Fig. 1: Capturing signatures of brain states estimating the inside out balance of intrinsic and extrinsic dynamics.

The arrow of time in brain signals contains a precise signature of the reversibility, reflecting how the brain is driven by the level of interaction with the environment. a In thermodynamics, the arrow of time in physical systems can be precisely estimated to provide a measure of non-reversibility and non-equilibrium, i.e. how the system is driven by external forces. The subpanel shows an example of a non-equilibrium system given by a Brownian particle in a moving potential, whose position is controlled externally, moving from one position to another in the forward process. The top of the subpanel shows examples the forward evolving trajectories (light grey lines) with the average (solid grey line). The average trajectory lags behind the centre of the potential (grey dashed line). In contrast (bottom of right subpanel) shows sample backward trajectories (light red lines) and their average (solid red line). Here the average trajectory leads the potential’s centre (red dashed line). This asymmetry in time of forward and backward trajectories (i.e. the differences between the lines for average and the potential’s centre) provides the exact level of non-reversibility/non-equilibrium of the system [adapted from¹⁸]. b The present framework uses this key idea from thermodynamics to extract the arrow of time in brain signals in order to capture the level of interaction between the brain and the environment. c The framework estimates the asymmetry by using pairwise comparisons of forward signals across the whole brain. d This is accomplished by constructing the backward brain signal from the forward signal by creating a reversal of the backward time series in each brain region. The panel shows the notation used for describing this process for a pair of regions. e The panel shows the main principle of the framework for measuring the level of non-reversibility/non-equilibrium through the pairwise level of asymmetry using a time-shifted measure of their correlation. The subpanels show how the shifted correlation captures the causal interactions between two time series where the top example shows strong time dependency while the bottom example shows weak time dependency. This is clearly seen by how the shifted correlation (as a function of the time shift, ∆t) decays more rapidly for signals with weak compared to strong time dependency. The middle subpanels show the same method but now used for comparing forward and reversed regionals pairs of brain signals. The right subpanel shows examples of the time series for a given shift ∆t = T and how the level of non-reversibility is computed as the absolute quadratic difference between the time-shifted correlations between forward and the reversal time series averaged over all pairs. f We applied this framework to the multidimensional time series covering the whole brain. g We created two time-shifted correlation matrices for the forward and reversed time series (at a given shift time point ∆t = T). h The level of non-reversibility/non-equilibrium is given exactly by the distance between the two matrices (see Methods).