Heteroclinic networks for brain dynamics

Abstract

Heteroclinic networks are a mathematical concept in dynamic systems theory that is suited to describe metastable states and switching events in brain dynamics. The framework is sensitive to external input and, at the same time, reproducible and robust against perturbations. Solutions of the corresponding differential equations are spatiotemporal patterns that are supposed to encode information both in space and time coordinates. We focus on the concept of winnerless competition as realized in generalized Lotka–Volterra equations and report on results for binding and chunking dynamics, synchronization on spatial grids, and entrainment to heteroclinic motion. We summarize proposals of how to design heteroclinic networks as desired in view of reproducing experimental observations from neuronal networks and discuss the subtle role of noise. The review is on a phenomenological level with possible applications to brain dynamics, while we refer to the literature for a rigorous mathematical treatment. We conclude with promising perspectives for future research.

The time-evolving epileptic brain network: concepts, definitions, accomplishments, perspectives

“…Similarly, a better understanding of the longer time scales of brain dynamics that govern the recurrence of seizures would profit from considering mechanisms that can give rise to various long term, fluctuating behavior. We here mention switching phenomena related to the different types of intermittency ( Perez Velazquez et al, 1999 ; Rizzi et al, 2016 ; Pisarchik et al, 2018 ), switching in fast-slow systems ( Kuehn, 2011 ) and in heteroclinic networks ( Kirst and Timme, 2008 ; Aguiar et al, 2011 ; Bick and Field, 2017 ; Morrison and Young, 2022 ; Meyer-Ortmanns, 2023 ) multistability ( Lopes da Silva et al, 2003 ; Takeshita et al, 2007 ; Rothkegel and Lehnertz, 2009 ; Breakspear, 2017 ; Pisarchik and Hramov, 2022 ), and metastability ( Kelso, 2012 ; Tognoli and Kelso, 2014 ; Rossi et al, 2023 ). The validity of such models could be tested if continuous long-term recordings of brain dynamics—covering weeks to months [see, e.g., Weisdorf et al (2019) ; Duun-Henriksen et al (2020) ]—would be publically available.…”