Effects of symmetry on the structural controllability of neural networks: A perspective

Abstract

The controllability of a dynamical system or network describes whether a given set of control inputs can completely exert influence in order to drive the system towards a desired state. Structural controllability develops the canonical coupling structures in a network that lead to un-controllability, but does not account for the effects of explicit symmetries contained in a network. Recent work has made use of this framework to determine the minimum number and location of the optimal actuators necessary to completely control complex networks. In systems or networks with structural symmetries, group representation theory provides the mechanisms for how the symmetry contained in a network will influence its controllability, and thus affects the placement of these critical actuators, which is a topic of broad interest in science from ecological, biological and man-made networks to engineering systems and design.

A practical guide to methodological considerations in the controllability of structural brain networks

“…Many (but perhaps not all) of these developments could be helpful in the study of the mind and brain. Efforts have recently revealed a relation between controllability and symmetry [89][90][91], which could prove useful in determining the impact of bilateral and other symmetries on neural dynamics. The field has begun considering multiobjective functions, tradeoffs, and constraints in control [92,93], in addition to probing a system's potential for control via local topological information [94].…”

An overview of structural systems theory

“…In Whalen et al (2015), we can find a numerical and group representational framework to quantify the observability and controllability of nonlinear networks with explicit symmetries, showing the relationship between symmetries and nonlinear measures of observability and controllability. The authors apply this work to neural networks in Whalen et al (2016). In Menara et al (2019), the authors show that (symmetric) structural controllability can be assessed by graph-theoretic elements similar to those previously proposed to verify (classic) structural controllability.…”

“…In Murota (2009aMurota ( , 2012, the parametric dependencies have been accounted for, using the notion of mixed matrices, where the entries could be zero/nonzero or fixed constant, often capturing the network dependencies (or, generally speaking, losses of degrees of freedom). More recently, in Menara, Bassett, and Pasqualetti (2019), Mousavi, Haeri, and Mesbahi (2018), Whalen, Brennan, Sauer, and Schiff (2015) and Whalen, Brennan, Sauer, and Schiff (2016), the authors examine the structural controllability criterion when the dependency of the parameters is given by the symmetry of the system's autonomous matrix. In Whalen et al (2015), we can find a numerical and group representational framework to quantify the observability and controllability of nonlinear networks with explicit symmetries, showing the relationship between symmetries and nonlinear measures of observability and controllability.…”

Structural Systems Theory: an overview of the last 15 years

“…In [196,197,198,199], the authors examine the structural controllability criterion when the dependency of the parameters is given by the symmetry of the system's autonomous matrix. For instance, in [198], the authors show that (symmetric) structural controllability can be assessed by graph theoretic elements similar to those previously proposed to verify structural controllability.…”