Entrainment of competitive threshold-linear networks

Abstract

Neuronal oscillations are ubiquitous in the brain and emerge from the combined activity of the participating neurons (or nodes), the connec- tivity and the network topology. Recent neurotechnological advances have made it possible to interrogate neuronal circuits by perturbing one or more of its nodes. The response to periodic inputs has been used as a tool to identify the oscillatory properties of circuits and the flow of information in networks. However, a general theory that explains the underlying mechanisms and allows to make predictions is lacking beyond the single neuron level. Threshold-linear network (TLN) models describe the activity of con- nected nodes where the contribution of the connectivity terms is lin- ear above some threshold value (typically zero), while the network is disconnected below it. In their simplest description, the dynamics of the individual nodes are one-dimensional and linear. When the nodes in the network are neurons or neuronal populations, their activity can be interpreted as the firing rate, and therefore the TLNs represent fir- ing rate models [1]. Competitive threshold-linear networks (CTLNs) are a class of TLNs where the connectivity weights are all negative and there are no self- connections [2,3]. Inhibitory networks arise in many neuronal systems and have been shown to underlie the generation of rhythmic activity in cognition and motor behavior [4,5]. Despite their simplicity, TLNs and CTLNs produce complex behavior including multistability, peri- odic, quasi-periodic and chaotic solutions [2,3,6]. In this work, we consider CTLNs with three or more nodes and cyclic symmetry in which oscillatory solutions are observed. We first assume that an external oscillatory input is added to one of the nodes and, by defining a Poincaré map, we numerically study the response proper- ties of the CTLN networks. We determine the ranges of input ampli- tude and frequency in which the CTLN is able to follow the input (1:1 entrainment). For this we define local and global entrainment measures that convey different information. We then study how the entrainment properties of the CTLNs is affected by changes in (i) the time scale of each node, (ii) the number of nodes in the network, and (iii) the strength of the inhibitory connections. Finally, we extend our results to include other entrainment scenarios (e.g., 2:1) and other net- work topologies.