Dec 2 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 ID: 533141
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Slide1
MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology
Dec 2, 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.htmlSlide2
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) Slide3
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).Slide4
http://en.wikipedia.org/wiki/File:Gray739-emphasizing-
hippocampus.png
http://en.wikipedia.org/wiki/
File:Hippocampus.gifhttp://en.wikipedia.org/wiki/File:Hippocampal-pyramidal-
cell.png
place cells = neurons in the hippocampus that are involved in spatial navigationSlide5
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?Slide6
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?Slide7
http://
www.ploscompbiol.org
/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205Slide8
http://
www.ploscompbiol.org
/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205
Place field
= region in space where the firing rates are
significantly above baselineSlide9
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. Slide10
Creating
the
Čech
simplicial complex
1
.) B
1
… B
k
+1
≠ ⁄ , create k-simplex {v
1
, ... , v
k+1
}.
U
U
0Slide11
Creating the
Čech
simplicial complex
1
.) B
1
… B
k
+1
≠ ⁄ , create k-simplex {v
1
, ... , v
k+1
}.
U
U
0Slide12
Consider
X an arbitrary topological space.
Let
V = {V
i
| i = 1, …, n
} where Vi
X , T
he 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.pdfSlide13
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.pdfSlide14
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 periodSlide15
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 - 1Slide16
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.Slide17
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.Slide18
http://
www.ploscompbiol.org
/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205
For
each of 300 trials, N= 70 place fields
were generated
as disks of radii 0.1 L to 0.15 L, with radii and
centers chosen uniformly at random.
Centers were chosen initially uniformly at random from uncovered space. Once all space
covered
,
remaining centers chosen
at
random.Slide19
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.Slide20
Add noise.
r% spikes were
deleted from
the spike train, and then added back to the spike train atrandom times, irrespective of position along trajectory, so as to preserve overall firing rate.
Control : ‘Shuffled
’ data sets were constructed
by randomly choosing cells from each of the five environments, and pooling them together to yield population spiking activity that
did not come from a single environment.Slide21
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.1000205Slide22
Assumptions about Place
Fields
Place
fields are omni-directional
Correct assumption in open field.
Not correct in linear track.
Place cells can also
encode
angles and directions
http://
www.ploscompbiol.org
/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205Slide23
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. Slide24
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. Note: the hippocampus can undergo rapid context dependent remapping.Slide25
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. Note: the hippocampus can undergo rapid context dependent remapping.Slide26
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. Note: the hippocampus can undergo rapid context dependent remapping.Slide27
http://
www.nature.com
/
nature/journal/v416/n6876/fig_tab/416090a_F2.html
Assumptions about Place
Fields
Place fields are omni
-directional
(2) Place fields have been previously formed and are stable.
Note: the hippocampus can undergo rapid context dependent remapping.
Nature
2002
, Long-term plasticity in hippocampal place-cell representation of environmental
geometry, Colin Lever, Tom Wills, Francesca
Cacucci
, Neil Burgess, John O'KeefeSlide28
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.Slide29
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
.
Biologically: place cells multiply cover the environment.Slide30
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
H
n
, we need up to (n+1)-fold intersections to be detectable via cell groups.Slide31
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.Slide32
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 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].Slide33
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.Slide34
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 - 1Slide35
http://
www.ploscompbiol.org
/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000205
Homology calculated via
the GAP software
package:
http
://www.cis.udel.edu/linbox/gap.html
Simplices correspond to cell groups.
dimension of simplex = number of place cells in cell group - 1Slide36
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 topologySlide37
Geometry???