In recent years, ideas from TDA have been increasingly adopted in order to analyze neuroscientific data. Of particular interest are TDA techniques which can summarize the information contained in the joint activation patterns of ensmebles of neurons. This topic course will overview the main ideas in this field.
Figure 1. A filtration on a data set.
Basic familiarity with data analysis principles and with linear algebra. Familiarity with topological data analysis will be beneficial.
Here are some useful papers for reference:
(1) Cell groups reveal structure of stimulus space. https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1000205
(2) A Topological Paradigm for Hippocampal Spatial Map Formation Using Persistent Homology. https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002581
(3) The importance of forgetting: Limiting memory improves recovery of topological characteristics from neural data. (2018) https://journals.plos.org/plosone/article/comments?id=10.1371/journal.pone.0202561
(4) Evaluating State Space Discovery by Persistent Cohomology in the Spatial Representation System. https://www.frontiersin.org/articles/10.3389/fncom.2021.616748/full
(5) Flexible Memory Networks. https://link.springer.com/article/10.1007/s11538-011-9678-9
(6) What can topology tell us about the neural code? https://www.ams.org/journals/bull/2017-54-01/S0273-0979-2016-01554-0/
(7) Clique topology reveals intrinsic geometric structure in neural correlations. https://www.pnas.org/content/112/44/13455
(8) Towards a new approach to reveal dynamical organization of the brain using topological data analysis. https://www.nature.com/articles/s41467-018-03664-4
(9) Cliques and cavities in the human connectome. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5769855/
(10) Homological scaffolds of brain functional networks. https://royalsocietypublishing.org/doi/10.1098/rsif.2014.0873
(11) Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function. https://www.frontiersin.org/articles/10.3389/fncom.2017.00048/full#h10
(12) The importance of forgetting. (2019). https://media.nature.com/original/magazine-assets/d41586-019-02211-5/d41586-019-02211-5.pdf
(13) Some more papers: A bibliography by Chad Giusti, http://www.chadgiusti.com/bib.html .
Class 1. 1/11/2022. Brief discussion and introduction. Lecture by Facundo Mémoli. Lecture 1 Slides.
Class 2. 1/18/2022. Introduction to Persistence. Lecture by Facundo Mémoli.
Class 3. 1/25/2022. Multiparameter persistence. Lecture by Alex McCleary. Lecture 3 Slides.
Class 4. 2/1/2022. Introduction to RIVET and ZigZag Persistence. Lecture by Nate Clause. Lecture 4 Slides.
Class 5. 2/8/2022. The importance of forgetting. Lecture by Brantley Vose. Lecture 5 Slides.
Class 6. 2/15/2022. Persistent cohomology and an application in neuroscience (Evaluating State Space Discovery by Persistent Cohomology in the Spatial Representation System). Lecture by Ling Zhou. Lecture 6 Slides.
Class 7. 2/22/2022. Flexible Memory Networks. Lecture by Viktor Giordano. Lecture 7 Slides.
Class 8. 3/1/2022. Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function. Lecture by Amber Wu. Lecture 8 Slides.
Class 9. 3/8/2022. Cliques and cavities in the human connectome. Lecture by Aziz Burak Guelen. Lecture 9 Slides.
Class 10. 3/22/2022. The topology of the directed clique complex as a network invariant. Lecture by Shreeya Behera. Lecture 10 Slides.
Class 11. 3/29/2022. Homological scaffolds of brain functional networks. Lecture by Jesse He. Lecture 11 Slides.
Class 12. 4/5/2022. Towards a new approach to reveal dynamical organization of the brain using topological data analysis. Lecture by Jay Patel. Lecture 12 Slides.
Class 13. 4/12/2022. Hyperbolic geometry of the olfactory space. Lecture by Daiyuan Li. Lecture 13 Slides.
Class 14. 4/19/2022. Mapper Platforms for Computational Neuroimaging. Guest lecture by Dr. Samir Chowdhury.