Moshe Silverstein | Searcharxiv

arXiv2024

Large Deviations of Piecewise-Deterministic-Markov-Processes with Application to Stochastic Calcium Waves

We prove a Large Deviation Principle for Piecewise Deterministic Markov Processes (PDMPs). This is an asymptotic estimate for the probability of a trajectory in the large size limit. Explicit Euler-Lagrange equations are determined for computing optimal first-hitting-time trajectories. The results are applied to a model of stochastic calcium dynamics. It is widely conjectured that the mechanism of calcium puff generation is a multiscale process: with microscopic stochastic fluctuations in the opening and closing of individual channels generating cell-wide waves via the diffusion of calcium and other signaling molecules. We model this system as a PDMP, with $N \gg 1$ stochastic calcium channels that are coupled via the ambient calcium concentration. We employ the Large Deviations theory to estimate the probability of cell-wide calcium waves being produced through microscopic stochasticity.

math.PR↗

arXiv2024

Population Level Activity in Large Random Neural Networks

We determine limiting equations for large asymmetric `spin glass' networks. The initial conditions are not assumed to be independent of the disordered connectivity: one of the main motivations for this is that allows one to understand how the structure of the limiting equations depends on the energy landscape of the random connectivity. The method is to determine the convergence of the double empirical measure (this yields population density equations for the joint distribution of the spins and fields). The limiting dynamics is expressed in terms of a fixed point operator. It is proved that repeated applications of this operator must converge to the limiting dynamics (thus yielding a relatively efficient means of numerically simulating the limiting equations,

cond-mat.dis-nn↗