Existence and properties of solutions for neural field equations

The first goal of this work is to study the solvability of the neural field equation (known as ‘Amari equation’) which is an integro-differential equation in m+ 1 dimensions. In particular, we show the existence of global solutions for smooth activation functions f with values in [0, 1] and L1 kernels w via the Banach fixpoint theorem. We note that this setting is much more general than in most related studies, e.g. Ermentrout and McLeod (Proceedings of the Royal Society of Edinburgh 1993; 123A:461–478). For a Heaviside-type activation function f, we show that the approach above fails. However, with slightly more regularity on the kernel function w (we use Holder continuity with respect to the argument x) we can employ compactness arguments, integral equation techniques and the results for smooth nonlinearity functions to obtain a global existence result in a weaker space. Finally, general estimates on the speed and durability of waves are derived. We show that compactly supported waves with directed kernels (i.e. w(x, y)⩽0 for x⩽y ) decay exponentially after a finite time and that the field has a well-defined finite speed. Copyright © 2009 John Wiley & Sons, Ltd.

[1]  W S McCulloch,et al.  A logical calculus of the ideas immanent in nervous activity , 1990, The Philosophy of Artificial Intelligence.

[2]  Jürgen Kurths,et al.  Simulating global properties of electroencephalograms with minimal random neural networks , 2008, Neurocomputing.

[3]  S. Schiff,et al.  Control of traveling waves in the Mammalian cortex. , 2005, Physical review letters.

[4]  S. Amari.  Dynamics of pattern formation in lateral-inhibition type neural fields , 1977, Biological Cybernetics.

[5]  J. Kurths,et al.  Neural and Cognitive Modeling with Networks of Leaky Integrator Units , 2007 .

[6]  P. Nunez.  Toward a quantitative description of large-scale neocortical dynamic function and EEG , 2000, Behavioral and Brain Sciences.

[7]  J A Kelso,et al.  Spatiotemporal pattern formation in neural systems with heterogeneous connection topologies. , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[8]  Paul C. Bressloff,et al.  Traveling Pulses and Wave Propagation Failure in Inhomogeneous Neural Media , 2008, SIAM J. Appl. Dyn. Syst..

[9]  James J. Wright,et al.  Propagation and stability of waves of electrical activity in the cerebral cortex , 1997 .

[10]  Axel Hutt,et al.  Analysis of nonlocal neural fields for both general and gamma-distributed connectivities , 2005 .

[11]  J. Griffith.  A field theory of neural nets: I. Derivation of field equations. , 1963, The Bulletin of mathematical biophysics.

[12]  P. Graben,et al.  Foundations of Neurophysics , 2007 .

[13]  Roland Potthast,et al.  Language processing with dynamic fields , 2008, Cognitive Neurodynamics.

[14]  P. Matthews,et al.  Dynamic instabilities in scalar neural field equations with space-dependent delays , 2007 .

[15]  Anders Krogh,et al.  Introduction to the theory of neural computation , 1994, The advanced book program.

[16]  Christian Igel,et al.  Optimization of dynamic neural fields , 2001, Neurocomputing.

[17]  P. Graben,et al.  Fock space representations in neural field theories , 2007 .

[18]  Peter A. Robinson,et al.  Unified neurophysical model of EEG spectra and evoked potentials , 2002, Biological Cybernetics.

[19]  J. Cowan,et al.  Excitatory and inhibitory interactions in localized populations of model neurons. , 1972, Biophysical journal.

[20]  W. Freeman.  Simulation of chaotic EEG patterns with a dynamic model of the olfactory system , 1987, Biological Cybernetics.

[21]  J. Cowan,et al.  The visual cortex as a crystal , 2002 .

[22]  John R. Terry,et al.  A unifying explanation of primary generalized seizures through nonlinear brain modeling and bifurcation analysis. , 2006, Cerebral cortex.

[23]  W. Pitts,et al.  A Logical Calculus of the Ideas Immanent in Nervous Activity (1943) , 2021, Ideas That Created the Future.

[24]  R. Palmer,et al.  Introduction to the theory of neural computation , 1994, The advanced book program.

[25]  K. V. Leung,et al.  Improved neuronal models for studying neural networks , 1974, Kybernetik.

[26]  Paul C Bressloff,et al.  Bloch waves, periodic feature maps, and cortical pattern formation. , 2002, Physical review letters.

[27]  J. Cowan,et al.  A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue , 1973, Kybernetik.

[28]  G. Ermentrout,et al.  Existence and uniqueness of travelling waves for a neural network , 1993, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[29]  J. Griffith.  A field theory of neural nets. II. Properties of the field equations. , 1965, The Bulletin of mathematical biophysics.

[30]  G. Lord,et al.  Waves and bumps in neuronal networks with axo-dendritic synaptic interactions , 2003 .

[31]  H. Haken,et al.  Field Theory of Electromagnetic Brain Activity. , 1996, Physical review letters.

[32]  Viktor K. Jirsa,et al.  Information Processing in Brain and Behavior Displayed in Large-Scale Scalp Topographies such as EEG and MEG , 2004, Int. J. Bifurc. Chaos.

[33]  H. Haken,et al.  A derivation of a macroscopic field theory of the brain from the quasi-microscopic neural dynamics , 1997 .

[34]  Axel Hutt,et al.  Effects of distributed transmission speeds on propagating activity in neural populations. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.