Tutorial on Neural Field Theory
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
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Tutorial on Neural Field Theory Stephen Coombes, Peter beim Graben, and Roland Potthast
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.
1.1 Background Ever since Hans Berger made the first recording of the human electroencephalogram (EEG) in 1924 [8] there has been a tremendous interest in understanding the physiological basis of brain rhythms. This has included the development of mathematical models of cortical tissue – which are often referred to as neural field models. One of the earliest of such models is due to Beurle [9] in the 1950s, who developed a continuum description of the proportion of active neurons in a randomly connected network. This was followed by work of Griffith [40, 41]
S. Coombes () School of Mathematical Sciences, University of Nottingham, NG7 2RD Nottingham, UK e-mail: [email protected] P.B. Graben Bernstein Center for Computational Neuroscience, 10115 Berlin, Germany R. Potthast Department of Mathematics, University of Reading, RG6 6AX Reading, UK Deutscher Wetterdienst, Offenbach, Germany S. Coombes et al. (eds.), Neural Fields, DOI 10.1007/978-3-642-54593-1__1, © Springer-Verlag Berlin Heidelberg 2014
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in the 1960s, who also published two books that still make interesting reading for modern practitioners of mathematical neuroscience [42, 43]. However, it were Wilson and Cowan [88,89], Nunez [67] and Amari [3] in the 1970s who provided the formulations for neural field models that is in common use today (see Chaps. 2 and 3 in this book). Usually, neural field models are conceived as neural mass models describing population activity at spatiotemporally coarse-grained scales [67, 89]. They can be classified as either activity-based [89] or voltage-based [3, 67] models (see [14, 64] for discussion). For their activity-based model Wilson and Cowan [88,89] distinguished between excitatory and inhibitory sub-populations, as well as accounted for refractoriness. This seminal model can be written succinctly in terms of the pair of partial integrodifferential equations: @E D E C .1 rE E/SE ŒwEE ˝ E wEI ˝ I ; @t @I D I C .1 rI I /SI ŒwIE ˝ E wII ˝ I : @t
(1.1)
Here E D E.r; t / is a temporal coarse-grained variable describing the proportion of excitatory cells firing per unit time at positio
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