# Adaptation of oscillatory systems in networks — a learning signal approach

## Abstract

We consider a network of coupled periodic stable signals (PSS) interacting through the gradient of a coupling potential. Each PSS has its own set of parameters, characterizing the time scale of the signal and its shape. The are allowed to modify their values (i.e. to adapt) by introducing adaptive mechanisms on them. Together with the state variable interactions, the adaptive mechanisms drive all PSS towards a consensual oscillatory state where they all have a common, constant set of parameters Once reached, the consensual oscillatory state remains invariant to the interactions. This implies that if the interactions are removed, all PSS continue to deliver the consensual signal. This situation is to be contrasted with classical synchronization problems where common dynamical patterns are attained and maintained thanks to the interactions. Hence, if the interactions are removed, all PSS converge back towards their individual behavior. The resulting value is analytically calculated. It does not depend on the network’s topology. However, the conditions for convergence do depend on the connectivity of the network and on the coupling potential.

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