Figure 1.
Place fields can be derived from place cell spike trains.
(a) As a rat explores a given environment, various place cells will fire in spatially discrete locations. Here, for the sake of simplicity, we depict three place fields as they might arise from spike trains from three place cells, as in the next panel. (b) Schematic representation of spike trains fired from three different place cells as a rat explores an environment. Note that there is contemporaneous spiking activity, or co-firing. (c) The place fields derived from the three place cells in (b): the co-firing patterns indicates areas of overlap of the place fields. When the rat makes a straightforward trajectory through an explored environment, different place cells will be activated and their place fields can overlap.
Figure 2.
Distributions of firing rate and place field size collected by recording place cell firing as a rat explores a linear track.
A typical place cell fires at a rate of ∼10–20 Hz and place fields typically range from 10 to 30 cm across. These experimentally derived distributions serve as realistic constraints on our simulated data by providing proportionality coefficients a and b so that the shape of the distributions P(f) and P(s) mimics those derived from the experiments. From the data depicted here, a = 1.2 and b = 1.7.
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).
Figure 4.
Variations in topology place different demands on hippocampal state.
The top row depicts three experimental configurations, each two meters square, for our computational simulations; note that the second and third scenarios (B and C) are topologically equivalent but geometrically different, and that scenario C will force our simulated rat to adopt a quasi-linear trajectory. The dense network of gray lines represents the simulated trajectories. Second row: Point cloud approximations that reveal mean map formation times for each space configuration. Each dot represents a hippocampal state as defined by the three parameters (,
, and N); the size of the dot reflects the proportion of trials in which a given set of parameters produced the correct outcome; the color of the dot is the mean time over ten simulations. If, for example, one set of parameters produced the correct topological information in 6 out of 10 trials, the dot will be 60% of the size of the largest dot, and the color will reflect the mean map formation time for the correct trials. (Blue represents success within the first 25% of the total time; green within the first 50%, yellow-orange within the first 75%, and red means success took nearly the whole time period. The maximal observed time was 4.3 minutes for configuration A, 11.7 minutes for B, and 9.3 minutes for C.) Note how the third scenario (C) contains a preponderance of blue dots, indicating that the majority of hippocampal states easily mapped this environment. This is because the two holes are so large that a rat is virtually forced into a straight-line trajectory. Third row: Each dot represents the relative standard deviation of map formation times
for successful trials where
is small (<0.3). Fourth row: Combining the mean map formation times (second row) with the robustness requirement
(third row) reveals a domain of stable, robust map formation times that we call the core of the region L in the text.
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.
Figure 6.
Ergodicity times for the three environments shown in Figure 4.
For each environment, the graph shows how much time is required to cover a certain percentage of the 3×3 cm spatial bins. This ergodic time scale shows that it takes approximately ten minutes for a rat to cover 80% of the environment; by comparison, the topological map formation time for stable regimes is much lower.
Figure 7.
Examples of low-dimensional manifolds and their Betti numbers with some of the corresponding loops.
(a) A point is a 0D loop; no higher dimensional loops are present. Thus, each manifold containing at least one point has a 0D loop, so every list of Betti numbers starts with a “1”. (b) A circle is a 1D loop, with no other loops in higher dimensions. (c) A 2D torus with two examples of non-contractible (red) 1D loops, and an example of a 1D loop contractible into a point (green). The 2D surface of the torus is the 2D loop listed. (d) A 2D sphere, with two exemplary contractible 1D loops. The 2D surface of the sphere “loops” onto itself.