Analyzing asymmetry in brain hierarchies with a linear state-space model of resting-state fMRI data

This study challenges the traditional focus on zero-lag statistics in resting-state functional magnetic resonance imaging (rsfMRI) research. Instead, it advocates for considering time-lag interactions to unveil the directionality and asymmetries of the brain hierarchy. Effective connectivity (EC), the state matrix in dynamical causal modeling (DCM), is a commonly used metric for studying dynamical properties and causal interactions within a linear state-space system description. Here, we focused on how time-lag statistics are incorporated within the framework of DCM resulting in an asymmetric EC matrix. Our approach involves decomposing the EC matrix, revealing a steady-state differential cross-covariance matrix that is responsible for modeling information flow and introducing time-irreversibility. Specifically, the system's dynamics, influenced by the off-diagonal part of the differential covariance, exhibit a curl steady-state flow component that breaks detailed balance and diverges the dynamics from equilibrium. Our empirical findings indicate that the EC matrix's outgoing strengths correlate with the flow described by the differential cross covariance, while incoming strengths are primarily driven by zero-lag covariance, emphasizing conditional independence over directionality. Modeling large-scale brain dynamics offers insight into the main principles of brain self-organization. In particular, the identification of traces of nonequilibrium steady-state dynamics also at the mascroscale level has been recently linked to the presence of intrinsic brain networks. Quantifying these aspects is generally limited by numerical difficulties. However, for resting-state BOLD data, a linear stochastic state-space model has demonstrated efficacy, simplifying analysis. Specifically, the asymmetric structure of effective connectivity, that is, the state interaction matrix, directly reflects nonequilibrium steady-state dynamics and time-irreversibility. By disentangling this asymmetry, we quantified departure from equilibrium and discerned primary directions of information propagation, identifying brain regions as sources or sinks.