Research: Mathematical and Analytic Studies

Analytic programs complement the computational studies by developing theoretical tools for analyzing and better understanding experimental data. Such tools are central in revealing common mechanisms and principles arising from different neural systems and brain regions, and in understanding how complex systems of synaptic and cellular events produce specified behaviors. One such project involves analysis of the effects of dendritic branching, spine distribution and spine morphology on firing patterns of cortical and brainstem neurons involved in producing versions of working memory. In collaboration with colleagues at the Courant Institute of Mathematical Sciences, NYU, this program aims to deduce the salient dynamical and topological features contributing to variations in working memory from biophysically based modeling of active dendritic structures.

Structural inhomogeneties and irregularities are ubiquitous in biology, yet classical dynamical systems theory assumes a homogeneous medium with a regular, differentiable geometry. In collaboration with colleagues at the Department of Applied Mathematics, University of New South Wales, Sydney, we are using the fractional calculus and fractional order dynamical systems to derive new models of reaction-diffusion processes in inhomogeneous media, and processes involving non-differentiable functions.