Third Order Edge Statistics Reveal Curvature Dependency

While neural circuits in vivo connect many thousands of different cells, statistical and measurement complexity limit functional data to pairwise interactions. This is especially important in visual cortex, superficial V1, where pairwise edge co-occurances have supported an association field model for long-range horizontal connections (Fig.1(c)). However, how well do such low-order models capture the (higher-order) neural structure? Computationally it is known that such second-order (pairwise) models can account for the mean of the connection distribution, but fail to predict the variance in connections across cells. We developed a method for estimating a third-order statistic for edge element interactions by conditioning the second-order interaction on a third element. Diffusion maps are used to reveal a global organization of the data, and embedded points that cluster together model the connections. A significant asymmetry emerges: (i) Excitatory (third-order) connections depend on curvature. This dependence models co-circularity and predicts both the mean and the variance in population statistics of excitatory connections (Fig.1(i)). (ii) Inhibitory connections are more uniformly distributed across orientaton and position. Consistent with axonal projections of inhibitory interneurons in V1, there is no dependency on curvature (Fig. 1(g)).