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This paper reveals the dynamics of a neural network of four neurons with multiple time delays and a short-cut connection through a combined study of theoretical analysis, numerical simulations, and experiments. The first step of the study is to derive the sufficient conditions for the stability and instability of the network equilibrium, and the second step is to determine the properties of the periodic response bifurcating from a Hopf bifurcation of the network equilibrium on the basis of the normal form and the center manifold reduction. Afterwards, the study turns to the validation of theoretical results through numerical simulations and a circuit experiment. The case studies show that both numerical simulations and circuit experiment get a nice agreement with theoretical results.

This paper studies the nonlinear dynamics of coupled ring networks each with an arbitrary number of neurons. Different types of time delays are introduced into the internal connections and couplings. Local and global asymptotical stability of the coupled system is discussed, and sufficient conditions for the existence of different bifurcated oscillations are given. Numerical simulations are performed to validate the theoretical results, and interesting neuronal activities are observed, such as rest state, synchronous oscillations, asynchronous oscillations, and multiple switches of the rest states and different oscillations. It is shown that the number of neurons in the sub-networks plays an important role in the network characteristics.

This paper reveals the dynamics of a delayed neural network of four neurons, with a short-cut connection through a theoretical analysis and some case studies of both numerical simulations and experiments. It presents a detailed analysis of the stability and the stability switches of the network equilibrium, as well as the Hopf bifurcation and the bifurcating periodic responses on the basis of the normal form and the center manifold reduction. Afterwards, the study focuses on the validation of theoretical results through numerical simulations and circuit experiments. The numerical simulations and the circuit experiments not only show good agreement with theoretical results, but also show abundant effects of the short-cut connection on the network dynamics.