of the markers of the imbalanced relation of excitation and inhibition [16, 71]thatinvolves excitability variations. In Fig. 12, a narrow band activity of theta-band of 3.5 Hz is followed by a large amplitude epileptic discharge between two sporadic discharges as we advance towards sustained pre- ictal discharges. Such narrow band activity may be a signature of canard-mediated regions where slowly varying system’s parameters and/or remote interactions lead to transitions between small-amplitude-low-frequency oscillations and large amplitude discharges. In Fig. 13 a narrow band activity of about 7 Hz is followed by a large amplitude epileptogenic Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 23 of 27 discharge. We also notice that the form of the epileptic discharges in Fig. 12 and Fig. 13 are diﬀerent, which may indicate that the systems would have diﬀerent characteristics. Interestingly, we have identiﬁed parameter regions for the corresponding frequency bands in the model (see simulated LFPs and PSPs in Fig. 10 and Fig. 11,respectively).Hence, we think that the properties of transient narrow band oscillations may be related to the excitability properties and level of synaptic projections (scaled by the coupling coeﬃcients in the model) of the epileptogenic zone. 5Conclusion In this article, we extended the multiple time-scale analyses previously initiated in [48]. Here, we both investigated canard transitions present in a neurophysiologically-relevant NMM and analyzed their consequences in terms of subsequent signatures in LFPs. In this three-time-scale model, the canard transitions occur in the 6-dimensional two-time- scale reduced system of slow and super-slow variables. They are associated with degen- erate FSN II singularities and singular Hopf bifurcations. They organize initiation of re- laxation/bursting oscillations from harmonic oscillations of 4.5–6 Hz or from equilibrium points, and determine the boundaries between them. We showed that the system switches between type-I and type-II excitability near the transitions between the equilibrium points and relaxation/bursting oscillations. We further noticed that the canard regimes of type-II excitability (and partially of type-I) yield low-frequency (near 3.5 Hz) oscillations in the LFP under stochastic input. These model predictions motivated a close analysis of SEEG recordings performed in epileptic patients. In this paper, results illustrative of both signatures are reported only in two patients. Interestingly, in brain structures clearly involved in the transition from in- terictal to ictal activity, we observed a narrow band activity between sporadic discharges before the seizure initiation, which strongly diﬀered from the preceding background ac- tivity. Although the parameter set used in this paper was not aimed for modeling these recordings speciﬁcally, it is striking to see such a matching between the mathematical analysis and the actual recordings. It has been evidenced that impaired excitation–inhibition balance shapes the activity of neural networks and, therefore, causes the emergence of “pathological” electrophysio- logical patterns such as pre-ictal spikes and seizures in the context of epilepsy (see for a review [72]). Indeed, epileptogenic brain regions are typical example of such excitation– inhibition imbalanced networks [73]. We showed that the level of EPSP on the subpop- ulation of SOM+ interneurons determines the type of the excitability. In particular, the system is type-I excitable if the average number of synaptic contacts from the excitatory pyramidal cells to the GABAergic SOM+ interneurons is low, and type-II excitable if the average number of synaptic contacts from the excitatory pyramidal cells to the GABAer- gic SOM+ interneurons is high. It is then the decreasing GABAergic inhibition (modeled by decreasing inhibitory drive by the subpopulation of the SOM+ interneurons) that is re- sponsible for transitions from background to epileptiform discharges. Interestingly, such model parameter variations are plausible and linked to the failure of inhibitory barrages observed in epileptic tissues [74] and generation of slow waves preceding the fast activity [75, 76]. Properties of emerging epileptic discharges (e.g. their shape and frequencies), and possible “silent” phases in between are strongly connected to the type of the excitability. In the context of epilepsy, transitory regimes between the background activity and epilep- tic discharges are crucial for understanding the underlying mechanisms [11, 77]. Epileptic Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 24 of 27 biomarkers during such regimes, such as high-frequency oscillations [78], shape features of epileptic spikes [79] or maybe frequency-speciﬁc oscillations reported here, are essen- tial for identiﬁcation of epileptogenic networks and for further development of therapeutic procedures. Veriﬁcation of the presence of such oscillations across diﬀerent patients and accurate modeling of the inter-ictal activity are needed, of course, for suggesting them as biomarkers. This is the topic of future investigations. As epilepsy can be considered as a dynamic disease [73, 80, 81], mathematical mod- els of diﬀerent cellular levels inherit multiple time-scale thinking [82–84]. We note a few studies on the slow–fast transitions in NMMs. Desroches et al. extended NMMs [85]by considering the synaptic gain of SOM+ interneurons as a slowly changing variable. They showed that this conﬁguration introduced regions of torus canards. Jafarian et al. [86]pro- posed a NMM which incorporates slow variations in ionic currents leading to spontaneous paroxysmal activity. Hebbink et al. [87] investigated response of the NMM of Wendling et al. [16] to slowly varying inputs under which the systems yields bursting oscillations. Weigenand et al. remarked the role of canard solutions in fast transitions in sleep wake patterns of K-complexes in a NMM of sleep-wake patterns [43]. Our paper shows that canard-mediated solutions are naturally present in the NMM of Wendling et al. [16]. Im- portantly, as this model implements two main sub-types of interneurons (dendrite- and soma-projecting), it is generic and can be considered for studying the dynamics of other regions than hippocampus, such as neocortical areas, and in diﬀerent contexts, such as consciousness [47] and Alzheimer’s disease [20]. Furthermore, canard regimes reported in this study are governed by the interactions between the pyramidal cell and SOM+ in- terneuron subpopulations that follow a two-time-scale structure. It would be natural to observe canard-mediated transitions in another generally used NMM of Jansen and Rit [13] for modeling the brain activity. Hence, the canard-mediated ﬁne structures we have demonstrated here could be relevant for a number of situations and lead to markers of sub- sequent critical transitions. The reported degenerate FSN II singularity leading to canard trajectories is due to the general structure of NMMs, which are deﬁned via second-order diﬀerential equations. The dynamics associated with the degenerate FSN II singularity merits further investigations and will be considered as a future work. Finally, organisation of homoclinic canard orbits, possible codimension-two bifurcations and interactions with the fold points will be studied in forthcoming works. Acknowledgements We would like to thank Prof. Fabrice Bartolomei (APHM, Timone Hospital, Clinical Neurophysiology, Marseille, France) for providing the clinical data, Mathieu Desroches (Inria Sophia Antipolis - Méditerranée, MathNeuro Team, Sophia Antipolis, France) for his helpful comments and referees for their useful suggestions. Funding EKE was supported by NIH (Application number: R01 NS092760-01A1; 18/02/2019 to 31/12/2020). She is member of the Galvani project (ERC-SyG 2019; grant agreement No 855109) since 01/01/2021. Abbreviations NMM, Neural Mass Model; EEG, Electroencephalography; LFP, Local Field Potential; GSPT, Geometric Singular Perturbation Theory; SOM+, Somatostatin positive; PV+, Parvanium positive; CA1, Cornu Ammonis 1; PSP, Post-Synaptic Potential; EPSP, Excitatory Post-Synaptic Potential; IPSP, Inhibitory Post-Synaptic Potential; SEEG, Stereoelectroencephalography; FSN II, Folded Saddle-Node type II; DSRS, Desingularized Slow Reduced System; LP, Limit Point; H, Hopf; HOM, Homoclinic; CP, Cusp Point; BT, Bogdanov–Takens; GH, Generalized Hopf; SNIC, Saddle-Node of Invariant Cycle; SNPO, Saddle-Node of Periodic Orbits. Availability of data and materials The codes used for numerical analysis are available from the GitHub database (https://github.com/elifkoksal/NMM_BurstingDynamics). Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 25 of 27 Ethics approval and consent to participate The SEEG recordings were carried out as part of normal clinical care of patients. Patients were informed that their data may be used for research purposes. Competing interests The authors declare that they have no competing interests. Consent for publication Not applicable. Authors’ contributions EKE developed the theoretical framework, conducted the mathematical analysis, interpreted the results and was a major contributor in writing the manuscript. FW contributed to interpreting the results and writing the manuscript. All authors read and approved the ﬁnal manuscript. Authors’ information Université de Rennes 1, INSERM, Laboratoire Traitement du Signal et de L’Image (LTSI) - U1099, Campus de Beaulieu - Batiment 22, 35042 Rennes, France. 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Publisher: Springer Journals
Copyright: Copyright © The Author(s) 2021
eISSN: 2190-8567
DOI: 10.1186/s13408-021-00109-z
Publisher site: See Article on Publisher Site

Abstract

Univ Rennes, INSERM, LTSI-U1099, Mathematical models at multiple temporal and spatial scales can unveil the Campus de Beaulieu, F - 35000 Rennes, France fundamental mechanisms of critical transitions in brain activities. Neural mass models (NMMs) consider the average temporal dynamics of interconnected neuronal subpopulations without explicitly representing the underlying cellular activity. The mesoscopic level oﬀered by the neural mass formulation has been used to model electroencephalographic (EEG) recordings and to investigate various cerebral mechanisms, such as the generation of physiological and pathological brain activities. In this work, we consider a NMM widely accepted in the context of epilepsy, which includes four interacting neuronal subpopulations with diﬀerent synaptic kinetics. Due to the resulting three-time-scale structure, the model yields complex oscillations of relaxation and bursting types. By applying the principles of geometric singular perturbation theory, we unveil the existence of the canard solutions and detail how they organize the complex oscillations and excitability properties of the model. In particular, we show that boundaries between pathological epileptic discharges and physiological background activity are determined by the canard solutions. Finally we report the existence of canard-mediated small-amplitude frequency-speciﬁc oscillations in simulated local ﬁeld potentials for decreased inhibition conditions. Interestingly, such oscillations are actually observed in intracerebral EEG signals recorded in epileptic patients during pre-ictal periods, close to seizure onsets. MSC: 34E15; 34E17; 37N25; 92B25 Keywords: Multiple time-scale systems; Canards; Bursting; Excitability; Epilepsy; Neural mass model 1 Introduction Brain dynamics emerges from neural entities interacting at diﬀerent levels, from sin- gle neurons to large-scale neural networks. At each level, transitions between diﬀerent regimes, such as ﬁring/resting states in single neurons and up/down states in neural net- works, are associated with both physiological functions and pathological activity [1–3]. One of the features of the system that determines how these transitions would occur is excitability. The concept of neural excitability for single neurons was introduced initially by Louis Lapique in 1907 [4, 5]. Alan Hodgkin, who then re-introduced the concept [6], classiﬁed the neural excitability with respect the ﬁring rate of neurons in response to in- jected steps of currents. Excitability properties of neural systems can vary with internal © 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/. Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 2 of 27 dynamics, leading to diﬀerent physiological and pathological behavior [7–10]. At the cor- tical scale, for instance, variations in excitability [11] and loss of network resilience [12] are associated with epileptic seizures. Yet, what may be as important as a transition itself is the dynamics preceding the transition. In the context of epilepsy, for example, identi- ﬁcation of the dynamic features along the path to a transition is crucial for intervention and prevention of seizures. Mathematical models of brain activity range from microscopic level of single cell dy- namics to macroscopic level of interactions between large scale neural systems. Neural mass models (NMMs) consider the average temporal dynamics of interconnected neural subpopulations without explicitly representing the underlying mechanisms at the level of single cells. The mesoscopic level oﬀered by the neural mass formulation has been used to model brain signals, from local ﬁeld potentials (LFPs) to global electroencephalo- graphic (EEG) recordings, and to investigate various cerebral rhythms [13–15]. NMMs have also been used extensively to study pathological dynamics such as in epilepsy [16– 19], Alzheimer’s disease [20] and Parkinson’s disease [21, 22]. Interactions between slow and fast components of neural systems, hence, of their math- ematical models, result in multiple time-scale complex oscillations, such as relaxation, bursting and mixed-mode oscillations. Geometric singular perturbation theory (GSPT) is a key tool for understanding the interaction between the geometry of the system and the emerging multiple time-scale dynamics. In particular, canard solutions, which can exist in multiple time-scale systems with a folded geometry, appear as building blocks of complex oscillations in both phenomenological and neurophysiologically plausible models ranging from single cell [23–26] to neural networks [27, 28]. The canard phenomenon in such sys- tems has been related to neural excitability [29], excitability thresholds [23, 30–34], and boundaries between diﬀerent type of solutions, such as subthreshold oscillations and large amplitude spiking/bursting oscillations [24, 28, 35–43]. While such canard-organized ﬁne structures have been shown in a wide range of two-time-scale models, recent studies started to explore canard-mediated processes in systems with three or more time-scales [44–46]. In this study we investigate critical regimes in the NMM initially presented in [16]. This physiologically-grounded model has been extensively used for modeling structural and functional changes leading to epileptic activity observed in intracranial (stereoelec- troencephalography, SEEG) signals. The model includes four interacting neuronal sub- populations: two interconnected subpopulations of glutamatergic pyramidal neurons and GABAergic inhibitory interneurons (somatostatin positive (SOM+), and parparvalbunim positive (PV+), also called dendrite-projecting slow and soma-projecting fast interneu- rons, respectively). Although the model was introduced for the CA1 region of the hip- pocampus, implementation of these four subpopulations mediating glutamatergic and GABAergic signaling makes it generic enough to be considered for many other cortical regions [47]. Activity of each subpopulation is given by the corresponding average post- synaptic potential (PSP) that is determined by two functions: 1) a “pulse to wave” function, S(v)=5/(1 + exp(0.56(6 – v))), transforming the incoming synaptic potentials into a ﬁring rate; and 2) a “wave to pulse” converting the input average ﬁring rate into a mean PSP at the input of each subpopulation, that is, h(t)= Wt/τ exp(–t/τ ), where W represents the w w average synaptic gain and τ is the average synaptic time constant mimicking the rise and w Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 3 of 27 decay of actual PSPs. The system schematized in Fig. 1areads y ˙ = y,(1a) 0 5 A 2 1 y ˙ = S[y – y – y ]– y – y,(1b) 5 1 2 3 5 0 τ τ τ a a y ˙ = y (1c) 1 6 A 2 1 y ˙ = p(t)+ C S[C y ] – y – y,(1d) 6 2 1 1 6 1 τ τ τ a a y ˙ = y , (1e) 2 7 B 2 1 y ˙ = C S[C y ]– y – y,(1f) 7 4 3 0 7 2 τ τ τ b b y ˙ = y,(1g) 3 8 G C 2 1 y ˙ = C S C y – y – y – y.(1h) 8 7 5 0 4 8 3 τ C τ τ g 4 g The variables y stand for the excitatory PSPs mediated by two pyramidal neuron sub- 0,1 populations, y and y are inhibitory PSPs mediated by the SOM+ and PV+ interneuron 2 3 subpopulations, respectively. Variables y (j ∈{5, 6, 7, 8}) are the auxiliary variables that are introduced to convert the second-order diﬀerential equations describing the wave to pulse functions to ﬁrst-order diﬀerential equations [13]. The parameters A, B, G are the synaptic gains, the C are the connectivity constants representing the average number of synaptic contacts, p(t) is the external (noisy) cortical input (p(t)= p + ξ,where p is the mean of external input ξ is a random variable following a normal distribution with N (0, σ )). The synaptic time constants are given by τ , τ , τ . The major contribution to LFPs (as recorded a b g by intracranial electrodes in patients candidate to surgery) corresponds to the PSPs sum- mated at the level of pyramidal neurons, which are geometrically aligned “in palisades”, i.e. one relative to the other and perpendicular to the plane of the cortical layers. In the model, the LFP is given by the sum of excitatory PSP (EPSP) and inhibitory PSPs (IPSPs) received by the glutamatergic pyramidal cells, hence LFP = y – y – y . 1 2 3 As introduced in [48], under the following variable conversion: t y y y y 0 1 2 3 , , , , , y , y , y , y ⇒ (t, v , v , v , v , y , y , y , y ), 5 6 7 8 0 1 2 3 5 6 7 8 τ τ τ τ τ g a a b g with δ = τ /τ and ε = τ /τ ,system(1a)–(1h) can be written in the following deterministic g a a b (σ =0) slow–fast form: dv = y := F (y ), (2a) 8 3 8 dt dy = GS[C τ v – C τ v ]– v –2y := F (v , v , v , y ), (2b) 5 a 0 6 b 2 3 8 8 0 2 3 8 dt dv = δy := δF (y ), (2c) 5 0 5 dt Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 4 of 27 dy = δ AS[Aτ p + C τ v – C τ v – C τ v ]– v –2y (2d) a 2 a 1 4 b 2 7 g 3 0 5 dt := δF (v , v , v , v , y ), 5 0 1 2 3 5 dv = δy := δF (y ), (2e) 6 1 6 dt dy = δ AS[C τ v ]– v –2y := δF (v , v , y ), (2f) 1 a 0 1 6 6 0 1 6 dt dv = δεy := δεF (y ), (2g) 7 2 7 dt dy = δε BS[C τ v ]– v –2y := δεF (v , v , y ). (2h) 3 a 0 2 7 7 0 2 7 dt In this manuscript, we will consider system (2a)–(2h) for the slow–fast analysis and (1a)–(1h) for simulations under the stochastic input. We will be using the parameter set given in Table 1,unlessotherwise stated,for which δ =0.3 and ε =0.2. Numerical bi- furcation analysis is performed in AUTO-07p software [49]. The stochastic diﬀerential equations were integrated using the Euler–Murayama method with a step size dt = 10e–4 second in XPPAUT software [50]. As noticed in [48], system (2a)–(2h) is a three-time-scale system written in fast form with (v , y )being fast,(v , y , v , y )slow and (v , y ) super-slow variables. Köksal Ersöz et 3 8 0 5 1 6 2 7 al. [48] have focused on electrophysiological pre-ictal bursting patterns recorded in hu- man patients just before the onset of seizure. Pre-ictal bursting patterns are characterized by fast oscillatory discharges (which will be referred as spikes) followed by a slower oscil- lation (a simulated pattern with the parameter set in Table 1 is exempliﬁed in Fig. 1b). The authors have reproduced pre-ictal bursting and unveiled the mechanism yielding these so- lutions by decorticating the three-time-scale structure of the model. They have discussed appropriate stimulation strategies for aborting of the pre-ictal bursting, hence, for pre- venting a subsequent epileptic seizure. However, they did not focus on possible slow–fast transitions. Here we extend the slow–fast analysis initiated in [48] by investigating the role of slow-manifolds in transitions to relaxation and bursting type of solutions. We will focus on how canard trajectories shape the diﬀerent routes from physiological to patho- logical brain activity. In what follows, we will go brieﬂy through the multiple-time-scale analysis presented in [48], and then show diﬀerent canard structures present in the model and how they take a part in critical transitions. Finally, we will see the system’s response to stochastic inputs near critical regimes, and make a remark on the slow oscillations ob- served along the path to seizure in SEEG signals recorded during pre-surgical evaluation of two patients with drug-resistant epilepsy. Table 1 Parameter values for the bursting-type discharges A (mV) B (mV) G (mV) p (Hz) C C C C C C C τ (s) τ (s) τ (s) 1 2 3 4 5 6 7 a b g 5 5 35 90 135 108 80 25 450 121 121 0.01 0.05 0.003 Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 5 of 27 Figure 1 Model diagram and time series of a bursting solution. (a) Model diagram showing excitatory (red connections) and inhibitory (blue connections) interactions between subpopulations of pyramidal neurons (PYR and PYR’) and inhibitory interneurons (PV+ and SOM+). The post-synaptic potential of a subpopulation, which is the output of h(t), is multiplied by a synaptic coeﬃcient C before being received by another subpopulation. (b) Time series of a bursting solution for the parameter set in Table 1. The panels from the top to the bottom show the time courses of post-synaptic potentials of PV+ (v ), PYR (v ), SOM+ (v )and thelocal 3 0 2 ﬁeld potential (LFP), i.e. Aτ p + C τ v – C τ v – C τ v a 2 a 1 4 b 2 7 g 3 2Preliminaries System (2a)–(2h) expressed in the fast time t is called a fast system.The slow system is ˜ ˜ obtained by deﬁning t = δt, dv δ = F (y ), (3a) 3 8 dt dy δ = F (v , v , v , y ), (3b) 8 0 2 3 8 dt dv = F (y ), (3c) 0 5 dt dy = F (v , v , v , v , y ), (3d) 5 0 1 2 3 5 dt dv = F (y ), (3e) 1 6 dt dy = F (v , v , y ), (3f) 6 0 1 6 dt dv = εF (y ), (3g) 2 7 dt dy = εF (v , v , y ), (3h) 7 0 2 7 dt where the functions F (·) are as deﬁned in (2a)–(2h). The super-slow system is obtained by ˜ ˜ ˜ deﬁning t = εt = εδt: ss s dv δε = F (y ), (4a) 3 8 dt ss dy δε = F (v , v , v , y ), (4b) 8 0 2 3 8 dt ss Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 6 of 27 dv ε = F (y ), (4c) 0 5 dt ss dy ε = F (v , v , v , v , y ), (4d) 5 0 1 2 3 5 dt ss dv ε = F (y ), (4e) 1 6 dt ss dy ε = F (v , v , y ), (4f) 6 0 1 6 dt ss dv = F (y ), (4g) 2 7 dt ss dy = F (v , v , y ). (4h) 7 0 2 7 dt ss Systems (2a)–(2h), (3a)–(3h)and (4a)–(4h) describe diﬀerent dynamics in the singular limits ε → 0 and/or δ → 0, although they are equivalent for ε =0 and δ = 0. Letting δ → 0 in (2a)–(2h)yieldsthe fast layer problem (2a)–(2b) which describes the dynamics of the fast variables (v , y ) for ﬁxed values of the slow (v )and super-slow (v )variables.The 3 8 0 2 critical manifold is deﬁned by the equilibrium points of the fast layer problem, that is, 0 8 S = (v , y , v , y , v , y v , y ) ∈ R | GS[C τ v – C τ v ]– v =0,(5) 3 8 0 5 1 6 2 7 5 a 0 6 b 2 3 which is eventually in the (y = 0)-space. Since the eigenvalues of the Jacobian matrix of the fast layer problem deﬁned by (2a)–(2b)withrespect to (v , y )are λ =–1, the 6- 3 8 1,2 dimensional critical manifold S is normally hyperbolic and stable, thus, it is perturbed to local slow manifolds for suﬃciently small δ > 0. Therefore, the fast dynamics can be approximated by slow dynamics as suggested by the Fenichel theorem [51]. Setting δ → 0in (3a)–(3h) gives an algebraic-diﬀerential slow reduced problem, 0= F (y ), (6a) 3 8 0= F (v , v , v , y ), (6b) 8 0 2 3 8 dv = F (y ), (6c) 0 5 dt dy = F (v , v , v , v , y ), (6d) 5 0 1 2 3 5 dt dv = F (y ), (6e) 1 6 dt dy = F (v , v , y ), (6f) 6 0 1 6 dt dv = εF (y ), (6g) 2 7 dt dy = εF (v , v , y ), (6h) 7 0 2 7 dt which describes the slow dynamics restricted to S .System(6a)–(6h) is a two-time-scale system of 4slow/2super-slowvariables with ε being the time-scaling parameter. The equi- libria of the slow layer problem in the ε → 0 limit deﬁnes the super-slow manifold L ,which Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 7 of 27 is reduced to 0 0 L = (v , y , v , y , v , y v , y ) ∈ S | (7) 3 8 0 5 1 6 2 7 AS Aτ p + C τ AS[C τ v ]– C τ v – C τ v – v =0 , a 2 a 1 a 0 4 b 2 7 g 3 0 and restricted to S by the algebraic condition v = GS[C τ v –C τ v ]= K(v , v )in (6a)– 3 5 a 0 6 b 2 0 2 0 0 (6b). The super-slow dynamics restricted to the 2-dimensional manifold L ,hence to S , are given by the super-slow reduced system in the ε → 0 limit of (4a)–(4h). In order to investigate the super-slow ﬂow on L , we consider the two-time-scale system (6a)–(6h)withthe fast variable v on S ,i.e. v = K(v , v ), and rewrite the slow reduced 3 3 0 2 system (6a)–(6h)as dv = F (y ), (8a) 0 5 dt dy = F v , v , v , K(v , v ), y,(8b) 5 0 1 2 0 2 5 dt dv = F (y ), (8c) 1 6 dt dy = F (v , v , y ), (8d) 6 0 1 6 dt dv = εF (y ), (8e) 2 7 dt dy = εF (v , v , y ). (8f) 7 0 2 7 dt Applying the time-scaling τ˜ = εt and taking the singular limit ε → 0 give the algebraic- s s diﬀerential system 0= F (y ), (9a) 0 5 0= F v , v , v , K(v , v ), y,(9b) 5 0 1 2 0 2 5 0= F (y ), (9c) 1 6 0= F (v , v , y ), (9d) 6 0 1 6 dv = F (y ), (9e) 2 7 dτ˜ dy = F (v , v , y ). (9f) 7 0 2 7 dτ˜ The algebraic conditions (9a)–(9d) deﬁne the ‘critical manifold’ of (8a)–(8f)which is 0 0 equivalent to L given by (7). Notice that L is restricted in the zero plane of the (y , y )- 5 6 0 0 space. Assuming that v is some function of v on L ,i.e. v = M(v ), the fold points on L 2 0 2 0 are deﬁned by ∂M(v ) F = (v , v , v , v , y , y , y , y ) ∈ L v = M(v ), =0 . (10) 0 1 2 3 5 6 7 8 2 0 ∂v 0 Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 8 of 27 Figure 2 Critical manifold, slow manifold and folded singular points. (a) Critical manifold S (green surface), 0 0 super-slow manifold L (red curve) and a bursting orbit in the (v , v , v )-space. The curve L is divided into 0 2 3 0 0 three branches at F and F (red dots) where it changes stability. The middle branch of the L (L )curve 1 2 0 0 between F and F is unstable (dashed). The stable left-hand side branch (L ), F , L and F are entirely on 1 2 1 2 l m 0 0 the almost horizontal part of S (approximately on the (v ≈ 0)-plane). The stable right-hand side branch of L 0 0 (L ) on expands both on the horizontal and vertical parts of S . Arrows indicate the corresponding time-scale (single-headed for super-slow, double-headed for slow dynamics). (b) Super-slow manifold L (red surface), fold curves F (black lines) and folded singular points p (red dots)inthe (y , v , v )-space. Arrows indicate 1,2 1,2 7 2 0 the corresponding time-scale 0 0 0 Figure 2ashows S and L in the (v , v , v )-space, and Fig. 2b L in the (y , v , v )-space. 0 2 3 7 2 0 The super-slow manifold L expands between the lower horizontal and vertical planes of 0 0 0 S .The part of curve L on the lower horizontal plane of S is folded with respect to v at along the fold curves F and F deﬁned by (10), i.e. F = F ∪ F .Inthisprojection, 1 2 1 2 0 0 0 the 1-D fold curves divide L into two stable (left-hand side L and right-hand side L ) l r and one unstable (middle L ) branches on the (v , v , v )-space. We also verify that four 0 2 3 eigenvalues of L (two real and two complex conjugate) have negative real parts along the stable parts of L . One of the real eigenvalues changes sign along F ,hence theunstable 1,2 middle branch is of saddle type. Along the stable and unstable branches L is normally hyperbolic, so L is perturbed to local super-slow manifolds for small values of ε >0 within (6a)–(6h); see the extension of Fenichel theory for systems with more than two time-scales [52]. On the other hand, the dynamics near the non-hyperbolic fold curves F should be 1,2 investigated by using the elements of GSPT. As being the usual strategy, we consider the desingularized version of the super-slow dynamics on L is given by the desingularized slow reduced system (DSRS), reading dv =–y , (11a) dτˆ dy ∂M(v ) 7 0 =– F v , M(v ), y , (11b) 7 0 0 7 dτˆ ∂v s 0 ∂M(v ) where τ˜ = τˆ .The equilibria of (11a)–(11b)onthe fold set F are located at s s ∂v p1 p1 p2 p2 (v , y ) = (1.2343, 0) and (v , y ) = (9.9976, 0) for the parameter set given in Table 1. 0 7 0 7 These equilibrium points, which are not generally the true equilibria of (8a)–(8f), are re- lated to the folded singularities of (8a)–(8f), hence of (2a)–(2h). On the other hand, equi- F F F F 7 7 7 7 librium points (v , y ), i.e. F (v , y ) = 0, are ordinary singularities since they are also 0 7 0 7 equilibria of (8a)–(8f), hence of (2a)–(2h). Figure 2bshows L ,foldcurves F and folded 1,2 singular points p in the (y , v , v )-space. 1,2 7 2 0 Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 9 of 27 Stability of the equilibrium points of the desingularized (slow) reduced system on the fold set determines the type of the folded singularities of the original system. Classiﬁca- tion of these equilibrium points is based on the linear stability analysis. When the desin- gularized (slow) reduced system is planar, this analysis can be done using the trace and the determinant of the Jacobian matrix at the fold equilibrium. If both are diﬀerent from zero, the fold equilibrium can be a folded saddle, a folded node or a folded focus. If the determinant equals zero but not the trace, then the desingularized ﬂow has a degenerate equilibrium point, which is a folded saddle-node. A folded saddle-node is either related to a saddle-node bifurcation of the folded equilibria or a transcritical bifurcation of a folded equilibrium with an ordinary equilibrium. The latter case refers to the folded saddle-node type II (FSN II) singularity [53, 54], where a folded node becomes a folded saddle and a regular saddle becomes a regular node. The original system exhibits a singular Hopf bi- furcation close to a FSN II singularity [55, 56]. The Jacobian matrix of (11a)–(11b) has the following general form: ⎡ ⎤ 0–1 ⎣ ⎦ 2 ∗ ∗ ∗ ∗ ∗ J = , (12) ∂ M(v ) ∂M(v ) ∂F (v ,y ) ∂M(v ) 0 ∗ ∗ 0 0 7 0 – F (v , y )– 2 2 7 0 7 ∂v ∂v ∂v ∂v 0 0 0 ∗ ∗ where (v , y ) stands for the equilibrium point of interest. Since on the folded equilibria 0 7 p1,p2 ∂M(v ) 2 = 0, the trace and determinant of (12) on the folded equilibria read ∂v p1,p2 ∂ M(v ) p1,p2 p1,p2 p1,p2 p1,p2 0 tr J =0, det J =– F v , y . (13) 2 0 7 ∂v The trace and determinant of (12) on the regular equilibria read F F F F 7 7 7 7 ∂M(v ) ∂M(v ) ∂F (v , y ) F 0 F 0 0 7 7 7 tr J =2 , det J =– . (14) ∂v ∂v ∂v 0 0 0 Notice that the generic folded singularity condition is violated due to the fact that ∂(F v ˙ +F y ˙ ) v 2 y 7 2 7 p1,p2 =0 in (12), and tr(J )=0 in (13). Therefore, the folded singularities de- ∂v termined by (11a)–(11b) are not generic and a folded equilibrium is one of the following p1,p2 p1,p2 p1,p2 types: a saddle for det(J )<0, a center for det(J )>0, a nilpotent for det(J )=0. The latter degenerate type corresponds to a point in the parameter space at which a F F 7 7 folded singularity and a regular singularity meet, i.e. tr(J )=0 and det(J )=0 in (14). Consequently, the equilibrium points of (11a)–(11b) related to the folded and regu- lar singularities undergo degenerate transcritical bifurcations where (12)has twozero- eigenvalues. Figure 3 shows the bifurcation diagram of (11a)–(11b)withrespect to B in the region of p1 p1 p2 p2 interest. Two straight lines of the equilibria (v , y )and (v , y ) intersect with the regu- 0 7 0 7 lar equilibria curve F (v , y )at two bifurcation points, BP at B ≈ 16.7817 and BP at 7 0 7 1 BP1 2 B ≈ 5.4817, which are degenerate transcritical bifurcations. For B < B ,the equilib- BP2 BP1 p1 p1 rium (v , y ) is a center with two complex conjugate eigenvalues. After the bifurcation at 0 7 p1 p1 BP ,(v , y ) becomes a saddle. Consequently, the system (8a)–(8f)(and(2a)–(2h)) has a 0 7 p2 p2 folded-saddle singularity near p for B > B .The equilibrium (v , y ) is of a saddle type 1 BP1 0 7 for B < B and becomes a center with two complex conjugate eigenvalues at B = B . BP2 BP2 Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 10 of 27 Figure 3 Bifurcation diagram of (11a)–(11b)withrespect to B. p1 p1 p1 Equilibrium points (v , y ) lie on the lower red horizontal line (v , 0 7 0 p2 p2 p2 p2 y ), and (v , y ) on the upper red horizontal line (v , y ). Dashed 0 7 0 7 parts of the red lines represent saddle type, solid parts represent center type solutions. The true equilibrium points line on the black curve, F . Dashed part of the black curve represents saddle type, solid parts stable focus type solutions. The saddle type equilibrium points p1 p1 p2 p2 along (v , y )and (v , y ) change to center at the intersections 0 7 0 7 with F at BP and BP , respectively 7 1 2 Hence, the system (8a)–(8f)(and(2a)–(2h)) has a folded-saddle singularity near p for B B . Similarly, in a neigh- BP1 BP1 borhood of BP , the equilibrium points along the F (v , y )curve are of stable focus type 2 7 0 7 for B B . BP2 BP2 As mentioned above, a generic transcritical bifurcation of regular and folded singular- ities is related to a FSN II singularity. In our case, a folded saddle becomes a folded cen- terand astablefocus becomesasaddleatthe degenerate transcritical bifurcation points BP and BP . Furthermore, system (2a)–(2h) can undergo the (singular) Hopf bifurcations 1 2 close to BP and BP in the parameter space (see Fig. 4d), as it will be detailed in the follow- 2 1 ing sections. Hence, the interaction of regular and non-generic folded singularities can be referred as a degenerate FSN II singularity. A degenerate FSN II singularity in (2a)–(2h) stems from the structure of the NMM, which is deﬁned as a second-order system that violates the generic folded singularity condition tr(J) =0. Folded singularities can lead to canard solutions in the original system. In a planar slow– fast system, a singular Hopf bifurcation can occur near a folded singularity, which is then called a canard point. In such case, the amplitude of the periodic orbits bifurcated at the singular Hopf point increases stiﬄy in a narrow interval of the parameter (scaled by the time-scale separation parameter) that controls the transition from small amplitude to relaxation oscillations [57]. This phenomenon is known as canard explosion [26, 58]. A canard-explosive branch hosts small canards following the unstable branch of the criti- cal manifold and one stable branch (so-called canard-without-head solutions), large ca- nards following the unstable branch of the critical manifold and two stable branches (so-called canard-with-head solutions), and a maximal canard solution that follows the longest the repelling branch. In planar multiple time-scale systems, canard solutions are tightly connected to excitability and ﬁring thresholds [30, 31]. In higher dimensional multiple time-scale systems with at least two slow variables, the folded-singularities are generic, hence they are robust to small parameter perturbations, and canard solutions as- sociated with folded singularities connect stable and unstable branches of a folded critical manifold [36, 53, 59–61]. Canards of folded node and FSN II singularities support mixed mode oscillations [27, 36, 44]. FSN II singularities have been identiﬁed in neuronal models where the exchange from an excitable to a relaxation oscillatory state is accompanied by subthreshold oscillations [24, 28, 42, 62]. Folded-saddle canards have been shown to be sculpting ﬁring threshold manifolds, as well [33, 34, 63–65]. In our problem, the critical manifold S (5) is hyperbolic, whereas the super-slow man- ifold L (8a)–(8f) has a folded structure. Thus, the critical transitions occur mainly in the 6-dimensional reduced system given by (8a)–(8f). As the analysis above have shown, (8a)– Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 11 of 27 Figure 4 Bifurcation diagrams of (2a)–(2h) with as a function of (B, C ). (a) Bifurcation diagram of (2a)–(2h)on the (B, C ) plane. Curves are named, respectively, after the limit point (LP, black curves), Hopf (H, red curves) and homoclinic (HOM, blue curves) bifurcations in panels (b–f). Only the LP bifurcations interacting with canard solutions are plotted. Black squares indicate cusp (CP), red circles indicate Bogdanov–Takens (BT) and red squares indicate generalized Hopf (GF) bifurcations. The regions marked by black, green and purple boxes are zoomed in black, green and purple framed insets. The region where the homoclinic curve tips to the LP is zoomed inside the green inset. (b–f) Bifurcation diagrams of (2a)–(2h) as a function of B for diﬀerent values of C . The limit point bifurcations of interest are marked by black squares, Hopf bifurcations by red circles, and homoclinic connections by blue stars. Stable and unstable solutions are represented by continuous and dashed curves, respectively. Along the curves of equilibrium points, (2a)–(2h) undergoes four Hopf bifurcations (H )for C = {50, 80, 145} (c, d, e) and three Hopf bifurcations (H )for C =15 (b)and 1,2,3,4 3 1,2,3 3 (H )for C = 200 (f) 1,2,4 3 (8f) has degenerate folded singularities along the fold curve at p and p .Noticethat, since 1 2 the system has neither a folded node nor a FSN II, small amplitude oscillations do not exist near p or p . But the folded saddle, degenerate FSN II and singular Hopf bifurcations can 1 2 lead to canard solutions governing the critical transitions in (8a)–(8f)(hencein(2a)–(2h)). On the other hand, the bursting behavior cannot be captured by (8a)–(8f)because (8a)– (8f) is restricted in the critical manifold S , whereas the fast oscillations of the bursting orbits leave oﬀ S . So the bursting solutions exist in the full system (2a)–(2h)(see[48]for a detailed analysis of the bursting solutions). As a result, our problem yields both three and two time-scaled behaviors. In the next section, we investigate canard dynamics near p and p . 1 2 3 Multiple time-scale oscillations and canard transitions Transitions near the folded singularities of (2a)–(2h) which lead to canard solutions de- pend on the system parameters. The reader may refer to Table 1 forthe parametervalues, Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 12 of 27 unless otherwise stated. The connectivity strength from the pyramidal cell population on the subpopulation of the SOM+ interneurons, C , and their synaptic gain, B,appearas two crucial parameters controlling the transitions by aﬀecting the curve F in (11a)–(11b) (see Fig. 3). Figure 4a shows a 2-parameter bifurcation diagram in the plane (B, C ). De- pending on C ,system(2a)–(2h) undergoes several Hopf bifurcations as a function of B. The ﬁrst two Hopf bifurcations, H and H , yield harmonic oscillations, whereas the pe- 1 2 riodic branches appearing at H and/or H connect to multiple time-scale oscillations. 3 4 Under the variations in (B, C ), H and H persist; and the emerging periodic orbits do 3 1 2 not change qualitatively. On the other hand, H and H undergo Bogdanov–Takens (BT) 3 4 bifurcations BT and the corresponding periodic branches vary qualitatively. The peri- 1,2 odic orbits emerging at H and H can end on homoclinic connections, namely HOM . 3 4 1,2,3,4 Figures 4b–f exemplify qualitative variations in (2a)–(2h) as a function of B for diﬀerent values of C .For C < C ≈ 18.9, the system undergoes three Hopf bifurcations, for in- 3 3 3,BT1 stance in Fig. 4bfor C = 15. The branch of periodic solutions starting at H terminates at 3 3 a homoclinic connection, HOM .As C increases, HOM and LP get closer while the am- 1 3 1 1 plitude and the number of spikes of the periodic orbits increase. The spike adding occurs as the HOM curve folds back and forth in the (B, C )-space(seethe blackframedinset 1 3 in Fig. 4a for an example folding). At C = C another Hopf bifurcation, H ,appears 3 3,BT1 4 yielding a new branch of periodic orbits making a second homoclinic connection, HOM (green framed inset in Fig. 4a). Consequently, HOM and HOM points coexist in a nar- 1 2 row range of (B, C ). The HOM curve touches the LP curve at (B, C ) ≈ (23.98, 22.43), 3 1 1 3 foldsbackand continuesinthe parameterspace,which then we name as HOM curve (dashed zone in the green framed inset in Fig. 4a). The curves HOM and HOM stay 1 3 very close to each other in (23.98 < B < 24.46, 21.94 < C < 22.43), before HOM bends 3 3 in the C direction at (B, C ) ≈ (24.46, 22.43). As it happens, the branch of periodic orbits 3 3 curls below LP in the B-space and eventually connects to HOM . With increasing C ,this 1 3 3 branch of periodic orbits advances further towards the stable equilibrium points while in- troducing a region of multi-attractors of nodes, saddles, unstable small oscillations and stable large amplitude bursting oscillations (see Fig. 4cfor C = 50, dynamics will be de- tailed in Sect. 3.2). Concurrently, H moves away from LP and HOM approach LP . 4 1 2,3 2 In (20.09 54.43, 2 3 the branch of periodic orbits initiated at H connects to the branch of large amplitude multiple time-scale oscillations (Fig. 4d). System (2a)–(2h) does not have any LPs between H and H for C ≤ C ≤ C .At 3 4 3,CP1 3 3,CP2 C = C ≈ 141.4, as the lower branch of equilibrium points curls below H ,the con- 3 3,CP2 3 nection between the large amplitude orbits and H is broken up on a saddle-saddle homo- clinic bifurcation (the equilibrium points in a neighborhood of LP for B ≥ B are saddles 3 LP3 [66, 67]). As a consequence, the branch of periodic orbits starting from H terminates on a homoclinic connection, HOM (for C = 145 in Fig. 4e). This homoclinic connection 4 3 remains until H disappears at C = C ≈ 157. Beyond C > C ,the largeamplitude 3 3 3,BT2 3 3,BT2 bursting orbits introduced by H terminate on a saddle-node homoclinic connection (for instance at C = 200 in Fig. 4f). The Hopf bifurcations H and H occur close to the folded singularities p and p ,re- 3 4 2 1 spectively. System (2a)–(2h) can yield canard solutions close to these points in the param- eter space of B,suchas B ≈ B and B ≈ B . In the following section, we will show the H3 H4 Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 13 of 27 canard-mediated transition from sinusoidal oscillations initiated by H to large amplitude bursting/relaxation type solutions. Subsequently, Sect. 3.2 will detail the canard dynamics and related excitability near H , in particular, the type-I excitability for C = 50 and type-II 4 3 excitability for C =80. 3.1 Canard-mediated transitions between sinusoidal and multiple time-scale oscillations Köksal Ersöz et al. [48] have realized that the number of spikes of a bursting solution of (2a)–(2h) depends on the amount of the PSP received by the PV+ interneuron subpop- ulation, hence on the EPSP coming from the pyramidal cell subpopulation and the IPSP from the SOM+ interneurons. For instance, increasing the IPSP on the PV+ interneurons by increasing B decreases the number of spikes while driving the oscillations one peak to the next one in the parameter space (see Fig. 4c–f and 5). The connectivity constant from the pyramidal cell subpopulation to the PV+ interneuron subpopulation, C , directly scales the EPSP on this subpopulation, therefore determines the maximum number of fast spikes of the bursting oscillations, or more generally, the type of the multiple time-scale oscillations. Figure 5 exempliﬁes how C modulates the large amplitude oscillations between H and 5 3 H on the bifurcation diagram of (2a)–(2h)for C =80. Increasing C decreases the ampli- 4 3 5 tude of the oscillations, moves H and H slightly to the right, but does not aﬀect consider- 1 2 ably the locations H and H with respect to B (Fig. 5a). The supercritical Hopf bifurcation 3 4 at H yields a branch of sinusoidal periodic oscillations (Fig. 5b) that folds back and forth as B varies and enters in a regime of multiple time-scale periodic oscillations. These os- cillations are of relaxation type for small values of C , and of bursting type for large values of C . Furthermore, the stable sinusoidal and multiple time-scale periodic oscillations can coexist depending on the values of C (see Fig. 5bat B ≈ 4.8). The form of the branch of periodic solutions between H and H in Fig. 5a indicates 3 4 the type of the multiple time-scale oscillations for a certain parameter combination. For C = 80 (black diagram in Fig. 5) the smoothly decreasing amplitude of v with B in- 5 0 dicates that the corresponding orbits are of relaxation type (exempliﬁed in Fig. 6). The Figure 5 Variation of large amplitude solutions with respect to (B, C ). (a) Bifurcation diagrams of (2a)–(2h) with respect to B for C = 80 and diﬀerent values of C . Curves and Hopf bifurcations (H , dots) are colored 3 5 1,2,3,4 with respect to the color codes of C values. Stable and unstable solutions are represented by continuous and dashed curves, respectively. For the sake of simplicity, the periodic solutions between H and H are not 1 2 shown. (b) Zoom into the region of transitions between sinusoidal and large amplitude multiple-time-scale solutions in B ∈ [4.7, 4.9] Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 14 of 27 Figure 6 Example canard orbits along the transition from sinusoidal oscillations to relaxation oscillations. Zoom near the bifurcation diagram for C =80, C =80 and B ∈ [4.7, 4.9] (see Fig. 5a for the whole diagram). 3 5 Continuous and dash curves represent stable and unstable solutions, respectively. Hopf (H ,red dot) is marked on the diagram. Numbered orbits from 1-7 are given in panels (b–d). The orange curve traces the frequency of the oscillations emerging at H The orange curve traces the frequency of the oscillations emerging at H .(b) Periodic orbits marked in panel (a), L (red curve), fold curves F (red points) and the 3 1,2 critical surface S (green surface) are projected on the (v , v , v )-space. Arrows indicate the corresponding 0 2 3 time-scale (single-headed for super-slow, double-headed for slow dynamics). (c) Periodic orbits marked in panel (a), L (red surface), fold curves F (black curves) and folded singular points p (red dots) are 1,2 1,2 projected on the (y , v , v )-space. Arrows indicate the corresponding time-scale. (d) Time series of the 7 2 0 ˜ ˜ ˜ periodic orbits on panels (b, c) with respective color codes. Period is normalized to 1 (t/T =1, where T represents period of a cycle) horizontal zigzags along the upper part of the periodic branches obtained for greater values of C indicate the presence of bursting solutions along these periodic branches and the number of their fast spikes. For instance, the 5 peaks that we count between H and H for C = 350 (blue diagram in Fig. 5) signify that the maximum number of fast 4 5 spikes for C = 350 is 4. Such a bursting orbit is obtained for suﬃciently small values of B (B = 5, for instance). Then as B increases, the bursting orbits lose their fast spikes one by one through the peaks of the horizontal branch. They become relaxation cycles (B = 16, for instance), before shrinking and disappearing via a subcritical Hopf bifurcation at H . 4 Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 15 of 27 The periodic solutions connected to H and H do notinteractwiththe singular fold 1 2 points of L , p and p ,but theonesnear H and H do because H and H take place 1 2 3 4 3 4 close to the degenerate FSN II singularities on the fold curve F. As a consequence, the multiple time-scale orbits emanating at singular Hopf bifurcations H and H can un- 3 4 dergo canard explosion along which canard trajectories sculpt the periodic oscillations. Figure 6 shows example orbits along the periodic branch that follow H for C =80 and 3 3 C = 80. As the periodic branch folds with respect to B and becomes unstable (Fig. 6a), the sinusoidal orbits of 4.5–6 Hz start to interact with p . In particular, they move along 0 0 0 theunstablebranchof L , L , before jumping back to the stable branch L .Hence,the m r periodic orbits become canard orbits (the 1st orbit). As B varies along the periodic branch in the parameter space, the canard orbits grow in amplitude along L (the second, third and fourth orbits) until they stretch out between F and F (the ﬁfth orbit). The canard 1 2 0 0 orbits that oscillate between L and L can be interpreted as canard-without-head orbits, m r and the ﬁfth orbit as the maximal canard since it has the largest period of the canard fam- ily of the periodic branch under consideration. Soon after the ﬁfth orbit, the trajectories jump to the attracting branch L , get a shape of canard-with-head solutions, and become stable (the sixth orbit). As B increases, the relaxation cycles appear with parts exclusively following the attracting branches of L and jumping close to the fold points. As mentioned in the introduction, canard solutions play a fundamental role in sepa- rating diﬀerent dynamical regimes. The unstable canard orbits in Fig. 6 (from the ﬁrst to the ﬁfth) appear as an other example of this phenomenon by accompanying the transi- tion from sinusoidal oscillations to relaxation oscillations. For instance, sinusoidal oscil- lations and large amplitude canard-with-head cycles coexist for B ∈ (4.75, 4.81) and the canard-without-head cycles form the boundary between them, as seen clearly in Fig. 6a. While increasing C introduces bursting type of solutions, it can preserve the bistability between the bursting and sinusoidal oscillations, for example for C = {250, 350, 425} in Fig. 5. Notice that with increasing C , the initially smooth branch of periodic orbits be- comes steeper, gains vertical zigzags that move to the right along the B-axis, and the region of bistability decreases. Figure 7 zooms into the region of canard orbits following the sinusoidal solutions of 4.8–6 Hz started at H for C = 450. As the stable sinusoidal oscillations grow in amplitude 3 5 with increasing B, they start to interact with p and to follow the bits of L (the ﬁrst orbit). Soon after, the orbits undergo a LP bifurcation (where the branch of the periodic orbits folds back at B ≈ 4.821) and become unstable. As B varies in the parameter space along the periodic branch, the orbits moving along L in the super-slow time-scale grow in 0 0 amplitude and they start to interact with the vertical panel of S as they jump to L in the slow time-scale. So, the orbits become canard orbits. As the part of the trajectory along L grows in amplitude, the trajectory gets attracted 0 0 0 by L along the vertical panel of S and it spirals around L before landing on the horizon- r r 0 0 tal plane of S . This interaction with the vertical panel of S occurs in the fast time-scale, and eventually yields fast spikes, i.e., bursting-type canard oscillations. For instance, the second orbit in Fig. 7cand 7d has one fast spike. The number of spikes increases as the trajectory stays longer and longer along L while B varies. More precisely, the number of spikes changes by one as we pass from one fold to another on the same side of along the snaking periodic branch with respect to B (Fig. 7b). For instance, solution-2 has 1 spike, solution-3, which is two fold below, has 3 spikes and so on. The spike adding continues Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 16 of 27 Figure 7 Example canard orbits along the transition from sinusoidal oscillations to bursting oscillations (a) Bifurcation diagram for C =80, C = 450 and B ∈ [4.7, 4.9] (see Fig. 5a for the whole diagram). Stable and 3 5 unstable solutions are represented by continuous and dashed curves, respectively. Hopf (H ,red dot) is marked on the diagram. The rectangular region is zoomed in panel (b). Numbered orbits from 1-7 are given in panels (c–d). The orange curve traces the frequency of the oscillations emerging at H .(c) Periodic orbits 0 0 marked in panels (a, b), L (red curve), fold curves F (red points) and the critical surface S (green surface) 1,2 are projected on the (v , v , v )-space. Arrows indicate the ﬂow direction and its time-scale (single-head for 0 2 3 super-slow, double-head for slow, triple-head for fast). (d) Periodic orbits marked in panel (a), L (red surface), (black curves) and folded singular points p (red dots) are projected on the (y , v , v )-space. fold curves F 1,2 1,2 7 2 0 Arrows indicate the corresponding time-scale. (e) Time series of the periodic orbits on panels are (b, c)shown ˜ ˜ ˜ with respective color codes. Period is normalized to 1 (t/T =1, where T represents period of a cycle) until the canard orbits (analogous to canard-without-head orbits) expands between two folded singularities, hence till the occurrence of the maximal canard of the family (ap- proximated by the sixth orbit). After the maximal canard, the canard cycles start to follow L in the super-slow time-scale (analogous to canard-with-head orbits) and they become stable (the seventh orbit). The part of the trajectory along L decreases as B increases further. Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 17 of 27 For C = 80, the system undergoes a complete canard explosion along the periodic branch following H since it visits the whole canard family from small to large (Figs. 6 and 7). However, what we may observe for small values of C is an incomplete canard ex- plosion terminating at a homoclinic connection. For instance, for C =15 (see Fig. 4b), the sinusoidal oscillations along the periodic branch initiated at H change qualitatively by interacting with L as B varies and we observe homoclinic canard-without-head orbits at HOM . The orbits terminating on HOM are homoclinic canard-with-head orbits sur- 1 3 rounded by stable large amplitude oscillations. Increasing C completes the canard explo- sion and the system enters into an excitable regime which will be detailed in the following section. 3.2 Canard-mediated transitions and excitability According to Hodgkin’s [6] classiﬁcation of neural excitability, type-I excitable neurons have continuous frequency-injected current curves, whereas type-II excitable neurons have discontinuous frequency-injected current curves. Rinzel and Ermentrout [68, 69] linked the type-I excitability to a SNIC bifurcation and the type-II excitability to a Hopf bifurcation. De Maesschalck and Wechselberger [29] explained the transition between the two excitability types via an intermediate regime of type-I excitability associated with a codimension-2 Bogdanov–Takens (BT) bifurcation in a planar system. They showed the existence of incomplete canard transitions in this transitory regime. Later on, transitions between the neuronal excitability types was shown to be induced by the inhibitory and excitatory autapse in the Morris-Lecar model [70]. Folded singularities and correspond- ing canard solutions in higher dimensional systems also have been shown to be shaping systems’ excitability properties [24, 28, 33, 34, 63–65]. System (2a)–(2h) can yield large amplitude oscillations in response to certain forms of stimulation (due to stochastic inputs, for instance) after being initiated from an equilib- rium point for a B value close to H , LP and LP in Figs. 4c–4f. Hence, system (2a)– 4 1 2 (2h) is excitable in these regions and the excitability properties of (2a)–(2h)determined by the parameter C (see Fig. 4). Indeed, the local pictures in these regions are similar to the ones investigated in [29, 70]. In particular, system (2a)–(2h)istype-Iexcitable for C ∈ (22.43, 54.43), basically between the homoclinic/saddle-saddle interactions near LP 3 1 and LP . In this parameter region, the large amplitude oscillations terminate on a homo- clinic orbit for which the ﬁring frequency is zero. System (2a)–(2h)istype-II excitablefor C > 54.43 for which the termination is issued via a Hopf bifurcation. In both cases, canard solutions shape the resulting dynamics. Figure 8 zoomsinnearthe excitableregionfor C =50 (see Fig. 4c for the whole dia- gram). For a particular value of B for B < B , the only attractor is the large amplitude LP1 bursting oscillation (the 1st orbit). In B < B < B theunstableattractorsofthe equi- LP1 H4 librium points appear. The subcritical Hopf bifurcation at B = B initiates a branch of H4 periodic orbits that terminates on the homoclinic point HOM , which bounds the canard explosion near H .For B , a homoclinic canard-without-head orbit (the second or- 4 HOM2 bit) coexists with a large stable bursting orbit of canard type (the third orbit). At B a HOM3 homoclinic canard-with-head orbit (the ﬁfth orbit) appears together with an outer large amplitude canard cycle (the fourth orbit). The large amplitude canard cycle grows in am- plitude and disappears on a saddle-node of periodic orbits (SNPO) at B = B (the sixth SNPO orbit). We also notice that, as HOM gets closer to LP for C ≈ 54.4 and B ≈ B ,the 3 2 3 LP2 canard orbits on the HOM become of without-head type. 3 Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 18 of 27 Figure 8 Example canard orbits near the type-I excitable regime. (a) Bifurcation diagram for C =50, C = 450 3 5 and B ∈ [18, 22.5] (see Fig. 4c for the whole diagram). Stable and unstable solutions are represented by continuous and dashed curves, respectively. Limit point (LP ,black squares),Hopf(H , red dot), homoclinic 1,2 4 (HOM , blue stars) bifurcations and saddle-node bifurcation of periodic orbits (SNOP, orange purple square) 2,3 are marked on the diagram. Numbered solutions are presented in panels (b–d). The orange curves trace the frequency of the oscillations. (b) Periodic orbits marked in panel (a), L (red curve), fold curves F (red points) 1,2 and the critical surface S (green surface) are projected on the (v , v , v )-space. Arrows indicate the ﬂow 0 2 3 direction and its time-scale (single-headed for super-slow, double-headed for slow dynamics. The homoclinic points HOM and HOM are marked by cyan and dark blue stars. (c) Periodic orbits marked in panel (a), L (red 2 3 surface), fold curves F (black curves) and folded singular points p (red dots) are projected on the 1,2 1,2 (y , v , v )-space. Arrows indicate the corresponding time-scale. (d) Time series of the periodic orbits on panels 7 2 0 ˜ ˜ ˜ (b, c) with respective color codes. Period is normalized to 1 (t/T =1, where T represents period of a cycle) Foraparametersetensuringthetype-IIexcitability(C = 80, for instance), the fast spikes of the bursting oscillations disappear and the ﬁnal oscillation turns out to be a relaxation type running in slow and super-slow timescales. These relaxation oscillations terminate via a complete canard-explosion near the singular Hopf bifurcation point H .Thishap- pens in a similar manner for all C values under consideration. Figure 9 provides an ex- ample for C = 450. As the large amplitude periodic solutions decrease in amplitude, they start to follow L and take the shape of canard-with-head solutions (the second and third m Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 19 of 27 Figure 9 Example canard orbits near the type-II excitable regime. (a) Bifurcation diagram for C =80, C = 450 and B ∈ [16, 18] (see Fig. 5a for the whole diagram). Stable and unstable solutions are represented by continuous and dashed curves, respectively. Hopf bifurcation (H , red dot) is marked on the diagram. Numbered orbits on the lower branch of periodic solutions are presented in panels (c–d). The orange curve traces the frequency of the oscillations emerging at H .(b) Periodic orbits marked in panel (a), L (red curve), fold curves F (red points) and the critical surface S (green surface) are projected on the (v , v , v )-space. 1,2 0 2 3 Arrows indicate the ﬂow direction and its time-scale (single-headed for super-slow, double-headed for slow dynamics). (c) Periodic orbits marked in panel (a), L (red surface), fold curves F (black curves) and folded 1,2 singular points p (red dots) are projected on the (y , v , v )-space. Arrows indicate the corresponding 1,2 7 2 0 time-scale. (d) Time series of the periodic orbits on panels (b, c) are shown with respective color codes. Period ˜ ˜ ˜ is normalized to 1 (t/T =1, where T represents period of a cycle) orbits). The maximal canard of this canard family is the fourth orbit that stays along to the super-slow manifolds as long as possible. After the fourth orbit, we observe canard- without-head orbits (the 5th and the 6th orbits) that shrink to p . The frequency of the oscillations along the canard explosion ranges in 1.8–3.5 Hz. We also notice a region of bistability between large amplitude bursting oscillations and equilibrium points. Once again the canard solutions construct the boundary between them. For a parameter set giving relaxation oscillations in this region (e.g. C = {80, 250, 350} in Fig. 5), the relax- ation oscillations shrink H via a ‘classical’ canard explosion, similar to the one in the 2D van der Pol system, without having any fast component in v . 3 Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 20 of 27 4 Local ﬁeld potential in critical regimes In the previous section we have shown two diﬀerent regions in parameter space where canard solutions determine boundaries and organize transitions between diﬀerent dy- namical regimes. The narrow-band sinusoidal activity of 4.5–6 Hz emerging near H and of 1.8–3.5 Hz emerging near H are connected to large amplitude periodic multiple- timescale solutions through canard orbits. System (1a)–(1h) emits aperiodic large ampli- tude epileptic discharges under stochastic input (p(t)= p + ξ,with ξ = N (0, 2 )) when it is initialized near the critical regions of H and H (Fig. 10–11). A parameter setting ensur- 3 4 ing the type-I excitability without any canard solutions near H gives a board band activity between the large amplitude spikes (Fig. 10a1–a3). On the other hand, taking the system to type-II excitability near H introduces transient small amplitude oscillations of ≈ 3.5 Hz due to the presence of the canard cycles in this region (Fig. 10b1–b3). We observe transi- tions between large amplitude discharges and harmonic oscillations of ≈ 6 Hz when the system is initialized close to the Hopf bifurcation H (Fig. 10c1–c3). Simulated PSPs at the levelofthe pyramidalcellsubpopulation aregiven in Fig. 11. Figures 12 and 13 show LFPs recorded by the SEEG electrodes in two diﬀerent patients with drug-resistant focal epilepsy during presurgical evaluation (see Table 2 for the de- tails). Multiple-contact depth electrodes were implanted according to the SEEG technique as a standard clinical procedure in the care of patients who consented the possible use of data for research purpose. The positioning of the electrodes is determined in each patient from hypotheses about the localization of the epileptogenic areas. Implantation accuracy Figure 10 LFP traces of system (1a)–(1h) near critical transitions under stochastic input. (a1) Transitions between multiple time-scale oscillations and background regime for a type-I setting at B =23, C =35, C = 200. Panel (a2) zooms between two large amplitude discharges (blue) and panel (a3)shows the normalized power spectral density of the signal. (b1) Transitions between multiple time-scale oscillations and background regime with slow oscillations of ≈ 3.5 Hz for a type-II B = 17.8, C =80, C = 200. Panel (b2) 3 5 zooms between two large amplitude discharges (red) and panel (b3) shows the normalized power spectral density of the signal. (c1) Transitions between multiple time-scale oscillations and sinusoidal oscillations for B =4.7, C =80, C = 200. Panel (c2) zooms between two large amplitude discharges (cyan) and panel (c3) 3 5 shows the normalized power spectral density of the signal Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 21 of 27 Figure 11 Corresponding PSPs of the LFPs given in Fig. 10.(a1)–(a3) The EPSP, slow IPSP (IPSP ) and fast IPSP (IPSP ) for the type-I setting given in Fig. 10a. (b1)–(b3) The EPSP, slow IPSP (IPSP ) and fast IPSP (IPSP )for the f s f type-IIsetting giveninFig. 10b. (c1)–(c3) The EPSP, slow IPSP (IPSP ) and fast IPSP (IPSP ) for the setting given s f in Fig. 10c Figure 12 SEEG signals recorded in a patient with epilepsy during the inter-ictal and ictal transition. (a) Transition from inter-ictal to ictal period in the ﬁrst patient. Background activity (BKG) observed further away from seizure (panel b1) has a broad-band frequency distribution (normalized power density spectrum in panel b2). A sporadic spike is preceded by narrow band low amplitude resembling canard-mediated oscillations (CMO, marked in red, zoomed in panel (c1)) at ≈ 3.5 Hz (normalized power density spectrum in panel (c2)) peri-operatively is veriﬁed by an X-ray CT scan. A post-operative CT scan without con- trast product is then used to verify the precise 3D location of each electrode contact. After SEEG exploration, intracerebral electrodes are removed. An MRI is performed on which Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 22 of 27 Figure 13 SEEG signals recorded in a patient with epilepsy during the inter-ictal and ictal transition. (a) Transition from inter-ictal to ictal period in the second patient. Background activity (BKG) observed further away from seizure (panel b1) has a broad-band frequency distribution (normalized power density spectrum in panel b2). A sporadic spike is preceded by narrow band low amplitude resembling canard-mediated oscillations (CMO, marked in red, zoomed in panel (c1)) at ≈ 7 Hz (normalized power density spectrum in panel (c2)) Table 2 Summary of patients’ features Feature Patient 1 (Fig. 12)Patient2(Fig. 13) Age at SEEG 16 y 14 y Gender Female Female MRI Right occipito-temporal focal cortical Left hippocampal sclerosis dysplasia Syndrome Temporal lobe epilepsy (temporal Mesial temporal lobe epilepsy plus) Recorded Cerebral Medial part of the middle temporal Internal temporal pole (electrode Region gyrus (electrode contact: B4) contact: Pt’1) Surgical outcome II (cortectomy) IA (anterior temporal lobectomy) (Engel Class) the trajectory of each electrode remains visible. Finally, a CT-scan/MRI data fusion is per- formed to anatomically locate each contact along each electrode trajectory. The patient had electrodes implanted in the temporal region. For this study, signals were selected as they exhibited clear transitions in electrophysiological patterns. In particular, we selected pre-ictal events followed by a fast discharge typical of the seizure onset, which is one of the markers of the imbalanced relation of excitation and inhibition [16, 71]thatinvolves excitability variations. In Fig. 12, a narrow band activity of theta-band of 3.5 Hz is followed by a large amplitude epileptic discharge between two sporadic discharges as we advance towards sustained pre- ictal discharges. Such narrow band activity may be a signature of canard-mediated regions where slowly varying system’s parameters and/or remote interactions lead to transitions between small-amplitude-low-frequency oscillations and large amplitude discharges. In Fig. 13 a narrow band activity of about 7 Hz is followed by a large amplitude epileptogenic Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 23 of 27 discharge. We also notice that the form of the epileptic discharges in Fig. 12 and Fig. 13 are diﬀerent, which may indicate that the systems would have diﬀerent characteristics. Interestingly, we have identiﬁed parameter regions for the corresponding frequency bands in the model (see simulated LFPs and PSPs in Fig. 10 and Fig. 11,respectively).Hence, we think that the properties of transient narrow band oscillations may be related to the excitability properties and level of synaptic projections (scaled by the coupling coeﬃcients in the model) of the epileptogenic zone. 5Conclusion In this article, we extended the multiple time-scale analyses previously initiated in [48]. Here, we both investigated canard transitions present in a neurophysiologically-relevant NMM and analyzed their consequences in terms of subsequent signatures in LFPs. In this three-time-scale model, the canard transitions occur in the 6-dimensional two-time- scale reduced system of slow and super-slow variables. They are associated with degen- erate FSN II singularities and singular Hopf bifurcations. They organize initiation of re- laxation/bursting oscillations from harmonic oscillations of 4.5–6 Hz or from equilibrium points, and determine the boundaries between them. We showed that the system switches between type-I and type-II excitability near the transitions between the equilibrium points and relaxation/bursting oscillations. We further noticed that the canard regimes of type-II excitability (and partially of type-I) yield low-frequency (near 3.5 Hz) oscillations in the LFP under stochastic input. These model predictions motivated a close analysis of SEEG recordings performed in epileptic patients. In this paper, results illustrative of both signatures are reported only in two patients. Interestingly, in brain structures clearly involved in the transition from in- terictal to ictal activity, we observed a narrow band activity between sporadic discharges before the seizure initiation, which strongly diﬀered from the preceding background ac- tivity. Although the parameter set used in this paper was not aimed for modeling these recordings speciﬁcally, it is striking to see such a matching between the mathematical analysis and the actual recordings. It has been evidenced that impaired excitation–inhibition balance shapes the activity of neural networks and, therefore, causes the emergence of “pathological” electrophysio- logical patterns such as pre-ictal spikes and seizures in the context of epilepsy (see for a review [72]). Indeed, epileptogenic brain regions are typical example of such excitation– inhibition imbalanced networks [73]. We showed that the level of EPSP on the subpop- ulation of SOM+ interneurons determines the type of the excitability. In particular, the system is type-I excitable if the average number of synaptic contacts from the excitatory pyramidal cells to the GABAergic SOM+ interneurons is low, and type-II excitable if the average number of synaptic contacts from the excitatory pyramidal cells to the GABAer- gic SOM+ interneurons is high. It is then the decreasing GABAergic inhibition (modeled by decreasing inhibitory drive by the subpopulation of the SOM+ interneurons) that is re- sponsible for transitions from background to epileptiform discharges. Interestingly, such model parameter variations are plausible and linked to the failure of inhibitory barrages observed in epileptic tissues [74] and generation of slow waves preceding the fast activity [75, 76]. Properties of emerging epileptic discharges (e.g. their shape and frequencies), and possible “silent” phases in between are strongly connected to the type of the excitability. In the context of epilepsy, transitory regimes between the background activity and epilep- tic discharges are crucial for understanding the underlying mechanisms [11, 77]. Epileptic Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 24 of 27 biomarkers during such regimes, such as high-frequency oscillations [78], shape features of epileptic spikes [79] or maybe frequency-speciﬁc oscillations reported here, are essen- tial for identiﬁcation of epileptogenic networks and for further development of therapeutic procedures. Veriﬁcation of the presence of such oscillations across diﬀerent patients and accurate modeling of the inter-ictal activity are needed, of course, for suggesting them as biomarkers. This is the topic of future investigations. As epilepsy can be considered as a dynamic disease [73, 80, 81], mathematical mod- els of diﬀerent cellular levels inherit multiple time-scale thinking [82–84]. We note a few studies on the slow–fast transitions in NMMs. Desroches et al. extended NMMs [85]by considering the synaptic gain of SOM+ interneurons as a slowly changing variable. They showed that this conﬁguration introduced regions of torus canards. Jafarian et al. [86]pro- posed a NMM which incorporates slow variations in ionic currents leading to spontaneous paroxysmal activity. Hebbink et al. [87] investigated response of the NMM of Wendling et al. [16] to slowly varying inputs under which the systems yields bursting oscillations. Weigenand et al. remarked the role of canard solutions in fast transitions in sleep wake patterns of K-complexes in a NMM of sleep-wake patterns [43]. Our paper shows that canard-mediated solutions are naturally present in the NMM of Wendling et al. [16]. Im- portantly, as this model implements two main sub-types of interneurons (dendrite- and soma-projecting), it is generic and can be considered for studying the dynamics of other regions than hippocampus, such as neocortical areas, and in diﬀerent contexts, such as consciousness [47] and Alzheimer’s disease [20]. Furthermore, canard regimes reported in this study are governed by the interactions between the pyramidal cell and SOM+ in- terneuron subpopulations that follow a two-time-scale structure. It would be natural to observe canard-mediated transitions in another generally used NMM of Jansen and Rit [13] for modeling the brain activity. Hence, the canard-mediated ﬁne structures we have demonstrated here could be relevant for a number of situations and lead to markers of sub- sequent critical transitions. The reported degenerate FSN II singularity leading to canard trajectories is due to the general structure of NMMs, which are deﬁned via second-order diﬀerential equations. The dynamics associated with the degenerate FSN II singularity merits further investigations and will be considered as a future work. Finally, organisation of homoclinic canard orbits, possible codimension-two bifurcations and interactions with the fold points will be studied in forthcoming works. Acknowledgements We would like to thank Prof. Fabrice Bartolomei (APHM, Timone Hospital, Clinical Neurophysiology, Marseille, France) for providing the clinical data, Mathieu Desroches (Inria Sophia Antipolis - Méditerranée, MathNeuro Team, Sophia Antipolis, France) for his helpful comments and referees for their useful suggestions. Funding EKE was supported by NIH (Application number: R01 NS092760-01A1; 18/02/2019 to 31/12/2020). She is member of the Galvani project (ERC-SyG 2019; grant agreement No 855109) since 01/01/2021. Abbreviations NMM, Neural Mass Model; EEG, Electroencephalography; LFP, Local Field Potential; GSPT, Geometric Singular Perturbation Theory; SOM+, Somatostatin positive; PV+, Parvanium positive; CA1, Cornu Ammonis 1; PSP, Post-Synaptic Potential; EPSP, Excitatory Post-Synaptic Potential; IPSP, Inhibitory Post-Synaptic Potential; SEEG, Stereoelectroencephalography; FSN II, Folded Saddle-Node type II; DSRS, Desingularized Slow Reduced System; LP, Limit Point; H, Hopf; HOM, Homoclinic; CP, Cusp Point; BT, Bogdanov–Takens; GH, Generalized Hopf; SNIC, Saddle-Node of Invariant Cycle; SNPO, Saddle-Node of Periodic Orbits. Availability of data and materials The codes used for numerical analysis are available from the GitHub database (https://github.com/elifkoksal/NMM_BurstingDynamics). Köksal Ersöz and Wendling Journal of Mathematical Neuroscience (2021) 11:11 Page 25 of 27 Ethics approval and consent to participate The SEEG recordings were carried out as part of normal clinical care of patients. Patients were informed that their data may be used for research purposes. Competing interests The authors declare that they have no competing interests. Consent for publication Not applicable. Authors’ contributions EKE developed the theoretical framework, conducted the mathematical analysis, interpreted the results and was a major contributor in writing the manuscript. FW contributed to interpreting the results and writing the manuscript. All authors read and approved the ﬁnal manuscript. Authors’ information Université de Rennes 1, INSERM, Laboratoire Traitement du Signal et de L’Image (LTSI) - U1099, Campus de Beaulieu - Batiment 22, 35042 Rennes, France. 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Published: Sep 16, 2021

Keywords: Multiple time-scale systems; Canards; Bursting; Excitability; Epilepsy; Neural mass model; 34E15; 34E17; 37N25; 92B25

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Canard solutions in neural mass models: consequences on critical regimes

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Canard solutions in neural mass models: consequences on critical regimes

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