The future shape of neuroimaging with Persistent Homology Courses... · 2017-06-19 · The future...

The future shape of neuroimaging with Persistent Homology

Taking Connectivity to a Skeptical Future: Challenges, Tools and TechniquesOHBM 2017, Educational Course

Ben Cassidy

BIOSTATISTICS

Motivations for network Topological Data Analysis

Local features Global featuresMesoscopic features

NodesLinks

Whole network

Clusters? ? ? ?

2BIOSTATISTICSBen Cassidy | [email protected]

Motivations for Topological Data Analysis

• Consistency and stability - We can make consistent conclusions when observing across scales

• Inference - Escape the problem of selecting an appropriate link strength threshold for weighted functional networks

• Signal to Noise Ratio - We can separate interesting from non-interesting patterns

• Feature Engineering

• Extract principled, unsupervised features from networks

• Separate basic and complicated patterns within whole-brain networks, more than just nodes and links

• Extract distributed and possibly overlapping patterns across a network

• Basic science - Find the `shape' of brain activity


TDA

• Persistent Homology

• Mapper algorithm

• Other related areas

• Graph Signal Processing

• Clustering

• Geometry

• Manifold learning

• Morphological Signal Processing

• …


Topology basics

5

• Topology is the study of

• shape properties that are preserved under continuous deformations

• qualitative features, e.g. Homology describes the holes at each dimension: 0 = connected components, 1 = loops, 2 = voids, …

• donuts and coffee mugs

BIOSTATISTICSBen Cassidy | [email protected]

Topology basics

• Topology is the study of

• shape properties that are preserved under continuous deformations

• qualitative features, e.g. Homology describes the holes at each dimension: 0 = connected components, 1 = loops, 2 = voids, …

• donuts and coffee mugs


• Throw away any idea of distances, what do we have left?

• The underlying shape regardless of (reasonable) sampling choices

• Features that cannot be trivialised to an arbitrarily small part of the shape

Topology basics


Shape in terms of sampling• We can calculate topological invariants

• e.g. Euler characteristic, = # nodes (vertices) - # links (edges) + # faces - …

• Here is the brain activity:

• No matter how we sample the shape of brain activity, = 2

�

�

0-simplex1-simplex2-simplex

8

Vertices 4 8 6 20 12

Edges 6 12 12 30 30

Faces 4 6 8 12 20

Euler Characteristic 2 2 2 2 2


• For more complicated shapes, it is more informative to describe shape in terms of invariants

• EC can be also defined using homology invariants

• = Betti 0 - Betti 1 + Betti 2 - …

• Betti 0 = # connected components

• Betti 1 = # loops

• Betti 2 = # voids

• Computational Homology gives us Betti numbers (and a whole lot more)

• This explains why topological features are ideal for comparing across scales.

Shape in terms of invariants

�

9

� = 2 � = 0 � = �2 � = �4BIOSTATISTICSBen Cassidy | [email protected]

Homology of a network

• Which link strength threshold do we pick?

• Some holes are more important than others

101IEEE SIGNAL PROCESSING MAGAZINE | March 2016 |

nonwheeze signal appears as chaotic, and its topology is also nicely captured at a minimal computational cost.

Sensor networksThe application of algebraic topology in sensor networks is very illustrative of its utility. Owing to the difficulties of obtaining location information of the sensors in the field and to the need for additional hardware to compute precise dis-tances between pairs of sensors, it may be prohibitively expensive to obtain geometric information. Interestingly,

problems like verifying coverage are purely topological in nature, and, as discussed previously, computational topology provides a coordinate-free solution to quantifying the cover-age status as topological information.

Consider a set of sensors randomly deployed in a region to be monitored. Two problems are often of interest:

■ verifying if the region being monitored is actually fully covered and accounted for

■ discovering uncovered regions and identifying their sur-rounding nodes.

FIGURE 10. A way to approximate the coverage area, shown in (b), in sensor networks using a complex constructed from the communication graph. If the communication radius is twice the individual sensor coverage radius, the Rips shadow is a good topological approximation to the coverage area, as shown in (d). (a) Sensors in a plane. (b) Sensor coverage. (c) The Cech complex. (d) The Rips shadow.

(a) (b) (c) (d)

FIGURE 9. The wheeze-detection process. In (a), the top row corresponds to normal signals, while the bottom two rows correspond to wheeze signals. (a) Delay embeddings. (b) Various samplings of a wheeze. (c) A triangulation of a wheeze. (d) A triangulation with large edges removed.

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X(t+τ)

X (t ) X (t )

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(a) (b)

(c) (d)

101IEEE SIGNAL PROCESSING MAGAZINE | March 2016 |

nonwheeze signal appears as chaotic, and its topology is also nicely captured at a minimal computational cost.

Sensor networksThe application of algebraic topology in sensor networks is very illustrative of its utility. Owing to the difficulties of obtaining location information of the sensors in the field and to the need for additional hardware to compute precise dis-tances between pairs of sensors, it may be prohibitively expensive to obtain geometric information. Interestingly,

problems like verifying coverage are purely topological in nature, and, as discussed previously, computational topology provides a coordinate-free solution to quantifying the cover-age status as topological information.

Consider a set of sensors randomly deployed in a region to be monitored. Two problems are often of interest:

■ verifying if the region being monitored is actually fully covered and accounted for

■ discovering uncovered regions and identifying their sur-rounding nodes.

FIGURE 10. A way to approximate the coverage area, shown in (b), in sensor networks using a complex constructed from the communication graph. If the communication radius is twice the individual sensor coverage radius, the Rips shadow is a good topological approximation to the coverage area, as shown in (d). (a) Sensors in a plane. (b) Sensor coverage. (c) The Cech complex. (d) The Rips shadow.

(a) (b) (c) (d)

FIGURE 9. The wheeze-detection process. In (a), the top row corresponds to normal signals, while the bottom two rows correspond to wheeze signals. (a) Delay embeddings. (b) Various samplings of a wheeze. (c) A triangulation of a wheeze. (d) A triangulation with large edges removed.

0.6

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(c) (d)


Persistent homology


PERSISTENT TOPOLOGY OF DATA 65

Figure 3. A sequence of Rips complexes for a point cloud dataset representing an annulus. Upon increasing ϵ, holes appear anddisappear. Which holes are real and which are noise?

assume a rudimentary knowledge of homology, as is to be found in, say, Chapter 2of [15].

Despite being both computable and insightful, the homology of a complex asso-ciated to a point cloud at a particular ϵ is insufficient: it is a mistake to ask whichvalue of ϵ is optimal. Nor does it suffice to know a simple ‘count’ of the number andtypes of holes appearing at each parameter value ϵ. Betti numbers are not enough.One requires a means of declaring which holes are essential and which can be safelyignored. The standard topological constructs of homology and homotopy offer nosuch slack in their strident rigidity: a hole is a hole no matter how fragile or fine.

2.1. Persistence. Persistence, as introduced by Edelsbrunner, Letscher, andZomorodian [12] and refined by Carlsson and Zomorodian [22], is a rigorous re-sponse to this problem. Given a parameterized family of spaces, those topologicalfeatures which persist over a significant parameter range are to be considered assignal with short-lived features as noise. For a concrete example, assume thatR = (Ri)N

1 is a sequence of Rips complexes associated to a fixed point cloud for anincreasing sequence of parameter values (ϵi)N

1 . There are natural inclusion maps

(2.1) R1ι

↪→ R2ι

↪→ · · · ι↪→ RN−1

ι↪→ RN .

Instead of examining the homology of the individual terms Ri, one examines thehomology of the iterated inclusions ι : H∗Ri → H∗Rj for all i < j. These mapsreveal which features persist.

As a simple example, persistence explains why Rips complexes are an acceptableapproximation to Cech complexes. Although no single Rips complex is an especiallyfaithful approximation to a single Cech complex, pairs of Rips complexes ‘squeeze’the appropriate Cech complex into a manageable hole.

Ghrist, 2008

Persistent homology

12

d2d1 d3


Persistent homology

• Data — Filtration — Persistence module — Barcode


Ghrist, 2008

PERSISTENT TOPOLOGY OF DATA 67

which come into existence at parameter ti and which persist for all future parame-ter values. The torsional elements correspond to those homology generators whichappear at parameter rj and disappear at parameter rj + sj . At the chain level,the Structure Theorem provides a birth-death pairing of generators of C (exceptingthose that persist to infinity).

2.3. Barcodes. The parameter intervals arising from the basis for H∗(C; F ) inEquation (2.3) inspire a visual snapshot of Hk(C; F ) in the form of a barcode. Abarcode is a graphical representation of Hk(C; F ) as a collection of horizontal linesegments in a plane whose horizontal axis corresponds to the parameter and whosevertical axis represents an (arbitrary) ordering of homology generators. Figure 4gives an example of barcode representations of the homology of the sampling ofpoints in an annulus from Figure 3 (illustrated in the case of a large number ofparameter values ϵi).

H0

H1

H2ϵ

ϵ

ϵ

Figure 4. [bottom] An example of the barcodes for H∗(R) in theexample of Figure 3. [top] The rank of Hk(Rϵi) equals the numberof intervals in the barcode for Hk(R) intersecting the (dashed) lineϵ = ϵi.

Theorem 2.3 yields the fundamental characterization of barcodes.

Theorem 2.4 ([22]). The rank of the persistent homology group Hi→jk (C; F ) is

equal to the number of intervals in the barcode of Hk(C; F ) spanning the parameterinterval [i, j]. In particular, H∗(Ci

∗; F ) is equal to the number of intervals whichcontain i.

A barcode is best thought of as the persistence analogue of a Betti number.Recall that the kth Betti number of a complex, βk := rank(Hk), acts as a coarsenumerical measure of Hk. As with βk, the barcode for Hk does not give any in-formation about the finer structure of the homology, but merely a continuously

13

Examples

• Human Connectome Project resting state data

• Three examples

★ Architecture (presence of links) vs dynamics (link strength)

★ Consistency between parcellations

★ Persistent homology of sparse networks

BIOSTATISTICSBen Cassidy | [email protected] 14

barcode

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Difference between architecture and dynamics

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bens-area


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Caudate

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parahippocampal

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superiortemporal

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Accum

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Cerebellum-Corte

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Cerebellum-Corte

posteriorcingulate

posteriorcingulate

Thalamus-Proper

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per

parahippocampal

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isthmuscingulate

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middletemporalmidd

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Cerebellum-Corte

posteriorcingulate

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Thalamus-Proper

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per

parahippocampal

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superiortemporal

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Full n

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arch

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ure


DISCOH Sparse Partial Correlation Partial Correlation Correlation

0 0.5 1

Filtration distance

1

87

fβ0(d)

0 0.5 1

Filtration distance

1

87

fβ0(d)

0 0.5 1

Filtration distance

1

87

fβ0(d)

0 0.5 1

Filtration distance

1

87

fβ0(d)

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0 0.5 1

Filtration distance

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0 0.5 1

Filtration distance

0

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No

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d d

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0 0.5 1

Filtration distance

1

30

60

90

β0

0 0.5 1

Filtration distance

1

30

60

90

β0

barcodeH0

Consistency between parcellationsFRO

INS

LIM

TEM

PAR

OCC

SBC

CEBBSTCEBSBC

OCC

PAR

TEM

LIM

INS

FRO

SupOcS/TrOcS

SupOcS/TrO

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MPosCg

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LoInG/CInSLoInG

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TrFPoG/S Tr

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MACgG/S

MACgG

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PosTrCoS

PosTrCoS

SupCirInSSupCirIn

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CoS/LinS

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InfFGTrip

InfFGOpp

PosDCgG

InfFGTrip

InfFGOpp

PosDCgG

SupTGLp

InfOcG/S

SupTGLp

InfOcG/S

FMarG/S

PosVCgG

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PosVCgG

InfFG

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InfFGOrp

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ALSHorpALSHorp

SupPrCS

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ALSVerpALSVerp

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SupOcG

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PaCL/S

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InfPrCS

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PerCaS

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LOcTS

LOcTS

SbCaG

SbCaG

SupTS

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SupFG

PosCG

SupFG

PosCG

SupPL

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SupFS

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SupFS

PosCS

MOcG

PosCS

MOcG

PosLSPosLS

PrCun

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InfTG

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POcS

POcSAngG

OcPo

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PrCG

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LOrS

InfFS

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InfFS

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MFG M

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MFS M

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medialorbitofrontal

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transversetemporal

transve

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posteriorcingulate

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Thalamus-Proper

Thalamus-Pro

per

parahippocampal

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isthmuscingulate

superiortemporal

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Accum

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Accum

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middletemporalmidd

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superiorparietal

superiorparietal

parstrian

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parsop

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Hippocampus

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Brain-Stem

paracentral

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paracentral

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InfFGOrp In

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Putamen

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bankssts

Pallidum

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Caudate

Caudate

fusiformfus

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cuneus

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lingual

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lingual


0.97 0.98 0.99 1

50

100

150

200

Dataset 4

0.97 0.98 0.99 1

50

100

150

200Dataset 3

0.97 0.98 0.99 1

Filtration distance

50

100

150

200

No

rma

lise

d d

eriva

tive

Dataset 1

87 nodes

167 nodes

0.97 0.98 0.99 1

50

100

150

200Dataset 2

Persistent homology of a sparse network

0.988 0.99 0.992 0.994 0.996 0.998 1

Filtration Distance

0

50

100

Be

tti 1

0.99997 0.99998 0.99999 1

Filtration Distance

0

50

100

150

Be

tti 2

0.999995 0.999996 0.999997 0.999998 0.999999 1

Filtration Distance

0

50

100

150

Be

tti 3


H1

H2 H3

FRO

INS

LIM

TEM

PAR

OCC

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CEBBSTCEBSBC

OCC

PAR

TEM

LIM

INS

FRO

SupOcS/TrOcS

SupOcS/TrO

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MPosCg

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MOcS/LuS

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TrFPoG/S Tr

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MACgG/S

MACgG

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CoS/LinS

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InfFGTrip

InfFGOpp

PosDCgG

InfFGTrip

InfFGOpp

PosDCgG

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InfOcG/S

SupTGLp

InfOcG/S

FMarG/S

PosVCgG

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PosVCgG

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InfFGOrp

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ALSVerpALSVerp

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SbCG/S

SupO

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SupOcG

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PaCL/S

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InfPrCS

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PaHipG

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PerCaS

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ShoInGShoIn

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LOcTS

LOcTS

SbCaG

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SupTS

SupTS

SupFG

PosCG

SupFG

PosCG

SupPL

SupPL

SbOrS

SupFS

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SupFS

PosCS

MOcG

PosCS

MOcG

PosLSPosLS

PrCun

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AOcS

AOcS

InfTG

InfTG

POcS

POcSAngG

OcPo

AngG

OcPo

TrTS

InfTS

TrTS

InfTS

PrCG

SbPS

PrCG

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NAcc

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LOrS

InfFS

MTG

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InfFS

MTG

MFG M

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VDC

VDC

MFS M

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PoPl

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Amg

LinG

LinG

TPo

CaN

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CaN

OrG O

rG

CeB

CeB

FuG

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OrS

CcS

Tha

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CcS

Tha

Cun

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TPl

BSt

TPl

Hip

Hip

HG

HG

RG

Pal

RG

Pal

CS

CS

Pu

Pu

JS

JS0 0.2 0.4 0.6 0.8 1

Filtration Distance

0

20

40

60

80

100

Be

tti 0

H0

Persistent homology of a sparse network

0.988 0.99 0.992 0.994 0.996 0.998 1

Filtration Distance

0

50

100

Be

tti 1

H1

x400 networks

Ben Cassidy | [email protected] 22

Reading list• Chad Giusti’s website for TDA in neuroscience

• http://www.chadgiusti.com/bib.html

• Introductions to TDA:

• Ghrist, 2008, “Barcodes: the persistent topology of data”

• Carlsson, 2009, “Topology and data”

• Krim et al., 2016, “Discovering the Whole by the Coarse: A topological paradigm for data analysis”

• A few recent neuroimaging/neuroscience applications:

• Memory gating Hahm et al. 2017, “Gating of memory encoding of time-delayed cross-frequency MEG networks revealed by graph filtration based on persistent homology”

• Brain arteries Bendich et al., 2016, “Persistent homology analysis of brain artery trees”

• Time varying connectivity Yoo et al., 2016, “Topological Persistence Vineyard for Dynamic Functional Brain Connectivity during Resting and Gaming Stages”

• Multimodal networks Lee et al., 2016, “Integrated multimodal network approach to PET and MRI based on multidimensional persistent homology”

• Anatomical networks Chung et al., 2015, “Persistent homology in sparse regression and its application to brain morphometry”

• TDA for Neuroscience Curto, 2015, “What can topology tell us about the neural code?”


Posters at OHBM

• 1855, Tuesday Lee et al., “Homological changes of metabolic connectivity during the transition to Alzheimer's disease”

• 4182, Wednesday Suryadevara et al., “Topological data analysis of fMRI signals in the hippocampus during learning: Function to structure”

• 4176, Wednesday Riaño et al., “TDA barcodes to identify topological features of resting state fMRI time courses in healthy subjects”

• 4189, Thursday Garg et al., “Persistence homology of brain geometry: a marker for preterm birth”

• 4174, Wednesday BC et al., “Exploratory Multidimensional Persistent Homology of Functional Connectivity Networks”

• 4061, Thursday, BC et al., “Sparse Functional Connectivity”


The future shape of neuroimaging with Persistent Homology Courses... · 2017-06-19 · The future...

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