Tuesday December 10, 2013 2 :00pm-2:50pm

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 http://www.math.uiowa.edu/~idarcy/AppliedTopology.html

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)

http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205 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).

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

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

http://arxiv.org/abs/q-bio/0702052

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 http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205

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

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 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. Proof of concept: Data obtained via computer simulations of mouse trajectories using biologically relevant parameters. http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205

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. http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205

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.

Remodeling http://arxiv.org/abs/q-bio/0702052

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

Assumptions about Place Fields Place fields are omni-directional I.e. direction does not affect the firing rate. http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205

Assumptions about Place Fields Place fields are omni-directional (2) Place fields have been previously formed and are stable. 2008 http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205

Assumptions about Place Fields Place fields are omni-directional (2) Look at formation of stable place fields 2012 http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205

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. http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205

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. http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205

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. http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205

Assumptions about Place Fields (5) Each (connected) component field of a single or multipeakedplace 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]. http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205

Assumptions about Place Fields (5) Each (connected) component field of a single or multipeakedplace 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 http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205

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.

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 http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205

Recovering the topology 2008 Trial is correct if Hi correct for i = 0, 1, 2, 3, 4. http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205

2012

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.

4.3 min, 11.7 min, 9.3 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.

4.3 min, 11.7 min, 9.3 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

4.3 min, 11.7 min, 9.3 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.

Geometry???

http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205http://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).

m1 = 1 mk = 1 – p √(k-1)/N where N = number of cells. http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205

Normalize distance so mean pairwise distances same for both simulated and calculated data. http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205

Multipeaked place fields http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205

Tuesday December 10, 2013 2 :00pm-2:50pm