1. Dec 4, 2013: Hippocampal spatial map formation
MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology
Dec 4, 2013: Hippocampal spatial map formation
Fall 2013 course offered through the
University of Iowa Division of Continuing Education
Isabel K. Darcy, Department of Mathematics
Applied Mathematical and Computational Sciences, University of Iowa

2. Tuesday December 10, 2013
2:00pm-2:50pm
A Topological Model of the Hippocampal Spatial Map, Yuri Dabaghian (Rice University)
Wednesday December 11, 2013
9:00am-9:50am
Topological Structures of Ensemble Neuronal Codes in the Rat Hippocampus, Zhe (Sage) Chen (Massachusetts Institute of Technology)
3:15pm-4:05pm
Topological tools for detecting hidden geometric structure in neural data, Carina Curto (University of Nebraska)

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

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

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

7. Simplices correspond to cell groups.
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

8. 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.

9. Simplices correspond to cell groups.
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

10. Idea: Can recover the topology of the space traversed by the mouse by looking only at the spiking activity of place cells.
Proof of concept: Data obtained via computer simulations of mouse trajectories using biologically relevant parameters.

11. The total duration of each simulated trajectory was 50 minutes.
a smoothed random-walk trajectory was generated, with speed = 0.1 L/s, which was constrained to ‘‘bounce’’ off boundaries and stay within the environment.
The total duration of each simulated trajectory was 50 minutes.

12. For each place cell in each trial, an average firing rate was chosen uniformly at random from the interval 2–3 Hz.
A spike train was generated from the trajectory and corresponding place field as an inhomogeneous Poisson process with constant rate when the trajectory passed inside the place field, and zero outside, so that the overall firing rate was preserved.

13. Remodeling

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

16. Assumptions about Place Fields Place fields are omni-directional
I.e. direction does not affect
the firing rate.

17. Assumptions about Place Fields Place fields are omni-directional
(2) Place fields have been previously formed and are stable.
2008

18. Assumptions about Place Fields Place fields are omni-directional
(2) Look at formation of stable place fields
2012

19. Assumptions about Place Fields Place fields are omni-directional
(2) Place fields have been previously formed and are stable.
(3) The collection of place fields corresponding to observed cells covers the entire traversed environment.
I.e., the trajectory must be dense enough to sample the majority of cell groups.

20. Assumptions about Place Fields Place fields are omni-directional
(2) Place fields have been previously formed and are stable.
(3) The collection of place fields corresponding to observed cells covers the entire traversed environment.
For accurate computation of the nth homology group Hn, we need up to (n+1)-fold intersections to be detectable via cell groups.

21. Assumptions about Place Fields Place fields are omni-directional
(2) Place fields have been previously formed and are stable.
(3) The collection of place fields corresponding to observed cells covers the entire traversed environment.
(4) The holes/obstacles are larger
than the diameters of place fields.

22. Assumptions about Place Fields
(5) Each (connected) component field of a single or multipeaked place field is convex.
(6) Background activity is low compared to the firing inside the place fields.
(7) Place fields are roughly circular and have similar sizes, as is typical in dorsal hippocampus [41,42].

23. Assumptions about Place Fields
(5) Each (connected) component field of a single or multipeaked place field is convex.
(6) Background activity is low compared to the firing inside the place fields.
(7) Place fields are roughly circular and have similar sizes, as is typical in dorsal hippocampus [41,42].
ellipsoidal

24. For each place cell in each trial, an average firing rate was chosen uniformly at random from the interval 2–3 Hz.
A spike train was generated from the trajectory and corresponding place field as an inhomogeneous Poisson process with constant rate when the trajectory passed inside the place field, and zero outside, so that the overall firing rate was preserved. Noise added.

25. time period = 2 theta cycles
Cell group = collection of place cells that co-fire within a specified time period (above a specified threshold) .
time period = 2 theta cycles
Simplices correspond to cell groups.
dimension of simplex = number of place cells in cell group - 1

26. Recovering the topology
2008
Recovering the topology
Trial is correct if Hi correct for i = 0, 1, 2, 3, 4.

27. 2012

29. N = number of cells (40 – 400 cells)
S = size of place field (10 – 90 cm)
f = firing rate (2 – 40 Hz)
10 runs for each of 10 choices for N, s, f = 10,000 trials
blue = success in first 25% of max time.
red = success required almost the full time.

30. 4.3 min, min, min.
N = number of cells (40 – 400 cells)
S = size of place field (10 – 90 cm)
f = firing rate (2 – 40 Hz)
10 runs for each of 10 choices for N, s, f = 10,000 trials
blue = success in first 25% of max time.
red = success required almost the full time.

31. 4.3 min, min, min.
N = number of cells (40 – 400 cells)
S = size of place field (10 – 90 cm)
f = firing rate (2 – 40 Hz)
10 runs for each of 10 choices for N, s, f = 10,000 trials
blue = success in first 25% of max time.
red = success required almost the full time.
the characteristic minimal map formation time is
< 2–5 mins, which is comparable to the biological learning time in rats and mice in simple environments

32. 4.3 min, min, min.

33. N = number of cells (40 – 400 cells)
S = size of place field (10 – 90 cm)
f = firing rate (2 – 40 Hz)
10 runs for each of 10 choices for N, s, f = 10,000 trials
blue = success in first 25% of max time.
red = success required almost the full time.

39. Geometry???

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

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

42. where N = number of cells.
mk = 1 – p √(k-1)/N
where N = number of cells.

43. Normalize distance so mean pairwise distances same for both simulated and calculated data.

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

45. Normalize distance so mean pairwise distances same for both simulated and calculated data.

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

47. Multipeaked place fields

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