Synchronization For Two Coupled Oscillators With Inhibitory Connection

In this paper we present an oscillatory neural network composed of two cou-pled neural oscillators with inhibitory connection. Each of the oscillators describes the dynamics of average activities of excitatory and inhibitory populations of neu-rons. We first investigate the absolute synchronization properties and criteria and then analyze the possible patterns of equilibria existing in the system. We have de-rived both patterns and asymptotically stability criteria of nonlinear synchronous (including in-phase and anti-phase) oscillations. The main points are organized as follows:Firstly,by means of Lyapunov function,we find some sufficient conditions en-suring absolute synchronization. In addition, we also analyze the possible patterns of equilibria with the non-trivial in-phase equilibrium does not exist, non-trivial anti-phase equilibrium exists. Secondly, we discuss the stability of the equilib-ria through analyzing the associated characteristic equation of linear DDEs. We investigate the distribution of zeros of the characteristic equation. Moreover,we derive some sufficient conditions ensuring the stability of the equilibria. About the stability of non-trivial equilibrium solution, through linear coordinate trans-formation, we can transform the original system into a new system, so we need only, consider the stability of trivial equilibrium solution in the new system. Next we make a systematic overview about bifurcation types the system may appear. Thirdly, as the system has symmetry, by the equivariant Hopf bifurcation theory, without the detailed dynamic system equations,we can get solution of the model in the Nonlinear oscillator system. By regarding the transmission delay of the system as bifurcation parameter, we discuss codimension one bifurcations (including fold bifurcations and Hopf bifurcations) and codimension two bifurcations (including fold-Hopf bifurcations and Hopf-Hopf bifurcations) near the trivial equilibrium. Based on the normal form theory and center manifold reduction, we obtain some criteria to determine the bifurcation direction and the stability of the bifurcated periodic solutions. Fourthly, numerical simulation is also given to support the the-oretical results. Let the activation function f(x)=tanh(x), this function satisfies the given conditions on the f(x) in the paper, the simulation results fit perfectly with the theoretical results. Finally, we summarize the paper and note some sub-jects which need further study.