Coarse-grained analysis of neural networks
There now exist a plethora of mathematical models, both biophysically realistic and phenomenological, describing the behaviour of individual neurons. The complexity of the brain arises not from the action of a single neuron, but from the complex spatio-temporal interactions between these cells. Due to long reaching processes of neurons, they can interact with cells over distances far greater than their cell body. Indeed, at the level of entire brains, even cells in differnt hemispheres can communicate with each other. Even at much smaller scales, these interactions offer neural tissue a diverse range of dynamic response.
It is often activity at much coarser temporal and spatial scales that gives us information about how the brain performs tasks, both because individual cells are thought to convey little information by themselves, but also because many of the imaging techniques available to record human brain rhythms only give us temporally or spatially averaged data. Whilst mathematical models of neural tissue at these coarser scales exists, they are typically phenomenological in nature. Critically, the averaged models do not incorporate subtle effects at microscopic scales that may have drastic consequences at macroscopic scales.
A key goal of mathematical biology then, is to identify meaningful ways to connect these scales. Our research analyses relatively small networks of coupled neural models with a view to understanding how emergent behaviour in the form of spatio- and temporally patterned cortical activity arises. Where possible, we use mathematical machinery and approximations to reduce the complexity of the whole network problem to one in which we can make quanitative predictions. Where this is not possible, we use numerical techniques, such as equation-free modelling and polynomial chaos expansions to achieve the same goals.
One key question we are interested in answeing is what role heterogeneity and noise play in these neural networks. For simplicity, mathematical models typically treat cells as being homogenoeus, though this is clearly not the case. Noise, whilst typically thought of as being detrimental to neural computation, has recently shown instead to be beneficial in certain cases. Understanding how these two features impact upon neural coding is thus an important avenue to explore.