Complex-valued Neuron to describe the Dynamics after Hopf Bifurcation: an Example of CPG Model for a Biped Locomotion

Complex-valued Hopfield network is used to model the dynamics of a network of limit cycle oscillators, each of which emerges via Hopf bifurcation, to investigate the dependency of the network dynamics on a bifurcation parameter. As an application, a network of two complex-valued neurons is used as a central pattern generator model for a biped locomotion. A bifurcation parameter is a constant input from higher motor centers. Numerical calculations show the system successfully expresses some characteristic behaviors, which were obtained by more complicated Fitzhugh-Nagumo oscillator model, and which were found in clinical data of disordered interlimb coordination caused by Parkinson´s disease. The observed results of symmetric anti -phase synchronization, asymmetric synchronization, and breakdown of the synchronization can be explained by the existence condition of the energy function of the complex-valued neural network, and by the synchronization condition of a coupled system of phase oscillators.