Capturing the emergent dynamical structure in biophysical neural models
Authors: Milinkovic Borjan , Barnett Lionel , Carter Olivia , Seth Anil K. , Andrillon Thomas 
Published: October 2024

Complex neural systems can display structured emergent dynamics. Capturing this structure remains a significant scientific challenge. Using information theory, we applyDynamical Independence(DI) to uncover the emergent dynamical structure in a minimal 5-node biophysical neural model, shaped by the interplay of two key aspects of brain organisation: integration and segregation. In our study, functional integration within the biophysical neural model is modulated by a global coupling parameter, while functional segregation is influenced by adding dynamical noise, which counteracts global coupling. DI defines a dimensionally-reducedmacroscopic variable(e.g., a coarse-graining) as emergent to the extent that it behaves as an independent dynamical process, distinct from the micro-level dynamics. We measure dynamical dependence (a departure from dynamical independence) for macroscopic variables across spatial scales. Our results indicate that the degree of emergence of macroscopic variables is relatively minimised at balanced points of integration and segregation and maximised at the extremes. Additionally, our method identifies to which degree the macroscopic dynamics are localised across microlevel nodes, thereby elucidating the emergent dynamical structure through the relationship between microscopic and macroscopic processes. We find that deviation from a balanced point between integration and segregation results in a less localised, more distributed emergent dynamical structure as identified by DI. This finding suggests that a balance of functional integration and segregation is associated with lower levels of emergence (higher dynamical dependence), which may be crucial for sustaining coherent, localised emergent macroscopic dynamical structures. This work also provides a complete computational implementation for the identification of emergent neural dynamics that could be applied both in silico and in vivo.

Author summary

Understanding how complex neural systems give rise to emergent macroscopic patterns is a central challenge in neuroscience. Emergence, where macroscopic structures appear from underlying microscopic interactions, plays a crucial role in brain function, yet identifying the specific dynamics involved remains elusive. Traditionally, methods have quantified the extent of emergence but have struggled to pinpoint the emergent dynamical structure itself. In this study, we develop and apply a method, based on a quantity called Dynamical Independence (DI), which simultaneously captures the extent of emergence and reveals the underlying dynamical structure in neurophysiological data. Using a minimal 5-node biophysical neural model, we explore how a balance between functional integration and segregation—two key organisational principles in the brain—affects emergent macroscopic dynamics. Our results show that a finely balanced system produces highly localised, coherent macroscopic structures, while extreme deviations lead to more distributed, less localised dynamics. This work provides a computational framework for identifying emergent dynamical structure in both theoretical models and potentially in empirical brain data, advancing our understanding of the brain’s complex organisation across higher-order scales.