Properties of the axial current of retinal ganglion cells at spike initiation

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


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 and 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 and Dan, 2008). It is also important for long-term intrinsic plasticity 30 since the soma holds the genetic material (Daoudal and 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 and Brette, 2018). The fact that the AIS, 41 a 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 and 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 and Pan, 2002), which could be an artifact due to the anchoring of 46 channels 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 (n = 20) followed by ankyrin-G-antibody labeling to measure the regeneration threshold as the potential when the acceleration d 2 V/dt 2 is maximal (Fig. 1D). In n = 67 10 cells with a stable reference potential, we observed that the spike threshold of spontaneous APs 68 was -49 ± 3.8 mV (s.d.) while the regeneration threshold is -16 ± 4.6 mV (s.d.) (Fig. 1F), about 33 ± 5 69 mV (s.d.) higher (Fig. 1F). This is similar to previous measurements in layer 5 cortical pyramidal cells 70 . We measured the axial current with whole-cell voltage clamp by stepping the command potential from 82 V0 = -60 mV to a variable potential V (Fig. 2). Voltage steps above a threshold value evoke large spikes 83 of inward current ( Fig. 2A). When the peak current is plotted against voltage, a sharp discontinuity is 84 seen (Fig. 2B). Similar recordings have been reported in several cell types in whole-cell patch (Diwakar 85 et al., 2009;Magistretti et al., 2006;Milescu et al., 2010), and also in two-electrode voltage clamp 86 recordings of cat motoneurons (Barrett and Crill, 1980). As argued by Milescu et al. (2010), this abrupt 87 increase in current most likely reflects the axial current produced by the AIS AP. Indeed, the current-88 voltage curve shows a plateau reflecting the all-or-none axonal spike, followed by an increase at higher 89 potential, most likely reflecting the somatic sodium current. These currents were eliminated by 1 μM 90 tetrodotoxin, a potent sodium channel blocker (2 ±0.4 % current remaining, n = 4 cells, paired t3 = 4.5, 91 p = 0.02). 92  Together with the membrane capacitance C, the series resistance forms a RC circuit that low-pass 108 filters the current with a characteristic time constant = ! . Thus, part of the axial current is lost as 109 a capacitive current that charges the somatic capacitance. This capacitive current should equal 110 − / . We correct the recorded current by subtracting this current, as described by Traynelis 111 (1998), with the time constant directly estimated from a passive response to a small voltage step ( Fig.  112 2C) (see Methods). Figure 2D shows a recording from a retinal ganglion cell (black). After correction, 113 the peak current is larger (red). 114 We tested the effect of series resistance and the correction in a simple biophysical model of a RGC with 115 an extended AIS, with the electrode modeled as a resistance (see Methods). Figure 2E shows that the 116 peak recorded current decreases substantially when increasing Rs, but this error is well corrected by 117 the method described above. The spike threshold is only marginally affected by the series resistance 118 ( Fig. 2F; the exact value of the threshold depends on model parameters). We selected cells with 119 residual series resistance smaller than 5 MΩ (n = 20). There was a correlation between measured peak 120 axial current Ip and Rs (Fig. 2G, Pearson correlation r = 0.43, p = 0.06), but it remained small. We 121 observed no correlation between spike threshold and Rs (Fig. 2H, p = 0.25, Pearson test). Therefore, 122 the impact of Rs on our measurements should be moderate. 123

124
Transmission of the axial current to the soma 125 When the soma is not voltage-clamped, the axial current at spike initiation charges the somatic 126 capacitance (and proximal dendrites). Therefore, we expect that the axial current measured in voltage 127 clamp is approximately equal to the capacitive current C.dV/dt during the initial rise of an AP recorded 128 in current clamp (more precisely, C.dV/dt = -Ip, as Ip is the current from the amplifier to the cell). This 129 equality assumes that the other currents are negligible, which is plausible given the typical amplitude 130 of axial currents (about 6.7 ± 1.8 nA). Another caveat is that the capacitance is only well defined for an 131 isopotential cell. We estimated the effective capacitance of cells on the first ms of the response to a 132 small current pulse, and measured dV/dt in the initial phase of a spontaneous AP (see Methods). 133 We found that for most cells, the axial current measured in voltage clamp was indeed close to the 134 capacitive current of a spontaneous AP ( Fig.3A; 5 cells were excluded because spontaneous APs or 135 capacitance were not measured). Two cells had a substantially larger capacitive current than expected. 136 As dV/dt was also high in these cells (162 and 201 V/s compared to an average of 142 ± 34 V/s, s.d.), 137 it is plausible that AIS Na + channels were less inactivated in these spontaneous APs than in the voltage 138 clamp measurement at -60 mV (see last section). The fact that the magnitude of the axial current is 139 generally consistent with the depolarization observed in current clamp suggests that our axial current 140 measurements are reasonably accurate. 141 From a functional viewpoint, since the axial current must charge the somatodendritic capacitance, we 142 may expect that Ip and C are linearly correlated. Such a correlation was inferred in layer 5 pyramidal 143 cortical cells by Hamada et al. (2016), who observed that the placement of the AIS was consistent with 144 the axial current being proportional to the capacitance. In our electrophysiological data, we observed 145 a correlation between axial current and capacitance, but it was not very strong statistically ( Fig. 3B; 146 Pearson correlation r= 0.47, p = 0.06). A more robust way to assess the transmission of the AP to the 147 soma is to examine the total charge Q transmitted to the soma, i.e., the integral of the axial current, 148 since the somatic depolarization due to the axial current is Δ = / . Technically, the measurement 149 of Q is more reliable than that of Ip because the integrated current is not affected by the filtering issue 150 discussed above, and because integration reduces noise. 151 We found a linear correlation between Q and C ( Fig. 3C; Pearson correlation r = 0.56, p = 0.02), with a 152 slope Δ = 31 mV. This is remarkably close to the difference between spike threshold and 153 regeneration threshold we observed on spontaneous APs (33 mV; Fig. 1G). This correlation was not 154 due to a correlation between current duration and C (current duration at 50% of the peak was t50 = 155 0.36 ± 0.05 ms, excluding one clear outlier; Fig. 3D The axial current was on average Ip = -6.7 nA (s.d. 1.8 nA). From this value, we deduce a lower bound 169 on Na + conductance density gmin in the AIS by three methods. The first method uses the fact that the 170 axial current is not greater than the maximum current that can pass through each Na + channel. Thus, 171 it produces a lower bound that depends only on axial current and AIS area. The second method consists 172 in calculating the maximum current that a cylindrical axon with uniform conductance density can pass 173 to the soma. Thus, it produces a lower bound that depends on axial current, axon diameter and 174 intracellular resistivity. The third method calculates the axial current based on all biophysical 175 parameters using resistive coupling theory. It is more precise but depends on accurate measurements 176 of AIS geometry and intracellular resistivity. 177 First, the axial current cannot be greater than the maximum current that all Na + channels can pass. 178 This maximum Na + current is ( "# − ), where ENa » 70 mV is the reversal potential of Na + , G is the 179 total Na + conductance and V » -15 mV is the local membrane potential at which the current through a 180 Na + channel is maximal (based on Na + channel properties measured at the AIS of cortical pyramidal 181 cells ). Therefore, a lower bound for the total Na + conductance is To estimate the corresponding minimum conductance density, we measured the geometry of the AIS 184 of n = 14 cells by immunolabeling ankyrin-G, while identifying recorded cells using biocytin in the 185 patch pipette (see Methods) ( Fig. 4A). The AIS was on average 31 µm long (± 6 µm s.d.) and started at 186 8.6 ± 3.3 µm from the soma (Fig. 4B), with no statistically significant correlation between the two 187 measurements (p = 0.59, Pearson test). We also measured the axon diameter at the proximal and distal 188 ends of the AIS. However, it should be kept in mind that these measurements cannot be accurate 189 because of the limitations of conventional light microscopy, and therefore must be considered as rough 190 estimates. The proximal and distal diameters were 0.9 ± 0.3 µm and 0.5 ± 0.2 µm, respectively. For  for d = 0.8 µm (as shown below, it is mostly the geometry of the proximal side that matters for this 220 calculation). With a higher value for Ri, the lower bound on conductance density would be 221 proportionally higher. 222 Finally, a more precise relation between axial current and conductance density can be estimated using 223 the measured AIS geometry (Fig. 5) (Hamada et al., 2016). Suppose first that conductance density is 224 very high, such that the AIS is clamped at ENa when sodium channels open. Then by Ohm's law, the AIS 225 will produce an axial current Ip = (ENa-Vt)/Ra, where Ra is axial resistance between the soma and the 226 proximal end of the AIS. Thus, we obtain an inverse relation between axial current and AIS position, 227 independent of AIS length. However, conductance density is finite, which implies that the proximal 228 side of the AIS is pulled towards the somatic potential (Fig. 5A). This is equivalent to shifting the AIS 229 distally by an amount : 230 where D is the distance of the AIS from the soma, ra is the axial resistance per unit length, and 232 = / 4 % 233 Here, AIS length L can be neglected provided that it is substantially larger than (see Methods). This is 234 clearly the case because L was 31 µm on average, while a higher estimate of using d = 1.2 µm and g = 235 1000 S/m 2 is 17 µm. Thus, in our cells, AIS length should have no impact on axial current. The formula 236 above agrees well with simulations of a simplified model with non-inactivating Na + channels (Fig. 5B), 237 except when the AIS is very proximal, where it gives an overestimation. 238 This analysis shows that it is the proximal geometry of the AIS that matters for the calculation of the 239 axial current. Using d = 1 µm, we find that the error between the predicted and the measured current 240 varies with g, with a broad minimum at about 5500 S/m 2 (Fig. 5C). This is close to the value that Guo  Figure 5D  242 shows the axial current measured in our cells as a function of AIS position, together with the theoretical 243 predictions using g = 5500 S/m 2 with diameters d = 0.8 µm, d = 1 µm and d = 1.2 µm. There is no 244 significant correlation between axial current and AIS position (p = 0.66, Pearson test), but this may 245 simply reflect the variability of AIS diameter, which has a strong impact on this relation. 246 Overall, this analysis shows that the strong axial current produced at spike initiation requires a Na + 247 conductance density in the AIS of at least about 1000 S/m 2 using the most conservative estimates, and 248 plausibly several thousand S/m 2 based on our measurements of AIS location and standard values of 249 Ri. In a model where the AIS is reduced to a single point, theory predicts that spikes initiate when the 263 sodium current, and therefore the axial current, reaches a threshold 0 = / # , where k is the 264 activation slope factor of sodium channels ( ≈ 5 mV) (Brette, 2013). This makes spike initiation distal 265 from the soma efficient because the Na + flux below threshold is low. We show in the Methods that the 266 formula is approximately correct in an extended AIS model, if Ra is measured between soma and the 267 middle of the AIS. Thus, the threshold axial current is determined by AIS geometry. We tried to 268 estimate It in our cells. 269 To give an order of magnitude, with d = 1 µm and given that the middle position of the AIS is 24 µm on 270 average, we obtain # ≈ 31 MΩ, which gives 0 ≈ 160 pA (assuming k = 5 mV), a small current. Figure  271 6A shows a recording of the axial current at threshold, which is noisy. We measure the peak current 272 after smoothing. In addition, theory predicts that the axial current increases very steeply near 273 threshold (dI/dV is infinite at threshold, see Methods), as shown in Figure 6B. This makes the threshold 274 current difficult to measure, and likely leads to an underestimation of the threshold current. We 275 measured the current at different step voltages in steps of 0.5 mV (n = 12). In the example shown in 276 Fig. 6C, a small but noticeable current appears at 3 mV below threshold, which increases at higher 277 subthreshold potentials. More precisely, theory predicts that V-Vt is proportional to (I/It -1) 2 . This 278 relation is shown in a biophysical model in Figure 6D. In a simplified model (no sodium channel 279 inactivation or potassium channels), the slope is predicted to be equal to k/2 (see Methods). This 280 slope is found to be larger in the more realistic model shown in Figure 6D, ≈ 4.7 mV, close to k. Our 281 data fitted this quadratic relation well (  Both the voltage and axial current at threshold are predicted to depend on AIS geometry, namely to 294 decrease when the AIS is shifted away from the soma, all else being equal. We analyzed these relations 295 in n = 10 cells (cells were excluded either because AIS geometry was not measured or reference 296 potential drifted). There was no significant linear correlation in our data between voltage threshold 297 and either AIS start position (Fig. 7A, p = 0.54, Pearson test) or length (Fig. 7B, p = 0.14, Pearson test). 298 However, voltage threshold varies theoretically with both quantities as − log( 1/+ ). The correlation 299 was stronger with log( 1/+ ) , although still weak (Pearson correlation r = 0.62, p = 0.06). The 300 regression slope was k = 4.3 mV, a plausible value (Fig. 7C). We note that diameter and perhaps 301 conductance density, which both contribute to the voltage threshold, may also vary across cells. 302 We observed an inverse correlation between axial current threshold and AIS position (Fig. 7D, p = 0.04, 303 Pearson test). Theory makes a quantitative prediction: 0 = / # , with Ra measured from the soma to 304 the middle of the AIS. This may differ by a constant factor in a complex biophysical model (Fig. 7E,  305 compare dashed and solid lines). Measured currents are lower than predicted and the inverse 306 correlation is barely significant (Pearson correlation r = -0.65, p = 0.08). As explained above, 307 underestimation and limited precision were expected. Nonetheless, the magnitude of measured 308 currents was reasonably close to theoretical estimations (92 pA vs. 160 pA on average, with all but two 309 cells between 70 and 150 pA). 310

Properties of adaptation 319
We observed that the axial current at spike initiation has just the right magnitude to depolarize the 320 soma to the somatic regeneration threshold. What would happen if the availability of sodium channels 321 varied? In many neurons, sodium channels can inactivate substantially below threshold, producing We examined this issue by holding the neuron at different potentials V0 before measuring the voltage 327 threshold. We observed that the threshold increases substantially with V0 (Fig. 8A). The relation 328 between Vt and V0 follows the theoretical expectation for threshold adaptation due to sodium channel factors, respectively). By fitting the theoretical relation, we find % ≈ −55.8 ± 3.1 mV (Fig. 8B), % − 333 $%& ≈ −0.7 ± 2.9 mV (Fig. 8C), # ≈ 4.1 ± 2.2 mV (Fig. 8D), and % / # ≈ 0.9 ± 0.18 (Fig. 8E). These 334 values are consistent with expectations if threshold adaptation is due to sodium channel inactivation. 335 We then measured the axial current at spike initiation (just above threshold) as a function of V0 (note 336 that there are fewer data points because current recordings were discarded when Rs changed by more 337 than 30%). We observed that the current decreased considerably with increasing V0 (Fig. 8F). On 338 average, it attenuates by a factor 12.3 ± 5.1 when V0 increases from -60 to -40 mV (Fig. 8G). At Vi, the 339 current is 32 ± 10 % smaller than the maximum current (Fig. 8H). 340 If adaptation of voltage threshold and axial current are both due to sodium channel inactivation, then 341 axial current and voltage threshold should co-vary with V0. Theoretically, Vt varies with available 342 conductance g as − log (Platkiewicz and Brette, 2011). For low g, the axial current Ip is proportional 343 to J . Therefore, Vt should vary with Ip as − log ' + , or equivalently, −2 log | ' |. 344 We first note that the potential % * at which the axial current is attenuated by √2 is indeed close to the 345 half-inactivation voltage Vi estimated from threshold adaptation ( % * = −56.5 ± 1.9 mV vs. -55.8 mV) 346 (Fig. 8I). Then when we compare Vt with Ip, we find a logarithmic relation (Fig. 8J While the axial current above threshold is strongly modulated by the available Na + conductance, 364 resistive coupling theory predicts that the threshold axial current depends on AIS geometry but not on 365 sodium conductance (It = k/Ra). Figure 9A shows current-voltage relations for different V0 in the same 366 cell. The curves appear to shift horizontally when V0 is changed, so that the voltage threshold increases 367 with V0 but the axial current at threshold does not, as shown specifically on Figure 9B. Over all 368 measured cells (n = 6; voltage threshold and current threshold were only measurable with a stable Rs 369 in a few cells), It varied by a factor smaller than 2.5 (1.2 ± 0.6) between -60 and -40 mV (Fig. 9C), 370 whereas Ip varied by a factor 12.3 on average. 371 between -60 and -40 mV. If the axial current at spike initiation attenuates by a factor 7, then we expect 379 the induced somatic depolarization to also attenuate by a factor 7, to about 4 mV, which seems 380 insufficient to reach the threshold for somatic regeneration. However, this is not what we found. In 381 this cell, the total transmitted charge, obtained by integrating the current, attenuates only by a factor 382 1.7 (Fig. 10B). This occurs because current duration increases at high V0 (Fig. 10C). Over all measured 383 cells (n = 7), transmitted charge attenuated by a factor 3.1 ± 1.4 from -60 to -40 mV, compared to 12.3 384 ± 5.1 for the axial current (Fig. 10D). The increase in current duration was observed consistently 385 above -50 mV (Fig. 10E). 386 Indeed, we could occasionally observe spontaneous bursts on a top of a depolarizing wave, with APs 387 triggered at potentials up to -40 mV, with no sign of transmission failure. An example is shown in 388 Figure 10F, with phase plots from the first 31 APs shown in Figure 10G (only one in 3 APs is plotted 389 for readability). During this burst, spike onset increased up to about -40 mV while the somatic 390 regeneration threshold was stable ( Fig. 10H; for the APs initiated at the highest potentials, the 391 recordings were too noisy for accurate measurement of the regeneration threshold -by eye, between 392 -30 and -20 mV). 393 Thus, detailed properties of the axial current appear to be such as to ensure reliable AP transmission 394 to the soma in changing conditions. 395 In summary, we have observed that the AIS of RGCs produces a large axial current at spike initiation 407 (about 7 nA), which requires a high Na + conductance density (most likely several thousand S/m 2 ). The 408 charge that this current transmits to the soma co-varies with somatic capacitance, in such a way as to 409 produce a depolarization of about 30 mV, the amount necessary to bring the somatic potential to spike 410 regeneration threshold. Theory shows that the axial current is mainly determined by AIS position and 411 diameter, and to some extent by Na + conductance density, but perhaps counter-intuitively not by AIS 412 length. 413 In agreement with resistive coupling theory (Brette, 2013;Kole and Brette, 2018), the axial current is 414 small below threshold (on the order of 100 pA at threshold, and undetectable a few mV below) and 415 decreases when the AIS is further away from the soma, which reduces energy consumption. 416 We have also observed that the voltage threshold for spike initiation adapts to depolarization, in a way 417 compatible with Na + channel inactivation. Consistently, the axial current at spike initiation also 418 decreases when the threshold adapts. This attenuation can reach a factor of 10 or more for large 419 depolarizations, which could potentially compromise spike transmission to the soma. However, we 420 found that this attenuation is compensated by a broadening of the axial current. 421 Overall, our results are in good agreement with predictions of resistive coupling theory. The inferred 422 Na + channel activation slope factor, which was found consistently to be ≈ 4 mV in several distinct 423 data sets, may seem to be on the low end of Boltzmann fits to patch-clamp recordings, typically 4-8 mV 424 (Angelino and Brenner, 2007). However, this is likely because this parameter is generally obtained 425 from fits on a broad voltage range, while an exponential fit around the spike initiation voltage yields 426 lower values (Platkiewicz and Brette, 2010, Fig. 10). For example, Hodgkin and Huxley (1952) found 427 that the Na + current-voltage curve of the squid axon was well fitted by an exponential of slope 4 mV; 428 Baranauskas and Martina (2006) also noted that in cortical pyramidal cells, the slope was lower when 429 estimated around spike threshold than on a broader range (5.4 mV vs. 6.4 mV). One of the main technical limitations to interpret the results of this study is that axonal diameter d 433 cannot be measured precisely with conventional optical microscopy. This is an important limitation 434 because theory shows that key properties are very sensitive to diameter. Specifically, axial resistance 435 is inversely proportional to d 2 . This results in an error in resistance estimation of around 50% for a 436 200 nm error in axon diameter estimation (assuming ≈ 1 µm). This translates to comparable errors 437 in axial current predictions. This limitation should also be kept in mind when interpreting other 438 studies where changes in AIS geometry are observed (see below). The best way to overcome this 439 limitation would be to measure axonal diameter precisely using either electron microscopy or super-440 resolution microscopy. 441 Axial resistance is proportional to intracellular resistivity Ri, but this parameter is difficult to estimate. 442 Ideally, it should be measured by simultaneous recordings in the axon and soma, and a precise estimate 443 requires a precise measurement of axon diameter. Stuart  constraint on Na + conductance density, the axial current that the AIS generates at spike initiation. 475 By just considering the area of the AIS, to produce a current of 6.7 nA requires a conductance density 476 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 477 upper estimate of d = 1 µm. This is a lower bound that neglects considerations of cable theory, namely 478 the fact that the axial current flows from the distal end of the AIS to the soma. 479 It is possible to calculate the maximum axial current produced by an axon of diameter d and 480 conductance density g. This calculation shows that, to account for a current of 6.7 nA, g must be at least 481 1200 S/m 2 if d = 1 µm, and about 2500 S/m 2 if d = 0.8 µm, independently of AIS position. Here the 482 relevant diameter is the diameter of the proximal AIS (about 0.9 µm in our data). Finally, taking into 483 account measured AIS position, the data are consistent with g around 5000 S/m 2 (with d = 1 µm), 484 although the minimum is broad. Overall, this analysis indicates that g should be several thousand S/m 2 . 485 This estimate is independent of Na + channel kinetics, and in particular it holds even if Na + channels 486 cooperate (Naundorf et al., 2006). Our analysis provides an estimation that is less dependent on model specifics, and confirms these 494 previous studies. 495 The theoretical analysis indicates that a high conductance density is likely a necessary condition to 496 transmit the AIS spike to the soma in a variety of cell types, due to the drastic geometrical variation at 497 the axosomatic boundary. The minimum conductance density to produce an axial current I is 498 proportional to I 2 /dAIS 3 . If we assume that the current must scale with the area of the soma, then the 499 minimum g is proportional to dsoma 4 /dAIS 3 . This ratio appears to be approximately conserved across 500 cell types (Goethals and Brette, 2020), and therefore most neurons should face the same constraint 501 requiring a similar conductance density in the AIS. 502 It should be noted that, despite a high conductance density at the AIS, the total Na + influx through the 503 AIS should theoretically have the same order of magnitude as through the soma and proximal 504 dendrites, as observed (Fleidervish et al., 2010). Indeed, the total Na + influx at the AIS should match 505 the charge necessary to depolarize the soma by about 30 mV, while the total Na + influx at the soma 506 (and proximal dendrites) should account for a further depolarization of a few tens of mV (about 45 mV 507 in our cells). The AIS influx should occur preferentially in the proximal AIS, even if conductance density 508 is uniform, because the driving force of the Na + channel is larger there (see Fig. 5A). This has indeed 509 been observed in cortical pyramidal cells (Baranauskas et al., 2013). proportionality relation between the axial current produced by the AIS and the somatodendritic 515 capacitance. Here we showed more directly that, in RGCs, the charge transmitted by the AIS covaries  516 with the somatodendritic capacitance, in such a way as to depolarize the soma to the threshold for 517 somatic spike regeneration. 518 Overall, our measurements are in line with quantitative predictions of resistive coupling theory. 519 However, we did not observe a correlation between AIS position and capacitance. Theoretically, the 520 structural parameters that determine the axial current are AIS position and diameter (and not AIS 521 length, at least not in the range of observed lengths). Therefore, one would expect a negative 522 correlation between AIS position and capacitance if diameter were homogeneous across cells, or at 523 least uncorrelated to capacitance. In fact, Raghuram  scaled with soma size. The authors did observe a positive correlation between soma size and the 526 diameter of the proximal axon (we note that observing such correlations for the AIS proper, which is 527 below 1 µm in diameter, may not be feasible). It cannot be excluded that the lack of significant inverse 528 correlation between AIS position and capacitance is due to the limited precision of our measurements, 529 especially as we did observe an inverse correlation between AIS position and threshold axial current. 530 It is possible that the availability of Na + channels varied across cells. In any case, we stress that axon 531 diameter is a key structural parameter in setting the axial current as well as excitability, and therefore 532 it must be considered to correctly interpret experimental results. 533 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 541 a homeostatic regulation not of excitability per se, but of the axial current required to transmit the AIS 542 spike to the soma. For example, Grubb  Threshold adaptation has been observed in current-clamp recordings of salamander RGCs (Mitra and 552 Miller, 2007), as well as in many other cell types (reviewed in (Platkiewicz and Brette, 2011)). We also 553 observed it in mice RGCs and quantified it precisely. The voltage threshold starts increasing when the 554 membrane is depolarized above ≈ −56 mV, and for large depolarizations the slope of the relation 555 between potential and threshold is close to 1. That is, the threshold tracks the membrane potential so 556 as to remain a few mV above it. These observations are consistent with theoretical expectations based 557 on Na + channel inactivation (Platkiewicz and Brette, 2011). In layer 5 pyramidal cells, half-inactivation 558 voltage of AIS Na + channels is about -61 mV , which is in line with our 559 observations. 560 To our knowledge, adaptation of the axial current had not been reported before. Our quantitative 561 analysis shows that the co-variation of axial current and threshold is consistent with AIS Na + channel 562 inactivation being the cause of both phenomena. The axial current attenuated by a factor 12 on average 563 over a 20 mV depolarization. This would reduce the charge transmitted to the soma by the same factor 564 if the current spike shape were unchanged, possibly compromising spike transmission to the soma. 565 However, we observed that this attenuation was largely compensated by a broadening of axial 566 currents. This means that the AP at the AIS broadens when the soma is depolarized. In fact, such 567 broadening has been observed in layer 5 pyramidal cells and attributed to the inactivation of Kv1 568 channels (Kole et al., 2007). As Kv1.2 is expressed in the distal AIS of RGCs (Van Wart et al., 2007), this 569 might explain our observations. 570 In conclusion, our observations indicate that structural and channel properties of the AIS are 571 functionally organized in such a way as to ensure reliable transmission of the AP to the soma. 572 573

586
Mice were taken at postnatal day 10-12 (P10-12). The pup was rapidly decapitated, the eyes were 587 removed and placed in Ringer's medium containing (in mM): 119 NaCl, 2.5 KCl, 1.0 KH2PO4, 11 glucose, 588 26.2 NaHCO3, 2 CaCl2 and 1 MgCl2 (290-295 mOsm), bubbled with carbogen (95% O2/5% CO2). The 589 retina was dissected and fixed on filter paper over a small hole (N8895, Sigma-Aldrich) with the RGC 590 layer upwards and continuously perfused with Ringer's solution warmed to 32 degrees Celsius. 10. High-resistance patch seals (>1 GΩ) were obtained before breaking into the cell. Recordings with 599 a series resistance Rs above 25 MΩ, or with a residual Rs (after compensation) above 5 MΩ, were 600 discarded. The resting membrane potential of the cell was recorded in the first minute after breaking 601 in. 602 Passive cell properties were recorded by stepping from -70 to -80 mV in voltage-clamp mode without 603 whole-cell compensation. Series resistance was electronically compensated 80-95% with a lag of 18 604 μs. Between protocols we repeated the voltage step without compensation to monitor changes in 605 series resistance, and series resistance compensation was adjusted if necessary. Passive currents were 606 subtracted using a P/n protocol (5 steps of 5 mV) that preceded each protocol. The P/n protocol was 607 missing for a few recordings; we then subtracted the passive response using a 10 mV step. 608 Adaptation protocols started with a long adaptation step at V0 (0.5 s, V0 varied by steps of 5 mV) 609 followed by an activation step (resolution of 1 mV) to elicit an AIS spike. We ensured that the 610 adaptation step was long enough by varying the step duration in a few cells. 611 In current-clamp mode, bridge balance and pipette capacitance cancellation (6.2-7.1 pF) were used. 612 Hyperpolarizing current pulses were injected to measure the cell's capacitance (see 613 Electrophysiological data analysis). Next, for n = 16 cells, 5-20 minutes of spontaneous activity were 614 recorded to analyze spontaneous APs. 615 At the end of the experiment, the pipette was retracted to obtain an outside-out patch. Outside the 616 retina the tip was cleaned with brief, positive pressure to remove the remaining membrane patch and 617 the potential offset was noted to check for any drift in the reference potential.

Estimation of passive properties 643
The raw series resistance Rs * was measured from responses to a test pulse in voltage clamp: ! = 644 Δ / 4 , where DV is the voltage pulse amplitude and I0 is the amplitude of the first transient peak. The 645 residual series resistance ! during a given recording is ! = ! * − 567 % 78$' where Rrec is the series 646 resistance used for compensation during the experiment and %comp is the amount of compensation. 647 Effective capacitance is estimated from the response to current pulses, by fitting an exponential to the 648 first ms, which is the time scale of the axial current. This estimation was done in n = 17 cells. 649 650

Analysis of APs 651
The first AP recorded during spontaneous activity was used to measure AP features (Fig. 1).

652
Spontaneous activity was recorded in 16 cells; 6 of them were excluded from this analysis because the 653 reference potential drifted by more than 3 mV. To compute the phase plots (dV/dt vs V), we ensured 654 that the plotted points are isochronic, by considering that dV/dt corresponds to the derivative midway 655 between two consecutive points, and interpolating the values of V at that midpoint. Spike onset was 656 defined as the potential when dV/dt crosses 20 mV/ms for the last time before the AP peak. The value 657 of dV/dt for the initial segment component was defined as the first local maximum between spike onset 658 and the global maximum of dV/dt. In a few cells, this was equal to the global maximum. The 659 regeneration threshold is defined as the potential at the point of maximal acceleration d 2 V/dt 2 after 660 the initial segment component. 661 662

Correction of series resistance error 663
Axial currents were corrected using a minor adjustment of the method described by Traynelis (1998).

664
The presence of the series resistance results in an error in clamping the somatic potential equal to -665 Rs.Ie, where Ie is the current through the electrode. This produces a capacitive current through the 666 somatic membrane equal to C.dV/dt = -RsC dIe/dt, which results in filtering the axial current through a 667 low-pass filter with time constant = ! . We correct the recorded current by subtracting this 668 capacitive current: where I* is the corrected current. The time constant is estimated directly by fitting an exponential to 671 the first 0.5 ms of the response to a voltage step (with the same amplifier tunings as for subsequent 672 recordings). We used the steps from the P/n protocol, except for a few cells with no P/n protocol, 673 where we used a -10 mV test pulse before the axial current recording. We then correct for the loss in 674 driving force due to imperfect clamping as in Traynelis (1998): 675 where Vc is the command potential. In practice, this was a minor correction. 677

678
Threshold 679 The voltage threshold Vt is defined as the highest command potential where no axonal spike is elicited. 680 The membrane potential at the soma differs slightly from the command potential by − 56! 6 . As the 681 axial current at threshold is about 100 pA, this error is smaller than 0.5 mV. 6 cells for which the 682 reference potential drifted by more than 3 mV during the recordings were discarded from voltage 683 threshold analyses. 684 As the current at threshold is small, we used only the recordings with P/n protocol (n = 15) to ensure 685 accurate leak subtraction. Three additional cells were excluded from the analysis of threshold current 686 because the recordings were either too noisy or with unstable baseline current. Thus, n= 12 cells were 687 used. The current traces below threshold were smoothed with a sliding window (half-window size is 688 50 points, 1 ms) before peak detection (Fig. 6A). The threshold current was then measured as the 689 largest peak current for the data points between Vt -1mV and Vt. 690 691

Charge and current duration 692
The charge Q transferred to the soma at spike initiation is estimated as the integral of Ie in the time 693 window where the current is greater than 10% of its peak value (to avoid integrating noise). Current 694 duration t50 is the duration during which the current is greater than 50% of the peak value. For the current adaptation analysis, cells were discarded if Rs increased by more than 30% during the 701 protocol. In a few cells, the threshold could not be clearly measured at -40 mV and therefore I40 is 702 missing. finer spacing corresponding to the pixel size (including in the z direction also). Interpolation was 710 performed with B-splines using Scipy (Virtanen et al., 2020) to evaluate the spline at each pixel 711 comprised in the axon profile, and coordinates were rounded at the nearest pixel. The ankyrin-G 712 images were then loaded as a 3D stack to get the fluorescence intensity at the interpolated coordinates 713 along the axon profile. The intensity profile was smoothed with a sliding mean (half-width 15 pixels). 714 The AIS start and end position were manually defined using the normalized intensity profile, the 3D 715 stacks and the maximal intensity projection in Fiji (Schindelin et al., 2012). Several cells for which the 716 start or end position were considered too unclear to be determined accurately, were discarded from 717 the analyses, so that morphological measurements were available for n = 14 cells. where Δ is AIS distance from the soma, L is AIS length, d is AIS diameter, g is Na + conductance density, 735 Ri is intracellular resistivity and 736 is axial resistance per unit length. The derivation makes the following assumptions: all Na + channels 738 are open; channel kinetics are neglected; capacitive and leak currents are considered negligible. These 739 assumptions all tend to overestimate the axial current, but the approximation is generally good (see 740 Fig. 5B). When L is much greater than ′ (which was the case in our measurements), ≈ ′ and the 741 axial current is essentially insensitive to L. 742 The maximum current across all possible AIS geometries can be calculated by setting Δ =0 and = ∞ 743 (AIS of infinite length starting from the soma): 744 Therefore, the minimum conductance density necessary to produce an axial current I is: 746 / 747 748

Axial current at threshold 749
In a model where the spatial extent of the AIS is neglected (all axonal Na + channels clustered at a single 750 point), the axial current at threshold is k/Ra, where k is the Boltzmann activation slope of Na + channels 751 and Ra is the axial resistance between soma and AIS (Brette, 2013). It is possible to calculate this 752 current for an AIS of length L starting from the soma. 753 The axial current at threshold is: 754 where ra is resistance per unit length. To obtain V'(0), we solve the cable equation in a simple axon 756 model where only the axial current and the Na + current are considered, as in (Goethals and Brette, 757 2020). We consider a cylindrical axon of diameter d. The AIS has length L and starts from the somatic 758 end. It has a uniform density of Nav channels. The total Nav conductance is 759 where g is the surface conductance density. We neglect leak and K + currents, Nav channel inactivation, Here the driving force ( "# − ) has been approximated by ( "# − 1/+ ) as in (Brette, 2013). We now 768 write the following change of variables: where 1/+ = Δ + /2 is the middle position of the AIS, relative to the soma. In simulations, we find 793 that this is a good approximation (Fig. 11). We calculate the axial current just below threshold as a function of somatic voltage in a point AIS 801 model. We consider a cylindrical axon of diameter d where all the Nav channels are located at a single 802 location. The AIS contains a total Na + conductance G. The axial current is 803 where V is the axonal voltage, Vs is the somatic voltage, and R is the axial resistance between the soma 805 and the AIS. It must equal the Na + current: 806 "# = ( "# − ) exp i − 1 + j 807 which is the exponential approximation near threshold. Near threshold, we have ( "# − ) ≈ 808 ( "# − ! ). We consider this driving force as a constant Δ . We then absorb 1/+ into G and take k as where ! * is the somatic voltage threshold and * is the axial current at threshold. We divide the two 816 previous equations and obtain: 817 In a point AIS, the axial current at threshold is: 819 * = 1/ 820 Therefore: 821 This can be rewritten as: 823 ! = ! * + 1 − * + log * 824 A Taylor expansion gives: 825 In original voltage units, we then obtain: 827

Relation between voltage threshold and axial current at spike initiation 830
Theoretically, voltage threshold varies as − log , where g is the available Na + conductance (Brette,831 2013; Goethals and Brette, 2020; Platkiewicz and Brette, 2011). The axial current at spike initiation 832 also depends on g, and therefore voltage threshold and axial current co-vary when g is varied. The 833 general relation is complicated, but a simple approximated relation can be obtained by considering the 834 equation for the maximum current ' max . Since ' max ∝ J , it follows that with this approximation the 835 threshold varies as − log ' + , i.e., as −2 log ' . 836 837 Simplified model 838 In Fig. 5A-B and Fig. 10, we used a simplified model with only non-inactivating Nav channels to check 839 analytical expressions, similar to Brette (2013). A spherical soma of diameter 30 µm is attached to an 840 axonal cylinder of diameter 1 µm and length 500 µm (soma diameter is in fact irrelevant as the soma 841 is voltage-clamped). Specific membrane capacitance is Cm = 0.9 µF/cm 2 ; specific membrane resistance 842 is Rm = 15 000 Ω.cm 2 ; leak reversal potential is EL = -75 mV; intracellular resistivity is Ri = 100 Ω.cm. If 843 not specified, Nav channels are placed from 5 µm to 35 µm on the axon. In Fig. 11, the length ranges 844 from 10 µm to 30 µm and the start position from 0 µm to 20 µm. We used simple single gate activation 845 dynamics with fixed time constant: where ENa = 70 mV, k = 5 mV, V1/2 = -35 mV and $ = 53.6 µs (corresponding to 150 µs before 850 temperature correction, see (Goethals and Brette, 2020)). For Fig. 5A and B, Na + conductance density 851 was g = 5000 S/m 2 . For Fig. 11, the total Na + conductance was fixed (G = 350 nS) to keep the total 852 number of Na + channels fixed when AIS length is varied. This corresponds to conductance densities of 853 about 11 100 and 3700 S/m 2 for a 10 and 30 µm long AIS, respectively. The model is simulated in 854 voltage-clamp and the threshold is measured with the bisection method. We used the Brian 2 simulator 855 (Stimberg et al., 2019) with 10 µs time step and 1 µm spatial resolution. 856 857 Biophysical model 858 In Fig. 2E, 2F, 6B, 6D and 7E, we used a biophysical model of an AP with inactivating Nav channels and 859 non-inactivating Kv channels, similar to (Goethals and Brette, 2020). The biophysical model has a 860 simple geometry, consisting of a spherical soma (30 µm diameter), a long dendrite (diameter: 6 µm, 861 length: 1000 µm) and a thin unmyelinated axon (diameter: 1 µm, length; 500 µm). The dendrite is 862 irrelevant to most simulations because the soma is voltage-clamped, electrically isolating the dendrites 863 from the axon. It only contributes an additional somatodendritic capacitance when an electrode model 864 is added (Fig. 2). When not specified, the AIS extends from 5 µm to 35 µm from the soma. Specific 865 membrane capacitance is Cm = 0.9 µF/cm 2 ; specific membrane resistance is Rm = 15 000 Ω.cm 2 ; leak 866 reversal potential is EL = -75 mV; intracellular resistivity is Ri = 100 Ω.cm. 867 In Fig. 2E-F, we inserted an electrode model, which consists of a resistance Rs (0 to 5 M Ω) between the 868 amplifier and the soma, such that a current (Vc-V)/Rs is injected into the soma (where Vc is the voltage 869 command). The Na + conductance density g was 7400 S/m 2 , to obtain peak axonal currents and 870 thresholds comparable to measurements in RGCs. 871 In Fig. 6B-D, the AIS start position was 10 µm, close to the mean AIS start position in our cell population. 872 The threshold was approached with 0.01 mV precision using the bisection method. In Fig. 7E, the AIS 873 start position was varied from 0 to 20 µm and the AIS length was 30 µm. In these three panels, the Na + 874 conductance density was g = 3700 S/m 2 . 875

Passive properties
Rm