Figure 2.

Connectivity reconstruction via detection of correlation triangles and classification of indirect and apparent links. (1) Given three neurons 1, 2, and 3, our algorithm searches, in time, for all correlations between them by computing the pairwise correlation functions R_1,2, R_2,3, and R_3,1. In this representative example, the algorithm detects two correlation peaks for each pair of neurons and associates the corresponding delays of interactions (${\tau }_{1,2}^{-1}$, ${\tau }_{1,2}^{+1}$, ${\tau }_{2,3}^{-1}$, ${\tau }_{2,3}^{-2}$, ${\tau }_{3,1}^{+1}$, ${\tau }_{3,1}^{+2}$), which are defined by the location of the peaks with respect to the origin of the x axis. Eight possible combinations of interactions can occur in time between the three pairs of neurons. These correspond to the eight correlation triangles shown on the right side of the panel. (2) Among the correlation triangles, the algorithm detects 2 critical cases ($|{\tau }_{1,2}^{-1}+{\tau }_{2,3}^{-2}+{\tau }_{3,1}^{+1}|<ϵ$ and $|{\tau }_{1,2}^{+1}+{\tau }_{2,3}^{-1}+{\tau }_{3,1}^{+2}|<ϵ$) and identifies the peaks relative to an indirect (multi-neurons path) and an apparent (common output) connection by searching for the ones with smaller amplitude. The smallest peaks (${\tau }_{3,1}^{+1}$ and ${\tau }_{3,1}^{+2}$) are discarded from the analysis. (3) The correlation triangles are functional to the estimation of the direct connections in the network. Because no more correlation exists between neuron 1 and 3, the estimated effective connectivity includes only the direct links for (1, 2) and (3, 1): two connections with opposite directionality exist between neuron 1 and 2 because positive and negative correlation peaks are detected in R_1,2 (${\tau }_{1,2}^{-1}$ and ${\tau }_{1,2}^{+1}$); one link connects neuron 2 to neuron 3 as a result of the positive correlation peaks in R_2,3 (${\tau }_{3,1}^{+1}$ and ${\tau }_{3,1}^{+2}$).