Abstract Minimal experiments, such as head-fixed wheel-running and sleep, offer experimental advantages but restrict the amount of observable behavior, making it difficult to classify functional cell types. Arguably, the grid cell, and its striking periodicity, would not have been discovered without the perspective provided by free behavior in an open environment. Here, we show that by shifting the focus from single neurons to populations, we change the minimal experimental complexity required. We identify grid cell modules and show that the activity covers a similar, stable toroidal state space during wheel running as in open field foraging. Trajectories on grid cell tori correspond to single trial runs in virtual reality and path integration in the dark, and the alignment of the representation rapidly shifts with changes in experimental conditions. Thus, we provide a methodology to discover and study complex internal representations in even the simplest of experiments. 

Uncovering 2-D toroidal representations in grid cell ensemble activity during 1-D behavior

Living organisms are limited in the range of values of ecological factors they can explore. This defines where animals exist (or could exist) and forms an ecological fingerprint that explains species' distribution at global scales. Species' ecological fingerprints can be represented as a n-dimensional hypervolume - known as Hutchinson's niche hypervolume. This concept has enabled significant progress in our understanding of species' ecological needs and distributions across environmental gradients. Nevertheless, the properties of Hutchinson's n-dimensional hypervolumes can be challenging to calculate and several methods have been proposed to extract meaningful measurements of hypervolumes' properties. One key property of hypervolumes are holes, which provide important information about the ecological occupancy of species. However, to date, current methods rely on volume estimates and set operations to identify holes in hypervolumes. Yet, this approach can be problematic because in high-dimensions, the volume of region enclosing a hole tends to zero. We propose the use of persistence homology (PH) to identify holes in hypervolumes and in ecological datasets more generally. PH allows for the estimates of topological properties in n-dimensional niche hypervolumes independent of the volume estimates of the hypervolume. We demonstrate the application of PH to canonical datasets and to the identification of holes in the hypervolumes of five vertebrate species with diverse niches, highlighting the potential benefits of this approach to gain further insights into animal ecology. Overall, our approach enables the study of a yet unexplored property of Hutchinson's hypervolumes, and thus, have important implications to our understanding of animal ecology. 

/pdf/holey-niche-finding-holes-in-niche-hypervolumes-using-2godfrow.pdf

‘Holey’ niche! finding holes in niche hypervolumes using persistence homology

When analysing neural activity, one often studies either the neural correlations or the state space of population vectors. However, the reason for choosing one over the other is seldom discussed. Here, with methods from the mathematical field of topology, we compare these approaches and propose using both for unsupervised inference of neural representations. If the recorded neurons have convex receptive fields on a single covariate space, there is a duality between the topological signatures derived from correlations on the one hand and population vectors on the other hand. However, in the presence of multiple neural modules with non convex receptive fields, this duality breaks down. We explain how to leverage complementary information derived from both approaches to sucessfully characterize the represented covariate spaces directly from the data also under these challenging circumstances. Furthermore, we prove appropriate reconstruction results and showcase applications to multiple neural datasets from various brain regions and diverse neural modules.

A topological perspective on the dual nature of the neural state space and the correlation structure

Abstract Neuroscience is pushing toward studying the brain during naturalistic behaviors with open-ended tasks. Grid cells are a classic example, where free behavior was key to observing their characteristic spatial representations in two-dimensional environments. In contrast, it has been difficult to identify grid cells and study their computations in more restrictive experiments, such as head-fixed wheel running. Here, we challenge this view by showing that shifting the focus from single neurons to the population level changes the minimal experimental complexity required to study grid cell representations. Specifically, we combine the manifold approximation in UMAP with persistent homology to study the topology of the population activity. With these methods, we show that the population activity of grid cells covers a similar two-dimensional toroidal state space during wheel running as in open field foraging, with and without a virtual reality setup. Trajectories on the torus correspond to single trial runs in virtual reality and changes in experimental conditions are reflected in the internal representation, while the toroidal representation undergoes occasional shifts in its alignment to the environment. These findings show that our method can uncover latent topologies that go beyond the complexity of the task, allowing us to investigate internal dynamics in simple experimental settings in which the analysis of grid cells has so far remained elusive. 

Abstract Hutchinson’s niche hypervolume concept has enabled significant progress in our understanding of species’ ecological needs and distributions across environmental gradients. Nevertheless, the properties of Hutchinson’s n -dimensional hypervolumes can be challenging to calculate and several methods have been proposed to extract meaningful measurements of hypervolumes’ properties (e.g., volume). One key property of hypervolumes are holes, which provide important information about the ecological occupancy of species. However, to date, current methods rely on volume estimates and set operations to identify holes in hypervolumes. Yet, this approach can be problematic because in high-dimensions, the volume of region enclosing a hole tends to zero. Here, we propose the use of the topological concept of persistence homology (PH) to identify holes in hypervolumes and in ecological datasets more generally. PH allows for the estimates of topological properties in n -dimensional niche hyper-volumes and is independent of the volume estimates of the hypervolume. We demonstrate the application of PH to canonical datasets and to the identification of holes in the hypervolumes of five vertebrate species with diverse niches, highlighting the potential benefits of this approach to gain further insights into animal ecology. Overall, our approach enables the study of an yet unexplored property of Hutchinson’s hypervolumes (i.e., holes), and thus, have important implications to our understanding of animal ecology. 

‘Holey’ niche! Finding holes in niche hypervolumes using persistence homology

Several persistent homology software libraries have been implemented in R. Specifically, the Dionysus, GUDHI, and Ripser libraries have been wrapped by the TDA and TDAstats CRAN packages. These software represent powerful analysis tools that are computationally expensive and, to our knowledge, have not been formally benchmarked. Here, we analyze runtime and memory growth for the 2 R packages and the 3 underlying libraries. We find that datasets with less than 3 dimensions can be evaluated with persistent homology fastest by the GUDHI library in the TDA package. For higher-dimensional datasets, the Ripser library in the TDAstats package is the fastest. Ripser and TDAstats are also the most memory-efficient tools to calculate persistent homology.

/pdf/benchmarking-r-packages-for-calculation-of-persistent-2nnm0eegsn.pdf

Benchmarking R packages for Calculation of Persistent Homology.

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Benchmarking R packages for Calculation of Persistent Homology.