Maximal switchability of centralized networks

TL;DR

This paper studies a class of centralized recurrent networks that can switch between different stable states, including chaotic ones, by adjusting a controller hub's response time, with applications in genetics and neuroscience.

Contribution

It introduces the concept of n/Ns-centrality in networks and demonstrates how slow hub dynamics enable maximal switchability between various attractor states.

Findings

Networks can switch from a rest state to complex dynamics by changing hub response time.

Number of attractors can grow exponentially with network size.

Algorithm provided for designing or learning switchable networks.

Abstract

We consider continuous time Hopfield-like recurrent networks as dynamical models for gene regulation and neural networks. We are interested in networks that contain n high-degree nodes preferably connected to a large number of Ns weakly connected satellites, a property that we call n/Ns-centrality. If the hub dynamics is slow, we obtain that the large time network dynamics is completely defined by the hub dynamics. Moreover, such networks are maximally flexible and switchable, in the sense that they can switch from a globally attractive rest state to any structurally stable dynamics when the response time of a special controller hub is changed. In particular, we show that a decrease of the controller hub response time can lead to a sharp variation in the network attractor structure: we can obtain a set of new local attractors, whose number can increase exponentially with N, the total number of nodes of the nework. These new attractors can be periodic or even chaotic. We provide an algorithm, which allows us to design networks with the desired switching properties, or to learn them from time series, by adjusting the interactions between hubs and satellites. Such switchable networks could be used as models for context dependent adaptation in functional genetics or as models for cognitive functions in neuroscience.