Mathematical framework for testing wiring specificity in cortical connectomes

We present a mathematical framework for formulating and testing rules of synaptic organization on both sparse and dense connectomics data. Our approaches will make it possible to implement hypotheses of synaptic organization in terms of mathematically formulated rules. We generated a structurally dense model of the rat barrel cortex and formulated a null hypothesis rule that synaptic wiring is based on axo-dendritic overlap. This null hypothesis states that synapses form (1) proportional to the locally available pre- and postsynaptic target structures, (2) locally random and (3) globally independent. The rule predicts distributions of pair-wise connectivity that are non-Gaussian and non-Poisson. We show that (sparse) pair-wise connectivity measurements obtained with different experimental methods cannot reject the null hypothesis. The rule predicts a wide range of 2nd and higher order connectivity patterns. These predictions can be used in the future to reject the null hypothesis and to identify wiring specificity that cannot be explained by axo-dendritic overlap. The framework will make it possible to (1) interpret sparse or dense connectivity measurements in a rule-based context, (2) identify which structural features are predictive of synaptic connections, (3) quantify how well a connectivity rule is constrained by data and (4) provide unbiased statistical tools for determining which set of rules is most consistent with empirical data