Figure 1.
Sharpness of spike initiation with distal initiation (Na channels clustered at a single point).
A, Proportion of open Na channels as a function of the voltage of an isopotential neuron. B, Current-voltage relationship in the isopotential neuron with Na and leak channels. C, Current-voltage relationship in a ball-and-stick model, with Na channels at the soma (dark blue), and at 20 µm (green), 40 µm (red) and 100 µm (light blue) away from the soma. D, Proportion of open Na channels as a function of somatic voltage of the ball-and-stick model (color code as in C). E, Voltage across the axon as the soma is depolarized by steps of 3 mV (4 values from −64 mV to −55 mV), with Na channels at 40 µm from the soma. F, Voltage at the initiation site as a function of somatic voltage, showing the loss of voltage control around −55 mV.
Figure 2.
Spike initiation sharpness due to a bifurcation.
A, Simplified electrical model of the axon initial segment (used for theoretical analysis but not for simulations). The initiation site is coupled to the soma through an axial resistance Ra. Sodium current f(Va) equals axial current (Va−Vs)/Ra. B, Sodium current (red) and lateral current (black) as a function of voltage at the initiation site when Na channels are placed at 20 µm away from the soma. The three lines correspond to somatic voltages of −60 mV, −55 mV and −50 mV. C, Same as B, with Na channels at 40 µm from the soma. D, Same as A at the critical point, with channels at 27 µm away from the soma. E, Predicted somatic spike threshold, defined as the bifurcation point, as a function of location of the initiation site (logarithmic scale) for the full formula (black) and its approximation (blue). The red curve shows the somatic voltage at which half of the sodium channels are activated in the numerical simulation of the ball-and-stick model. The dashed line is the predicted spike threshold at the critical point.
Figure 3.
A, A pulse of current is injected at time 20(black), with Na channels opening almost all at once (dashed black: proportion of open channels), resulting in a discontinuity in the voltage derivative at the soma (the “kink”). B, Phase plot (voltage derivative vs. voltage) at the initiation site (black) and at the soma (red), showing a (modest) kink at the soma of about 5 mV/ms. C, Same as A, but Nav1.2 channels are placed at 15 µm from the soma, in addition to the Nav1.6 channels at 40 µm. D, Proportion of open Na channels as a function of somatic voltage, for both channel subtypes (Nav1.6 at initiation site, Nav1.2 closer to the soma). The two activation curves are shown in dashed with the same colors. E, Same as B, but with the additional Nav1.2 channels. The dashed red curve shows the somatic phase plot when Nav1.2 channels are also added at the soma. F, Enlargement of panel F, Onset rapidness is measured as the slope of the trajectories (green segments) when dVm/dt reaches 10 mV/ms (dashed line).
Figure 4.
Spatially distributed channels.
A, Voltage across the axon as the soma is depolarized by steps of 3(5 values from −64 mV to −52 mV), with Na channels uniformly distributed between 25 and 40 µm away from the soma (compare with Fig. 1E, same total Na conductance). B, Same as A, but with Na channel density linearly decreasing between 25 and 40 µm. C, Spike threshold defined as the somatic voltage at which half of the sodium channels are activated as function of the location of Na channels when they are clustered at a single point (black). Red dots show the spike threshold when Na channels are uniformly distributed on a segment, as a function of the “effective” location, defined as a point inside the segment (x = 0.6 xstart+0.4 xend). Start points are taken among 1, 10, 20, 25, 30 and 35 µm; end points are taken among 40, 50 and 60 µm. D, Same as C but for initiation sharpness. E, Same as A but a tapering piece of axon is inserted at the beginning (the initial segment is moved), with length 10 µm and diameter linearly decreasing from 4 µm to 1 µm. F, Same as B with the initial tapering.
Figure 5.
Energy consumption with spike initiation at the soma and in the axon.
A, Response of the neuron model with Na channels at 40 µm away from the soma to a fluctuating current injected at the soma. Since there are only Na channels in the model, the membrane potential is reset when half of Na channels are open (spikes are added to the trace for readability). B, Firing rate as a function of maximum Na conductance, when channels are in the axon (red, 40 µm from the soma) and at the soma (black). C, Average Na current as a function of output firing rate, for both cases. D, Same as C, but normalized by the maximum Na conductance (essentially reflecting the average proportion of open channels).