Random innervation of cerebellar Purkinje cells as a substrate for diverse representational learning

Although the cerebellar microcircuit is among the most well-characterized systems for studying neural computation, recent connectomic analyses highlight deviations from canonical models, raising fundamental questions about connectivity, function, and learning in the system. A key feature of the circuit is the convergence of a vast number of parallel fibers (PFs) onto Purkinje cells (PCs). Prevailing models of cerebellar computation assume all-to-all connectivity at this intersection, whereby a single PC "samples" from all PFs. However, experimental evidence suggests that each PC is innervated by only a subset of accessible PFs. This partial sampling creates differential connectivity that may reflect an organizational principle of cerebellar learning, but its implications are poorly understood. Based on electron microscopy (EM) reconstructions, we show that PF innervation of PCs is largely consistent with a Bernoulli model where connections are randomly and independently distributed within anatomical constraints. In further support of a random model, we observe that connections of the ascending branches of granule cells are not predictive of connections of their PFs, nor is connectivity correlated across separate spatial encounters of a PF with the same PC. In a model of the cerebellar circuit, we then address the possibility that partial connectivity is a substrate of learning, i.e., a fixed, random mask that enforces diversity between PCs. We find that when considering "ensembles" of PCs, random partial connectivity can indeed outperform all-to-all connectivity. Our results provide a theoretical framework for understanding the role of partial connectivity between cerebellar PFs and PCs and may have implications for cerebellum-like systems and beyond.