Dynamics of four-neuron recurrent inhibitory loop with state-dependent time delays

In previous works, recurrent inhibitory loops with state-dependent propagation delays, firing process and absolute refractory period have been successfully employed to solve capacity-simplicity dilemma in associative memory attractors networks. But in realistic networks, usually there are more than two or three neurons. In order to disclose their natures, in this paper, we consider the dynamics of periodic patterns mainly in a four-neuron recurrent inhibitory loop. We explicitly address how to give rise to its enormous periodic patterns and obtain their existence conditions. At last, we further execute numerical simulations to address the difference from the five-neuron loop and demonstrate similar periodic patterns. Even for smaller τ, the coexistence of three periodic patterns is found in four-neuron loop and that of four periodic patterns is discovered in five-neuron loop. New periodic patterns are also generated and the maximum values of τ for the existence of possible periodic patterns also decrease with the increment of the number of neurons in loops. They will improve loops performance greatly. Our research shows that delays are significant and remarkable for the dynamics of recurrent inhibitory loops, since loops with more neurons have more complicated dynamical behavior and loops performance are also enhanced due to the the increase of synaptic delay and propagation delay.