## College

College of Arts & Sciences

## Department

Mathematics & Statistics

## Rank

Assiciate Professor

## Type of Work

Journal Article

## Publication Information

COMMUNICATIONS IN STATISTICSâ€”THEORY AND METHODS

2018, VOL 47, NO.2, 289-306

## Description of Work

Approximation of stochastic differential equations (SDEs) with parametric noise plays an important role in a range of application areas, including engineering, mechanics, epidemiology, and neuroscience. A complete understanding of SDE theory with perturbed noise requires familiarity with advanced probability and stochastic processes. In this paper, we derive an asymptotic estimate of variance, and it is shown that numerical method gives a useful step toward solving SDEs with perturbed noise. Our goal is to diffuse the results to an audience not entirely familiar with functional notations or semi-group theory, but who might nonetheless be interested in the practical simulation of dynamical systems with fast noise or a slow manifold. In this article a matrix representation of a limit of variance for circular process is given. It is shown that the variance is asymptotically measured by the decrease in spectral energy in one step of a Markov chain. Then we apply this result to a stochastic differential equation with parametric noise (which arises in mathematical neuroscience) and demonstrate how the results can be used to analyze propagation of a signal in sound mechanism.

#### Included in

Dynamic Systems Commons, Ordinary Differential Equations and Applied Dynamics Commons, Other Applied Mathematics Commons

Asymptotic estimate of variance with applications to stochastic differential equations arises in mathematical neuroscience

Approximation of stochastic differential equations (SDEs) with parametric noise plays an important role in a range of application areas, including engineering, mechanics, epidemiology, and neuroscience. A complete understanding of SDE theory with perturbed noise requires familiarity with advanced probability and stochastic processes. In this paper, we derive an asymptotic estimate of variance, and it is shown that numerical method gives a useful step toward solving SDEs with perturbed noise. Our goal is to diffuse the results to an audience not entirely familiar with functional notations or semi-group theory, but who might nonetheless be interested in the practical simulation of dynamical systems with fast noise or a slow manifold. In this article a matrix representation of a limit of variance for circular process is given. It is shown that the variance is asymptotically measured by the decrease in spectral energy in one step of a Markov chain. Then we apply this result to a stochastic differential equation with parametric noise (which arises in mathematical neuroscience) and demonstrate how the results can be used to analyze propagation of a signal in sound mechanism.

## Biographical Statement

I am interested in the applications of nonlinear dynamics to biological problems. My main focus is in the area of mathematical neuroscience where I try to understand the propagation of wave in networks of neurons. I model approximations of stochastic differential equations with applications to mathematical neurosciences. I am also interested in problems from population biology, epidemiology, immunology, and cell biology for which I have also developed an upper level courses entitled as mathematical modeling and a graduate level mathematical biology.