Can symmetric whole-brain models reproduce asymmetric summary statistics?

Determine whether whole-brain dynamical models with symmetric coupling matrices (A_ij = A_ji), such as those constrained by undirected structural connectivity, can reproduce asymmetric summary statistics of neuroimaging time series (e.g., time-lagged correlations or directed functional connectivity measures), and, if so, specify the precise conditions and mechanisms under which this is possible; otherwise, prove impossibility results.

Background

In standard whole-brain modeling pipelines, the structural connectivity used to constrain interactions is typically undirected, leading to symmetric coupling and time-reversible dynamics. However, many empirical summary statistics of neural recordings (e.g., time-lagged correlations, directed functional metrics) are asymmetric and reflect non-equilibrium dynamics.

This raises the fundamental question of whether symmetric models can, in principle, fit or generate asymmetric summary statistics or whether directionality must be explicitly encoded in the coupling structure. The answer has direct implications for model selection and inference in human neuroimaging where directed structural data are often unavailable.