  Recent advancements in measurement techniques have resulted in an increasing
amount of data on neural activities recorded in parallel, revealing largely
heterogeneous correlation patterns across neurons. Yet, the mechanistic origin
of this heterogeneity is largely unknown because existing theoretical
approaches linking structure and dynamics in neural circuits are restricted to
population-averaged connectivity and activity. Here we present a systematic
inclusion of heterogeneity in network connectivity to derive quantitative
predictions for neuron-resolved covariances and their statistics in spiking
neural networks. Our study shows that the heterogeneity in covariances is not a
result of variability in single-neuron firing statistics but stems from the
ubiquitously observed sparsity and variability of connections in brain
networks. Linear-response theory maps these features to the effective
connectivity between neurons, which in turn determines neuronal covariances.
Beyond-mean-field tools reveal that synaptic heterogeneity modulates the
variability of covariances and thus the complexity of neuronal coordination
across many orders of magnitude.
