We consider a network of two coupled neurons with delayed feedback. We show that the connection topology of the network plays a fundamental role in classifying the rich dynamics and bifurcation phenomena. Regarding eigenvalues of the connection matrix as bifurcation parameters, we obtain codimension 1 bifurcations (including a fold bifurcation and a Hopf bifurcation) and codimension 2 bifurcations (including fold-Hopf bifurcations and Hopf-Hopf bifurcations). We also give concrete formulae for the normal form coefficients derived via the center manifold reduction that give detailed information about the bifurcation and stability of various bifurcated solutions. In particular, we obtain stable or unstable equilibria, periodic solutions, quasi-periodic solutions, and sphere-like surfaces of solutions.We also show how to evaluate critical normal form coefficients from the original system of delay-differential equations without computing the corresponding center manifolds.

In this paper, we consider a simple discrete-time single-directional network of four neurons. The characteristics equation of the linearized system at the zero solution is a polynomial equation involving very high-order terms. We first derive some sufficient and necessary conditions ensuring that all the characteristic roots have modulus less than 1. Hence, the zero solution of the model is asymptotically stable. Then, we study the existence of three types of bifurcations, such as fold bifurcations, flip bifurcations, and Neimark-Sacker (NS) bifurcations. Based on the normal form theory and the center manifold theorem, we discuss their bifurcation directions and the stability of bifurcated solutions. In addition, several codimension two bifurcations can be met in the system when curves of codimension one bifurcations intersect or meet tangentially. We proceed through listing smooth normal forms for all the possible codimension 2 bifurcations. © 2007 Elsevier B.V. All rights reserved.

In this paper, we study the effect of synaptic delay of signal transmission on the pattern formation and some properties of non-linear waves in a ring of identical neurons. First, linear stability of the model is investigated by analyzing the associated characteristic transcendental equation. Regarding the delay as a bifurcation parameter, we obtained the spontaneous bifurcation of multiple branches of periodic solutions and their spatio-temporal patterns. Second, global continuation conditions for Hopf bifurcating periodic orbits are derived by using the equivariant degree theory developed by Geba et al. and independently by Ize & Vignoli. Third, we show that the coincidence of these periodic solutions is completely determined either by a scalar delay differential equation if the number of neurons is odd, or by a system of two coupled delay differential equations if the number of neurons is even. Fourth, we summarize some important results about the properties of Hopf bifurcating periodic orbits, including the direction of Hopf bifurcation, stability of the Hopf bifurcating periodic orbits, and so on. Fifth, in an excitatory ring network, solutions of most initial conditions tend to stable equilibria, the boundary separating the basin of attraction of these stable equilibria contains all of periodic orbits and homoclinic orbits. Finally, we discuss a trineuron network to illustrate the theoretical results obtained in this paper and conclude that these theoretical results are important to complement the experimental and numerical observations made in living neurons systems and artificial neural networks, in order to understand the mechanisms underlying the system dynamics better.

In this paper, we consider a ring of identical neurons with self-feedback and delays. Based on the normal form approach and the center manifold theory, we derive some formula to determine the direction of Hopf bifurcation and stability of the Hopf bifurcated synchronous periodic orbits, phase-locked oscillatory waves, standing waves, mirror-reflecting waves, and so on. In addition, under general conditions, such a network has a slowly oscillatory synchronous periodic solution which is completely characterized by a scalar delay differential equation. Despite the fact that the slowly oscillatory synchronous periodic solution of the scalar equation is stable, we show that the corresponding synchronized periodic solution is unstable if the number of the neurons is large or arbitrary even. © 2007 Elsevier Inc. All rights reserved.

In this paper, some sufficient conditions for the local and global exponential stability of the discrete-time Hopfield neural networks with general activation functions are derived, which generalize those existing results. By means of Mmatrix theory and some inequality analysis techniques, the exponential convergence rate of the neural networks to the equilibrium is estimated, and for the local exponential stability, the basin of attraction of the stable equilibrium is also characterized. © 2004 Elsevier Ltd. All rights reserved.