A Topological Paradigm for Hippocampal Spatial Map Formation Using Persistent Homology
Figure 5
2D sections highlight dependence of map formation times on hippocampal state.
These 2D sections are based on the point cloud data in Fig. 4C, second row (far right). Dot sizes and colors represent the same characteristics as described in Fig. 4 (i.e., the larger and bluer the dot, the more successful and more rapid the map formation). Graph A fixes the mean place field size at 50 cm, and shows that robust map formation in this case requires a larger number of cells firing at a higher rate. Graph B shows that, at a mean firing rate of 17 Hz, any ensemble size between 100 and 400 neurons can fairly rapidly form a correct topological map as long as the place fields are between 50 and 80 cm. Graph C shows that an ensemble of 325 cells can have mean firing rates from 10 to 35 Hz and form maps quickly and accurately with place field sizes from 40–80 cm. In short, smaller place cell ensembles, with low mean firing rates (<10 Hz) and too small (
<20 cm) or too large (
>100 cm) mean place field sizes, fail to produce the correct topological signature. In contrast, sufficiently large place cell ensembles with higher firing rate neurons and well-tuned place fields reliably capture the topological structure of the environment in a time frame comparable to the experimentally observed map formation period.