Theory of spike-train power spectra for multidimensional integrate-and-fire neurons

Multidimensional stochastic integrate-and-fire (IF) models are a standard spike-generator model in studies of firing variability, neural information transmission, and neural network dynamics. Most popular is a version with Gaussian noise and adaptation currents that can be described via Markovian embedding by a set of $d+1$ stochastic differential equations corresponding to a Fokker-Planck equation (FPE) for one voltage and $d$ auxiliary variables. For the specific case $d=1$, we find a set of partial differential equations that govern the stationary probability density, the stationary firing rate, and, central to our study, the spike-train power spectrum. We numerically solve the corresponding equations for various examples by a finite-difference method and compare the resulting spike-train power spectra to those obtained by numerical simulations of the IF models. Examples include leaky IF models driven by either high-pass-filtered (green) or low-pass-filtered (red) noise (surprisingly, already in this case, the Markovian embedding is not unique), white-noise-driven IF models with spike-frequency adaptation (deterministic or stochastic) and models with a bursting mechanism. We extend the framework to general $d$ and study as an example an IF neuron driven by a narrow-band noise (leading to a three-dimensional FPE). The many examples illustrate the validity of our theory but also clearly demonstrate that different forms of colored noise or adaptation entail a rich repertoire of spectral shapes. The framework developed so far provides the theory of the spike statistics of neurons with known sources of noise and adaptation. In the final part, we use our results to develop a theory of spike-train correlations when noise sources are not known but emerge from the nonlinear interactions among neurons in sparse recurrent networks such as found in cortex. In this theory, network input to a single cell is described by a multidimensional Ornstein-Uhlenbeck process with coefficients that are related to the output spike-train power spectrum. This leads to a system of equations which determine the self-consistent spike-train and noise statistics. For a simple example, we find a low-dimensional numerical solution of these equations and verify our framework by simulation results of a large sparse recurrent network of integrate-and-fire neurons.

Figure 1

Features of the integrate-and-fire model. Spike train $x\left(t\right)$ of a generalized two-dimensional neuron model (top); here $f(v,a)=\mu -v-a$ and $g(v,a)=Av-a$. When the voltage $v\left(t\right)$ (middle) crosses the threshold $v_{th}$ (red dashed line), the neuron fires a spike and the voltage is set to the constant $v_{\mathrm{ref}}$ for a certain refractory period. Afterward, the voltage is set to $v_r$ and evolves according to its Langevin equation. The voltage dynamics depends on the auxiliary stochastic variable $a\left(t\right)$ (third panel from top). If the neuron spikes, the auxiliary variable is incremented by ${\delta }_a$. Parameters: ${\tau }_m=20\mathrm{ms},{\delta }_a=10\mathrm{mV},{\tau }_{\mathrm{ref}}=10\mathrm{ms},v_{\mathrm{ref}}=50\mathrm{mV},\mu =30\mathrm{mV},A=0.2,{\beta }_1^{*}=1\mathrm{mV}\sqrt{s},{\beta }_1=0$, and ${\beta }_2=2\mathrm{mV}\sqrt{s}$.

Figure 2

Illustration of the fire-and-reset rules for the probability in the Fokker-Planck equation. (a) Fire-and-reset rule without refractory period or spike-triggered adaptation: the flux of probability that is absorbed at the threshold $v_{th}$ (red line) is reinserted at the reset voltage $v_r$ (red arrows) without a change in the auxiliary variable $a$. (b) Fire-and-reset rule with refractory period but in the absence of spike-triggered adaptation: the absorbed probability is transferred to the refractory voltage $v_{\mathrm{ref}}$ (red arrows). The probability density (red line directly after firing) evolves in the auxiliary variable until the refractory period has passed (blue line after the refractory period). Subsequently, the evolved density is reinserted at $v_r$ (blue arrows). (c) Spike-triggered adaptation without refractory period: the probability is reinserted at the reset voltage and, simultaneously, shifted by ${\delta }_a$ along the $a$ axis (red arrows). (d) Combination of spike-triggered adaptation and refractory period: the probability is shifted along $a$ and inserted at $v_{\mathrm{ref}}$. After the refractory period has passed, the evolved probability is reinserted at $v_r$. In addition to the stationary probability currents (strength indicated by arrow density and background color), we also show the stationary probability densities in the four situations.

Figure 3

Ambiguity of the Markovian embedding. Two Markovian embeddings exist that generate the same colored noise [see spectra in (a)] and yield the same spike-train power spectrum (d), although the corresponding Fokker-Planck equations and their solutions (b) and (c) are different. The first panel (a) shows the spectrum of noise at zero frequency as a function of the parameter ratio $\beta /{\beta }^{*}$. At the intersections with the white dashed line, we generate white noise. As an example, we used $S\left(0\right)/{{\beta }^{*}}^2=0.1$, that is plotted as yellow dashed-dotted line. Contour plots in (b) and (c) show stationary solutions of the corresponding FPEs, where (b) has a smaller value of $\beta$. The dashed white line and red line indicate the reset voltage $v_r$ and the threshold $v_{th}$, respectively. Spike-train power spectra with the two distinct colored-noise drivings (d) are nearly indistinguishable which is seen both in our theory (circles and crosses) and in the simulation results (lines). See Table 1 for parameters.

Figure 4

Green noise in the fluctuation-dominated regime. Spike-train power spectra (a)–(i). From left to right we decrease the time constant of the auxiliary variable ${\tau }_a$ and from top to bottom we increase the noise strength ${\beta }^{*}$. The ratio between the noise strengths is $\beta /{\beta }^{*}=\sqrt{0.1}-1$. The input spectrum $S_{\eta \eta }\left(f\right)$ in units of $\left({\mathrm{mV}}^2\mathrm{s}\right)$ is presented in orange, results of the neuron simulation in blue, and the result of our theory by red dots. The thin vertical lines represent $f_{\infty }={\left(2{\tau }_a\right)}^{-1}$ in green and the firing rates $r_0$ in red. Stationary probability density corresponding to the parameters used in (f) is shown in (j). For all parameters, see Table 1.

Figure 5

Green noise in the mean-driven regime. Stationary density in (j) corresponds to parameters in (c). See caption of Fig. 4 for detailed description and Table 1 for all parameters.

Figure 6

White-plus-red noise in the fluctuation-dominated regime. Stationary density in (j) corresponds to parameters in (f). See caption of Fig. 4 for detailed description and Table 1 for all parameters.

Figure 7

White-plus-red noise in the mean-driven regime. Stationary density in (j) corresponds to parameters in (c). See caption of Fig. 4 for detailed description and Table 1 for all parameters.

Figure 8

Effects of the refractory period. Spike-train power spectra for LIF neuron driven by (a) green noise with $S_{\eta \eta }\left(0\right)/S_{\eta \eta }\left(\infty \right)=0.1$ and (b) white-plus-red noise with $S_{\eta \eta }\left(0\right)/S_{\eta \eta }\left(\infty \right)=2$. We increase the refractory period from 0 to 100 ms, which yields in both cases more regular spike trains and a reduced firing rate. The vertical dashed lines represent the inverse refractory period at $f=1/{\tau }_{\text{ref}}$.

Figure 9

Exponential IF neuron in the fluctuation-dominated regime with deterministic adaptation. We vary the time constant of the auxiliary process ${\tau }_a$ in (a), and the adaptation strength ${\delta }_a$ in (b). The vertical lines represent the inverse of the time constants $1/{\tau }_a$ as solid lines and the firing rates as dotted lines, consistent with the color encoding. In (b), the inverse time constant is the black line. See Table 1 for other parameters.

Figure 10

Exponential IF neuron with stochastic adaptation in the mean-driven regime. We vary the time constant ${\tau }_a$ in (a) and the noise intensity in the auxiliary process in (b). The vertical lines represent the inverse of the time constant $1/{\tau }_a$ as solid line and the firing rates as dotted lines, consistent with the color encoding. In (b), the inverse time constant is the black line. See Table 1 for all parameters.

Figure 11

Exponential IF neuron with spike-triggered and subthreshold adaptation. Various values of $A$ as indicated are used. For comparison, we show the same data for $A=8$ in both panels. The vertical black line represents the inverse of the time constant $1/{\tau }_a$ and the dotted lines represent the firing rate consistent with the color encoding. See Table 1 for parameters.

Figure 12

Exponential IF neuron with adaptation in a bursting regime. Time evolution of the membrane voltage and adaptation variable (a), effective threshold (green), and threshold (red) shown as dashed lines; stationary solution as contour plot (b) with $v$ null cline (brown line) and $a$ null cline (gray line); spike-train power spectrum (c), firing rate is indicated by vertical line. See Table 1 for parameters.

Figure 13

Leaky IF neuron with a narrow-band noise, represented by higher-dimensional Markovian embedding. Time evolution of the voltage (a) and driving harmonic noise (b); spike-train power spectrum (c) with peaks at firing rate, the noise peak frequency, and side bands; input noise spectrum (d). Parameters in Table 1. Important frequencies are indicated by vertical lines.

Figure 14

Self-consistent solutions of the spike-train power spectrum for low-dimensional Markovian embeddings and comparison to network simulation. Black line represents spike-train power spectrum in the network (cf. Table 2 for parameters), obtained by averaging raw spectra over all neurons. Colored lines are self-consistent input $S_{\eta \eta }\left(\omega \right)/\varphi$ (dashed) and output (solid) power spectra and colors indicate dimensionality of Markovian embedding. Frequency values at which the self-consistency is enforced are indicated by vertical dotted lines. See Tables 3 and 2 for parameters.