The effects of geometry and dynamics on biological pattern formation

This project examines the influence of geometry and dynamics on pattern formation in biological development . Since the work of Turing (1952) it has been known that patterns can form spontaneously given certain relatively simple conditions. The Turing mechanism involved a symmetry-breaking bifurcation from a stable spatially homogeneous state. However the development of patterns in developing organisms does not take place from such simple conditions, biological development causes pattern formation to occur within geometric structures which are complex and the environment is very noisy. This thesis examines the effects of such complexity and noise on pattern formation. The biological situations modelled in this thesis relate to the development of the mammalian cortex. The cortex is a very thin sheet, and there is evolutionary and developmental pressure to utilise cortical space to the maximum. This promotes the formation of spatial superstructures encompassing regions serving different functions. Also cortical development produces two types of pattern, one in the actual physical structure, this is common to much biological pattern formation, but also in terms of patterns of neural response which can be viewed as a feature mapping and is specific to cortical function. We examine the first type of pattern formation within the barrel field of the rat cortex, a geometric superstructure that has the properties of a Voronoi tessellation and apply a dynamical constraint from the observation that the patterns are sparse. We show that these constraints produce a distribution of patterns closer to what is observed than predictions derived from studies in a single domain of perfect circular shape. We also discover a novel effect of geometric alignment of patterns in neighbouring domains, without any physical communication between them, in a wide class of tessellations. This effect is confirmed by analysis of actual images of the subbarrel patterns in the developing rat cortex. The effect of geometry and dynamics of the second type of pattern formation is investigated in the patterns of orientation preference of neuronal response in the visual cortex of certain mammals. Where the domains are sufficiently small so that topological defects (pinwheels) cannot form the behaviour is similar to the reaction-diffusion equations. However, when there are many defects in the region alignment at the boundaries disappears.