Figure 4
Topological dynamics in maps with multiple firing fields. (A) Left panel shows three examples of convex place fields used to obtain the results illustrated in Fig. 2. Allowing a cell to spike in several (\(2-3\)) locations produces multiply connected place fields (middle panel; clusters of dots of a given color correspond to spikes produced by a single simulated neuron). Right panel shows a \(\varpi =50\) second long fragment of the trajectory \(\gamma _{\varpi }\) covering a segment \(\chi _{\varpi }\) of the environment (reddened area). (B) The Leray dimensionality of the detector-complex, evaluated for the same place cell population as in Fig. 2A, can reach \(D({\mathcal {T}}_{\sigma })=4\) if we allow \(30\%\) of multiply connected place fields (\(2-3\) components each). (C) In a clique coactivity complex, the spurious loops in dimensions \(D=2\) and lower may persist indefinitely, implying either that the firing fields are 3D-representable or that they may be multiply connected. Note that the number of spurious loops in both \({\mathcal {T}}_{\sigma }\) and in \({\mathcal {T}}_{\varsigma }\) is higher than in the case with convex firing fields (Fig. 2A,B). (D) The persistence bars computed for the flickering complex \({\mathcal {F}}_{\varpi }\) with spike integration window \(\varpi =1\) minute, indicate stable mean Leray dimensionality \(\langle D({\mathcal {F}}_{\varpi })\rangle =1\), implying that the local charts \(\chi _{\varpi }\) are planar and hence that the firing fields are two-dimensional.
