Networks of Mixed Canonic-Dissipative Systems and Dynamic Hebbian Learning

We consider a collection {O_k}_{k=1}^N of interacting parametric mixed canonical-dissipative systems, (MCD). Each individual Ok, exhibits, in absence of interaction, a limit cycle L_k on which the orbit circulation is parameterized by w_k(t). The underlying network defining the interactions between the O_k’s is assumed to possess a diffusive Laplacian matrix. For each O_k, we construct a class of position- and velocity-dependent interactions which lead to a dynamic learning process of the Hebbian type (DHL). More precisely, the interactions affect the circulation parameterization w_k(t) and the DHL mechanisms manifests itself by asymptotically driving the system towards a consensual (oscillatory) global state in which all O_k share a common circulation parameterization !c. It is remarkable that for our class of interactions, we are able to analytically calculate w_c which, in our case, is independent of the topology of the connecting network. However, the coupling network topology explicitly controls the relaxation rate via the spectral gap of the underlying adjacency matrix (i.e. the so called Fiedler number of the associated graph). Finally, we report several numerical illustrations which enable to observe the DHL mechanisms at work and confirm our theoretical assertions.