Electrical match between initial segment and somatodendritic compartment for action potential backpropagation in retinal ganglion cells

12 The action potential of most vertebrate neurons initiates in the axon initial segment (AIS), and is then 13 transmitted to the soma where it is regenerated by somatodendritic sodium channels. For successful 14 transmission, the AIS must produce a strong axial current, so as to depolarize the soma to the threshold 15 for somatic regeneration. Theoretically, this axial current depends on AIS geometry and Na+ 16 conductance density. We measured the axial current of mouse RGCs using whole-cell recordings with 17 post-hoc AIS labeling. We found that this current is large, implying high Na+ conductance density, and 18 carries a charge that co-varies with capacitance so as to depolarize the soma by ~30 mV. Additionally, 19 we observed that the axial current attenuates strongly with depolarization, consistent with sodium 20 channel inactivation, but temporally broadens so as to preserve the transmitted charge. Thus, the AIS 21 appears to be organized so as to reliably backpropagate the axonal action potential. 22


Introduction 25
In most vertebrate neurons, action potentials (APs) initiate in the axon initial segment (AIS), a highly 26 organized structure near the soma (Bender & Trussell, 2012), then propagate forward to the axon 27 terminals and backward to the soma and dendrites (Debanne et al., 2011). This backward transmission 28 is functionally important for synaptic plasticity, which requires a precisely timed signal of firing 29 activity at the synapse (Caporale & Dan, 2008). It is also important for long-term intrinsic plasticity 30 since the soma holds the genetic material (Daoudal & Debanne, 2003), and presumably also for 31 structural plasticity of the AIS, which depends on somatic voltage-gated calcium channels (Evans et al., 32 2013). 33 At spike initiation, the soma receives an axial current from the AIS, which depolarizes the membrane. 34 When the somatic membrane potential is depolarized about 30 mV above firing threshold, the AP is 35 regenerated by somatic sodium channels . Hamada et al. (2016) found indirect 36 evidence that the axial current matches the capacitance of the somatodendritic compartment, as they 37 observed that larger cortical pyramidal cells tend to have a more proximal AIS, which should 38 theoretically produce a stronger current. However, the axial current was not directly measured. 39 Such a measurement could also allow estimating the conductance density of AIS sodium channels, in 40 particular using resistive coupling theory (Brette, 2013;Kole & Brette, 2018). The fact that the AIS, a 41 small structure, must produce a current able to charge a much larger piece of membrane (soma and 42 proximal dendrites), suggests that conductance density is high, in agreement with immunochemical 43 observations (Lorincz & Nusser, 2010). However, this has remained a somewhat contentious issue 44 (Fleidervish et al., 2010) because direct patch-clamp measurements in the intact AIS indicate low 45 conductance density (Colbert & Pan, 2002), which could be an artifact due to the anchoring of channels 46 to the cytoskeleton . 47 Finally, it is known that sodium channels can inactivate substantially below threshold, resulting in This suggests that the axial current at spike initiation may also vary substantially. If this is the case, 50 then how can spikes be reliably transmitted to the soma? 51 To address these questions, we measured the axial current and spontaneous action potentials in 52 ganglion cells of isolated mouse retina followed by ankyrin-G-antibody labeling to measure AIS 53 geometry. We examined the axial current at spike initiation, just below threshold, and with threshold 54 adaptation, and compared these results to theoretical predictions. 55 56 Results 57 threshold as the potential when the acceleration d 2 V/dt 2 is maximal. In n = 10 cells (see Methods for 67 the inclusion criteria), we observed that the spike threshold of spontaneous APs was -49 ± 3.8 mV (s.d.) 68 while the regeneration threshold is -16 ± 4.6 mV (s.d.), about 33 ± 5 mV (s.d.) higher (Fig. 1C). This is 69 similar to previous measurements in layer 5 cortical pyramidal cells . The double arrow shows the difference ΔV between spike onset and regeneration threshold. C, Statistics of ΔV. 75 76

Transmission of the axial current to the soma 77
We measured the axial current with whole-cell voltage clamp by stepping the command potential from 78 V0 = -60 mV to a variable potential V. Voltage steps above a threshold value evoke large spikes of 79 inward current ( Fig. 2A). When the peak current is plotted against voltage, a sharp discontinuity is 80 seen (Fig. 2B). Similar recordings have been reported in several cell types in whole-cell patch 81 (Magistretti et al., 2006;Diwakar et al., 2009;Milescu et al., 2010), and also in two-electrode voltage 82 clamp recordings of cat motoneurons (Barrett & Crill, 1980). As argued by Milescu et al. (2010), this 83 abrupt increase in current most likely reflects the axial current produced by the AIS AP. Indeed, the 84 current-voltage curve shows a plateau reflecting the all-or-none axonal spike, followed by an increase 85 at higher potential, most likely reflecting the somatic sodium current. These currents were eliminated 86 by 1 μM tetrodotoxin, a potent sodium channel blocker (2 ±0.4 % current remaining, n = 4 cells, paired 87 t3 = 4.5, p = 0.02). For further analysis, the current measurements were corrected for the errors 88 introduced by the series resistance, as described by Traynelis (1998) (see Methods). 89

101
When the soma is not voltage-clamped, the axial current at spike initiation charges the somatic 102 capacitance (and proximal dendrites). Therefore, we expect that the axial current measured in voltage 103 clamp is approximately equal to the capacitive current C.dV/dt during the initial rise of an AP recorded 104 in current clamp. We estimated the effective capacitance on the first ms of the response to a small 105 current pulse, and measured dV/dt in the initial phase of a spontaneous AP (see Methods). We found 106 that for most cells, the axial current measured in voltage clamp was indeed close to the capacitive 107 current of a spontaneous AP (Fig. 2C). The somatic depolarization due to the axial current is Δ = / , 108 where Q is the total charge Q transmitted to the soma, i.e., the integral of the axial current. We found a 109 linear correlation between Q and C ( Fig. 2D; Pearson correlation r = 0.56, p = 0.02), with a slope Δ = 110 31 mV. This is remarkably close to the difference between spike threshold and regeneration threshold 111 . CC-BY 4.0 International license perpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for this this version posted December 10, 2020. This correlation was likely due to a linear relation between axial current and capacitance, because axial 115 current increased with capacitance ( Fig. 2E) while current duration was essentially constant (Fig. 2F). 116 However, the relation with current was statistically less strong than with charge (Pearson correlation 117 r= 0.47, p = 0.06), possibly because the measurement of Q is more reliable than that of Ip, as integration 118 reduces noise and charge is not affected by electrode filtering. 119 120 Axial current and AIS geometry 121 Theoretically, axial current depends on AIS geometry (Hamada et al., 2016). We measured the 122 geometry of the AIS of n = 14 cells by immunolabeling ankyrin-G, while identifying recorded cells using 123 biocytin in the patch pipette (see Methods) ( Fig. 3A). The AIS was on average 31 µm long (± 6 µm s.d.) 124 and started at 8.6 ± 3.3 µm from the soma (Fig. 3B), with no statistically significant correlation between 125 the two measurements (p = 0.59, Pearson test). We also measured the axon diameter at the proximal 126 and distal ends of the AIS. Keeping in mind that these measurements approach the limits of 127 conventional light microscopy, the proximal and distal diameters were 0.9 ± 0.3 µm and 0. The axial current was on average Ip = -6.7 nA (s.d. 1.8 nA). A conservative estimate of Na + conductance 138 density gmin in the AIS can be estimated by noting that the axial current cannot be greater than the 139 maximum current that all Na + channels can pass. This maximum Na + current is ( !" − ), where ENa 140 » 70 mV is the reversal potential of Na + , G is the total Na + conductance and V » -15 mV is the local 141 membrane potential at which the current through a Na + channel is maximal (based on Na + channel 142 properties measured at the AIS of cortical pyramidal cells ). Therefore, a lower bound 143 for the total Na + conductance is #$% = − & /( !" − ) ≈ 79 nS. Using the high estimate of 1 µm for 144 the diameter, the average AIS area was therefore 97 µm 2 (± 19 µm 2 s.d.), which implies a minimum 145 conductance density of 814 S/m 2 . With a diameter of 0.7 µm, we find a minimum of 1159 S/m 2 . 146 147 Figure 4. Predictions  This is a conservative estimate because it ignores the conditions of propagation of the action potential. 156 Resistive coupling theory provides a quantitative estimate of the axial current as a function of AIS 157 geometry and g, by assuming that current entering the axonal membrane flows resistively towards the 158 soma, which acts as a current sink (Brette,  Brette, 2020). Suppose first that conductance density is very high, such that the AIS is clamped at ENa 160 when sodium channels open (Fig. 4A, dark green). Then by Ohm's law, the AIS will produce an axial 161 current Ip = (ENa-Vt)/Ra, where Ra is axial resistance between the soma and the proximal end of the AIS. 162 Thus, we obtain an inverse relation between axial current and AIS position. However, conductance 163 density is finite, which implies that the proximal side of the AIS is pulled towards the somatic potential 164 (Fig. 4A, light green). This is electrically equivalent to shifting the AIS distally by an amount : 165 where D is the distance of the AIS from the soma, ra is the axial resistance per unit length, Vt is the 167 somatic spike threshold ( !" − ' ≈ 120 mV) and 168 = / 4 % 169 . CC-BY 4.0 International license perpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for this this version posted December 10, 2020. ; https://doi.org/10.1101/2020.09.15.297937 doi: bioRxiv preprint where Ri is intracellular resistivity. Here, AIS length L can be neglected provided that it is substantially 170 larger than (see Methods). This is clearly the case because L was 31 µm on average, while an upper 171 estimate of using d = 1.2 µm and g = 1000 S/m 2 is 17 µm. Thus, in our cells, AIS length should have no 172 impact on axial current. The formula above agrees well with simulations of a simplified model with 173 non-inactivating Na + channels (Fig. 4B), except when the AIS is very proximal, where it gives an 174 overestimation. 175 This analysis shows that it is the proximal geometry of the AIS that matters for the calculation of the 176 axial current. Using d = 1 µm, we find that the error between the predicted and the measured current 177 varies with g, with a broad minimum at about 5500 S/m 2 (Fig. 4C). This is close to the value that Guo  Figure 4D  179 shows the axial current measured in our cells as a function of AIS position, together with the theoretical 180 predictions using g = 5500 S/m 2 with diameters d = 0.8 µm, d = 1 µm and d = 1.2 µm. Although the 181 magnitude of the recorded axial currents is consistent with the theoretical prediction, the correlation 182 between axial current and AIS position did not reach significance (p = 0.66, Pearson test). This may 183 simply reflect the variability of AIS diameter, which has a strong impact on this relation. 184 This estimate relies on the precision of AIS geometry measurements. However, even if AIS position and 185 length were different, high conductance density would still be necessary to account for the measured 186 axial current. This can be shown by calculating the maximum axial current that can be generated across 187 all possible AIS geometries for a given conductance density, and then deducing the minimum 188 conductance density necessary to produce a given axial current I (see Methods). Taking Ri = 100 W.cm, 189 we find gmin = 1263 S/m 2 for d = 1 µm and gmin = 2467 S/m 2 for d = 0.8 µm (as argued above, it is mostly 190 the geometry of the proximal side that matters for this calculation). With a higher value for Ri, the 191 lower bound on conductance density would be proportionally higher. 192 Overall, this analysis shows that the strong axial current produced at spike initiation requires a Na + 193 conductance density in the AIS of at least about 1000 S/m 2 using the most conservative estimates, and 194 plausibly several thousand S/m 2 based on our measurements of AIS location and standard values of 195 Ri . 196 197 The threshold axial current 198

Variation of axial current near threshold 199
In a model where the AIS is reduced to a single point, theory predicts that spikes initiate when the 200 sodium current, and therefore the axial current, reaches a threshold ' = / " , where k is the 201 activation slope factor of sodium channels ( ≈ 5 mV) (Brette, 2013). This makes spike initiation distal 202 from the soma efficient because the Na + flux below threshold is low. We show in the Methods that the 203 formula is approximately correct in an extended AIS model, if Ra is measured between soma and the 204 middle of the AIS. Thus, the threshold axial current is determined by AIS geometry. We tried to 205 estimate It in our cells. 206 To give an order of magnitude, with d = 1 µm and given that the middle position of the AIS is 24 µm on 207 average, we obtain " ≈ 31 MΩ, which gives ' ≈ 160 pA (assuming k = 5 mV), a small current. Figure  208 5A shows a recording of the axial current at threshold, which is noisy. We measure the peak current 209 after smoothing. In addition, theory predicts that the axial current increases very steeply near 210 threshold (dI/dV is infinite at threshold, see Methods), as shown in Figure 5B. This makes the threshold 211 current difficult to measure, and likely leads to an underestimation of the threshold current. We 212 . CC-BY 4.0 International license perpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for this this version posted December 10, 2020. ; https://doi.org/10.1101/2020.09.15.297937 doi: bioRxiv preprint measured the current at different step voltages in steps of 0.5 mV (n = 12). In the example shown in 213 Fig. 5C, a small but noticeable current appears at 3 mV below threshold, which increases at higher 214 subthreshold potentials. More precisely, theory predicts that V-Vt is proportional to (I/It -1) 2 . This 215 relation is shown in a biophysical model in Figure 5D. In a simplified model (no sodium channel 216 inactivation or potassium channels), the slope is predicted to be equal to k/2 (see Methods). This 217 slope is found to be larger in the more realistic model shown in Figure 5D, ≈ 4.7 mV, close to k. Our 218 data fitted this quadratic relation well (Fig. 5E), with slopes ≈ 4.2 mV (± 1.7 mV), in the expected 219 range (Fig. 5F). Thus, the axial current increases steeply just below threshold, in agreement with 220 theory. 221 222 Figure 5. Axial  CC-BY 4.0 International license perpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for this this version posted December 10, 2020. ; https://doi.org/10.1101/2020.09.15.297937 doi: bioRxiv preprint Both the voltage and axial current at threshold are predicted to depend on AIS geometry, namely to 231 decrease when the AIS is shifted away from the soma, all else being equal. We analyzed these relations 232 in n = 10 cells (cells were excluded either because AIS geometry was not measured or reference 233 potential drifted). There was no significant linear correlation in our data between voltage threshold 234 and either AIS start position (Fig. 6A, p = 0.54, Pearson test) or length (Fig. 6B, p = 0.14, Pearson test). 235 However, voltage threshold varies theoretically with both quantities as − log( (/* ). The correlation 236 was stronger with log( (/* ) , although still weak (Pearson correlation r = 0.62, p = 0.06). The 237 regression slope was k = 4.3 mV, a plausible value (Fig. 6C). We note that diameter and perhaps 238 conductance density, which both contribute to the voltage threshold, may also vary across cells. 239 We observed an inverse correlation between axial current threshold and AIS position (Fig. 6D, p = 0.04, 240 Pearson test). Theory makes a quantitative prediction: ' = / " , with Ra measured from the soma to 241 the middle of the AIS. This may differ by a constant factor in a complex biophysical model (Fig. 6E, 242 compare dashed and solid lines). Measured currents are lower than predicted and the inverse 243 correlation is barely significant (Pearson correlation r = -0.65, p = 0.08). As explained above, 244 underestimation and limited precision were expected. Nonetheless, the magnitude of measured 245 currents was reasonably close to theoretical estimations (92 pA vs. 160 pA on average, with all but two 246 cells between 70 and 150 pA). CC-BY 4.0 International license perpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for this this version posted December 10, 2020. ; https://doi.org/10.1101/2020.09.15.297937 doi: bioRxiv preprint

Properties of adaptation 256
We observed that the axial current at spike initiation has just the right magnitude to depolarize the 257 soma to the somatic regeneration threshold. What would happen if the availability of sodium channels 258 varied? In many neurons, sodium channels can inactivate substantially below threshold, producing Threshold adaptation has been observed in current-clamp recordings of salamander RGCs (Mitra & 261 Miller, 2007). If this phenomenon reflects the inactivation of AIS sodium channels, it may compromise 262 the transmission of the AIS spike to the soma. 263 We examined this issue by holding the neuron at different potentials V0 before measuring the voltage 264 threshold. We observed that the threshold increases substantially with V0 (Fig. 7A). The relation 265 between Vt and V0 follows the theoretical expectation for threshold adaptation due to sodium channel We then measured the axial current at spike initiation (just above threshold) as a function of V0 (note 273 that there are fewer data points because current recordings were discarded when Rs changed by more 274 than 30%). We observed that the current decreased considerably with increasing V0 (Fig. 7F). On 275 average, it attenuates by a factor 12.3 ± 5.1 when V0 increases from -60 to -40 mV ( Fig. 7G). At Vi, the 276 current is 32 ± 10 % smaller than the maximum current (Fig. 7H). We first note that the potential $ * at which the axial current is attenuated by √2 is indeed close to the 282 half-inactivation voltage Vi estimated from threshold adaptation ( $ * = −56.5 ± 1.9 mV vs. -55.8 mV) 283 (Fig. 7I). Then when we compare Vt with Ip, we find a logarithmic relation (Fig. 7J, K) with half-slope 284 " = 2.4 ± 2.7 mV (Fig. 7L). Note that this average includes one outlier; the median ka is 3.4 mV (the 285 smaller number of points is due to the fact that exclusion criteria for both Vt and Ip are applied). This 286 strongly suggests that both threshold and axial current adaptation are due to the same phenomenon, 287 sodium channel inactivation. 288 289 . CC-BY 4.0 International license perpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for this this version posted December 10, 2020. While the axial current above threshold is strongly modulated by the available Na + conductance, 301 resistive coupling theory predicts that the threshold axial current depends on AIS geometry but not on 302 sodium conductance (It = k/Ra). Figure 8A shows current-voltage relations for different V0 in the same 303 cell. The curves appear to shift horizontally when V0 is changed, so that the voltage threshold increases 304 with V0 but the axial current at threshold does not, as shown specifically on Figure 8B. Over all 305 measured cells (n = 6; voltage threshold and current threshold were only measurable with a stable Rs 306 in a few cells), It varied by a factor smaller than 2.5 (1.2 ± 0.6) between -60 and -40 mV (Fig. 8C), 307 whereas Ip varied by a factor 12.3 on average. 308 . CC-BY 4.0 International license perpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for this this version posted December 10, 2020. ; https://doi.org/10.1101/2020.09.15.297937 doi: bioRxiv preprint between -60 and -40 mV. If the axial current at spike initiation attenuates by a factor 7, then we expect 316 the induced somatic depolarization to also attenuate by a factor 7, to about 4 mV, which seems 317 insufficient to reach the threshold for somatic regeneration. However, this is not what we found. In 318 this cell, the total transmitted charge, obtained by integrating the current, attenuates only by a factor 319 1.7 (Fig. 9B). This occurs because current duration increases at high V0 (Fig. 9C). Over all measured 320 cells (n = 7), transmitted charge attenuated by a factor 3.1 ± 1.4 from -60 to -40 mV, compared to 12.3 321 ± 5.1 for the axial current (Fig. 9D). The increase in current duration was observed consistently above 322 -50 mV (Fig. 9E). . CC-BY 4.0 International license perpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for this this version posted December 10, 2020. ; https://doi.org/10.1101/2020.09.15.297937 doi: bioRxiv preprint Indeed, we could occasionally observe spontaneous bursts on top of a depolarizing wave, with APs 330 triggered at potentials up to -40 mV, with no sign of transmission failure. An example is shown in 331 Figure 10A, with a selection of individual APs shown in Figure 10B, and corresponding phase plots in 332 Figure 10C. During this burst, spike onset increased up to about -40 mV while the somatic regeneration 333 threshold was stable (Fig. 10D). In summary, we have observed that the AIS of RGCs produces a large axial current at spike initiation 346 (about 7 nA), which requires a high Na + conductance density (most likely several thousand S/m 2 ). The 347 charge that this current transmits to the soma co-varies with somatic capacitance, in such a way as to 348 produce a depolarization of about 30 mV, the amount necessary to bring the somatic potential to spike 349 regeneration threshold. Theory shows that the axial current is mainly determined by AIS position and 350 diameter, and to some extent by Na + conductance density, but perhaps counter-intuitively not by AIS 351 length. 352 In agreement with resistive coupling theory (Brette, 2013;Kole & Brette, 2018), the axial current is 353 small below threshold (on the order of 100 pA at threshold, and undetectable a few mV below) and 354 decreases when the AIS is further away from the soma, which reduces energy consumption. 355 . CC-BY 4.0 International license perpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for this this version posted December 10, 2020. ; https://doi.org/10.1101/2020.09.15.297937 doi: bioRxiv preprint We have also observed that the voltage threshold for spike initiation adapts to depolarization, in a way 356 compatible with Na + channel inactivation. Consistently, the axial current at spike initiation also 357 decreases when the threshold adapts. This attenuation can reach a factor of 10 or more for large 358 depolarizations, which could potentially compromise spike transmission to the soma. However, we 359 found that this attenuation is compensated by a broadening of the axial current. 360 Overall, our results are in good agreement with predictions of resistive coupling theory. The inferred 361 Na + channel activation slope factor, which was found consistently to be ≈ 4 mV in several distinct 362 data sets, may seem to be on the low end of Boltzmann fits to patch-clamp recordings, typically 4-8 mV 363 (Angelino & Brenner, 2007). However, this is likely because this parameter is generally obtained from 364 fits on a broad voltage range, while an exponential fit around the spike initiation voltage yields lower 365 values (Platkiewicz & Brette, 2010). For example, Hodgkin and Huxley (1952) found that the Na + 366 current-voltage curve of the squid axon was well fitted by an exponential of slope 4 mV; Baranauskas 367 and Martina (2006)  In the theory, we did not take into account axonal tapering. However, in RGCs, axon diameter decreases 391 from soma to AIS, then decreases again along the AIS (Raghuram et al., 2019). The theory assumes a 392 uniform diameter, because general analytical solutions do not exist with variable diameter. For the 393 calculation of the axial current at spike initiation, taking into account tapering would tend to reduce 394 the axial resistance between soma and AIS, as if the AIS were closer to the soma. The diameter in the 395 calculation of should be the diameter of the proximal AIS. 396 We were not able to confirm the inverse relation between AIS position and axial current at spike 397 initiation that theory predicts. A plausible reason is that the axial current is very sensitive to diameter, 398 which might have varied substantially across cells. It is also possible that the available Na + conductance 399 . CC-BY 4.0 International license perpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for this this version posted December 10, 2020. on Na + conductance density, the axial current that the AIS generates at spike initiation. 415 By just considering the area of the AIS, to produce a current of 6.7 nA requires a conductance density 416 of about 1150 S/m 2 with d = 0.7 µm (average diameter across the AIS in our data) or 800 S/m 2 with an 417 upper estimate of d = 1 µm. This is a lower bound that neglects considerations of cable theory, namely 418 the fact that the axial current flows from the distal end of the AIS to the soma. 419 It is possible to calculate the maximum axial current produced by an axon of diameter d and 420 conductance density g. This calculation shows that, to account for a current of 6.7 nA, g must be at least This estimate is independent of Na + channel kinetics, and in particular it holds even if Na + channels 426 cooperate (Naundorf et al., 2006). 427 Lorincz and Nusser (2010) counted 187 Nav1.6 channels per µm 2 in the AIS of CA1 pyramidal cells.

428
Assuming a unitary conductance of 10-20 pS per channel (Hille, 2001), this amounts to 1870-3740 429 S/m 2 . However, this is an estimate of the structural density, not necessarily of the functional density.

430
In a computational model of layer 5 pyramidal cells, a density of 2500 S/m 2 was necessary to account 431 for the measured initial depolarization speed of somatic APs (Kole et al., 2008). Similarly, optimization 432 of a model of RGCs for AP shape yielded a conductance density of about 5000 S/m 2 (Guo et al., 2013).

433
Our analysis provides an estimation that is less dependent on model specifics, and confirms these 434 previous studies. 435 The theoretical analysis indicates that a high conductance density is likely a necessary condition to 436 transmit the AIS spike to the soma in a variety of cell types, due to the drastic geometrical variation at 437 the axosomatic boundary. The minimum conductance density to produce an axial current I is 438 proportional to I 2 /dAIS 3 . If we assume that the current must scale with the area of the soma, then the 439 minimum g is proportional to dsoma 4 /dAIS 3 . This ratio appears to be approximately conserved across 440 cell types (Goethals & Brette, 2020), and therefore most neurons should face the same constraint 441 requiring a similar conductance density in the AIS. 442 . CC-BY 4.0 International license perpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for this this version posted December 10, 2020. ; https://doi.org/10.1101/2020.09.15.297937 doi: bioRxiv preprint It should be noted that, despite a high conductance density at the AIS, the total Na + influx through the 443 AIS should theoretically have the same order of magnitude as through the soma and proximal 444 dendrites, as observed (Fleidervish et al., 2010). Indeed, the total Na + influx at the AIS should match 445 the charge necessary to depolarize the soma by about 30 mV, while the total Na + influx at the soma 446 (and proximal dendrites) should account for a further depolarization of a few tens of mV (about 45 mV 447 in our cells). The AIS influx should occur preferentially in the proximal AIS, even if conductance density 448 is uniform, because the driving force of the Na + channel is larger there (see Fig. 4A). This has indeed 449 been observed in cortical pyramidal cells (Baranauskas et al., 2013). proportionality relation between the axial current produced by the AIS and the somatodendritic 455 capacitance. Here we showed more directly that, in RGCs, the charge transmitted by the AIS covaries  456 with the somatodendritic capacitance, in such a way as to depolarize the soma to the threshold for 457 somatic spike regeneration. 458 Overall, our measurements are in line with quantitative predictions of resistive coupling theory. 459 However, we did not observe a correlation between AIS position and capacitance. Theoretically, the scaled with soma size. The authors did observe a positive correlation between soma size and the 466 diameter of the proximal axon (we note that observing such correlations for the AIS proper, which is 467 below 1 µm in diameter, may not be feasible). It cannot be excluded that the lack of significant inverse 468 correlation between AIS position and capacitance is due to the limited precision of our measurements, 469 especially as we did observe an inverse correlation between AIS position and threshold axial current. 470 It is possible that the availability of Na + channels varied across cells. In any case, we stress that axon 471 diameter is a key structural parameter in setting the axial current as well as excitability, and therefore 472 it must be considered to correctly interpret experimental results. 473 It remains that, in order to produce an axial current of appropriate magnitude from an AIS of a given Brette, 2020), and a large effect on axial current. Therefore, it is conceivable that these changes reflect 481 a homeostatic regulation not of excitability per se, but of the axial current required to transmit the AIS 482 spike to the soma. For example, Grubb  Threshold adaptation has been observed in current-clamp recordings of salamander RGCs (Mitra & 492 Miller, 2007), as well as in many other cell types (reviewed in (Platkiewicz & Brette, 2011)). We also 493 observed it in mouse RGCs and quantified it precisely. The voltage threshold starts increasing when 494 the membrane is depolarized above ≈ −56 mV, and for large depolarizations the slope of the relation 495 between potential and threshold is close to 1. That is, the threshold tracks the membrane potential so 496 as to remain a few mV above it. These observations are consistent with theoretical expectations based 497 on Na + channel inactivation (Platkiewicz & Brette, 2011). In layer 5 pyramidal cells, half-inactivation 498 voltage of AIS Na + channels is about -61 mV , which is in line with our 499 observations. 500 To our knowledge, adaptation of the axial current had not been reported before. Our quantitative 501 analysis shows that the co-variation of axial current and threshold is consistent with AIS Na + channel 502 inactivation being the cause of both phenomena. The axial current attenuated by a factor 12 on average 503 over a 20 mV depolarization. This would reduce the charge transmitted to the soma by the same factor 504 and possibly compromise spike transmission to the soma, if the current spike shape were unchanged. 505 However, we observed that this attenuation was largely compensated by a broadening of axial 506 currents. This means that the AP at the AIS broadens when the soma is depolarized. In fact, such 507 broadening has been observed in layer 5 pyramidal cells and attributed to the inactivation of Kv1 508 channels (Kole et al., 2007 10. High-resistance patch seals (>1 GΩ) were obtained before breaking into the cell. Recordings with 539 a series resistance Rs above 25 MΩ, or with a residual Rs (after compensation) above 5 MΩ, were 540 discarded. The resting membrane potential of the cell was recorded in the first minute after breaking 541 in. 542 Passive cell properties were recorded by stepping from -70 to -80 mV in voltage-clamp mode without 543 whole-cell compensation. Series resistance was electronically compensated 80-95% with a lag of 18 544 μs. Between protocols we repeated the voltage step without compensation to monitor changes in 545 series resistance, and series resistance compensation was adjusted if necessary. Passive currents were 546 subtracted using a P/n protocol (5 steps of 5 mV) that preceded each protocol. The P/n protocol was 547 missing for a few recordings; we then subtracted the passive response using a 10 mV step. 548 Adaptation protocols started with a long adaptation step at V0 (0.5 s, V0 varied by steps of 5 mV) 549 followed by an activation step (resolution of 1 mV) to elicit an AIS spike. We ensured that the 550 adaptation step was long enough by varying the step duration in a few cells. 551 In current-clamp mode, bridge balance and pipette capacitance cancellation (6.2-7.1 pF) were used. 552 Hyperpolarizing current pulses were injected to measure the cell's capacitance (see 553 Electrophysiological data analysis). Next, for n = 16 cells, 5-20 minutes of spontaneous activity were 554 recorded to analyze spontaneous APs. 555 At the end of the experiment, the pipette was retracted to obtain an outside-out patch. Outside the 556 retina the tip was cleaned with brief, positive pressure to remove the remaining membrane patch and 557 the potential offset was noted to check for any drift in the reference potential.

Estimation of passive properties 584
The raw series resistance Rs * was measured from responses to a test pulse in voltage clamp: , = 585 Δ / -, where DV is the voltage pulse amplitude and I0 is the amplitude of the first transient peak. The 586 residual series resistance , during a given recording is , = , * − ./0 % 01#& where Rrec is the series 587 resistance used for compensation during the experiment and %comp is the amount of compensation. 588 Effective capacitance is estimated from the response to current pulses, by fitting an exponential to the 589 first ms, which is the time scale of the axial current. This estimation was done in n = 17 cells. 590 591

Analysis of APs 592
The first AP recorded during spontaneous activity was used to measure AP features (Fig. 1). 593 Spontaneous activity was recorded in 16 cells; 6 of them were excluded from this analysis because the 594 reference potential drifted by more than 3 mV. To compute the phase plots (dV/dt vs V), we ensured 595 that the plotted points are isochronic, by considering that dV/dt corresponds to the derivative midway 596 between two consecutive points, and interpolating the values of V at that midpoint. Spike onset was 597 defined as the potential when dV/dt crosses 20 mV/ms for the last time before the AP peak. The value 598 of dV/dt for the initial segment component was defined as the first local maximum between spike onset 599 and the global maximum of dV/dt. In a few cells, this was equal to the global maximum. The 600 regeneration threshold is defined as the potential at the point of maximal acceleration d 2 V/dt 2 after 601 the initial segment component. 602 603

Correction of series resistance error 604
Axial currents were corrected using a minor adjustment of the method described by Traynelis (1998).

605
The presence of the series resistance results in an error in clamping the somatic potential equal to -606 Rs.Ie, where Ie is the current through the electrode. This produces a capacitive current through the 607 somatic membrane equal to C.dV/dt = -RsC dIe/dt, which results in filtering the axial current through a 608 . CC-BY 4.0 International license perpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for this this version posted December 10, 2020. ; https://doi.org/10.1101/2020.09.15.297937 doi: bioRxiv preprint low-pass filter with time constant = , . We correct the recorded current by subtracting this 609 capacitive current: where I* is the corrected current. The time constant is estimated directly by fitting an exponential to 612 the first 0.5 ms of the response to a small voltage step (with the same amplifier tunings as for 613 subsequent recordings) (Fig. 11A). Note that as the whole cell compensation circuit of the amplifier 614 was used, this must be understood as a residual time constant, which can be negative. We used the 615 steps from the P/n protocol, except for a few cells with no P/n protocol, where we used a -10 mV test 616 pulse before the axial current recording. We then correct for the loss in driving force due to imperfect 617 clamping as in Traynelis (1998): 618 where Vc is the command potential. In practice, this was a minor correction. 620 621 Figure 11. Correction Figure 11B shows a recording from a retinal ganglion cell (black). After correction, the peak current is 631 larger (red). We tested the effect of series resistance and the correction in a simple biophysical model 632 of a RGC with an extended AIS, with the electrode modeled as a resistance (see Biophysical Model). 633 Figure 11C shows that the peak recorded current decreases substantially when increasing Rs, but this 634 error is well corrected by the method described above. The spike threshold is only marginally affected 635 by the series resistance ( Fig. 11D; the exact value of the threshold depends on model parameters). We 636 selected cells with residual series resistance smaller than 5 MΩ (n = 20). There was a correlation 637 between measured peak axial current Ip and Rs (Fig. 11E, Pearson correlation r = 0.43, p = 0.06), but it 638 remained small. We observed no correlation between spike threshold and Rs (Fig. 11F, p = 0.25, 639 Pearson test). Therefore, the impact of Rs on our measurements should be moderate. 640

641
Threshold 642 The voltage threshold Vt is defined as the highest command potential where no axonal spike is elicited. 643 The membrane potential at the soma differs slightly from the command potential by − ./, / . As the 644 axial current at threshold is about 100 pA, this error is smaller than 0.5 mV. 6 cells for which the 645 reference potential drifted by more than 3 mV during the recordings were discarded from voltage 646 threshold analyses. 647 As the current at threshold is small, we used only the recordings with P/n protocol (n = 15) to ensure 648 accurate leak subtraction. Three additional cells were excluded from the analysis of threshold current 649 because the recordings were either too noisy or with unstable baseline current. Thus, n= 12 cells were 650 used. The current traces below threshold were smoothed with a sliding window (half-window size is 651 50 points, 1 ms) before peak detection (Fig. 5A). The threshold current was then measured as the 652 largest peak current for the data points between Vt -1mV and Vt. 653 654

Charge and current duration 655
The charge Q transferred to the soma at spike initiation is estimated as the integral of Ie in the time 656 window where the current is greater than 10% of its peak value (to avoid integrating noise). Current 657 duration t50 is the duration during which the current is greater than 50% of the peak value. finer spacing corresponding to the pixel size (including in the z direction also). Interpolation was 673 performed with B-splines using Scipy (Virtanen et al., 2020) to evaluate the spline at each pixel 674 comprised in the axon profile, and coordinates were rounded at the nearest pixel. The ankyrin-G 675 images were then loaded as a 3D stack to get the fluorescence intensity at the interpolated coordinates 676 along the axon profile. The intensity profile was smoothed with a sliding mean (half-width 15 pixels). 677 The AIS start and end position were manually defined using the normalized intensity profile, the 3D 678 stacks and the maximal intensity projection in Fiji (Schindelin et al., 2012). Several cells for which the 679 start or end position were considered too unclear to be determined accurately, were discarded from 680 the analyses, so that morphological measurements were available for n = 14 cells. 681 682

683
The box plots display the distribution of data in the following way. The central bar shows the median. 684 The lower and upper limit of the box shows the first and third quartiles (Q1 and Q3), respectively. The 685 lower and upper whisker bars show Q1-1.5 IQR (interquartile range, the range between Q1 and Q3) 686 and Q3 + 1.5 IQR, respectively. The data points outside the whiskers are outliers and indicated by 687 diamonds marker. assumptions all tend to overestimate the axial current, but the approximation is generally good (see 703 Fig. 4B). When L is much greater than ′ (which was the case in our measurements), ≈ ′ and the 704 axial current is essentially insensitive to L. Intracellular resistivity has not been measured directly in 705 the RGC axons. In dendrites of cortical pyramidal cells, it was estimated to be Ri = 70-100 W.cm (Stuart 706 & Spruston, 1998). Modeling studies in RGCs assume somewhat higher values, up to about 150 W.cm 707 (Sheasby & Fohlmeister, 1999;Fohlmeister et al., 2010), but these are based on model optimization. 708 We chose Ri = 100 W.cm (higher values would yield a higher estimate of Na + channel conductance 709 density). 710 The maximum current across all possible AIS geometries can be calculated by setting Δ =0 and = ∞ 711 (AIS of infinite length starting from the soma): 712 Therefore, the minimum conductance density necessary to produce an axial current I is: 714 Axial current at threshold 717 In a model where the spatial extent of the AIS is neglected (all axonal Na + channels clustered at a single 718 point), the axial current at threshold is k/Ra, where k is the Boltzmann activation slope of Na + channels 719 and Ra is the axial resistance between soma and AIS (Brette, 2013). It is possible to calculate this 720 current for an AIS of length L starting from the soma. 721 The axial current at threshold is: 722 where ra is resistance per unit length. To obtain V'(0), we solve the cable equation in a simple axon 724 model where only the axial current and the Na + current are considered, as in (Goethals & Brette, 2020). 725 We consider a cylindrical axon of diameter d. The AIS has length L and starts from the somatic end. It 726 has a uniform density of Nav channels. The total Nav conductance is 727 where g is the surface conductance density. We neglect leak and K + currents, Nav channel inactivation, 729 as well as all time-varying phenomena. The cable equation then becomes: 730 * * = − " ( !" − )e (?@A !/# )/C 731 . CC-BY 4.0 International license perpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for this this version posted December 10, 2020. Here the driving force ( !" − ) has been approximated by ( !" − (/* ) as in (Brette, 2013). We now 736 write the following change of variables: which defines c ( as an implicit function of -. We look for a bifurcation, that is, a value of U0 when the 750 number of solutions changes. This is obtained by setting the derivative of the right hand-side to 0, 751 which gives: 752 The solution can be calculated: √ c ( /2 ≈ 1.2, giving c ( ≈ 5.8. We have 3 (0) =

757
. CC-BY 4.0 International license perpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for this this version posted December 10, 2020. ; https://doi.org/10.1101/2020.09.15.297937 doi: bioRxiv preprint A simple extrapolated formula consistent with the formulae for both point AIS away from the soma 758 and extended AIS starting from the soma is: 759 where (/* = Δ + /2 is the middle position of the AIS, relative to the soma. In simulations, we find 761 that this is a good approximation (Fig. 12). We calculate the axial current just below threshold as a function of somatic voltage in a point AIS 769 model. We consider a cylindrical axon of diameter d where all the Nav channels are located at a single 770 location. The AIS contains a total Na + conductance G. The axial current is 771 where V is the axonal voltage, Vs is the somatic voltage, and R is the axial resistance between the soma 773 and the AIS. It must equal the Na + current: 774 which is the exponential approximation near threshold. Near threshold, we have ( !" − ) ≈ 776 ( !" − , ). We consider this driving force as a constant Δ . We then absorb (/* into G and take k as At the bifurcation (threshold), we have (differentiation with respect to I): 782 . CC-BY 4.0 International license perpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for this this version posted December 10, 2020. ; https://doi.org/10.1101/2020.09.15.297937 doi: bioRxiv preprint 1 = Δ exp( , * + * ) 783 where , * is the somatic voltage threshold and * is the axial current at threshold. We divide the two 784 previous equations and obtain: 785 In a point AIS, the axial current at threshold is: 787 * = 1/ 788 Therefore: 789 This can be rewritten as:

Relation between voltage threshold and axial current at spike initiation 798
Theoretically, voltage threshold varies as − log , where g is the available Na + conductance 799 (Platkiewicz & Brette, 2011;Brette, 2013;Goethals & Brette, 2020). The axial current at spike initiation 800 also depends on g, and therefore voltage threshold and axial current co-vary when g is varied.  Fig. 4A-B and Fig. 12, we used a simplified model with only non-inactivating Nav channels to check 807 analytical expressions, similar to Brette (2013). A spherical soma of diameter 30 µm is attached to an 808 axonal cylinder of diameter 1 µm and length 500 µm (soma diameter is in fact irrelevant as the soma 809 is voltage-clamped). Specific membrane capacitance is Cm = 0.9 µF/cm 2 ; specific membrane resistance 810 is Rm = 15 000 Ω.cm 2 ; leak reversal potential is EL = -75 mV; intracellular resistivity is Ri = 100 Ω.cm. If 811 not specified, Nav channels are placed from 5 µm to 35 µm on the axon. In Fig. 12, the length ranges 812 from 10 µm to 30 µm and the start position from 0 µm to 20 µm. We used simple single gate activation 813 dynamics with fixed time constant: 814 . CC-BY 4.0 International license perpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for this this version posted December 10, 2020. ; https://doi.org/10.1101/2020.09.15.297937 doi: bioRxiv preprint where ENa = 70 mV, k = 5 mV, V1/2 = -35 mV and # = 53.6 µs (corresponding to 150 µs before 818 temperature correction, see (Goethals & Brette, 2020)). For Fig. 4A and B, Na + conductance density 819 was g = 5000 S/m 2 . For Fig. 12, the total Na + conductance was fixed (G = 350 nS) to keep the total 820 number of Na + channels fixed when AIS length is varied. This corresponds to conductance densities of 821 about 11 100 and 3700 S/m 2 for a 10 and 30 µm long AIS, respectively. The model is simulated in In Fig. 5B, 5D, 6E, 11C and 11D, we used a biophysical model of an AP with inactivating Nav channels 827 and non-inactivating Kv channels, similar to (Goethals & Brette, 2020). The biophysical model has a 828 simple geometry, consisting of a spherical soma (30 µm diameter), a long dendrite (diameter: 6 µm, 829 length: 1000 µm) and a thin unmyelinated axon (diameter: 1 µm, length; 500 µm). The dendrite is 830 irrelevant to most simulations because the soma is voltage-clamped, electrically isolating the dendrites 831 from the axon. It only contributes an additional somatodendritic capacitance when an electrode model 832 is added (Fig. 2). When not specified, the AIS extends from 5 µm to 35 µm from the soma. Specific 833 membrane capacitance is Cm = 0.9 µF/cm 2 ; specific membrane resistance is Rm = 15 000 Ω.cm 2 ; leak 834 reversal potential is EL = -75 mV; intracellular resistivity is Ri = 100 Ω.cm. 835 In Fig. 5B and 5D, the AIS start position was 10 µm, close to the mean AIS start position in our cell 836 population. The threshold was approached with 0.01 mV precision using the bisection method. In Fig.  837 6E, the AIS start position was varied from 0 to 20 µm and the AIS length was 30 µm. In these three 838 panels, the Na + conductance density was g = 3700 S/m 2 . 839 In Fig. 11C-D, we inserted an electrode model, which consists of a resistance Rs (0 to 5 M Ω) between 840 the amplifier and the soma, such that a current (Vc-V)/Rs is injected into the soma (where Vc is the 841 voltage command). The Na + conductance density g was 7400 S/m 2 , to obtain peak axonal currents and 842 thresholds comparable to measurements in RGCs. . CC-BY 4.0 International license perpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for this this version posted December 10, 2020.