Dynamics of neuronal and genetic networks

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Mathematics is increasingly valuable in studying complex physiological systems, such as the genetic control of cell functions and control of movement by the brain and nervous system. This research project deals with mathematical models of such 'networks', both to help understand and explain their dynamical behaviour, and to discover underlying mathematical principles at work in any networks of interacting components.One aspect of the research program is to use neuronal network descriptions of parts of the brain's motor circuitry and methods of nonlinear dynamics to study movement abnormalities resulting from Parkinson's disease such as tremor, and the means by which high-frequency electrical stimulators implanted in particular brain structures alleviate these symptoms. (Remarkably, how they work is not well understood, despite their common use.) The network of brain structures involved in motor control is complex, and it is not yet clear how these stimulators operate physically, why they can be placed effectively in one of several different brain structures and how these structures communicate.Genes control the production of proteins that carry out cell functions, but these proteins regulate the activity of other genes. Thus, networks of interacting genes underly cell development and metabolism. One gene-regulated process that is dynamically interesting is the circadian (daily) oscillations of some proteins in cells. Such oscillations are common across many organisms but involve different biochemical components and genes, so a challenge is to understand how they arise and operate and why they are so strongly persistent (jet lag, for example, is only a temporary disruption in our own circadian rhythms). This is a question involving interacting network loops, and another main thrust of my research program is to understand from a mathematical point of view how the pattern of interconnection determines dynamical behaviour in general networks with components that switch on or off. This framework permits analytic methods to probe quite deeply into complex dynamics, and thus contributes to the mathematics of complex systems as well as providing insight into physiology.