Nonequilibrium physics of brain dynamics

TL;DR

This article surveys how nonequilibrium physics informs brain dynamics, arguing that time-irreversibility and entropy production are intrinsic to neural activity and vary with conscious state and cognitive load. It articulates a dual approach: model-based analyses (linear Langevin, Kuramoto, Hopf, neural-field theories) and model-free measures (time-lagged correlations, ML arrows-of-time, symbolization, and visibility graphs) to quantify irreversibility in continuous and discrete neural data. It further develops higher-order and information-decomposition methods (DiMViGI, PID-based Phi-ID) and explores spike-train dynamics via asymmetric kinetic Ising models, including exact and mean-field inferences and irreversibility decompositions, linking nonequilibrium dynamics to neural computation. The review culminates with nonequilibrium neural computation framed through Bayesian mechanics and the free-energy principle, offering a cohesive, multi-scale framework with potential to improve understanding of cognition and consciousness beyond equilibrium models.

Abstract

Information processing in the brain is coordinated by the dynamic activity of neurons and neural populations at a range of spatiotemporal scales. These dynamics, captured in the form of electrophysiological recordings and neuroimaging, show evidence of time-irreversibility and broken detailed balance suggesting that the brain operates in a nonequilibrium stationary state. Furthermore, the level of nonequilibrium, measured by entropy production or irreversibility appears to be a crucial signature of cognitive complexity and consciousness. The subsequent study of neural dynamics from the perspective of nonequilibrium statistical physics is an emergent field that challenges the assumptions of symmetry and maximum-entropy that are common in traditional models. In this review, we discuss the plethora of exciting results emerging at the interface of nonequilibrium dynamics and neuroscience. We begin with an introduction to the mathematical paradigms necessary to understand nonequilibrium dynamics in both continuous and discrete state-spaces. Next, we review both model-free and model-based approaches to analysing nonequilibrium dynamics in both continuous-state recordings and neural spike-trains, as well as the results of such analyses. We briefly consider the topic of nonequilibrium computation in neural systems, before concluding with a discussion and outlook on the field.

Figures (13)

• Figure 1: Nonequilibrium brain dynamics. A. Nonequilibrium dynamics are characterised by stationary probability currents, which occur when joint transition probabilities contain asymmetry i.e. $P_{ij}\neq P_{ji}$, that cause a system to violate detailed balance and produce irreversible trajectories. B. To estimate irreversibility, one must compare the statistical properties of a stochastic trajectory with its time-reversal . C. When this is applied to neural recordings, differences between experimental conditions appear, such as an increase in irreversibility and entropy production in task compared to rest. Adapted from nartallokaluarachchi2024broken.
• Figure 2: Neurons, populations and dynamics. A. A schematic diagram of a single neuron. Adapted from Notjim by CC BY-SA 3.0 (https://commons.wikimedia.org/w/index.php?curid=4824168). B. Action potentials from different dynamic regimes. These are simulated using the Izhikevich model Izhikevich2006dynamical. C. Neurons are organised into networks of connected units, which produce collective activity. This can be visualised as a spike-train in a raster plot. Drawing by Santiago Ramón y Cajal (Public Domain). D. The average activity of a population or network or neurons can be approximated with a mean-field model. These models have an array of dynamic regimes. These are simulated from the Jansen-Rit neural-mass model Coombes2023neurodynamics.
• Figure 3: Symmetric whole-brain modelling. A. In order to model the brain, we select a parcellation into discrete brain areas. B. From functional imaging data, such as fMRI, nodal activity in the form of multivariate time-series can be extracted. From these data, we obtain a functional connectivity matrix. C. Anatomical fibre tracing, such as diffusion tensor imaging, allows the reconstruction of connectivity between brain areas. This yields the structural connectivity matrix. D. A whole-brain model is then defined by selecting a network dynamical system, such as a neural mass model or Hopf oscillator network. This model is constrained by the empirical structural network. The parameters of each regional dynamic model are fit by minimising the difference between the functional connectivity simulated from the model, and that calculated from the empirical data. Adapted from patow2024wholebrain.
• Figure 4: The Helmholtz-Hodge decomposition. Stationary processes admit a decomposition of their drift field, and by extension their trajectories, into irreversible and reversible components. The irreversible component drives rotation around the stationary distribution whilst the reversible component maintains the process at stationarity. The top row shows the decomposition of the Ornstein-Uhlenbeck process which is a linear process with a Gaussian stationary distribution. The bottom row shows the decomposition of the stochastic planar limit cycle which is a nonlinear process with a 'Mexican-hat-type' stationary distribution. We show examples of decomposed trajectories depicted over the drift components (purple arrows) and the stationary density. On the right, we show the stationary density alongside the stationary probability flux (arrows). Details on the processes and decomposition are in Ref. nartallokaluarachchi2024decomposing.
• Figure 5: Asymmetric effective networks. Using nonequilibrium model-based approaches, one can infer an asymmetric effective network of connections between brain regions. This network can then be studied using the tools of directed networks such as trophic analysis to discern the hierarchical organisation of brain regions and functional sub-networks. Analysis of hierarchical structure in directed networks is also closely linked to the discrete HHD defined earlier mackay2020directedstrang2022networkhhd. Figure is adapted from results in Ref. nartallokaluarachchi2024broken.