Dynamics of Pulse-Coupled Inhibitory Networks

Abstract

In this study, we try to characterize the role inhibition plays in the generation of spatiotemporal patterns that underlie the encoding of odours by the olfactory system and most of the information processing in the cerebral cortex. In the study of network dynamics, the effect of inhibition can be divided into 2 interdependent parts: the connectivity among various neurons (i.e. the structure of the network) and the magnitude of inhibition in the network. We build our work on earlier studies which draw a connection between the dynamics of inhibitory networks and a structural property i.e. the vertex colorings of the inhibitory graph. We find an ideal model for our study in the game of Sudoku. The Sudoku puzzle can be mapped to a graph coloring problem so that the solutions to the empty Sudoku grid are mapped to the 9-colorings of the Sudoku graph. There are O(10^9) solutions of the Sudoku puzzle and thus, there are O(10^9) 9-colorings. We wire a network of pulse-coupled neurons that in principle could generate all the 9-colorings of the Sudoku graph. We show empirically that the asymptotic dynamics of the Sudoku network can be classified in terms of the solutions of the empty Sudoku puzzle or equivalently the 9-colorings of the Sudoku graph. We find that the dynamical system has a measure of closeness between colorings i.e. perturbations to the state representing a Sudoku solution lead you to other near by solutions. We propose that these dynamical patterns of excitatory-inhibitory pulse coupled networks which are characterized by the colorings of the graph underlying the inhibitory network are the spatiotemporal patterns involved in odour representation and memory. The huge number of such patterns found in our system point to a very high encoding capacity of neuronal networks. The sense of locality in response to perturbations to the state in our system is similar to the associative nature of memory. Lastly, we focus our attention on the role of the magnitude of inhibition on the dynamics. We vary the coupling strengths systematically to find that our system shows a maximization of encoding capacity for a certain ratio of excitation to inhibition.