Waves in random neural media

Stephen Coombes (March 23, 2011)

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Abstract

The propagation of waves of neural activity across the surface of the brain is known to subserve both natural and pathological neurobiological phenomena. An example of the former is spreading excitation associated with sensory processing, whilst waves in epilepsy are a classic example of the latter. There is now a long history of using integro-differential neural field models to understand the properties of such waves. For mathematical convenience these models are often assumed to be spatially translationally-invariant. However, it is hard even at a first approximation to view the brain as a homogeneous system and so there is a pressing need to develop a set of mathematical tools for the study of waves in heterogeneous media that can be used in brain modeling. Homogenization is one natural multi-scale approach that can be utilized in this regard, though as a perturbation technique it requires that modulation on the micro-scale be both small in amplitude and rapidly varying in space. In this talk I will present novel techniques that improve upon this standard approach and can further tackle cases where the inhomogeneous environment is modeled as a random process.