Backtracking activation impacts the criticality of excitable networks

TL;DR

This paper investigates how backtracking activation influences the criticality of excitable networks, revealing that inhibition strength and refractory states determine whether the system's critical point aligns with eigenvalues of the adjacency or non-backtracking matrices.

Contribution

It introduces a novel analysis of backtracking activation effects on network criticality, highlighting the role of eigenvalues and inhibition in excitable systems.

Findings

Critical state occurs when the largest eigenvalue of the WNB matrix is near one.

Inhibition strength modulates the impact of backtracking activation on criticality.

Numerical simulations confirm the theoretical predictions across different network types.

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, $\lambda^E_{NB}$, 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, $\lambda^E_W$, to the other where the critical state is reached when $\lambda_{NB}^E$ is close to one. For systems in between, we find that $\lambda^E_{NB}<1$ and $\lambda^E_W>1$ 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.