Dependence of Excitability Indices on Membrane Channel Dynamics

Noname manuscript No. (will be inserted by the editor)

Dependence of excitability indices on membrane channel dynamics, myelin impedance, electrode location and stimulus waveforms in myelinated and unmyelinated ﬁbre models

Thomas Tarnaud · Wout Joseph · Luc Martens · Emmeric Tanghe

Received: date / Accepted: date

Abstract Neuronal excitability is determined in a com- Keywords Electrostimulation · A-fibres · C-fibres · plex way by several interacting factors, such as mem- strength-duration · rheobase brane dynamics, fibre geometry, electrode configuration, myelin impedance, neuronal terminations,. . . This study aims to increase understanding in excitability, 1 Introduction by investigating the impact of these factors on different models of myelinated and unmyelinated fibres (five Two different approaches have been investigated in lit- well known membrane models are combined with three erature to model neuronal activity: mathematical IF electrostimulation models, that take into account the (“integrate-and-fire”) models and detailed biophysical spatial structure of the neuron). Several excitability models of the Hodgkin-Huxley (HH) type [12]. The for- indices (rheobase, polarity ratio, bi/monophasic ratio, mer type of model has a low computational cost [16], al- time constants,. . . ) are calculated during extensive pa- lowing for large scale simulations, such as performed by rametersweeps, allowing us to obtain novel findings on Izhikevich and Edelman [17]. It was furthermore noted how these factors interact, e.g. how the dependency of in literature that the IF-model is more realistic than excitability indices on the fibre diameter and myelin a single-compartment Hodgkin-Huxley model, in which impedance is influenced by the electrode location and channel properties are constrained by patch-clamp mea- membrane dynamics. surements [3]. However, a multi-compartmental model It was found that excitability is profoundly impacted of the HH-type has more empirical content (e.g. descrip- by the used membrane model and the location of the tion of ionic currents) and is potentially more realistic neuronal terminations. The approximation of infinite than IF-models. Furthermore, computational models of myelin impedance was investigated by two implemen- the Hodgkin-Huxley type have biophysical meaning- tations of the spatially extended nonlinear node model. ful and measurable parameters. E.g., in the blue brain The impact of this approximation on the time constant project HH-type models are used to reproduce in vivo of strength-duration plots is significant, and most im- and in vitro experiments without parameter tuning [19]. portant in the Frankenhaeuser-Huxley membrane model In this paper, multi-compartmental HH-type mod- for large electrode-neuron separations. Finally, a multi- els are used to investigate the dependence of excitability compartmental model for C-fibres is used to determine indices on the used equation set for the membrane dy- the impact of the absence of internodes on excitability. namics and on the electrode location. Each membrane model is combined with one of the multi-compartmental models listed in Table1, yielding a total of 15 differ- T. Tarnaud ent numerical electrostimulation models. An overview Tel.: +32-471-85-30-53 of the five used membrane models is shown in Table2. E-mail: [email protected] The Hodgkin-Huxley (HH) model was the first mathe- T. Tarnaud · W. Joseph · L. Martens · E. Tanghe matical (physiological) model to describe the excitable Ghent University-IMEC, Technologiepark 15, 9052 Zwij- membrane and was obtained from voltage-clamp ex- naarde, Belgium periments on the squid giant axon [13]. Since then, var- 2 Thomas Tarnaud et al.

Model Description Applicability SENN-MA Crank-Nicolson implementation of spatially-extended non-linear node Myelinated axons model [30]. All nodes non-linear. Myelin capacitance and conductance neglected. SENN-M Compartmental representation of myelin included Myelinated axons C-FIBRE Unmyelinated ﬁbre model Unmyelinated axons

Table 1 Spatially extended electrostimulation models ious researchers have derived similar electrodynamical dynamics, fibre diameter, myelin impedance, electrode models from different types of excitable tissue. Franken- location and stimulus waveform on excitability indices haeuser and Huxley [10] (FH-model), Chiu et al. [7], (e.g., strength-duration and rheobase) is important. and Sweeney et al. [38] (CRRSS-model), Schwarz and It was noted by McNeal that the approximation Eikhof [35] (SE-model), and Schwarz et al. [36] (SRB- of negligible conductivity and capacity values of the model) derived their models from myelinated frog, rab- myelin layer is the cause of the most serious error in bit, rat, and human axons respectively. These mem- his model [22]. This assumption, that we will call the brane models (summarized in Table2) are selected, be- myelin approximation (MA), is implicit to the SENN- cause they are the “classical” models of membrane dy- model as proposed by Reilly et al. [30]. One of the goals namics, that are often used for applications in which no of this paper is to assess the impact of the myelin ap- specialized model exists or is accepted. Because of their proximation on excitability indices. To this end, a sec- widespread use in computational neuroscience (e.g., in ond model (SENN-M) is used that incorporates addi- computational studies of the cochlear nerve [5,6, 23] or tional sets of differential equations to account for the for modeling conduction block [39, 42, 43]), these mem- finite myelin parameters, that are distributed along the brane models are especially interesting for this study, internodes. In this model, the capacity and conductivity which aims to determine the influence of membrane dy- values of myelin reported by Tasaki are used (Appendix namics on neuronal excitability. A)[40]. The first multi-compartmental model (SENN-MA) We note that also the SENN-MA model departs listed in Table1, describes a myelinated axon and is from the original SENN-model in this study, after sub- an implementation of the spatially extended nonlinear stituting the Frankenhaeuser-Huxley equation set with node (SENN) model [30]. It extends the original elec- one of the membrane dynamical models from Table2. trostimulation model proposed by McNeal, by including This procedure was for instance used by Cartee to com- non-linear FH-equations for all nodes of Ranvier [22]. pare the effect between different membrane models on The SENN-model has been widely used by researchers the summation and refraction properties of the modeled to study electrostimulation of myelinated fibres and is neuron [5]. used by IEEE C95.6 to derive basic restrictions for elec- Finally, a model for unmyelinated fibres (termed “C- tromagnetic exposure [15]. As a consequence, ICNIRP FIBRE” in Table1) is obtained by carefully altering the (International Commission on Non-Ionizing Radiation geometrical and electrophysiological parameters of the Protection) has implicitly adopted results of the SENN SENN-M model. Models of unmyelinated fibres, similar model as well, by using basic restriction parameters to the C-FIBRE model used in this study, have been from the IEEE standard [14, 31]. used by researchers to study the excitation properties In the context of exposure assessment and electro- of C-fibres [39, 43, 11]. Model equations for the different magnetic safety, a comparison between excitability in- fibre and membrane models (Table1-2) used in this dices obtained by different fibre and membrane models study are summarized in AppendixA. is especially relevant. A recent publication on the research agenda of the IEEE International Committee on Electromagnetic Safety has listed 25 issues when using Model Abbr. Experiments numerical models to derive electromagnetic basic re- Hodgkin-Huxley HH Squid axon strictions, one of these on the consistency of excitation Frankenhaeuser-Huxley FH Frog node Chiu-Ritchie-Rogart-Stagg- CRRSS Rabbit node models [31]. Furthermore, a survey has been published, Sweeney comparing the excitation thresholds obtained by neu- Schwarz-Eikhof SE Rat node rostimulation models of different researchers [28]. As it Schwarz-Reid-Bostock SRB Human nerve was observed that excitation threshold can vary significantly between different electrostimulation models, a Table 2 Models of membrane channel dynamics and thorough investigation of the effect of the membrane their corresponding abbreviations (abbr.) Comparison of numerical electrostimulation models 3

The first goal of this paper is to investigate the dependence of excitability indices on the used equation set for the neuronal membrane, the electrode location, and stimulus properties. Simulation results of the strength-duration relation, rheobase, polarity ratio and bi/monophasic ratio obtained by the different models (summarized in Table1 and Table2) are presented and compared in the following sections. Correspondence of strength-duration (SD) plots, obtained by different membrane models and electrode locations, with Weiss- Fig. 1 Schematic representation of the simulation Lapicque and Lapicque-Blair fits is subsequently inves- configuration. A current I is injected by a spherical tigated. Furthermore, the impact of the presence of neu- electrode into the extracellular fluid. The electrode is ronal terminations on excitability is considered. The placed at longitudinal distance x and radial distance second goal is to determine the impact of the myelin E y approximation in the SENN-model on the excitability E indices, for different electrode locations and models of membrane dynamics. The third goal is to compare excitability indices among different fibre models (SENN- conductance are obtained from Tasaki [40]. Other phys- MA, SENN-M, and C-FIBRE). The impact of the ab- iological and geometrical parameters are taken from the sence of internodes in the C-FIBRE model on neuronal original McNeal publication and the appropriate papers excitability is determined. To this end, all geometrical describing the active membrane models [22, 13, 10,7, and physiological parameters (i.e. all parameters, ex- 38, 35, 36]. cept for the spatial and temporal discretisation step and For all models (A-fibres and C-fibres) a 10 µm neu- simulation time) are first held constant, to allow com- ron of length L = 4 cm is used, except when the SD parison between the SENN-M and C-FIBRE model for time constant τ or rheobase I are explicitly plotted as the FH, CRRSS, SE, and SRB membrane models (see e b function of the diameter D. As the propagation speed also next section). In a second step, the impact of the of action potentials is lower in C-fibres than in A-fibres, fibre diameter on the rheobase and strength-duration while activation thresholds are higher, a larger (maxi- time constant τe in C-fibres is determined. mal) temporal discretization step of 10 µs could be used in the C-FIBRE model. For A-fibres, a maximal tempo- 2 Methods ral discretisation step of 1.26 µs was used. Furthermore, the temporal discretization step is altered for low pulse Computational models for a myelinated axon (SENN- durations tp, in order to ensure a minimum of 10, 50, MA, SENN-M) and an unmyelinated axon (C-FIBRE) and 70 samples per pulse in the C-FIBRE, SENN-MA, are implemented in Matlab. In each of these models, and SENN-M model respectively. The total simulation- the corresponding system of equations is discretized by time Tsim is at least six times the phase duration (i.e. a Crank-Nicholson scheme (see AppendixA)[8], while Tsim ≥ 6τp, where τp is the duration of a single phase). the ends of the neuron are assumed to be sealed (i.e. Finally, the simulation time is not allowed to be lower no axonal current at the neuronal terminations). An than 1 ms for A-fibres and 5 ms for C-fibres. These re- overview of the used parameter values is shown in Ta- quirements are necessary to ensure sufficient simulation ble3. All other parameters, not listed in Table3, are time to allow the propagation of the action potential obtained from literature and summarized in Appendix and were determined by simulation of generic electrode A. Experimental values for the myelin capacitance and configurations. Higher simulation times are required for

Variable Description A-ﬁbre C-ﬁbre ∆x Spatial discretisation step 0.1 mm 0.1 mm ∆t Temporal step 1.26 µs 0.01 ms D Outer radius 10 µm 10 µm d Inner radius 0.7D D L Neuron length 4 cm 4 cm Tsim Simulation time ≥ max(6τp, 1 ms) ≥ max(6τp, 5 ms) T Temperature 18.5 ◦C 18.5 ◦C

Table 3 Values and description of parameters of the A-ﬁbre and C-ﬁbre model 4 Thomas Tarnaud et al.

Type Pulse-duration τp Rad. distance yE Long. distance xE SD-plot I0(τp) 5µs − 5ms (31,log.) 2mm − 1cm 2cm 5µs − 5ms (31,log.) 5mm 0.5cm − 2cm (4,lin.) Rbm 5µs − 5ms (31,log.) 2mm − 1cm 2cm Rheobase Ib 5ms 1.5mm − 10mm (10,lin.) 0.5cm − 2cm (4, lin.) Polarity ratio Rp 5ms 2mm − 1cm 0.5cm − 2cm (10, lin.)

Table 4 Simulation range of electrode and waveform parameters used in diﬀerent types of plots. The number of simulation points and the used distribution for these points (linear (lin.) or logarithmic (log.)) is indicated between brackets. I0 is the threshold current, Rbm is the bi/monophasic ratio, Ib is the rheobase, and Rp is the polarity ratio

C-fibres due to low propagation speeds of the action po- schematic representation of the simulation configura- tential. tion, defining the longitudinal and radial electrode dis- As the Hodgkin-Huxley model was obtained from tance xE and yE, is shown in Fig.1. The minimal stim- an unmyelinated squid nerve, axonal currents are too ulation current I required to obtain activation is de- weak to sustain the propagation of an action potential termined by a titration procedure alternating between in the framework of the SENN-model [5,9]. This phe- activation and absence of activation [29]. The titration nomenon is caused by the higher axonal resistance be- procedure terminates after reaching an accuracy of 1% tween neighbouring compartments in the SENN-model, or better. Neuronal activation was defined as depolar- as compared with the C-FIBRE model. However, com- ization of the active membrane at subsequent active bination of the Hodgkin-Huxley with a myelinated fi- compartments. Electrode parameters that are varied bre model has proven useful, e.g. in modeling the audi- are the electrode location, the pulse duration τp and tory nerve [23]. Consequentially, the Hodgkin-Huxley the sign of the current (anodic or cathodic). Further- equations were modified to allow compatibility with more, both monophasic and biphasic square waves are the SENN-model. The modifications were performed, used as stimulation pulses. following the procedure used by Cartee [5], which is The excitability indices, used in this study, are now based on Fitzhugh [9]. To allow for conduction of an ac- defined. Strength-duration plots represent the relation potential when the Hodgkin-Huxley model is com- tion between the pulse duration τp and the threshold bined with the SENN-model, a scaling factor Asc = current I0 required for neuronal activation. An empiri- 0.003 mm2 is used. The nodal conductivity and capaci- cal strength-duration relationship was first obtained by tance for the SENN-model with HH membrane dynam- Weiss [41] and by Lapicque [18]. The Weiss-Lapicque ics is scaled with Asc/(πdln) and Asc/(20πdln) respec- relation between the stimulus strength and the pulse tively. Here, ln = 2.5 µm is the nodal length. It is noted duration can be formulated as: that Asc has no physical significance and should be in- τe terpreted as a scaling factor to match model parame- I0 = Ib(1 + ) (1) τp ters with the experimentally measured conductivity and capacitance. This modification of the Hodgkin-Huxley Here, Ib is the rheobase current and τe is the SD time equations, when applied to the SENN-model, was the constant. only deviation of the original membrane parameters Another strength-duration relation is based on the- and is only applied when the HH-equations are used oretical considerations, and was obtained by Lapicque in combination with the SENN-model. and Blair [18,1,4]. The Lapicque-Blair relation has an A different phenomenon observed by Rattay and exponential form: Aberham is known as “heat block”. Unmodified HH- I I = b (2) equations stop to conduct action potentials for tem- 0 1 − exp(− τp ) peratures greater than 31◦C [27]. The modified HH- τe equations that are used in the SENN-model will not These equations define the rheobase current as the be hindered by heat block [5]. However, unmodified threshold current for a square-wave pulse of long du- Hodgkin-Huxley equations are still used in this study, ration. The rheobase is obtained in this study by eval- in the C-FIBRE model. Because it is desired to com- uating the SD-plots at τp = 5 ms, similarly as is done pare all models at the same temperature, a universal in Reilly [28]. This procedure results in a negligible er- temperature of 18.5◦C is used for all models. ror on the rheobase, which is defined as the asymptotic A spherical micro-electrode is used to probe the ex- minimum of the SD-plot and will be slightly lower for citation properties of the model under consideration. A longer pulse durations. Comparison of numerical electrostimulation models 5

Similarly, the strength-duration time constant τe was approximated by taking the ratio of the charge Qe = 5µs · I0(τp = 5µs) with respect to the rheobase Ib. The bi/monophasic ratio Rbm is defined as the ratio of the biphasic and monophasic threshold current. To determine these thresholds, a cathodal monophasic square wave is used, while the biphasic square wave will start with a cathodal initial phase. Note that the time τp indicates the duration of the initial phase in a biphasic square wave and not the duration of the pulse. Finally, the polarity ratio Rp is the ratio of the anodic and cathodic rheobase. The polarity ratio was approximated by evaluating the anodic and cathodic threshold current at pulse durations of τp = 5 ms [28]. Values of Rp lower than one indicate that anodic stimulation is more effective than cathodic stimulation to initiate the propagation of an action potential, and vice versa for Rp > 1. The simulation range of the electrode and waveform parameters, that are used to generate the plots in this paper, is summarized in Table4. The table also indicates the number of samples in the simulation range and whether a linear (lin.) or logarithmic (log.) distribution Fig. 2 Illustration of propagation of AP for the SGE- of points is used. mode (a,c,e) and NT-mode (b,d,f) of activation (see text for definitions). The anode is placed at longitudinal distance x = 20 mm. (a)-(b) Colour map of membrane 3 Results E potential. Yellow indicates depolarization, dark blue in- An example of an action potential in the SENN-model dicates hyperpolarization. The abscissa represents the with FH-dynamics is shown in Fig.2. The anode is time t passed after the onset of the stimulus, while the ordinate xl represents the length along the neuron. (c)- placed at the centre of the axon, at xE = 20 mm. The activating function f allows for a straightforward inter- (d) Spatial distribution of membrane voltage along the pretation of the location of the excitation node, which is neuron for 7 snapshots, equally spaced in time (snap- defined as the first node to reach the voltage threshold shots at the four earliest times are indicated). (e)-(f) [25]. The activating function is shown in Fig.2(e)-(f) Spatial distribution of activating function f and is given by:

1 h ∗ 2 h i f(l) = ∗ Ga,f (l − ∆l)(Ve(l − ∆l) − Ve(l))+ ∗ Cm(l) f(L) = ∗ Ga,f (L − ∆l)(Ve(L − ∆l) − Ve(L)) (5) Cm(L) ∗ i Ga,b(l + ∆l)(Ve(l + ∆l) − Ve(l)) (3) Here, the factor 2 is a consequence of the fact that sealed-end boundary conditions are applied at the cen- Here C (l)∗ is the membrane capacitance of the com- m tre of the compartments at the neuronal terminations. partment at x = l. G∗ (l) is the “backward” axial con- a,b Negative values of the activating function will fa- ductance between the compartments at x = l − ∆l vor hyperpolarization of the active membrane, while and x = l. Similarly, G∗ (l) is the “forward” axial a,f the membrane will tend to be depolarized at compart- conductance between the compartments at x = l and ments with positive values of f. It can be observed from x = l + ∆l. Equation (3) is valid for all nodes, except Fig.2(e)-(f), that the envelope of the activating func- the termination nodes at x = 0 mm and x = 40 mm. tion reaches a central minimum at x = 20 mm and At the neuronal terminals, the activation function f is E two maxima at the so-called “virtual cathodes”. These given by [26]: virtual cathodes are positioned symmetrically, at both 2 h i sides of the anode. Furthermore, positive peaks at both f(0) = G∗ (∆l)(V (∆l) − V (0)) (4) ∗ a,b e e neuronal terminations can be observed. Cm(0) 6 Thomas Tarnaud et al.

80 (x =2cm) 80 (x =2cm) Cathodal E (a) Anodal E (b) 103 Cathodal (a) 104 Anodal (b) FH 60 SRB 60

[mA] SE [mA] FH SRB 0 0 3 SRB CRRSS 10 40 102 CRRSS

SE [%] 40 [%] SRB CRRSS 2 SE 20 10 SE SD 20 SD ǫ ǫ 0 101 HH CRRSS 101 HH HH HH FH 0 -20 FH 0 2 4 6 8 10 0 2 4 6 8 10 (x =2cm,y =5mm) (x =2cm,y =5mm) 0 E E 0 E E Radial distance y [mm] Radial distance y [mm]

Activation current I 10 10 0.001 0.01 0.1 1 10 Activation current I 0.001 0.01 0.1 1 10 E E [mA]

Phase duration τ [ms] Phase duration τ [ms] b 2 Cathodal (x =2cm) (c) 150 Cathodal (x =2cm,y =5mm) (d) p p E E E ∆

SRB [%]

0 CRRSS 0 I ǫ 120 (x =2cm) (c) 60 Anodal (x =2cm) (d) 100 Cathodal E E SE -2 SRB 100 50 HH CRRSS CRRSS -4 HH SE 50 [mA] 80 [mA] 40 b b FH FH SRB SE -6 CRRSS SE 60 30 SRB

FH FH -8 Cathodal error 0 40 HH 20 2 4 6 8 10 0 1 2 3 4 5 Radial distance y [mm] Phase duration τ [ms]

20 10 Cathodal rheobase error E p Rheobase I Rheobase I HH 0 0 0 2 4 6 8 10 0 2 4 6 8 10 Fig. 4 Eﬀect of the myelin approximation on excitabil- Radial distance y [mm] Radial distance y [mm] E E ity indices. (a)-(b) Relative error on τe. (c) Relative 12 (x =2cm,y =5mm) (e) 5 (y =5mm) E E E (f) 10 error on cathodal threshold current. (d) Absolute error 4 CRRSS 8 on cathodal rheobase. Electrode location is xE = 2 cm 3 FH [-]

6 [-] SRB in (a)-(d), and y = 5 mm in (d). FH E p

bm 2 SE 4 SE R CRRSS R HH SRB 1 2 HH

0 0 3.1 A-fibres 0 100 200 300 0 5 10 15 20 Phase duration τ [µs] Longitudinal distance x [mm] p E The effect of the myelin approximation and the mem- Fig. 3 Myelinated (A-fibre) simulation results. (a) brane channel dynamics on excitability indices in the cathodal activation current, (b) anodal activation cur- SENN-MA and SENN-M model is shown in Fig.3. rent, (c) cathodal rheobase, (d) anodal rheobase, (e) Fig.3(a) and Fig.3(b) are the cathodal and anodal SD- bi/monophasic rheobase ratio, (f) polarity ratio. Full plots obtained by running a parameter sweep over 31 lines indicate the use of the myelin parameters reported logarithmically distributed pulse-durations in the inter- by Tasaki [40], while the stars indicate the use of the val from 5 µs to 5 ms. Fig.3(c)-(d) depict cathodal and myelin approximation anodal rheobase-distance plots respectively, obtained by simulation of 10 linearly distributed radial (rad.) dis- When the anode is placed closed to the axon, e.g. tances, while the longitudinal (long.) distance is kept at yE = 3 mm in Fig.2(a),(c),(e), neuronal activation constant. Fig.3(e)-(f) show the bi/monophasic ratio will be determined by the virtual cathodes. Because the Rbm and the polarity ratio in 31 logarithmically and 10 activating function f is proportional to the spatial gra- linearly distributed points respectively. dient of the electric field (SGE) in equation (3), we will call this case the SGE-mode of activation. The exci- Impact of the myelin approximation on excitability in- tation nodes are roughly located at the virtual cath- dices odes and the action potential propagates orthodromi- cally and antidromically. Propagation to the axon cen- The influence of the myelin approximation on excitabil- tre (xE = 20 mm) is inhibited by the hyperpolarized ity indices was determined by Richardson et al. [32], for region at the position of the anode. neurons described by a modification of the Schwarz- Conversely, if the anode is placed at larger radial dis- Reid-Bostock model [20]. We expand some of their re- tance, e.g. at yE = 6 mm in Fig.2(b),(d),(f), then exci- sults in this subsection, to include the electrode range, tation will be driven by the large peaks of the activating excitability indices and membrane models that are con- function f at the neuronal terminations. The action po- sidered in this study. tential in Fig.2(b),(d) starts at the neuronal termina- Fig.3 and Fig.4 are used to illustrate the impact tions and propagates towards the centre (xE = 20 mm). of the myelin approximation. Strength-duration curves We will refer to this mode of activation as the neuronal shift to lower threshold current values, when the myelin termination (NT) mode. conductance and capacitance are set to zero. This effect For cathodal stimulation the sign of the activating is most significant for shorter time durations (Fig.3(a)- function f is reversed and the excitation node coincides (b), Fig.4(d)). The observation that shorter time du- with the location of the cathode. rations are associated with greater errors, is a conse- Comparison of numerical electrostimulation models 7

105 Cathodal (a) 105 Anodal (b) 6 HH (a) 8 CRRSS (b) CRRSS 6 [mA] 4 [mA] 4 0 10 0 4

10 [-] SRB [-] 4 p SE SRB p 3 2

10 R FH CRRSS R 2 103 SE 102 HH 0 0 0 5 10 15 20 0 5 10 15 20 2 Longitudinal length x [mm] Longitudinal length x [mm] 1 10 FHHH E E 10 HHHH 40 8000 (x =2cm,y =5mm) (x =2cm,y =5mm) HH(-) (c) CRRSS(-) (d) 0 E E 1 E E 30 [mA] 6000 10 10 [mA] Activation current I Activation current I b 1 10 100 1000 10000 1 10 100 1000 10000 b Phase duration τ [µs] Phase duration τ [µs] 20 4000 p p 10 2000 (x =2cm) (x =2cm) Anodal (d) 10000 E Cathodal (c) 600 E

0 Rheobase I 0 Rheobase I 0 2 4 6 8 10 0 2 4 6 8 10 500 8000 SE Radial distance y [mm] Radial distance y [mm] E E CRRSS [mA] [mA] 400 30 HH(+) 600 CRRSS(+) (f) b 6000 b (e)

300 SRB [mA] b SRB 20 [mA] 400 4000 FH b CRRSS 200 SE FH 10 200 2000 100 Rheobase I Rheobase I HH HH 0 0 Rheobase I

0 0 0 2 4 6 8 10 Rheobase I 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 Radial distance y [mm] Radial distance y [mm] E E Radial distance y [mm] Radial distance y [mm] 15 4 E E HH (g) CRRSS (h) (x =2cm,y =5mm) 5 (y =5mm) 3 10 E E (e) E (f) 10 [-] [-] 2

4 bm 5 bm 8 1 R R

3 0 0

[-] 6 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 HH [-] HH

p phase duration τ [µs] phase duration τ [µs] bm 4 2 p p R R FH 1 2 SRB SRBFH SE CRRSS CRRSSSE Fig. 6 C-ﬁbre model simulations with the HH and 0 0 0 50 100 150 200 250 300 0 5 10 15 20 CRRSS membrane models. (a)-(b) polarity ratio. (c)- Phase duration τ [µs] Longitudinal distance x [mm] p E (f) anodal (+) or cathodal (-) rheobase. (g)-(h) Rbm ratio. Legend. (g)-(h) (long. distance) xE = 2 cm. Fig. 5 Unmyelinated (C-ﬁbre) simulation results. (a) (a)-(b),(g)-(h) (rad. distance): ( ) 2mm, ( ) 4mm, cathodal activation current, (b) anodal activation cur- ( ) 6mm, ( ) 8mm, (∗) 1cm. (c)-(f) (long. dis- rent, (c) cathodal rheobase, (d) anodal rheobase, (e) tance): ( ) 0.5cm, ( ) 1cm, ( ) 1.5cm, ( ) 2cm bi/monophasic rheobase ratio (f) polarity ratio

Ib is decreased by −6.6 mA by the myelin approxima- quence of a decrease in chronaxie or strength-duration tion in the Frankenhaeuser-Huxley model, if the radial time constant when using a model with infinite myelin electrode distance is equal to 10 mm. impedance. This decrease in time constant was theoret- ically predicted by Rubinstein [33]. Richardson et al. reports an increase of SD = 11% 3.2 C-fibres of the chronaxie when abandoning the myelin approximation, for the electrode configuration and model pa- Simulation results in the C-FIBRE model are shown rameters used in their study. It can be observed from in Fig.5. Subsequently, Fig.6 and Fig.7 show more Fig.4(a)-(b) that the error on the strength-duration detail on the effect of the radial and longitudinal elec- time constant is dependent on the electrode position, trode position on the excitability indices in the HH and polarity and the used membrane model. Both the error CRRSS models. Lower computational cost in these two on the time constant, SD, as the error on the rheobase conditionally linear models, allowed for more extensive b, due to the myelin approximation, increase with ra- parameter sweeps. dial electrode distance (Fig.4). The error is furthermore determined by the mode of activation (stimulation at neural terminations or on the axon), as can be observed 4 Discussion from Fig.4(b). These results show that the impact of the myelin 4.1 Strength-duration relation approximation on the SD time constant and rheobase is most significant in the Frankenhaeuser-Huxley model Strength-duration relations at constant electrode loca- at large electrode-neuron separations. For instance, the tion, but for different membrane models, are shown in time constant τe is increased by 70% and the rheobase Fig.3(a)-(b) and Fig.5(a)-(b) for A-fibres and C-fibres 8 Thomas Tarnaud et al.

90 105 (rad.,HH,-) (a) 104 (rad.,HH,+) (b) 80

102 70 SE [mA] 0 I 60 CRRSS 100 100 1 100 10000 1 100 10000 50

40 SRB (rad.,CRRSS,+) (d) (rad.,CRRSS,-) (c) 104 30 FH 105 C/A Multiplier [-] 20 103 HH [mA]

0 10 I

100 102 0 1 100 10000 1 100 10000 1 2 3 4 5 6 7 8 9 10 Radial distance y [mm] E 104 (long.,HH,-) (e) 104 (long.,HH,+) (f) Fig. 8 Dependence of rheobase C/A multiplier on ra- 102 102 dial distance and membrane dynamics. The cathode is [mA] 0 I placed at xE = 20 mm 100 100 1 100 10000 1 100 10000

4 (h) (long.,CRRSS,-) (g) 10 (long.,CRRSS,+) by the model to the rheobase that would be obtained

104 if the membrane model would be substituted by the

[mA] 2 Frankenhaeuser-Huxley system. Consequentially, RR-

0 10 I values obtained by the FH-system are equal to unity. 102 1 100 10000 1 100 10000 An overview of RR-values is presented in Table5. Mod- τ [µs] τ [µs] p p els with RR-values greater than unity are more difficult to excite than the FH system at long pulse durations, Fig. 7 C-fibre anodal (+) and cathodal (-) strength- while models with RR-values smaller than unity are duration plots with the HH and CRRSS membrane more easy to excite. models. Legend. (a)-(d) (rad. distance): ( ) 2mm, The relative magnitude of required threshold cur- ( ) 4mm, ( ) 6mm, ( ) 8mm, (∗) 1cm. (long. rent at long pulse durations to initiate an action poten- distance): xE = 20 mm. (e)-(h) (radial distance): yE = tial, as compared to the Frankenhaeuser-Huxley sys- 5 mm. (long. distance): ( ) 0.5cm, ( ) 1cm, ( ) tem, is dependent on the electrode position, the elec- 1.5cm, ( ) 2cm trode polarity and the type of multi-compartmental electrostimulation model (A-fibre or C-fibre). However, as can be seen from the values in Table5, this depen- respectively. We observe that both A-fibres and C-fibres dency is not strong enough to change the order of ex- have a fixed order of excitability at long pulse durations. citability of the different models of membrane channel To compare the relative magnitude of stimulus current dynamics. In order of increasing threshold, we find: HH, required to initiate an action potential, we introduce FH, SRB, CRRSS and SE. E.g., for cathodal stimula- the rheobase ratio (RR). The rheobase ratio of a tion of A-fibres, we find average RR-values of 0.22, 1, model is defined as the ratio of the rheobase calculated 2.40, 2.94 and 3.35 for the different membrane models

Membrane Model Min. RR Max. RR Average RR A-fibre (-) 0.20; (+) 0.06 (-) 0.27; (+) 0.25 (-) 0.22; (+) 0.14 HH C-fibre (-) 0.02; (+) 0.02 (-) 0.04; (+) 0.10 (-) 0.02; (+) 0.08 A-fibre (-) 2.00; (+) 1.84 (-) 3.46; (+) 2.66 (-) 2.94; (+) 2.26 CRRSS C-fibre (-) 3.31; (+) 1.97 (-) 3.52; (+) 3.44 (-) 3.47; (+) 2.13 A-fibre (-) 2.48; (+) 2.30 (-) 3.83 ;(+) 3.09 (-) 3.35; (+) 2.75 SE C-fibre (-) 3.69; (+) 2.46 (-) 3.88; (+) 3.80 (-) 3.84; (+) 2.60 A-fibre (-) 1.94; (+) 1.78 (-) 2.64; (+) 2.23 (-) 2.40; (+) 2.10 SRB C-fibre (-) 2.52; (+) 1.95 (-) 2.60; (+) 2.60 (-) 2.60; (+) 2.02

Table 5 Minimum, maximum and average rheobase ratio (RR) obtained by the different models for cathodal (-) and anodal (+) stimulation. RR-values are defined by taking the FH-system as reference. A-fibre values are calculated with the SENN-MA model Comparison of numerical electrostimulation models 9 in the aforementioned order of excitation. This order of Furthermore, the strength-duration relation was ob- excitability is reported as well by Rattay and Aberham tained for the Hodgkin-Huxley and CRRSS models, [27], excluding the SRB-system, for a pulse duration while performing extensive parameter sweeps of the elec- of 100 µs and for current directly injected into a single trode location and waveform polarity. The result is shown compartmental neuron. However, the order of excitabil- in Fig.7, and allows us to determine the dependency ity among the different membrane models is altered at of the SD-relation on the electrode and stimulus pa- short pulse durations, as the SRB-model becomes rela- rameters. Fig.7(a),(c) show the dependence of the SD- tively more difficult to excite (Fig.3 and Fig.5). plots on the radial cathode distance yE in the Hodgkin- Huxley and CRRSS models respectively. Increasing the For cathodal stimulation (Fig.3(c) and Fig.5(c)), cathode-neuron separation, will shift the SD-plot to rheobase ratios first diverge from unity with increasing higher values. Similarly, Fig7(b)-(d) shows the impact radial electrode distance, after which they level off. For of anode-neuron separation on SD-plots. Here, increas- anodal stimulation (Fig.3(d) and Fig.5(d)), rheobase ing the anode-neuron separation will lead to a tran- ratios reach a steady level when stimulation at the neu- sition from the SGE-mode to the NT-mode of activa- ron terminations becomes the dominant mode of exci- tion. Consequentially, the strength-duration plot does tation. not depend on the anode-neuron separation, for y > yc. The impact on threshold current of the presence of Here, yc is the critical distance, at which the transition internodes is shown in Fig.8. Here, the C/A-multiplier from SGE-mode to NT-mode of activation occurs. In is the ratio of the rheobase obtained by the C-FIBRE Fig.7(e)-(h) it is shown that the longitudinal electrode model and the SENN-MA model. The C/A-multiplier distance will only impact the strength-duration plot for values obtained by the Hodgkin-Huxley model are not anodal stimulation. In this case, smaller anode-terminal in line with the other models, because the HH param- separations result in lower threshold current values. eters are altered in the SENN-model (see section2). Most of the dependency of the SD-plots in Fig.7 C/A-multiplier values increase with electrode-neuron on electrode location and polarity, can be explained separation. At the largest simulated radial electrode through the rheobase current Ib and the SD time con- distance, the FH, CRRSS, SE, and SRB model have stant τe. Indeed, both the rheobase and the time con- similar C/A-multiplier values between 77 and 80. stant do depend on the electrode location and waveform polarity. Furthermore, it is possible to normalize the threshold current and pulse duration by using these 2 HH (-): (x =20mm,y =2mm) 10 E E excitability indices (rheobase and SD time constant). HH (+): (x =20mm,y =4mm) I τp E E Subsequently plotting the 0 ( )-relation reveals that HH (+): (x =20mm,y =6mm) Ib τe E E CRRSS (-): (x =20mm,y =2mm) all normalized SD-plots coincide, under the constraint E E CRRSS (+): (x =20mm,y =4mm) E E of identical membrane dynamics and dominant mode CRRSS (+): (x =20mm,y =6mm) E E of excitation (results not shown). Indeed, it is observed Weiss-Lapicque Lapicque-Blair that the shape of the normalized strength-duration re- [-] 1 b 10 lation is altered, when stimulation at neuronal termina- /I 0 I tions becomes the dominant mode of excitation. This is an extension of a result obtained by Sarolićetˇ al.: it was determined by Sarolićetˇ al. that the normalized strength-duration curve does not depend on the electrode location and spatial waveform (wire electrode, 100 10-2 10-1 100 101 102 bipolar electrode setup or homogeneous E-field) in the τ /τ [-] p e SENN-model [34]. However, the effect of neuronal terminations, waveform polarity, and membrane dynamics Fig. 9 Normalized SD-relations in the Hodgkin-Huxley was not yet taken into account. and CRRSS models for three generic electrode loca- The observation that, of the studied independent tions. Cathodal (-) stimulation at a radial distance of parameters, normalized SD-plots are only dependent 2 mm, represents the case in which the neuron is stim- on membrane dynamics and the dominant mode of ex- ulated by the gradient of the electric field. Anodal (+) citation, is illustrated in Fig.9. For both the HH and stimulation at a radial distance of 6 mm represents the the CRRSS models three generic electrode configura- case in which stimulation at neuronal terminations is tions are used. First, a cathode at a radial distance dominant. Anodal (+) stimulation at yE = 4 mm is yE = 2 mm is used to represent stimulation by the spa- shown to illustrate an intermediate case tial gradient of the electric field (SGE-mode). Second, 10 Thomas Tarnaud et al.

Weiss-Lapicque Lapicque-Blair Membrane A-fibres C-fibre A-fibre C-fibre HH 14.5863 93.3663 2.4681 56.3922 FH 3.8529 8.05448 1.0775 0.93236 CRRSS 3.5515 3.31010 1.8848 1.57979 SE 5.1951 7.20470 0.6878 0.76276 SRB 2.8468 2.33073 8.1539 3.78514

Table 6 Sum of squares of residuals (SSres[−]). A-ﬁbre values are calculated with the SENN-MA model

an anode at yE = 6 mm is used, to illustrate stimulation SGE-mode, are compared with the Weiss-Lapicque and at neuronal terminations (NT-mode). It is indeed clear Lapicque-Blair equations in Fig. 10. Furthermore, sum from Fig.9 that normalized strength-duration curves of squares of residuals (SSres) values are summarized in do depend on the dominant mode of excitation, as well Table6. It is observed, that fitting the SD-relation by as the membrane dynamics. An intermediate case is the Lapicque-Blair equation is superior for the HH, FH, represented by an anode at a radial distance yE = CRRSS and SE models. Conversely, SD-plots obtained 4 mm. For the Hodgkin-Huxley model, the transition by the SRB-model within the SGE-mode of excitation, from the SGE-mode to the NT-mode occurs at anodal are better approximated by the Weiss-Lapicque equa- radial distances of yE = 4.33 mm(±0.95 mm) in the tion. used C-FIBRE model. As a consequence, an intermediate strength-duration curve is observed, indicating a continuous transition of the normalized SD-curve with 4.2 Dependence on fibre diameter increasing radial distance. In contrast, this transition between dominant modes of excitation is observed at The dependence of the rheobase current and SD time a radial anodal distance of 2.44 mm(±0.95 mm) for all constant on the fibre diameter in the C-FIBRE model other membrane models (see also the subsequent sub- is shown in Fig. 11(a)-(b). In Fig. 11(a) the rheobase section). Consequentially, the normalized SD-plots for multiplier M = Ib(D)/Ib(D = 10 µm) is shown for di- ameters between 1 µm and 10 µm. Rheobase current is yE = 4 mm and yE = 6 mm coincide in the CRRSS- model. inversely proportional with fibre diameter. The slope The strength-duration relations, obtained with the dif- of the log M-log D plot fluctuates around −1 in the ferent membrane models and by excitation wihin the FH, CRRSS, SE, and SRB model (min: −1.1, max: −0.9), indicating that approximately Ib ∝ 1/D. In the

Cathodal SD (SENN-MA): (x =20mm,y =5mm) Cathodal SD (C-FIBRE): (x =20mm,y =5mm) E E E E 102 103 Lapicque-Blair Weiss-Lapicque Schwarz-Reid-Bostock Schwarz-Eikhof CRRSS Frankenhaeuser-Huxley 2 10 Hodgkin-Huxley

[-] 1 b 10 /I 0 I

101

100 100 10-2 100 102 10-2 100 102 τ /τ [-] τ /τ [-] p e p e

Fig. 10 Correspondence of normalized strength-duration plots with the Weiss-Lapicque and Lapicque-Blair relations. Electrode location (xE = 2 cm, yE = 5 mm) and polarity (cathodal) is chosen to ensure that neuronal excitation is caused by the spatial gradient of the electric ﬁeld at the centre of the axon Comparison of numerical electrostimulation models 11

Hodgkin-Huxley model the slope in the log M-log D Reference: D=10 µm 10 (a) 1500 (b) plot lies between −0.95 and −0.8 for xE = 20 mm and 8 FH HH yE = 5 mm (Fig. 11(a)). Fig. 11(c) investigates the role 1000 6 CRRSS s] of cathode-neuron separation on the rheobase-diameter SE µ

4 [ SRB e 500 relation. In the CRRSS model, a I ∝ 1/D relation τ b SRB FH SE CRRSS is found between rheobase and ﬁbre diameter, for all 2 HH 0 0 simulated radial distances 1.5 cm ≤ yE ≤ 10 cm. It is 0 1 2 3 4 5 6 7 8 9 10 0 2 4 6 8 10 Rheobase multiplier M [-] Diameter [µm] observed from Fig. 11(c) that the Ib ∝ 1/D relation Diameter [µm] will be valid for the Hodgkin-Huxley model as well, for 12 Reference: D=1 µm (c) 1500 (d) 10 large cathode-neuron separations. We conclude that for CRRSS 8 HH 1000 HH C-ﬁbres, a Ib ∝ 1/D relation holds for large radial elec- s]

6 µ trode distances and that the Hodgkin-Huxley model re- CRRSS [ e

4 τ 500 quires a larger radial distance than the other simulated HH CRRSS HH CRRSS models. This is in contrast with A-fibres models, for 2 0 0 which the slope in the log M-log D plot decreases with Rheobase multiplier [-] 0 2 4 6 8 10 0 2 4 6 8 10 Radial distance y [mm] Radial distance y [mm] increasing diameter (McNeal [22]). E E Similarly, Fig. 11(b) shows that the SD time con- Fig. 11 Impact of diameter on rheobase ((a),(c)) and stant τe is approximately constant over the simulated diameter range in the FH, CRRSS, SE, and SRB model. SD time constant ((b),(d)) in C-FIBRE. (a) Rheobase In the Hodgkin-Huxley model, the time constant de- multipliers (reference indicated in figure) and SD time creases with diameter. In Fig 11(d) it is shown that the constants are plotted against diameter ((a)-(b)) and ra- sensitivity of the time constant on the fibre diameter is dial electrode distance ((c)-(d)). Cathode location is reduced at large cathode-neuron separations. (xE = 20 mm, yE = 5 mm) in (a)-(b) and xE = 20 mm in (c)-(d). Legend. (diameter) (c) ( ) 0.5µm, ( ) 5µm, ( ) 10µm. (d) (diameter): ( ) 0.5µm, ( ) 4.3 Polarity ratio 1µm, ( ) 5µm, ( ) 10µm. Method. Plot was obtained by first order Crank-Nicolson discretisation with If the neuron is excited by the spatial gradient of the time step ∆t = min(τp/50, 0.01 ms) electric field rather than at the neuronal terminations, then cathodal stimulation is more effective than anodal stimulation and Rp > 1. However, if the radial electrode 1 cm) are represented by five curves. The dependency distance yE is increased or the longitudinal electrode of the polarity ratio on the radial distance yE is es- distance xE is decreased, stimulation at the neuronal timated by comparing the values indicated by the five terminations becomes the dominant mode of excitation. curves at the constant longitudinal distance xE = 2 cm. This is expressed by a decrease in the polarity ratio Rp. We observe that for the Hodgkin-Huxley membrane The polarity ratio is shown for A-fibres and C-fibres model, the polarity ratio starts to decrease at a radial in Fig.3(f) and Fig.5(f) respectively at ( xE, yE) = distance between 4 mm and 6 mm. In contrast, in the (2 cm, 5 mm). The polarity ratio is furthermore shown CRRSS model (Fig.6(b)) the critical distance lies be- for different radial and longitudinal electrode positions tween 2 mm and 4 mm. Another procedure to determine in the C-FIBRE model in Fig.6(a)-(b). These results the critical radial electrode distance is by investigating show that the polarity ratio drops as the location of for the discontinuity in the slope of anodal rheobase- the virtual electrode approaches the neural terminal. distance curves, as will be described below. Furthermore, it can be observed from Fig.6(a)-(b) that the minimum in the Rp(xE) plots shifts to higher values of xE, for greater radial electrode distances. 4.4 Bi/monophasic ratio The transition from the SGE-mode to the NT-mode of excitation occurs when the polarity ratio Rp begins If cathodal stimulation is more effective than anodal to decrease. This transition is dependent on the used stimulation in initiating an action potential (Rp > 1), membrane model. For example, consider an electrode then the bi/monophasic ratio Rbm will always be larger at a longitudinal distance xE = 2 cm of an unmyeli- than one. This is the case for the SENN-M and SENN- nated fibre. We investigate the critical radial distance MA models at (xE, yE) = (2 cm, 5 mm), see Fig.3(e). If of the electrode yc for which the transition between the phase duration τp is less than the strength-duration these two modes of excitation occurs. In Fig.6(a), dif- time constant τe, Rbm-values are significantly greater ferent radial distances (2 mm, 4 mm, 6 mm, 8 mm and than unity. This is due to the fact that the reversed cur- 12 Thomas Tarnaud et al. rent in the second phase of the biphasic pulse, can stop end, simulations are restricted to three different rep- the development of the action potential ([29], pp 132- resentations of a multi-compartmental axon (SENN- 41). For long pulse durations however, the two phases MA, SENN-M and C-FIBRE). A fourth axonal model of the waveform do not negate each other’s influence, (MRG) was developed by Blight, and aims to unravel and Rbm = 1 [28]. the mechanism for depolarising afterpotentials [2]. Only This reasoning does not hold true if Rp < 1, see the MRG-model has the important feature of repro- Fig.6(g)-(h). It is now possible that the first (cathodal) ducing the depolarising afterpotential (DAP) and after- phase is not able to initiate an AP, while the subsequent hyperpolarisation (AHP). However, the DAP and AHP (anodal) phase is. Consequentially, in this case biphasic have no direct impact on the excitability indices, which stimulation would be more effective than monophasic are determined by the first depolarisation in the ac- stimulation, resulting in Rbm < 1. tion potential. Nevertheless, excitability is still influenced by the double cable structure. Several researchers have used similar models in their computational stud- 4.5 Rheobase-distance relation and critical electrode ies [37, 32, 21]. However, these double-cable models are distance more computationally expensive than the used SENN and C-FIBRE model. Because of this reason, the impact The critical radial electrode distance yc for the tran- of double-cable structures and explicit modeling of the sition between the SGE-mode and NT-mode of excita- paranodes and juxtaparanodes on excitability indices tion, can be obtained from the anodal rheobase-distance was not included in this study. (RD) relations (Fig.3(d), Fig.5(d) and Fig.6(e)-(f)). Similarly, several specialized models of the active This critical distance y is the abscissa, associated with c membrane have been proposed in literature. E.g., at the “kink” in the anodal RD-curve. In contrast, catho- present, Channelopedia lists 52 different HH-type mod- dal RD-relations are smooth and do not exhibit a kink els for active currents [24]. Consequentially, we restricted (Fig.3(c), Fig.5(c), Fig.6(c)-(d)). The discontinuity in this comparative study to five often used models, de- the spatial derivative w.r.t. y of the anodal RD-curve E scribing the active membrane. at the critical electrode distance yc is a consequence of Finally, a relatively low model temperature of 18.5◦C the polarity ratio Rp, that starts to decrease at yc. In- (+) was chosen, to prevent heat-block in the unmodified deed, the anodal rheobase I can be expressed as a b HH-equation set in the C-FIBRE model. Higher model function of R and the cathodal rheobase I(−): p b temperatures result in faster membrane gate dynam- (+) (−) ics, leading to shorter action potentials, faster propa- I (y ) = R (y )I (y ) (6) b E p E b E gation of the action potential, shorter summation, and (−) excitation time constants, . . . [5]. Because temperature The derivative of Ib (yE) does not exhibit a discontinuity. Consequentially, the “kink” in the anodal RD- alters the membrane dynamics (through multiplication curve is related with a discontinuity in slope in the with a temperature dependent factor k of the model- dependent rate constants α and β, see Eq. (11)-(10)) Rp(yE)-curve, i.e. when the polarity ratio drops at the and because in this study it is observed that membrane critical distance yc. channel dynamics has a significant impact on excitation As an example, the critical radial distance yc in C- threshold, a similar impact of temperature on excitation fibres at xE = 2 cm is obtained by looking for the dis- is expected. However, modulation of membrane dynam- continuity in slope in Fig.5(d). The uncertainty on yc is determined by the size of the step in the parameter- ics by changing the model temperature is qualitatively similar to substituting the model of membrane dynam- sweep. The critical radial distance yc, that indicates the transition between the dominant modes of excitation ics. E.g., in a study by Cartee, the temperature is used for stimulation with long pulse durations, is dependent to alter the membrane dynamics in order to obtain a on the used membrane model. For the Hodgkin-Huxley better fit to the summation and refraction properties of the cochlear nerve [5]. However, changing the tem- model we find yc = 4.33 mm(±0.95 mm), while all other perature will have a more modest impact on the mem- models have yc = 2.44 mm(±0.95 mm). brane dynamics, than substituting the equation set. As a consequence of this similarity between substituting 5 Limitations the equation set for membrane dynamics and changing the temperature, it is expected that the conclusions of The goal of this study is to compare excitability in our study will not be qualitatively altered by a different different models of spatially extended neurons, for a model temperature. For example, we observed that the variety of stimulus and electrode parameters. To this strength-duration time constant is independent of the Comparison of numerical electrostimulation models 13

fibre diameter for large electrode-neuron separations. dependent on the membrane channel dynamics, only This result is valid for all studied membrane models, for short cathode-neuron separations. although larger electrode-neuron separations are necessary for the HH-model. As a consequence, it is unlikely The impact of stimulation at neuronal terminations that a different temperature would have a significant is investigated. Sealed-end terminals impact anodal RD impact on this result. plots (resulting in a discontinuity at the critical electrode distance yc), impact the bi/monophasic ratio and polarity ratio (Rbm < 1 is possible if the neuron is ac- tivated by the NT-mode) and lead to a change in the 6 Conclusion normalized SD-plots (Fig.3–6,9). We compared the strength-duration and rheobase- Threshold currents, rheobase, polarity ratio, bi/mono- distance plots between the SENN-M and SENN-MA phasic ratio and SD time constants are widely used in models, in order to investigate the effect of the myelin literature to quantize neuronal excitability. In this pa- approximation on neuronal excitability. Neglecting the per, three multi-compartmental models are combined myelin admittance always leads to a reduction in the with five models of membrane dynamics to generate rheobase and SD time constant. However, we found that plots of these excitability indices for a wide range of the significance of this approximation error is strongly electrode locations and stimulus properties. Excitabil- dependent on several interacting factors (electrode lo- ity in different membrane and spatial models is com- cation, stimulus polarity, membrane channel dynamics, pared by the rheobase ratio RR and C/A-multiplier, de- etc.). As a consequence, specification of these factors is fined in this study. Furthermore, SD-plots obtained by necessary when adressing questions about the justifica- the different computational models are compared with tion of the myelin approximation in a simulation study Weiss-Lapicque and Lapicque-Blair fits. We summarise or about the effect of demyelination on neuronal ex- the following conclusions (Fig.3,5,7–10, Table5–6). citability. In particular, we found the following (Fig.4).

– The used membrane model has a significant impact – The reduction in rheobase and SD time constant, on the observed excitation threshold. In order of due to the myelin approximation, is more significant increasing excitation threshold, we found average for larger electrode-neuron separations cathodal rheobase ratios of 0.22, 1, 2.40, 2.94 and – The impact of the myelin impedance on the 3.35 for the HH, FH, SRB, CRRSS and SE mod- rheobase current is dependent on the selected els respectively. E.g., this means that for an average membrane model. E.g., a maximal error in rheobase cathode position the SE-model is 3.35 times more of 57% was found for a model with Hodgkin-Huxley difficult to excite with long pulses than the FH- dynamics. In contrast, maximal rheobase errors are model. smaller than 3.2% for SE or SRB models. – The difference in excitability between the membrane – The impact of the myelin impedance on the models becomes more pronounced for larger separa- SD time constant is dependent on the selected tion between the electrode and the neuron, while membrane model. E.g., a maximal error in time con- the neuron is stimulated by the spatial gradient of stant of 70% and 20% is observed for a model with the electric field (SGE-mode). FH-dynamics and HH-dynamics, respectively. – The rheobase ratio is independent of the electrode- – The error in strength-duration time constant SD is neuron separation for neurons stimulated at the ter- independent of the electrode-neuron separation for minations (NT-mode). neurons in which NT-mode is the dominant (i.e., – Membrane channel dynamics determine the shape for short neurons or large electrode-neurons separa- of the strength-duration plot. HH, FH, CRRSS, and tions). SE models are better approximated by the Lapicque- Finally, the impact of electrode location and mem- Blair formula, while the Weiss-Lapicque equation is brane dynamics on the Ib(D) and τe(D) relations was a better fit for the SRB-model. Normalized SD-plots investigated. In our C-FIBRE models, we observed that are independent of the electrode position, stimulus (Fig. 11): polarity and waveform. – Excluding the HH-model, it is observed from com- – For sufficiently large electrode-neuron separations, parison between the SENN-MA and C-FIBRE model the rheobase Ib is inversely proportional to the fibre that for large cathode-neuron separations C-fibre diameter D (i.e., Ib ∝ 1/D). This result is indepen- models are about 77 to 80 times more difficult to dent of the membrane channel dynamics. Further- excite than A-fibre models. This C/A-multiplier is more, this result is in contrast with A-fibre models, 14 Thomas Tarnaud et al.

in which the slope of the log M-log D plot decreases And similar expressions for the other gate parameters n, h, with increasing diameter. and p. α and β are model specific rate constants, that de- – For sufficiently large electrode-neuron separations, pend non-linearly on transmembranevoltage V . Finally, k is a temperature constant, given by: the SD time constant τSD is independent of the fibre (T −T0)/10 diameter D. k = Q10 (11) ◦ ◦ These proportionalities (Ib ∝ 1/D and τSD ∝ 1) are The temperature T0 is 6.3 C for the HH-model, 20 C for the valid for sufficiently large electrode-neuron separations. FH-model, and 37◦C for the other membrane models. The Consequentially, deviations are found if the electrode is temperature constant Q10 = 3, except for: too close to the neuron. Larger electrode-neuron sepa- Q10,F H (αm) = 1.8; Q10,F H (αn) = 3.2; Q10,F H (αh) = 2.8 rations are required in the Hodgkin-Huxley model, as Q10,F H (βm) = 1.7; Q10,F H (βn) = 2.8; Q10,F H (βh) = 2.9 compared to the other simulated membrane models. Q10,SE (m) = 2.2; Q10,SE (h) = 2.9

Q10,SRB (m) = 2.2; Q10,SRB (h) = 2.9

A Summary of model equations Here Q10,I (j) refers to the temperature coeﬃcient for αj and βj of membrane model I. Model equations and parameters are summarized in this ap- Previous equations can be cast in two sets of matrix equa- pendix. All equations are taken from literature: references are tions, in the case of the conditionally linear membrane mod- given in section2. Systems of equations are discretised accord- els (HH and CRRSS models). Consequentially, a second order ing to a ﬁrst (SENN-M) or second (SENN-MA, C-FIBRE) correct solution is obtained without iteration for the Hodgkin- order staggered Crank-Nicolson scheme (time step ∆t and Huxley and CRRSS model. The other models (FH, SE, and spatial step ∆x). Discretisation of the cable equation in the SRB) are not conditionally linear and iteration of the equa- most general case yields: tions is necessary, to obtain a solution that is correct to second order in the discretisation time. ∗ 2Cm(l) ∆t ∆t Each model of membrane channel dynamics contains a (V (l, t + ) − V (l, t)) + Is(l, t + ) ∆t 2 2 loss current expressed by: ∗ ∆t ∆t = G (l − ∆l)(V (l − ∆l, t + ) − V (l, t + ))+ iL = gL(V − VL) (12) a,f 2 2 2 ∗ ∆t ∆t Here, gL is the leak conductance and is equal to 0.015 mS/cm , G (l + ∆l)(V (l + ∆l, t + ) − V (l, t + ))+ 2 2 2 2 a,b 2 2 0.3 mS/cm , 30.3 mS/cm , 128 mS/cm , 86 mS/cm , and 60 mS/cm2 for the myelinated internodes, HH, FH, CRRSS, SE, ∗ ∆t ∆t G (l − ∆l)(Ve(l − ∆l, t + ) − Ve(l, t + ))+ and SRB model respectively. Furthermore, the bias voltage a,f 2 2 VL is −59.4 mV, −69.974 mV, −80.01 mV, −78 mV, −84 mV ∗ ∆t ∆t Ga,b(l + ∆l)(Ve(l + ∆l, t + ) − Ve(l, t + )) (7) for the HH, FH, CRRSS, SE, and SRB model respectively. 2 2 An example of the spatiotemporal response, generated by the ∗ Here Cm(l) is the membrane capacitance of the compart- used membrane models, is shown in Fig. 12. ment at x = l. It is obtained from the capacitance per unit 2 2 2 area cm, which is 0.0073 µF/cm , 1 µF/cm , 2 µF/cm , 2.5 2 2 2 µF/cm , 2.8 µF/cm , and 2.8 µF/cm for the myelinated in- A.1 Hodgkin-Huxley model ternodes, HH, FH, CRRSS, SE, and SRB model respectively. G∗ (l) is the “backward” axial conductance between the a,b The ionic membrane current I is related to the ionic mem- compartments at x = l − ∆l and x = l. Similarly, G∗ (l) is s a,f brane current per area, by the expression isπd∆l = Is. The the “forward” axial conductance between the compartments Hodgkin-Huxley model consists of a natrium current i , at x = l and x = l + ∆l. V (l, t) is the transmembranepo- Na potassium current iK , and a loss current iL: tential, while Ve stands for the external potential. Finally, 3 Is is the ionic membrane current. The external potential is iNa = gNam h(V − VNa) (13) calculated analytically:

ρeI Ve = (8) 4 4πr iK = gK n (V − VK ) (14)

Here, ρe = 3 Ωm is the extracellular resistivity and r is the 2 2 gK = 36 mS/cm and gNa = 120 mS/cm are maximal con- distance from the electrode. The membrane voltage at V (t + ductances per area. VNa = 45 mV and VK = −82 mV are ∆t) is then calculated by: Nernst-potentials for sodium and potassium respectively. The ∆t Hodgkin-Huxley rest potential is Vr = −70 mV. V (t + ∆t) = 2V (t + ) − V (t) (9) 2 The rate coeﬃcients are given by (with V in millivolts):

The sodium gate parameter is progressed in time by: V˜ = V − Vr 3 1 ˜ ˜ m(t + 2 ∆t) − m(t + 2 ∆t) 2.5 − 0.1V V αm = ; βm = 4 exp − ∆t exp (2.5 − 0.1V˜ ) − 1 18 m(t + 3 ∆t) + m(t + 1 ∆t) ˜ ˜ = kα (t + ∆t)(1 − 2 2 ) 0.1 − 0.01V V m αn = ; βn = 0.125 exp − 2 exp (1 − 0.1V˜ ) − 1 80 m(t + 3 ∆t) + m(t + 1 ∆t) 2 2 ˜ − βm(t + ∆t) (10) V ˜ −1 2 αh = 0.07 exp − ; βh = (exp (3 − 0.1V ) + 1) 20 Comparison of numerical electrostimulation models 15

A.2 Chiu-Ritchie-Rogert-Stagg-Sweeney model A.4 Schwarz-Eikhof model

The CRRSS model consists of a sodium and loss current: The SE-model constains a sodium, potassium and loss current: 2 is = gNam h(V − VNa) + gL(V − VL) (15) 2 VF 3 VF [Na]0 − [Na]i exp RT iNa = PNam h VF (19) 2 RT 1 − exp Here, gNa = 1445 mS/cm , VNa = 35 mV, and Vr = −80 mV. RT The equations for the rate coeﬃcients are: 2 VF VF [K]0 − [K]i exp ˜ 2 RT 97 + 0.363V 15.6 iK = PK n VF (20) αm = ; βh = RT 1 − exp 31−V˜ 24−V˜ RT 1 + exp 5.3 1 + exp 10 1.87(V˜ − 25.41) 3.97(21 − V˜ ) α = ; β = m ˜ m ˜ 15.6 1 − exp 25.41−V 1 − exp V −21 α = 6.06 9.41 h h ˜ ih ˜ i 1 + exp 24−V exp V −5.5 10 5 0.13(V˜ − 35) 0.32(10 − V˜ ) αn = ; βn = ˜ 35−V˜ V˜ −10 97 + 0.363V 1 − exp 10 1 − exp 10 βm = h ih i 31−V˜ V˜ −23.8 ˜ 1 + exp 5.3 exp 4.17 0.55(V + 27.74) 22.6 αh = − ; βh = V˜ +27.74 56−V˜ 1 − exp 9.06 1 + exp 12.5

The rest potential of the SE-model is Vr = −78 mV. The A.3 Frankenhaeuser-Huxley model other constants are: P = 0.00328 cm/s; P = 0.000134 cm/s The Frankenhaeuser-Huxley model contains a sodium, potas- na K sium, non-speciﬁc and loss current. The rest potential is Vr = Na = 154 mmol/l; Na = 8.71 mmol/l −70 mV. 0 i K = 5.9 mmol/l; K = 155 mmol/l 0 i

2 VF 2 VF [Na]0 − [Na]i exp RT iNa = PNam h (16) RT VF A.5 Schwarz-Reid-Bostock model 1 − exp RT The SRB-model contains a sodium, fast and slow potassium

2 VF and loss current: 2 VF [K]0 − [K]i exp RT iK = PK n VF (17) 2 VF RT 1 − exp 3 VF [Na]0 − [Na]i exp RT RT iNa = PNam h (21) RT VF 1 − exp RT

2 VF 2 VF [Na]0 − [Na]i exp RT 4 ip = Ppp (18) iK,f = gK,f n (V − VK ) (22) RT VF 1 − exp RT

iK,s = gK,sp(V − VK ) (23) 0.36(V˜ − 22) 0.4(13 − V˜ ) α = ; β = m ˜ m ˜ Here, g = 30 mS/cm2, g = 60 mS/cm2, V = −84 mV 1 − exp 22−V 1 − exp V −13 K,f K,s K 3 20 and Vr = −84 mV. 4.6(V˜ − 65.6) 0.33(61.3 − V˜ ) 0.02(V˜ − 35) 0.05(10 − V˜ ) α = ; β = α = ; β = m ˜ m ˜ n ˜ n ˜ −V +65.6 V −61.3 35−V V −10 1 − exp 10.3 1 − exp 9.16 1 − exp 10 1 − exp 10 ˜ 0.0517(V˜ + 9.2) 0.092(8 − V˜ ) 0.1(V + 10) 4.5 αn = ; βn = αh = − ; βh = −V˜ −9.2 V˜ −8 V˜ +10 45−V˜ 1 − exp 1.1 1 − exp 10.5 1 − exp 6 1 + exp 10 −0.21(V˜ + 27) 14.1 0.006(V˜ − 40) 0.09(V˜ + 25) αh = ; βh = V˜ +27 55.2−V˜ αp = ; βp = − 1 − exp 1 + exp −V˜ +40 V˜ +25 11 13.4 1 − exp 10 1 − exp 20 0.0079(V˜ − 71.5) −0.00478(V˜ − 3.9) Here, the constants are: αp = ; βp = 71.5−V˜ V˜ −3.9 1 − exp 23.6 1 − exp 21.8 Pna = 0.008 cm/s; PK = 0.0012 cm/s; PP = 0.00054 cm/s The other constants are:

Na = 114.5 mmol/l; Na = 13.7 mmol/l Pna = 0.00704 cm/s 0 i Na = 154 mmol/l; Na = 30 mmol/l K = 2.5 mmol/l; K = 120 mmol/l 0 i 0 i 16 Thomas Tarnaud et al.

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Thomas Tarnaud is a Ph.D. stu- Luc Martens is a professor at dent in wireless, acoustics, environ- UGent. His interests are in mod- ment and expert systems (UGent- eling and measurement of electro- imec). His current research involves magnetic channels and exposure computational modeling of neu- and energy consumption rostimulation.

Wout Joseph is a professor in Emmeric Tanghe is a postdoc- the domain of experimental char- toral fellow of the FWO-V in radio acterization of wireless communica- propagation and parttime professor tion systems: UGent. He specializes in medical applications of electro- in EM exposure, propagation, and magnetic ﬁelds in and around the wireless performance analysis. human body.