Patterns, Memory and Periodicity in Two-Neuron Delayed Recurrent Inhibitory Loops

J. Ma¹ and J. Wu^2*

¹ Department of Mathematics, University of Houston, Houston TX 77204-3008, USA
² Center for Disease Modeling; Laboratory for Industrial and Applied Mathematics, Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3, Canada

^* Corresponding author. E-mail: wujh@mathstat.yorku.ca

Abstract

We study the coexistence of multiple periodic solutions for an analogue of the integrate-and-fire neuron model of two-neuron recurrent inhibitory loops with delayed feedback, which incorporates the firing process and absolute refractory period. Upon receiving an excitatory signal from the excitatory neuron, the inhibitory neuron emits a spike with a pattern-related delay, in addition to the synaptic delay. We present a theoretical framework to view the inhibitory signal from the inhibitory neuron as a self-feedback of the excitatory neuron with this additional delay. Our analysis shows that the inhibitory feedbacks with firing and the absolute refractory period can generate four basic types of oscillations, and the complicated interaction among these basic oscillations leads to a large class of periodic patterns and the occurrence of multistability in the recurrent inhibitory loop. We also introduce the average time of convergence to a periodic pattern to determine which periodic patterns have the potential to be used for neural information transmission and cognition processing in the nervous system.