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, 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.

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 two-neuron network model with three delays. The characteristic equation of the linearized system at the zero solution is a polynomial equation involving very high order terms. We derive some sufficient and necessary conditions on the asymptotic stability of the zero solution. Regarding the eigenvalues of connection matrix as the bifurcation parameters, we also consider the existence of three types of bifurcations: Fold bifurcations, Flip bifurcations, and Neimark-Sacker bifurcations. The stability and direction of these three kinds of bifurcations are studied by applying the normal form theory and the center manifold theorem. Our results are a very important generalization to the previous works in this field. © 2007 Published by Elsevier Ltd.

We study the existence and branching patterns of wave trains in the one-dimensional in nite Fermi-Pasta-Ulam (FPU) lattice. A wave train Ansatz in this Hamiltonian lattice leads to an advance-delay di erential equation on a space of periodic functions, which carries a natural Hamiltonian structure. The existence of wave trains is then studied by means of a Lyapunov Schmidt reduction, leading to a nite-dimensional bifurcation equation with an inherited Hamiltonian structure. While exploring some of the additional symmetries of the FPU lattice, we use invariant theory to nd the bifurcation equations describing the branching patterns of wave trains near p∶q resonant waves. We show that at such branching points, a generic nonlinearity selects exactly two two-parameter families of mixed-mode wave trains.

In this paper, we consider a ring of neurons with self-feedback and delays. As a result of our approach based on global bifurcation theorems of delay differential equations coupled with representation theory of Lie groups, the coexistence of its asynchronous periodic solutions (i.e. mirror-reflecting waves, standing waves and discrete waves), bifurcated simultaneously from the trivial solution at some critical values of the delay, will be established for delay not only near to but also far away from the critical values. Therefore, we can obtain wave solutions of large amplitudes. In addition, we consider the coincidence of these periodic solutions.

Without assuming boundedness and differentiability of the activation functions and any symmetry of interconnections, we employ Lyapunov functions to establish some sufficient conditions ensuring existence, uniqueness, global asymptotic stability, and even global exponential stability of equilibria for the Cohen–Grossberg neural networks with and without delays. Our results are not only presented in terms of system parameters and can be easily verified and also less restrictive than previously known criteria and can be applied to neural networks, including Hopfield neural networks, bidirectional association memory neural networks, and cellular neural networks.

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 study a class of neural networks with variable coefficients which includes delayed Hopfield neural networks, bidirectional associative memory networks and cellular neural networks as its special cases. By matrix theory and inequality analysis, we not only obtain some new sufficient conditions ensuring the existence, uniqueness, global attractivity and global exponential stability of the periodic solution but also estimate the exponentially convergent rate. Our results are less restrictive than previously known criteria and can be applied to neural networks with a broad range of activation functions assuming neither differentiability nor strict monotonicity. Moreover, these conclusions are presented in terms of system parameters and can be easily verified for the globally Lipschitz and the spectral radius being less than 1. Therefore, our results have an important leading significance in the design and applications of periodic oscillatory neural circuits for neural networks with delays. © 2005 Elsevier Ltd. All rights reserved.

In this paper, we consider a ring of neurons with self-feedback and delays. The linear stability of the model is investigated by analyzing the associated characteristic transcendental equation. Based on the normal form approach and the center manifold theory, we derive the formula for determining the properties of Hopf bifurcating slowly oscillating periodic orbits for a ring of neurons with delays, including the direction of Hopf bifurcation, stability of the Hopf bifurcating slowly oscillating periodic orbits, and so on. Moreover, by means of the symmetric bifurcation theory of delay differential equations coupled with representation theory of standard dihedral groups, we not only investigate the effect of synaptic delay of signal transmission on the pattern formation, but also obtain some important results about the spontaneous bifurcation of multiple branches of periodic solutions and their spatio-temporal patterns.