Turing instability induced by random network in FitzHugh-Nagumo model.

• The approximate range of eigenvalue of the network matrix is shown. • The instability condition about diffusion and the connection probability in the network-organized system are derived. • The estimated range of connection probability is also deduced. • Mechanism of spiking of the neuron can be explained by using diffusion network and matrix theory. Although there is general agreement that the network plays an essential role in the biological system, how the connection probability of network affects the natural model(Especially neural network) is poorly understood. In this paper, we show the impact of the network on Turing instability in the FitzHugh-Nagumo(FN) model. Then we obtain the condition of how the Turing bifurcation, saddle-node bifurcation, and Turing instability occur. By using the Gershgorin circle theorem, we investigate the relationship between degree and eigenvalue of the network matrix, and obtain the approximate range of eigenvalue of the network matrix. Also, We derive the instability condition about diffusion and the connection probability in the network-organized system. And then we obtain the estimated range of connection probability. Meanwhile we apply these results to explaining the spiking of neuron and find this system has rich dynamics behavior. Finally, the numerical simulation verifies our analytical results. [ABSTRACT FROM AUTHOR]