Neural manifolds - The Geometry of Behaviour

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

Special thanks to Crimson Ghoul for providing English subtitles!
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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 23, 260–270 (2020).
3.Bernardi, S. et al. The Geometry of Abstraction in the Hippocampus and Prefrontal Cortex. Cell 183, 954-967.e21 (2020).
4.Shine, J. M. et al. Human cognition involves the dynamic integration of neural activity and neuromodulatory systems. Nat Neurosci 22, 289–296 (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... (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 18, 569–575 (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 102, 614–635 (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)

Icons by flaticon.com and biorender.com
Relevant resources:

Definition of a hole:
mathworld.wolfram.com/Hole.html

Cup-doughnut transformation:
   • Coffee Cup Donut  

Voltage traces and spike trains were obtained from atrificially simulated neurons using BRIAN2 python package:
Stimberg, M, Brette, R, Goodman, DFM. “Brian 2, an Intuitive and Efficient Neural Simulator.” eLife 8 (2019): e47314. doi: 10.7554/eLife.47314.

Generated spike trains were later analysed to obtain rate curves with the help of ELEPHANT:
Denker M, Yegenoglu A, Grün S (2018) Collaborative HPC-enabled workflows on the HBP Collaboratory using the Elephant framework. Neuroinformatics 2018, P19. doi: 10.12751/incf.ni2018.0019

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Socials:
VK: vk.com/atpsynthase

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