Stability and bifurcation of a simple neural network with multiple time delays. (English) Zbl 0926.92003

Campbell, Sue Ann

Ruan, Shigui (ed.) et al., Differential equations with applications to biology. Proceedings of the international conference, Halifax, Canada, July 25–29, 1997. Providence, RI: American Mathematical Society. Fields Inst. Commun. 21, 65-79 (1999).

Summary: A system of delay differential equations representing a simple model for a ring of neurons with time delayed connections between the neurons is studied. Conditions for the linear stability of fixed points of this system are represented in a parameter space consisting of the sum of the time delays between the elements and the product of the strengths of the connections between the elements. It is shown that both Hopf and steady state bifurcations may occur when a fixed point loses stability. Codimension two bifurcations are shown to exist and numerical simulations reveal the possibility of quasiperiodicity and multistability near such points.
For the entire collection see [Zbl 0903.00038].

MSC:

92B20	Neural networks for/in biological studies, artificial life and related topics
34K20	Stability theory of functional-differential equations
34K18	Bifurcation theory of functional-differential equations

Keywords:

delay differential equations; neurons