^{1}

^{2}

^{1}

^{2}

^{*}

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

The authors have declared that no competing interests exist.

Neurons spike when their membrane potential exceeds a threshold value. In central neurons, the spike threshold is not constant but depends on the stimulation. Thus, input-output properties of neurons depend both on the effect of presynaptic spikes on the membrane potential and on the dynamics of the spike threshold. Among the possible mechanisms that may modulate the threshold, one strong candidate is Na channel inactivation, because it specifically impacts spike initiation without affecting the membrane potential. We collected voltage-clamp data from the literature and we found, based on a theoretical criterion, that the properties of Na inactivation could indeed cause substantial threshold variability by itself. By analyzing simple neuron models with fast Na inactivation (one channel subtype), we found that the spike threshold is correlated with the mean membrane potential and negatively correlated with the preceding depolarization slope, consistent with experiments. We then analyzed the impact of threshold dynamics on synaptic integration. The difference between the postsynaptic potential (PSP) and the dynamic threshold in response to a presynaptic spike defines an effective PSP. When the neuron is sufficiently depolarized, this effective PSP is briefer than the PSP. This mechanism regulates the temporal window of synaptic integration in an adaptive way. Finally, we discuss the role of other potential mechanisms. Distal spike initiation, channel noise and Na activation dynamics cannot account for the observed negative slope-threshold relationship, while adaptive conductances (e.g. K+) and Na inactivation can. We conclude that Na inactivation is a metabolically efficient mechanism to control the temporal resolution of synaptic integration.

Neurons spike when their combined inputs exceed a threshold value, but recent experimental findings have shown that this value also depends on the inputs. Thus, to understand how neurons respond to input spikes, it is important to know how inputs modify the spike threshold. Spikes are generated by sodium channels, which inactivate when the neuron is depolarized, raising the threshold for spike initiation. We found that inactivation properties of sodium channels could indeed cause substantial threshold variability in central neurons. We then analyzed in models the implications of this form of threshold modulation on neuronal function. We found that this mechanism makes neurons more sensitive to coincident spikes and provides them with an energetically efficient form of gain control.

Action potentials are initiated when the membrane potential exceeds a threshold value, but this value depends on the stimulation and can be very variable

Among the mechanisms that can modulate the spike threshold

We analyzed the influence of Na inactivation on spike threshold in a model, in which we were able to express the spike threshold as a function of Na channel properties and variables

We previously derived a formula, the threshold equation, which relates the instantaneous value of the spike threshold to ionic channels properties _{a} is the half-activation voltage of Na channels, k_{a} is the activation slope factor, g_{Na} is the total Na conductance, g_{L} is the leak conductance, E_{Na} is the Na reversal potential, h is the inactivation variable (1-h is the fraction of inactivated Na channels). Here the spike threshold is defined as the voltage value at the minimum of the current-voltage function in the membrane equation (we compared various threshold definitions in _{T} is a constant term, corresponding to the minimum spike threshold (when Na channels are not inactivated). We call the EIF model with Na inactivation the inactivating exponential integrate-and-fire model (iEIF; see _{L}, and

We start by studying the steady-state threshold, which is the value _{0}. It corresponds to the threshold measured with the following experiment. The cell is clamped at a voltage V_{0} (

A, The membrane potential is clamped at a given voltage _{0})/C are shown in the phase plane _{0} (solid lines; V_{0} = −80 mV and −26 mV). The steady-state threshold _{0}. C, Steady-state threshold (red lines) of a cortical neuron model

One way to understand threshold adaptation is to look at how the excitability curve changes with h (and therefore with depolarization). The excitability curve (

The membrane potential V is always below threshold, unless the cell spikes. Therefore the observable threshold values cannot be larger than the intersection between the threshold curve and the diagonal line _{T} and the solution of

Using the threshold equation, we can calculate the steady-state threshold as a function of V: _{T}) for large negative potentials and a linear asymptote for large positive potentials, because the inactivation function is close to exponential (

A, The steady-state threshold curve (red curve) is well approximated by a piecewise linear curve determined by Na channel properties (top dashed black curve), where V_{i} is the half-inactivation voltage and V_{T} is the non-inactivated threshold. The slope of the linear asymptote is k_{a}/k_{i} (resp. activation and inactivation slope parameters). Na channel properties in this figure were taken from Kuba et al. (2009). The spike threshold is variable only when _{T} is determined by the maximum Na conductance g_{Na}, relative to the leak conductance g_{L}. As the ratio _{th} = 5 mV/ms). Note that the membrane potential can exceed threshold without triggering a spike because the threshold is soft (unlike in integrate-and-fire models).

In other words, the minimum threshold is V_{T}, which is determined by the maximum Na conductance (_{i}, and the slope is the ratio of activation and inactivation slope factors. Regarding threshold variability, we can distinguish three cases, depending on Na channel properties:

if

if

if

_{i} in this dataset, which is rather wide (−90 mV to −25 mV). Central neuron channel types, i.e., Nav1. _{T} depends on the maximal Na conductance, it cannot be deduced from channel properties alone. Considering that V_{T} should lie between −55 and −45 mV _{a}>k_{i}. _{a}>k_{i}. Thus, it seems that all three cases occur in similar proportions for channel types expressed in central neurons.

A, Distribution of half-inactivation voltage (V_{i}) of Na channels expressed in exogenous systems (from a database of 40 Na channels reported in Angelino and Brenner, 2007 _{i}) vs. activation slope (k_{a}) for the same dataset. Channels with V_{i}<−50 mV (variable threshold) are indicated by a black contour. These channels have high threshold variability when k_{a}>k_{i} (right of the dashed line). C, Distribution of V_{i} for Na channels expressed in central neurons _{i}) vs. activation slope (k_{a}) for the same dataset. High threshold variability is predicted in about half cases.

However, not all Na channels are involved in spike initiation. In particular, in central neurons, spike initiation is mediated by Nav1.6 channels while Nav1.2 channels are involved in axonal backpropagation _{i}<−50 mV in all cases (−61±8.4 mV), suggesting significant threshold variability, but this is a small sample. Besides, this first dataset was somewhat artificial, because channels, some of which had mutations, were artificially expressed in an exogenous system, which might alter their properties. Therefore we looked at a second dataset, consisting of _{a}>k_{i} (

We have shown that Na channel properties, i.e., parameters

Before we describe threshold dynamics in more details, we need to make an important remark. As is seen in _{T}≈1 mV _{a} ≈ 6 mV), meaning that spike initiation is almost as sharp as in an integrate-and-fire model. This phenomenon is well captured by multicompartmental models _{L}, and the threshold is increased - see

The threshold also increases with each action potential

Quantitatively, the relationship between average membrane potential and threshold depends on the steady-state threshold function _{i}. In these simulations, the slope of the steady-state threshold curve was k_{a}/k_{i} = 1, close to experimental values, but we note that the average threshold only increases as about 2/3 the average membrane potential in the depolarized region. This is because the membrane potential is very variable (about 6 mV in this figure) and therefore the threshold is not constantly in the sensitive region (V>V_{i}). This is consistent with previous measurements in the visual cortex

To calculate the relationship between the slope of depolarization and the threshold, we consider a linear depolarization with slope s (i.e., V(t) = V_{0}+st) and calculate the intersection with the threshold

We simulated the iLIF model (see _{i} (0.64 from linear regression, red line) than below (0.23).

A, The neuron is linearly depolarized with a given slope s (V(t) = E_{L}+st) until the membrane potential (black) reaches threshold (red) and the neuron spikes. The intersection of the black and red traces (red dots) can be calculated (see _{a} = k_{i} (dashed line). C, Slope-threshold relationship for different values of the half-inactivation voltage V_{i} (V_{i} = −63 mV in panels A,B). D, Slope-threshold relationship for different values of the inactivation time constant (

Unfortunately, this implicit equation does not give a closed formula for

In this particular case, the threshold diverges to infinity at _{T}, i.e., to the lowest possible threshold, and it increases for smaller _{i} and on the threshold time constant _{i} is low compared to the minimum threshold V_{T} (_{T} was −55 mV). The role of the threshold time constant can be seen as a scaling factor for slopes, i.e., the threshold depends on the product

These dynamical properties of the threshold imply that the threshold should be variable for fluctuating inputs (typical of

An iLIF model was stimulated by fluctuating inputs with different means and standard deviations and the threshold distribution was measured. A, Average threshold (color-coded) as a function of the mean (<V>) and standard deviation (σ_{V}) of the membrane potential. The average threshold depends primarily on the average membrane potential. White areas correspond to parameter values that were not tested (top) or that elicited no spike (bottom). B, Standard deviation of the threshold as a function of membrane potential statistics. Threshold variability depends on both the average and the standard deviation of the membrane potential.

These results have two main implications for synaptic integration: 1) threshold adaptation reduces the impact of the input mean, relative to its variance, and 2) the negative correlation between threshold and depolarization rate shortens the timescale of synaptic integration.

When V>V_{i}, the steady-state threshold increases with the voltage (_{i}:

An iLIF model was simulated in the same way as in _{a}/k_{i}, which controls threshold adaptation. A, Output firing rate vs. mean input with threshold adaptation (solid line, k_{a}/k_{i} = 1), with mild threshold adaptation (dashed line, k_{a}/k_{i} = 0.5) and without threshold adaptation (mixed line, k_{a}/k_{i} = 0). The horizontal axis is the input resistance R times the mean input <I>, i.e., the mean depolarization in the absence of spikes. The input standard deviation was chosen so that the neuron fires at 10 Hz when the mean depolarization is 10 mV. B, Firing rate (color-coded) vs. mean and standard deviation of the input, without adaptation (k_{a}/k_{i} = 0). The standard deviation is shown in voltage units to represent the standard deviation of the membrane potential in the absence of spikes, i.e., _{I} is the input standard deviation (in current units) and _{a}/k_{i} = 0.5). D, Same as B, but with normal threshold adaptation (k_{a}/k_{i} = 1).

It was remarked in previous studies that the negative relationship between threshold and depolarization rate should make the neuron more sensitive to coincidences _{i}), then the threshold

A, Top: Normalized postsynaptic potential (PSP, solid line) and threshold PSP, i.e., effect of the PSP on the threshold (dashed line). Bottom: The effective PSP is the difference between the PSP and the threshold PSP. It is briefer and can change sign. B, The effect of the PSP on spike threshold depends on how the threshold changes with voltage (dθ/dV, bottom), which depends on the membrane potential V and is determined by the Na inactivation curve (top; dashed line: half-inactivation). At high voltage, dθ/dV = k_{a}/k_{i} ( = 1 here). C, Half-width of the effective PSP (color-coded) as a function of threshold sensitivity dθ/dV and the threshold time constant

Thus, it is a linear superposition of

In other words, threshold adaptation acts as a simultaneous inhibition with slower time constant (than the excitatory PSP), or as a simultaneous excitation for inhibitory PSPs. As a result, the temporal width of effective PSPs is smaller than that of PSPs, so that the timescale of synaptic integration is shorter (_{i}, i.e., when the threshold varies linearly with the membrane potential, the threshold PSP is proportional to k_{a}/k_{i}, which is close to 1 in experimental measurements. Closer to V_{i}, the threshold PSP is proportional to dθ/dV, which lies between 0 and k_{a}/k_{i} (_{i} (with k_{a} = k_{i}), threshold adaptation reduces the half-width of the PSP by a factor greater than 2 (intersection of the two lines in

Similar properties are seen when synaptic filtering is taken into account, that is, when the synaptic current is an exponentially decaying function rather than an instantaneous pulse (Dirac), giving biexponential PSPs (

A, Normalized biexponential PSPs obtained with non-instantaneous synaptic currents (i.e., postsynaptic currents are exponentially decaying with time constant τ_{s} between 1 ms and 20 ms). B, As for exponential PSPs (_{s}, but less rapidly for ePSPs than for PSPs. D, ePSP peak time vs. PSP peak time. Threshold adaptation makes peak times shorter and compressed.

As is illustrated in _{i} (_{i} in the dataset of Na channels in central neurons

A, The iLIF model was simulated with random inputs (exponentially decaying PSPs), temporally distributed according to a Poisson process. Top: Spikes are produced when the membrane potential V (black) exceeds the threshold θ (red). Bottom: This is equivalent to a model with fixed zero threshold (red) and potential V-θ (black), which is the sum of effective PSPs. Effective PSPs are sharper than PSPs. B, Top: The threshold is more adaptive when the neuron is depolarized (right) than near resting potential (left). Bottom: When the mean input is increased (4 different levels shown), effective PSPs become sharper and their negative part cancels the input mean (see

For inhibitory PSPs (IPSPs), threshold adaptation is equivalent to simultaneous excitation with a slower time constant. Thus, in some cases, the later part of the effective PSP can be positive (

Finally, while we have previously ignored the voltage dependence of the time constant of Na inactivation, we show in _{i} (see

Based on voltage clamp measurements of Na channel properties, we have found that Na inactivation can produce by itself large threshold variability, as observed in experiments

The criterion for large threshold variability (_{a}) and half-inactivation voltages (k_{i}), obtained from Boltzmann fits. However, the relevant voltage range for these fits is the spike initiation range, and reported experimental values generally correspond to fits over the entire voltage range. This could contribute a significant measurement error in these values, as we previously showed _{a} _{a}/k_{i} in Na channels.

One consequence of threshold adaptation is to reduce the sensitivity of neurons to their mean input, and to make them more sensitive to fluctuations.

Threshold adaptation implies that a presynaptic spike has an effect on both the membrane potential (the classical PSP) and the spike threshold. We defined an

Although Na channel inactivation can account for all the properties that have been experimentally observed, other mechanisms could potentially contribute to threshold variability: somatic measurement when spikes are initiated in the axon, channel noise and other ionic mechanisms. We discuss below these alternative mechanisms and evaluate whether they may account for threshold adaptation.

A recent debate about the validity of the Hodgkin-Huxley model for cortical neurons has highlighted the fact that, for central neurons, spikes are initiated in the axon while

To address this question, we consider a simplified situation where spikes are initiated in the axon hillock when the potential is above a fixed threshold V_{T} (

A, Illustration of the effect of depolarization slope _{T} is reached (dashed red line). The spike is backpropagated to the soma with delay

We confirmed this reasoning by simulating the response of the multicompartmental model of Yu et al. (2008)

The Hodgkin-Huxley formalism describes the dynamics of the macroscopic average of many sodium channels, but individual channels have stochastic dynamics

When depolarization is very slow, spikes will be initiated lower than

The spike threshold increases with the total non-sodium conductance, because spike initiation requires more Na channels to be open in order to counteract a larger total conductance. Thus, fluctuating synaptic conductances could be a source of threshold variability. We previously estimated the effect of total conductance on spike threshold through the following formula _{e}) and inhibitory (g_{i}) conductances, and we ignored the effects of Na inactivation. Threshold variability is determined by the variability of total conductance at spike time. In low-conductance states (_{e} _{e}, and therefore with the threshold. Besides threshold variability can only be mild because the total conductance is low (relative to the leak conductance). In high-conductance states (up states _{i} _{i}, and therefore with the threshold. Therefore, in high-conductance states but not in low-conductance states, the slope-threshold relationship induced by synaptic conductances is qualitatively consistent with experimental observations

In our analysis, we assumed that Na activation is instantaneous. Voltage clamp measurements indeed show that its time constant is only a fraction of millisecond

In the same way as synaptic conductances, voltage-gated channels may also modulate the spike threshold _{K} through the following formula:_{a}^{Na}/k_{a}^{K}). The impact on synaptic integration is also different, because the conductance impacts not only the threshold but also the PSPs and effective membrane time constant.

Finally, we discuss below the possible interactions of several Na channel subtypes and of slow and fast Na inactivation.

We assumed that a single Na channel type (e.g. Nav1.6) was present. It is possible to extend our analysis to the case of multiple subtypes. Suppose the Na current is made of two components corresponding to two channel types:

To simplify, we assumed that the two channels have the same activation Boltzmann factor k_{a}, which is not unreasonable. Then the Na current can be equivalently expressed as:

In other words, when several subtypes are present, inactivation in the threshold equation is replaced by a linear combination of inactivation variables of all subtypes. For example, Nav1.2 and Nav1.6 are both found in the axon initial segment _{2} for Nav1.2 is less voltage-dependent and its threshold is higher); at more depolarized voltages (assuming the threshold has not been reached), Nav1.6 channels inactivate (h_{1}≈0) and threshold modulation is then determined by Nav1.2 channels. Note however that with several channel subtypes, it is not possible to express threshold dynamics as a single kinetic equation for _{1} and h_{2}).

In the present study, we focused on fast Na inactivation. We have briefly mentioned that the threshold equation applies when Na inactivation is slow, and implies that the threshold increases after each spike, which induces a negative correlation between threshold and preceding inter-spike interval. This effect is expected, but it gets more interesting when the interaction between slow and fast components is considered. One way to model this interaction is to consider two Na currents, as in the previous section. But since inactivation in the same channel can show slow and fast components, it might be more relevant to include this interaction in the gating variables. The simplest way is to consider these components as independent gating processes, that is:_{slow} and h_{fast} have slow and fast dynamics, respectively

In this case, it is possible to write a kinetic equation for each component of the threshold (_{i}). Suppose for example that V_{T}<V_{i}. At low firing rates (when interspike intervals are larger than the slow inactivation time constant),

In summary, many mechanisms may contribute to the variability of the spike threshold, but only two can account for its observed adaptive properties: Na inactivation and adaptive conductances (most likely K channels). Although threshold dynamics is qualitatively similar for both mechanisms, they can be distinguished by the fact that Na inactivation has no subthreshold effect on the membrane potential. Specifically, if the threshold is mainly modulated by adaptive conductances, then we can make two predictions:

The relationship between membrane potential and threshold should be determined by the I-V curve in the region where Na channels are closed:

The effective membrane time constant

In a few experimental studies, the application of α-dendrotoxin, a pharmacological blocker of low-voltage-activated potassium channels, greatly reduces threshold variability _{T} (see the threshold equation with voltage-gated channels), possibly below half-inactivation voltage V_{i}, where there is no threshold adaptation due to Na inactivation. Thus, it could be that threshold adaptation was due to Na inactivation, but that suppressing K conductances shifted the minimum threshold out of the operating range of this mechanism. This hypothesis could be tested by simultaneously injecting a fixed conductance in dynamic clamp, to compensate for the reduction in total conductance of the cell.

Although we cannot draw a universal conclusion at this point, and while it is possible that either or both mechanisms are present in different cells, we observe that Na inactivation is a metabolically efficient way for neurons to shorten and regulate the time constant of synaptic integration. Indeed, Na inactivation implies no charge movement across the membrane while K+ conductances modulate the threshold by counteracting the Na current, which implies a large transfer of charges across the membrane (Na+ inward and K+ outward) in the entire region where the threshold is variable. Recently, it was found in hippocampal mossy fibers that K+ channels open only after spike initiation, in a way that minimizes charge movements

All numerical simulations were implemented with the Brian simulator

Near spike initiation, the Na current can be approximated by an exponential function of the voltage _{T} is the threshold when Na channels are not inactivated, _{T} = −58 mV, k_{a} = 5 mV, _{i} = −63 mV and k_{i} = 6 mV.

A very good approximation of the Na current is an exponential function of V

We refer to the differential equation of _{L}. Refractoriness is implemented either by maintaining V at resting potential for 5 ms (_{a} = 6 mV (see _{T} = −55 mV, V_{i} = −63 mV (average value in the _{a}/k_{i} = 1 (average in the dataset: 1.05).

In

Fluctuating inputs (

To measure spike onset in models with no explicit threshold (

To calculate the relationship between the slope of depolarization and the threshold, we consider a linear depolarization with slope s: V(t) = st, and we calculate the intersection with the threshold

For low values of s, this equation may have no solution (i.e., the neuron does not spike). Using the piecewise linear approximation of the steady-state threshold, we obtain:

This is also an implicit equation for

Slope-threshold relationship in the multicompartmental model of Hu et al. (2009), measured with linear regression over 5 ms (black dots), superimposed on the calculated relationship (red dashed line), using the Na channel properties of the model (as in Platkiewicz and Brette, 2010,

(0.17 MB PDF)

Properties of Na channels of central neurons

(0.16 MB PDF)

Impact of sodium channel inactivation on spike threshold dynamics and synaptic integration.

(0.53 MB PDF)