Abstract
Networks of excitable elements are widely used to model real-world biological and social systems. The dynamic range of an excitable network quantifies the range of stimulus intensities that can be robustly distinguished by the network response, and is maximized at the critical state. In this study, we examine the impacts of backtracking activation on system criticality in excitable networks consisting of both excitatory and inhibitory units. We find that, for dynamics with refractory states that prohibit backtracking activation, the critical state occurs when the largest eigenvalue of the weighted non-backtracking (WNB) matrix for excitatory units, , is close to one, regardless of the strength of inhibition. In contrast, for dynamics without refractory state in which backtracking activation is allowed, the strength of inhibition affects the critical condition through suppression of backtracking activation. As inhibitory strength increases, backtracking activation is gradually suppressed. Accordingly, the system shifts continuously along a continuum between two extreme regimes—from one where the criticality is determined by the largest eigenvalue of the weighted adjacency matrix for excitatory units, , to the other where the critical state is reached when is close to one. For systems in between, we find that and at the critical state. These findings, confirmed by numerical simulations using both random and synthetic neural networks, indicate that backtracking activation impacts the criticality of excitable networks.
Highlights
Excitable networks have been used to model a range of phenomena in biological and social systems including signal propagation in neural networks [1,2,3,4,5,6], information processing in brain networks [7,8,9], epidemic spread in human population and information diffusion in social networks [10,11,12,13]
We first analyze the model dynamics of EI networks in two extreme conditions where backtracking activation is allowed without restrictions or entirely prohibited. In the former case, the critical state is better characterized by the largest eigenvalue of the weighted adjacency matrix for excitatory nodes, λEW, while in the latter case, the criticality is more related to the largest eigenvalue of the weighted non-backtracking (WNB) matrix for excitatory nodes, λENB
We explore the impact of backtracking activation on the criticality of excitable networks with both excitatory and inhibitory nodes
Summary
Excitable networks have been used to model a range of phenomena in biological and social systems including signal propagation in neural networks [1,2,3,4,5,6], information processing in brain networks [7,8,9], epidemic spread in human population and information diffusion in social networks [10,11,12,13]. It was found that, for a number of excitable network models, the dynamic range is maximized at the critical state [1, 14,15,16,17]. For more general network structures, the criticality for dynamics without refractory state is characterized by the unit largest eigenvalue of the weighted adjacency matrix [14]. It was shown that for dynamics with refractory states, the critical state is governed by the largest eigenvalue of the weighted non-backtracking (WNB) matrix [17]. In these studies, the largest eigenvalue of the weighted adjacency matrix or WNB matrix is used to define the critical state of excitable networks. When inhibitory nodes are introduced, it is unclear how criticality is related to the largest eigenvalues of these two matrices
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