Approximation of Neural Network Dynamics by Reaction‐Diffusion Equations

The equations of the Hopfield network, without the constraint of symmetry, can have complex behaviours. Cottet borrowed techniques from particle methods to show that a class of such networks with symmetric, translation-invariant connection matrices may be approximated by a reaction-diffusion equation. This idea is extended to a wider class of network connections yielding a slightly more complex reaction-diffusion equation. It is also shown that the approximation holds rigorously only in certain spatial regions (even for Cottet's special case) but the small regions where it fails, namely within transition layers between regions of high and low activity, are not likely to be critical.

[1]  J. Cahn,et al.  A microscopic theory for antiphase boundary motion and its application to antiphase domain coasening , 1979 .

[2]  M. Ward,et al.  Resolving weak internal layer interactions for the Ginzburg–Landau equation , 1994, European Journal of Applied Mathematics.

[3]  W. Freeman,et al.  How brains make chaos in order to make sense of the world , 1987, Behavioral and Brain Sciences.

[4]  P. Raviart.  An analysis of particle methods , 1985 .

[5]  Shui-Nee Chow,et al.  Spatially Discrete Nonlinear Diffusion Equations , 1995 .

[6]  A.N. Michel,et al.  Associative memories via artificial neural networks , 1990, IEEE Control Systems Magazine.

[7]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[8]  Very slow phase separation in one dimension , 1989 .

[9]  A. Coolen,et al.  Image evolution in Hopfield networks. , 1988, Physical review. A, General physics.

[10]  S. Amari.  Dynamical stability of formation of cortical maps , 1988 .

[11]  S. Amari,et al.  A Mathematical Foundation for Statistical Neurodynamics , 1977 .

[12]  J. Carr,et al.  Invariant manifolds for metastable patterns in ut = ε2uxx—f(u) , 1990, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[13]  P. Degond,et al.  The weighted particle method for convection-diffusion equations , 1989 .

[14]  U. Heiden.  Analysis Of Neural Networks , 1980 .

[15]  J. Carr,et al.  Metastable patterns in solutions of ut = ϵ2uxx − f(u) , 1989 .

[16]  J. Cowan,et al.  Large Scale Spatially Organized Activity in Neural Nets , 1980 .

[17]  P. Fife.  Sigmoidal Systems and Layer Analysis , 1982 .

[18]  Sommers,et al.  Chaos in random neural networks. , 1988, Physical review letters.

[19]  M. Ward.  Metastable Patterns, Layer Collapses, and Coarsening for a One‐Dimensional Ginzburg‐Landau Equation , 1994 .

[20]  S. Renals,et al.  A study of network dynamics , 1990 .

[21]  Perspectives on neural network models and their relevance to neurobiology , 1989 .

[22]  G. Ermentrout.  Asymptotic Behavior of Stationary Homogeneous Neuronal Nets , 1982 .

[23]  J J Hopfield,et al.  Neural networks and physical systems with emergent collective computational abilities. , 1982, Proceedings of the National Academy of Sciences of the United States of America.

[24]  Steve Renals,et al.  Chaos in Neural Networks , 1990, EURASIP Workshop.

[25]  Richard P. Lippmann,et al.  An introduction to computing with neural nets , 1987 .

[26]  J J Hopfield,et al.  Neurons with graded response have collective computational properties like those of two-state neurons. , 1984, Proceedings of the National Academy of Sciences of the United States of America.

[27]  J. Cowan,et al.  Secondary Bifurcation in Neuronal Nets , 1980 .

[28]  Misha Tsodyks,et al.  Chaos in Highly Diluted Neural Networks , 1991 .

[29]  Reinhard Illner,et al.  On information-processing abilities of chaotic dynamical systems , 1992 .

[30]  J. Cowan,et al.  Temporal oscillations in neuronal nets , 1979, Journal of mathematical biology.

[31]  Aggregation and Segregation Phenomena in Reaction-Diffusion Equations , 1982 .

[32]  G. Parisi,et al.  Asymmetric neural networks and the process of learning , 1986 .

[33]  J S Denker,et al.  Neural network models of learning and adaptation , 1986 .

[34]  G.-H. Cottet,et al.  Diffusion approximation on neural networks and applications for image processing , 1992 .

[35]  Stephen Grossberg,et al.  Nonlinear neural networks: Principles, mechanisms, and architectures , 1988, Neural Networks.

[36]  O. A. Ladyzhenskai︠a︡,et al.  Linear and Quasi-linear Equations of Parabolic Type , 1995 .