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The author has declared that no competing interests exist.

Conceived and designed the experiments: RB. Performed the experiments: RB. Analyzed the data: RB. Wrote the paper: RB.

In cortical neurons, spikes are initiated in the axon initial segment. Seen at the soma, they appear surprisingly sharp. A standard explanation is that the current coming from the axon becomes sharp as the spike is actively backpropagated to the soma. However, sharp initiation of spikes is also seen in the input–output properties of neurons, and not only in the somatic shape of spikes; for example, cortical neurons can transmit high frequency signals. An alternative hypothesis is that Na channels cooperate, but it is not currently supported by direct experimental evidence. I propose a simple explanation based on the compartmentalization of spike initiation. When Na channels are placed in the axon, the soma acts as a current sink for the Na current. I show that there is a critical distance to the soma above which an instability occurs, so that Na channels open abruptly rather than gradually as a function of somatic voltage.

Spike initiation determines how the combined inputs to a neuron are converted to an output. Since the pioneering work of Hodgkin and Huxley, it is known that spikes are generated by the opening of sodium channels with depolarization. According to this standard theory, these channels should open gradually when the membrane potential increases, but spikes measured at the soma appear to suddenly rise from rest. This apparent contradiction has triggered a controversy about the origin of spike “sharpness.” This study shows with biophysical modelling that if sodium channels are placed in the axon rather than in the soma, they open all at once when the somatic membrane potential exceeds a critical value. This work explains the sharpness of spike initiation and provides another demonstration that morphology plays a critical role in neural function.

Action potentials are generated in central neurons by the opening of sodium channels in the axon initial segment (AIS)

However, this explanation misses an important part of the story, because it focuses on the shape of action potentials, rather than on spike initiation per se. Indeed several lines of evidence indicate that spike initiation is very sharp, and not only the initial shape of spikes seen at the soma. First, cortical neurons can reliably transmit frequencies up to 200–300 Hz, and respond to input changes at the millisecond timescale

These remarks imply that sharpness is a functionally relevant property of spike initiation rather than a measurement artifact. In fact, there are two distinct sets of observations. The first set focuses on the shape of spikes at onset, the “kink” seen at the soma in the temporal waveform of the action potential. I will simply refer to this phenomenon as the “kink” at spike onset, that is, the abrupt voltage transition seen at the soma at spike onset. Observations of the second set do not refer to the shape of spikes, but rather to the input-output properties of the spike initiation process. Sharpness of spike initiation refers to the abrupt opening of Na channels at the initiation site when a threshold somatic voltage is exceeded. Thus, it can be quantified as the somatic voltage interval over which available Na channels switch from mostly closed to mostly open: in a single-compartment Hodgkin-Huxley model, it would be on the order of 6 mV; in an integrate-and-fire model, it would be 0 mV (no Na current flows until a spike is generated); in a cooperative Na channel model, it would be in between. Thus, to claim that spike initiation is sharp essentially means that spikes are initiated as in an integrate-and-fire model: a negligible amount of Na current flows until a threshold somatic voltage value is reached and a spike is suddenly produced.

Sharpness of spike initiation and the “kink” at spike onset are directly related in a single-compartment Hodkgin-Huxley model, but they are not necessarily equivalent in a spatially extended neuron. Using a simple geometrical model consisting of a sphere and a thin cylinder, I give a parsimonious account of these observations by showing that spike initiation sharpness arises from the geometrical discontinuity between the soma and the AIS, rather than from backpropagation of spikes. When Na channels are placed in the thin axon, they open abruptly rather than gradually as a function of somatic voltage, as an all-or-none phenomenon. I further show that the phenomenon is governed by equations (a bifurcation) that are mathematically almost equivalent to the cooperativity model of Na channels

In order to clearly demonstrate the phenomenon and avoid confounding factors, I consider a neuron model with only passive leak channels and Na channels (low-threshold Kv1 channels are considered in the _{1/2} = −40 mV and slope factor k_{a} = 6 mV (_{1/2} by the sum of a linear part, representing the leak current, and of an exponential part, representing the Na current _{T}: this is the maximum voltage that can be reached with a constant current injection without triggering a spike. This voltage is set by V_{1/2} and by the maximal conductance of the Na channel, relative to the leak conductance _{a}, and sets the sharpness of spike initiation, as assessed for example by the cut-off frequency of signal transmission

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.

However, neurons are not isopotential and spikes are initiated in the AIS, not in the soma. In cortical neurons, the AIS is about 1 µm in diameter, and extends from the axon hillock over an unmyelinated length of 10–60 µm

The increased sharpness due to the lateral current coming from the initiation site has been attributed to the active backpropagation of the distally initiated spike _{a}, the Boltzmann slope factor of the Na activation curve (6 mV in this model, in accordance with patch-clamp measurements

This phenomenon is not due to active backpropagation, since sharpness is defined for Na channels at the initiation site; in addition there are no Na channels between the soma and the initiation site in the present model.

Mathematically, this loss of voltage control corresponds to a bifurcation, that is, a sudden change in the equilibrium points of the system when a parameter (here somatic voltage) is changed by a small amount. The Na current is a function f(V_{a}) of the voltage V_{a} at the initiation site. If the soma acts as a current sink, then the lateral current must be equal to the Na current. This corresponds to a simplified electrical model that approximates the spatially extended model, in which the initiation site and the soma are connected by a resistor, and the Na current is inserted at the initiation site (_{a}−V_{s})/R_{a}, where V_{s} is the somatic voltage and R_{a} is the axial resistance between the soma and the initiation site, which is proportional to the distance. Thus at any time the axonal voltage V_{a} is determined by the somatic voltage V_{s} through a non-linear equation, which expresses the equality of the lateral and Na currents: (V_{a}−V_{s})/R_{a} = f(V_{a}). This equation, which I shall call the current equation, is almost equivalent to the cooperativity model of Na channels _{a} for an initiation site at 20 µm from the soma, corresponding to the green curves in _{a} for a somatic voltage V_{s} of −60, −55 and −50 mV. The value of V_{a} is determined by the intersection of the red and black curves: −59, −52 and −40 mV. Thus V_{a} is amplified compared to the somatic voltage but still varies continuously with it. When the initiation site is at 40 µm from the soma, corresponding to the red curves in _{a} is twice larger. But now as the somatic voltage is increased, the intersection point suddenly jumps from about −55 mV to −25 mV. This occurs because the number of solutions to the current equation changes from 3 to 1 when V_{s} is increased, that is, a bifurcation occurs with respect to variable V_{s}. It corresponds to the loss of voltage control seen in

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(V_{a}) equals axial current (V_{a}−V_{s})/R_{a}. 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.

Graphically, this bifurcation occurs when the line representing the lateral current (black) is tangent to the curve representing the Na current (red) at the intersection point. This can only happen if the slope of the line (1/R_{a}) is smaller than the maximum slope of the Na curve. Thus there is a critical value of R_{a} above which spike initiation becomes sharp. At this point, represented in _{a}.g_{Na}>0.27, where R_{a} is the axial resistance to the initiation site and g_{Na} is the total maximal conductance of Na channels. The axial resistance is determined by the geometry of the AIS and by the intrinsic resistivity R_{i} (150 Ω.cm in this model). The condition can then be written R_{i}.g_{Na}.x/d^{2}>0.21, where d is the axon diameter and x is the distance of the initiation site away from the soma. For the present model, the critical point occurs when the Na channels are placed at distance x = 27 µm away from the soma (see

The spike threshold can be defined as the voltage at the bifurcation point, which is when the line representing the lateral current (black) is tangent to the curve representing the Na current (red) at the intersection point. Mathematically, this is obtained by differentiating the current equation with respect to V_{a}: 1/R_{a} = f′(V_{a}). A simple calculation shows that the threshold is higher at the initiation site than at the soma by an amount k_{a} (see _{L}) replaced by the axial resistance R_{a}. This equation implies that the spike threshold decreases logarithmically with the distance of the initiation site.

Many empirical discussions have focused on the shape of action potentials at onset: a “kink” is indeed observed in cortical neurons, as if the action potential were suddenly rising from nowhere _{a}, where ΔV = V_{a}−V_{s} is the voltage difference between the somatic voltage and the voltage at the initiation site, i.e., the discontinuous voltage change seen in _{a}_{a}) = 7.5 mV/ms.

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).

Two remarks are in order. First, from the formula, it can be seen that this kink becomes more pronounced when Na conductance increases, which is expected, but it becomes _{a} increases). This latter fact is more surprising, because it means that the sharpness of the “kink” at the soma is

Increasing the Na conductance would lower the threshold (by about 4 mV for every doubling), and in any case it cannot push the lateral current above

Therefore, to explore the conditions for a significant “kink” at the soma, we now consider Na channels between the soma and initiation site, again clustered at a single location (distributed channels are considered later). Immunostaining in the AIS of cortical neurons shows that low-threshold Nav1.6 channels accumulate at the distal end of the AIS, while high-threshold Nav1.2 channels accumulate at the proximal end

To quantify the initial sharpness of spikes, previous studies have used a measure named “onset rapidness”, defined as the slope of the trajectory in the phase plot when a fixed value α of dV/dt is reached, typically of the order of α = 10 mV/ms ^{−1} (_{s}, and as a function of time. But as a function of the _{a} at the initiation site, the opening of Na channels follows the Na activation function _{a}/dt essentially reflects the Na activation function, and “onset rapidness” at the initiation site is essentially a measure of this function. This point can be demonstrated analytically: at the initiation site, as in an isopotential neuron, onset rapidness equals α/k_{a}, independently of all other properties (^{−1}/6 mV≈1.7.ms^{−1}, close to the numerical value. In contrast, onset rapidness is about four times larger at the soma (7.7 ms^{−1}). Indeed the voltage trajectory at the soma is determined by the lateral current, and in particular should correlate with the total Na conductance at the initiation site (

As this is a rather subtle point, I will try to rephrase this result, in the context of previous results. At the initiation site, the voltage derivative dV_{a}/dt reflects the Na current, and therefore is a smooth function of V_{a}, as determined by the Na activation curve (_{s} at the soma, Na channels open abruptly (_{a} is a discontinuous function of somatic voltage V_{s}, due to loss of voltage control (

All the previous results were obtained with channels clustered at a single location, but Na channels are rather distributed along the AIS _{1} varied between 1 µm and 35 µm and the end point x_{2} varied between 40 and 60 µm. Empirically, it appeared that both spike threshold (_{1}+0.4 x_{2}. This formula was empirically determined and may depend on other parameters.

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 _{start}+0.4 x_{end}). 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.

In the ball-and-stick model, the axon is geometrically modelled as a cylinder. However, at the hillock near the soma, the diameter is larger than in the initial segment _{a}, by an amount that can be analytically calculated (see

It has been proposed that a potential benefit of spike initiation in the distal AIS is to make it more energetically efficient, because the AIS has a smaller capacitance than the soma and therefore requires less transfer of charge to produce a spike, consistently with the fact the current threshold is lower in the axon than in the soma

In fact, spike initiation in the distal AIS is indeed more energetically efficient, not because of the smaller axonal capacitance, but because it reduces the flow of Na current below threshold, which is proportional to the rate of ATP consumption

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).

The claim that spike initiation is much sharper in cortical neurons than expected from isopotential Hodgkin-Huxley models is supported by different lines of evidence: 1) the initial shape of spikes recorded at the soma is very sharp, with a distinct “kink”

Two hypotheses have been proposed to explain this phenomenon. One is that Na channels cooperate in the AIS, which would make their collective activation curve much sharper

I have proposed a parsimonious explanation of the sharpness of both spike initiation and somatic spike shape, which is compatible with standard biophysics and empirical measurements. The explanation is based on compartmentalization. Because the soma is large compared to the axon diameter, it acts as a current sink for the axonal initiation site. If follows that the Na current equals the resistive axial current at all times, which results in an instability when a voltage threshold is crossed (a bifurcation in dynamical systems theory). This instability manifests itself as a loss of voltage control between the soma and the initiation site. As a result, Na channels open abruptly as a function of somatic voltage. The mathematics of this phenomenon are very close to the cooperativity model, although channels are independent. Spike initiation sharpness does not require active backpropagation, but the “kink” at spike onset at the soma requires proximal Na channels to transmit the spike (Nav1.2).

This simple hypothesis accounts for a number of empirical observations: 1) spike initiation is sharp, even though the “onset rapidness” measure is low at initiation site

What is the functional benefit of spike initiation in the axon? Since in cortical neurons a full spike is also seen at the soma, it cannot be the sole fact that the capacitance of the AIS is lower than that of the soma. In fact, spike initiation in the AIS is more energetically efficient because fewer Na channels are open below threshold than if spikes were initiated at the soma. This is in line with other known mechanisms that enhance the efficiency of action potential propagation

Incidentally, the fact that spike initiation is very sharp and that Na channels open abruptly above a critical voltage has important implications for neural modeling. Indeed, it is commonly assumed that biophysical models of the Hodgkin-Huxley family are more realistic than simpler phenomenological models such as the integrate-and-fire model, and that the latter are only used for their computational and theoretical simplicity. However, these results show that the simplified model shown in

All neuron models were simulated with the Brian simulator _{m} = 30,000 Ω.cm^{2}, specific membrane capacitance is C_{m} = 0.75 µF.cm^{−2}, giving a membrane time constant of 22.5 ms. Intracellular resistivity is R_{i} = 150 Ω.cm. The leak reversal potential is E_{L} = −75 mV. Apart from leak channels, only Na channels were included, so as to demonstrate the phenomenon without confounding factors (see _{Na} = 60 mV, k_{a} = 6 mV, V_{1/2} = −40 mV, τ_{m} = 100 µs _{Na}) uses m^{1} rather than m^{3}, which is used in the original Hodgkin-Huxley model. Indeed recent evidence shows that this is more accurate for Na channels of central neurons in mammalians

Unless specified, the maximum total Na conductance g_{Na} is twice the somatic leak conductance. Note that, since the model did not include inactivation, this represents the conductance of available channels (i.e., maximal conductance would be larger). The model is simulated in somatic voltage-clamp mode in

In _{1/2} = −25 mV), and the maximal conductance is 20 times larger than for Nav1.6 channels (again this represents the ratio for available channels; considering that Nav1.6 channels are partially inactivated, the ratio would be lower). In

In _{I} = 5 ms, I_{0} = 18.3 pA, σ_{I} = 31.4 pA and

Theoretical calculations are based on a simplified model of the axon initial segment (AIS). The analysis is described in more detail in the supplementary methods (_{Na} = f(V_{a}) as specified in the previous section, where V_{a} is the voltage at the initiation site (_{a}. In this model, the lateral current equals the Na current, which means:_{a} is determined as an implicit function of the somatic voltage V_{s} through a fixed point equation. This equation exhibits a bifurcation when R_{a} is above a critical value determined by model parameters, in particular by the geometry (_{a} to this equation changes discontinuously (discrete increase) when the somatic voltage V_{s} exceeds a certain value (the bifurcation point). We define the spike threshold as this value.

The spike threshold can be calculated as the bifurcation point of the above equation, as a function of model parameters (_{s} is given by the following approximated formula:

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