Turing instability in the reaction-diffusion network

It is an established fact that a positive wave number plays an essential role in Turing instability. However, the impact of a negative wave number on Turing instability remains unclear. Here, we investigate the effect of the weights and nodes on Turing instability in the FitzHugh-Nagumo model, and theoretical results reveal genesis of Turing instability due to a negative wave number through the stability analysis and mean-field method. We obtain the Turing instability region in the continuous media system and provide the relationship between degree and eigenvalue of the network matrix by the Gershgorin circle theorem. Furthermore, the Turing instability condition about nodes and the weights is provided in the network-organized system. Additionally, we found chaotic behavior because of interactions between I Turing instability and II Turing instability. Besides, we apply this above analysis to explaining the mechanism of the signal conduction of the inhibitory neuron. We find a moderate coupling strength and corresponding number of links are necessary to the signal conduction.