Statistics of a leaky integrate-and-fire model of neurons driven by dichotomous noise

The behavior of a stochastic leaky integrate-and-fire model of neurons is considered. The effect of temporally correlated random neuronal input is modeled as a colored two-level (dichotomous) Markovian noise. Relying on the Riemann method, exact expressions for the output interspike interval density and for the serial correlation coefficient are derived, and their dependence on noise parameters (such as correlation time and amplitude) is analyzed. Particularly, noise-induced sign reversal and a resonancelike amplification of the kurtosis of the interspike interval distribution are established. The features of spike statistics, analytically revealed in our study, are compared with recently obtained results for a perfect integrate-and-fire neuron model.

Figure 1

The behavior of the regular part $W\left(t\right)$ of the ISI distribution computed from Eq. (14) for various values of the noise switching rate $\lambda$. System parameter values: $\mu =2$ and $a=0.1$. All quantities are dimensionless with time and voltage scaling determined by $\gamma =1$ and $v_c=1$, respectively. (a) Solid line, $\lambda =0.1$ and $r=10$; dashed line, $\lambda =1.2$ and $r=1$; dotted line, $\lambda =1.9$ and $r=0.75$. (b) Solid line, $\lambda =100$ and $r=0.2$; dashed line, $\lambda =10$ and $r=0.5$; dotted line, $\lambda =2.1$ and $r=1$. The arrows mark the positions of $\delta$ peaks [see Eq. (13)]. The points labeled as squares are obtained by means of computer simulations of Eq. (1); the integration time step was set to 0.001 for ${10}^6$ trajectories of $v\left(t\right)$.

Figure 2

The skewness ${\gamma }_s$ and the coefficient of variation $C_v$ of the ISI distribution as functions of the noise switching rate $\lambda$. The curves are computed from Eqs. (6), (13), and (14) for $\mu =2$. As for Fig. (1), all quantities are dimensionless with $\gamma =v_c=1$. Solid lines, $a=0.9$; dashed lines, $a=0.5$; dotted lines, $a=0.2$. At high values of the noise switching rate, $\lambda \to \infty$, all curves tend to zero.

Figure 3

Dependence of the kurtosis $\kappa$ of the ISI distribution on the noise switching rate $\lambda$ computed from Eqs. (6), (13), and (14) at different values of the noise amplitude $a$. System parameter values: $\mu =2$ and $\gamma =v_c=1$. (a) The case of $a<a_{c1}$. Solid line, $a=0.1$; dashed line, $a=0.3$; dotted line, $a=0.5$. (b) The case of $a>a_{c1}$. Solid line, $a=0.82$; dashed line, $a=0.74$; dotted line, $a=0.66$. The thin vertical lines mark the positions of the sign reversals of $\kappa$. In the limit of $\lambda \to \infty$ the kurtosis $\kappa$ tends to zero. The critical noise amplitudes are $a_{c1}\approx 0.54$ and $a_{c2}\approx 0.769$. More details are in the text.

Figure 4

The rescaled critical noise amplitude $a_c^{*}$ versus the system parameter $\beta$ [see Eqs. (30) and (31)]. Note that for the PIF model $\beta =0$ and $a_c^{*}=1/\sqrt{3}$. More details are in the text.

Figure 5

The regular part $W\left(t\right)$ of the ISI distribution [Eqs. (32) and (33)] as a function of time $t$ for different values of the noise switching rate $\lambda$. System parameter values: $\mu =2, \gamma =v_c=1$. Solid line, $\lambda =50$ and $r=1$; dashed line, $\lambda =10$ and $r=1$; dotted line, $\lambda =1$ and $r=10$. At high values of the time, $t\to \infty$, all curves tend to zero. The arrow marks the position of the $\delta$ peak [see Eq. (32)].

Figure 6

Dependence of the SCC with lag 1, ${\rho }_1$, and the quantity $1-g$ on the noise switching rate $\lambda$ for various values of the noise amplitude $a$ [see also Eqs. (19), (20), and (B11)]. Solid line, $a=0.1$; dashed line, $a=0.6$; dotted line, $a=0.9$. Other system parameters are the same as in Fig. (3). More details are in the text.

Figure 7

Serial correlation coefficient ${\rho }_1$ (for lag 1) and the factor $1-g$ in Eq. (19) plotted over the noise amplitude $a$ for different values of the noise switching rate $\lambda$ [Eqs. (20) and (B11)]. Solid line, $\lambda =0.1$; dashed line, $\lambda =1$; dotted line, $\lambda =2.5$. Other system parameters are the same as in Fig. 3. Note that, in the case of a large noise correlation time, ${\tau }_c=1/\lambda =10$, a rapid decrease of both ${\rho }_1$ and $1-g$ occurs at the noise amplitude values $a\approx 1$.

Figure 8

Comparison of the behavior of the SCC ${\rho }_n$ and the Fano factor $F\left(\infty \right)$ for the LIF model [see Eqs. (19) and (22)] with that of the same parameters for the PIF model. System parameter values: $\mu =2, v_c=1$, and $a=0.8$. Solid lines, the leak-coefficient $\gamma =0$ (i.e., the PIF model); dashed lines, $\gamma =0.5$; dotted lines, $\gamma =1.0$. (a) ${\rho }_n$ vs the lag $n$: $\lambda =0.1$. The solid dots denote the actual discrete values of lags $n$. (b) At $\lambda =0$ the Fano factor tends to infinity, $F\left(\infty \right)\to \infty$. Note the increase of ${\rho }_n$ and the decrease of $F\left(\infty \right)$ by decreasing $\gamma$.