Abstract
In a series of articles (Leung et al., 1973, 1974; Ogŭztöreli, 1972, 1975, 1978, 1979; Stein et al., 1974) we have investigated some of the physiologically significant properties of a general neural model. In these papers the nature of the oscillations occuring in the model has been briefly analyzed by omitting the effects of the discrete time-lags in the interaction of neurons, although these time-lags were incoporated in the general model. In the present work we investigate the effects of the time-lags on the oscillations which are intrinsic to the neural model, depending on the structural parameters such as external inputs, interaction coefficients, self-inhibition, self-excitation and selfadaptation coefficients. The numerical solution of the neural model, the computation of the steady-state solutions and the natural modes of the oscillations around the steady-state solutions are described.
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References
Amari, S.: A mathematical approach to neural systems. In: Systems neuroscience. Metzler, J. (ed.), pp. 67–117, New York, London: Academic Press 1977
Amari, S., Arbib, M.A.: Competition and cooperation in neural nets. In: Systems neuroscience. Metzler, J. (ed.), pp. 119–165. New York, London: Academic Press 1977
Bellman, R., Cooke, K.L.: Differential-difference equations. New York, London: Academic Press 1963
Caianiello, E.R. (ed.): Neural networks. New York: Springer 1968
Collatz, L.: The numerical treatment of differential equations. Berlin, Heidelberg, New York: Springer 1966
Culbertson, J.T.: The minds of robots. Urbana: Univ. of Illinois Press 1963
Deutsch, S.: Models of the nervous systems. New York, London, Sydney: John Wiley & Sons 1967
El'sgol'ts, L.E., Norkin, S.B.: Introduction to the theory and applications of differential equations with deviating arguments. New York, London: Academic Press 1973
Heiden, U. an der. Existence of periodic solutions of a nerve equation. Biol. Cybernetics 21, 37–39 (1976)
Leung, K.V., Mangeron, D., Ogŭztöreli, M.N., Stein, R.B.: On a class of nonlinear integro-differential equations. I. Rend. Accad. Naz. Lincei 54, 522–528 (1978a)
Leung, K.V., Mangeron, D., Ogŭztöreli, M.N., Stein, R.B.: On a class of nonlinear integro-differential equations. II. Rend. Accad. Naz. Lincei 54, 699–705 (1973b)
Leung, K.V., Mangeron, D., Ogŭztöreli, M.N., Stein, R.B.: On a class of nonlinear integro-differential equations. III. Bull. Acad. R. Sci. Belgique, Ser. 5, 59, 492–499 (1973c)
Leung, K.V., Mangeron, D., Ogŭztöreli, M.N., Stein, R.B.: On the stability and numerical solutions of two neural models. Utilitas Mathematica 5, 167–212 (1974)
Levin, B.Ja.: Distribution of zeros of entire functions. American Mathematical Society. Translation series, Vol. 5, 1964
Miller, S., Scott, P.D.: The spinal locomotor generator. Exp. Brain Res. 30, 387–403 (1977)
Ogŭztöreli, M.N.: Time-lag control systems. New York, London: Academic Press 1966
Ogŭztöreli, M.N.: On the neural equations of Cowan and Stein (I). Utilitas Mathematica 2, 305–317 (1972)
Ogŭztöreli, M.N.: On the activities in a continuous neural network. Biol. Cybernetics 21, 41–48 (1975)
Ogŭztöreli, M.N.: Activities in a one-dimensional continuous neural network. Arch. Math. 3, Scripta Fac. Sci. Nat. Ujep Brunensis 14, 161–169 (1978)
Ogŭztöreli, M.N.: Activity propagation in a continuous neural network. “Jubilee Volume” of the Institute of Mathematics of the Soviet Academy of Sciences, Moscow-USSR (in press)
Pontryagin, L.S.: On the zeros of some elementary transcendental functions. Izv. Akad. Nauk SSR, Ser. Mat. 6, 115–134 (1942)
Ratliff, F.: Mach bands: quantitative studies on neural networks in the retina. San Francisco, London, Amsterdam: Holden-Day 1965
Sansone, G., Gerretsen, J.: Lectures on the theory of functions of a complex variable, Vol. 1. Groningen: Noordhoff 1960
Stein, R.B.: A theoretical analysis of neuronal variability. Biophys. J. 5, 173–194 (1965)
Stein, R.B.: Some models of neuronal variability. Biophys. J. 7, 37–68 (1967)
Stein, R.B., Leung, K.V., Ogŭztöreli, M.N., Williams, D.W.: Properties of small neural networks. Kybernetik 14, 223–230 (1974)
Stein, R.B., Leung, K.V., Mangeron, D., Ogŭztöreli, M.N.: Improved neuronal models for studying neural networks. Kybernetik 15, 1–9 (1974)
Ward, D.S.: Stochastic relationships for neurons and neuron pair networks. Ph.D. Dissertation, Univ. of Calif., Los Angeles, 1975
Ward, D.S., Rubin, I., Sclabassi, R.J.: Applications of queuing theory to some problems in neural circuity. Appl. Math. Comp. 3, 227–245 (1977)
Wilson, H.R., Cowan, J.D.: A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik 13, 55–80 (1972)
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This work was partly supported by the Natural Sciences and Engineering Research Council of Canada under Grant NRC-A-4345 through the University of Alberta
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Ogŭztöreli, M.N. Activity analysis of neural networks. Biol. Cybernetics 34, 159–169 (1979). https://doi.org/10.1007/BF00336967
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DOI: https://doi.org/10.1007/BF00336967