Statistics of a neuron model driven by asymmetric colored noise

Irregular firing of neurons can be modeled as a stochastic process. Here we study the perfect integrate-and-fire neuron driven by dichotomous noise, a Markovian process that jumps between two states (i.e., possesses a non-Gaussian statistics) and exhibits nonvanishing temporal correlations (i.e., represents a colored noise). Specifically, we consider asymmetric dichotomous noise with two different transition rates. Using a first-passage-time formulation, we derive exact expressions for the probability density and the serial correlation coefficient of the interspike interval (time interval between two subsequent neural action potentials) and the power spectrum of the spike train. Furthermore, we extend the model by including additional Gaussian white noise, and we give approximations for the interspike interval (ISI) statistics in this case. Numerical simulations are used to validate the exact analytical results for pure dichotomous noise, and to test the approximations of the ISI statistics when Gaussian white noise is included. The results may help to understand how correlations and asymmetry of noise and signals in nerve cells shape neuronal firing statistics.

Figure 1

Illustration of the voltage dynamics of the PIF model. (a) Realization of dichotomous noise. (b) Simulation of the voltage driven by this realization. (c) As (b) but with additional weak Gaussian white noise ($D=0.01$). The spikes (horizontal lines exceeding $v_T$, red) were added for a better illustration of spike times. Parameters: $\sigma =0.7,\lambda =\mu =v_T=1,u=-0.5$.

Figure 2

Different asymmetries of the dichotomous noise process. (a) Positive asymmetry $u=0.6$. (b) Symmetric dichotomous noise $u=0$. (c) Negative asymmetry $u=-0.6$. Remaining parameters as in Fig. 1.

Figure 3

Illustration of the first-passage-time problem in state space at one point in time. The probability distributions $P_{-}$ and $P_{+}$ evolve in time according to Eqs. (10) and (11). Apart from drifting in the $v$ direction, probability can also flow from $P_{+}$ to $P_{-}$ and vice versa. The first-passage-time distribution corresponds to the probability current at the threshold $n{v}_T$.

Figure 4

ISI distribution for different mean transition rates $\lambda$ and dichotomous input only ($D=0$). The theory (solid lines) is compared to stochastic simulations (black circles). Note that we use a temporally discretized histogram version of Eq. (24) (bin width equal to that of simulation data), in which $\delta$ peaks turn into peaks of finite height. Parameters: $\mu =v_T=1.0$, $u=-0.4$, $\sigma =0.5$.

Figure 5

As Fig. 4 but for different asymmetries $u$ and a mean transition rate $\lambda =10$. When holding $\mu$ constant, a decreasing asymmetry shifts the distribution to the right.

Figure 6

ISI distribution for different asymmetries $u$ and a mean transition rate $\lambda =10$. To illustrate only the effect of the asymmetry, we keep the mean and the variance of the input current constant: $\mu +u\sigma =1$ and ${\sigma }^2(1-u^2)=0.25$.

Figure 7

ISI distribution for a PIF neuron driven by Gaussian white noise and dichotomous noise. Stochastic simulation results (black circles) are compared to the approximation of Eq. (27). The $\delta$ peaks broaden with increasing noise intensity $D$. The plot shows examples for $D=0.005$ and $0.05$ and the model parameters $\lambda =0.1$, $\mu =v_T=1.0$, $\sigma =0.5$, and $u=-0.6$.

Figure 8

Skewness of the ISI distribution ${\gamma }_s$ as a function of the asymmetry $u$ of the dichotomous noise. The theory (colored lines) is compared to simulations (black circles) for different $\lambda$. $u$ is varied while holding the mean $\mu +u\sigma =1$ and the variance ${\sigma }^2(1-u^2)=0.25$ of the input constant. Note that there is an upper limit for $u$ because of the constraint of the theory $\mu >\sigma$.

Figure 9

Skewness of the ISI distribution ${\gamma }_s$ as a function of the input skewness ${\gamma }_{\text{DN}}$ and the effect of additional white noise on it (theory, continuous line; simulations, black circles for $D=0$; black squares, otherwise). The mean $\mu +u\sigma =1$ and the variance ${\sigma }^2(1-u^2)=0.25$ of the input are kept constant. The theory requires $\mu >\sigma$, which results in a minimal input skewness.

Figure 10

The rescaled skewness ${\alpha }_s$ plotted over the constant input current $\mu$ for different values of the noise asymmetry $u$. The black circles indicate simulations without white noise, which are in perfect agreement with the theory. The black squares are simulations with additional white noise $D=0.01$. Remaining parameters: $\lambda =v_T=1.0$, $\sigma =0.5$.

Figure 11

Serial correlation coefficient obtained by simulations (black circles) compared with the theory (colored lines) for pure dichotomous noise ($D=0$) [Eq. (44)]. The curves show the SCC for different mean transition rates $\lambda$. Remaining model parameters: $\mu =v_T=1.0$, $\sigma =0.5$, and $u=0.8$.

Figure 12

Serial correlation coefficients (for lag $k=1$, 5, 10, and 20) normalized by the respective SCC without white noise and plotted over different white noise intensities. The dots represent stochastic simulations, whereas the solid line shows the approximation of Eq. (46) with model parameters $\lambda =0.01$, $\mu =v_T=1.0$, $\sigma =0.5$, and $u=0.6$.

Figure 13

The power spectrum $S\left(\omega \right)$ from simulations (blue dots) compared with the theory [red (light gray) solid line], Eqs. (47) and (79), for different mean transition rates $\lambda$ and purely dichotomous input. Remaining parameters: $\mu =v_T=1.0$, $\sigma =0.7$, $u=-0.4$.

Figure 14

The simulated power spectrum $S\left(\omega \right)$ of a PIF neuron driven by dichotomous noise and Gaussian white noise for different white-noise intensities $D$ as indicated. Remaining parameters as in Fig. 13.