Figure 7
Cliques and simplexes. (A) Pairwise interlinked place fields produce cliques of the coactivity graph \({\mathcal {G}}\). Shown is a vertex \(\varsigma ^{(0)}\) (0-clique), a link \(\varsigma ^{(1)}\) (1-clique), a three-vertex \(\varsigma ^{(2)}\) and a four-vertex \(\varsigma ^{(3)}\) clique. (B) Geometric simplexes: a 0D dot (\(\kappa ^{(0)}\)), a 1D link (\(\kappa ^{(1)}\)), a 2D triangle (\(\kappa ^{(2)}\)) and a 3D tetrahedron (\(\kappa ^{(3)}\)). (C) The corresponding complexes: a simplicial coactivity complex \({\mathcal {T}}_{\sigma }\) whose simplexes (1) are detected as singular coactivity events (left) may topologically differ from the clique coactivity complexes \({\mathcal {T}}_{\varsigma }\), assembled from the cliques of a coactivity graph \({\mathcal {G}}\) (right) over a spike integration period \(\varpi \). A simplicial complex K is a combination of matching simplexes. The set of vertexes and black lines highlight the 1D-skeleton of \(sk_1(K)\).
