A Topological Paradigm for Hippocampal Spatial Map Formation Using Persistent Homology
Figure 3
Persistent cycles form a topological barcode.
Top and bottom graphs show which 0D and 1D cycles, respectively, persist in this cell ensemble. Each colored horizontal line represents one 0D cycle (top panel) or one 1D cycle (bottom panel). Initially, until cells begin co-firing, each 0D cycle corresponds to one cell. At later times, both 0D and 1D cells are emergent phenomena, produced by co-firing of groups of cells. The dotted red vertical line at 5.84 minutes marks the moment when the correct number of loops appears in both 0D and in 1D, which is the minimal map formation time . The series of short horizontal bars in both panels (some quite miniscule) and the longer lines that disappear before
represent topological noise, i.e., cycles that fail to persist. The one persistent 1D cycles indicates that the environment in question has one physical (topological) loop, and the single 0D cycle indicates that the space is connected, of one piece. Together, this pattern of stable bars forms a barcode that can be ‘scanned’ to discern the topological structure of the environment (see Methods).