Texture-like neural representations

An almost ubiquitous approach taken in systems neuroscience is that of devising stimulus response functions (SRFs), which specify how a stimulus is encoded into a neural response, e.g. place fields, tuning curves. However, it is clear that the brain itself is able to infer properties of the environment in real time via neural activity alone, without having to resort to performing experiments on its own response to stimuli. A central aim in this work is to explore the relationship between the geometric/topological structure of the stimulus space (also animal behaviour) and neural activity and to investigate to what extent the former can be derived from the latter without having to perform the standard operation of constructing dictionaries between the two as provided by an SRF. In the last decade, successful attempts have been made to eliminate this albeit very useful middleman using topological data analysis (TDA).

We build on an approach initiated in, which uses clique topology, a form of TDA. Common statistics related to neural activity and connectivity derived from experimental data are often presented in the form of a matrix of correlations or connectivity strengths between pairs of neurons, voxels, etc. Analogous statistics can be obtained from stimuli presented to the animal, e.g. textures both visual, auditory, images of natural scenes, olfaction. Clique topology enables us to test whether signatures of structures of stimulus spaces and environments are detectable in the correlation structures of the raw data obtained from neuronal recordings. The advantage of using clique topology over traditional eigenvalue-based methods is that the latter is badly distorted by monotone nonlinearities, whereas the information encoded in the `order complex’ is invariant under such transformations.

The statistical topological approach in could determine whether correlations resulting from both the stimulus and response side were random or induced by a geometric process (e.g. pairwise distances obtained via sampling points from a unit cube in Rd). Here we introduce a new regime of complexes that are derived instead from textures. In experimental scenarios were considered where neurons were tuned to features lying in a continuous coding space where correlations decrease with distance, e.g. hippocampal place cells. Textures are in many ways the antithesis of this and also exhibit both repetitive and random features. Clique topology techniques on textures have led us to a menagerie of order complexes which have very small values for Betti numbers throughout the filtration as compared to same-sized ‘random’ and ‘geometric’ order complexes (Fig. 1). Analogous Betti curves have been shown to have been induced by order complexes derived from low-rank matrices by Curto (unpublished). The matrices that emerge from textures, however, do not generally have low-rank and it is an open question as to whether the two can be related.

We were surprised to find that datasets from a wider range of modalities appear to exhibit texture-like rather than ‘geometric’ structure. We have extracted texture-like order complexes from olfactory datasets as well as a simulated dataset from a spiking neural network modelling speech recognition.