Neural Field Models: A mathematical overview and unifying framework

Mathematical modelling of the macroscopic electrical activity of the brain ishighly non-trivial and requires a detailed understanding of not only theassociated mathematical techniques, but also the underlying physiology andanatomy. Neural field theory is a population-level approach to modelling thenon-linear dynamics of large populations of neurons, while maintaining a degreeof mathematical tractability. This class of models provides a solid theoreticalperspective on fundamental processes of neural tissue such as state transitionsbetween different brain activities as observed during epilepsy or sleep.Various anatomical, physiological, and mathematical assumptions are essentialfor deriving a minimal set of equations that strike a balance betweenbiophysical realism and mathematical tractability. However, these assumptionsare not always made explicit throughout the literature. Even though neuralfield models (NFMs) first appeared in the literature in the early 1970's, therelationships between them have not been systematically addressed. This maypartially be explained by the fact that the inter-dependencies between thesemodels are often implicit and non-trivial. Herein we provide a review of keystages of the history and development of neural field theory and contemporaryuses of this branch of mathematical neuroscience. First, the principles of thetheory are summarised throughout a discussion of the pioneering models byWilson and Cowan, Amari and Nunez. Upon thorough review of these models, wethen present a unified mathematical framework in which all neural field modelscan be derived by applying different assumptions. We then use this framework toi) derive contemporary models by Robinson, Jansen and Rit, Wendling, Liley, andSteyn-Ross, and ii) make explicit the many significant inherited assumptionsthat exist in the current literature.Comment: 67 pages, 10 figures, 1 tabl