On two diffusion neuronal models with multiplicative noise: The mean first-passage time properties

Exact solutions and asymptotic behaviors for the reflected Wiener, Ornstein-Uhlenbeck and Feller diffusion processes

We analyze the transition probability density functions in the presence of a zero-flux condition in the zero-state and their asymptotic behaviors for the Wiener, Ornstein Uhlenbeck and Feller diffusion processes. Particular attention is paid to the time-inhomogeneous proportional cases and to the time-homogeneous cases. A detailed study of the moments of first-passage time and of their asymptotic behaviors is carried out for the time-homogeneous cases. Some relationships between the transition probability density functions for the restricted Wiener, Ornstein-Uhlenbeck and Feller processes are proved. Specific applications of the results to queueing systems are provided.

Expediting Feller process with stochastic resetting

We explore the effect of stochastic resetting on the first-passage properties of the Feller process. The Feller process can be envisioned as space-dependent diffusion, with diffusion coefficient D(x)=x, in a potential U(x)=x(x/2-θ) that owns a minimum at θ. This restricts the process to the positive side of the origin and therefore, Feller diffusion can successfully model a vast array of phenomena in biological and social sciences, where realization of negative values is forbidden. In our analytically tractable model system, a particle that undergoes Feller diffusion is subject to Poissonian resetting, i.e., taken back to its initial position at a constant rate r, after random time epochs. We addressed the two distinct cases that arise when the relative position of the absorbing boundary (x_{a}) with respect to the initial position of the particle (x_{0}) differ, i.e., for (a) x_{0}<x_{a} and (b) x_{a}<x_{0}. Utilizing the Fokker-Planck description of the system, we obtained closed-form expressions for the Laplace transform of the survival probability and hence derived the exact expressions of the mean first-passage time 〈T_{r}〉. Performing a comprehensive analysis on the optimal resetting rate (r^{★}) that minimize 〈T_{r}〉 and the maximal speedup that r^{★} renders, we identify the phase space where Poissonian resetting facilitates first-passage for Feller diffusion. We observe that for x_{0}<x_{a}, resetting accelerates first-passage when θ<θ_{c}, where θ_{c} is a critical value of θ that decreases when x_{a} is moved away from the origin. In stark contrast, for x_{a}<x_{0}, resetting accelerates first-passage when θ>θ_{c}, where θ_{c} is a critical value of θ that increases when x_{0} is moved away from the origin. Our study opens up the possibility of a series of subsequent works with more case-specific models of Feller diffusion with resetting.

A Symmetry-Based Approach for First-Passage-Times of Gauss-Markov Processes through Daniels-Type Boundaries

Symmetry properties of the Brownian motion and of some diffusion processes are useful to specify the probability density functions and the first passage time density through specific boundaries. Here, we consider the class of Gauss-Markov processes and their symmetry properties. In particular, we study probability densities of such processes in presence of a couple of Daniels-type boundaries, for which closed form results exit. The main results of this paper are the alternative proofs to characterize the transition probability density between the two boundaries and the first passage time density exploiting exclusively symmetry properties. Explicit expressions are provided for Wiener and Ornstein-Uhlenbeck processes.

The Jacobi diffusion process as a neuronal model

The Jacobi process is a stochastic diffusion characterized by a linear drift and a special form of multiplicative noise which keeps the process confined between two boundaries. One example of such a process can be obtained as the diffusion limit of the Stein's model of membrane depolarization which includes both excitatory and inhibitory reversal potentials. The reversal potentials create the two boundaries between which the process is confined. Solving the first-passage-time problem for the Jacobi process, we found closed-form expressions for mean, variance, and third moment that are easy to implement numerically. The first two moments are used here to determine the role played by the parameters of the neuronal model; namely, the effect of multiplicative noise on the output of the Jacobi neuronal model with input-dependent parameters is examined in detail and compared with the properties of the generic Jacobi diffusion. It appears that the dependence of the model parameters on the rate of inhibition turns out to be of primary importance to observe a change in the slope of the response curves. This dependence also affects the variability of the output as reflected by the coefficient of variation. It often takes values larger than one, and it is not always a monotonic function in dependency on the rate of excitation.

Statistics of inverse interspike intervals: The instantaneous firing rate revisited

The rate coding hypothesis is the oldest and still one of the most accepted and investigated scenarios in neuronal activity analyses. However, the actual neuronal firing rate, while informally understood, can be mathematically defined in several different ways. These definitions yield distinct results; even their average values may differ dramatically for the simplest neuronal models. Such an inconsistency, together with the importance of "firing rate," motivates us to revisit the classical concept of the instantaneous firing rate. We confirm that different notions of firing rate can in fact be compatible, at least in terms of their averages, by carefully discerning the time instant at which the neuronal activity is observed. Two general cases are distinguished: either the inspection time is synchronised with a reference time or with the neuronal spiking. The statistical properties of the instantaneous firing rate, including parameter estimation, are analyzed, and compatibility with the intuitively understood concept is demonstrated.