Tutorial on Neural Field Theory

Abstract

The tools of dynamical systems theory are having an increasing impact on our understanding of patterns of neural activity. In this tutorial chapter we describe how to build tractable tissue level models that maintain a strong link with biophysical reality. These models typically take the form of nonlinear integrodifferential equations. Their non-local nature has led to the development of a set of analytical and numerical tools for the study of spatiotemporal patterns, based around natural extensions of those used for local differential equation models. We present an overview of these techniques, covering Turing instability analysis, amplitude equations, and travelling waves. Finally we address inverse problems for neural fields to train synaptic weight kernels from prescribed field dynamics.

Robust stabilization of delayed neural fields with partial measurement and actuation

“…This success is due to technological advances, including electrode arrays and brain imaging, that allow recordings of unprecedented resolution in both time and space. The possibility to represent the spatiotemporal evolution of neural activity constitutes a crucial feature to deepen our understanding of cerebral functions or diseases that involve propagating waves or pattern formation, such as memory, epilepsy, or Parkinson's disease (Bressloff, 2012;Coombes et al, 2014).…”

Incremental stability of spatiotemporal delayed dynamics and application to neural fields

“…The spatiotemporal dynamics of brain oscillations can be captured by neural fields models [9]. These models take the form of integro-differential equations that abstract the brain structure to a continuous medium.…”

Sequences by Metastable Attractors: Interweaving Dynamical Systems and Experimental Data

“…To this end, we identify the MS as equilibria in the underlying dynamical system model. Each pattern, i.e., column of V (c) can be seen as a spatial discretization of a continuous spatial neural field v(x) [35] serving as a saddle in a heteroclinic sequence of field activations [36][37][38].…”