Firing Time Statistics for Driven Neuron Models: Analytic Expressions versus Numerics

Analytical expressions are put forward to investigate the forced spiking activity of abstract neuron models such as the driven leaky integrate-and-fire model. The method is valid in a wide parameter regime beyond the restraining limits of weak driving (linear response) and/or weak noise. The novel approximation is based on a discrete state Markovian modeling of the full long-time dynamics with time-dependent rates. The scheme yields excellent agreement with numerical Langevin and Fokker-Planck simulations of the full nonstationary dynamics, not only for the first-passage time statistics, but also for the important interspike interval (residence time) distribution.

Figure 1

First-passage time density for parameters $U_{+}/D=8$, $U_{-}/D=5$, and $\omega =0.05$. The jagged line depicts the histogram obtained from iterations of the Langevin equation in (16). Note that fluctuations in the histogram depend on the total number of events and the width of the histogram bins. These fluctuations stay completely within their expected range which is indicated as the gray shaded area in the inset. The height of this area is twice the expected standard deviation of the histogram levels. The solid line shows the analytic first-passage time density from Eq. (12), with the rate used in (9). Both lines are in excellent agreement.

Figure 2

Residence time density vs time for the same parameters as in Fig. 1. The jagged line shows the histogram obtained from iterations of (16). Again, the numerics practically coincides within the linewidth with the analytic expression in (13) evaluated with the rate in (9) (solid line).

Figure 3

Testing extreme limits. First-passage time density for an extremely small lower barrier $U_{-}/D=3$. The remaining parameters are as in Fig. 1. Langevin simulation results (jagged line), analytical result in (12) with Eq. (9) (solid line), likewise, with Eq. (10) (dashed line). The inset compares the rate $\kappa (t)$ obtained from simulations of (16) with the analytic results from Eqs. (9, 10), respectively.

Figure 4

Testing extreme limits. Probability density of the first-passage time for a very fast driving, $\omega =0.5$. The other parameters are as in Fig. 1. The analytic approximation [Eqs. (12, 9), dashed line] still depicts maxima that are approximately located at $1,2,\dots$ while the numerical results are shifted to later times. The jagged line presents the Langevin-iterations from (16). The solid curve presents within its linewidth both the numerical solution to the FP Eq. (2) and to the integral equation from 10.