1
2
STRENGTH–DURATION RELATIONSHIP FOR INTRA- VERSUS
3
EXTRACELLULAR STIMULATION WITH MICROELECTRODES
4Q1 F. RATTAY,a*L. P. PAREDESaAND R. N. LEAOb,c
5 aInstitute for Analysis and Scientific Computing, Vienna University 6Q2 of Technology, A-1040 Vienna, Austria
7 bNeurodynamics Laboratory, Department of Neuroscience, 8 Uppsala University, Uppsala, Sweden
9 cBrain Institute, Federal University of Rio Grande do Norte, Natal-RN, 10 Brazil
11 Abstract—Chronaxie, a historically introduced excitability time parameter for electrical stimulation, has been assumed to be closely related to the time constant of the cell mem-brane. Therefore, it is perplexing that significantly larger chronaxies have been found for intracellular than for extra-cellular stimulation. Using compartmental model analysis, this controversy is explained on the basis that extracellular stimulation also generates hyperpolarized regions of the cell membrane hindering a steady excitation as seen in the intracellular case. The largest inside/outside chronaxie ratio for microelectrode stimulation is found in close vicinity of the cell. In the case of monophasic cathodic stimulation, the length of the primarily excited zone which is situated between the hyperpolarized regions increases with electrode–cell distance. For distant electrodes this results in an excitation process comparable to the temporal behav-ior of intracellular stimulation. Chronaxie also varies along the neural axis, being small for electrode positions at the nodes of Ranvier and axon initial segment and larger at the soma and dendrites. As spike initiation site can change for short and long pulses, in some cases strength–duration curves have a bimodal shape, and thus, they deviate from a classical monotonic curve as described by the formulas of Lapicque or Weiss.Ó 2012 Published by Elsevier Ltd on behalf of IBRO.
Key words: chronaxie, strength–duration curve, electrical stimulation, microelectrode, activating function.
12
13 INTRODUCTION
14 Selective neural stimulation is a great challenge in the
15 development of neural prostheses. As an example, active 16 contacts of an electrode array implanted at the retina or
17 other structures along the visual pathway should stimulate 18 elements that elicit visual sensations corresponding to the
19 place in the array (Brindley, 1955; Zrenner, 2002; Dowling,
20 2005; Ferna´ndez et al., 2005; Fried et al., 2006; Cohen,
21
2007; Sekirnjak et al., 2008; Horsager et al., 2009; Pezaris
22
and Reid, 2009; Tehovnik et al., 2009)while avoiding side Q3 23
effects like co-stimulation of bypassing axons arising from
24
distant locations (Greenberg et al., 1999; Rattay and
Re-25
satz, 2004). Considering the fact that neural substructures
26
have different strength–duration characteristics may give
27
an opportunity for more selective stimulation. For example,
28
in the treatment of chronic pain during spinal cord
stimula-29
tion smaller dorsal column fibers can only be activated
30
when pulse width is sufficiently large (Holsheimer et al.,
31
2011).
32
Recently, the strength–duration relationship for
extra-33
cellular neural stimulation was analyzed under the
34
assumption of a constant electrical field in flat, spherical
35
and cylindrical cells (Boinagrov et al., 2010). These
inves-36
tigations provided a biophysical basis for the stimulation
37
with large electrodes and also explained effects of
stimula-38
tion with ultrashort pulses (<5
l
s). In accordance withret-39
inal ganglion cell experiments (Jensen et al., 2005), for
40
pulse durations <2 ms, it was found that a 500
l
mdiam-41
eter electrode placed above the soma caused excitation
42
with significantly lower thresholds compared to a position
43
above the axon. This result is unexpected given the
44
assumption that the axon is the most excitable part of a
45
neuron externally stimulated (Nowak and Bullier, 1998;
46
Rattay, 1999). Differences in experimental setups are the
47
reason for the controversy about such observations.
Stim-48
ulation of the retina with large electrodes generates a
49
rather constant field and consequently transverse currents
50
depolarize the cell membrane at one side and
hyperpolar-51
ize the cell at the opposite side. On the other hand, external
52
excitation with small electrodes is mainly based on
stimu-53
lating effects resulting from extracellular potential
varia-54
tions along the neural structure (Rushton, 1927; Ranck,
55
1975; Rattay, 1986, 1987, 1999). These facts vary the
ex-56
pected excitation paradigm and should be carried in mind.
57
Voltage sensitive ion channel types and densities are
58
other important elements for neuron excitability. In
rela-59
tion to retinal implant, electrode positions applied at a
60
dense two-dimensional grid around the soma region of
61
rabbit ganglion cells showed the lowest thresholds along
62
a small section of the axon, about 40
l
m from the soma.63
At this axonal section, immunochemical staining revealed
64
a dense band of voltage-gated sodium channels (Fried et
65
al., 2009).Similarly, high sodium channel densities at the Q4 66
axon initial segments (AIS) of cortical cells define sites for
67
action potential initiation under natural conditions (Stuart
68
et al., 1997; Yu et al., 2008; Hu et al., 2009) as well as
69
spike initiation candidates for external stimulation with
70
microelectrodes (Rattay and Wenger, 2010).
0306-4522/12 $ - see front matterÓ 2012 Published by Elsevier Ltd on behalf of IBRO.
http://dx.doi.org/10.1016/j.neuroscience.2012.04.004
*Corresponding author. Tel: +43-1-58801-10114. E-mail address:frank.rattay@tuwien.ac.at(F. Rattay).
Abbreviations: AIS, axon initial segments; VSD, voltage sensitive dye. Neuroscience xxx (2012) xxx–xxx
71 A strength–duration curve describes electrode thresh-72 old current as function of stimulus pulse duration.
Chron-73 axie is defined as the time on such a strength–duration 74 curve for twice the minimum (rheobase) current needed
75 for very long pulses. Established by experimental
find-76 ings, this excitability parameter was assumed to be rather
77 independent from the distance between the current
78 source and the excited cell (Weiss, 1901; Lapicque,
79 1907, 1929; Blair, 1932).
80 As different neural structures have different
chronax-81 ies, stimulus pulse duration is an important element for
82 selective stimulation as well as for electrophysiological 83 classification. For example, the shorter chronaxie of
mye-84 linated motorneurons compared to that of unmyelinated
85 fibers of the myocardium allows safe stimulation with
86 short pulses during artificial respiration. Motoneurons
87 then become activated with little disturbance of the
88 myocardial function (Voorhees et al., 1992). In
electro-89 physiological studies, extracellular gray matter stimulation 90 shows larger chronaxies for somas than for axons,
sup-91 porting the idea that spike initiation occurs in the axon
92
(Nowak and Bullier, 1998). Additionally, in cochlear
im-93
plants, significantly shorter chronaxies have been
de-94
scribed in long-term (compared to acute) deafened
95
cochleae, indicating the loss of peripheral processes in
96
many spiral ganglion cells (Shepherd et al., 2001).
97
Based on theoretical considerations, chronaxie has
98
been suggested to be about 0.7 times the time constant
99
of the cell membrane (Blair, 1932; Ranck, 1975; Reilly,
100
1992). Therefore, some authors assume chronaxie as
101
independent of in- or outside stimulation of the cell
102
(Nowak and Bullier, 1998). This hypothesis contrasts with
103
reports of considerably larger chronaxies for intracellular
104
than for extracellular stimulation (Ranck, 1975).
105
Discrepancy among different works prompts
specula-106
tions that experimental differences could be caused by
107
artifacts like sampling errors or neuronal damage during
108
recordings (Ranck, 1975). To address this controversy,
109
we systematically reproduced experimental findings using
110
computer simulations of a pyramidal cell and its simplified
111
rectified version for electrode positions at the dendrite, the
112
soma, the lateral ending of the AIS and the myelinated
0.0001 0.001 0.01 0.1 1 µA 10 100 0.01 0.1 1 10 ms SOMA soma soma ais ais AIS
dend is not excitable for shorter pulses
DEND AIS NODE NODE NODE soma ais internode dend SOMA DEND AIS dend dend DEND SOMA PULSE DURATION CHRONAXIE NEURON 2 NEURON 1 extracellular stimulation intracellular stimulation node node node basal dendrite main dendrite d~4,7µm electrode positions 50 µm above first internode naked (thin) axon N.R. = node of Ranvier 200µm dendrite unmyel. axon myelin. axon N.R. N.R. N.R. N.R. rectified neuron soma AIS naked axon
axon hillock AIS
NODE soma dend node apical dendrite N.R. N.R. N.R. N.R. N.R. N.R. N.R. N.R. N.R. N.R. N.R. N.R. N.R. x N.R.
B
A
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.
113 axon (Fig. 1). In all cases shown inFig. 1B, the intracellu-114 lar/extracellular chronaxie ratio is two or higher. It is
115 important to note that these ratios depend on cell proper-116 ties as well as on the distance and type of electrodes.
117 Such variants could be the reason for the diversity in
118 chronaxie data.
119 Here, we demonstrate that (i) hyperpolarized regions
120 are responsible for the shorter extracellular chronaxies, 121 (ii) for short electrode distances, excitability and chronaxie
122 differ essentially along a selected neuron and conse-123 quently (iii) strength duration curves are expected to
devi-124 ate from the classical form when short and long pulses 125 cause action potential initiation at sites with different
elec-126 trical membrane properties.
127 EXPERIMENTAL PROCEDURES
128 An analysis of two model neurons is presented. In both models 129 excitation is based on recently measured densities of high thresh-130 old sodium channels Nav1.2 in dendrites and the soma and low 131 threshold sodium channels Nav1.6 in the axon (Hu et al., 132 2009). The phenomenon of short chronaxies for extracellular 133 stimulation was also tested in simulations with other neural struc-134 tures (not shown) and did not depend on the specific examples. 135 Model Neuron 1 (Fig. 1A left,Rattay and Wenger, 2010), a 136 simplification of Neuron 2 concerning geometry and ion channel 137 types, is straightforward to analyze and with less computational 138 cost. It has a straight axis and consists of a single non-branching 139 dendrite (500lm, d = 5lm), spherical soma (d = 20lm), axon 140 hillock (10lm, d = 3.1lm), AIS (50lm, d = 1.22lm), naked 141 axon (unmyelinated, 200lm, d = 1lm), myelinated axon 142 (500lm, d = 1lm) and unmyelinated terminal (50lm, 143 d = 1lm). Assumptions for ion channel distribution and ion cur-144 rent computations are quite similar as inHu et al. (2009): the 145 same constant Nav1.2 channel density for the dendrite and soma 146 (gna = 8 mS/cm2), but 40 times higher sodium channel density 147 in hillock and AIS with a change to the low threshold type 148 Nav1.6 in the axon. Intracellular resistivity is 150Xcm, mem-149 brane capacity c = 1lF/cm2. Using ACSL (Advanced Continu-150 ous Simulation Language) software, Neuron 1 is simulated by 151 evaluating the compartment model equations (1)–(3) as de-152 scribed below.
153 Model Neuron 2 (Fig. 1A right) is based on geometrical 154 parameters of a traced cortical layer 5 pyramidal cell with ion 155 channel assumptions as available in the NEURON (Carnevale 156 and Hines, 2006) Model DB (Hines et al., 2004; Hu et al., 157 2009). In contrast to Neuron 1, soma diameter is increased to 158 30lm, naked axon diameter is reduced to 0.4lm and the capac-159 ity of the cell membrane of all compartments with exception of the 160 soma and internodes is reduced to 0.5lF/cm2. Moreover, 161 Neuron 2 incorporates branching dendrites, tapering diameters, 162 uneven ion channel distribution within compartments of a single 163 type and a membrane capacity of 0.02lF/cm2in the axonal inter-164 nodes. High threshold sodium Nav1.2, low threshold Nav1.6 and 165 fast voltage-gated K+of Neuron 1 are complemented by slow 166 non-inactivating potassium current, high-voltage activated Ca2+ 167 and calcium dependent K+ in dendritic and somatic compart-168 ments. All parameters are used as inHu et al. (2009). However, 169 in order to demonstrate current–distance relations in a clear way, 170 the axial 3d structure of the cell was compressed into the x–y 171 plane. Neuron 2 is simulated with NEURON.
172 In more detail, Neuron 1 soma and dendrite were imple-173 mented with the same constant maximum conductances (in mS/ 174 cm2) g
Nav1.2= 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= 176 100, gNav1.6= 320, gKv= 100, the unmyelinated axon with 177 gNav1.2= 0, gNav1.6= 300, gKv= 150 and the nodes of Ranvier 178 with gNav1.2= 0, gNav1.6= 160, gKv= 20. Sodium current kinetics
179 are calculated by INav1.j= gNav1.jm3h (V ENa) with j equals to
180 either 2 or 6 and ENa= 60 mV. Details on the differential equations
181 of the different variables are shown inMainen et al. (1995). The
182 values for the half (in)activation voltages V1/2, the slopes k and
183 the coefficients A were obtained from a previously published
mod-184 el in the NEURON Model DB (Hu et al., 2009) after subtracting the
185 corresponding value for the shift of voltage dependence of the
186 kinetics. Consequently, the currents INav1.jhave the same values
187 for A, i.e., A(am) = 0.182, A(bm) = 0.124, A(ah) = 0.024, A(bh
188 ) = 0.0091, and the slope of inactivation, i.e., k(sh) = 5 and
189 k(h1) = 6.2, in contrast to altered slope of activation, i.e.,
190 k(sm) = k(m1) = 7 for Nav1.2 but k(sm) = k(m1) = 6 for
191 Nav1.6. To account for the reduced threshold of Nav1.6 channels
192 V1/2(m) is decreased to41 mV compared to the calculated value
193 of28 mV for activation of Nav1.2 channels. The corresponding
194 values in mV for the inactivation of Nav1.2/Nav1.6 channels are
195 V1/2(ah) =35/41, V1/2(bh) =60/73, V1/2(h1) =57/70.
196 The potassium currents are determined by IK= gKn(V EK) with
197 EK =90 mV. To be consistent the corresponding values of
198 A(a) = 0.02, A(b) = 0.002, V1/2(a) = V1/2 (b) = 25 mV and
199 k(a) = k(b) = 9 were also obtained from the NEURON Model
200 DB (Hu et al., 2009). Internodes are simulated with 17 sheets of
201 membrane with a conductance of 1 mS/cm2and C = 1lF/cm2
202 per sheet (Rattay, 1999). The presented results are simulated
203 for 37°C.
204 In the first modeling step, the extracellular potential Ve
gener-205 ated by an electrode tip is approximated considering a monopolar
206 spherical electrode in an infinite homogeneous extracellular
med-207 ium with resistivityqe= 300Xcm. This is equivalent to a point
208 source stimulation resulting in spherical isopotentials with Ve=
209 qeIel/4pr when a current pulse with amplitude Ielis applied; r is
210 the distance from a point of interest to the point source.
211 In the second step, the response of a neuron is simulated by
212 a compartment model. That is a network of resistances and
213 capacitances, where the current to the centre of compartment n
214 consists of the following components: a capacitive current, ion
215 currents across the membrane and intracellular currents to
neigh-216 bored compartments. Applying Kirchhoff’s law for compartment n
217 results in 218 dðVi;n Ve;nÞ dt Cnþ Iion;nþ Vi;n Vi;n1 Rn=2þ Rn1=2 þ Vi;n Vi;nþ 1 Rn=2þ Rnþ1=2 ¼ 0 ð1Þ 220 220 221 with intracellular potential Vi, axial resistance R and membrane
222 capacity C. The following system of differential equations is deduced
223 by introducing the transmembrane voltage V¼ Vi Veto compute
224 the time courses of Vnin every compartment (Rattay, 1999):
225 dVn dt ¼ Iion;nþ Vn1 Vn Rn1=2þ Rn=2 þ Vnþ1 Vn Rnþ1=2þ Rn=2 þ Ve;n1 Ve;n Rn1=2þ Rn=2 þ Ve;nþ1 Ve; n Rnþ1=2þ Rn=2 Cn: ð2Þ 227227 228 The direct stimulating influence of the extracellular potential on
229 compartment n is defined by the activating function (Rattay, 1999)
230 fn¼ Ve;n1 Ve;n Rn1=2þ Rn=2 þ Ve;nþ1 Ve;n Rnþ1=2þ Rn=2 Cn: ð3Þ 232 232 233 For a fiber with constant diameter d, constant compartment length
234 Dx, intracellular resistivityqiand specific capacity c,(3)appears
235 in a simpler form 236 fn¼ d 4c
q
iVe;nþ1 2Ve;nþ Ve;nþ1
Dx2 : ð3aÞ 238238
239 The value within the brackets of(3)corresponds to a virtual
in-240 jected current applied to compartment n. In regions where this
cur-241 rent is positive, the membrane depolarizes and where it is
242 negative, it tends to hyperpolarize.
243 VOLTAGE SENSITIVE DYE IMAGING 244 Transverse hippocampal slices were obtained from P21
245 C57B6 mice as previously described (Leao et al., 2009)
246 and according to the rules of Animal Experimentation of
247 the Uppsala University. Slices were maintained in artificial 248 ACSF (in mM: 124 NaCl, 3.5 KCl, 1.25 NaH2PO4, 1.5
249 MgCl2, 1.5 CaCl2, 30 NaHCO3, 10 glucose), constantly
250 bubbled with 95% O2and 5% CO2. Recordings/imaging
251 were obtained at 25°C, in the presence of 10
l
M CNQX, 252 30l
M dAP5 and 10l
M bicuculline methochloride tomin-253 imize the effect of synaptic currents (Leao et al., 2005).
254 Voltage sensitive dye (VSD) loading (JPW3027 obtained 255 from Prof Leslie Loew, University of Connecticut, USA)
256 was performed exactly as described inPalmer and Stuart 257Q5 (2006).Images were acquired using a EM-CCD camera
258 (Luca, Andor, Ireland). Excitation was produced by a 259 200W metal-halide lamp through a bypass filter centered
260 at 535 nm (510 to 560 nm pass) and emission was
low-261 pass filtered at 590 nm. Image acquisition and extracellular
262 stimulation were synchronized by a National Instruments
263 digital device and to guarantee time precision, we recorded 264 the stimulator and the camera ‘fire’ outputs (that flags
im-265 age acquisition) using a National Instruments DAQ card. 266 Current clamp recordings were also obtained from the
im-267 aged cell using winWCP (Dr John Dempster, Strathclyde 268 University, UK). The stimulating electrode (tungsten,
269 10
l
m tip) was placed 50 and 75l
m above axons. Pulses270 had 100
l
s durations and extracellular current intensities 271 were adjusted to 75% of the minimum threshold current.272 Images were taken 100
l
s after the stimulus using273 100
l
s exposure. Pixel intensity was measured in 4 2274 pixel region of interest and averaged from 10 images ac-275 quired with 1s delay between each other.
276 RESULTS
277 In the following several computer experiments are
278 proposed to analyze step by step characteristics of
279 strength–duration curves and related historical formulas.
280 Intracellular stimulation: space clamp versus cable
281 model
282 In order to describe the neural excitation process
inde-283 pendently from the intracellular current needed for spike 284 conduction,Hodgkin et al. (1952)used the space clamp
285 technique. They stimulated with a long inserted wire that
286 clamped the squid axon uniformly along its entire length. 287 An equivalent model situation is created by current
injec-288 tion into the soma of Neuron 1 by cutting all soma pro-289 cesses. Stimulation of the resting soma with a rather
290 long weak pulse results in a transmembrane voltage
291 curve V(t) with an asymptotic exponential increase during
292 the subthreshold response (Fig. 2A). The time constant
293 for the exponential voltage of the passive membrane is 294 the product of the ohmic resistance and the capacity of
295 the cell membrane (
s
= R C). The time constants
of 296 such an exponential increase can be found graphically297 by linear extrapolation at stimulus onset as the intersec-298 tion of the tangent in V at pulse onset with the steady state
299 indicated by the horizontal dashed line inFig. 2A. At the
300
end of the 8 ms pulse, the subthreshold approach of V
301
is very close to its stationary value.
302
The shape of the subthreshold membrane voltage V(t)
303
changes considerably when the stimulus current is
304
allowed to flow not only across the membrane but also
305
intracellularly from the stimulated soma into the dendrite
306
and the axon. Finding
s
by the graphical method of307
Fig. 2A becomes contradictory when applied in Fig. 2B
308
and C as it results in shorter time values t1 while needing
Iel=50pA Iel=50pA I =50pAel I =150pAel 1ms 1ms 1ms 10mV 2mV 0.63*Vstationary 0.63*Vstationary 0.63*Vstationary 1ms 1mV 30mV transmembrane voltage V =3.4ms t1=1.4ms, t 2=4.3ms t1=1.4ms, t 2=5.3ms STIMULUS CURRENT STIMULUS CURRENT STIMULUS CURRENT Multiplication of subthreshold response with a factor 3 corresponding to Iel=150pA
Subthreshold response Iel=50pA Soma without processes, intracellular stimulation
Soma with dendrite, intracellular stimulation
C
Soma with axon and dendrite, intracellular stimulationA
B
Fig. 2.Transmembrane voltage V of Neuron 1 for intracellular stimulation at the soma. (A) Space clamp condition with no axial current flow. The top graph demonstrates the coincidence of the passive and active membrane response in the subthreshold regime. Linear extrapolation of V at stimulus onset up to the steady state value of the single RC circuit (gray arrow) defines the time constant of the passive membrane response ass= R C (lower graph of A). With the same electrode current Iel= 50 pA, the steady state value of V is reduced by a factor5 when axial current flows into the dendritic branch (B) and by a factor10 for conductance into dendrite and axon (C). In comparison with the space clamp condition, intracellular current flow into dendrite and axon results in shorter times t1 but longer times to reach the steady state of the subthreshold membrane.
309 longer times for reaching the steady state (pulse dura-310 tions >20 ms). Obviously, the fit by a single exponential
311 function corresponding to a single RC element (as in case 312 A, where 63% of Vstationary is determined by
s
) cannot313 describe the subthreshold response of the cable model
314 in an acceptable way. We have to change the method
315 and define the time constant as the time t2 when the cable
316 model reaches 63% (1 1/e) of the stationary membrane 317 voltage V (Fig. 2B and C). Note that the time constant of
318 the cable model is not an average value of the time
con-319 stants of the compartments. This is demonstrated by
320 Fig. 2B where both the dendrite and the soma of Neuron1 321 have quite the same electric membrane properties. The
322 main difference between
s
and its replacing values t1323 and t2 inFig. 2A and B is a consequence of a distinctly 324 reduced steady state membrane voltage when the greater
325 part of the stimulus current flows into the dendrite. The dif-326 ferent temporal excitation profiles shown inFig. 2 have
327 consequences on the corresponding strength–duration
328 curves.
329 The middle solid line in Fig. 3 shows the strength–
330 duration relation under the space clamp condition where
331 chronaxie is independent from the soma area. Allowing
332 part of the injected current to flow into dendrites and the 333 axon demands for stronger threshold currents for all pulse
334 durations (Fig. 3, upper solid line). The intracellular
335 strength–duration curves for Neuron 1 and its space
336 clamped version deviate in their shapes for long pulses. 337 With the same electrical cell properties, the calculated
338 intracellular chronaxie ratio 2.5/1.79 = 1.4 (Fig. 3)
339 becomes smaller by enlargement of the soma surface
340 or by reducing the number or diameters of processes.
341 The variable chronaxie ratio decreases to 1 if the diame-342 ters of the soma processes converge to zero, which is the
343 space clamp condition.
344 This dependence of chronaxie on diameter disproves
345 the often applied rule mentioned in the introduction, 346 namely chronaxie0.7
s
withs
= R C, where theprod-347 uct R C is independent of membrane size. However,
348 even more surprising is the huge deviation from this for-349 mula for extracellular stimulation. With the time constant
350 of the soma membrane (Fig. 2A) we obtain chronaxie 351 0.7 3.4 ms = 2.38 ms instead of 0.38 ms for the case 352 presented inFig. 3.
353 In contrast to intracellular stimulation, in most
applica-354 tions extracellular stimulations are optimally achieved with
355 cathodic currents (Ranck, 1975; Rattay, 1986, 1999). 356 Hence, we compare anodic inside with cathodic outside
357 stimulation in order to explain in the next subsections 358 the large chronaxie differences between intra- and
extra-359 cellular stimulation shown in Figs.1B and3.
360 Extracellularly activated region increases with
361 electrode distance
362 A straight fiber with the properties of the naked axon of 363 Neuron 1 is shown inFig. 4A–C for external stimulation
364 with a cathodic 100
l
s pulse at threshold intensity. In con-365 trast to intracellular stimulation, the externally positioned366 microelectrode causes in every compartment an injected
367 virtual current. As second important effect, outside
368 stimulation with a monophasic pulse produces virtual
369
currents of both polarities where the sum of all the virtual
370
currents is zero.1The driving forces of excitation are the
371
currents in the region with positive activating function
val-372
ues (red arrows2inFig. 4C).
373
The length of the region where a cathodic point source
374
causes positive activating function ispffiffiffi2 zel, where zelis
375
the electrode distance. This formula is correct for a
376
straight nonmyelinated fiber with constant diameter in a
377
homogeneous medium and the relation can be described
378
by a 70-degree angle (Fig. 4A, tan(35°) =pffiffiffi2=2 zel=zel),
379
(Rattay, 1986). According to the activating function
380
concept, this angle is independent of fiber diameter and
381
electrical membrane properties and can also be used as
382
an approach for myelinated axons (Rattay, 1986). Large
383
positive and negative isolated activating functions’ values
384
appear at locations with considerable diameter changes,
385
in branching or in fiber endings. As axial intracellular
cur-386
rents level such local effects, the 70° rule is a rough
ap-387
proach even for these cases. A specific chronaxie value
388
of a region can be expected as long as the zone defined
389
by the 70° seen from the point source is concentrated in a
390
cell region with common electrical properties.
391
The length of the activated portion flanked by the
hyper-392
polarized regions increases with electrode–neuron
dis-393
tance, and consequently, the influence of the
394
hyperpolarized region on the excitation process is
gradu-395
ally reduced when moving the current source away. This 1 10 100 1000 µA 0.01 0.1 1 10ms electrode current Iel rheobase I0 2 rheobase* pulse duration extracellular chronaxie = 0.38ms space clamp chronaxie = 1.79ms intracellular chronaxie = 2.5ms
Fig. 3.Strength–duration curves of the somatic membrane of Neuron 1 for stimulation with square pulses. Chronaxie defined as the pulse duration needed for twice the rheobase I0 (horizontal arrows) is different for the space clamp condition (central group of three curves) and the case of current flowing into dendrites and the axon (upper curves). Injected anodic current amplitudes (upper and central groups) are shown as current densities for 1 cm2 membrane. Extracellular stimulation (lower curves) is with cathodic currents for a point source 50lm above soma center. Approximation of the strength–duration curves according to the classical formulas of Lapicque (1907): Iel= I0/(1 2k) (dashed lines) andWeiss (1901): Iel= I0(1 + 1/k) (dotted lines) where k is pulse duration/chronaxie.
1 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 compartment (n ? n + 1) includes the ‘old’ right neighbor as new left neighbor, but the current flow has changed signs. Thus, the sum of these two currents is 0 and 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 seeRattay (1990).
2 For interpretation of color in Figs. 1, 4 and 10, the reader is referred to the web version of this article.
396
effect could be also demonstrated using VSD imaging. The
397
length of the depolarization detected by the intracellular
398
VSD increases with electrode distance (Fig. 4D). The area
399
under the curve in fluorescence (normalized by the
maxi-400
mum) versus length graph was used as a measurement
401
of depolarization ‘spread’ caused by the extracellular
stim-402
ulus. When the stimulating electrode was placed 50
l
m403
(zel= 50
l
m) above the axon, the fluorescence (DF/F0,404
normalized) versus length integral was equal to 0.038 ±
405
0.002
l
m1 and 0.058 ± 0.004l
m1 for zel= 75l
m406
(n = 5 cells, p = 0.008, paired t test).
407
Intracellular versus extracellular stimulation
408
The spatio-temporal evolution of transmembrane voltage
409
profiles is significantly different for a microelectrode
posi-410
tioned either above or in the soma (Fig. 5A–D). InFig. 5A
411
and B, pulse duration is 100
l
s and membrane voltages412
along the neural axis are compared for threshold intensity.
413
At the end of the pulse, extracellular stimulation shows a
414
larger voltage maximum, but a shorter depolarized region.
415
For 1 ms threshold pulses the same voltage profiles
ap-416
pear again with smaller amplitudes (due to lower
stimula-417
tion currents) as the firsts of ten advancing 100
l
s time418
steps (thick lines inFig. 5C and D). The rather constant
in-419
crease of the maximum and spatial extension of the
volt-420
age profile for the case of intracellular stimulation in the
421
advancing 100
l
s time steps demonstrate that intracellular422
stimulation is more effective as the injected current is
avail-423
able to load the membrane capacity in the vicinity of the
424
electrode (Fig. 5D). Virtual negative injected currents
dur-425
ing pulse application (Eq.(3)) cause strong
hyperpolariza-426
tion (Fig. 5A and C). An essential part of the virtual positive
427
current, the driving force for excitation, is lost as a result of
428
the counterbalancing axial current flow between the
depo-429
larized and the two hyperpolarized regions. Note that half
430
of the positive voltage profile (its extension and maximum
431
value) is already reached at approx. 10% of the 1 ms
stim-432
ulation pulse (thick line inFig. 5C). This is contrary to the
433
gradual increase of the voltage profile seen inFig. 5D.
434
A second important effect is demonstrated in the
435
10 ms pulse example (Fig. 6). The subthreshold response
436
for extracellular stimulation has a maximum at half of the
437
stimulus pulse time with a decay related to the inactivation
438
gating variable h (Mainen et al., 1995) that act as a factor
439
for the sodium current3(Fig. 6A). The quicker
extracellu-440
larly elicited voltage increase affects the lower h values.
441
The maximum appears earlier for extracellular electrode
442
positions above AIS and node of Ranvier (Fig. 6C). In
con-443
trast,Fig. 6B and D shows a rather constant increase in
444
membrane voltages indicating that stimulation is still
possi-445
ble with longer and weaker pulses.
446
Chronaxie increases with electrode distance
447
In the previous section and with Fig. 3, it was
demon-448
strated that axial current flow from the activated area into
A
B
C
D
hyperpolarized region activated region hyperpolarized region z =50µmel 70° 2 x zel * z axon 0 -25 -25mV -50mV -100mV -50 -75 -100 750 500 250 -250 0 electrode I =-25µA, Vel el=-500mV extracellular potential Ve along the fibreactivating function and virtual injected currents 100µm 0.2% F/F0 ED=50 m ED=75 m ED=75 m ED=50 m Soma
Arbitrary Pixel Intensity
Axon Rest 1 0 mV/ms mV
Fig. 4.Extracellular stimulation. (A) Geometry and isopotentials for a point source 50lm above a fiber positioned at the x-axis. (B) Extracellular potential Ve=qeIel/4pr with r¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx xelÞ2þ z2el q
is used to calculate the activating function. (C) A fiber with d = 1lm and compartment lengthDx = 10lm results in a peak activating function value of 740 mV/ms. According to(2)this is the slope of the membrane voltage in the compartment below the electrode at the beginning of the25lA pulse. According to(3), the virtual currents are fn Cn. With Cn= dp Dx c = 3.14159 * 107lF and specific membrane capacity c = 1lF/cm2, the maximum injected current is 232 pA at the center of the activated region. As the length of the 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), 100ls after the stimulus for electrode distance zel= 50lm (middle photomicrograph) and zel= 75lm (lower photomicrograph) (scale bar=10lm;⁄
electrode position in the horizontal axis). Fluorescent traces for zel= 50 and 75lm versus axon position is shown below the photomicrographs and are aligned to the region in the axon activated by the extracellular stimulus. Each point in the traces represents average fluorescence of an axon region of approx. 10lm length in five images.
3 Sodium current, the driving component in excitation, is modeled as g Nam3 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.
449 the hyperpolarized regions causes shorter chronaxies for 450 cathodic extracellular than for intracellular stimulation. It
451 was also shown that the activated length is related to an
452 angle of 70° at the electrode and this activated length in-453 creases with microelectrode distance (Fig. 4).
Conse-454 quently, stimulating current loss along the neural axis 455 into the side lobes as defined by the activating function
456
shows a reduction when intracellular resistance between
457
depolarized and hyperpolarized regions increases by
458
increasing electrode distance. Comparison of excitation
459
profiles for electrodes at 50 and 200
l
m above the soma460
(Fig. 7) emphasizes the trend to longer chronaxies for
lar-461
ger electrode distances. As seen in the case at 200
l
m462
above the soma, the larger distance between the primarily 500 µm
20 mV extracellular stimulation intracellular stimulation
pulse duration 0.1 ms pulse duration 1 ms
A
B
C
D
Iel =-37 µA I =-8.6 µAel I =1.0 nAel I =7.5 nAel resting voltage -70 mVdendrite axon dendrite axon
soma soma
electrode electrode
t=100 µs t=100 µs
Fig. 5.Spatial transmembrane voltage profiles for extra- and intracellular stimulation computed for Neuron 1. Extracellular stimulation 50lm 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 100ls stimulation pulse (A, B) and in ten advancing 100ls time steps for a 1 ms stimulation pulse (C, D).
20 mV 2 ms
extracellular stimulation intracellular stimulation pulse duration 10 ms pulse duration 5 ms
A
C
D
B
I =-7.1 µAel I =-7.0 µAel I =0.27 nAel I =0.28 nAel AIS soma soma AIS node of Ranvier node of Ranvier h=1 h=0Fig. 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 50lm 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.
463 depolarized and hyperpolarized regions results in a more 464 constant increase of the voltage profile which becomes
465 comparable to that of the intracellular case.
466 Strength–duration curves for electrode positions
467 above the soma and a node of Ranvier underline the
468 association of increasing chronaxies with increasing elec-469 trode distance (Fig. 8). Chronaxie was determined by the
470 intersection of strength–duration curves with the horizon-471 tal line 2⁄rheobase, with electrode currents normalized to
472 rheobase (Fig. 8). The lower part of the figure uses the 473 chronaxie values of the shown strength–duration curves
474 for electrode distances of 50, 100, 200 and 300
l
m to475 quantitatively describe chronaxie versus distance rela-476 tionships for electrodes above the third node of Ranvier
477 and the soma. For the extracellular stimulation cases, 478 when increasing electrode distance, different functional
479 neural parts become excited. For this reason, and more 480 remarkably for the soma case, the curves containing the
481 circles deviate from the direction pointed by the arrows
482 (Fig. 8, bottom). Note the quite large difference between 483 intra- and extracellular chronaxies.
484 A shift of the spike initiation zone by electrode dis-485 tance increase is analyzed in the next example with two
486 electrode positions at zel= 50
l
m and zel= 200l
m487 above the center of the naked axon of Neuron 1
488 (Fig. 9). In contrast to the 50
l
m case (A) the depolarized 489 region predicted by the activating function with the70-de-490 gree angle (Fig. 4) exceeds the 200
l
m long naked axon491
for zel= 200
l
m. There is a change in the sign of the492
slopes at the onset of the 200
l
s stimulus within493
the naked axon region in (A), whereas for zel= 200
l
m494
the slopes are always positive in the larger region marked
495
by the gray rectangle indicating the activated region
496
(Fig. 9B). In case B, the activating function values at the
497
border between AIS and the naked axon are essentially
498
smaller than below the electrode. However, the high
so-499
dium channel density of AIS supports spike initiation at
500
the naked axon, acting as a favorable neighboring
com-501
partment. The activating function values for the AIS are
502
very similar to those values for the naked axon close to
503
it, but its position next to the soma results unfavorable.
504
The convenient neighboring compartments together with
505
axial current flow are crucial components for excitation
506
during intra- and extracellular stimulation.
507
In the next section we show that in some cases axial
508
current flow initiates spikes rather far away from the
stim-509
ulating electrode when long pulses are applied whereas
510
for short pulses spike initiation is close to the electrode.
511
By systematic evaluation and analysis of computer
simu-512
lations we discovered this phenomenon with remarkable
513
consequences for strength–duration curves.
514
Deviations from classical strength–duration curves
515
In order to smoothen recording errors, strength–duration
516
curves are usually fitted by one of the classical
electrode 50µm above soma Iel=-8.6 µA
electrode 200µm above soma Iel=-55 µA
A
B
intracellular soma stimulation Iel=1.0 nA 250 µm 500 µs 10 mV 24% 25% 56% 100% 100% 100%
C
Fig. 7.Temporal evolution of transmembrane voltage profiles in 100ls 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 100ls. Increasing the electrode distance from zel= 50lm (A) to zel= 200lm (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 100ls (C) Left figures A and C are replications ofFig. 5C and D.
517 approaches, the formulas of Weiss and Lapicque (Fig. 3).
518 However, for a constant electrode position our computer
519 simulations demonstrate that deviations from these
520 curves have to be expected when spikes are initiated at 521 different functional parts for short and long pulses.
Intra-522 cellular stimulation of the main dendrite of Neuron 2
re-523 sults in a composite strength–duration curve. This curve 524 consists of two segments of ‘classical curves’ that are
525 connected with a transition part (thick red curve marked 526 as DEND in the lower part ofFig. 1B, the case of
intracel-527 lular stimulation).
528 The occurring phenomena for this main dendrite and
529 other neural structures are explained extensively in 530 Fig. 10. For short pulses, spikes are generated within
531 the dendrite at the site of current injection (green curve
532 inFig. 10A). As expected, but not shown, the spike bifur-533 cates there with one part conducted toward the axon and
534 the other toward peripheral dendritic regions. For a longer 535 stimulation pulse of 5 ms, during the first part of
stimula-536 tion spatial transmembrane voltage distributions similar 537 to the green curve are produced at subthreshold intensity,
538 but with smaller amplitudes (e.g., the two thick blue lines
539 inFig. 10A). During a 5 ms pulse just above threshold 540 intensity, the peak value at this electrode position is not
541 strong enough to produce a dendritic spike, but axial cur-542 rent flow into the axon causes enough sodium current via
543 the low threshold sodium channels Nav1.6 in AIS. As a 544 result a spike initiates at the beginning of the thin part of
545 the axon (t = 5.7 ms, lower thin blue line in Fig. 10A).
546
Thereafter, this spike bifurcates in an asymmetric way
be-547
cause of asymmetric cell properties.
548
In the presented examples of extracellular stimulation,
549
the effect of combined strength–duration curves is difficult
550
to observe for cathodic currents, although the
phenome-551
non is present in a weak form for the cases SOMA and
552
DEND (upper part of Fig. 1B). In contrast, combined
553
strength–duration curves occurs clearly for anodic pulses
554
(Fig. 10B), especially for an electrode above the soma or
555
a node of Ranvier. In contrast to cathodic stimulation,
556
anodic stimulation generates a center of hyperpolarization
557
at a region of the neuron closest to the electrode, and the
558
side lobes of the activating function (marked inFig. 4by
559
green arrows) became the driving forces for excitation.
1 electr ode 50µm 100µm 200µm 300µm
abo ve node of Ran vier
electr ode 50µm 100µm 200µm 300µm above soma 2 3
curr ent inject ed in the soma in a node of Ran vier 4 rheobase*
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 Fig. 8.Strength–duration curves for Neuron 1 normalized to rheo-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 lower part of the figure help to identify the strength–duration curves. The curves containing the circles show a trend toward larger chronaxies with increasing electrode distances.
200µm dendrite unmyel. axon myelin. axon N.R. N.R. N.R. N.R. soma AIS naked axon axon hillock electrode activated region activated region N.R. dendrite unmyel. axon myelin. axon N.R. N.R. N.R. N.R. soma AIS naked axon axon hillock N.R. electrode 70° 100mV 200µs
STIMULUS Iel =-15µA (A) Iel =-150µA (B)
B
A
Fig. 9. Stimulation of Neuron 1 by a microelectrode 50lm (A) and 200lm (B) above the center of the naked axon. Every line on the right side shows the membrane voltage of one compartment in accordance with the cell geometry (left). Short compartment lengths in the unmyelinated axon are because of numerical reasons. Thick gray lines represent the membrane voltage where spikes are initiated which is at the center of the naked axon corresponding to the electrode position in A but is shifted into AIS in B.
560 Therefore, the first reaction describing the membrane 561 voltage after 10
l
s of anodic stimulation for an electrode562 at 50
l
m above the soma has two positive peaks563 (Fig. 10C, black line). Short pulses cause spike initiation
564 at the distal end of the AIS (Fig. 10C, green curves),
565 whereas spikes arise at the centre of the thin axonal
seg-566 ment for long pulses (Fig. 10C, blue curves). A similar
567 shift phenomenon occurs when the electrode is above
568
the third node of Ranvier (Fig. 10D): spikes are initiated
569
at the distal end of the thin segment and at a position
clo-570
ser to its center by short and long pulses, respectively.
571 DISCUSSION
572
The time constant of a patch of cell membrane is defined
573
as
s
= R C, with capacity C and resistance R. Chronaxie500 1000 µm -40 0 40mV short pulse ( Iel=20nA t=50µs 50µs) long pulse (5ms) Iel=0.8nA t=6.18ms t=5.7ms t=5 ms x x x x soma AIS main dendrite thin axon myelinated axon current injection membrane voltage 1 10 100 1000 SOMA AIS DEND NODE OF RANVIER NAKED AXON µA 0.01 0.1 1 10 ms PULSE DURATION ELECTRODE CURRENT 50µs pulse Iel=60µA 5ms pulse I=el13.3µA 500 1000 µm x x x x soma AIS main dendrite thin axon myelinated axon membrane voltage -100 0 50 mV t=4.66ms t=200µs t=50µs t=5ms 5ms pulse I=11.4µAel 50µs pulse Iel=80µA -50 50 mV t=5.14ms t=50µs t=10µs t=300µs t=5ms membrane voltage electrode -250 -150 500 1000 µm x x x x soma AIS main dendrite thin axon myelinated axon
A
B
C
D
C
D
ANODIC STIMULATION ANODIC STIMULATION ABOVE SOMA ANODIC STIMULATION ABOVE N.R. INTRACELLULAR STIMULATION t=1 ms electrodeFig. 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 50ls (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 50ls 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 (300lm) at the thin segment of the axon. (B–D) Extracellular anodic stimulation of Neuron 2 for electrode positions as defined inFig. 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 10ls response of the short pulse stimulation is shown in black. (D) Stimulus electrode above node of Ranvier.
574 is about 0.7
s
for such a patch (Ranck, 1975; Reilly, 1992) 575 which is independent of cell geometry as long asintracel-576 lular axial current flow is negligible. The large variety in
577 membrane resistance is the main reason for specific
578 chronaxie values for different functional unmyelinated cell
579 regions. In contrast to the variance found in membrane
580 resistance, membrane capacity is rather uniform with a
581 value close to 1
l
F/cm2(Cole, 1968). However, myelin-582 ated regions have considerably smaller chronaxies as583 the total membrane capacity is inversely proportional to 584 the number of myelin sheets.
585 Strength–duration characteristics for external stimula-586 tion of myelinated peripheral axons can be derived with
587 linear methods from the intracellular relationship
588 (Warman et al., 1992). In contrast to peripheral nerve 589 applications, microelectrodes for central nervous system
590 stimulation operate often in or in the vicinity of the soma, 591 where additional effects appear as consequences of the
592 variation of the electrical and geometrical parameters 593 along the neural axis.
594 Here, we have demonstrated a radical switch to short
595 chronaxies when the stimulating electrode position
596 changes from inside to outside the cell. The inside/outside
597 chronaxie ratio with a value close to 6.5 was the largest 598 for the smallest investigated cell–electrode distance of
599 50
l
m (Fig. 8). This ratio depends essentially on the 600 length of the excited zone positioned betweenhyperpolar-601 ized regions for the cathodic case. These hyperpolarizing 602 lobes hinder the continuous excitation process seen by
603 intracellular stimulation. Consequently, the inside/outside
604 chronaxie ratio becomes smaller for larger electrode–cell 605 distances. Increasing electrode distance usually will
606 cause the excitation of other functional cell regions with 607 other chronaxic properties. This phenomenon is
espe-608 cially obvious for the electrode position above a soma with
609 highly excitable neighbored axonal regions showing
610 shorter chronaxies (upper part ofFig. 1B), and therefore, 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 614 chronaxie values for intracellular stimulation as marked
615 by gray arrows in the lower part ofFig. 8.
616 The presented theory predicts the experimentally
617 observed trend to larger chronaxies for increased dis-618 tances (West and Wolstencroft, 1983). An analytical
ap-619 proach is available for non-myelinated nerve and muscle
620 fibers that predicts different chronaxies for intra- and 621 extracellular stimulation and the increase of chronaxie
622 with electrode–fiber distance (Suarez-Antola, 2005).
623 Examples of combined strength–duration curves have
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 627 approaches of Weiss or Lapique may be equivocal. Some
628 combined strength–duration curves resulting from retinal
629 ganglion cell recordings can be found in Fig. 9 ofGerhardt 630 et al. (2011).
631 Our results help explain trends of strength–duration
632 characteristics. For example, Neuron 1 and Neuron 2
633 have comparable strength–duration curves with similar 634 excitability sequences (node-ais-soma-dend) both for
635
intra- and extracellular stimulation (Fig. 1B). However,
636
unexpected differences between both neurons occur,
637
e.g., only Neuron 2 shows a bimodal intracellular
638
strength–duration curve (red thick curve in the lower part
639
ofFig. 1B).
640
It is important to notice that many variables like
synap-641
tic activity (Spruston, 2008; Sjo¨stro¨m et al., 2008),
refrac-642
tory behavior (Miocinovic and Grill, 2004), inhomogeneity
643
in ion channel density (Migliore and Shepherd, 2002;
644
Keren et al., 2009), branching (Manita and Ross, 2009),
645
curvature of axons (Rattay et al., 2000; Iles, 2005), pulse
646
shape (Wongsarnpigoon et al., 2010), electrode
configu-647
rations (Smith and Finley, 1997), the inhomogeneity and
648
anisotropy in the tissue (Roth, 1995), pulse trains and
649
neuromodulation (De Vries et al., 2007; Minassian et al.,
650
2007) can influence recruitment and strength–duration
651
relationships. Nevertheless, by applying careful analysis,
652
the large variance in chronaxie, even within one cell
653
should be helpful for selective stimulation, especially
654
when microelectrodes are used to activate a specific
655
region like the AIS, a part of the dendritic tree or the
mye-656
linated axon. As an example, short pulse widths
selec-657
tively activate cells with their somas close to the
658
electrode in epiretinal stimulation (Behrend et al., 2011).
659
One should be aware that chronaxie is also the time
660
where stimulus pulse duration needs a minimum of
661
energy (Geddes, 2004), an important fact for neural
pros-662
theses. Moreover, charge injection (electrode current
663
times pulse duration) from electrode surface limits neural
664
prosthetic applications. The left (quite) linear part of
dou-665
ble logarithmic strength–duration curves, e.g. inFig. 1B,
666
predicts (nearly) constant charges when thresholds of
667
very short pulses are compared (half pulse duration
668
needs double threshold current). Minimum charge occurs
669
with infinitely short pulse duration. With the approach of
670
Weiss (Fig. 3) the factor for the additional charge costs
671
are pulse duration divided by chronaxie.
672 CONCLUSIONS
673
A recent review and several often cited articles sustain
674
the wrong dogma that chronaxie is the same for
intra-675
and extracelluar stimulation (Borchers et al., 2012; Nowak
676
and Bullier, 1998; Geddes, 2004). On the contrary,
evalu-677
ations of our compartment model of a cortical pyramidal
678
cell demonstrate up to 20 times longer chronaxies for
679
intracellular stimulation than for extracellular stimulation
680
(Fig. 1B). This fact as well as the occurrence of combined
681
strength–duration curves are supported by theoretical
682 investigations. 683 UNCITED REFERENCE 684 Butson et al. (2007). Q6 685 Acknowledgments—We thank Dr. Katarina Leao for meaningful
686 comments in earlier versions of this manuscript. This work was
687 supported by the Austrian Science Fund , Grant No.
21848-688 N13. R.N.L. is supported by a Kjell and Ma¨rta Beijers Foundation
689 Grant.
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867 (Accepted 5 April 2012)
