Desynchronization and Oscillatority in Excitable FitzHugh-Nagumo Networks

Study of dynamics of complex networked systems is one of the relevant problems. Networked systems can be in various states, ranging from complete synchronization, when all systems in the network are coherent, to complete desynchronization, i.e. complete incoherence in the functioning of systems. Synchronization phenomenon has already been well studied, namely, the mathematical definitions of synchronization are introduced, algorithms of studying synchronization are proposed, and synchronization conditions of various types of networked systems are established. Whereas a few works are devoted to the study of desynchronization nowadays. This paper introduces output desynchronization notion for networks of nonlinear systems. The definitions about Yakubovich oscillatority are considered and the link between oscillatority and desynchronization in networks of excitable nonlinear systems is established. Excitable systems are stable; therefore, they do not generate oscillations. Adding couplings between such systems can lead to occurrence of oscillations. The conditions about oscillatority in diffusively coupled networks of FitzHugh-Nagumo systems, which are the simplest neuron models, are derived. Firstly, the case of the simplest network of two coupled systems is considered, and afterwards, obtained result is generalized for the case of several systems. Laplace matrix spectrum plays crucial role in dynamics of such networks. The condition that connects the parameters of the uncoupled system in the network and the eigenvalues of the Laplace matrix, is obtained which determines whether the network is oscillatory or not. The number of systems that generate oscillations in such a network depends on the number of eigenvalues of the Laplace matrix that satisfy the obtained conditions. Obtained analytical results are confirmed by simulation. The results of simulation of complete desynchronization in the network, when all systems begin to oscillate, as well as a chimera-like state, in which only a part of the systems oscillates, while the other part are rest, are presented.