A Topological Paradigm for Hippocampal Spatial Map Formation Using Persistent Homology

Abstract

An animal's ability to navigate through space rests on its ability to create a mental map of its environment. The hippocampus is the brain region centrally responsible for such maps, and it has been assumed to encode geometric information (distances, angles). Given, however, that hippocampal output consists of patterns of spiking across many neurons, and downstream regions must be able to translate those patterns into accurate information about an animal's spatial environment, we hypothesized that 1) the temporal pattern of neuronal firing, particularly co-firing, is key to decoding spatial information, and 2) since co-firing implies spatial overlap of place fields, a map encoded by co-firing will be based on connectivity and adjacency, i.e., it will be a topological map. Here we test this topological hypothesis with a simple model of hippocampal activity, varying three parameters (firing rate, place field size, and number of neurons) in computer simulations of rat trajectories in three topologically and geometrically distinct test environments. Using a computational algorithm based on recently developed tools from Persistent Homology theory in the field of algebraic topology, we find that the patterns of neuronal co-firing can, in fact, convey topological information about the environment in a biologically realistic length of time. Furthermore, our simulations reveal a “learning region” that highlights the interplay between the parameters in combining to produce hippocampal states that are more or less adept at map formation. For example, within the learning region a lower number of neurons firing can be compensated by adjustments in firing rate or place field size, but beyond a certain point map formation begins to fail. We propose that this learning region provides a coherent theoretical lens through which to view conditions that impair spatial learning by altering place cell firing rates or spatial specificity.

How Does the Brain Solve the Computational Problems of Spatial Navigation?

“…The map could in principle either encode detailed metric (distance) and geometric (angle) information relating different locations or encoding more qualitative topological information that preserves relative distances and other topological features (Muller et al 1996;Balakrishnan et al 1999). Recent mathematical analysis of the CA3 code suggests the latter, that the CA3 map appears to be more topological than geometric and metric (Dabaghian et al 2012).…”

“…This map may be compressed in the sense that it lacks geometric and metric information about the environment (angles and distance between locations and landmarks) (Dabaghian et al 2012). This retrieved map generates predictions about location based on learned knowledge of commonly taken past routes and relative locations, which may then be compared, may then be compared in CA1 against the sensory-based inputs arriving from the EC (Hasselmo and Wyble 1997), to perform self-localization and possibly influence, via feedback, the map in CA3 (Sik et al 1994), Fig.…”

What can topology tell us about the neural code?

“…The place cell code thus naturally reflects the topology of the represented space. These and related observations have led some researchers to speculate that the hippocampal place cell code is fundamentally topological in nature [6,14], while others (including this author) have argued that considerable geometric information is also present and can be extracted using topological methods [11,20]. In order to disambiguate topological and geometric features, Dabaghian et al performed an elegant experiment using linear tracks with flexible joints [13].…”

Multilevel Integration Entropies: The Case of Reconstruction of Structural Quasi-Stability in Building Complex Datasets

“…The most important tool of TDA is the persistent homology method [12,13], which is proven as useful in many real-world applications. The abundance of applications covers a broad range of phenomena in biological and medical science, like breast cancer research [14], brain science [15][16][17][18][19][20][21], biomolecules [22][23][24], evolution [25] and bacteria [26], followed by the applications in sensor networks [27,28], signal analysis [29], image processing [30], musical data [31], text mining [32], phase space reconstruction of dynamical systems [33,34], as well as complex networks related to either dynamics taking place on networks [35] or structural properties [36,37].…”