Feedback-induced stationary localized patterns in networks of diffusively coupled bistable elements

TL;DR

This paper explores how feedback mechanisms influence the formation of localized stationary patterns in networks of bistable elements, revealing the emergence of active subnetworks and the impact of feedback strength and node degree.

Contribution

It develops an approximate analytical theory for localized patterns in regular trees and demonstrates their existence in large random networks, highlighting the role of feedback in pattern organization.

Findings

Localized stationary activation patterns are observed in various network types.

The size of active subnetworks decreases with increasing feedback strength.

Strong feedbacks organize active nodes into tree-like subnetworks.

Abstract

Effects of feedbacks on self-organization phenomena in networks of diffusively coupled bistable elements are investigated. For regular trees, an approximate analytical theory for localized stationary patterns under application of global feedbacks is constructed. Using it, properties of such patterns in different parts of the parameter space are discussed. Numerical investigations are performed for large random Erd\"os-R\'enyi and scale-free networks. In both kinds of systems, localized stationary activation patterns have been observed. The active nodes in such a pattern form a subnetwork, whose size decreases as the feedback intensity is increased. For strong feedbacks, active subnetworks are organized as trees. Additionally, local feedbacks affecting only the nodes with high degrees (i.e. hubs) or the periphery nodes are considered.