Stability and instability of a neuron network with excitatory and inhibitory small-world connections.
This study considers a delayed neural network with excitatory and inhibitory shortcuts. The global stability of the trivial equilibrium is investigated based on Lyapunov's direct method and the delay-dependent criteria are obtained. It is shown that both the excitatory and inhibitory shortcuts decrease the stability interval, but a time delay can be employed as a global stabilizer. In addition, we analyze the bounds of the eigenvalues of the adjacent matrix using matrix perturbation theory and then obtain the generalized sufficient conditions for local stability. The possibility of small inhibitory shortcuts is helpful for maintaining stability. The mechanisms of instability, bifurcation modes, and chaos are also investigated. Compared with methods based on mean-field theory, the proposed method can guarantee the stability of the system in most cases with random events. The proposed method is effective for cases where excitatory and inhibitory shortcuts exist simultaneously in the network.
College of Mathematics, Jilin University, 2699 Qianjin Street, Changchun 130012, PR China.
Dongyuan
D
Yu
College of Mathematics, Jilin University, 2699 Qianjin Street, Changchun 130012, PR China. Electronic address: xuxu@jlu.edu.cn.
Xu
X
Xu
College of Mathematics, Jilin University, 2699 Qianjin Street, Changchun 130012, PR China.
Jing
J
Zhou
Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Hong Kong, PR China.
Eric
E
Li
D009433:Neural Inhibition; D016571:Neural Networks, Computer; D009474:Neurons / Q000502:physiology
Bifurcation; Delay-dependent global stability; Matrix perturbation theory; Mean-field theory; Small-world network