Cage effect for the velocity correlation functions of a Brownian particle in viscoelastic shear flows

Statistical moments of the interspike intervals for a neuron model driven by trichotomous noise

The influence of a colored three-level input noise (trichotomous noise) on the spike generation of a perfect integrate-and-fire (PIF) model of neurons is studied. Using a first-passage-time formulation, exact expressions for the Laplace transform of the output interspike interval (ISI) density and for the statistical moments of the ISIs (such as the coefficient of variation, the skewness, the serial correlation coefficient, and the Fano factor) are derived. To model the anomalous subdiffusion that can arise from, e.g., the trapping properties of dendritic spines, the model is extended by including a random operational time in the form of an inverse strictly increasing Lévy-type subordinator, and exact formulas for ISI statistics are given for this case as well. Particularly, it is shown that at some parameter regimes, the ISI density exhibits a three-modal structure. The results for the extended model show that the ISI serial correlation coefficient and the Fano factor are nonmonotonic with respect to the input current, which indicates that at an intermediate value of the input current the variability of the output spike trains is minimal. Similarities and differences between the behavior of the presented models and the previously investigated PIF models driven by dichotomous noise are also discussed.

Effects of transient subordinators on the firing statistics of a neuron model driven by dichotomous noise

The behavior of a stochastic perfect integrate-and-fire (PIF) model of neurons is considered. The effect of temporally correlated random activity of synaptic inputs is modeled as a combination of an asymmetric dichotomous noise and a random operation time in the form of an inverse strictly increasing Lévy-type subordinator. Using a first-passage-time formulation, we find exact expressions for the output interspike interval (ISI) statistics. Particularly, it is shown that at some parameter regimes the ISI density exhibits a multimodal structure. Moreover, it is demonstrated that the coefficient of variation, the serial correlation coefficient, and the Fano factor display a nonmonotonic dependence on the mean input current μ, i.e., the ISI's regularity is maximized at an intermediate value of μ. The features of spike statistics, analytically revealed in our study, are compared with previously obtained results for a perfect integrate-and-fire neuron model driven by dichotomous noise (without subordination).

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 α_{c}≈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.

Memory effects for a stochastic fractional oscillator in a magnetic field

The problem of random motion of harmonically trapped charged particles in a constant external magnetic field is studied. A generalized three-dimensional Langevin equation with a power-law memory kernel is used to model the interaction of Brownian particles with the complex structure of viscoelastic media (e.g., dusty plasmas). The influence of a fluctuating environment is modeled by an additive fractional Gaussian noise. In the long-time limit the exact expressions of the first-order and second-order moments of the fluctuating position for the Brownian particle subjected to an external periodic force in the plane perpendicular to the magnetic field have been calculated. Also, the particle's angular momentum is found. It is shown that an interplay of external periodic forcing, memory, and colored noise can generate a variety of cooperation effects, such as memory-induced sign reversals of the angular momentum, multiresonance versus Larmor frequency, and memory-induced particle confinement in the absence of an external trapping field. Particularly in the case without external trapping, if the memory exponent is lower than a critical value, we find a resonancelike behavior of the anisotropy in the particle position distribution versus the driving frequency, implying that it can be efficiently excited by an oscillating electric field. Similarities and differences between the behaviors of the models with internal and external noises are also discussed.

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.

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.