Lifeboat Foundation

This video is my take on 3B1B’s Summer of Math Exposition (SoME) competition.

It explains in pretty intuitive terms how ideas from topology (or “rubber geometry”) can be used in neuroscience, to help us understand the way information is embedded in high-dimensional representations inside neural circuits.

OUTLINE:
00:00 Introduction.
01:34 — Brief neuroscience background.
06:23 — Topology and the notion of a manifold.
11:48 — Dimension of a manifold.
15:06 — Number of holes (genus)
18:49 — Putting it all together.

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Main paper:
Chaudhuri, R., Gerçek, B., Pandey, B., Peyrache, A. & Fiete, I. The intrinsic attractor manifold and population dynamics of a canonical cognitive circuit across waking and sleep. Nat Neurosci 22, 1512–1520 (2019).

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Other relevant references:
1. Jazayeri, M. & Ostojic, S. Interpreting neural computations by examining intrinsic and embedding dimensionality of neural activity. arXiv:2107.04084 [q-bio] (2021).
2. Gallego, J. A., Perich, M. G., Chowdhury, R. H., Solla, S. A. & Miller, L. E. Long-term stability of cortical population dynamics underlying consistent behavior. Nat Neurosci 23260–270 (2020).
3. Bernardi, S. et al. The Geometry of Abstraction in the Hippocampus and Prefrontal Cortex. Cell 183954–967.e21 (2020).
4. Shine, J. M. et al. Human cognition involves the dynamic integration of neural activity and neuromodulatory systems. Nat Neurosci 22289–296 (2019).
5. Remington, E. D., Narain, D., Hosseini, E. A. & Jazayeri, M. Flexible Sensorimotor Computations through Rapid Reconfiguration of Cortical Dynamics. Neuron 98, 1005–1019.e5 (2018).
6. Low, R. J., Lewallen, S., Aronov, D., Nevers, R. & Tank, D. W. Probing variability in a cognitive map using manifold inference from neural dynamics. http://biorxiv.org/lookup/doi/10.1101/418939 (2018) doi:10.1101/418939.
7. Elsayed, G. F., Lara, A. H., Kaufman, M. T., Churchland, M. M. & Cunningham, J. P. Reorganization between preparatory and movement population responses in motor cortex. Nat Commun 7, 13239 (2016).
8. Peyrache, A., Lacroix, M. M., Petersen, P. C. & Buzsáki, G. Internally organized mechanisms of the head direction sense. Nat Neurosci 18569–575 (2015).
9. Dabaghian, Y., Mémoli, F., Frank, L. & Carlsson, G. A Topological Paradigm for Hippocampal Spatial Map Formation Using Persistent Homology. PLoS Comput Biol 8, e1002581 (2012).
10. Yu, B. M. et al. Gaussian-Process Factor Analysis for Low-Dimensional Single-Trial Analysis of Neural Population Activity. Journal of Neurophysiology 102614–635 (2009).
11. Singh, G. et al. Topological analysis of population activity in visual cortex. Journal of Vision 8, 11–11 (2008).

The majority of animations in this video were made using Manim — an open source python library (github.com/ManimCommunity/manim) and brainrender (github.com/brainglobe/brainrender)