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Bioelectronic Medicines that modulate the activity patterns on peripheral nerves have promise as a new way of treating diverse medical conditions from epilepsy to rheumatism. Progress in the field builds upon time consuming and expensive experiments in living organisms. To reduce experimentation load and allow for a faster, more detailed analysis of peripheral nerve stimulation and recording, computational models incorporating experimental insights will be of great help. We present a peripheral nerve simulator that combines biophysical axon models and numerically solved and idealised extracellular space models in one environment. We modelled the extracellular space as a three-dimensional resistive continuum governed by the electro-quasistatic approximation of the Maxwell equations. Potential distributions were precomputed in finite element models for different media (homogeneous, nerve in saline, nerve in cuff) and imported into our simulator. Axons, on the other hand, were modelled more abstractly as one-dimensional chains of compartments. Unmyelinated fibres were based on the Hodgkin-Huxley model; for myelinated fibres, we adapted the model proposed by McIntyre et al. in 2002 to smaller diameters. To obtain realistic axon shapes, an iterative algorithm positioned fibres along the nerve with a variable tortuosity fit to imaged trajectories. We validated our model with data from the stimulated rat vagus nerve. Simulation results predicted that tortuosity alters recorded signal shapes and increases stimulation thresholds. The model we developed can easily be adapted to different nerves, and may be of use for Bioelectronic Medicine research in the future. Keywords Simulation · Peripheral nerve · Finite element model · Biophysics · Bioelectronic medicines Introduction being applied in patients suffering from refractory epilepsy (Milby et al. 2010), Alzheimer’s disease (Sjogren et al. Manipulations of the peripheral nervous system (PNS) by 1997), anxiety (George et al. 2008), obesity (Krzysztof et al. implanted devices might soon serve as a treatment for 2011), chronic heart failure (Rousselet et al. 2014), and various medical conditions. Such Bioelectronic Medicines to evoke anti-inflammatory effects (Meregnani et al. 2011; (Birmingham et al. 2014) can be seen as a permanent, Borovikova et al. 2000). The more localised targeting of organs e.g. of the heart (Pohl et al. 2015) has also shown highly localised alternative to molecular medicines with promising results in animal experiments. less side effects. Already today, vagus nerve stimulation is Current devices operate in open-loop mode and stimula- tion selectivity is low. Future Bioelectronic Medicines will Carl H. Lubba need more precise stimulation interfaces and the capabil- email@example.com ity to analyse (or ‘decode’) nerve activity to stimulate in Simon R. Schultz an adaptive manner. First advances towards a decoding of firstname.lastname@example.org information from peripheral nerves have been successfully undertaken (Citi et al. 2008; Lubba et al. 2017). To both Department of Bioengineering, Imperial College London, South Kensington, London SW7 2AZ, UK accelerate the design of interfaces and to further develop 2 decoding algorithms, computational peripheral nerve models Department of Mathematics, Imperial College London, South Kensington, London SW7 2AZ, UK that integrate physiological insights acquired in experiments 3 at different levels (i.e. properties of axons, extracellular Department of Medicine, Imperial College London, South Kensington, London SW7 2AZ, UK media, spontaneous activity patterns, organ responses) will 4 be of great merit to predict stimulation efficiency and Department of Electrical and Electronic Engineering, Imperial College London, South Kensington, London SW7 2AZ, UK recording selectivity and as a source of surrogate data. 64 Neuroinform (2019) 17:63–81 Previous efforts to simulate peripheral nerves date back simulation is run for each point in time even though the qua- approximately twenty years. In 1997, Struijk (1997)devel- sistatic approximation of the Maxwell equations allows for oped a 2D model of recordings from myelinated peripheral a separation of time and space. Static FEM simulations for axons. Models for stimulation were also proposed at the each source position are sufficient (see Methods). time (Veltink et al. 1989; Goodall et al. 1995). One main More detailed models that go beyond the simplifying difficulty in peripheral nerve simulations, already appreci- assumptions of a constant extracellular potential during ated at that time, is the calculation of extracellular potentials axon simulation and a quasistatic extracellular space were from membrane currents in the inhomogeneous medium proposed as well. In particular, electrical feedback of neu- surrounding the axons. As a major difference from record- ronal activity on membrane processes, ephaptic coupling, ings in the central nervous system (CNS), the recording is neglected by the methods mentioned so far. However, it method (e.g. cuff electrode, oil bath) often shapes nature can be significant (Tveito et al. 2017;Bokiletal. 2001). of the extracellular space in the PNS. Early simulations To incorporate such feedback, the entire model (including therefore often concentrated on modelling the extracellular the intracellular space and membranes) can be simulated medium whilst approximating axons with simplified mod- in a FEM solver as in Agudelo-Toro and Neef (2013)and els such as the Fitz-Hugh-Nagumo equations (Plachta et al. Tveito et al. (2017). Even if elegantly capturing intra-, 2012, e.g.) or the McNeal model (McNeal 1976; Veltink extracellular, and membrane effects in a combined, self- et al. 1989). Only recently precise biophysical axon models contained model, the calculation becomes more expensive and detailed, numerically solved models of the surround- and is only suitable for simple geometries (Tveito et al. ing medium have been combined (Grinberg et al. 2008; 2017). An even higher degree of accuracy can be attained by Raspopovic et al. 2011), thanks to increasing availability of going beyond the quasistatic Maxwell equations and incor- computational resources. porating electrodiffusive effects (diffusion of charge ions) For the general task of modelling extracellular potentials in Poisson-Nernst-Planck solvers, see Pods et al. (2013)and caused by cells (axons), many choices at different levels Halnes et al. (2016) or the simplified version (electroneu- of detail and computational cost exist. The most simple tral model) (Pods 2017). Whilst offering great degree of approach is based on volume conductor theory: the extra- detail, the computation becomes very expensive in those cellular space is modelled as being resistive, homogeneous, formulations. and isotropic (Holt and Koch 1999; Linden ´ et al. 2014)so In face of those choices, our open-source toolbox that the extracellular potential becomes an analytic func- PyPNS aims at realising an compromise between usability, tion of source (membrane) currents. The latter are obtained computational efficiency, and accuracy for a bundle of many axons. Our approach is in principle very comparable from compartmental simulations of the cell membrane in which usually the extracellular potentials are assumed to to a model that precomputes membrane currents in a be constant. However, in peripheral nerves, the surround- compartmental simulator and imports them into an FEM ing medium is not homogeneous or isotropic, requiring simulation. We use the NEURON simulator (Hines and a more complex approach. To accommodate conductiv- Carnevale 1997) to model axon membrane processes at the ity inhomogeneities, precomputed membrane currents from scale of ion channels. Standard models for both myelinated compartmental cell simulators can be imported into a finite and unmyelinated axons in the diameter range found in the element model (FEM) solver (as a point source or boundary periphery were implemented. The extracellular space was condition) where the potential over the entire space and time governed by the electro-quasistatic approximation of the span is computed based on the quasistatic Maxwell equa- Maxwell equations. Ephaptic coupling and electrodiffusion tions (cf. McIntyre and Grill 2001; Lempka and McIntyre were neglected for the sake of computational efficiency. Importantly however, PyPNS improves the efficiency 2013;Nessetal. 2015). This costlier method was employed in the recent aforementioned works on peripheral nerves and usability of previous approaches by avoiding to (Grinberg et al. 2008; Raspopovic et al. 2011). It has dis- run FEM simulations repeatedly for every simulation. advantages, however. The model needs to be defined in two Instead, we took advantage of the quasistatic approximation different environments, the compartmental axon simulator of the Maxwell equations to separate time and space and the FEM solver. Both environments need to be coor- and simplified the nerve geometry to create symmetries. dinated in terms of geometry, coordinate systems, units, Potential distributions could thereby be precomputed in a and so on and setting up a model this way is a time con- reusable way for arbitrary axon shapes and were imported suming and error prone process. There exist commercial into PyPNS. In this way, the accuracy of hybrid FEM- simulation solutions that combine a compartmental simula- based solutions was reached at the computational cost of the tor and a FEM solver in a single framework, e.g. Sim4Life (Zurich MedTech AG) but no openly available simulators. See information sharing statement at the end for accessing the As a further limitation of current hybrid solutions, the FEM toolbox. Neuroinform (2019) 17:63–81 65 simple volume conductor method. In addition, PyPNS adds RHD2000 system, using a 16-channel bipolar ended ampli- detail compared to previous simulations by letting the user fier (RHD221). The obtained recordings were averaged over choose the degree of axon tortuosity. Tortuosity is expected 10 repeated stimulations in the same animal. to be particularly relevant for peripheral nerves. An increased axon length can act as a buffer against mechanical Imaging of Peripheral Nerve Tortuosity influences only present in the periphery; this buffer allows curvier axons. Lastly, our simulator is embedded into the All procedures were carried out in accordance with Python ecosystem. For the entire simulation, the user can the Animals (Scientific Procedures) Act 1986 (United stay in Python to stimulate nerves and record from them in Kingdom) and Home Office (United Kingdom) approved silico. project and personal licences, and experiments were approved by the Imperial College Animal Welfare Ethical Review Board under project licence PPL 70/7355. To Methodology reproduce the morphology of axons, we imaged the vagus and sciatic nerves in mice using two photon fluorescence Nerve Stimulation Experiments imaging. In the experiment, ChAT-Cre FLEX-VSFP 2.3 mice were euthanised by intraperitoneal overdose of −1 Experiments were carried out in accordance with the Ani- pentobarbital (150 mg kg ). The pre-thoracic left and mals (Scientific Procedures) Act 1986 (United Kingdom) right vagus nerves were surgically exposed and 0.5 cm sections were removed and placed in phosphate buffered and Home Office (United Kingdom) approved project and personal licences, and experiments were approved by the saline (155.1 mmol NaCl, 2.96 mmol Na2HPO4, 1.05 mmol Imperial College Animal Welfare Ethical Review Board KH2PO4) adjusted to 8.0 pH with 1 mol NaOH. Sections under project licence PPL 70/7365. A male Wistar rat (body of the left and right sciatic nerves of between 1 and 2 cm weight 350–400 g) was initially anaesthetised with isoflu- from above the knee were also removed. To prepare for rane. Urethane was then slowly administered through a microscopy, the nerves were placed on microscope slides, −1 tail vein (20 mg kg ). The left cervical vagus nerve was stretched until straight, and the nerve ends were fixed exposed and contacted with a stainless steel pseudo-tripolar with super glue. The preparation was covered with PBS. hook electrode of pole distance 1–2 mm for stimulation. Distortions potentially caused by the stretching of the nerves To record from the nerve, a bipolar platinum hook elec- were assumed to lie within the physiological range of trode (pole distance 2–3 mm) was then wrapped around the movement-induced deformations the nerve undergoes in the anterior branch of the subdiaphragmatic vagus nerve with living organism. A commercial 2P microscope was used an Ag/AgCl ground electrode placed in the abdominal cav- for imaging (Scientifca, emission blue channel: 475/50 nm, ity. Distance between recording and stimulating electrodes yellow channel 545/55 nm, 511 nm dichroic, Semrock) was 8–10 cm. See Fig. 1. Mineral oil was applied to each whilst exciting at 950 nm using a Ti-Sapphire laser (Mai Tai site to insulate the electrodes from environmental and prox- HP, Spectra-Physics). imal noise sources. Stimulation of the cervical vagus nerve was performed using a Keithley 6221 current source, con- PyPNS Overview trolled by Standard Commands for Programmable Instru- ments (SCPI) via a custom built Matlab interface. Bipolar Every PyPNS simulation describes one peripheral nerve cuff recordings were achieved with an Intan Technology consisting of an arbitrary number of unmyelinated and myelinated axons, each with a certain diameter and trajectory. Axons can be activated by synaptic input, stimulation intra- and/ or extracellular stimulation. For extracellular recordings, electrodes are positioned along the nerve. The module is organised as several core classes mapped recording to the physiological entities found in a peripheral nerve (shown in Fig. 2 along with the data flow). All objects are managed by the main class Bundle. This is the central object in the Python domain and represents the vagus nerve whole nerve. It contains instances of the Axon-class that define properties needed by the NEURON simulations. Fig. 1 The validation data were obtained through stimulation of a rat Unmyelinated and Myelinated are derived from vagus nerve. A pseudotripolar electrode excited axons at the cervical the parent Axon-class. Each axon is characterised by its vagus nerve, signals were picked up at the subdiaphragmatic vagus diameter and trajectory. To activate axons, Excitation nerve with a bipolar electrode 66 Neuroinform (2019) 17:63–81 .Bundle .spikeTrain .Upstream synaptic input Generation Spiking mem V , V .signal intra current .Recording SFAP CAP .Axon .StimIntra V mem Generation Mechanism extra potential .StimField .create electrode position electrode position Geometry s, r, i(t)) s, r, i(t)) .Extracellular .Excitation .homogeneous Mechanism field .precomputedFEM bundle guide .analytic Fig. 2 The Axon-class is the central object of PyPNS’s r. They are used by both StimField for extracellular stimula- internal information flow. Together with its associated tion and by RecordingMechanism for recording. All classes ExcitationMechanisms it defines the NEURON simulation. are managed in the Bundle-class and supported by helper mod- Extracellular-objects allow the calculation of extracellular ules spikeTrainGeneration, signalGeneration and potentials given current i(t), source position s and receiver position createGeometry. Mechanisms are added to the Bundle. Those can be of Maxwell’s equations governed the extracellular space, either synaptic input (UpstreamSpiking), intracellular neglecting magnetic induction: stimulation (StimIntra) or extracellular stimulation ∂B ∇× E =− 0(1) (StimField). Similarly for recording, electrodes can be ∂t added to the whole nerve as a RecordingMechanism. Further, all media were assumed to be purely resistive, For all interactions with the extracellular space, i.e. so that all changes in current affected the potentials of extracellular stimulation or recording, a model of the the entire space immediately. In Maxwell’s equations this medium defined in a Extracellular-class has to be results in neglecting displacement currents: set. This can be either homogeneous (homogeneous), ∂D an FEM result (precomputedFEM) or an analytically ∇× H = J + J (2) ∂t defined potential distribution (analytic). In the simulation step, the definition of each axon For the brain and in the considered frequency range, the in Bundle is sequentially transmitted to NEURON electro-quasistatic approximation is assumed to be valid via the Python-NEURON-Interface (Hines et al. 2009) (Ham ¨ al ¨ ainen ¨ et al. 1993; Bossetti et al. 2008); previous alongside its associated ExcitationMechanisms. After peripheral nerve simulation studies have built on both the calculation of single axon membrane processes is quasistatic and purely resistive approximations (Raspopovic finished in NEURON, PyPNS computes the extracellular et al. 2012; Struijk 1997; Veltink et al. 1989; Goodall single fibre action potential (SFAP) for the associated et al. 1995). Layers of tissue surrounding the nerve were RecordingMechanisms from membrane currents. Once modelled with a circular symmetry and only one fascicle all axons have been processed, their contributions to the was considered. Extracellular recordings and stimulation overall compound action potential (CAP) are added. did not take into account the electrode-electrolyte interface (see Cantrell et al. (2008) for its effect on stimulation Assumptions and Simpliﬁcations efficiency). Several assumptions were required for the computational Axon Models feasibility and efficiency of our model. Axons were assumed to be independent from each other in their We used the original Hodgkin-Huxley parameters (Hodgkin activity (no ephaptic coupling). Properties such as diameter, and Huxley 1952) for unmyelinated axons. Myelinated ones myelination, and channel densities stayed constant along were based on the model of McIntyre et al. (2002)that the axon length. The electro-quasistatic approximation has originally been developed for peripheral motor fibres Neuroinform (2019) 17:63–81 67 with thicker diameters (5.7–16.0 μm). To match the thinner i(s, t) axons found in the PNS (0.2–3 μm), we extrapolated all diameter dependent parameters to smaller diameters as showninFig. 3. Extrapolated parameters were: (1) the diameters of the different segments – nodes, MYSA (myelin attachment segment), FLUT (paranode main segment), r, t) STIN (internode segment), (2) node distance and (3) the Fig. 4 Axon segments can be interpreted as current point sources. The number of myelin sheaths. Neither model is claimed to extracellular potential φ(r,t) at position r caused by a current i(s,t) exactly match the properties of single neurons found in at position s is determined by current time course scaled with a static the PNS. We aimed to implement a generalised framework potential depending on the extracellular space and the spatial relation in which parameters can be fine-tuned to match specific between source and receiver position datasets. To fit our axon placement method to realistic axon Generation of Axonal Geometry trajectories, fibres in microscopy images were manually traced and segmented into straight sections of length 15 μm. Axons in peripheral nerves are not perfectly straight, but For all traced axons of one nerve, the normalised difference instead follow the nerve path with a certain degree of in direction between consecutive segments c =||a − a || i i+1 tortuosity. To model this in our simulation without defining was calculated. We then compared the c-distribution of the geometry for each fibre manually we iteratively placed imaged, traced axons to the ones obtained from artificial straight axon segments along a previously defined bundle fibres placed at different tortuosity coefficients α and guide, itself composed of longer straight segments. In each ||w||-distributions to select the best fit. For details see step, the axon segment direction a was calculated as Appendix B. a + (1.1 − α) · b + α · w i−1 k i a = , (3) ||a + (1.1 − α) · b + α · w || i−1 k i Extracellular Potentials based on the corresponding bundle guide segment direction Recordings from peripheral nerves capture changes in the b (k ≤ i as bundle guide segments were longer than potential of the extracellular medium caused by membrane axon segments), the previous axon segment direction a i−1 currents. To calculate those changes in PyPNS, axon and a random component perpendicular to the bundle guide segments were interpreted as point current sources, each segment direction w . All vectors have unit length. The causing a potential change in the entire medium. See Fig. 4. parameter α ∈[0, 1] regulates the tortuosity of the axon Potentials generated by all current sources were superposed. and can, together with the distribution of ||w||, be fit to From the electro-quasistatic approximation of the Maxwell geometries measured by microscopy. The factor (1.1 − α), equations, combined with pure resistivity, time and space rather than (1 − α), was chosen to maintain forward axon can be separated in the compound action potential (CAP) growth. See Appendix A for the exact implementation of w calculation: which insures that axons stay within the nerve. φ (s , r,I ) static i ref φ (r,t) = · i(s ,t).(4) CAP i Node, MYSA ref FLUT, STIN 20 200 B The extracellular potential over time at receiver position A C 160 r, φ (r,t), was calculated as the sum over single axon CAP segment contributions. The contribution of one segment 10 120 at position s to the potential recorded at position r was obtained from a known static potential φ (s , r,I ) at static i ref 0 40 reference current I that was then scaled by the temporally ref 10 20 0 10 20 0 10 20 varying membrane current of the segment i(s ,t). diameter (µm) Extracellular stimulation follows exactly the same prin- Fig. 3 Linear and quadratic fits were used to extrapolate the ciple, with stimulation electrodes modelled as assemblies parameters of myelinated axons to smaller diameters. a Diameters of point current sources and axon segments as potential of all segments – nodes, MYSA (myelin attachment segments), receivers. and paranodal elements FLUT (paranode main segment) and STIN (internode segment, see McIntyre et al. (2002) for more information on the model)–were fit quadratically to prevent negative values. Node Point sources were given preference over the line source approxima- distance (b) and number of myelin sheaths (c) were extrapolated tion to enable our efficient precomputation of extracellular potentials. linearly µm 68 Neuroinform (2019) 17:63–81 To further clarify the implications of Eq. 4 on extra- For one t = t , the instantaneous currents i(z | t = cellular recordings, consider a single straight axon on the t ) = i (t − z /CV ) of all segments shown in Fig. 5b 0 i z-axis, so that φ (s, r,I ) becomes φ (z, I ) with are multiplied by the static potential corresponding to their static ref static ref z = (s − r) · e . The translation of membrane current to spatial displacement (Fig. 5c) and added up. recorded single fibre action potential (SFAP) in the extra- If one assumes, as an extreme example, the Kronecker cellular medium is then solely determined by the profile of delta as a profile (φ(z) = δ(z)), the SFAP would have the static potential over longitudinal distance: exactly the same time course as the membrane current. On the other hand a constant profile φ(z) = c will make φ (z ,I ) static i ref φ (t ) = · i (z ,t ).(5) SFAP i the resulting SFAP vanish because of charge conservation ref ( i(t)dt = 0 ⇒ i(z/CV )dz/CV = 0). The recorded action potential is maximal if positive and negative peaks of As Fig. 5 demonstrates, the membrane current of each axon membrane current add up constructively. To quantify when segment is temporally displaced according to its distance z this happens, an active length l of an axon can be defined as and the conduction velocity CV (Fig. 5a): a z z i i l = t · CV , (7) a a i(z ,t) = i t − | z = 0 := i t − .(6) i 0 CV CV with t denoting the time during which an axon segment emits current of constant sign and CV the conduction propagation direction velocity. Membrane current is of the same sign over length l . The match between this length and the range of the profile (z = z − z with φ(z) > 0for z in [z ,z ]) 2 1 1 2 axon will determine the amplitude of the SFAP – in addition to a segment scaling factor depending on the absolute values of φ (z) static in Eq. 4. segment Homogeneous Media membrane current If the medium is assumed to be homogeneous with a constant conductivity σ , the potential φ(r,t) at r caused by a point source of current i(s,t) at s can be analytically t’ z / CV written (see Malmivuo and Plonsey 1995, Chapter 8 or Linden ´ et al. 2014 for reference) as Δt = z / CV 1 i(s,t) φ(r,t) = .(8) 4πσ |s − r| Compared to the formulation in Eq. 4, the static potential term that translates current to voltage here became φ (s, r,I ) 1 static ref = .(9) I 4πσ |s − r| ref z PyPNS implements the homogeneous case as PyPNS.Extracellular.homogeneous. Radially Inhomogeneous Media As the medium surrounding the axons in peripheral nerves is anisotropic and inhomogeneous, the homogeneous assumption is not appropriate. Consequently, no exact (t’) analytical solution for the potential caused by a point current SFAP source exists and numerical methods become necessary. Fig. 5 The impact of the longitudinal profile φ (z) on SFAPs can SFAP In order to reduce computational load, we precomputed be understood by studying the potential caused by a perfectly straight axon recorded at z = 0for t = t . Axon segments of length z exhibit A homogeneous but anisotropic medium can in fact be modelled the exact same current time course except for a delay t = z /CV analytically using a conductivity tensor (Nicholson and Freeman 1975; (a). The potential φ at t = t is then obtained as the sum over SFAP Goto et al. 2010). A combination of inhomogeneities and anisotropy is membrane currents i(z | t = t ) shown in (b), multiplied by the static not feasible, however. potential φ (z ,I )/I (c) static i ref ref z = -2·Δz -2 z = -Δz -1 z = 0 z = Δz z = 2·Δz (z) / I i (t’ - z / CV) static ref i Neuroinform (2019) 17:63–81 69 Table 1 Conductivity of different tissues contained in the simulated potential fields once in a finite element model (FEM) and peripheral nerve; colours correspond to Fig. 6 (Capogrosso et al. 2013; then imported and reused them in PyPNS. This means that Struijk 1997) the computationally expensive field calculation only had −1 to be carried out once per extracellular medium geometry. Tissue Conductivity S m To insure the feasibility of this approach, the extracellular Axons (light blue) 0.5 longitudinal, 0.8 transversal space was modelled using the simplified geometry shown in Epineurium (darker blue) 0.1 isotropic Fig. 6a, with conductivities set to the values given in Table 1. Saline (white) 2.0 isotropic By making the conductivity a function of radius only (i.e. conductivity boundaries were circularly symmetric), a very limited number of unique point source positions exists, each for a different radius (dots in Fig. 6a). We refer to this setup as a radially inhomogeneous medium. Voltage fields φ(x, y, z, r) for different radial point In the FEM solver COMSOL 4.3, the nerve had a length source displacements r were computed. Due to our assump- of 10 cm and was placed in a cubic volume of equal edge tions concerning the medium, steady state simulations were length. The inner nerve radius was set to 190 μm, the sufficient (separation of time and space). The static volt- endoneurium thickness to 50 μm. All inner boundaries had age fields were exported on a grid of x ∈−[1.5, 1.5] mm von Neumann boundary conditions, the potential of the with a step of 0.015 mm, y ∈ [0, 1.5 mm] with a outer border of the cubic volume was set to zero (Dirichlet step of 0.015 mm, z ∈[0, 30] mm with a step size boundary condition). The current entered the mesh at a of 0.03 mm where z is the longitudinal nerve axis and single point. source positions are displaced along x. The fields were imported in PyPNS as a linear 4D spline interpolator. PyPNS afterwards scales the static potentials with cur- rent time courses as given in Eq. 4 with I setto1nA ref in COMSOL. The corresponding mechanism in PyPNS is PyPNS.Extracellular.precomputedFEM.When using an imported potential field, attention has to be paid saline to the source radii used in the FEM precomputation step. The radius selected in PyPNS needs to lie within the pre- computed range. E.g. for stimulation, radii might be larger axons than the nerve radius whereas for recording the precom- epineurium point source puted source radii have to lie within the nerve. Of course, different precomputed fields can be used for recording and stimulation respectively. Longitudinally Inhomogeneous Media In electrophysiological experiments, the nerve does not usually lie within its natural surrounding tissue. Instead, to improve stimulation and recording performance, a cuff or a mineral oil bath increases the extracellular resistivity. recording The medium is in this case no longer longitudinally electrode homogeneous, and any longitudinal shift in current source oil/ cuff position will result in a different potential field. For stimulation, the current source (stimulation electrode) position can be fixed and the precomputation of very few potential fields, each for one electrode radius, characterises the effect of the electrode completely. For recordings, Fig. 6 A circularly symmetric geometry makes it possible to import however, the longitudinal source position necessarily varies, precomputed potential fields. The nerve is modelled as axons (white matter) surrounded by the epineurium. The positions of exemplary as the axon segments extend through the nerve. Therefore, current point sources, each generating one potential field, are shown. to cover all unique axon segment potential fields, a 2D- For radially inhomogeneous media, a line of sources does characterise array of source positions distributed along both radial and all unique fields. For longitudinal inhomogeneities (a), potential fields longitudinal direction must be precomputed, as shown in for a two-dimensional array of point current sources need to be precomputed (b) Fig. 6b. 70 Neuroinform (2019) 17:63–81 Note that without circular symmetry, a volume of A B 0.5 source positions would need to be simulated, making the precomputation infeasible. In this case, the most efficient 0 approach would be to fix the axon geometries for one node -0.5 particular case, perform an FEM simulation for each axon MYSA -1.0 segment position and either export the potential fields for FLUT sum of all the whole space or also fix the electrode positions and -1.5 export the potentials only at the electrodes. This method, 0 0.5 1.0 1.5 0.25 0.5 0.75 1.0 1.25 however, is less universal, much more computationally time (ms) expensive, and involves a lot more coordination between -2 FEM simulation and compartmental axon model. We found that for recording, a reasonable number of current source positions (∼ 20, each using about 40MB of memory) could not abolish interpolation errors between unmyelinated -3 myelinated fields from longitudinally adjacent source positions, causing artefacts in the extracellular action potentials. To generate recordings without artefacts, a smoothed transfer function 0.5 1.0 1.5 2.0 2.5 3.0 3.5 between point current source position and potential in the diameter (µm) cuff was fit to FEM model results. Details are given in Fig. 7 Unmyelinated axons (a) produce a smoother membrane time Appendix C. This transfer function served in PyPNS as a course than myelinated (b) ones. Both axons had a diameter of 3 μm. c variant of PyPNS.Extracellular.analytic. Unmyelinated axons produce a higher current output per distance. The integrated absolute current during a single action potential over axon length is shown Results Axon Models one node of Ranvier (Fig. 7b). In Fig. 7c, the integrated current output is plotted over diameters. Importantly, For thin (< 1 μm) myelinated axons, extrapolated parame- unmyelinated axons emitted more current per distance and ters from the McIntyre model (McIntyre et al. 2002) yielded the signal shapes differed considerably. The unmyelinated bursting behaviour as soon as the fibres were activated current time course was smooth, whereas the myelinated through either synaptic input or stimulation. To prevent this, one was more complex with a sharp peak and a long the potassium channel density at the nodes was increased by lasting recovery. The latter axons contain different segments a factor of 1.5. Node size reduction with diameter achieved (node, myelin attachment segment (MYSA), paranodal the same effect but is not observed (Tuisku and Hildebrand main segments (FLUT)) which all contribute to the overall 1992; Berthold and Rydmark 1983). Potassium channels in current output and thereby caused the more complex shape. the paranodal regions (not included in the original model) See the model of McIntyre et al. (2002) for more details on have been observed physiologically (Poliak and Peles 2003; section types. Roper ¨ and Schwarz 1989) but their integration in the model could not abolish bursting. Myelinated conduction velocity Proﬁles of Extracellular Media −1 (CV) fit experimental data well (CV [ms ] ∼ 5 · d with diameter d in μm). Unmyelinated axons based on Hodgkin- In “Extracellular Potentials” we described the impact of Huxley channels had very low conduction velocities, CV the longitudinal profile φ (z) on the single fibre action static ∼ 0.4 · d , in comparison with expected values of around potentials (SFAPs). Building on these considerations, the 2 · d (Waxman 1980). This is an inherent property of the normalised φ (z)-profiles of our media can be compared. static Hodgkin-Huxley axon model. Figure 8 shows the normalised static potentials over distance As membrane current directly shapes extracellular for all three media and makes the strong impact of the cuff potential recordings, Fig. 7 compares the membrane current insulation obvious. The potential profile became smooth, in time for one unmyelinated axon segment (Fig. 7a) and stretched out in space. The thin nerve surrounded by an insulation acted as two parallel resistors, causing a For a nerve of radius 200 μm, a longitudinal length coverage of linear characteristic. For radial displacements of the current 20.000 μm and a source position grid step of 20 μm this would mean source towards the electrode, a sharp peak emerged (see approximately 300,000 simulations, each taking at least 30 min on a also Fig. 18). We expect fast conducting axons with long single core of a state of the art workstation, totaling to a computation time of over 17 years. The result would occupy 12 TB of RAM. active length l to best match this large range profile. integrated` current (nAmS / µm) membrane current (nA) Neuroinform (2019) 17:63–81 71 axons) a negative main and two positive entrance and exit 1 homogeneous displacement 0 µm radial inhom. displacement 180 µm peaks. cuff Figure 9c compares the SFAP amplitude for unmyeli- nated and myelinated axons over diameters and media. Whilst the SFAP amplitude of unmyelinated axons was 0.4 similar and even higher than myelinated SFAPs in homo- -0.4 0 0.4 geneous and radially inhomogeneous media, myelinated -15 -10 -5 0 510 15 longitudinal distance z (mm) A B homogeneous Fig. 8 Compared to the homogeneous and radially inhomogeneous radially inhomo. extracellular media the cuff insulation caused a much softer and cuff strikingly linear characteristic. Electrode radius was 235 μm. The profiles are shown for two radial axon displacements in solid and -2 dashed lines respectively -4 The other two media had a different, much narrower -6 characteristic. Radial inhomogeneities produced a slightly 10 20 30 0.5 1.0 1.5 2.0 2.5 smoother potential profile compared to the homogeneous 0.1 medium but differences remained small. Both profiles 0.2 decayed a lot steeper with longitudinal distance than in 0.1 -0.1 the cuff and were therefore expected to better suit slower -0.2 0 conducting axons with a shorter l . -0.3 -0.1 -0.4 Extracellular Single Fibre Action Potentials -0.2 -0.5 -0.6 20 0.6 0.8 1.0 1.2 1.4 16 18 In Fig. 9, the effect of different extracellular media on the resulting SFAPs can be compared. The axons were time (ms) activated by intracellular stimulation and recorded with a 5 1 C monopolar circular electrode at radius 235 μm. In the cuff 10 medium, the electrode was placed centrally as shown for one point electrode in Fig. 6b. Figure 9a shows extracellular potentials from a single unmyelinated fibre. Between the three different media, mostly amplitude varied with only slight differences in shape. Insulating the nerve with a -1 cuff increased the potential by a factor of about ten and caused a narrower signal shape. In addition, an entrance homogeneous -2 and an exit peak at the sides of the cuff arose that were unmyelinated radial inhom. myelinated cuff not present in the two longitudinally homogeneous media. The radially inhomogeneous medium slightly stretched the 0.5 1.0 1.5 2.0 2.5 3.0 3.5 action potential in time which can be explained by the diameter (µm) preference of current to flow along the nerve rather than transversally (compare to profile in Fig. 8). Fig. 9 Unmyelinated and myelinated SFAPs showed different sen- sitivities towards the extracellular space. In the upper plots (a, b), The SFAP of myelinated axons in Fig. 9b was much diameters were set to 3 μm. a The main peak of unmyelinated fibres more strongly affected than the unmyelinated fibres when mostly varied in amplitude over media, not in shape. In cuff insulated insulating the nerve. Whilst the difference between homo- nerves, additional side peaks emerged. b Myelinated fibres produced geneous and radially inhomogeneous medium remained much higher and longer lasting SFAPs in the cuff insulated medium. Both axons had diameter 3 μm, were placed centrally within the nerve small, myelinated SFAP amplitude increased by a factor of and recorded by a circular monopolar electrode with radius 235 μm. about 20 in the cuff and shape was changed radically. The −1 Conductivity of the homogeneous medium was set to 1 S m .Lower recorded signal lasted longer and had (as for unmyelinated row shows zoomed-in plots. c The amplitude boost achieved by cuff insulation was stronger for myelinated than for unmyelinated axons Electrode radius was chosen to be slightly smaller than nerve radius over the whole diameter range. For the other two media, unmyelinated to maintain a small distance to the non-conducting insulation layer SFAPs produced stronger SFAP amplitudes at diameters above 0.5 and surrounding the nerve. 1 μm respectively normalized (z) static extracellular voltage peak to peak (µV) extracellular voltage (µV) 0.15 0.45 0.30 0.75 0.60 72 Neuroinform (2019) 17:63–81 fibres achieved much stronger amplitudes following cuff their lower conduction velocity and therefore shorter active insulation – even though their membrane current output is length produce the strongest signals for (theoretic) cuff substantially lower compared to unmyelinated axons (see lengths of about 1 mm. Whilst those are most likely not Fig. 7). This difference in reaction to the cuff medium achievable, medium lengths of about 1 cm seem reasonable between fibre types can be explained by the two different according to our simulation. The amplitude of myelinated mechanisms through which cuff insulation changed SFAP axons keeps rising until the investigated maximum cuff amplitude. The first one is the increased extracellular resis- length of 10 cm but starts saturating at about 1 cm. PyPNS tance. Current cannot freely dissipate into the surrounding therefore predicts an ideal cuff length in this order. Results tissue but needs to flow along the thin nerve. As membrane will vary for a more accurate unmyelinated axon model, current was modelled to be independent of the medium, an where higher conduction velocities would be expected to increase in extracellular resistance equaled an increase in increase the ideal cuff length. extracellular potential. This effect increases SFAP ampli- tude equally for both fibre types. The second one – that can Compound Action Potentials explain the difference in amplitude gain between fibre types – is the match of active length (as defined in Eq. 7)and For validation, we aimed at reproducing experimental cuff dimension (equal to range of the profile; 20 mm in this recordings from the stimulated rat vagus nerve in PyPNS. case) as detailed in “Extracellular Potentials”. For a myeli- To this end we obtained diameter distributions and fibre nated axon of diameter 3 μm the active length evaluated to counts from microscopy images (Prechtl and Powley 1990) −1 = 7.5 mm, an unmyeli- approximately 0.5 ms · 15 ms as summarised in Table 2 and set the geometry of the nerve nated axon of this diameter only had an active length of and the recording electrodes so as to match the experimental −1 about 0.5 ms · 1ms = 0.5 mm. Figure 9c demonstrates set-up. Outer and inner radius were set to 240 μmand the matching effect between myelinated axons and the cuff 190 μm respectively; a circular bipolar electrode of radius over all diameters. 235 μm and pole distance 3 mm (20 recording positions per pole) surrounded the nerve. Axons were placed centrally Effects of Varying the Cuff Length and were activated intracellularly; due to the difference in stimulation threshold between fibres types, the entire As a tool for Bioelectronic Medicines, PyPNS should help population of myelinated and only a small fraction of the design of peripheral nerve interfaces. Here we take a unmyelinated axons (∼ 20% of 10,000) was triggered. As look at the impact of cuff electrode length on the recorded unmyelinated fibres based on Hodgkin-Huxley channels signal amplitude. Figure 10 demonstrates how unmyelinated had very low conduction velocities, we corrected their and myelinated fibres require different cuff lengths for SFAP timings. The nerve was insulated with mineral oil a maximal SFAP amplitude. Unmyelinated fibres with in the experimental recording. Therefore only the cuff medium should produce similar extracellular signals in the simulation. The results from homogeneous and radially 0.20 0.10 A B inhomogeneous media are presented as well in the following 0.25 0.14 0.32 0.21 for comparison. 0.41 0.30 0.53 0.43 Figure 11 plots simulation results in all media against the 0.67 0.62 experimental data and demonstrates a reasonable agreement 0.86 0.89 1.1 1.3 between simulation and experiment in the time domain. This 1.4 1.8 match naturally only held for the cuff insulated medium 1.8 2.6 2.3 3.8 – homogeneous and radially inhomogeneous media led to 2.9 5.5 very low extracellular potential amplitudes as expected from 3.7 7.9 4.7 11.0 their lower tissue resistance. The signal segment between 5.9 16.0 7.6 23.0 A- and C-fibres from 25 to 40 ms can be attributed to B- 34.0 9.7 fibres and was not compared to the simulation as PyPNS 48.0 12.0 70.0 16.0 only models A- and C-fibres. 100.0 20.0 Especially the signal portion caused by myelinated fibres (Fig. 11b) matches the experiment well in peak amplitudes, axon diameter (µm) area, zero crossings and overall duration. See Table 3 for Fig. 10 Unmyelinated and myelinated axons have different ideal a quantitative comparison. Unmyelinated axons (Fig. 11c) cuff lengths. For unmyelinated fibres (a), very short ranges around also produced a CAP comparable to the experiment in 1 mm produce the maximal amplitude. For myelinated ones (b), the both amplitude and timing although the comparison is more amplitude only rises with length. Contour lines show the peak-to-peak difficult as the signal to noise ratio in the experimental data amplitude in μV 4.0 6.0 2.0 8.0 10.0 12.0 14.0 cuff width (mm) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.1 1.2 1.3 1.4 1.5 1.0 1.4 1.8 2.2 2.6 3.0 3.4 3.8 Neuroinform (2019) 17:63–81 73 Table 2 Axon number and Type # diameter (μm) properties set in the simulation for comparing model results Unmyelinated 2000 ∈ (0.2, 1.52) distribution from Prechtl and Powley (1990) with experimental recordings Myelinated 200 ∼ N (1.7, 0.4) (Prechtl and Powley 1990) is lower for unmyelinated than for myelinated fibres. Table 3 signal proportion in our experimental data (black lines in summarises how area and amplitude of the experimental Fig. 12a) had an overall flat profile with a main peak recording are larger than in the simulation and that there (lower plot) at around 500 Hz. This characteristic was occur considerably more zero crossings in the experiment. approached to a certain extent by our model. The spectra The noise present in the experiment will be accountable for in all three media have slighly earlier peaks below 500 Hz a share of those crossings. Of course, the Hodgkin-Huxley but homogeneous and cuff medium result followed the model of the unmyelinated axons did not to exactly match characteristic of the experiment well between 0 and 2 kHz the properties of the rat vagus nerve C-fibres, therefore before decaying further below − 20 dB from there. We differences in the extracellular recordings were expected. surmise that the high frequency content of the experimental In Fig. 12 simulation and experiment can be compared data may be be caused by high frequency noise from the in the frequency domain for both fibre types. The similarity recording process. Meaningful, spike-event related signal between simulated and experimental data was comparable components from experimental recordings usually stay to the match in time domain for both myelinated and below 2 kHz (Diedrich et al. 2003). unmyelinated fibres. The spectrum of the unmyelinated The experimental spectrum of myelinated fibres (Fig. 12b) was dominated by low frequency power below 2 kHz with a peak at about 500 Hz. Our simulation result in experiment the cuff medium matched this characteristic for frequencies homogeneous over the whole frequency range, although showing a later radial inhom. peak around 1 kHz. The other two media led to a flat char- cuff acteristic with a larger amount of high frequency power and 0 less low frequency power. This could be predicted from the SFAPs in Fig. 9 where the myelinated SFAP is much wider -20 in the cuff than in the other media. -40 In conclusion, the experimentally obtained frequency characteristic of both axon types was reasonably matched 04 20 0 60 80 100 120 by our simulation for the cuff medium. B C Fitting Axon Tortuosity to Experimental Data 0 0 In order to obtain axon shapes close to reality, we compared -20 the distributions of axon segment direction changes c -40 -10 as detailed in methods “Imaging of Peripheral Nerve 5 15 20 30 50 70 90 110 time (ms) Table 3 Quantitative comparison between compound action potentials Fig. 11 a The simulated compound action potential in the cuff medium from experiment and simulation (cuff medium) approaches the experimental recording well in the relevant signal segments. As expected, homogeneous and radially inhomogeneous Feature Experiment Simulation media lead to much weaker signal amplitudes. For the experimental recording, the grey underlying area indicates the standard deviation Myelinated axons over the 10 stimulation repetitions. See Table 2 for axon properties. Distance between stimulation site and bipolar electrode (3 mm pole Area (μV ms) 115 83.2 distance, 235 μm radius) was 8 cm. All axons were activated by Peak-to-peak voltage (μV) 57.5 50.3 intracellular stimulation. The timing of unmyelinated SFAPs was Zero crossings 50 45 adapted to regular conduction velocity values assumed in mammalian −1 peripheral nerves (CV = 1.4 · d,CVinms ,din μm). b The Unmyelinated axons signal from myelinated fibres, which arrive first, appears similar to the Area (μV ms) 147 93.9 experiment. c The unmyelinated signal segment matches the amplitude Peak-to-peak voltage (μV) 25.2 14.0 and duration of the experimental recording as well. The signal-to-noise ratio of the recordings is much worse for unmyelinated fibres, however, Zero crossings 271 133 as the amplitude of their SFAPs is low extracellular voltage (µV) 74 Neuroinform (2019) 17:63–81 0 all ten runs is plotted in the lower row. Unmyelinated SFAPs (Fig. 15a) were especially sensitive to tortuosity. -20 They developed complex, long lasting signals, especially -40 A B in homogeneous and radially inhomogeneous media. When insulating the nerve, the amplitude of the main SFAP peak 02468 10 02468 10 became very weak at high tortuosity whilst many small side peaks arose, giving the signal a noisy appearance. -10 recording -20 Myelinated fibres (Fig. 15b) were more robust to tortuosity hom. radial inhom. -30 – their SFAP shape remained invariant at low and medium cuff 01.0 2.0 3.0 α-values. Only high degrees of tortuosity could change 0 0.5 1.0 1.5 2.0 frequency (kHz) signal timing and shape; as for unmyelinated axons, the cuff isolated medium let the signal become noisy. Fig. 12 In the frequency domain, simulation and experiment did not The overall effect of tortuosity to change SFAP shape match equally well for both fibre types. a For unmyelinated axons, the simulation did not perfectly approach the experimental spectrum in can be understood by looking at Eq. 10 (same as Eq. 5) any medium with best results for the cuff. b The simulated frequency and changing it as in Eq. 11 where s is the distance along characteristic of myelinated axons in the cuff insulated medium was the axon. The longitudinal distance z(s) along the nerve close to reality becomes a function of s, shaped by tortuosity. Differences in the potential φ depending on the radial displacement Tortuosity” for imaged mouse sciatic and vagus nerve. See of the axon were neglected here. The potential profiles Fig. 13 for fluorescence microscopy images and traced of the extracellular media (see Fig. 8) are then both axons. stretched (z(s) ≤ s) and distorted in a degree dependent on In Fig. 14, the obtained direction change distributions tortuosity. Different axons show different susceptibilities to from microscopy (Fig. 14a) are compared to the ones this distortion because of their different active lengths. If of simulated axons (Fig. 14b) alongside a few example the active length is large compared to the spatial frequency axons in space (Fig. 14c). In Fig. 14a, the higher of the tortuosity-induced profile distortion, variabilities in tortuosity observed in the vagus nerve is visible from φ(z(s)) are shadowed. Axons with shorter active length the wider distribution of segment direction changes (c- respond to those variabilities making their SFAPs noisier. values) compared to the sciatic nerve. A set of direction This explains the difference in susceptibility between axon change distribution obtained at different parameters (||w||- types. distribution and α)in PyPNS is shown in Fig. 14b. When comparing to Fig. 14a, a Gaussian ||w||-distribution z φ = φ(z ) · i t − (10) SFAP i produced c-distributions the most similar to microscopy CV data. The sciatic nerve then corresponded to an α-value of about 0.6, the vagus nerve had a wider c-distribution as ⇒ φ = φ(z(s )) · i t − (11) SFAP i CV its axons were curvier, corresponding to a higher α.When comparing the trajectories in Fig. 14c from uniform (upper To quantify the influence of α on the heterogeneity of SFAP plot) and Gaussian (lower plot) c-distributions, it can be shape, we calculated the pairwise cross-correlation seen how the normal distribution of random vector length ||w|| leads to both a slightly smoother trajectory and rare strong direction changes, especially for high α-values. (f g)(τ ) = f(t) · g(t + τ) dt (12) −∞ Recording from Tortuous Axons between normalised SFAP waveforms s from repeated α,i A more complex axon trajectory caused more complex simulation runs whilst keeping α, fibre type, and medium SFAPs, as it can be seen in Fig. 15. Upper plots of Fig. 15 unchanged. The mean maximum cross-correlation over all show superposed SFAP shapes for ten individual axons in waveform pairs described shape homogeneity: both radially inhomogeneous (upper row) and cuff medium (second row). A summary of SFAP similarity between n−1 n−1 c = max(s s ). (13) α α,k α,l n · (n − 1) k=0 l=k+1 The homogeneous medium produced very similar SFAP shapes in time as the radially inhomogeneous medium. Therefore the traces are Figure 15 confirms that a higher α caused higher not shown. The similarity summary in the lower row was plotted for all three media. differences in shape (lower c ). As expected from the time normalised power (dB) Neuroinform (2019) 17:63–81 75 Fig. 13 Fluorescence microscopy images of the mouse A B sciatic and vagus nerve both show slight tortuosity in their axon trajectories. a The thick myelinated fibres in the sciatic nerve appear very parallel. b The thinner axons in the vagus take a more curvy trajectory. Several manually traced fibres used to fit the model are highlighted in orange 1 1 100µm 100µm course, myelinated SFAPs remained similar even for large regimes, very tortuous unmyelinated axons were the most α whilst unmyelinated ones lost their similarity. Note that consistently triggered. Stimulation of myelinated axons on this measure does take into account differences in timing or the other hand was successful from low amplitudes of about amplitude. 150 nA and at almost any higher current at all degrees of tortuosity. In Fig. 16c a minor increase in stimulation Stimulation of Tortuous Axons threshold with tortuosity becomes visible. Therefore, tortuosity affected the activation ratio of unmyelinated Not only the recording from but also the stimulation axons stronger than it did for myelinated ones. of axons is influenced by their trajectory. Figure 16 plots the activation ratio of unmyelinated and myelinated fibres for different degrees of tortuosity and different Discussion stimulation amplitudes. It shows that firstly, regardless of tortuosity, unmyelinated axons had much higher stimulation The open-source simulation framework that we have thresholds than did myelinated ones. Second, unmyelinated proposed here for the first time integrates compartmental fibres had an optimal stimulation current with a smooth axon models and numerically solved extracellular space decrease in stimulation efficiency for higher and lower models into a single environment. To make the import current amplitudes. In the low amplitude range (< 3mA), of precomputed voltage fields feasible and efficient, the perfectly straight axons are activated best. In higher current modelled media needed to fulfil certain constraints. One B C 1 1 1 sciatic uniform vagus Gaussian -200 0 0 0 0 .126.96.36.199 1 0 0.5 1 0 0.5 1 0 0.5 1 1 1 1 1 -200 0 0 0 0 .188.8.131.52 1 0 0.5 1 0 0.5 1 0 0.5 1 c (unitless) c (unitless) length (µm) Fig. 14 The axon placing algorithm result (b, c) was fit to tortuos- the random component ||w|| in Eq. 3 for αs of 0.2, 0.6, and 1.0. c ity of microscopy imaged fibres (a). a Direction change distributions Example axon trajectories in space for uniform (upper) and Gaussian (c-distributions) for vagus and sciatic nerve. b c-distributions in the (lower) ||w||-distributions at α-values of 0.2, 0.6, and 1.0 simulation for both normally and uniformly distributed amplitude of normalised count height (µm) 76 Neuroinform (2019) 17:63–81 0.1AB 0.1 -0.1 -0.2 -0.3 -0.1 -0.4 21 22 23 24 24 26 45 50 55 60 0.5 1.0 1.5 1.0 1.5 0.5 1.0 1.5 0.1 0.2 -0.1 -0.2 -0.2 -0.4 -0.3 -0.6 -0.4 02 02 02 4 21 22 23 24 20 40 25 50 75 time (ms) 1.00 1.0 0.8 0.98 0.6 homogeneous radial inhomogeneous cuff 0.4 0.96 0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 random direction coefficient Fig. 15 Unmyelinated axons were more sensitive to tortuousity in their main SFAP peak almost disappeared for α = 1.0. b Myelinated axons SFAP shapes than myelinated ones. Tortuosity parameter α was set mostly differed in timing in the radially inhomogeneous extracellular to 0.2, 0.6, and 1.0 for the signals shown in the first two rows. Grey space, and not as much in shape. In the cuff, noisiness only arose at lines correspond to SFAPs of different trials (axon geometries) at the high tortuosity values. In the lower plots, the mean maximum pairwise same parameters. a Unmyelinated axons produced SFAPs differing cross-correlation gives a quantitative confirmation of the higher sus- both in timing and shape for the non-insulated nerve (radially inhomo- ceptibility of unmyelinated axons to change their SFAP shape in the geneous medium), even for small α-values of 0.2. In the cuff-insulated presence of tortuosity. Note the different ordinate scales nerve (middle row) their signals became noisy at low α-values and the was the geometry that had to be circularly symmetric. In terms of axon geometry, we implemented a simple Whilst presenting a strong simplification of the extracellular iterative placement mechanism that was fit to microscopy medium, this implementation can be seen as a generic data. To our knowledge this is the first implementation peripheral nerve in which axons can still cluster to of such automated shape generation for peripheral nerve fascicles. Another constraint concerned material properties. models. It enabled us to investigate the influence of tortu- Displacement currents and therefore frequency dependence osity on recordings and stimulation efficiency and indicated of the tissues was not accounted for. Such frequency that perfectly straight axons are an oversimplification. Our dependence certainly exists to a certain extent. It can arise simulation predicted that SFAPs become more complex from macroscopic structures at constant material properties with increasing tortuosity – an effect that is exploited by (dielectric constant and conductivity σ ) – the epineurium spike sorting algorithms which differentiate single units can for instance act as a capacitor. In addition, polarisation from their SFAP shape. For now, axons were positioned at different microscopic levels (Bedard ´ and Destexhe 2009; independently from another. As a next step, fibre tra- Martinsen et al. 2002) can render the material properties jectories could be correlated as observed in microscopy and σ themselves frequency dependent. Such dielectric images. dispersion is observed in most biological tissues (Gabriel The modular nature of our model allows for an easy et al. 1996). Ephaptic coupling and neurodiffusive effects comparison of different extracellular media. Building on were neglected as well. this functionality, we identified an ideal cuff length for pairwise correlation extracellular voltage (mV) Neuroinform (2019) 17:63–81 77 intracellular resistivity could be adapted to reach the A 100% expected conduction velocity but we chose to leave them .05 at their physiological values. If more accurate axon models become available, they can be integrated into PyPNS. Several steps to improve the model beyond the mentioned limitations are imaginable. First, axons are currently simulated sequentially. For the simulation of closed loop 0% systems interacting with peripheral nerves, the simultaneous simulation of all nerves would be preferable. Second, axon membrane sections only need to be simulated if they are either stimulated or recorded from extracellularly, otherwise the calculation of their highly uniform membrane processes is unnecessary and time consuming. In order to eliminate computational overhead, one could introduce an abstract layer into the simulation in which the position change of spikes along axons is computed based on a known conduction velocity profile. Only for axon segments relevant to stimulation or recording, would the full membrane process be simulated. In conclusion, a unified computer model of a generic .70 peripheral nerve was developed. It combined an efficient calculation of extracellular potentials in inhomogeneous media from precomputed potential fields with compart- stimulation current amplitude (mA) mental axon models in a convenient Python module. The Fig. 16 Unmyelinated axons have higher stimulation thresholds model was validated against experimental data and used to and are activated less reliably than myelinated ones. Both bundles investigate the effects of conductivity inhomogeneities on consisted of 15 axons with diameter 3 μm and were stimulated with a bipolar electrode of radius 235 μm and pole distance 1 mm using a amplitude and frequency content as well as the influence biphasic pulse of frequency 1 kHz, duration 1 ms and duty cycle 0.5. of axon tortuosity on both recording and stimulation. We The extracellular medium was a nerve of diameter 240 μmbathedin hope that the simulation framework presented here, PyPNS, oil. a Unmyelinated axons started to be activated at 1 mA and showed becomes a useful tool for researchers working on periph- a peak in activation ratio at about 3 mA. b Myelinated fibres had a sharp activation threshold at a much lower current of about 0.15 mA eral nerves, nerve stimulation, and its medical applications, and stayed activated for higher currents. Only when incrementing the and envision that the toolbox could be augmented by mul- stimulation current in very small steps of about 10 nA c a slight tiple branches, organ models, and a variety of specific axon tortuosity-induced increase in stimulation threshold became visible for models matched to fibre types found in different parts of the them as well peripheral nervous system, to facilitate this. peripheral nerve interfaces. We also showed how the Information Sharing Statement long temporal extent of SFAPs in cuff-insulated media – especially for myelinated axons – makes differentiation of The latest version of our toolbox PyPNS (RRID:SCR 016336) single fibre contributions difficult as overlaps are probable. can be accessed over GitHub: github.com/chlubba/PyPNS. Overall a cuff therefore increased amplitude but reduced The version this paper is based on is stored on Zenodo: recording precision. https://doi.org/10.5281/zenodo.1204836. Scripts for the fi- One limitation of the current NEURON simulation is gures are as well maintained on GitHub: github.com/chlubba/ the unmyelinated axon model. Its conduction velocity was PyPNS-PaperFigures. too low compared to that reported for mammalian axons. For the overall CAP, the velocity needed to be corrected. Acknowledgements This work was funded by EPSRC grant Still, the Hodgkin-Huxley parameters are the accepted EP/L016737/1 and Galvani Bioelectronics; we further acknowledge support from grant EP/N014529/1. We would also like to thank Peter standard model for unmyelinated axons and more detailed Quicke, Subhojit Chakraborty, and June Kyu Hwang for the two- C-fibre models (e.g. Sundt et al. 2015) do not achieve photon microscopy images and Thomas Knopfel ¨ for the ChAT-Cre significantly higher conduction velocities either. Parameters FLEX-VSFP 2.3 mice used to obtain them. We further thank Siwoo of the current model such as membrane capacitance or Jeong for testing the framework. random direction coefficient 0.03 1.0 0.04 1.3 0.05 1.7 0.06 2.2 0.08 2.8 0.10 3.6 0.13 4.6 0.16 6.0 0.21 7.7 0.27 10.0 0.34 13.0 0.43 17.0 0.55 22.0 0.70 28.0 0.89 36.0 1.1 46.0 1.5 60.0 1.8 77.0 2.4 100.0 3.0 78 Neuroinform (2019) 17:63–81 Compliance with Ethical Standards Conﬂict of interests CH Lubba received funding from Galvani Bioelectronics. i+1 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons. org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to a (1.1 - ) b i k the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Fig. 17 If bundle b and current axon segment a have a fixed relation, k i e.g. parallel, the expected distribution of segment direction differences c =||a − a || can be easily obtained from the distribution of ||w|| i i+1 (P ) by their geometrical relation Appendix A: Calculation of the Random Component of the Axon Placing Algorithm The random vector w in Eq. 3 is split into an inward Appendix C: Fitted Cuﬀ Transmission pointing radial w and a tangential component w (14), rad tan Function both weighted independently with a weight drawn from a distribution P (15, 16). P can be either a uniform For extracellular recording in a cuff, a transfer function distribution U (−1, 1) between −1and1oranormal between current point source position and the potential at an distribution N (μ, σ ) with μ = 0and σ = 0.33 (sigma electrode longitudinally centrally placed in the cuff was fit. chosen to have 99.7% of all values in the range [−1, 1]). Input variables describe the spatial relation between source When the radial distance between axon segment and bundle and receiver position. As apparent from Fig. 18, the relation guide d approaches the bundle radius r , the radial bundle is strongly linear with an additional peak for low distances component w becomes more inward directed (16)and rad between current source and potential receiver – facilitating thereby ensures that the axon stays inside the nerve. One linear implementation of the correction factor e is shown the fit of a transfer function. in Eq. 17. The parameter r defined the relative radius The static potential was therefore described as a corr from which on the correction should begin, set to 0.7 in our linear component f (z) plus a non-linear peak f (z). lin peak simulation; e , set to 2 by default in PyPNS, limits the Equations 21–23 implement this characteristic for φ in max correction. mV with variables r radial axon displacement in m, α axe angle between axon displacement direction and electrode β · w + β · w rad rad tan tan perpendicular on the nerve centre in rad and z longitudinal w = (14) ||β · w + β · w || rad rad tan tan distance between electrode and axon in m. The transfer β ∼ P (15) tan function is parametrised with r for the inner radius of the β ∼ P − e (16) rad nerve in m, a and b for maximum peak amplitude and d/r − r bundle corr steepness, c for maximum of triangular component and d e = min(1, max(0, )) · e (17) max 1 − r half the cuff length in m. The left and right borders of the corr f (z)-function were smoothed with a moving average of lin width c/20. Appendix B: Generation of Simulated c-Distributions f (z) = max(0,c · (1 −|z/d|)) (19) lin To directly translate ||w||-distributions (P ) to distributions of the normed difference in direction of consecutive axon f (z) = (20) peak |z|+b segments c =||a − a || projected onto a 2D-plane, we i i+1 made the simplifying assumption that b and a are aligned. k i By doing so || a + (1.1 − α) · b ||) (see Eq. 3) becomes i k f (r ) = min(1,(r /r ) ) (21) peak,r axe axe 1 (2.1 − α) ·||a ||. Following Fig. 17, it is easily shown that then ||w|| relates to c as f (α) = max(0,(1−| mod (α +π, 2π)−π |)/π·5) (22) peak,α 1 ||w|| α c = 2 ·||a|| · sin · arctan . (18) 2 ||a|| 2.1 − α φ(z, α, r ) = tr(z) + p(z) · pf (r ) · pf (α) (23) axe r axe α Neuroinform (2019) 17:63–81 79 Fig. 18 An analytic 45° transmission function 15° implements the relation between 0° current source position and potential for recording in a cuff electrode. In the shown case, a nerve of diameter 480 μmina cuff of 2 cm length was 0.1 simulated in the FEM model. Functions are displayed for three different angles between 0.05 the perpendiculars of source and electrode position onto the bundle guide respectively 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 radius = 0 mm FEM 0.12 radius = 0.09 mm fit radius = 0.18 mm 0.1 0.08 0.06 0 0.1 0.2 0 0.1 0.2 0 0.1 0.2 distance from cuff center (cm) Table 4 Parameters of the fitted transmission function for cuff Nature Reviews Drug Discovery, 13(6), 399–400. https://doi.org/ recordings 10.1038/nrd4351. Bokil, H., Laaris, N., Blinder, K., Ennis, M., Keller, A. (2001). Parameter Value Ephaptic interactions in the mammalian olfactory system. 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