Figure 1.
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.
Figure 2.
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 (dashed curve). With I = 0, the lower equilibrium (filled circle) corresponds to the resting potential Vr, while the higher equilibrium (open circle) corresponds to the spike threshold with short pulses
(as in Fig. 1A): if the membrane potential is quickly shifted above
, the membrane potential blows up and the neuron spikes (thus, this corresponds to the case when
, i.e., an impulse current). Slowly increasing the input current amounts to vertically shifting the excitability curve, and the membrane potential follows the resting equilibrium until it disappears, when
. The voltage VT at that point corresponds to the minimum of the excitability curve. The empirical threshold
(with the first derivative method) is the voltage at the intersection of the excitability curve with the horizontal line dV/dt = kth (dashed line). The slope threshold
corresponds to the radius of curvature at VT. B, Threshold for short pulses
(solid line) and empirical threshold
(blue dashed line) as a function of the threshold for slow inputs VT (black dashed line is the identity line): the definitions are quantitatively different but highly correlated. C, Dependence of empirical threshold on derivative criterion kth: spike onsets are measured on a voltage trace (as in Fig. 1C) with derivative criterion kth = 7.5 mV/ms (blue dots), 10 mV/ms (black), 12.5 mV/ms (green) and 15 mV/ms (red). D, Empirical threshold measured with kth = 7.5 mV/ms (blue dots), 12.5 mV/ms (green) and 15 mV/ms (red) vs. threshold measured with 10 mV/ms, and linear regression lines. The dashed line represents the identity. The value of the derivative criterion (kth) impacts the threshold measure but not its relative variations.
Figure 3.
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].
Figure 4.
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).
Figure 5.
A, Voltage trace of the fluctuating conductance-based model (black line) and predicted threshold according to our threshold equation (, red line), calculated continuously as function of h, gK, ge and gi. Black dots represent spike onsets (empirical threshold with the first derivative method). B, Predicted threshold vs. membrane potential for the trace in A. Trajectories lie above theoretical threshold on the right of the dashed line (
). C, Zoom on the second spike in A. Colored lines represent increasingly complex threshold predictions: using Na activation characteristics only (blue,
), with Na channel inactivation (green,
), with potassium channel activation (purple,
) and with synaptic conductances (red,
). Here the threshold varies mainly after spike onset.
Figure 6.
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 ().
Figure 7.
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).
Figure 8.
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).
Figure 9.
Fitting the Na activation curve to a Boltzmann function.
A, The Na channel activation curve of the conductance-based model (black line) was fit to a Boltzmann function on the entire voltage range (dashed blue line) and on the spike initiation range only (−60 mV to −40 mV, red line). The green line shows the exponential fit on the spike initiation range. B, In the hyperpolarized region (zoom of the dashed rectangle in A), the global Boltzmann fit (dashed blue line) is not accurate, while the local Boltzmann fit and the local exponential fit better match the original curve. C, For hyperpolarized voltages (<−50 mV), the resulting excitability curve is closer to the original curve (black) with a local Boltzmann fit (red) than with a global fit (dashed blue), yielding more accurate threshold estimations (dots). D, The estimated Boltzmann slope ka is very sensitive to the position of the fitting window and varies between 2 mV and 6 mV.
Figure 10.
Experimental difficulties in the measurement of Na activation curves.
A, Estimation of the activation slope ka from simulated voltage-clamp data as a function of the inactivation time constant τh. The model was of a membrane with only Na channels, and activation and inactivation curves were Boltzmann functions (see Materials and Methods). The activation slope was measured by a Boltzmann fit in the hyperpolarized region (<−40 mV). The activation slope ka was 6 mV in the model (dashed line), but the measurement overestimated it when the inactivation time constant was very close to the activation time constant. B, Na activation curve measured in vitro (dots, digitized from [86]) and Boltzmann fits over the entire voltage range (dashed curve) and over the hyperpolarized range (V<−40 mV, solid curve).