1. ©2008 I.K. Darcy. All rights reserved This work was partially supported by the Joint DMS/NIGMS Initiative to Support Research in the Area of Mathematical Biology (NSF 0800285). Isabel K. Darcy Mathematics Department Applied Mathematical and Computational Sciences (AMCS) University of Iowa http://www.math.uiowa.edu/~idarcy

2. http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205 First paper to use only the spiking activity of place cells to determine the topology (and geometry) of the environment using homology (and graphs). 2008

3. http://www.nature.com/news/nobel-prize-for-decoding-brain-s-sense-of-place-1.16093 Edvard Moser May-Britt Moser John O’Keefe

4. http://www.nature.com/news/nobel-prize-for-decoding-brain-s-sense-of-place-1.16093 http://www.nature.com/news/neuroscience-brains-of- norway-1.16079 John O’Keefe Edvard Moser May-Britt Moser

5. http://www.ntnu.edu/kavli/research/grid-cell-data

6. http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205 First paper to use only the spiking activity of place cells to determine the topology (and geometry) of the environment using homology (and graphs). 2008

7. http://en.wikipedia.org/wiki/File:Gray739-emphasizing-hippocampus.png http://en.wikipedia.org/wiki/File:Hippocampus.gif http://en.wikipedia.org/wiki/File:Hippocampal-pyramidal-cell.png place cells = neurons in the hippocampus that are involved in spatial navigation

8. http://www.nytimes.com/2014/10/07/science /nobel-prize-medicine.html

9. http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1002581

10. 2012 Ignoble Prize The Ig Nobel Prizes honor achievements that make people LAUGH, and then THINK. http://www.improbable.com/ig/

11. False Positives will occur

12. http://upload.wikimedia.org/wikipedia/en/5/5e/Place_Cell_Spiking_Activity_Example.png How can the brain understand the spatial environment based only on action potentials (spikes) of place cells?

13. http://upload.wikimedia.org/wikipedia/en/5/5e/Place_Cell_Spiking_Activity_Example.png How can the brain understand the spatial environment based only on action potentials (spikes) of place cells?

14. http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205 Idea: Can recover the topology of the space traversed by the mouse by looking only at the spiking activity of place cells.

15. v2v2 e2e2 e1e1 e3e3 v1v1 v3v3 2-simplex = triangle = {v 1, v 2, v 3 } Note that the boundary of this triangle is the cycle e 1 + e 2 + e 3 = {v 1, v 2 } + {v 2, v 3 } + {v 1, v 3 } 1-simplex = edge = {v 1, v 2 } Note that the boundary of this edge is v 2 + v 1 e v1v1 v2v2 0-simplex = vertex = v Building blocks for a simplicial complex

16. 3-simplex = {v 1, v 2, v 3, v 4 } = tetrahedron boundary of {v 1, v 2, v 3, v 4 } = {v 1, v 2, v 3 } + {v 1, v 2, v 4 } + {v 1, v 3, v 4 } + {v 2, v 3, v 4 } n-simplex = {v 1, v 2, …, v n+1 } v4v4 v3v3 v1v1 v2v2 Building blocks for a simplicial complex v4v4 v3v3 v1v1 v2v2 Fill in

17. Creating a simplicial complex 0.) Start by adding 0-dimensional vertices (0-simplices)

18. Creating a simplicial complex 1.) Next add 1-dimensional edges (1-simplices). Note: These edges must connect two vertices. I.e., the boundary of an edge is two vertices

19. Creating a simplicial complex 2.) Add 2-dimensional triangles (2-simplices). Boundary of a triangle = a cycle consisting of 3 edges.

20. Creating a simplicial complex 3.) Add 3-dimensional tetrahedrons (3-simplices). Boundary of a 3-simplex = a cycle consisting of its four 2-dimensional faces.

21. Creating a simplicial complex n.) Add n-dimensional n-simplices, {v 1, v 2, …, v n+1 }. Boundary of a n-simplex = a cycle consisting of (n-1)-simplices.

22. http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1002581 Place field = region in space where the firing rates are significantly above baseline

23. Creating a simplicial complex

24. 1.) Adding 1-dimensional edges (1-simplices) Add an edge between data points that are “close”

25. Creating the Čech simplicial complex 1.) B 1 … B k+1 ≠ ⁄, create k-simplex {v 1,..., v k+1 }. UU 0

26. Creating the Čech simplicial complex 1.) B 1 … B k+1 ≠ ⁄, create k-simplex {v 1,..., v k+1 }. UU 0

27. Consider X an arbitrary topological space. Let V = {V i | i = 1, …, n } where V i X, The nerve of V = N(V) where The k -simplices of N(V) = nonempty intersections of k +1 distinct elements of V. For example, Vertices = elements of V Edges = pairs in V which intersect nontrivially. Triangles = triples in V which intersect nontrivially. http://www.math.upenn.edu/~ghrist/EAT/EATchapter2.pdf

28. Consider X an arbitrary topological space. Let V = {V i | i = 1, …, n } where V i X, The nerve of V = N(V) where The k -simplices of N(V) = nonempty intersections of k +1 distinct elements of V. For example, Vertices = elements of V Edges = pairs in V which intersect nontrivially. Triangles = triples in V which intersect nontrivially. http://www.math.upenn.edu/~ghrist/EAT/EATchapter2.pdf Čech complex = Mathematical nerve, not biological nerve

29. Creating the Čech simplicial complex 1.) B 1 … B k+1 ≠ ⁄, create k-simplex {v 1,..., v k+1 }. UU 0

30. Nerve Lemma: If V is a finite collection of subsets of X with all non-empty intersections of subcollections of V contractible, then N(V) is homotopic to the union of elements of V. http://www.math.upenn.edu/~ghrist/EAT/EATchapter2.pdf Mathematical

31. http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205 Idea: Can recover the topology of the space traversed by the mouse by looking only at the spiking activity of place cells. Vertices = place cells Add simplex if place cells co-fare within a specified time period

32. http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205 Cell group = collection of place cells that co-fire within a specified time period (above a specified threshold). Simplices correspond to cell groups. dimension of simplex = number of place cells in cell group - 1

33. 2012 Ignoble Prize The Ig Nobel Prizes honor achievements that make people LAUGH, and then THINK. http://www.improbable.com/ig/ fMRI of dead salmon The salmon was shown images of people in social situations, either socially inclusive situations or socially exclusive situations. The salmon was asked to respond, saying how the person in the situation must be feeling. http://blogs.scientificamerican.com/sc icurious-brain/2012/09/25/ignobel- prize-in-neuroscience-the-dead- salmon-study/ Activated compared to other voxels

34. http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205 Trial is correct if H i correct for i = 0, 1, 2, 3, 4. Recovering the topology

35. Remodeling: the hippocampus can undergo rapid context dependent remapping. http://arxiv.org/abs/q-bio/0702052

36. http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1002581 2012

46. Cycles Time

47. Cycles Time

48. Cycles Time

49. Cycles Time

50. Cycles Time

51. Cycles Time

52. Cycles Time

53. Cycles Time

54. Cycles Time

55. Cycles Time

56. Cycles Time

57. Cycles Time

58. Cycles Time

59. 2012

61. Data obtained via computer simulations

62. http://www.ntnu.edu/kavli/research/grid-cell-data

63. Note the above examples use the Čech complex to determine the topology of the mouse environment. But often in topological data analysis for computational efficiency, one uses the Rips complex instead of the Čech complex. Unfortunately there is no nerve lemma for the Rips complex.

64. 0.) Start by adding 0-dimensional data points Note: we only need a definition of closeness between data points. The data points do not need to be actual points in R n Creating the Vietoris Rips simplicial complex

65. Step 0.) Start by adding data points = 0-dimensional vertices (0-simplices) Creating the Vietoris Rips simplicial complex

66. 1.) Adding 1-dimensional edges (1-simplices) Add an edge between data points that are “close” Creating the Vietoris Rips simplicial complex

67. 2.) Add all possible simplices of dimensional > 1.

68. Vietoris Rips complex = flag complex = clique complex 2.) Add all possible simplices of dimensional > 1.

69. Creating the Čech simplicial complex 1.) B 1 … B k+1 ≠ ⁄, create k-simplex {v 1,..., v k+1 }. UU 0

70. Creating the Čech simplicial complex 1.) B 1 … B k+1 ≠ ⁄, create k-simplex {v 1,..., v k+1 }. UU 0