Response to a periodic stimulus in a perfect integrate-and-fire neuron model driven by colored noise

The output interspike interval statistics of a stochastic perfect integrate-and-fire neuron model driven by an additive exogenous periodic stimulus is considered. The effect of temporally correlated random activity of synaptic inputs is modeled by an additive symmetric dichotomous noise. Using a first-passage-time formulation, exact expressions for the output interspike interval density and for the serial correlation coefficient are derived in the nonsteady regime, and their dependence on input parameters (e.g., the noise correlation time and amplitude as well as the frequency of an input current) is analyzed. It is shown that an interplay of a periodic forcing and colored noise can cause a variety of nonequilibrium cooperation effects, such as sign reversals of the interspike interval correlations versus noise-switching rate as well as versus the frequency of periodic forcing, a power-law-like decay of oscillations of the serial correlation coefficients in the long-lag limit, amplification of the output signal modulation in the instantaneous firing rate of the neural response, etc. The features of spike statistics in the limits of slow and fast noises are also discussed.

Figure 1

The behavior of the regular part $Q_1\left(T_1\right)$ of the ISI distribution computed from Eqs. (27) and (28) for various values of the noise-switching rate $\nu$ and the frequency $\mathrm{\Omega }$. System parameter values: $a=0.4$ and $A=0.3$. All quantities are dimensionless, with time and voltage scaling determined by $\mu =1$ and $v_c=1$, respectively. (a) Dashed line, $\nu =50$ and $\mathrm{\Omega }=1$; solid line, $\nu =1$ and $\mathrm{\Omega }=1$. The points marked by open triangles and circles are obtained by means of computer simulations of Eq. (1); the integration time step was set to 0.001 for ${10}^6$ trajectories of $v\left(t\right)$. (b) Dashed line, $\nu =50$ and $\mathrm{\Omega }=200$; solid line, $\nu =1$ and $\mathrm{\Omega }=200$. The arrows mark the positions of $\delta$ peaks [see Eq. (26)].

Figure 2

The mean $\langle T_1\rangle$ and the variance $D_1$ of the ISI distribution as functions of the input driving frequency $\mathrm{\Omega }$. The curves are computed from Eqs. (7) and (26, 27, 28) for $A=0.3$ and $a=0.4$. As for Fig. 1, all quantities are dimensionless with $\mu =v_c=1$. For $\langle T_1\rangle$ vs $\mathrm{\Omega }$: solid line, $\nu =0.1$; dotted line, $\nu =50$. For $D_1$ vs $\mathrm{\Omega }$: solid line, $\nu =50$; the inset corresponds to the case $\nu =0.1$. At large values of the driving frequency, $\mathrm{\Omega }\to \infty$, all curves tend to the same values as in the limit $\mathrm{\Omega }\to 0$ determined by Eqs. (37) and (38).

Figure 3

The dependence of the ratio of the ISI variance $D_1$ to the noise intensity $\sigma =a^2/\nu$ on the driving frequency $\mathrm{\Omega }$. System parameter values: $\mu =v_c=1,A=0.3,a=0.4$, and $\nu =10000$. The solid line is computed from the exact formulas (7), (26), and (27). The dots are computed from the asymptotical Eq. (46).

Figure 4

The oscillatory part $Q_1^{\left(P\right)}$ of the ISI distribution vs the firing time $T_1$ at $\mu =v_c=1,a=0.4,A=0.3$, and $\mathrm{\Omega }=400$. The curves are computed from Eq. (51). The noise-switching rate (a) $\nu =100$ and (b) $\nu =50$. Note the amplification of the oscillations by increasing $\nu$.

Figure 5

Serial correlation coefficient ${\rho }_{1,1}$ (for lag 1) plotted vs the driving frequency $\mathrm{\Omega }$ and vs the noise-switching rate $\nu$ [Eqs. (28, 29, 30, 31, 32)]. System parameter values: $v_c=\mu =1,a=0.4$, and $A=0.3$. (a) ${\rho }_{1,1}$ vs $\mathrm{\Omega }$. Solid line, $\nu =10$; dotted line, $\nu =0.2$. (b) ${\rho }_{1,1}$ vs $\nu$. Solid line, $\mathrm{\Omega }=2.53$, exhibits a minimum ${\rho }_{1,1}\approx 0.349$, at $\nu \approx 6.1$; at high values of the noise-switching rate, $\nu \to \infty ,{\rho }_{1,1}$ tends to 0.364. Dotted line, $\mathrm{\Omega }=4.3$. Note the sign reversals of ${\rho }_{1,1}$ in panel (a) (solid line) and in panel (b) (dotted line).

Figure 6

The SCC ${\rho }_{k,1}$ vs lags $k$ at various values of the noise-switching rate $\nu$ [Eqs. (28, 29, 30, 31, 32)]. The system parameter values: $v_c=\mu =1,a=0.4,A=0.3$, and $\mathrm{\Omega }=2.23$ (a) The case of a low switching rate, $\nu =0.01$. (b) The case of a moderate switching rate, $\nu =10$. The filled dots denote the actual discrete values of lags $k\ge 1$. Note the sign reversals of ${\rho }_{k,1}$ in panel (b). The thin lines are supplemented for the eye.

Figure 7

The SCC ${\rho }_{k,1}$ versus lags $k\ge 1$ in the fast-noise regime. System parameter values: $v_c=\mu =1,a=0.4,A=0.3$, and $\mathrm{\Omega }=0.3$. The solid line is computed from the asymptotic formula (57). The dots correspond to the exact Eqs. (29, 30, 31, 32) at $\nu =500$.

Figure 8

The dependence of the SCC ${\rho }_{k,j}$ in the fast-noise regime on the lag $k$ at various values of the index $j$ and input amplitude $A$. The dots are computed from Eq. (57) for $v_c=\mu =1$, and $\mathrm{\Omega }=2$. Other parameters are (a) $j=2$ and $A=0.05$; (b) $j=2$ and $A=0.9$; (c) $j=100A=0.05$; (d) $j=100$ and $A=0.9$. The lines are supplemented for the eye.