A Topological Model of the Hippocampal Cell Assembly Network

It is widely accepted that the hippocampal place cells' spiking activity produces a cognitive map of space. However, many details of this representation's physiological mechanism remain unknown. For example, it is believed that the place cells exhibiting frequent coactivity form functionally interconnected groups—place cell assemblies—that drive readout neurons in the downstream networks. However, the sheer number of coactive combinations is extremely large, which implies that only a small fraction of them actually gives rise to cell assemblies. The physiological processes responsible for selecting the winning combinations are highly complex and are usually modeled via detailed synaptic and structural plasticity mechanisms. Here we propose an alternative approach that allows modeling the cell assembly network directly, based on a small number of phenomenological selection rules. We then demonstrate that the selected population of place cell assemblies correctly encodes the topology of the environment in biologically plausible time, and may serve as a schematic model of the hippocampal network.


SUPPLEMENTARY FIGURES
Supplementary Figure 1. Simplicial complexes. (A) A schematic representation of an irregular simplicial complex K, in which the number of maximal simplexes is larger than the number of vertexes (black dots). The maximal 1D simplexes are shown as blue segments, the 2D simplexes as gray triangles and the 3D simplexes as pink tetrahedrons. (B) A simplicial "quasi-manifold," Q, which has a similar number of vertexes and maximal simplexes of different dimensionalities. If each maximal simplex, e.g. (1, 2, 3) or (10,7,8,9), corresponds to an assembly of place cells driving a readout neuron (red dots), then Q is a cell assembly complex. Dotted lines represent synaptic connections from the place cells to the readout neuron.
Supplementary Figure 2. The simplicial complexes T 0 (θ) constructed by direct selection of coactive combinations, for four different values of θ. (A) The appearance rates of simplexes, arranged from left to right according to their dimension. Each dot corresponds to a maximal simplex whose dimension is color-coded according to the colorbar on the right. (B) Spatial distribution of the dimensionalities of the selected simplexes. (C) Spatial distribution of the appearance rates of the selected simplexes. (D) The histograms of the lapse times, fit to double exponential distribution (blue line), and the value of the fitted distribution's rate β. (E) Spatial projections of the 2D skeletons of the T 0 (θ). Data for all panels is computed for a specific place field map for illustrative purposes.
If the 2D simplexes are discarded and their 1D faces are retained, then the second-order simplex fields are produced, shown here as overlapping shaded regions. The original simplex fields a, b, c and d are now represented by the coactivity of three pairs, e.g., a is represented by . The three place fields on the left exhibit pairwise, but not triple overlap. In the generic spatial configuration shown on the right, pairwise overlapping place fields also produce a triple overlap. (D) Four pairwise overlapping convex regions in 2D produce all the higher order (triple and quadruple) overlaps.   , (b 0 , b 1 , b 2 , b 3 ). For the low rate, f σ = 0.04 Hz, T 0 (θ) encodes the correct spatial connectedness of the environment (b 0 = 1). For intermediate rates, 0.05 ≤ f σ ≤ 0.07 Hz, T 0 (θ) may occasionally break into two pieces (b 0 = 2) and for f σ ≥ 0.10 Hz and higher, T 0 (θ) fragments into multiple components. In higher dimensions D ≥ 1, T 0 (θ) contains over a hundred noncontractible topological loops. For high thresholds, f σ ≥ 0.10 Hz, T 0 (θ) fragments into multiple components. However, the connectivity of T 0 (θ) in higher dimensions, D ≥ 2, is correct for all cases, which implies that T 0 (θ) contracts into 2D. (B) After applying the correction algorithms, the selected complexes acquire correct topological signature for f σ ≤ 0.07 for all maps. The corresponding learning times T min are listed in Suppl. Table 4.  Table 3. Topological signatures of the complexes selected by the neighbor-selection algorithm. (A) If but one pair of closest vertexes is selected n 0 = 2, T 0 (n 0 ) breaks into multiple components. For n 0 ≥ 5, T 0 (n 0 ) has only one component, but path connectivity is compromised (b 0 = 1, b 1 1). In higher dimensions, T 0 (n 0 ) is contractible, b n>1 = 0. (B) After applying the correction algorithms, the selected complexes for almost all maps acquire correct topological signature in 1D and 2D (shown in boldface) for n 0 ≥ 7. The corresponding learning times, T min , are listed in Suppl. Table 4.  Table 4. The learning times, T min (in minutes) computed for the selected simplicial complexes T 0 (θ) and T 0 (n 0 ) are similar to the learning times computed via the full temporal nerve complex T . Thus, the information about the topological structure of the environment emerges from the cell assembly activity as fast as from the entire pool of place cell coactivities. However, note that the complex T 0 (n 0 ) sometimes fails to produce a finite leaning time; non-convergent cases are marked by ∞.

SUPPLEMENTARY MOVIES ILLUSTRATING THE FIRST OF THE TEN TESTED MAPS
Suppl. Movie 1. The grey dots represent centers of the place fields, viewed from above, similar to Figure  3B. The centers of the place fields that correspond to the coactive place cells are shown in red. The resulting activity packet moves in the environment following the simulated rat's trajectory.
Suppl. Movie 2. Selection of the cell assemblies by Method I (θ = 100) and assigning readout neurons to the cell assemblies.
Suppl. Movie 3. A side projection view of the activity packet propagating in the cell assembly network, selected via Method I. To emphasize that the coactive place cell combinations comprise a cell assembly complex T 0 (θ), the corresponding place field centers are schematically connected to the readout neurons.
Suppl. Movie 4. The same system shown in the same projection as the Figure 3B and Suppl. Movie 1.