Geometry of spiking patterns in early visual cortex: a topological data analytic approach

Abstract

In the brain, spiking patterns live in a high-dimensional space of neurons and time. Thus, determining the intrinsic structure of this space presents a theoretical and experimental challenge. To address this challenge, we introduce a new framework for applying topological data analysis (TDA) to spike train data and use it to determine the geometry of spiking patterns in the visual cortex. Key to our approach is a parametrized family of distances based on the timing of spikes that quantifies the dissimilarity between neuronal responses. We applied TDA to visually driven single-unit and multiple single-unit spiking activity in macaque V1 and V2. TDA across timescales reveals a common geometry for spiking patterns in V1 and V2 which, among simple models, is most similar to that of a low-dimensional space endowed with Euclidean or hyperbolic geometry with modest curvature. Remarkably, the inferred geometry depends on timescale and is clearest for the timescales that are important for encoding contrast, orientation and spatial correlations.

Topological features of spike trains in recurrent spiking neural networks that are trained to generate spatiotemporal patterns

“…To compare different neurons in terms of dissimilarities between their spike trains, we apply the method proposed by Victor and Purpura (see Guidolin et al, 2022 and works cited therein). Namely, this method endows a pair of spike trains with a notion of distance.…”

“…To get more insight into intricate structure of spike trains, following Giusti et al (2015) and Guidolin et al (2022), we transform the obtained matrices by rank ordering their entries. Namely, given a matrix of VP distances D ij with zeros on its main diagonal, we consider the entries of its above-diagonal part and replace them by natural numbers 0, 1, .…”

“…Victor-Purpura distance for spike trains Each spike train is considered to be a point in an abstract topological space. A spike train metric is defined according to the special rule, which assigns a non-negative number to pairs of spike trains and expresses how dissimilar they are (Guidolin et al, 2022). We use the variant of the spike time VP distance which is parametrized by the cost quantity q in units of the inverse time.…”

“…One of the promising approaches to characterize spiking patterns is methods of algebraic topology. Such tools as persistent homology analysis have been used in relating spike patterns with functions of neural networks (Dabaghian et al, 2012;Petri et al, 2014;Curto, 2017;Bardin et al, 2019;Santos et al, 2019;Sizemore et al, 2019;Naitzat et al, 2020;Billings et al, 2021;Guidolin et al, 2022)-both biological and artificial-and more widely for studying topological aspects of dynamical systems (Maletić et al, 2016;Stolz et al, 2017;Salnikov et al, 2018;Myers et al, 2019).…”

Topological Structure of Population Activity in Mouse Visual Cortex Encodes Visual Scene Rotations

“…al. [43], in which the authors try to understand the geometry of visual space by combining spiking metrics [44] with topological methods. Nevertheless, the reduction methods used in these examples rely on visual inspection and remove structural properties of the data, due to the fact that the data has to be projected to a low-dimensional subspace.…”