Neural Field Models: A mathematical overview and unifying framework

Abstract

Mathematical modelling of the macroscopic electrical activity of the brain is highly non-trivial and requires a detailed understanding of not only the associated mathematical techniques, but also the underlying physiology and anatomy. Neural field theory is a population-level approach to modelling the non-linear dynamics of large populations of neurons, while maintaining a degree of mathematical tractability. This class of models provides a solid theoretical perspective on fundamental processes of neural tissue such as state transitions between different brain activities as observed during epilepsy or sleep. Various anatomical, physiological, and mathematical assumptions are essential for deriving a minimal set of equations that strike a balance between biophysical realism and mathematical tractability. However, these assumptions are not always made explicit throughout the literature. Even though neural field models (NFMs) first appeared in the literature in the early 1970's, the relationships between them have not been systematically addressed. This may partially be explained by the fact that the inter-dependencies between these models are often implicit and non-trivial. Herein we provide a review of key stages of the history and development of neural field theory and contemporary uses of this branch of mathematical neuroscience. First, the principles of the theory are summarised throughout a discussion of the pioneering models by Wilson and Cowan, Amari and Nunez. Upon thorough review of these models, we then present a unified mathematical framework in which all neural field models can be derived by applying different assumptions. We then use this framework to i) derive contemporary models by Robinson, Jansen and Rit, Wendling, Liley, and Steyn-Ross, and ii) make explicit the many significant inherited assumptions that exist in the current literature.

arXiv comment67 pages, 10 figures, 1 table

References in corpus (14)

• Mathematical frameworks for oscillatory network dynamics in neuroscience
• Topological and Dynamical Complexity of Random Neural Networks
• Dynamics of Random Neural Networks with Bistable Units
• Intrinsically-generated fluctuating activity in excitatory-inhibitory networks
• Theta-nested gamma oscillations in next generation neural mass models
• Exact neural mass model for synaptic-based working memory
• A next generation neural field model: The evolution of synchrony within patterns and waves
• Neural Field Models: A mathematical overview and unifying framework
• Unified neural field theory of brain dynamics underlying oscillations in Parkinson's disease and generalized epilepsies
• Neural mass modeling of slow-fast dynamics of seizure initiation and abortion
• Fast and slow domino regimes in transient network dynamics
• Consequences of Dale's law on the stability-complexity relationship of random neural networks
• May-Wigner transition in large random dynamical systems
• Finite-Size Effects on Traveling Wave Solutions to Neural Field Equations

Cited by in corpus (5)

• Neural Field Models: A mathematical overview and unifying framework
• Lattice physics approaches for neural networks
• Reproducibility via neural fields of visual illusions induced by localized stimuli
• Analysis of the Geometric Structure of Neural Networks and Neural ODEs via Morse Functions
• Spot solutions to a neural field equation on oblate spheroids