Abstract
Focusing on systems of sinusoidally coupled active rotators, we study the emergence and stability of periodic collective oscillations for systems of identical excitable units with repulsive all-to-all interaction. Special attention is put on splay states and two-cluster states. Recently, it has been shown that one-parameter families of such systems, containing the parameter values at which the Watanabe–Strogatz integrability takes place, feature an instantaneous non-local exchange of stability between splay and two-cluster states. Here, we illustrate how in the extended families that circumvent the Watanabe–Strogatz dynamics, this abrupt transition is replaced by the “gradual transfer” of stability between the 2-cluster and the splay states, mediated by mixed-type solutions. We conclude our work by recovering the same kind of dynamics and transfer of stability in an ensemble of voltage-coupled Morris–Lecar neurons.
Highlights
One of the most prolific applications of the theory of dynamical systems lies in the field of neuroscience
Class I excitability translates to a dynamical system, being close to a saddle-node homoclinic bifurcation
We reviewed previous results concerning two important types of generic collective oscillatory solutions: the splay states and the periodic 2-cluster states
Summary
One of the most prolific applications of the theory of dynamical systems lies in the field of neuroscience. If the identical excitable units are decoupled, there exists a stable collective equilibrium for the ensemble dynamics, with every element located at its state of rest. Attractive coupling binds these units stronger together, enhancing the stability of the collective equilibrium. We are interested in systems of N identical active rotators, coupled in such a way that the interaction (i) is pairwise, (ii) depends on the difference between the phases, (iii) is all-to-all, and (iv) is repulsive. Such a system can be described by the equation
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