Dynamics of a one-dimensional neural network with a “small world”—topology of synaptic connections

Dynamics of a one-dimensional neural network with external periodic stimulation are investigated numerically. Synaptic connections with constant and random values were assumed. Three ranges of network dynamics—periodic, intermediate and chaotic—were found, depending on the number of short-range synaptic connections k and the gain parameter g, which are the control parameters. The influence of the number of k and additional r long-range connections (shortcuts), typical for a “small-world” network, on the dynamics of the system is discussed. With an increase in the values of g and k, clusters of neurons which do not follow external stimulation appear in the network. For large enough values of g and k, a network has chaotic dynamics. The presence of shortcuts may have either a stabilizing or destabilizing effect on the dynamics of a network. In particular, the presence of shortcuts with certain locations may increase the susceptibility of network to external stimulation