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<div class="references" contenteditable="true"> <strong>References</strong><br> <p>[1] Chung, S., et al. (2018). “Topological insights into recurrent neural dynamics.” Neural Computation, 30(5), 1221-1251.</p> <p>[2] Gardner, R. J., et al. (2022). “Toroidal topology of population activity in grid cells.” Nature, 602, 123–128.</p> <p>[3] Vasquez, E., Thorne, A. (2024). “Persistent homology for high-dimensional neural manifolds.” Advances in Neural Information Processing Systems, 36, 689–704.</p> <p>[4] O’Brien, K., et al. (2023). “Topological data analysis in neuroscience: a review.” Network Neuroscience, 7(2), 410–441.</p> </div> <p style="font-size: 0.7rem; text-align: center; margin-top: 1.2rem; color: #5b6e8c;" contenteditable="true">© 2025 Cognitive Dynamics Lab. This work is licensed under CC BY-NC 4.0. Preprint version.</p> </div> </div> </div> </div> <footer> <i class="fas fa-edit"></i> Fully editable academic paper — click any paragraph, title, or author line. Then click <strong>Download PDF</strong> to generate a typeset document using html2pdf.js. </footer> -types html2pdf.js
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<h2 contenteditable="true">2. Methods & Topological Framework</h2> <h3 contenteditable="true">2.1 Neural Manifold Reconstruction</h3> <p contenteditable="true">Let <span class="math">\( \mathbfX = \\mathbfx_i\_i=1^N \subset \mathbbR^D \)</span> be the recorded neural activity after spike smoothing and dimensionality reduction via UMAP or PCA. We construct a Vietoris–Rips complex at varying scales <span class="math">\( \epsilon \)</span>. The persistence diagram tracks the birth and death of homological features. For each dimension <span class="math">\( k \)</span>, the Betti number <span class="math">\( \beta_k \)</span> is defined as the number of persistent <span class="math">\( k \)</span>-dimensional holes. We computed these using the GUDHI library. To quantify significance, we compare against null distributions generated by shuffling neural activity across trials.</p>