Cascades of Periodic Solutions in a Neural Circuit With Delays and Slow-Fast Dynamics - Ferrario, Andrea; Rankin, James (Frontiers in Applied Mathematics and Statistics, 2021)
We analyze periodic solutions in a system of four delayed differential equations forced by periodic inputs representing two competing neural populations connected with fast mutual excitation and and slow delayed inhibition. The combination of mechanisms generates a rich dynamical structure that we are able to characterize using slow-fast dissection and a binary classification of states. We previously proved the existence conditions of all possible states 1:1 locked to the inputs and applied this analysis to the tracking the rhythms perceived when listening to alternating sequences of low and high tones. Here we extend this analysis using analytical and computational tools by proving the existence a set of n:1 periodically locked states and their location in parameter space. Firstly we examine cycle skipping states and find that they accumulate in an infinite cascade of period-incrementing bifurcations with increasing periods and scaling constants for decreasing values of the local input strength. Secondly we analyze periodic solutions that alternate between 1:1 locked states that repeat after an integer multiple of the input period. We show that such states accumulate in similar bifurcation cascades with decreasing values of the lateral input strength. The periodic states reported here - emergent behaviours in the model - can be linked to known phenomena in auditory perception that are beyond the original scope of the model's design.