Interface dynamics in planar neural field models
1 School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK
2 School of Psychology (CN-CR), University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK
3 Centre for Neuroscience, Donders Institute for Brain, Cognition and Behaviour, Nijmegen, 6500 HB, The Netherlands
The Journal of Mathematical Neuroscience 2012, 2:9 doi:10.1186/2190-8567-2-9Published: 2 May 2012
Neural field models describe the coarse-grained activity of populations of interacting neurons. Because of the laminar structure of real cortical tissue they are often studied in two spatial dimensions, where they are well known to generate rich patterns of spatiotemporal activity. Such patterns have been interpreted in a variety of contexts ranging from the understanding of visual hallucinations to the generation of electroencephalographic signals. Typical patterns include localized solutions in the form of traveling spots, as well as intricate labyrinthine structures. These patterns are naturally defined by the interface between low and high states of neural activity. Here we derive the equations of motion for such interfaces and show, for a Heaviside firing rate, that the normal velocity of an interface is given in terms of a non-local Biot-Savart type interaction over the boundaries of the high activity regions. This exact, but dimensionally reduced, system of equations is solved numerically and shown to be in excellent agreement with the full nonlinear integral equation defining the neural field. We develop a linear stability analysis for the interface dynamics that allows us to understand the mechanisms of pattern formation that arise from instabilities of spots, rings, stripes and fronts. We further show how to analyze neural field models with linear adaptation currents, and determine the conditions for the dynamic instability of spots that can give rise to breathers and traveling waves.