Summary
Dynamics 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahn, S. M., Freeman, W. J.: Steady-state and limit cycle activity of mass of neurons forming simple feedback loops (II): Distributed parameter model, Kybernetik 16, 127–132 (1974)
Amari, S.: Homogeneous nets of neuron-like elements. Biol. Cybernetics 17, 211–220 (1975)
Amari, S.: Dynamics of pattern formation in lateral-inhibition type neural fields. Biol. Cybernetics 27, 77–87 (1977)
Amari, S.: Topographic organization of nerve fields, to appear
Amari, S., Arbib, M. A.: Competition and cooperation in neural nets. In: Systems Neuroscience (J. Metzler, ed.) pp. 119–165, New York-London: Academic Press, 1977
Beurle, R. L.: Properties of a mass of cells capable of regenerating pulses. Trans. Roy. Soc. London B-240, 55–94 (1956)
Coleman, B. D.: Mathematical theory of lateral sensory inhibition. Arch. Rational Mech. Anal., 43, 79–100 (1971)
Cowan, J. D., Ermentrout, G. B.: Some aspects of the eigenbehaviour of neural nets. MAA Studies in Mathematical Biology, to appear
Cronin, J.: Fixed Points and Topological Degree in Non-Linear Analysis. Providence, Rhode Island, AMS (1964)
Ellias, S. A., Grossberg, S.: Pattern formation, contrast control, and oscillations in the short term memory of shunting on-center off-surround networks. Biol. Cybernetics 20, 69–98 (1975)
Farley, B. G., Clark, W. A.: Activity in networks of neuron-like elements. Proc. 4th London Symp. on Inf. Theory (C. Cherry, ed.), London: Butterworths, 1961
Griffith, J. S.: A field theory of neural nets (I). Bull. Math. Biophys. 25, 111–120 (1963)
Griffith, J. S.: A field theory of neural nets (II). Bull. Math. Biophys. 27, 187–195 (1965)
Hadeler, K. P.: On the theory of lateral inhibition. Kybernetik 14, 161–165 (1974)
Krasnosel'skii, M. A.: Positive Solutions of Operator Equations. (Translated by R. E. Flaherty). Groningen, Noordhoff, 1964
Marr, D., Palm, G., Poggio, T.: Analysis of cooperative stereo algorithm. Biol. Cybernetics 28, 223–239 (1978)
Oguztöreli, M. N.: On the activities in a continuous neural network. Biol. Cybernetics 18, 41–48 (1975)
Okuda, M.: A dynamical behavior of active region in randomly connected neural network. J. Theor. Biol. 48, 51–73 (1974)
Ratliff, F.: Mach Bands: Quantitative Studies on Neural Networks in the Retina. San Francisco, Holden-Day, 1975
Reichardt, W., MacGinitie, G.: Zur Theorie der lateralen Inhibition. Kybernetik 1, 155–165 (1962)
Stanley, J. C.: Simulation studies of a temporal sequence memory model. Biol. Cybernetics 24, 121–137 (1976)
Tanaka, A., Noguchi, S.: Topological properties of continuous neural networks with analog neurons. Trans. IECE, Japan, J61-D, 9–16 (1978)
Tokura, T., Morishita, I.: Analysis and simulations of double-layer neural networks with mutually inhibiting interconnections. Biol. Cybernetics 25, 83–92 (1977)
Walter, H.: Inhibitionsfelder. Acta Informatica 1, 253–269 (1972)
Wiener, N., Rosenblueth, A.: The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle. Arch. Inst. Cardiol. (Mexico) 16, 205–265 (1946)
Wilson, H. R., Cowan, J. D.: A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik 13, 55–80 (1973)
Willshaw, D. J., Malsburg, C. von der: How patterned neural connections can be set up by self organization. Proc. Roy. Soc. London B194, 431–445 (1976)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kishimoto, K., Amari, S. Existence and stability of local excitations in homogeneous neural fields. J. Math. Biol. 7, 303–318 (1979). https://doi.org/10.1007/BF00275151
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00275151