A topological paradigm for hippocampal spatial map formation using persistent homology

Extent: 14 p.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.Y. Dabaghian, F. Mémoli, L. Frank, G. Carlsso

A survey of partial differential equations in geometric design

YesComputer aided geometric design is an area where the improvement of surface generation techniques is an everlasting demand since faster and more accurate geometric models are required. Traditional methods for generating surfaces were initially mainly based upon interpolation algorithms. Recently, partial differential equations (PDE) were introduced as a valuable tool for geometric modelling since they offer a number of features from which these areas can benefit. This work summarises the uses given to PDE surfaces as a surface generation technique togethe

Meshless geometric subdivision

C. Moenning, F. Mémoli, G. Sapiro, N. Dyn, N.A. Dodgso

Variations in topology place different demands on hippocampal state.

<p>The <i>top row</i> 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. <i>Second row</i>: 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. <i>Third row:</i> Each dot represents the relative standard deviation of map formation times for successful trials where is small (<0.3). <i>Fourth row</i>: 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 <i>L</i> in the text.</p

Examples of low-dimensional manifolds and their Betti numbers with some of the corresponding loops.

<p>(a) A point is a 0<i>D</i> loop; no higher dimensional loops are present. Thus, each manifold containing at least one point has a 0<i>D</i> loop, so every list of Betti numbers starts with a “1”. (b) A circle is a 1<i>D</i> loop, with no other loops in higher dimensions. (c) A 2<i>D</i> torus with two examples of non-contractible (red) 1<i>D</i> loops, and an example of a 1<i>D</i> loop contractible into a point (green). The 2<i>D</i> surface of the torus is the 2<i>D</i> loop listed. (d) A 2<i>D</i> sphere, with two exemplary contractible <i>1</i>D loops. The 2<i>D</i> surface of the sphere “loops” onto itself.</p

Ergodicity times for the three environments shown in <b>Figure 4</b>.

<p>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.</p

Place fields can be derived from place cell spike trains.

<p>(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.</p