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We describe a mode of signal transmission among neuronnodes in a finite deterministic network. In this mode, an activated node passes a signal to its nearest neighbors and becomes deactivated, unless it concurrently receives the signal from a nearest neighbor. By means of matrix representation, we show that a connected network equipped with this mode of signal transmission converges to one of two states: 1) System-Wide Synchronization (SWS), wherein all nodes are activated; and 2) Subgroup Alternation (SGA), wherein two subsets of nodes alternate on and off. Conditions on wiring configuration required for SWS are presented. We then focus on finite random networks in which the presence of wiring between any two nodes is stochastically determined. We consider two optimal design problems: 1) How can we allocate wiring probabilities subject to a budget constraint in order to maximize the probability of achieving SWS? 2) What impact does a robustness criterion have on the optimal wiring structure? We implement the simulated annealing algorithm to find such optimal probability allocations and present our results. Under a robustness requirement, we show that robust random networks require a larger budget and significantly more triangular-loop sub-structures.

Optimal and robust design for efficient system-wide synchronization in networks of randomly-wired neuron-nodes