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 by comparing multivariate response patterns elicited by sensory stimuli. However, traditional RSA relies solely on geometric properties, overlooking crucial topological information. This thesis introduces topological RSA (tRSA), a novel framework that combines 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 that are robust to noise and individual idiosyncrasies. This thesis introduces several key methodological advances: (1) Topological RSA (tRSA) identifies computational signatures as accurately as RSA while compressing unnecessary variation with capabilities to test topological hypotheses; (2) Adaptive Geo-Topological Dependence Measure (AGTDM) provides a robust statistical test for detecting complex multivariate relationships; (3) Procrustes-aligned Multidimensional Scaling (pMDS) aligns time-resolved representational geometries to illuminate processing stages in neural computation; (4) Temporal Topological Data Analysis (tTDA) applies spatio-temporal filtration techniques to reveal developmental trajectories in biological systems; and (5) Single-cell Topological Simplicial Analysis (scTSA) characterizes higher-order cell population complexity across different stages of development.
Through analyses of neural recordings, biological data, and simulations of neural network models, this thesis demonstrates the power and versatility of these new methods. By advancing RSA with topological techniques, this work provides a powerful new lens for understanding brains, computational models, and complex biological systems. These methods not only offer robust approaches for adjudicating among competing models but also reveal novel theoretical insights into the nature of neural computation.
This thesis lays the foundation for future investigations at the intersection of topology, neuroscience, and time series data analysis, promising to deepen our understanding of how information is represented and processed in biological and artificial neural networks. The methods developed here have potential applications in fields ranging from cognitive neuroscience to clinical diagnosis and AI development, paving the way for more nuanced understanding of brain function and dysfunction.
| Identifer | oai:union.ndltd.org:columbia.edu/oai:academiccommons.columbia.edu:10.7916/fb5z-3v10 |
| Date | January 2023 |
| Creators | Lin, Baihan |
| Source Sets | Columbia University |
| Language | English |
| Detected Language | English |
| Type | Theses |
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