Strength–Duration Relationship for Intra- Versus Extracellular Stimulation with Microelectrodes

NSC 13663 No. of Pages 13, Model 5G 29 April 2012

Please cite this article in press as: Rattay F et al. Strength–duration relationship for intra- versus extracellular stimulation with microelectrodes. Neu- roscience (2012), http://dx.doi.org/10.1016/j.neuroscience.2012.04.004

1 Neuroscience xxx (2012) xxx–xxx

2 STRENGTH–DURATION RELATIONSHIP FOR INTRA- VERSUS

3 EXTRACELLULAR STIMULATION WITH MICROELECTRODES

a a b,c 4 Q1 F. RATTAY, * L. P. PAREDES AND R. N. LEAO 2007; Sekirnjak et al., 2008; Horsager et al., 2009; Pezaris 21 5 a Institute for Analysis and Scientific Computing, Vienna University and Reid, 2009; Tehovnik et al., 2009) while avoiding side Q3 22 6 Q2 of Technology, A-1040 Vienna, Austria effects like co-stimulation of bypassing axons arising from 23 7 b Neurodynamics Laboratory, Department of Neuroscience, distant locations (Greenberg et al., 1999; Rattay and Re- 24 8 Uppsala University, Uppsala, Sweden satz, 2004). Considering the fact that neural substructures 25 9 c Brain Institute, Federal University of Rio Grande do Norte, Natal-RN, have different strength–duration characteristics may give 26 10 Brazil an opportunity for more selective stimulation. For example, 27 in the treatment of chronic pain during spinal cord stimula- 28 tion smaller dorsal column fibers can only be activated 29 11 Abstract—Chronaxie, a historically introduced excitability when pulse width is sufficiently large (Holsheimer et al., 30 time parameter for electrical stimulation, has been assumed to be closely related to the time constant of the cell mem- 2011). 31 brane. Therefore, it is perplexing that significantly larger Recently, the strength–duration relationship for extra- 32 chronaxies have been found for intracellular than for extracellular neural stimulation was analyzed under the 33 cellular stimulation. Using compartmental model analysis, assumption of a constant electrical field in flat, spherical 34 this controversy is explained on the basis that extracellular and cylindrical cells (Boinagrov et al., 2010). These inves- 35 stimulation also generates hyperpolarized regions of the tigations provided a biophysical basis for the stimulation 36 cell membrane hindering a steady excitation as seen in the with large electrodes and also explained effects of stimula- 37 intracellular case. The largest inside/outside chronaxie ratio tion with ultrashort pulses (<5 ls). In accordance with ret- 38 for microelectrode stimulation is found in close vicinity of inal ganglion cell experiments (Jensen et al., 2005), for 39 the cell. In the case of monophasic cathodic stimulation, pulse durations <2 ms, it was found that a 500 lm diam- 40 the length of the primarily excited zone which is situated between the hyperpolarized regions increases with eter electrode placed above the soma caused excitation 41 electrode–cell distance. For distant electrodes this results with significantly lower thresholds compared to a position 42 in an excitation process comparable to the temporal behav- above the axon. This result is unexpected given the 43 ior of intracellular stimulation. Chronaxie also varies along assumption that the axon is the most excitable part of a 44 the neural axis, being small for electrode positions at the neuron externally stimulated (Nowak and Bullier, 1998; 45 nodes of Ranvier and axon initial segment and larger at Rattay, 1999). Differences in experimental setups are the 46 the soma and dendrites. As spike initiation site can change reason for the controversy about such observations. Stim- 47 for short and long pulses, in some cases strength–duration ulation of the retina with large electrodes generates a 48 curves have a bimodal shape, and thus, they deviate from a rather constant field and consequently transverse currents 49 classical monotonic curve as described by the formulas of depolarize the cell membrane at one side and hyperpolar- 50 Lapicque or Weiss. Ó 2012 Published by Elsevier Ltd on behalf of IBRO. ize the cell at the opposite side. On the other hand, external 51 excitation with small electrodes is mainly based on stimu- 52 lating effects resulting from extracellular potential varia- 53 tions along the neural structure (Rushton, 1927; Ranck, 54 Key words: chronaxie, strength–duration curve, electrical 1975; Rattay, 1986, 1987, 1999). These facts vary the ex- 55 stimulation, microelectrode, activating function. pected excitation paradigm and should be carried in mind. 56 12 Voltage sensitive ion channel types and densities are 57 other important elements for neuron excitability. In rela- 58 13 INTRODUCTION tion to retinal implant, electrode positions applied at a 59 dense two-dimensional grid around the soma region of 60 14 Selective neural stimulation is a great challenge in the rabbit ganglion cells showed the lowest thresholds along 61 15 development of neural prostheses. As an example, active a small section of the axon, about 40 lm from the soma. 62 16 contacts of an electrode array implanted at the retina or At this axonal section, immunochemical staining revealed 63 17 other structures along the visual pathway should stimulate a dense band of voltage-gated sodium channels (Fried et 64 18 elements that elicit visual sensations corresponding to the al., 2009). Similarly, high sodium channel densities at the Q4 65 19 place in the array (Brindley, 1955; Zrenner, 2002; Dowling, axon initial segments (AIS) of cortical cells define sites for 66 20 ´ 2005; Fernandez et al., 2005; Fried et al., 2006; Cohen, action potential initiation under natural conditions (Stuart 67 et al., 1997; Yu et al., 2008; Hu et al., 2009) as well as 68 *Corresponding author. Tel: +43-1-58801-10114. spike initiation candidates for external stimulation with 69 E-mail address: [email protected] (F. Rattay). microelectrodes (Rattay and Wenger, 2010). 70 Abbreviations: AIS, axon initial segments; VSD, voltage sensitive dye.

0306-4522/12 $ - see front matter Ó 2012 Published by Elsevier Ltd on behalf of IBRO. http://dx.doi.org/10.1016/j.neuroscience.2012.04.004 1 NSC 13663 No. of Pages 13, Model 5G 29 April 2012

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71 A strength–duration curve describes electrode thresh- (Nowak and Bullier, 1998). Additionally, in cochlear im- 92 72 old current as function of stimulus pulse duration. Chron- plants, significantly shorter chronaxies have been de- 93 73 axie is defined as the time on such a strength–duration scribed in long-term (compared to acute) deafened 94 74 curve for twice the minimum (rheobase) current needed cochleae, indicating the loss of peripheral processes in 95 75 for very long pulses. Established by experimental find- many spiral ganglion cells (Shepherd et al., 2001). 96 76 ings, this excitability parameter was assumed to be rather Based on theoretical considerations, chronaxie has 97 77 independent from the distance between the current been suggested to be about 0.7 times the time constant 98 78 source and the excited cell (Weiss, 1901; Lapicque, of the cell membrane (Blair, 1932; Ranck, 1975; Reilly, 99 79 1907, 1929; Blair, 1932). 1992). Therefore, some authors assume chronaxie as 100 80 As different neural structures have different chronax- independent of in- or outside stimulation of the cell 101 81 ies, stimulus pulse duration is an important element for (Nowak and Bullier, 1998). This hypothesis contrasts with 102 82 selective stimulation as well as for electrophysiological reports of considerably larger chronaxies for intracellular 103 83 classification. For example, the shorter chronaxie of mye- than for extracellular stimulation (Ranck, 1975). 104 84 linated motorneurons compared to that of unmyelinated Discrepancy among different works prompts specula- 105 85 fibers of the myocardium allows safe stimulation with tions that experimental differences could be caused by 106 86 short pulses during artificial respiration. Motoneurons artifacts like sampling errors or neuronal damage during 107 87 then become activated with little disturbance of the recordings (Ranck, 1975). To address this controversy, 108 88 myocardial function (Voorhees et al., 1992). In electro- we systematically reproduced experimental findings using 109 89 physiological studies, extracellular gray matter stimulation computer simulations of a pyramidal cell and its simplified 110 90 shows larger chronaxies for somas than for axons, sup- rectified version for electrode positions at the dendrite, the 111 91 porting the idea that spike initiation occurs in the axon soma, the lateral ending of the AIS and the myelinated 112

A B dend SOMA apical soma dend is not excitable for shorter pulses dendrite DEND ais node AIS 100 NEURON 1 NEURON 2 NODE

rectified CHRONAXIE neuron NODE electrode positions node AIS 50 µm main 10 dendrite DEND dend above dendrite dend ais d~4,7µm extracellular stimulation SOMA soma soma soma basal AIS axon hillock AIS dendrite naked 1 µA axon 0.01 0.1 1 10 ms N.R. PULSE DURATION N.R. naked (thin) axon SOMA myelin. N.R. node axon first dend N.R. N.R. internode DEND 0.1 N.R. N.R. soma unmyel. NODE N.R. = node of Ranvier axon N.R. N.R. N.R. 0.01 ais N.R. AIS N.R. internode intracellular stimulation 200µm N.R. node NODE N.R. N.R. 0.001 N.R. N.R. x N.R.

0.0001

Fig. 1. Strength–duration curves of the investigated pyramidal cell. (A) Neuron 1 (left) and Neuron 2 (right) with selected electrode positions. (B) Strength–duration curves with chronaxie values for electrode positions as marked in A. Lower and upper case letters are used for electrode positions for Neuron 1 and Neuron 2, respectively. Color coding of lines in B corresponds to electrode positions in A. For better discrimination some extracellular strength–duration curves are shown as broken lines. Colors of break point markers are without meaning. Anodic and cathodic square pulses for intra- and extracellular stimulation, respectively.

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3 113 axon (Fig. 1). In all cases shown in Fig. 1B, the intracellu- are calculated by INav1.j = gNav1.jm h (V ENa) with j equals to 179 114 lar/extracellular chronaxie ratio is two or higher. It is either 2 or 6 and ENa = 60 mV. Details on the differential equations 180 of the different variables are shown in Mainen et al. (1995). The 181 115 important to note that these ratios depend on cell proper- values for the half (in)activation voltages V1/2, the slopes k and 182 116 ties as well as on the distance and type of electrodes. the coefficients A were obtained from a previously published mod- 183 117 Such variants could be the reason for the diversity in el in the NEURON Model DB (Hu et al., 2009) after subtracting the 184 118 chronaxie data. corresponding value for the shift of voltage dependence of the 185 119 Here, we demonstrate that (i) hyperpolarized regions kinetics. Consequently, the currents INav1.j have the same values 186 120 are responsible for the shorter extracellular chronaxies, for A, i.e., A(am) = 0.182, A(bm) = 0.124, A(ah) = 0.024, A(bh 187 ) = 0.0091, and the slope of inactivation, i.e., k(s ) = 5 and 188 121 (ii) for short electrode distances, excitability and chronaxie h k(h ) = 6.2, in contrast to altered slope of activation, i.e., 189 122 differ essentially along a selected neuron and conse- 1 k(sm )=k(m1 ) = 7 for Nav1.2 but k(sm )=k(m1 ) = 6 for 190 123 quently (iii) strength duration curves are expected to devi- Nav1.6. To account for the reduced threshold of Nav1.6 channels 191 124 ate from the classical form when short and long pulses V1/2(m) is decreased to 41 mV compared to the calculated value 192 125 cause action potential initiation at sites with different elec- of 28 mV for activation of Nav1.2 channels. The corresponding 193 126 trical membrane properties. values in mV for the inactivation of Nav1.2/Nav1.6 channels are 194 V1/2(ah)=35/41, V1/2(bh)=60/73, V1/2(h1)=57/70. 195 The potassium currents are determined by IK = gKn(V EK) with 196 127 EXPERIMENTAL PROCEDURES EK = 90 mV. To be consistent the corresponding values of 197 A(a) = 0.02, A(b) = 0.002, V1/2(a)=V1/2 (b) = 25 mV and 198 k( )=k(b) = 9 were also obtained from the NEURON Model 199 128 An analysis of two model neurons is presented. In both models a DB (Hu et al., 2009). Internodes are simulated with 17 sheets of 200 129 excitation is based on recently measured densities of high thresh- membrane with a conductance of 1 mS/cm2 and C =1 F/cm2 201 130 old sodium channels Nav1.2 in dendrites and the soma and low l per sheet (Rattay, 1999). The presented results are simulated 202 131 threshold sodium channels Nav1.6 in the axon (Hu et al., for 37 °C. 203 132 2009). The phenomenon of short chronaxies for extracellular In the first modeling step, the extracellular potential V gener- 204 133 stimulation was also tested in simulations with other neural struc- e ated by an electrode tip is approximated considering a monopolar 205 134 tures (not shown) and did not depend on the specific examples. spherical electrode in an infinite homogeneous extracellular med- 206 135 Model Neuron 1 (Fig. 1A left, Rattay and Wenger, 2010), a ium with resistivity = 300 X cm. This is equivalent to a point 207 136 simplification of Neuron 2 concerning geometry and ion channel qe source stimulation resulting in spherical isopotentials with V = 208 137 types, is straightforward to analyze and with less computational e I /4 r when a current pulse with amplitude I is applied; r is 209 138 cost. It has a straight axis and consists of a single non-branching qe el p el the distance from a point of interest to the point source. 210 139 dendrite (500 lm, d =5lm), spherical soma (d =20lm), axon In the second step, the response of a neuron is simulated by 211 140 hillock (10 lm, d = 3.1 lm), AIS (50 lm, d = 1.22 lm), naked a compartment model. That is a network of resistances and 212 141 axon (unmyelinated, 200 lm, d =1lm), myelinated axon capacitances, where the current to the centre of compartment n 213 142 (500 lm, d =1lm) and unmyelinated terminal (50 lm, consists of the following components: a capacitive current, ion 214 143 d =1lm). Assumptions for ion channel distribution and ion cur- currents across the membrane and intracellular currents to neigh- 215 144 rent computations are quite similar as in Hu et al. (2009): the bored compartments. Applying Kirchhoff’s law for compartment n 216 145 same constant Nav1.2 channel density for the dendrite and soma 2 results in 217 146 (gna = 8 mS/cm ), but 40 times higher sodium channel density 218 147 in hillock and AIS with a change to the low threshold type dðVi;n Ve;nÞ Vi;n Vi;n1 148 Nav1.6 in the axon. Intracellular resistivity is 150 X cm, mem- Cn þ Iion;n þ dt R =2 þ R =2 149 brane capacity c =1lF/cm2. Using ACSL (Advanced Continu- n n1 V V þ 1 150 ous Simulation Language) software, Neuron 1 is simulated by þ i;n i;n ¼ 0 ð1Þ 151 evaluating the compartment model equations (1)–(3) as de- Rn=2 þ Rnþ1=2 220 152 scribed below. 153 Model Neuron 2 (Fig. 1A right) is based on geometrical with intracellular potential Vi, axial resistance R and membrane 221 154 parameters of a traced cortical layer 5 pyramidal cell with ion capacity C. The following system of differential equations is deduced 222 155 channel assumptions as available in the NEURON (Carnevale by introducing the transmembrane voltage V ¼ Vi Ve to compute 223 156 and Hines, 2006) Model DB (Hines et al., 2004; Hu et al., the time courses of Vn in every compartment (Rattay, 1999): 224 225 157 2009). In contrast to Neuron 1, soma diameter is increased to dVn Vn1 Vn Vnþ1 Vn 158 30 lm, naked axon diameter is reduced to 0.4 lm and the capac- ¼I þ þ dt ion;n R =2 R =2 R =2 R =2 159 ity of the cell membrane of all compartments with exception of the n1 þ n nþ1 þ n 2 160 soma and internodes is reduced to 0.5 lF/cm . Moreover, Ve;n1 Ve;n Ve;nþ1 Ve; n þ þ Cn: ð2Þ 161 Neuron 2 incorporates branching dendrites, tapering diameters, Rn1=2 þ Rn=2 Rnþ1=2 þ Rn=2 227 162 uneven ion channel distribution within compartments of a single 163 type and a membrane capacity of 0.02 lF/cm2 in the axonal inter- The direct stimulating influence of the extracellular potential on 228 164 nodes. High threshold sodium Nav1.2, low threshold Nav1.6 and compartment n is defined by the activating function (Rattay, 1999) 229 230 165 fast voltage-gated K+ of Neuron 1 are complemented by slow 2+ Ve;n1 Ve;n Ve;nþ1 Ve;n 166 non-inactivating potassium current, high-voltage activated Ca fn ¼ þ Cn: ð3Þ + 232 167 and calcium dependent K in dendritic and somatic compart- Rn1=2 þ Rn=2 Rnþ1=2 þ Rn=2 168 ments. All parameters are used as in Hu et al. (2009). However, For a fiber with constant diameter d, constant compartment length 233 169 in order to demonstrate current–distance relations in a clear way, Dx, intracellular resistivity qi and specific capacity c, (3) appears 234 170 the axial 3d structure of the cell was compressed into the x–y in a simpler form 235 171 plane. Neuron 2 is simulated with NEURON. 236 172 In more detail, Neuron 1 soma and dendrite were imple- d Ve;nþ1 2Ve;n þ Ve;nþ1 fn ¼ : ð3aÞ 173 mented with the same constant maximum conductances (in mS/ 4c Dx2 238 2 qi 174 cm ) gNav1.2 =8,gNav1.6 = 0 and gKv = 10, the axon hillock with 175 gNav1.2 = 320, gNav1.6 =0, gKv= = 100, the AIS with gNav1.2 = The value within the brackets of (3) corresponds to a virtual in- 239 176 100, gNav1.6 = 320, gKv = 100, the unmyelinated axon with jected current applied to compartment n. In regions where this cur- 240 177 gNav1.2 =0,gNav1.6 = 300, gKv = 150 and the nodes of Ranvier rent is positive, the membrane depolarizes and where it is 241 178 with gNav1.2 =0,gNav1.6 = 160, gKv = 20. Sodium current kinetics negative, it tends to hyperpolarize. 242

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243 VOLTAGE SENSITIVE DYE IMAGING A Soma without processes, intracellular stimulation

244 Transverse hippocampal slices were obtained from P21 245 C57B6 mice as previously described (Leao et al., 2009) Iel =150pA 246 and according to the rules of Animal Experimentation of Multiplication of subthreshold 247 the Uppsala University. Slices were maintained in artificial response with a factor 3 corresponding to Iel =150pA 248 ACSF (in mM: 124 NaCl, 3.5 KCl, 1.25 NaH2PO4, 1.5 transmembrane 249 MgCl2, 1.5 CaCl2, 30 NaHCO3, 10 glucose), constantly voltage V 250 bubbled with 95% O2 and 5% CO2. Recordings/imaging Subthreshold response 251 were obtained at 25 °C, in the presence of 10 lM CNQX, Iel =50pA 30mV 252 30 lM dAP5 and 10 lM bicuculline methochloride to min- 253 imize the effect of synaptic currents (Leao et al., 2005). 1ms 254 Voltage sensitive dye (VSD) loading (JPW3027 obtained 255 from Prof Leslie Loew, University of Connecticut, USA) =3.4ms 256 was performed exactly as described in Palmer and Stuart 257 Q5 (2006). Images were acquired using a EM-CCD camera Iel =50pA 258 (Luca, Andor, Ireland). Excitation was produced by a 10mV 259 200W metal-halide lamp through a bypass filter centered 0.63*Vstationary 260 at 535 nm (510 to 560 nm pass) and emission was low- 1ms 261 pass filtered at 590 nm. Image acquisition and extracellular 262 stimulation were synchronized by a National Instruments STIMULUS CURRENT 263 digital device and to guarantee time precision, we recorded 264 the stimulator and the camera ‘fire’ outputs (that flags im- B Soma with dendrite, intracellular stimulation 265 age acquisition) using a National Instruments DAQ card. t1=1.4ms, t 2=5.3ms 266 Current clamp recordings were also obtained from the im- 267 aged cell using winWCP (Dr John Dempster, Strathclyde I =50pA 268 University, UK). The stimulating electrode (tungsten, el 2mV 269 10lm tip) was placed 50 and 75 lm above axons. Pulses 0.63*Vstationary 270 had 100 ls durations and extracellular current intensities 1ms 271 were adjusted to 75% of the minimum threshold current. 272 Images were taken 100 ls after the stimulus using STIMULUS CURRENT 273 100 ls exposure. Pixel intensity was measured in 4 2 274 pixel region of interest and averaged from 10 images ac- C Soma with axon and dendrite, intracellular stimulation 275 quired with 1s delay between each other. t1=1.4ms, t 2=4.3ms

Iel =50pA 276 RESULTS 1mV 277 In the following several computer experiments are 0.63*Vstationary 1ms 278 proposed to analyze step by step characteristics of 279 strength–duration curves and related historical formulas. STIMULUS CURRENT

Fig. 2. Transmembrane voltage V of Neuron 1 for intracellular 280 Intracellular stimulation: space clamp versus cable stimulation at the soma. (A) Space clamp condition with no axial 281 model current flow. The top graph demonstrates the coincidence of the passive and active membrane response in the subthreshold regime. 282 In order to describe the neural excitation process inde- Linear extrapolation of V at stimulus onset up to the steady state 283 pendently from the intracellular current needed for spike value of the single RC circuit (gray arrow) defines the time constant of 284 conduction, Hodgkin et al. (1952) used the space clamp the passive membrane response as s = R C (lower graph of A). 285 technique. They stimulated with a long inserted wire that With the same electrode current Iel = 50 pA, the steady state value of V is reduced by a factor 5 when axial current flows into the dendritic 286 clamped the squid axon uniformly along its entire length. branch (B) and by a factor 10 for conductance into dendrite and 287 An equivalent model situation is created by current injec- axon (C). In comparison with the space clamp condition, intracellular 288 tion into the soma of Neuron 1 by cutting all soma pro- current flow into dendrite and axon results in shorter times t1 but 289 cesses. Stimulation of the resting soma with a rather longer times to reach the steady state of the subthreshold membrane. 290 long weak pulse results in a transmembrane voltage 291 curve V(t) with an asymptotic exponential increase during end of the 8 ms pulse, the subthreshold approach of V 300 292 the subthreshold response (Fig. 2A). The time constant is very close to its stationary value. 301 293 for the exponential voltage of the passive membrane is The shape of the subthreshold membrane voltage V(t) 302 294 the product of the ohmic resistance and the capacity of changes considerably when the stimulus current is 303 295 the cell membrane (s = R C). The time constant s of allowed to flow not only across the membrane but also 304 296 such an exponential increase can be found graphically intracellularly from the stimulated soma into the dendrite 305 297 by linear extrapolation at stimulus onset as the intersec- and the axon. Finding s by the graphical method of 306 298 tion of the tangent in V at pulse onset with the steady state Fig. 2A becomes contradictory when applied in Fig. 2B 307 299 indicated by the horizontal dashed line in Fig. 2A. At the and C as it results in shorter time values t1 while needing 308

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309 longer times for reaching the steady state (pulse dura- electrode current Iel 310 tions >20 ms). Obviously, the fit by a single exponential extracellular 311 function corresponding to a single RC element (as in case 1000 chronaxie = 0.38ms µA 312 A, where 63% of Vstationary is determined by s) cannot space clamp 313 describe the subthreshold response of the cable model chronaxie = 1.79ms 314 in an acceptable way. We have to change the method intracellular 100 chronaxie = 2.5ms 315 and define the time constant as the time t2 when the cable 2 rheobase 316 model reaches 63% (1 1/e) of the stationary membrane * rheobase I 317 voltage V (Fig. 2B and C). Note that the time constant of 0 318 the cable model is not an average value of the time con- 10 319 stants of the compartments. This is demonstrated by 320 Fig. 2B where both the dendrite and the soma of Neuron1 pulse duration 321 have quite the same electric membrane properties. The 1 322 main difference between s and its replacing values t1 0.01 0.1 1 10ms 323 and t2inFig. 2A and B is a consequence of a distinctly 324 reduced steady state membrane voltage when the greater Fig. 3. Strength–duration curves of the somatic membrane of Neuron 325 part of the stimulus current flows into the dendrite. The dif- 1 for stimulation with square pulses. Chronaxie defined as the pulse duration needed for twice the rheobase I (horizontal arrows) is 326 ferent temporal excitation profiles shown in Fig. 2 have 0 different for the space clamp condition (central group of three curves) 327 consequences on the corresponding strength–duration and the case of current flowing into dendrites and the axon (upper 328 curves. curves). Injected anodic current amplitudes (upper and central 2 329 The middle solid line in Fig. 3 shows the strength– groups) are shown as current densities for 1 cm membrane. 330 duration relation under the space clamp condition where Extracellular stimulation (lower curves) is with cathodic currents for a point source 50 lm above soma center. Approximation of the 331 chronaxie is independent from the soma area. Allowing strength–duration curves according to the classical formulas of 332 k part of the injected current to flow into dendrites and the Lapicque (1907): Iel =I0/(1 2 ) (dashed lines) and Weiss (1901): 333 axon demands for stronger threshold currents for all pulse Iel = I0(1 + 1/k) (dotted lines) where k is pulse duration/chronaxie. 334 durations (Fig. 3, upper solid line). The intracellular 335 strength–duration curves for Neuron 1 and its space currents of both polarities where the sum of all the virtual 369 1 336 clamped version deviate in their shapes for long pulses. currents is zero. The driving forces of excitation are the 370 337 With the same electrical cell properties, the calculated currents in the region with positive activating function val- 371 2 338 intracellular chronaxie ratio 2.5/1.79 = 1.4 (Fig. 3) ues (red arrows in Fig. 4C). 372 339 becomes smaller by enlargement of the soma surface The length of the region where a cathodicpffiffiffi point source 373 340 or by reducing the number or diameters of processes. causes positive activating function is 2 zel, where zel is 374 341 The variable chronaxie ratio decreases to 1 if the diame- the electrode distance. This formula is correct for a 375 342 ters of the soma processes converge to zero, which is the straight nonmyelinated fiber with constant diameter in a 376 343 space clamp condition. homogeneous medium and the relation canpffiffiffi be described 377 344 This dependence of chronaxie on diameter disproves by a 70-degree angle (Fig. 4A, tan(35°)= 2=2 zel=zel), 378 345 the often applied rule mentioned in the introduction, (Rattay, 1986). According to the activating function 379 346 namely chronaxie 0.7s with s = R C, where the prod- concept, this angle is independent of fiber diameter and 380 347 uct R C is independent of membrane size. However, electrical membrane properties and can also be used as 381 348 even more surprising is the huge deviation from this for- an approach for myelinated axons (Rattay, 1986). Large 382 349 mula for extracellular stimulation. With the time constant positive and negative isolated activating functions’ values 383 350 of the soma membrane (Fig. 2A) we obtain chronaxie appear at locations with considerable diameter changes, 384 351 0.7 3.4 ms = 2.38 ms instead of 0.38 ms for the case in branching or in fiber endings. As axial intracellular cur- 385 352 presented in Fig. 3. rents level such local effects, the 70° rule is a rough ap- 386 353 In contrast to intracellular stimulation, in most applica- proach even for these cases. A specific chronaxie value 387 354 tions extracellular stimulations are optimally achieved with of a region can be expected as long as the zone defined 388 355 cathodic currents (Ranck, 1975; Rattay, 1986, 1999). by the 70° seen from the point source is concentrated in a 389 356 Hence, we compare anodic inside with cathodic outside cell region with common electrical properties. 390 357 stimulation in order to explain in the next subsections The length of the activated portion flanked by the hyper- 391 358 the large chronaxie differences between intra- and extra- polarized regions increases with electrode–neuron dis- 392 359 cellular stimulation shown in Figs. 1B and 3. tance, and consequently, the influence of the 393 hyperpolarized region on the excitation process is gradu- 394 395 360 Extracellularly activated region increases with ally reduced when moving the current source away. This 361 electrode distance

1 362 A straight fiber with the properties of the naked axon of Eq. (1) is a current balance where the last two terms describe the intracellular current flow to the left and right neighboring compartments. Shifting to the next 363 Neuron 1 is shown in Fig. 4A–C for external stimulation compartment (n ? n + 1) includes the ‘old’ right neighbor as new left neighbor, but 364 with a cathodic 100 ls pulse at threshold intensity. In con- the current flow has changed signs. Thus, the sum of these two currents is 0 and 365 trast to intracellular stimulation, the externally positioned consequently the sum of all axial currents is 0. Applying this principle in Eq. (2) one find that the sum of all virtual stimulating currents is 0. For details see Rattay 366 microelectrode causes in every compartment an injected (1990). 367 virtual current. As second important effect, outside 2 For interpretation of color in Figs. 1, 4 and 10, the reader is referred to the web 368 stimulation with a monophasic pulse produces virtual version of this article.

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A z effect could be also demonstrated using VSD imaging. The 396 -25mV length of the depolarization detected by the intracellular 397 -50mV electrode VSD increases with electrode distance (Fig. 4D). The area 398 -100mV Iel=-25µA, Vel=-500mV under the curve in fluorescence (normalized by the maxi- 399 mum) versus length graph was used as a measurement 400 zel=50µm axon B 0 of depolarization ‘spread’ caused by the extracellular stim- 401 70° x zel * 2 ulus. When the stimulating electrode was placed 50 lm 402 -25 (zel =50lm) above the axon, the fluorescence (DF/F0, 403 -50 normalized) versus length integral was equal to 0.038 ± 404 1 1 0.002 lm and 0.058 ± 0.004 lm for zel =75lm 405 -75 extracellular (n = 5 cells, p = 0.008, paired t test). 406 potential Ve -100 mV along the fibre Intracellular versus extracellular stimulation 407 C 750 mV/ms The spatio-temporal evolution of transmembrane voltage 408 500 profiles is significantly different for a microelectrode posi- 409 activating function tioned either above or in the soma (Fig. 5A–D). In Fig. 5A 410 250 100µm and virtual injected currents and B, pulse duration is 100 ls and membrane voltages 411 0 along the neural axis are compared for threshold intensity. 412 At the end of the pulse, extracellular stimulation shows a 413 -250 hyperpolarized activated hyperpolarized region region region larger voltage maximum, but a shorter depolarized region. 414 D For 1 ms threshold pulses the same voltage profiles ap- 415 Rest pear again with smaller amplitudes (due to lower stimula- 416 tion currents) as the firsts of ten advancing 100 ls time 417 steps (thick lines in Fig. 5C and D). The rather constant in- 418 crease of the maximum and spatial extension of the volt- 419 age profile for the case of intracellular stimulation in the 420 ED=50 m advancing 100 ls time steps demonstrate that intracellular 421 stimulation is more effective as the injected current is avail- 422 able to load the membrane capacity in the vicinity of the 423 electrode (Fig. 5D). Virtual negative injected currents dur- 424 ing pulse application (Eq. (3)) cause strong hyperpolariza- 425 ED=75 m tion (Fig. 5A and C). An essential part of the virtual positive 426 current, the driving force for excitation, is lost as a result of 427 the counterbalancing axial current flow between the depo- 428 larized and the two hyperpolarized regions. Note that half 429 0 1 of the positive voltage profile (its extension and maximum 430 Arbitrary Pixel Intensity value) is already reached at approx. 10% of the 1 ms stim- 431 ED=75 m ulation pulse (thick line in Fig. 5C). This is contrary to the 432 ED=50 m Soma 0.2% gradual increase of the voltage profile seen in Fig. 5D. 433 F/F Axon 0 A second important effect is demonstrated in the 434 10 ms pulse example (Fig. 6). The subthreshold response 435 Fig. 4. Extracellular stimulation. (A) Geometry and isopotentials for a for extracellular stimulation has a maximum at half of the 436 point source 50 lm above a fiber positionedq atffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the x-axis. (B) stimulus pulse time with a decay related to the inactivation 437 2 2 Extracellular potential Ve = qeIel/4pr with r ¼ ðx xelÞ þ zel is gating variable h (Mainen et al., 1995) that act as a factor 438 3 used to calculate the activating function. (C) A fiber with d =1lm for the sodium current (Fig. 6A). The quicker extracellu- 439 and compartment length Dx =10lm results in a peak activating larly elicited voltage increase affects the lower h values. 440 function value of 740 mV/ms. According to (2) this is the slope of the The maximum appears earlier for extracellular electrode 441 membrane voltage in the compartment below the electrode at the beginning of the 25 lA pulse. According to (3), the virtual currents positions above AIS and node of Ranvier (Fig. 6C). In con- 442 7 are fn Cn. With Cn = d p Dx c = 3.14159 * 10 lF and specific trast, Fig. 6B and D shows a rather constant increase in 443 2 membrane capacity c = 1 lF/cm , the maximum injected current is membrane voltages indicating that stimulation is still possi- 444 232 pA at the center of the activated region. As the length of the ble with longer and weaker pulses. 445 activated region is defined by an angle of 70°, this region increases with electrode distance. (D) VSD imaging of a pyramidal cell axon at rest (upper photomicrograph), 100 ls after the stimulus for electrode Chronaxie increases with electrode distance 446 distance zel =50lm (middle photomicrograph) and zel =75lm (lower photomicrograph) (scale bar=10 lm; ⁄ electrode position in In the previous section and with Fig. 3, it was demon- 447 the horizontal axis). Fluorescent traces for zel = 50 and 75 lm strated that axial current flow from the activated area into 448 versus axon position is shown below the photomicrographs and are aligned to the region in the axon activated by the extracellular 3 3 stimulus. Each point in the traces represents average fluorescence of Sodium current, the driving component in excitation, is modeled as gNam an axon region of approx. 10 lm length in five images. h(V ENa), with maximum conductance gNa, gating variables m and h, membrane voltage V and Nernst potential ENa. In the resting state m has a quite small value in contrast to the inactivation variable h which starts with a high value that decreases (with some delay) when V increases.

F. Rattay et al. / Neuroscience xxx (2012) xxx–xxx 7

extracellular stimulation intracellular stimulation

electrode electrode dendrite axon dendrite axon soma soma AB pulse duration 0.1 ms

Iel =-37 µA Iel =7.5 nA

resting voltage -70 mV

500 µm

t=100 µs 20 mV CDt=100 µs pulse duration 1 ms Iel =1.0 nA Iel =-8.6 µA

Fig. 5. Spatial transmembrane voltage proﬁles for extra- and intracellular stimulation computed for Neuron 1. Extracellular stimulation 50 lm above the soma (A, C) with cathodic threshold currents. Corresponding intracellular anodic threshold current cases at the right side (B, D). Voltage along the neural axis at the end of a 100 ls stimulation pulse (A, B) and in ten advancing 100 ls time steps for a 1 ms stimulation pulse (C, D).

extracellular stimulation intracellular stimulation A B pulse duration 10 ms h=1 I =0.28 nA Iel =-7.1 µA el

2 ms Iel =-7.0 µA 20 mV Iel =0.27 nA h=0 soma soma C D pulse duration 5 ms

node of node of AIS Ranvier AIS Ranvier

Fig. 6. Comparison of threshold and subthreshold membrane voltages as functions of time of the compartments closest to the electrode, computed for Neuron 1. Extracellular cathodic stimulation 50 lm above soma (A), AIS and node of Ranvier (C). Corresponding intracellular anodic stimulation at the right side. The dashed lines in A and B show the inactivation gating variable h for subthreshold cases; note its quicker decay and the lower minimum for extracellular stimulation. During pulse application there is a monotonous voltage increase in all intracellularly stimulated cases, whereas extracellular stimulation causes a maximum in every of the subthreshold voltage curves, most pronounced in the node of Ranvier example. As shown in A, the maximum is a consequence of the decreasing inactivation variable h. Pulses loose their stimulating properties as soon as the maximum is reached. Same scaling in all graphs.

449 the hyperpolarized regions causes shorter chronaxies for shows a reduction when intracellular resistance between 456 450 cathodic extracellular than for intracellular stimulation. It depolarized and hyperpolarized regions increases by 457 451 was also shown that the activated length is related to an increasing electrode distance. Comparison of excitation 458 452 angle of 70° at the electrode and this activated length in- proﬁles for electrodes at 50 and 200 lm above the soma 459 453 creases with microelectrode distance (Fig. 4). Conse- (Fig. 7) emphasizes the trend to longer chronaxies for lar- 460 454 quently, stimulating current loss along the neural axis ger electrode distances. As seen in the case at 200 lm 461 455 into the side lobes as deﬁned by the activating function above the soma, the larger distance between the primarily 462

8 F. Rattay et al. / Neuroscience xxx (2012) xxx–xxx

A electrode 50µm above soma 100% Iel =-8.6 µA

56%

10 mV 250 µm 500 µs

B electrode 200µm above soma 100% Iel =-55 µA 25%

C intracellular soma stimulation 100% Iel =1.0 nA 24%

Fig. 7. Temporal evolution of transmembrane voltage profiles in 100 ls time steps during a 1 ms pulse at threshold stimulation above the soma (left) and soma membrane voltage as a function of pulse time (right). In case A, 56% of the transmembrane voltage is already reached after the first 100 ls. Increasing the electrode distance from zel =50lm (A) to zel = 200 lm (B) causes a similar and consistent voltage increase as for intracellular stimulation. For both cases approx. 25% of the transmembrane voltages are reached after the first 100 ls (C) Left figures A and C are replications of Fig. 5C and D.

463 depolarized and hyperpolarized regions results in a more for zel = 200 lm. There is a change in the sign of the 491 464 constant increase of the voltage profile which becomes slopes at the onset of the 200 ls stimulus within 492 465 comparable to that of the intracellular case. the naked axon region in (A), whereas for zel = 200 lm 493 466 Strength–duration curves for electrode positions the slopes are always positive in the larger region marked 494 467 above the soma and a node of Ranvier underline the by the gray rectangle indicating the activated region 495 468 association of increasing chronaxies with increasing elec- (Fig. 9B). In case B, the activating function values at the 496 469 trode distance (Fig. 8). Chronaxie was determined by the border between AIS and the naked axon are essentially 497 470 intersection of strength–duration curves with the horizon- smaller than below the electrode. However, the high so- 498 ⁄ 471 tal line 2 rheobase, with electrode currents normalized to dium channel density of AIS supports spike initiation at 499 472 rheobase (Fig. 8). The lower part of the figure uses the the naked axon, acting as a favorable neighboring com- 500 473 chronaxie values of the shown strength–duration curves partment. The activating function values for the AIS are 501 474 for electrode distances of 50, 100, 200 and 300 lmto very similar to those values for the naked axon close to 502 475 quantitatively describe chronaxie versus distance rela- it, but its position next to the soma results unfavorable. 503 476 tionships for electrodes above the third node of Ranvier The convenient neighboring compartments together with 504 477 and the soma. For the extracellular stimulation cases, axial current flow are crucial components for excitation 505 478 when increasing electrode distance, different functional during intra- and extracellular stimulation. 506 479 neural parts become excited. For this reason, and more In the next section we show that in some cases axial 507 480 remarkably for the soma case, the curves containing the current flow initiates spikes rather far away from the stim- 508 481 circles deviate from the direction pointed by the arrows ulating electrode when long pulses are applied whereas 509 482 (Fig. 8, bottom). Note the quite large difference between for short pulses spike initiation is close to the electrode. 510 483 intra- and extracellular chronaxies. By systematic evaluation and analysis of computer simu- 511 484 A shift of the spike initiation zone by electrode dis- lations we discovered this phenomenon with remarkable 512 485 tance increase is analyzed in the next example with two consequences for strength–duration curves. 513 486 electrode positions at zel =50lm and zel = 200 lm 487 above the center of the naked axon of Neuron 1 Deviations from classical strength–duration curves 514 488 (Fig. 9). In contrast to the 50 lm case (A) the depolarized 489 region predicted by the activating function with the 70-de- In order to smoothen recording errors, strength–duration 515 490 gree angle (Fig. 4) exceeds the 200 lm long naked axon curves are usually fitted by one of the classical 516

F. Rattay et al. / Neuroscience xxx (2012) xxx–xxx 9

4* rheobase A

dendrite current inject ed in the soma in a node of Ran vier 3 soma AIS axon hillock naked axon 2 N.R. N.R.

myelin. N.R. electrode axon N.R. electr ode 1 electr ode N.R. activated region 50µm 50µm 100µm 100µm unmyel. 200µm axon 200µm STIMULUS Iel =-15µA (A) 300µm 300µm 200µm 100mV Iel =-150µA (B) above node of Ran vier above soma 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 ms B 200µs

Fig. 8. Strength–duration curves for Neuron 1 normalized to rheo- dendrite base for electrodes above the soma (solid lines) and the third node of Ranvier (broken lines). Chronaxies marked by vertical arrows increase with electrode–cell axis distance toward the value for inside stimulation (marked by thick lines) which is 1.1 ms for the third node of Ranvier (comp. Fig. 1B) and 2.6 ms for the soma. The circles in the soma AIS lower part of the ﬁgure help to identify the strength–duration curves. axon hillock The curves containing the circles show a trend toward larger naked chronaxies with increasing electrode distances. axon 70° N.R. N.R. 517 approaches, the formulas of Weiss and Lapicque (Fig. 3). myelin. N.R. 518 However, for a constant electrode position our computer axon 519

simulations demonstrate that deviations from these N.R. electrode 520 curves have to be expected when spikes are initiated at N.R. 521 diﬀerent functional parts for short and long pulses. Intra-

522 cellular stimulation of the main dendrite of Neuron 2 re- unmyel. activated region axon 523 sults in a composite strength–duration curve. This curve 524 consists of two segments of ‘classical curves’ that are Fig. 9. Stimulation of Neuron 1 by a microelectrode 50 lm (A) and 525 connected with a transition part (thick red curve marked 200 lm (B) above the center of the naked axon. Every line on the 526 as DEND in the lower part of Fig. 1B, the case of intracel- right side shows the membrane voltage of one compartment in 527 lular stimulation). accordance with the cell geometry (left). Short compartment lengths in the unmyelinated axon are because of numerical reasons. Thick 528 The occurring phenomena for this main dendrite and gray lines represent the membrane voltage where spikes are initiated 529 other neural structures are explained extensively in which is at the center of the naked axon corresponding to the 530 Fig. 10. For short pulses, spikes are generated within electrode position in A but is shifted into AIS in B. 531 the dendrite at the site of current injection (green curve 532 in Fig. 10A). As expected, but not shown, the spike bifur- Thereafter, this spike bifurcates in an asymmetric way be- 546 533 cates there with one part conducted toward the axon and cause of asymmetric cell properties. 547 534 the other toward peripheral dendritic regions. For a longer In the presented examples of extracellular stimulation, 548 535 stimulation pulse of 5 ms, during the first part of stimula- the effect of combined strength–duration curves is difficult 549 536 tion spatial transmembrane voltage distributions similar to observe for cathodic currents, although the phenome- 550 537 to the green curve are produced at subthreshold intensity, non is present in a weak form for the cases SOMA and 551 538 but with smaller amplitudes (e.g., the two thick blue lines DEND (upper part of Fig. 1B). In contrast, combined 552 539 in Fig. 10A). During a 5 ms pulse just above threshold strength–duration curves occurs clearly for anodic pulses 553 540 intensity, the peak value at this electrode position is not (Fig. 10B), especially for an electrode above the soma or 554 541 strong enough to produce a dendritic spike, but axial cur- a node of Ranvier. In contrast to cathodic stimulation, 555 542 rent flow into the axon causes enough sodium current via anodic stimulation generates a center of hyperpolarization 556 543 the low threshold sodium channels Nav1.6 in AIS. As a at a region of the neuron closest to the electrode, and the 557 544 result a spike initiates at the beginning of the thin part of side lobes of the activating function (marked in Fig. 4 by 558 545 the axon (t = 5.7 ms, lower thin blue line in Fig. 10A). green arrows) became the driving forces for excitation. 559

10 F. Rattay et al. / Neuroscience xxx (2012) xxx–xxx

560 Therefore, the ﬁrst reaction describing the membrane the third node of Ranvier (Fig. 10D): spikes are initiated 568 561 voltage after 10 ls of anodic stimulation for an electrode at the distal end of the thin segment and at a position clo- 569 562 at 50 lm above the soma has two positive peaks ser to its center by short and long pulses, respectively. 570 563 (Fig. 10C, black line). Short pulses cause spike initiation 564 at the distal end of the AIS (Fig. 10C, green curves), DISCUSSION 571 565 whereas spikes arise at the centre of the thin axonal seg- 566 ment for long pulses (Fig. 10C, blue curves). A similar The time constant of a patch of cell membrane is deﬁned 572 567 shift phenomenon occurs when the electrode is above as s = R C, with capacity C and resistance R. Chronaxie 573

A membrane voltage B ELECTRODE CURRENT INTRACELLULAR ANODIC 40mV 1000 short pulse ( 50µs) STIMULATION STIMULATION Iel=20nA µA t=50µs long pulse (5ms) Iel=0.8nA 100 0 t=6.18ms current injection t=5.7ms DEND SOMA t=5 ms C AIS 10 -40 NAKED AXON NODE OF RANVIER D

1 t=1 ms soma 500 1000 µm 0.01 0.1 1 10 ms xxx x PULSE DURATION main AIS thin myelinated dendrite axon axon

C membrane voltage ANODIC D membrane voltage ANODIC STIMULATION STIMULATION ABOVE SOMA 50 mV t=300µs ABOVE N.R. 50µs pulse 50 mV 5ms pulse Iel=80µA I=11.4µA t=200µs el t=5.14ms 50µs pulse 5ms pulse I= 13.3µA t=5ms Iel=60µA 0 el -50

t=5ms t=50µs t=4.66ms -150

t=10µs -100 t=50µs electrode electrode -250

soma 500 1000 µm soma 500 1000 µm xxx x xxx x main AIS thin myelinated main AIS thin myelinated dendrite axon axon dendrite axon axon

Fig. 10. Spike initiation sites change for short and long current pulses. Dashed vertical lines indicate electrode position, end of AIS and the beginning of myelination. Membrane voltages proﬁles are shown for 50 ls (green) and 5 ms (blue) pulse stimulation at selected times. (A) Intracellular stimulation of Neuron 2 at the main dendrite. At the end of a 50 ls pulse with threshold intensity (20 nA) a dendritic spike (green curve) is generated. Although during a 5 ms pulse just above the 0.79 nA threshold the maximum membrane potential occurs close to the electrode position (thick blue curves), the spike arises at a quite distant position (300 lm) at the thin segment of the axon. (B–D) Extracellular anodic stimulation of Neuron 2 for electrode positions as deﬁned in Fig. 1A. (B) Electrode positions above the soma (black line) and the third node of Ranvier (blue line) cause combined strength–duration curves as spike initiation site depends on pulse duration in these cases. (C) Stimulus electrode above soma. Thick lines show the situation at the end of the pulses. For better recognition the 10 ls response of the short pulse stimulation is shown in black. (D) Stimulus electrode above node of Ranvier.

F. Rattay et al. / Neuroscience xxx (2012) xxx–xxx 11

574 is about 0.7s for such a patch (Ranck, 1975; Reilly, 1992) intra- and extracellular stimulation (Fig. 1B). However, 635 575 which is independent of cell geometry as long as intracel- unexpected differences between both neurons occur, 636 576 lular axial current flow is negligible. The large variety in e.g., only Neuron 2 shows a bimodal intracellular 637 577 membrane resistance is the main reason for specific strength–duration curve (red thick curve in the lower part 638 578 chronaxie values for different functional unmyelinated cell of Fig. 1B). 639 579 regions. In contrast to the variance found in membrane It is important to notice that many variables like synap- 640 580 resistance, membrane capacity is rather uniform with a tic activity (Spruston, 2008; Sjo¨ stro¨ m et al., 2008), refrac- 641 2 581 value close to 1 lF/cm (Cole, 1968). However, myelin- tory behavior (Miocinovic and Grill, 2004), inhomogeneity 642 582 ated regions have considerably smaller chronaxies as in ion channel density (Migliore and Shepherd, 2002; 643 583 the total membrane capacity is inversely proportional to Keren et al., 2009), branching (Manita and Ross, 2009), 644 584 the number of myelin sheets. curvature of axons (Rattay et al., 2000; Iles, 2005), pulse 645 585 Strength–duration characteristics for external stimula- shape (Wongsarnpigoon et al., 2010), electrode configu- 646 586 tion of myelinated peripheral axons can be derived with rations (Smith and Finley, 1997), the inhomogeneity and 647 587 linear methods from the intracellular relationship anisotropy in the tissue (Roth, 1995), pulse trains and 648 588 (Warman et al., 1992). In contrast to peripheral nerve neuromodulation (De Vries et al., 2007; Minassian et al., 649 589 applications, microelectrodes for central nervous system 2007) can influence recruitment and strength–duration 650 590 stimulation operate often in or in the vicinity of the soma, relationships. Nevertheless, by applying careful analysis, 651 591 where additional effects appear as consequences of the the large variance in chronaxie, even within one cell 652 592 variation of the electrical and geometrical parameters should be helpful for selective stimulation, especially 653 593 along the neural axis. when microelectrodes are used to activate a specific 654 594 Here, we have demonstrated a radical switch to short region like the AIS, a part of the dendritic tree or the mye- 655 595 chronaxies when the stimulating electrode position linated axon. As an example, short pulse widths selec- 656 596 changes from inside to outside the cell. The inside/outside tively activate cells with their somas close to the 657 597 chronaxie ratio with a value close to 6.5 was the largest electrode in epiretinal stimulation (Behrend et al., 2011). 658 598 for the smallest investigated cell–electrode distance of One should be aware that chronaxie is also the time 659 599 50 lm(Fig. 8). This ratio depends essentially on the where stimulus pulse duration needs a minimum of 660 600 length of the excited zone positioned between hyperpolar- energy (Geddes, 2004), an important fact for neural pros- 661 601 ized regions for the cathodic case. These hyperpolarizing theses. Moreover, charge injection (electrode current 662 602 lobes hinder the continuous excitation process seen by times pulse duration) from electrode surface limits neural 663 603 intracellular stimulation. Consequently, the inside/outside prosthetic applications. The left (quite) linear part of dou- 664 604 chronaxie ratio becomes smaller for larger electrode–cell ble logarithmic strength–duration curves, e.g. in Fig. 1B, 665 605 distances. Increasing electrode distance usually will predicts (nearly) constant charges when thresholds of 666 606 cause the excitation of other functional cell regions with very short pulses are compared (half pulse duration 667 607 other chronaxic properties. This phenomenon is espe- needs double threshold current). Minimum charge occurs 668 608 cially obvious for the electrode position above a soma with with infinitely short pulse duration. With the approach of 669 609 highly excitable neighbored axonal regions showing Weiss (Fig. 3) the factor for the additional charge costs 670 610 shorter chronaxies (upper part of Fig. 1B), and therefore, are pulse duration divided by chronaxie. 671 611 the measured chronaxie cannot just reflect the somatic 612 value, but a value influenced by the excited axonal region. 613 This chronaxie mix disturbs the original trend toward CONCLUSIONS 672 614 chronaxie values for intracellular stimulation as marked A recent review and several often cited articles sustain 673 615 by gray arrows in the lower part of Fig. 8. the wrong dogma that chronaxie is the same for intra- 674 616 The presented theory predicts the experimentally and extracelluar stimulation (Borchers et al., 2012; Nowak 675 617 observed trend to larger chronaxies for increased dis- and Bullier, 1998; Geddes, 2004). On the contrary, evalu- 676 618 tances (West and Wolstencroft, 1983). An analytical ap- ations of our compartment model of a cortical pyramidal 677 619 proach is available for non-myelinated nerve and muscle cell demonstrate up to 20 times longer chronaxies for 678 620 fibers that predicts different chronaxies for intra- and intracellular stimulation than for extracellular stimulation 679 621 extracellular stimulation and the increase of chronaxie (Fig. 1B). This fact as well as the occurrence of combined 680 622 with electrode–fiber distance (Suarez-Antola, 2005). strength–duration curves are supported by theoretical 681 623 Examples of combined strength–duration curves have investigations. 682 624 been presented where short and long pulses cause spike 625 initiation in different cell regions. Thus, fitting strength– 626 duration data with a single curve according to the classical UNCITED REFERENCE 683 627 approaches of Weiss or Lapique may be equivocal. Some 628 combined strength–duration curves resulting from retinal Butson et al. (2007). Q6 684 629 ganglion cell recordings can be found in Fig. 9 of Gerhardt 630 et al. (2011). Acknowledgments—We thank Dr. Katarina Leao for meaningful 685 631 Our results help explain trends of strength–duration comments in earlier versions of this manuscript. This work was 686 632 characteristics. For example, Neuron 1 and Neuron 2 supported by the Austrian Science Fund , Grant No. 21848- 687 633 have comparable strength–duration curves with similar N13. R.N.L. is supported by a Kjell and Ma¨rta Beijers Foundation 688 634 excitability sequences (node-ais-soma-dend) both for Grant. 689

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