Figure 1.
A, The membrane potential is clamped at a given voltage , then a constant current I is injected (iEIF model). The steady-state threshold
is defined as the maximum voltage that can be reached without triggering an action potential. B, Two excitability curves dV/dt = F(V,V0)/C are shown in the phase plane
, for two different initial clamp values V0 (solid lines; V0 = −80 mV and −26 mV). The steady-state threshold
is the voltage at the minimum of the excitability curve for the initial voltage V0. C, Steady-state threshold (red lines) of a cortical neuron model [63] for the original maximal Na conductance (solid line) and for a higher and lower Na conductance (resp. bottom and top dashed line). When the cell is slowly depolarized, it spikes when
, i.e., the spike threshold is the intersection of the red and black dashed curves. If there is no intersection, the neuron cannot spike with slow depolarization. The top dashed line (low Na conductance) is interrupted because the threshold is infinite at high voltages (i.e., the cell is no longer excitable).
Figure 2.
Role of Na channel properties in threshold variability in the iEIF model.
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 Vi is the half-inactivation voltage and VT is the non-inactivated threshold. The slope of the linear asymptote is ka/ki (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 , and very variable when (additionally)
. B, The non-inactivated threshold VT is determined by the maximum Na conductance gNa, relative to the leak conductance gL. As the ratio
increases, the steady-state threshold curve
shifts downward (red curves; r = 0.4; 2; 10) and threshold variability is reduced. C, Trajectory of the model in the
phase plane (blue), superimposed on the steady-state threshold curve (red). Spikes are initiated when
(dashed line:
), but the empirical measurement overestimates the threshold. The spike threshold is highly variable in this example (−50 to −10 mV). D, Trajectory of the model in the
phase plane (blue), superimposed on the Na inactivation curve (black). The threshold is very variable when most Na channels are inactivated. E, Voltage trace (black curve) and spike threshold
(red curve;
) in the inactivating exponential model driven by a fluctuating input (see Methods), where black dots represent empirical measurement of spike onsets (first derivative method, kth = 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).
Figure 3.
Measured properties of Na channels and threshold variability.
A, Distribution of half-inactivation voltage (Vi) of Na channels expressed in exogenous systems (from a database of 40 Na channels reported in Angelino and Brenner, 2007 [25]), including central neuron channel types (red), sensory neuron channel types (blue) and muscular channel types (green). Assuming a minimum spike threshold between −55 mV and −45 mV (dashed lines), channels on the left have variable threshold while channels of the right have a constant threshold. B, Inactivation (ki) vs. activation slope (ka) for the same dataset. Channels with Vi<−50 mV (variable threshold) are indicated by a black contour. These channels have high threshold variability when ka>ki (right of the dashed line). C, Distribution of Vi for Na channels expressed in central neurons in situ (see Table S1). The threshold should be variable in most cases. D, Inactivation (ki) vs. activation slope (ka) for the same dataset. High threshold variability is predicted in about half cases.
Figure 4.
Predicted relationship between mean membrane potential and mean threshold.
We simulated the iLIF model (see Methods) with a fluctuating input current. The standard deviation was fixed while the mean current was varied between trials. The mean spike threshold () is plotted as a function of the mean membrane potential (
). The slope of the curve is larger above half-inactivation voltage Vi (0.64 from linear regression, red line) than below (0.23).
Figure 5.
Slope-threshold relationship in the adaptive threshold model.
A, The neuron is linearly depolarized with a given slope s (V(t) = EL+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 Results). B, Threshold vs. depolarization slope (solid line) and analytical formula when ka = ki (dashed line). C, Slope-threshold relationship for different values of the half-inactivation voltage Vi (Vi = −63 mV in panels A,B). D, Slope-threshold relationship for different values of the inactivation time constant ( in panels A,B). E, The iLIF model is driven by a fluctuating current and we measure the slope of depolarization before each spike over a duration
by linear regression. F, Slope-threshold relationship measured with linear regression in the noise-driven iLIF model (red dots), superimposed on the calculated relationship from panel B.
Figure 6.
Threshold distribution as a function of membrane potential statistics.
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 (σV) 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.
Figure 7.
Firing rate as a function of input statistics.
An iLIF model was simulated in the same way as in Figure 6, but with different values for the parameter ka/ki, which controls threshold adaptation. A, Output firing rate vs. mean input with threshold adaptation (solid line, ka/ki = 1), with mild threshold adaptation (dashed line, ka/ki = 0.5) and without threshold adaptation (mixed line, ka/ki = 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 (ka/ki = 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., , where σI is the input standard deviation (in current units) and
is the input time constant. The horizontal mixed line corresponds to the mixed line shown in panel A, and the vertical dashed line corresponds to the threshold for constant currents. C, Same as B, but with mild threshold adaptation (ka/ki = 0.5). D, Same as B, but with normal threshold adaptation (ka/ki = 1).
Figure 8.
The effective postsynaptic potential.
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 = ka/ki ( = 1 here). C, Half-width of the effective PSP (color-coded) as a function of threshold sensitivity dθ/dV and the threshold time constant . The black cross corresponds to the situation shown in panel A. The membrane time constant (
) is shown by a horizontal solid line. D, Zero crossing time of the effective PSP as a function of threshold sensitivity and threshold time constant. The white triangle corresponds to parameter values where the effective PSP is always positive.
Figure 9.
Effective postsynaptic potential with synaptic filtering.
A, Normalized biexponential PSPs obtained with non-instantaneous synaptic currents (i.e., postsynaptic currents are exponentially decaying with time constant τs between 1 ms and 20 ms). B, As for exponential PSPs (Figure 8), effective PSPs (ePSPs) are narrower and change sign (only the positive part is shown). The time to peak is also shorter. Threshold adaptation parameters were and dθ/dV = 1. C, The peak time increases with the synaptic filtering time constant τs, but less rapidly for ePSPs than for PSPs. D, ePSP peak time vs. PSP peak time. Threshold adaptation makes peak times shorter and compressed.
Figure 10.
Synaptic integration with adaptive threshold.
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 Figure 8). C, Random inhibitory PSPs are added to a depolarizing current ramp. Without inhibitory inputs (dashed), the threshold adapts and the neuron does not spike. With inhibitory inputs (solid), the sign change in effective PSPs (see Figure 8) acts as a rebound and triggers spikes. This phenomenon is often called postinhibitory facilitation [36]. D, When the voltage dependence of the Na inactivation time constant is taken into account (see Methods), effective PSPs become sharper as the neuron is more depolarized, which implies an adaptive coincidence detection property [2].
Figure 11.
Effect of distal spike initiation and channel noise on the slope-threshold relationship.
A, Illustration of the effect of depolarization slope s on somatic spike onset. In cortical neurons, spikes are initiated in the axon initial segment (AIS, black), then backpropagated to the soma (blue). Somatic depolarization is propagated forward to the spike initiation site in the axon with delay . A spike is initiated in the axon when the threshold VT is reached (dashed red line). The spike is backpropagated to the soma with delay
. During time
, the somatic voltage has increased by
and the spike onset is seen higher (red dot). B, Slope-threshold relationship in the multicompartmental model of Yu et al. (2008) [7] with fluctuating inputs (mean 0.7 nA, standard deviation 0.2 nA, time constant 10 ms), measured at the AIS (top) and at the soma (bottom). As expected, the slope-threshold relationship is less pronounced at the soma than at the AIS. C, The effect of channel noise is modeled by a stochastic threshold (red;
,
, and
) and the neuron is linearly depolarized. With slow depolarization (left), the threshold (at spike time) is lower than the average instantaneous threshold. With fast depolarization (right), the threshold distribution (at spike time) follows the distribution of θ. D, As a result, the threshold is positively correlated with the depolarization slope (blue dots: threshold vs. slope for all spikes in the simulations; black dots: average threshold for each slope).