Existence and stability of local excitations in homogeneous neural fields

SummaryDynamics of excitation patterns is studied in one-dimensional homogeneous lateral-inhibition type neural fields. The existence of a local excitation pattern solution as well as its waveform stability is proved by the use of the Schauder fixed-point theorem and a generalized version of the Perron-Frobenius theorem of positive matrices to the function space. The dynamics of the field is in general multi-stable so that the field can keep short-term memory.

[1]  A ROSENBLUETH,et al.  The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle. , 1946, Archivos del Instituto de Cardiologia de Mexico.

[2]  R. L. Beurle.  Properties of a mass of cells capable of regenerating pulses , 1956, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences.

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

[4]  Leo F. Boron,et al.  Positive solutions of operator equations , 1964 .

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

[6]  D. Jameson,et al.  Mach bands : quantitative studies on neural networks in the retina , 1966 .

[7]  Bernard D. Coleman,et al.  A mathematical theory of lateral sensory inhibition , 1971 .

[8]  M. Okuda.  A dynamical behaviour of active regions in randomly connected neural networks. , 1974, Journal of theoretical biology.

[9]  C. Malsburg,et al.  How patterned neural connections can be set up by self-organization , 1976, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[10]  Shun-Ichi Amari,et al.  Topographic organization of nerve fields , 1979, Neuroscience Letters.