In this experiment, which stimulus causes the membrane potential to reach the threshold value?

PLoS Comput Biol. 2010 Jul; 6[7]: e1000850.

Lyle J. Graham, Editor

Abstract

In central neurons, the threshold for spike initiation can depend on the stimulus and varies between cells and between recording sites in a given cell, but it is unclear what mechanisms underlie this variability. Properties of ionic channels are likely to play a role in threshold modulation. We examined in models the influence of Na channel activation, inactivation, slow voltage-gated channels and synaptic conductances on spike threshold. We propose a threshold equation which quantifies the contribution of all these mechanisms. It provides an instantaneous time-varying value of the threshold, which applies to neurons with fluctuating inputs. We deduce a differential equation for the threshold, similar to the equations of gating variables in the Hodgkin-Huxley formalism, which describes how the spike threshold varies with the membrane potential, depending on channel properties. We find that spike threshold depends logarithmically on Na channel density, and that Na channel inactivation and K channels can dynamically modulate it in an adaptive way: the threshold increases with membrane potential and after every action potential. Our equation was validated with simulations of a previously published multicompartemental model of spike initiation. Finally, we observed that threshold variability in models depends crucially on the shape of the Na activation function near spike initiation [about −55 mV], while its parameters are adjusted near half-activation voltage [about −30 mV], which might explain why many models exhibit little threshold variability, contrary to experimental observations. We conclude that ionic channels can account for large variations in spike threshold.

Author Summary

Neurons communicate primarily with stereotypical electrical impulses, action potentials, which are fired when a threshold level of excitation is reached. This threshold varies between cells and over time as a function of previous stimulations, which has major functional implications on the integrative properties of neurons. Ionic channels are thought to play a central role in this modulation but the precise relationship between their properties and the threshold is unclear. We examined this relationship in biophysical models and derived a formula which quantifies the contribution of various mechanisms. The originality of our approach is that it provides an instantaneous time-varying value for the threshold, which applies to the highly fluctuating regimes characterizing neurons in vivo. In particular, two known ionic mechanisms were found to make the threshold adapt to the membrane potential, thus providing the cell with a form of gain control.

Introduction

Spike initiation in neurons follows the all-or-none principle: a stereotypical action potential is produced and propagated when the neuron is sufficiently excited, while no spike is initiated below that threshold. The value of that threshold sets the firing rate and determines the way neurons compute, for example their coincidence detection properties [1], [2]. It is generally described as a voltage threshold: spikes are initiated when the neuron is depolarized above a critical value, when voltage-dependent sodium channels start to open. That biophysical mechanism is well understood since the studies of Hodgkin and Huxley in the squid giant axon [3] and subsequent modeling studies [4]–[7].

Recent findings have renewed the interest in the spike threshold. First, there is an intense ongoing debate about the origin of threshold variability observed in vivo [8]–[14]. In particular, it is unclear whether threshold variability is mainly due to experimental artifacts or molecular mechanisms, which might question the relevance of the Hodgkin-Huxley model for central neurons. Moreover, numerous experiments have shown that spike initiation does not only depend on the membrane potential but also on complex features of the inputs. For example, it depends on the preceding rate of depolarization [15]–[21] and on the preceding interspike intervals [12], [22]. Those properties are functionally important because they enhance the selectivity of neurons in several sensory modalities, in particular in audition [23], vision [24], and touch [21].

Developmental and learning studies have also shown that the threshold adapts to slow changes in input characteristics. This phenomenon is known as long-term plasticity of intrinsic excitability and may be involved in the regulation of cell firing, short term memory and learning [25]–[31]. The excitability threshold also varies with the distance to the soma in a given neuron and with cell type [2], [15], [32]–[35], which may explain functional differences.

The modulation of cell excitability might be explained by the activation of voltage-gated potassium channel Kv1 [36]–[41], inactivation of voltage-gated sodium channels [15], [16], [19], [21], fluctuations in sodium channel gating [42], inhibitory synaptic conductance [43]–[45] and the site of spike initiation [14]. To understand the origin of spike threshold variability, we examined the role of several candidate mechanisms in biophysical neuron models: activation and inactivation of the sodium channel, slow voltage-gated channels [e.g. Kv1], synaptic conductances and the site of spike initiation. Our analysis is based on a simplification of the membrane equation near spike initiation and results in a simple formula for the spike threshold that quantifies the contribution of all those mechanisms. The threshold formula provides an instantaneous time-varying value which was found to agree well with numerical simulations of Hodgkin-Huxley type models driven by fluctuating inputs mimicking synaptic activity in vivo, and with simulations of a realistic multicompartmental model of spike initiation [54].

Results

What is the spike threshold?

Spike threshold in vitro

In a typical in vitro experiment, one measures the response of the cell to a controlled stimulus, whose strength is defined by a parameter [e.g. current intensity]. The excitability threshold is then defined as the minimal value of this parameter above which a spike is elicited. Thus, the threshold is initially defined in stimulus space, for example as a charge threshold for short current pulses [Fig. 1A, simulated recording] or as a current threshold for current steps or ramps [Fig. 1B]. The stimulus threshold corresponds to a voltage value, which we call the voltage threshold, but that value depends on the type of stimulation [46]. Nevertheless, we are interested in the voltage threshold rather than in the stimulus threshold because only the voltage is usually available in intracellular recordings in vivo.

Spike threshold definitions.

All plots were generated using the single-compartment model described in the Materials and Methods. A, In vitro, the neuron is stimulated with short current pulses with increasing intensity [bottom] and the threshold is the minimal value of that intensity above which the neuron spikes [top]. The voltage threshold is the value of the membrane potential at that critical point. B, The threshold can be defined similarly with current steps [bottom] or other types of parameterized stimulations, yielding different values for the voltage threshold. C, In vivo, spike “threshold” is defined as a measure of the voltage at the onset of the action potential [black dots]. The plot shows a simulated trace of a conductance-based model with fluctuating conductances [see Materials and Methods] and threshold is measured with the first derivative method. D, Representation of the trace in [C] in phase space, showing dV/dt vs. V. The first derivative method consists in measuring the membrane potential V when the derivative crosses a predefined value [dashed line] shortly before an action potential. The trace is superimposed on the excitability curve dV/dt = [F[V]+I0]/C, which defines the dynamics of the model. I0 is the mean input current, so that trajectories in phase space fluctuate around this excitability curve.

Spike threshold in vivo

Since the input to the neuron is not directly controlled in vivo, the concept of spike threshold does not have exactly the same meaning as in vitro. Rather, it is defined as the voltage at the “onset” of action potentials [Fig. 1 C], as observed on an intracellular recording of the membrane potential. Therefore the spike threshold is an empirical quantity that hopefully captures the same concept as in vitro, i.e., the point above which an action potential is initiated. Several measures of spike onset have been used in experimental studies [47]. The first derivative method consists in measuring the membrane potential V when its derivative dV/dt crosses an empirical criterion [8], [34] [Fig. 1D]. The second and third derivative methods consist in measuring V when respectively d2V/dt2 and d3V/dt3 reach their maximum [12], [21]. Sekerli et al. [2004] compared those methods by asking electrophysiologists to identify spike onsets by eye on several membrane potential traces [47]. They found that visual inspection was best matched by the first derivative method, although that method critically relies on the choice of the derivative criterion [Fig. 2 C,D]. However, all methods produced the same relative variations of the measured threshold.

Relationships between spike threshold definitions.

A, Excitability curve of the neuron model [dV/dt = [F[V]+I]/C; see Materials and Methods] for DC input current I = 0 [solid curve] and

Spike threshold in models

It might seem confusing that the definition of the voltage threshold is ambiguous and that most modulation effects that have been reported in the literature seem to apply to spike onset rather than spike threshold. However, as remarked in [47], those measures differ in absolute value but they vary in the same way. We can relate those definitions with a simple one-dimensional neuron model, where the membrane potential is governed by a differential equation:

where C is the membrane capacitance, F[V] is the sum of all intrinsic voltage-dependent currents and I the input current. The dynamics of the membrane potential is determined by the excitability curve in phase space [dV/dt as a function of V, Fig. 2A]. With no DC injected current [I = 0, solid curve], the differential equation has two fixed points, which are solutions of F[V] = 0: the lower one is stable and corresponds to the resting potential and the higher one is unstable and corresponds to the threshold for fast depolarizations [short current pulses, i.e.,

The current-voltage function F[V] can be approximated by an exponential function near spike initiation [see Materials and Methods], leading to the exponential integrate-and-fire model [48]. In that model, we can calculate the relationship between the voltage threshold for slow inputs VT and the voltage threshold for fast inputs

where

Empirical threshold measures used in vivo can be analyzed in the same way. For example, the voltage threshold measured by the first derivative method is the value

where

In the remaining of this paper, we chose the voltage threshold for slow depolarizations VT as the definition of the spike threshold [i.e., the voltage at current threshold].

The threshold equation

Sodium channel activation

Cells excitability is generally due to the presence of voltage-gated sodium channels [49]. More precisely, Na channel activation gates mediate a positive feedback mechanism, which produces the instability phenomenon necessary to initiate an action potential. Activation is very fast compared to all other relevant time constants [a fraction of ms], in particular the membrane time constant [50]. We make the approximation that it is instantaneous, so that the proportion of open sodium channels at any time is

where gNa [resp. gL] is the maximum Na conductance [resp. leak conductance] and ENa [resp. EL] is the Na reversal potential [resp. leak reversal potential]. We neglect inactivation and other ionic channels for the moment [see below]. The activation function

where

is the threshold [defined for slow inputs]. The activation slope factor

Influence of Na activation characteristics on spike threshold.

A, Excitability curve of the model for different values of the ratio gNa/gL [maximum Na conductance over leak conductance], discarding inactivation [h = 1] and other ionic conductances. The resulting threshold is shown with a red dot. B, Excitability curve for different values of half-activation voltage Va. C, Excitability curve for different values of Boltzmann factor ka. D, Threshold as a function of the ratio gNa/gL for the 9 types of voltage-gated sodium channels [52] with characteristics reported in [Angelino and Brenner, 2007 [51]]. For each channel type, the mean threshold obtained across the dataset is plotted. Nav1.[1], [2], [3], [6] are expressed in the central nervous system, Nav1.[4], [5] are expressed in cardiac and muscle cells and Nav1.[7], [8], [9] are expressed in the peripheral nervous system. Nav1.6 is expressed at the action potential initiation site [53]–[55].

This formula provides some quantitative insight about the role of Na channel on cell excitability. For example, Pratt and Aizenman [2007] observed that during development, tectal neurons adapt their intrinsic excitability to changes in visual input so as to stabilize output firing [28]. They hypothesized that this adaptation was mediated by regulation of Na channel density, which could be quantitatively evaluated using the formula above. Our formula also explains differences in excitability between cells. There are 9 Na channel types, which are expressed in different regions of the nervous system [52], and each one has specific properties, in particular specific values of Va and ka. In Fig. 3D, we show how the threshold varies with channel density for each channel type, based on the dataset collected by Angelino and Brenner [2007] [51]. For the same channel density, the threshold can differ by up to 50 mV between channel types. Lowest threshold values were found for Nav1.5, expressed in cardiac cells, and highest ones were found for Nav1.8, expressed in dorsal root ganglion. Interestingly, among all channel types expressed in central neurons, the one with lowest threshold is Nav1.6, which is expressed in the spike initiation zone in the axon hillock [53]–[55].

Sodium channel inactivation and other conductances

The threshold can also be modulated by sodium channel inactivation and by the many other ion channels that can be found in neurons [56]–[58]. These factors might explain the effects of preceding spikes and membrane potential history on cell excitability [56]–[58]. To examine how they may modulate the threshold, we make two important assumptions: 1] inactivation is independent from activation, 2] these processes are slow compared to the timescale of spike initiation [about a millisecond]. We then consider the membrane equation near spike initiation:

where h is the inactivation variable and gi is the conductance of channel i, which may be voltage-gated [K+ channel] or synaptic. The contribution of additional ionic channels can be summed to yield an effective channel with conductance

where

is the threshold [mathematically, it satisfies F′[

Fig. 4 illustrates the dependence of threshold on Na inactivation and conductances. As expected, the threshold increases when h decreases, that is, when more Na channels inactivate. It also increases with the total non-sodium conductance, which is also intuitive: more Na conductance is required to produce a spike when the other conductances are larger. Threshold modulation is proportional to the slope factor

Influence of Na inactivation and ionic conductances on spike threshold in the conductance-based model [ka = 3.4 mV, see Materials and Methods].

A, Spike threshold θ as a function of Na+ inactivation variable h, with all other ionic conductances suppressed. B, Threshold as a function of K+ activation variable n, without inactivation [h = 1]. C, Threshold as a function of total synaptic conductance [excitatory ge and inhibitory gi], relative to the resting conductance gL [conductances are considered static].

The threshold equation predicts several effects. Spike threshold should be higher in vivo than in vitro because the total conductance is several times larger [59]. For the same reason, it should also be higher in up states than in down states. It is correlated with sodium inactivation, so that it should increase with the membrane potential, as observed in vitro and in vivo [15], [16], [54]. Besides, threshold modulation by inactivation is strongest when many Na channels are inactivated [h close to 0], that is, when the neuron is depolarized. Spike threshold is correlated with voltage-gated conductances such as those of K+ channels. For high-threshold K+ channels with large conductance, the spike threshold increases by

Threshold dynamics

To derive the threshold equation, we made a quasi-static approximation, assuming that all mechanisms that modulate the threshold are slow processes [compared to the timescale of spike initiation]. That threshold equation provides an instantaneous value of the spike threshold, as a function of modulating variables. Here we show how the dynamics of sodium inactivation, voltage-gated conductances and synaptic conductances translate into spike threshold dynamics.

Sodium inactivation

Several authors have hypothesized that Na inactivation is responsible for experimentally observed threshold variability in vivo [12], [15], [16], [21]. We have shown that the instantaneous value of the spike threshold depends on the value of the inactivation variable h [1-h is the proportion of inactivated channels]. We assume, as in the Hodgkin-Huxley model, that h evolves according to a first-order kinetic equation:

where

where

This equation allows us to predict the time-varying value of the threshold from the membrane potential trace [provided that Na inactivation properties are known]. The threshold time constant is given by the inactivation time constant [which is voltage-dependent]. Fig. 5 shows how the spike threshold varies in a biophysical model with fluctuating synaptic conductances.

Dynamical spike threshold.

A, Voltage trace of the fluctuating conductance-based model [black line] and predicted threshold according to our threshold equation [

The effect of previous spikes on spike threshold, which is presumably due to slow Na inactivation [12], can be understood by looking at how an action potential acts on the inactivation variable h. Typical equilibrium curves for Na inactivation

If we note

Voltage-dependent conductances

In the same way, the dynamics of voltage-dependent conductances translates into threshold dynamics. Potassium currents, in particular Kv1 delayed rectifier currents, are also thought to play a role in threshold modulation [36]–[41]. Let us consider a current with Hodgkin-Huxley-type kinetics:

where

Synaptic conductances

Finally, synaptic conductances fluctuate in vivo, which also impacts the instantaneous value of the threshold, through the following equation:

where we neglected Na inactivation and voltage-gated conductances to simplify the formula, and ge[t] [resp. gi[t]] is the excitatory [resp. inhibitory] synaptic conductance. This formula emphasizes the fact that the threshold equation defines an instantaneous value, which applies to realistic in vivo situations where synaptic activity fluctuates. However, we need to make the approximation that fluctuations are slow compared to spike initiation.

Spikes can be triggered either by an increase in the excitatory conductance or by a decrease in inhibitory conductance. In the former case, the total conductance increases and the threshold increases while in the latter case the threshold decreases. In high-conductance regimes [typical of cortical neurons in vivo], it has been argued that spikes are mainly triggered by inhibitory decrease because synaptic inhibition is dominant [59], [62]. It might imply that faster depolarization corresponds to lower inhibitory conductance and lower threshold, so that depolarization speed is inversely correlated with spike threshold, as observed experimentally [15]. However, this effect is fundamentally limited by the fact the inhibitory conductance cannot be negative.

Spike initiation site

Effect of neuronal morphology

Spikes are initiated in the axon initial segment [AIS] in spinal motoneurons [63] and in cortical neurons [64], about 35–50 µm from the soma [54], [55]. Our analysis relies on a single-compartment model of spike generation, but the axon hillock is connected with the soma through a large section and with the rest of the axon through a smaller section. To evaluate how electrotonically far the spike initiation site is from the soma, we can compare the length of the AIS to its electrotonic length, given by the following formula [65]:

where

For the threshold equation, these considerations imply that conductance values in the equation refer to total conductances over the surface of the soma, proximal dendrites and AIS. Since channel densities are different on these sites, the total conductance for a given ionic channel is

Na channel density in the AIS

Spikes could be initiated in the AIS rather than in the soma because of higher Na channel density [66], [69]–[71], or lower Na half-activation voltage Va [72] in the first segment. Recent experiments and computational modeling suggest that the former hypothesis is more plausible [34], [69], [70]. As an application of our analysis, we can estimate the Na channel density at the AIS using the parameter values reported in [70]. Since Na channels are mainly located in the AIS, we use

Using the values from Kole et al. [2008] [70] for the channel properties and neuron geometry [Va = −31.1 mV, ka = 6.5 mV, SAIS = 871.3 µm2, ENa = 55 mV], we find

Accuracy of the threshold equation

Threshold dynamics in a single-compartment model

To evaluate the quality of the threshold equation, we first simulated a biophysical single-compartment model with fluctuating synaptic conductances, mimicking the effect of synaptic activity in vivo. The instantaneous value of the threshold was measured by injecting brief current pulses of varying amplitude in repeated trials with the same synaptic inputs [Fig. 6A, B; see Materials and Methods], and we compared this time-varying value with the prediction from the threshold equation, including the effects of Na inactivation, voltage-gated channels and synaptic conductances. We used this particular stimulation protocol to measure the value of the threshold at any time, rather than only at spike time. We shifted the Na inactivation curve by −12.5 mV so as to obtain more threshold variability [the original model shows little variability]. The threshold equation predicted the variations of the measured threshold very well [83% of the variance], with a constant shift which can also be predicted [Fig. 6C, D]. This shift has two causes. Firstly, the threshold was measured with brief pulses whereas the predicted threshold corresponds to the definition with slow inputs. Using our formula relating the two definitions [Text S1] indeed reduced this shift from 13.5 mV to 7.4 mV [Fig. 6D]. Secondly, because we had to shift the inactivation curve to observe substantial threshold variability, spike threshold was depolarized closer to Na half-activation voltage [−30 mV] and the activation curve is less exponential in that region. Indeed, if VT is calculated as the minimum of the excitability curve rather than with the exponential formula, we find VT = −60.6 mV, which exactly compensates the 7.4 mV shift. When these two predictable biases were taken into account, both the mean and time course of the prediction matched the measured threshold [Fig. 6C, dashed red line]. When we did not shift Na inactivation as much, these biases were reduced but the model displayed little variability, which made the prediction less interesting. We address this point in more detail in the Discussion.

Predicted versus measured dynamical threshold.

A, Five superimposed voltage traces of the fluctuating conductance-based model [black traces] stimulated at different times with random depolarization [blue dots show the value of the membrane potential just after the stimulation]. Synaptic conductances are identical on all trials. In these examples the stimulations elicited spikes, in other cases [smaller depolarization] they did not. The theoretical threshold is shown in red. B, At a given time [here t = 50 ms], trials with varying depolarization are compared and the measured threshold is defined as the minimal depolarization that elicits a spike [blue dot]. C, Predicted threshold [red line] and measured threshold [blue] as a function of time. The shift is mainly due to the fact that the measured threshold is defined with fast inputs [charge threshold] whereas the theoretical threshold is defined with slow inputs: this bias can be calculated and corrected for, as shown by the dashed red line [see also text]. D, Measured threshold vs. theoretical threshold for the entire trace [blue dots; blue line: linear regression]. The dashed line represents the ideal relationship, taking into account the theoretical difference between threshold for fast inputs and for slow inputs [

In this single-compartment model, threshold variability is much lower than observed in vivo. However, the half-inactivation voltage Vi in the model is −42 mV, while experimental measurements suggest values around −60 mV in central neurons [e.g. Kole et al. [2008] [70]]. According to our analysis, this reduces threshold variability because Na channels do not inactivate below threshold [log h≈0]. In Fig. 7, we hyperpolarized Vi by 20 mV, giving Vi = −62 mV, close to experimental values, and measured the spike threshold with fluctuating inputs [Fig. 7A]. We found that the threshold varied over more than 10 mV and the standard deviation 2.2 mV [Fig. 7B], similar to values reported in vivo [15]. According to the threshold equation, most threshold variability was due to Na inactivation. A linear regression at spike times gave

Threshold variability and Na channel inactivation in a single-compartment model.

A, We simulated the same model as in Fig. 6, but the half-inactivation voltage Vi was shifted to −62 mV [instead of −42 mV in the original model] to increase threshold variability. As a result, spike height was also more variable. B, The threshold distribution [red] spanned a range of more than 10 mV [standard deviation 2.2 mV] and overlapped with the membrane potential distribution [black]. C, According to the threshold equation, most threshold variability was due to Na inactivation. Black dots show the measured threshold vs. the inactivation variable h [in log scale] at spike times. The linear regression [red line] gives a slope of 3.1 mV, close to the value of ka in this model [Fig. 9D].

Threshold prediction in a realistic multicompartmental model of spike initiation

We then checked the accuracy of the threshold equation with a realistic multicompartmental model of spike initiation, where action potentials are initiated in the axon [54]. We injected a fluctuating current in the soma and compared measured spike thresholds with our theoretical predictions [Fig. 8]. Spikes were initiated in the axon 400±60 µs earlier than observed at the soma [Fig. 8B]. When action potentials were removed from the voltage traces, the membrane potential was 1.8±0.6 mV higher at the soma than at the spike initiation site in the axon initial segment [AIS; Fig. 8C]. The threshold measured at the soma was −47.7±2.8 mV and varied between −52.1 mV and −42.2 mV [Fig. 8D]. Its distribution significantly overlapped the subthreshold distribution of the membrane potential, as observed in vivo. We estimated the activation properties of the Nav1.6 channel, which is responsible for spike initiation in this model, by fitting a Boltzmann function to the activation curve [

Accuracy of the threshold equation in a multicompartmental model of spike initiation [54].

A, Voltage trace at the soma [black] and at the spike initiation site in the axon initial segment [AIS, blue] in response to a fluctuating current. The spike threshold was measured at the soma when dV/dt exceeded 10 V.s−1 [red dots]. B. Zoom on an action potential: spikes were initiated at the AIS 400±60 µs before observed at soma. C. Between spikes, the membrane potential was slightly higher at the soma than at the AIS [1.8±0.6 mV]. D. The spike threshold [measured at the soma] was very variable [standard deviation 2.8 mV]: its distribution spanned 10 mV [−52 to −42 mV] and significantly overlapped the subthreshold distribution of the membrane potential [i.e., with spikes removed]. E, F. We fitted the activation curve of the Nav1.6 channel [black] to a Boltzmann function [red] in the spike initiation zone [rectangle and panel F], yielding Va = −33 mV and ka = 3.6 mV. G. Measured threshold [red: at the soma, black: at the AIS] vs. theoretical prediction for all spikes. The dashed line represents equality [measurement = prediction]. H. Somatic membrane potential vs. theoretical threshold at all times. Spikes are shown in black [defined as voltage trace 7 ms from spike onset], subthreshold trajectories in blue and spike times as red dots: spikes are indeed initiated when the membrane potential exceeds the theoretical threshold [inset: zoom on spike onsets].

Using these estimated values and the time-varying values of the channel variables [h, nKm, nKv 4] at the AIS, we calculated the theoretical threshold at all times, and compared the prediction with the measured threshold at spike times [Fig. 8G]. Values of the channel variables were taken at the time of spike initiation in the AIS and the threshold was measured at the AIS [black] and at the soma [red]. The prediction with the threshold equation was very good: the average error was 0.7 mV. The threshold prediction was on average only 0.49 mV higher than the measured threshold. However, this excellent match is probably fortunate because the value of the measured threshold is correlated with the measurement criterion [on dV/dt] and in general, we would expect a constant shift between prediction and measurement. When this shift was removed, the average prediction error was 0.53 mV. Among the different contributions to the threshold, we found that only Na inactivation had a significant impact. The fast K+ current [IKv] had a very high maximum conductance but was only activated after spike initiation, while the slow K+ current [IKm] had a small maximum conductance. According to the threshold equation, total conductance contributed only 0.07 mV to threshold variability. A linear regression gave

These results show that the value of the membrane potential at spike onset is well predicted by the threshold equation. However, to prove that our equation really defines a spike threshold, we also need to show that the membrane potential is always below the predicted threshold before spikes. In Fig. 8H, we plotted the membrane potential vs. the predicted threshold for the entire voltage trace [5 seconds]. It clearly appears that the neuron spikes when its membrane potential exceeds the predicted threshold, and that the potential is always below threshold between spikes.

Discussion

The spike threshold differs between cells and for different types of stimulations [2], [15], [32], [33], [35]. We have identified several modulation factors, whose quantitative influence is summarized by the threshold equation:

That formula relates the value of the threshold to the activation and inactivation properties of the Na channel, the properties of other voltage-gated channels such as Kv1 and synaptic conductances [gtot is total conductance, excluding Na conductance]. It consists of a static part [first two terms], determined by the properties of Na channel activation, and of a dynamic part, which depends on the proportion of inactivated Na channels [1-h] and on the total conductance of other channels.

It describes the voltage threshold at the site of spike initiation [rather than at the soma], and is correlated but not identical to empirical “threshold” measures, which measure spike onset rather than threshold [those normally overestimate the threshold]. From that formula, we were able to derive a dynamical equation for the instantaneous threshold, which explains the variability of the spike threshold in the same cell and predicts its relationship with previous membrane potential history. We found that the threshold equation was a good predictor of the time-varying threshold in biophysical models with fluctuating inputs [Fig. 6– ​8].

Mechanisms for threshold modulation and variability

Since Na channels are responsible for the generation of action potentials, the threshold is firstly determined by their activation characteristics. Activation curves for Na channels are well approximated by Boltzmann functions with similar slope factors [ka = 4–8 mV in neuronal channels]. The threshold is linearly related to the half-activation value Va and logarithmically related to the maximum Na conductance ga. The threshold also depends logarithmically on the Na inactivation variable h, so that it increases with the membrane potential and with every emitted spike. The modulating effect of inactivation is most pronounced when the half-activation value Vi is lowest [i.e., Na channels are partially inactivated at rest]. Finally, the threshold depends logarithmically on the total conductance, which includes the leak conductance, voltage-gated conductances and synaptic conductances. In particular, Kv1 channels, which are expressed with high density at the spike initiation site [37], [39], [41], increase the threshold in an adaptive manner [the threshold increases with the membrane potential]. This change in threshold occurs simultaneously with the effective membrane time constant, whereas threshold changes due to Na inactivation have no effect on the time constant, which might suggest a way to experimentally distinguish between the two effects. Indeed, the effective membrane time constant [as measured in vivo for example in Léger et al., 2003 [73]] is

We have not considered the effect of channel noise, i.e., fluctuations in Na channel gating [42], [75]–[78], which result in random threshold variations. Although dynamical equations of fluctuations in Na channel gating are well set [79], [80], they cannot be included in our theoretical framework because we neglected the time constant of Na activation [which leads to the exponential model].

There are two additional sources of variability which are artefactual: the fact that the threshold is not measured at the site of spike initiation, and threshold measurement methods. The latter source is difficult to avoid in vivo because only spike onsets can be measured. The former one also seems technically very difficult to avoid in vivo, since spikes are initiated in the axon hillock, which is only a few microns large. Although the soma and AIS are virtually isopotential below threshold, experimentally measured values of threshold differ between the two sites [34] because, as we previously remarked, in vivo measurements correspond to spike onset rather than threshold and therefore take place after spike initiation, when the two sites are not isopotential anymore. This experimental difficulty may introduce artefactual variability in threshold measurements [14].

Approximations in the threshold equation

To derive the threshold equation, we made several simplifying assumptions. First, we assumed that Na activation is instantaneous. It is indeed significantly faster than all other time constants but not instantaneous. The approximation is legitimate as long as the effective membrane time constant in the membrane equation is small [

We also assumed that Na activation and inactivation curves were Boltzmann functions. Experimental data is indeed well fitted by Boltzmann functions, but the reported parameter values [Va, ka] correspond to fits on the entire voltage range, whereas we are interested in hyperpolarized voltage regions where the activation values are small. When only the relevant part of the experimental data is considered, different parameter values might be obtained. For example, when analyzing previously published biophysical models, we found that better results were obtained when Na activation curves, which were not exactly Boltzmann functions, were fitted in the spike initiation region [−60 to −40 mV] rather than on the entire voltage range [Fig. 9]. We examined this issue in the biophysical model used in this paper [see Materials and Methods]. The Na activation curve of this model seemed to be well fit to a Boltzmann function [Fig. 9A], however the fit was poor in the spike initiation zone [−60 to −40 mV, Fig. 9C] where activation is close to zero, which makes fit errors relatively larger. Although the slope factor