Operator-valued free probability applied to recurrent neural networks with heterogeneity in local and global circuit structures

Random connectivity is a common assumption in models of recurrent neural networks, yet biological circuits combine heterogeneous synaptic connections with organization into distinct anatomical regions and functional groups. How does this global structure interact with cellular and local circuit heterogeneity to shape the dimensionality of population activity? In our work, we use operator-valued free probability to derive the covariance spectrum and effective dimension of recurrent networks with heterogeneous gains and block-structured random connectivity. This framework handles the noncommuting matrix operations through which cellular and circuit heterogeneity jointly determine population variability. We show that the linear relationship between effective dimension and firing rate uniformity persists despite global connectivity structure. At fixed total connectivity variance and with identical firing rate distributions across areas, concentrating variance within one area decreases dimension, whereas concentrating it in directed connections between areas increases dimension. Together, these results establish a mechanistic connection between circuit organization and population dimensionality, and demonstrate how operator-valued free probability unifies cellular, local, and global sources of heterogeneity in recurrent networks.