Topological Representational Similarity Analysis in Brains and Beyond

TL;DR

This work proposes Topological Representational Similarity Analysis (tRSA), extending classical RSA by integrating topology into the analysis of neural representations. It introduces geo-topological descriptors and a family of nonlinear intensity-transform statistics to capture both local topology and intermediate geometry, yielding robust model adjudication across brains and neural networks. Key contributions include Adaptive Geo-Topological Dependence Measure (AGTDM) for adaptive multivariate dependence testing, Procrustes-aligned MDS (pMDS) for temporal alignment, and temporal topological data analysis (tTDA) plus scTSA for developmental and single-cell dynamics. The framework is implemented in RSAToolbox and demonstrated across neural recordings, imaging, and neural network simulations, with broad implications for neuroscience, biology, and AI.

Abstract

Understanding how the brain represents and processes information is crucial for advancing neuroscience and artificial intelligence. Representational similarity analysis (RSA) has been instrumental in characterizing neural representations, but traditional RSA relies solely on geometric properties, overlooking crucial topological information. This thesis introduces Topological RSA (tRSA), a novel framework combining geometric and topological properties of neural representations. tRSA applies nonlinear monotonic transforms to representational dissimilarities, emphasizing local topology while retaining intermediate-scale geometry. The resulting geo-topological matrices enable model comparisons robust to noise and individual idiosyncrasies. This thesis introduces several key methodological advances: (1) Topological RSA (tRSA) for identifying computational signatures and testing topological hypotheses; (2) Adaptive Geo-Topological Dependence Measure (AGTDM) for detecting complex multivariate relationships; (3) Procrustes-aligned Multidimensional Scaling (pMDS) for revealing neural computation stages; (4) Temporal Topological Data Analysis (tTDA) for uncovering developmental trajectories; and (5) Single-cell Topological Simplicial Analysis (scTSA) for characterizing cell population complexity. Through analyses of neural recordings, biological data, and neural network simulations, this thesis demonstrates the power and versatility of these methods in understanding brains, computational models, and complex biological systems. They not only offer robust approaches for adjudicating among competing models but also reveal novel theoretical insights into the nature of neural computation. This work lays the foundation for future investigations at the intersection of topology, neuroscience, and time series analysis, paving the way for more nuanced understanding of brain function and dysfunction.

Figures (46)

• Figure 1: Application scenarios enabled by representational comparison. RSA relates representations across species, modalities, and models. (Extracted from kriegeskorte2008representational).
• Figure 2: Computing the representational dissimilarity matrix (RDM). Dissimilarities are computed between all pairs of activity patterns and assembled into a representational dissimilarity matrix (RDM). The RDM is symmetric about a diagonal of zeros. (Extracted from nili2014toolbox).
• Figure 3: Visualizing RDMs with multi-dimensional scaling (MDS). The three eample RDMs can be visualized in 2d arrangements via unsupervised learning methods such as MDS, with each dot as a representation pattern colored by their categorical information. (Extracted from nili2014toolbox).
• Figure 4: Inferential comparison of multiple model representations. Demonstrated here is an example model comparison results enabled by the Matlab RSAToolbox nili2014toolbox. The RDMs of 10 candidate models are compared with the reference RDM (computed from the brain-activity data) and ranked in terms of their performance to predict the data RDM. (Extracted from nili2014toolbox).
• Figure 5: Comparing representations between brains and models. To understand the degree to which a computational model can account for the cognitive process of a certain brain region, the same set of stimuli is presented to both the model and the biological system. The response patterns across measured response channels (e.g. neurons or voxels) are then characterized by a summary statistic, the representational dissimilarity matrix (RDM, center), which defines the metric configuration of the stimuli in the neural population response space. However, the metric configuration can be sensitive to measurement noise and idiosyncrasies of individual brains that do not reflect computational function. An alternative summary statistic that captures the topology would be the adjacency matrix (right), which defines the unweighted graph of neighborhood relationships in the population response space. This summary statistic promises to be more robust to noise and idiosyncrasies, but may discard too much information. Considering the geometry (RDM) and topology (adjacency matrix) as extremes of a continuum suggests that it may be possible to get the best of both (Fig. \ref{['trsa_fig2']}).