Memory-induced resonancelike suppression of spike generation in a resonate-and-fire neuron model

The behavior of a stochastic resonate-and-fire neuron model based on a reduction of a fractional noise-driven generalized Langevin equation (GLE) with a power-law memory kernel is considered. The effect of temporally correlated random activity of synaptic inputs, which arise from other neurons forming local and distant networks, is modeled as an additive fractional Gaussian noise in the GLE. Using a first-passage-time formulation, in certain system parameter domains exact expressions for the output interspike interval (ISI) density and for the survival probability (the probability that a spike is not generated) are derived and their dependence on input parameters, especially on the memory exponent, is analyzed. In the case of external white noise, it is shown that at intermediate values of the memory exponent the survival probability is significantly enhanced in comparison with the cases of strong and weak memory, which causes a resonancelike suppression of the probability of spike generation as a function of the memory exponent. Moreover, an examination of the dependence of multimodality in the ISI distribution on input parameters shows that there exists a critical memory exponent ${\alpha }_c\approx 0.402$, which marks a dynamical transition in the behavior of the system. That phenomenon is illustrated by a phase diagram describing the emergence of three qualitatively different structures of the ISI distribution. Similarities and differences between the behavior of the model at internal and external noises are also discussed.

Figure 1

Behavior of the relaxation function $H\left(t\right)$ computed from Eqs. (B6)–(B12) for various values of the memory exponent $\alpha$ and the damping constant $\gamma$: (a) $\alpha =0.2$ and $\gamma =0.5$ (solid line) and $\alpha =0.2$ and $\gamma =2.5$ (dashed line) and (b) $\alpha =0.7$ and $\gamma =0.5$ (solid line) and $\alpha =0.7$ and $\gamma =2.5$ (dashed line). Note the absence of sign reversals of $H\left(t\right)$ in (b) at $\gamma =2.5$ (dashed line). All quantities are dimensionless, with time scaling determined by $\omega =1$.

Figure 2

Phase diagram of the fractional oscillator (1). In the shaded region, the relaxation function $H\left(t\right)$ [see Eqs. (16) and (17)] is always non-negative. In the unshaded domain, sign reversals of $H\left(t\right)$ occur. The critical curve (solid line) ${\gamma }_c/{\omega }^{2-\alpha }=\kappa \left(\alpha \right)$ [see Eq. (38)] tends to infinity at $\alpha ={\alpha }_c\approx 0.402$. The thin dotted lines mark the positions of the critical exponent ${\alpha }_c$ and the minimal value of $\kappa \left(\alpha \right)$, respectively; min $\kappa \left(\alpha \right)=\kappa \left({\alpha }_m\right)\approx 1.461$ with ${\alpha }_m\approx 0.849$. The bold dashed line marks another critical curve ${\gamma }_{c1}/{\omega }^{2-\alpha }={\kappa }_1\left(\alpha \right)$ above which the ISI distribution (34) is, in the case of internal noise at system parameters $k_{B}T=0.15, \omega =\mu =1$, and $v_c=1.75$, unimodal.

Figure 3

The ISI distribution $w\left(t\right|v_c)$ versus time $t$ [see Eq. (34)]. All quantities are dimensionless, with time and voltage scaling determined by $\omega =1$ and $\mu =1$, respectively. The system parameter values are $\gamma =2.5, D=1$, and $v_c=1.75$. (a) Case of white driving noise $[\beta \to 1$ in Eq. (4)] for $\alpha =0.2$ (solid line) and $\alpha =0.8$ (dashed line). (b) Case of colored noise ($\beta \ne 1$) for $\beta =\alpha =0.6$ (internal noise) (solid line) and $\beta =0.3$ and $\alpha =0.6$ (external noise) (dashed line). At large values of time $t\to \infty$, all curves tend to zero.

Figure 4

Behavior of the ISI distribution $w\left(t\right|v_c)$ at $\omega =\mu =1, \alpha =0.2$, and $v_c=1.75$. (a) Case of internal noise ($\beta =\alpha =0.2$). The system parameter values are $\gamma =2.5$ and $k_{B}T=0.5$. The solid line is computed from Eqs. (A16, A17, A18). The dots represent the calculation data obtained after averaging over ${10}^2$ time steps from the numerical results of a computer simulation of Eq. (A10). The integration time step was set to 0.001 for ${10}^6$ trajectories of $v\left(t\right)$. Note that $w\left(t\right|v_c)=0$ at $t_1\approx 1.7$. (b) Case of an external noise, computed from Eq. (34) at $\beta =0.6$ and $D=1$. The parameters are $\gamma =0.5$ (solid line), $\gamma =2.5$ (dashed line), and $\gamma =4$ (dotted line). Note that in those parameter regimes the condition (36) is valid only in a finite-time interval.

Figure 5

Dependence of the scaling function ${\tilde{\kappa }}_1(\alpha ,\beta ,\tilde{\rho })$ [see also Eq. (42)] on the memory exponent $\alpha$ computed from Eqs. (34), (B6), (B15), and (C3) at different values of the noise exponent $\beta$ in the case of external noise. The time scaling is determined with $\omega =1$ and the chosen value of the parameter $\tilde{\rho }=0.281$. The vertical dashed line indicates the critical memory exponent ${\alpha }_c\approx 0.402$. The solid line shows $\beta =0.2$, the dashed line $\beta =0.6$, and the dotted line $\beta =0.9$.

Figure 6

Dependence of the survival probability $F\left(\infty \right)$, computed from Eqs. (32) and (C1), on the memory exponent $\alpha$. All quantities are dimensionless with time and voltage scaling determined by $\omega =1$ and $\mu =1$, respectively. The system parameter values are $D=1$ and $v_c=1.5$. (a) Case of white external noise for $\beta =1$ and $\gamma =6$ (solid line), $\gamma =2.5$ (dashed line), and $\gamma =1$ (dotted). (b) Behavior of $F\left(\infty \right)$ for various values of the noise exponent $\beta$ at $\gamma =3$: $\beta =0.9$ (solid line), $\beta =0.6$ (dashed line), and $\beta =0.3$ (dotted line). The thin vertical dashed line indicates the value of the memory exponent $\alpha \approx 0.52$, below which the condition (30) is not fulfilled for any $t\in (0,\infty )$. At $\alpha =0$ all curves tend to zero very fast.

Figure 7

Memory-induced suppression of the ISI distribution $w\left(t\right|v_c)$. The graphs of $w\left(t\right|v_c)$ versus time $t$ are computed from Eqs. (34), (B6), (C1), and (C3). All quantities are dimensionless with the time and voltage scaling determined by $\omega =1$ and $\mu =1$, respectively. The system parameter values are $D=1, \beta =1, \gamma =6, v_c=1.5$, and $\alpha =0.2$ (solid line), $\alpha =0.5$ (dashed line), and $\alpha =0.9$ (dotted line). Note the suppression of $w\left(t\right|v_c)$ at $\alpha =0.5$ (dashed line), i.e., the area under the dashed line is smaller than those under other lines.