Theoretical relation between axon initial segment geometry and excitability
- Sorbonne Université, INSERM, CNRS, Institut de la Vision, France
Figures
Figure 1
Steady-state passive response of the axon (ra=1.3 MΩ/μm, λ=612 μm, EL = -75 mV).
(A, B) With a very small soma; (C, D) With a large soma. (A) Voltage response along the axon for a 10 pA current injected at 20 µm (black) and at 100 µm (red). (B) Input resistance as a function of …
Figure 2
Time scale of responses to axonal current injection in a model with large soma (A-C) and in layer five pyramidal neurons (D-F).
(A, D) Voltage response at the axonal injection site 75 µm away from the soma (red) and at the soma (black). (B, E) Difference between the two responses. (C, F) Input resistance measured 300 µs …
Figure 3
AIS diameter vs. soma diameter in a variety of cell types.
Four points are averages over many neurons of the same type (light blue symbols), other points are individual measurements (dots). Electrical equivalence is represented by the dashed line.
Figure 4
Simple biophysical model of spike initiation.
Top, morphology of the neuron. (A) Equilibrium functions of the gating variables m, h, and n8. (B) Time course of the gating variables at the distal end of the AIS during an action potential. (C) …
Figure 5
Measuring excitability in the biophysical model.
(A) Rheobase (dark blue) and somatic voltage threshold (light blue) as a function of total Nav conductance in the AIS G (dS = 30 µm). The resting potential also changes slightly (dashed). (B) …
Figure 6
Spike threshold vs. AIS position and Nav conductance with a point AIS.
(A) Theoretical prediction of spike threshold vs. AIS position in logarithmic scale, for different total Nav conductances (from 200 to 600 nS). (B) Theoretical prediction of spike threshold vs. …
Figure 7
Relation between AIS geometry and voltage threshold in the biophysical model with constant Nav channel density.
(A) Threshold vs. AIS position (distance between soma and AIS start), for different AIS lengths between 10 and 40 µm. (B) Threshold vs. AIS length for different AIS positions between 10 and 30 …
Figure 8
Effect of compressing the AIS on spike threshold, with total Nav conductance held fixed.
(A) Theoretical spike threshold of an AIS of length L starting from the soma (dashed), compared to an equivalent point AIS placed at position = L (solid black) and at = x1/2 = L/2 (solid …
Figure 9
Dependence of spike threshold on AIS geometry and Nav conductance density in the biophysical model.
(A) Spike threshold vs. Nav conductance density g, for 4 combinations of AIS middle position x1/2 and length L (light blue, x1/2 = 20 µm, L = 20 µm, regression slope: 8.4 mV; light orange, x1/2 = 20 …
Figure 10
Excitability as a function AIS position with hyperpolarizing conductance (biophysical model).
The conductance has reversal potential E = −90 mV and is uniformly expressed in the distal half of the 30 µm long AIS (G = 500 nS). (A) Somatic threshold vs. AIS position for different conductance …
Figure 11
Effect of a hyperpolarizing axonal current in the biophysical model with a point AIS.
(A) Resting potential at the soma and AIS as a function of current intensity |I|, for an AIS placed 25 µm away from the soma. (B) Threshold at the AIS and soma as a function of |I|. (C) Difference …
Figure 12
Effect of axon morphology on the relation between AIS position and excitability (biophysical model).
Top: schematics of the different morphologies considered. (A) Somatic threshold vs. AIS position for a large neuron (light blue; diameter: 3 µm) compared to the original neuron (dark blue; diameter: …
Figure 13
Corrective term .
The threshold of an extended AIS differs from that of point AIS with the same total conductance placed at the midpoint by at most .
Tables
Table 1
Parameters values of the biophysical model.
Time constants corrected for temperature are indicated in brackets.
| Passive properties | Rm | 15 000 .cm2 |
| EL | −75 mV | |
| Ri | 100 Ω.cm | |
| Cm | 0.9 µF/cm2 | |
| Nav channels | gNa, soma | 250 S/m2 |
| gNa, dendrite and axon (non AIS) | 50 S/m2 | |
| gNa, AIS | variable (default: 3500 S/m2) | |
| ENa | 70 mV | |
| , soma | −30 mV | |
| , soma | −60 mV | |
| , AIS | −35 mV | |
| , AIS | −65 mV | |
| km | 5 mV | |
| kh | 5 mV | |
| 150 µs (corrected: 54 µs) | ||
| 5 ms (corrected: 1.8 ms) | ||
| Kv1 channels | gK, soma | 250 S/m2 |
| gK, dendrite and axon (non AIS) | 50 S/m2 | |
| gK, AIS | 1500 S/m2 | |
| EK | −90 mV | |
| −70 mV | ||
| kn | 20 mV | |
| 1 ms | ||
| Kv7 channels | gKv7 | variable |
| EK | −90 mV |
Table 2
Changes in AIS geometry and voltage threshold (ΔVs) in structural plasticity and development studies, with the theoretical expectation, assuming constant functional Nav channel density and everything else unchanged (e.g. axon diameter, channel properties).
| Neuron type | Initial AIS L (μm) | Initial AIS x1/2 (μm) | Final AIS L (μm) | Final AIS x1/2 (μm) | ΔVs (mV) | ΔVs theory (mV) | Reference |
|---|---|---|---|---|---|---|---|
| Plasticity | |||||||
| Chicks nucleus magnocelluaris | 9.6 | 13.3 | 19.5 | 18.4 | -4 | −5.2 | Kuba et al., 2010 |
| Hippocampal cultures (only excitatory) | 34.8 | 20.9 | 33.6 | 27.2 | 4.3 | −1.1 | Grubb and Burrone (2010) |
| Hippocampal dentate granule cells in cultures | 19.2 | 10.4 | 15.7 | 7.85 | −1.1 | 2.4 | Evans et al., 2015 |
| Olfactory bulb dopaminergic neurons in cultures | 11.7 | 21.1 | 14.2 | 15.5 | −0.4 | 0.6 | Chand et al., 2015 |
| Development* | |||||||
| Chicks nucleus laminaris, low | 30.3 | 24.8 | 23.9 | 19.9 | −12.7 | 2.3 | Kuba et al., 2014 |
| Chicks nucleus laminaris, middle | 28.8 | 24.8 | 14.4 | 28.3 | / | 2.8 | Kuba et al., 2014 |
| Chicks nucleus laminaris, high | 26.5 | 26.6 | 9.8 | 50.1 | −14.3 | 1.8 | Kuba et al., 2014 |
-
*Initial = E15, final = P3-7.
Table 3
Mean diameter of soma and AIS in 4 cell types, extracted from electron microscopy studies.
| Adult cat olivary cells | Soma | 21.7 µm ± 3.7 µm | (de Zeeuw et al., 1990) |
| AIS | 1.1 µm ± 0.3 µm | (Ruigrok et al., 1990) | |
| Adult rat CA3 pyramidal cells | Soma | 20.9 ± 3.2 µm | (Buckmaster, 2012) |
| AIS | 1.2 µm ± 0.4 µm | (Kosaka, 1980) | |
| Adult rat Purkinje cells | Soma | 21.9 µm ± 1.9 µm | (Takacs and Hamori, 1990) |
| AIS | 0.7 µm ± 0.2 µm | (Somogyi and Hámori, 1976) | |
| Adult mouse cerebellar granule cells | Soma | 5.9 µm ± 0.3 µm | (Delvendahl et al., 2015) |
| AIS | 0.2 µm (no s.d.) | (Palay and Chan-Palay, 2012) |
