Decomposing the modulation of interactions between neuronal populations.

Identifying subpopulations of neurons that interact with each other from simultaneous recordings of populations of many neurons is key for understanding across-brain communication with cellular resolution. Recent work identified communication subspaces, which capture additive interactions between pairs of high-dimensional neural populations through a small number of source and target activity patterns. However, no current method captures how a third, potentially multivariate variable - such as behavioral state or the activity of a third population - modulates these interactions. Here we extend the communication subspace framework by parameterizing modulation as a low-rank tensor. This identifies multiplicative interaction channels (MICs), defined as triplets of source, target, and modulator activity patterns, in which the modulator pattern gates the source-target interaction. We derive MICs as a bilinear perturbation of reduced-rank regression. We develop a hierarchical fitting pipeline and provide a closed-form decomposition that quantifies whether modulation reshapes the modulator-averaged baseline interaction, recruits private dimensions of one population, or opens new interactions. In simulations, MICs reliably recover the presence and geometry of ground-truth modulation even in the high-dimensional, low-sample regime. Applying MICs to simultaneous calcium imaging of prefrontal axons and interneurons in the visual cortex revealed that behavioral state asymmetrically modulates top-down interactions, reconfiguring the patterns of prefrontal projections that interact with a stable set of visual interneuron activity patterns. By providing an efficient and compact characterization of modulatory interactions, MICs enable asking new questions about how potentially high-dimensional variables shape interactions between neural populations.