Analysis Of Neural Population Dynamics Based On Bifurcation Theory

Objectives:Neural mass model (NMM) is a typical nonlinear system with rich and complex dynamics. Due to synaptic and intrinsic plasticity, the parameters of NMM are time varying, such as excitatory and inhibitory synapses gain, synapse connection numbers, and parameters of soma.. Up to now, most bifurcation researches concerning NMM focused on the impact of input or connection parameters between subpopulations on the dynamics of NMM. However, it still remains little known how excitatory and inhibitory synaptic plasticity and intrinsic plasticity exert an influence on the dynamics of NMM. On the context, we elucidate the regulation mechanism underlined by the plasticity of neural dynamics through bifurcation analysis.Methods:Through three output mathematical description of Synaptic mode of NMM, we can derive the mathematical description of NMM in the form of six first order differential equation. According to the differential equation, we can get the equilibrium of the model and the model output. By computing system characteristic root according to Jacobian matrix, we can analyze the equilibrium stability and estimate stable and unstable regions of the equilibrium curve of NMM. At the same time, based on mathematical description of NMM, we use bifurcation software matcont to study the dynamics of NMM.Results:In this study, through bifurcation analysis we determined the parameters regions of stability, bistablility, normal and abnormal oscillation, respectively, to illustrate how the model parameters of NMM regulate its dynamics, and reveal the oscillation transition mechanism. The results are composed of the following three parts.1. Codimension one bifurcation analysis of excitatory and inhibitory synaptic strengths and synapse connection number is conducted, through which the parameters regions of stability, bistablility, normal and abnormal oscillation are determined, respectively. Further more, through codimension two bifurcation analysis we gain insight into how the interaction of excitatory and inhibitory synaptic strengths and the number of synapse contacts exerts impact of on the dynamics of NMM. The validity of the bifurcation analysis is demonstrated by the simulation results.2. Codimension one bifurcation analysis of intrinsic plasticity parameters of neurons is conducted, respectively, through which the parameters regions of stability, bistablility, normal and abnormal oscillation are determined. Codimension two bifurcation analysis is conducted, which further provides insight into the impact of the interaction of intrinsic plasticity parameters of neurons on the dynamics of NMM.3. Codimension one bifurcation analysis of the input is derived, respectively, through which the parameters regions of stability, bistablility, normal and abnormal oscillation are determined. Codimension two bifurcation analysis further provide insight into the impact of the interaction of the input on the dynamics of NMM.Conclusion:Neural oscillation plays a key role in information transmission and processing in neural system. Through the bifurcation analysis of NMM, the study elucidates the regulating mechanism of the dynamics of NMM underlined by the synaptic and intrinsic plasticity, as well as, it reveals two kinds of oscillation mechanisms:limit cycle oscillation mechanism and input-induced transition between two states of the bistability. Moreover, the study has important value to understand the role of the parameters and input of NMM on the maintenance of brain function and the mechanism of various neurological diseases. Since NMM is typical neural unit, about the study is basis to understand the regulating mechanism of neural activities of higher level brain network.