Possibilities of introducing different functional circuits on top of a structural neuron triplet: Where do the gains lie?

Instigated by the research on clusterization phenomena in complex neural networks, we study a triplet of bursting Rulkov map neurons connected via inhibitory synapses with delay. It is demonstrated how on a background of structural motif one can build different types of functional circuits. The approach is based on utilizing the properties of the chemical synapses, whose gating is modeled by the fast threshold modulation, in conjunction with the phase plane analysis, allowing the system state to be represented in terms of maps the neurons reside on. For both the dynamical configurations, monitoring the layout of active neurons, and the functional motifs, following the maps where the synchronized neurons lie, we establish a one-on-one correspondence between sequences in the time series and the triads, making up the subgraphs of the original graph. By introducing the appropriate sets of quantities, one obtains not only the distributions of triads as a function of synaptic parameters, but is also able to identify a distinct triad whose presence may be viewed as a signature of the burst synchronization process. In another setup, the regularization of burst cycles for an arbitrary neuron is explained by classifying all the bursts as long or short, with their fractions linked to the abundances of triads under variation of synaptic parameters.