Abstract
The stability of the null solution in a system of coupled cells is investigated. Each cell evolves according to Hopfield's equation for an analog circuit, with a delay incorporated to account for finite switching speed of amplifiers. A necessary and sufficient condition on the connection matrix is obtained for delay-induced oscillations to be possible in a general (not necessarily symmetric) network.
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References
Bélair, J., and Mackey, M. C. (1989). Consumer memory and price fluctuations in commodity markets: An integrodifferential model.J. Dynam. Diff. Eq. 1, 299–325.
Braddock, R. D., and van den Driessche, P. (1976). On the stability of differential-difference equations.J. Austral. Math. Soc. (Ser. B) 19, 358–370.
Franke, J. M., and Stech, H. W. (1991). Extensions of an algorithm for the analysis of nongeneric Hopf bifurcations, with applications to delay-differential equations. In Busenberg, S., and Martelli, M. (eds.),Delay Differential Equations and Dynamical Systems, Springer, New York, pp. 161–175.
Haddock, J. R., and Terjéki, J. (1983). Liapunov-Razumikhin functions and an invariance principle for functional differential equation.J. Diff. Eq. 48, 95–122.
Hale, J. K. (1977).Theory of Functional Differential Equations, Springer, New York.
Hopfield, J. J. (1984). Neurons with graded response have collective computational properties like those of two-state neurons.Proc. Natl. Acad. Sci. 81, 3088–3092.
Kurzweil, J. (1967). Invariant manifolds for flows. In Hale, J., and LaSalle, J. (eds.),Differential Equations and Dynamical Systems, Academic Press, New York, pp. 431–468.
Lancaster, P., and Tismenetsky, M. (1985).The Theory of Matrices, 2nd ed., Academic Press, Orlando, FL.
Lémeray, E. M. (1987). Le quatrième algorithme naturel.Proc. Edinburgh Math. Soc. 16, 13–35.
Losson, J. (1991).Multistability and Probabilistic properties of Differential Equations, M.S. thesis (Physics), McGill University, Montreal.
Mackey, M. C., and Milton, J. (1987). Dynamical diseases.Ann. N.Y. Acad. Sci. 504, 16–32.
Mahaffy, J. (1982). A test for stability of linear differential delay equations.Q. Appl. Math. 40, 193–202.
Mallet-Paret, J. (1988). Morse decompositions for delay-differential equations.J. Diff. Eq. 23, 293–314.
Marcus, C. M., and Westervelt, R. M. (1989). Stability of analog neural networks with delay.Phys. Rev. A 39, 347–359.
Palmer, R. (1989). Neural nets. In Stein, D. (ed.),Lectures in the Sciences of Complexity, Addison-Wesley, Redwood City, CA.
Stech, H. (1985). Hopf bifurcation calculations for functional differential equations.J. Math. Anal. Appl. 109, 472–491.
Walther, H.-O. (1991). An invariant manifold of slowly oscillating solutions for x(t)=μx(t)+f(x(t−1)).J. reine angew. Math. 414, 67–112.
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Bélair, J. Stability in a model of a delayed neural network. J Dyn Diff Equat 5, 607–623 (1993). https://doi.org/10.1007/BF01049141
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DOI: https://doi.org/10.1007/BF01049141