Multistate and Multistage Synchronization of Hindmarsh-Rose Neurons With Excitatory Chemical and Electrical Synapses

The new phenomena of the multistate synchronization of Hindmarsh-Rose (HR) neurons with both excitatory chemical and electrical synapses over the complex network are analytically studied. The regions for coupling strengths to achieve local synchronization are explicitly obtained. Such regions are characterized by the second largest eigenvalue λ₂ of the electrical connection matrix and the number k of chemical signals each neuron receives. The dynamics of the multistate synchronization includes the coexistence of stable regular bursting and periodic/steady-state behaviors. Our theory predicts that recurrent networks formed by a certain cell types in layers 4 and 6 in cat area 17 could lead to multistate synchronization. These are in contrast with coupled oscillator systems or coupled map lattices where only single-state synchronization is found. It should also be noted that if the parameters of HR neurons are chosen resulting in an irregular (chaotic) bursting, then the coexistence state would contain chaotic attractor. Our method employed here is quite general. For instance, it can be immediately applied to other coupled nervous systems such as FitzHugh-Nagumo and Morris-Lecar nervous systems. The analytical tools and concepts needed include coordinate transformations, matrix measures, monotone dynamics and time averaging estimates.