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Networks, as efficient representations of complex systems, have appealed to scientists for a long time and now permeate many areas of science, including neuroimaging (Bullmore and Sporns 2009 Nat. Rev. Neurosci. 10, 186–198. (doi:10.1038/nrn2618)). Traditionally, the structure of complex networks has been studied through their statistical properties and metrics concerned with node and link properties, e.g. degree-distribution, node centrality and modularity. Here, we study the characteristics of functional brain networks at the mesoscopic level from a novel perspective that highlights the role of inhomogeneities in the fabric of functional connections. This can be done by focusing on the features of a set of topological objects—homological cycles—associated with the weighted functional network. We leverage the detected topological information to define the homological scaffolds, a new set of objects designed to represent compactly the homological features of the correlation network and simultaneously make their homological properties amenable to networks theoretical methods. As a proof of principle, we apply these tools to compare resting-state functional brain activity in 15 healthy volunteers after intravenous infusion of placebo and psilocybin—the main psychoactive component of magic mushrooms. The results show that the homological structure of the brain's functional patterns undergoes a dramatic change post-psilocybin, characterized by the appearance of many transient structures of low stability and of a small number of persistent ones that are not observed in the case of placebo.
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Research
Cite this article: Petri G, Expert P, Turkheimer
F, Carhart-Harris R, Nutt D, Hellyer PJ,
Vaccarino F. 2014 Homological scaffolds of
brain functional networks. J. R. Soc. Interface
11: 20140873.
http://dx.doi.org/10.1098/rsif.2014.0873
Received: 5 August 2014
Accepted: 3 October 2014
Subject Areas:
computational biology
Keywords:
brain functional networks, fMRI, persistent
homology, psilocybin
Author for correspondence:
P. Expert
e-mail: paul.expert@kcl.ac.uk
Electronic supplementary material is available
at http://dx.doi.org/10.1098/rsif.2014.0873 or
via http://rsif.royalsocietypublishing.org.
Homological scaffolds of brain functional
networks
G. Petri1, P. Expert2, F. Turkheimer2, R. Carhart-Harris3, D. Nutt3, P. J. Hellyer4
and F. Vaccarino1,5
1
ISI Foundation, Via Alassio 11/c, 10126 Torino, Italy
2
Centre for Neuroimaging Sciences, Institute of Psychiatry, Kings College London, De Crespigny Park,
London SE5 8AF, UK
3
Centre for Neuropsychopharmacology, Imperial College London, London W12 0NN, UK
4
Computational, Cognitive and Clinical Neuroimaging Laboratory, Division of Brain Sciences, Imperial College
London, London W12 0NN, UK
5
Dipartimento di Scienze Matematiche, Politecnico di Torino, C.so Duca degli Abruzzi no 24, Torino 10129, Italy
Networks, as efficient representations of complex systems, have appealed to
scientists for a long time and now permeate many areas of science, including
neuroimaging (Bullmore and Sporns 2009 Nat. Rev. Neurosci. 10, 186– 198.
(doi:10.1038/nrn2618)). Traditionally, the structure of complex networks has
been studied through their statistical properties and metrics concerned with
node and link properties, e.g. degree-distribution, node centralityand modular-
ity. Here, we study the characteristics of functional brain networks at the
mesoscopic level from a novel perspective that highlights the role of inhomo-
geneities in the fabric of functional connections. This can be done by focusing
on the features of a set of topological objectshomological cycles—associated
with the weighted functional network. We leverage the detected topological
information to define the homological scaffolds, a new set of objects designed to
represent compactly the homological features of the correlation network and
simultaneously maketheir homological properties amenable to networkstheor-
etical methods. As a proof of principle, we apply these tools to compare resting-
state functional brain activity in 15 healthy volunteers after intravenous infusion
of placebo and psilocybin—the main psychoactive component of magic mush-
rooms. The results show that the homological structure of the brain’s functional
patterns undergoes a dramatic change post-psilocybin, characterized by the
appearance of many transient structures of low stability and of a small
number of persistent ones that are not observed in the case of placebo.
1. Motivation
The understanding of global brain organization and its large-scale integration
remains a challenge for modern neurosciences. Network theory is an elegant frame-
work to approach these questions, thanks to its simplicity and versatility [1]. Indeed,
in recent years, networks have become a prominent tool to analyse and understand
neuroimaging datacoming from very diverse sources, such as functional magnetic
resonance imaging (fMRI), electroencephalography and magnetoencephalography
[2,3], also showing potential for clinical applications [4,5].
A naturalway of approaching these datasets is to devise a measure of dynami-
cal similarity between the microscopic constituents and interpret it as the strength
of the link between those elements. In the case of brain functional activity, this often
implies the use of similarity measures such as (partial) correlations or coherence
[6– 8], which generally yield fully connected, weighted and possibly signed adja-
cency matrices. Despite the fact that most network metrics can be extended to
the weighted case [9– 13], the combined effect of complete connectedness and
edge weights makes the interpretation of functional networks significantly
harder and motivates the widespread use of ad hoc thresholding methods
[7,14– 18]. However, neglecting weak links incurs the dangers of a trade-off
&2014 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution
License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original
author and source are credited.
between information completeness and clarity. In fact, it risks
overlooking the role that weak links might have, as shown for
example in the cases of resting-state dynamics [19,20], cognitive
control [21] and correlated network states [22].
In order to overcome these limits, Rubinov & Sporn
[13,23,24] recently introduced a set of generalized network and
community metrics for functional networks that among others
were used to uncover the contrasting dynamics underlying
recollection [25] and the physiology of functional hubs [26].
In this paper, we present an alternative route to the analy-
sis of brain functional networks. We focus on the combined
structure of connections and weights as captured by the hom-
ology of the network. A summary of all the keywords and
concepts introduced in this paper can be found in table 1.
2. From networks to topological spaces and
homology
Homology is a topological invariant that characterizes a topo-
logical space Xby counting its holes and their dimensions. By
hole, we mean a hollow region bounded by the parts of that
space. The dimension of a hole is directly related to the dimen-
sion of its boundary. The boundary of a two-dimensional
hole is a one-dimensional loop; the three-dimensional inner
part of a doughnut, where the filling goes, is bounded by
two-dimensional surface; for dimensions higher than 2, it
becomes difficult to have a mental representation of a hole,
but k-dimensional holes are still bounded by (k21) dimen-
sional faces. In our work, we start with a network and from
it construct a topological space. We now use figure 1 to
show how we proceed and make rigorous what we mean by
boundaries and holes.
In a network like that of figure 1a, we want the ring of nodes
(a,b,c,d) to bea good candidate fora one-dimensional boundary,
whereas the other rings of three nodes should not constitute
interesting holes. The reason for this choice comes from the for-
malization of the notion of hole. One way to formalize this is by
opposition that is we define what we mean by a dense subnet-
work in order to highlight regions of reduced connectivity, i.e.
holes. The most natural and conservative definition we can
adopt for a dense subnetwork is that of clique, a completely
jja
d
k
c
b
i
e
f
g
h
l
a
c
i
e
f
h
l
g
d
k
b
(a)(b)
Figure 1. Panels (a,b) display an unweighted network and its clique complex, obtained by promoting cliques to simplices. Simplices can be intuitively thought as
higher-dimensional interactions between vertices, e.g. as a simplex the clique (b,c,i) corresponds to a filled triangle and not just its sides. The same principle applies
to cliques—thus simplices—of higher order. (Online version in colour.)
Table 1. List of notations.
name symbol definition
graph Ga graph G¼(V,E) is a representation of a set Vof nodes iinterconnected by edges or links eij [E; this
interaction can be weighted, directional and signed
clique ca completely connected subgraph C¼(V’,E’) contained in an undirected and unweighted graph
G¼(V,E)(V0,V,eij [E08i,j[E0)
k-simplex
s
k
formally, a convex hull of kþ1 nodes [ p
0
,p
1
,...,p
k
], it is used here as a generalization to higher
dimensions of the concept of link, e.g. a 2-simplex is a triangle, a 3-simplex a tetrahedron. The faces f
of
s
k
are obtained as subsets of [ p
0
,p
1
,...,pk]
simplicial complex Ka topological space composed by attaching simplices
s
, with two conditions: (i) if
s
[Kthen all its
faces f[Kand (ii) the intersection of any two
s
i,
s
j[Kis empty or a face of both
s
i
and
s
j
.
clique complex l(G) a simplicial complex built from a graph Gby promoting every k-clique c,Gto a (k21)-simplex
defined by the nodes of c, e.g. a 3-clique becomes a 2-simplex (a full triangle)
kth homology group Hk(K) a group describing the holes of a simplicial complex Kbounded by k-dimensional boundaries, e.g. H
1
describes two-dimensional cycles bounded by 1-simplices, H
2
describes three-dimensional voids bounded
by 2-simplices, etc.
H
k
generator gan element of the generating set of H
k
, a subset of H
k
such that all elements can be expressed as
combination of generators
homological scaffold H(K) a weighted graph constructed from the persistent homology generators of H
1
of a simplicial complex K
rsif.royalsocietypublishing.org J. R. Soc. Interface 11: 20140873
2
connected subgraph [27]. Moreover, cliques have the crucial
property, which will be important later, of being nested, i.e. a
clique of dimension k(k-clique) contains all the m-cliques
defined by its nodes with m,k. Using this definition and filling
in all the maximal cliques, the network in figure 1acan be rep-
resented as in figure 1b: 3-cliques are filled in, becoming tiles,
and the only interesting structure left is the square (a,b,c,d). It
is important at this point to note that a k-clique can be seen as
ak21 simplex, i.e. as the convex hull of k-points. Our represen-
tation of a network can thus be seen as a topological space
formed by a finite set of simplices that by construction satisfy
the condition that defines the type of topological spaces called
abstract simplicial complexes [28]: each element of the space is
a simplex, and each of its faces (or subset in the case of
cliques) is also a simplex.
This condition is satisfied, because each clique is a sim-
plex, and subsets of cliques are cliques themselves, and the
intersection of two cliques is still a clique.
The situation with weighted networks becomes more
complicated. In the context of a weighted network, the
holes can be thought of as representing regions of reduced
connectivity with respect to the surrounding structure.
Consider, for example, the case depicted in figure 2a:the
network is almost the same as figure 1 with the two excep-
tions that it now has weighted edges and has an additional
very weak edge between nodes aand c. The edges in the
cycle [a,b,c,d] are all much stronger than the link (a,c)that
closes the hole by making (a,b,d) and (b,c,d) cliques and there-
fore fills them. The loop (e,f,g,h,i) has a similar situation, but
the difference in edge weights between the links along the
cycle and those crossing, is not as large as in the previous
case. It would be therefore useful to be able to generalize
the approach exposed earlier for binary networks to the case
of weighted networks in such a way as to be able to measure
the difference between the two cases (a,b,c,d)and(e,f,g,h,i). As
shown by figure 2b, this problem can be intuitively thought of
as a stratigraphy in the link-weight fabric of the network,
where the aim is to detect the holes, measure their depth
and when they appear as we scan across the weights’ range.
From figure 2b, it becomes clear that the added value of
this method over conventional network techniques lies in its
capability to describe mesoscopic patterns that coexist over
different intensity scales, and hence to complement the infor-
mation about the community structure of brain functional
networks. A way to quantify the relevance of holes is given
by persistent homology. We describe it and its application to the
case of weighted networks in full detail in §3.
3. A persistent homology of weighted networks
The method that we adopt was introduced in references
[29,30] and relies on an extension of the metrical persistent
homology theory originally introduced by references [31,32].
Technical details about the theory of persistent homology
and how the computation is performed can be found
in the works of Carlsson, Zomorodian and Edelsbrunner
[28,31– 35]. Persistent homology is a recent technique in com-
putational topology developed for shape recognition and
the analysis of high dimensional datasets [36,37]. It has been
used in very diverse fields, ranging from biology [38,39]
and sensor network coverage [40] to cosmology [41]. Similar
approaches to brain data [42,43], collaboration data [44] and
network structure [45] also exist. The central idea is the con-
struction of a sequence of successive approximations of the
original dataset seen as a topological space X. This sequence
of topological spaces X
0
,X
1
,...,X
N
¼Xis such that
Xi#Xjwhenever i,jand is called the filtration. Choosing
how to construct a filtration from the data is equivalent to
choosing the type of goggles one wears to analyse the data.
In our case, we sort the edge weights in descending
order and use the ranks as indices for the subspaces. More
specifically, denote by
V
¼(V,E,
v
) the functional network
with vertices V, edges Eand weights
v
:E!R. We then con-
sider the family of binary graphs G
v
¼(V,E
v
), where an edge
e[Eis also included in G
v
if its weight
v
e
is larger than
v
(e[E
v
,
v
(e)
v
).
To each of the G
v
, we associate its clique,orflag complex K
v
,
that is the simplicial complex that contains the k-simplex [n
0
,
n
1
,n
2
,...n
k21
] whenever the nodes n
0
,n
1
,n
2
,...n
k21
define
acliqueinG
v
[27]. As subsets of cliques and intersections of
cliques are cliques themselves, as we pointed out in §2, our
clique complex is thus a particular case of a simplicial
complex.
The family of complexes fK
v
gdefines a filtration, because
we have K
v
#K0
v
for
v
.
v
0. At each step, the simplices in K
v
inherit their configuration from the underlying network
structure and, because the filtration swipes across all weight
scales in descending order, the holes among these units con-
stitute mesoscopic regions of reduced functional connectivity.
Moreover, this approach also highlights how network
properties evolve along the filtration, providing insights
about where and when lower connectivity regions emerge.
This information is available, because it is possible to keep
track of each k-dimensional cycle in the homology group H
k
.
A generator uniquely identifies a hole by its constituting
elements at each step of the filtration process. The importance
of a hole is encoded in the form of ‘time-stamps’ recording its
birth
b
g
and death
d
g
along the filtration fK
v
g[31]. These two
time-stamps can be combined to define the persistence
p
g
¼
d
g
2
b
g
of a hole, which gives a notion of its importance in
terms of it lifespan. Continuing the analogy with stratigraphy,
b
g
and
d
g
correspond, respectively, to the top and the bottom
of a hole and
p
g
would be its depth. As we said above, a genera-
tor gk
i, or hole, of the kth homology group H
k
is identified by its
birth and death along the filtration. Therefore, gk
iis described by
the point (
b
g,
d
g)[R2. A standardway to summarize the infor-
mation about the whole kth persistent homology group is then
to consider the diagram obtained plotting the points corre-
sponding to the set of generators. The (multi)set f(
b
g
,
d
g
gis
called the persistence diagram of H
k
. In figure 2c, we show the
persistence diagram for the network shown in figure 2afor
H
1
. Axes are labelled by weights in decreasing order. It is easy
to check that the coordinates correspond exactly to the appear-
ance and disappearance of generators. The green vertical bars
highlight the persistence of a generator along the filtration.
The further a point is from the diagonal (vertically), the more
persistent the generator is. In §4, we introduce two objects, the
persistence and the frequency homological scaffolds, designed
to summarize the topological information about the system.
4. Homological scaffolds
Once one has calculated the generators {gk
i}iof the kth persist-
ent homology group H
k
, the corresponding persistence
rsif.royalsocietypublishing.org J. R. Soc. Interface 11: 20140873
3
diagram contains a wealth of information that can be used,
for example, to highlight differences between two datasets.
It would be instructive to obtain a synthetic description of
the uncovered topological features in order to interpret the
observed differences in terms of the microscopic components,
at least for low dimensions k. Here, we present a scheme to
obtain such a description by using the information associated
with the generators during the filtration process. As
each generator, gk
iis associated with a whole equivalence
class, rather than to a single chain of simplices, we need to
j10
appearance of
(a,b,c,d) hole
10
(a)
(b)
(c)
8
6
4
weights
2
0
0
1
2
3
4
5
death weight
6
7
8
9
98765
birth wei
g
ht
43210
closure of
(a,b,c,d) hole
appearance of
(e,f,g,h,i) hole
closure of
(e,f,g,h,i) hole
8
8
8
1
10
10
10
6
5
6
6
4
5
4
8
10
a
d
k
c
b
i
e
f
g
h
l
(a,b,c,d)
(e,f,g,h,i)
Figure 2. Panels (a–c) display a weighted network (a), its intuitive representation in terms of a stratigraphy in the weight structure according the weight filtration
described in the main text (b) and the persistence diagram for H
1
associated with the network shown (c). By promoting cliques to simplices, we identify network
connectivity with relations between the vertices defining the simplicial complex. By producing a sequence of networks through the filtration, we can study the
emergence and relative significance of specific features along the filtration. In this example, the hole defined by (a,b,c,d) has a longer persistence (vertical solid
green bars) implying that the boundary of the cycle are much heavier than the internal links that eventually close it. The other hole instead has a much shorter
persistence, surviving only for one step and is therefore considered less important in the description of the network homological properties. Note that the births and
deaths are defined along the sequence of descending edge weights in the network, not in time. (Online version in colour.)
rsif.royalsocietypublishing.org J. R. Soc. Interface 11: 20140873
4
choose a representative for each class, we use the representa-
tive that is returned by the javaplex implementation [46] of the
persistent homology algorithm [47]. For the sake of simplicity
in the following, we use the same symbol gk
ito refer to a
generator and its representative cycle.
We exploit thisto definetwo new objects, the persistence and
the frequency homological scaffolds Hp
Gand Hf
Gof a graph G.The
persistence homological scaffold is the network composed of all
the cycle paths corresponding to generators weighted by their
persistence. If an edge ebelongs to multiple cycles g
0
,g
1
,...,g
s
,
its weight is defined as the sum of the generators’ persistence:
vp
e¼X
gije[gi
p
gi:(4:1)
Similarly, we define the frequency homological scaffold Hf
Gas
the network composed of all the cycle paths corresponding
to generators, where this time, an edge eis weighted by the
number of different cycles it belongs to
v
f
e¼X
gi
1e[gi,(4:2)
where 1e[giis the indicator function for the set of edges com-
posing g
i
. By definition, the two scaffolds have the same edge
set, although differently weighted.
The construction of these two scaffolds therefore high-
lights the role of links which are part of many and/or long
persistence cycles, isolating the different roles of edges
within the functional connectivity network. The persistence
scaffolds encodes the overall persistence of a link through
the filtration process: the weight in the persistence scaffold
of a link belonging to a certain set of generators is equal to
the sum of the persistence of those cycles. The frequency scaf-
fold instead highlights the number of cycles to which a link
belongs, thus giving another measure of the importance of
that edge during the filtration. The combined information
given by the two scaffolds then enables us to decipher the
nature of the role different links have regarding the homolo-
gical properties of the system. A large total persistence for a
link in the persistence scaffold implies that the local structure
around that link is very weak when compared with the
weight of the link, highlighting the link as a locally strong
bridge. We remark that the definition of scaffolds we gave
depends on the choice of a specific basis of the homology
group, and the choice of a consistent basis is an open problem
in itself, therefore the scaffolds are not topological invariants.
Moreover, it is possible for an edge to be added to a cycle
shortly after the cycle’s birth in such a way that it creates a
triangle with the two edges composing the cycle. In this
way, the new edge would be part of the shortest cycle, but
the scaffold persistence value would be misattributed to the
two other edges. This can be checked, for example, by moni-
toring the clustering coefficient of the cycle’s subgraph as
edges are added to it. We checked for this effect and found
that in over 80% of the cases the edges do not create triangles
that would imply the error, but instead new cycles are cre-
ated, whose contribution to the scaffold is then accounted
for by the new cycle. Finally, we note also that, when a
new triangle inside the cycle is created, the two choices of
generator differ for a path through a third strongly connected
node, owing to the properties of boundary operators. Despite
this ambiguity, we show in the following that they can be
useful to gain an understanding of what the topological
differences detected by the persistent homology actually
mean in terms of the system under study.
5. Results from fMRI networks
We start from the processed fMRI time series (see Methods for
details). The linear correlations between regional time series
were calculated after covarying out the variance owing to all
other regions and the residual motion variance represented
by the 24 rigid motion parameters obtained from the pre-
processing, yielding a partial-correlation matrix
x
a
for each
subject. The matrices
x
a
were then analysed with the algorithm
described in the previous sections. We calculated the generators
g1
iof the first homological group H
1
along the filtration. As
mentioned before, each of these generators identifies a lack of
mesoscopic connectivity in the form of a one-dimensional
cycle andcan be represented in a persistence diagram. Weaggre-
gate together the persistence diagrams of subjects belonging to
each group and compute an associated persistence probability
density (figure 3). These probability density functions constitute
the statistical signature of the groups’ H
1
features.
We find that, although the number of cycles in the groups
are comparable, the two probability densities strongly differ
(Kolmogorov– Smirnov statistics: 0.22, p-value less than 10
210
).
The placebo group displays generators appearing and per-
sisting over a limited interval of the filtration. On the contrary,
most of the generators for the psilocybin group are situated in
a well-defined peak at small birth indices, indicating a shorter
average cycle persistence. However, the psilocybin distribution
is also endowed with a longer tail implying the existence of
a few cycles that are longer-lived compared with the placebo
condition and that influences the weight distribution of the psi-
locybin persistence scaffold. The difference in behaviour of the
two groups is made explicit when looking at the probability dis-
tribution functions for the persistence and the birth of
generators (figure 4), which are both found to be significantly
different (Kolmogorov–Smirnov statistics: 0.13, p-value ,
10
230
for persistence and Kolmogorov–Smirnov statistics:
0.14, p-value ,10
235
for births). In order to better interpret
and understand the differences between the two groups,
we use the two secondary networks described in §4, Hf
Pla and
Hp
Pla for the placebo group and Hf
Psi and Hp
Psi for the psilocybin
group. The weight of the edges in these secondary networks is
proportional to the total number of cycles an edge is part of, and
the total persistence of those cycles, respectively. They comp-
lement the information given by the persistence density
distribution, where the focus is on the entire cycle’s behaviour,
with information on single links. In fact, individual edges
belonging to many and long persistence cycles represent func-
tionally stable ‘hub’ links. As with the persistence density
distribution, the scaffolds are obtained at a group level by
aggregating the information about all subjects in each group.
These networks are slightly sparser than the original complete
x
a
networks
r
(Hf,p
pla )¼2m(Hp
pla)
n(n1) ¼0:92 (5:1)
and
r
(Hf,p
Psi )¼2m(Hp
Psi)
n(n1) ¼0:91 (5:2)
and have comparable densities. A first difference between the
rsif.royalsocietypublishing.org J. R. Soc. Interface 11: 20140873
5
two groups becomes evident when we look at the distributions
for the edge weights (figure 5a). In particular, the weights of
Hp
pla display a cut-off for large weights, whereas the weights
of Hp
psi have a broader tail (Kolmogorov– Smirnov statistics:
0.06, p-value ,10
220
;figure5a). Interestingly, the frequency
scaffold weights probability density functions cannot be distin-
guished from each other figure 5a(inset) (Kolmogorov–
Smirnov statistics: 0.008, p-value ¼0.72). Taken together,
these two results imply that while edges statistically belong to
the same number of cycles, in the psilocybin scaffold, there
exist very strong, persistent links.
The difference between the two sets of homological scaf-
folds for the two groups becomes even more evident when
one compares the weights between the frequency and
persistence scaffolds of the same group. Figure 5bis a scatter
plot of between the weights of edges from both scaffolds for
the two groups. The placebo group has a linear relationship
between the two quantities meaning that edges that are per-
sistent also belongs to many cycles (R
2
¼0.95, slope ¼0.23).
Although the linear relationship is still a reasonable fit for
the psilocybin group (R
2
¼0.9, slope ¼0.3), the data in this
case display a larger dispersion. In particular, it shows that
edges in Hf,p
psi can be much more persistent/longer-lived
than in Hf,p
pla but still appear in the same number of cycles,
i.e. the frequency of a link is not predictive of its persistence
or simply put: some connections are much more persistent in
the psychedelic state. Moreover, the slopes of linear fits of the
two clouds are statistically different ( p-value ,10
220
,n
pla
¼
1.0
0.9
0.8
0.7
0.6
0.5
death
death
0.4
0.3
0.2
birth birth
(a)(b)
0.1
0
–10.0 –9.5 –9.0 –8.5 –8.0 –7.5 –7.0 –6.5 –10.0 –9.6 –9.2 –8.8 –8.4 –7.6–8.0 –7.2 –6.8
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Figure 3. Probability densities for the H
1
generators. Panel (a) reports the (log-)probability density for the placebo group, whereas panel (b) refers to the psilocybin
group. The placebo displays a uniform broad distribution of values for the births– deaths of H
1
generators, whereas the plot for the psilocybin condition is very
peaked at small values with a fatter tail. These heterogeneities are evident also in the persistence distribution and find explanation in the different functional
integration schemes in placebo and drugged brains. (Online version in colour.)
0
0.01
0.02
P(p)
0.03
0.04
0.05
10–5
0.0001
P(b)
0.001
0.01
placebo
psilo
0.1
(a)(b)
0.2 0.4 0.6
persistence p
0.8 1.0 0 0.2 0.4 0.6
birth
b
0.8 1.0
Figure 4. Comparison of persistence
p
and birth
b
distributions. Panel (a) reports the H
1
generators’ persistence distributions for the placebo group (blue line) and
psilocybin group (red line). Panel (b) reports the distributions of births with the same colour scheme. It is very easy to see that the generators in the psilocybin
condition have persistence peaked at shorter values and a wider range of birth times when compared with the placebo condition. (Online version in colour.)
rsif.royalsocietypublishing.org J. R. Soc. Interface 11: 20140873
6
13 200 and n
psi
¼13 275 [48]) pointing to a starkly different
local functional structure in the two conditions.
The results from the persistent homology analysis and the
insights provided by the homological scaffolds imply that
although the mesoscopic structures, i.e. cycles, in the psilocy-
bin condition are less stable than in the placebo group, their
constituent edges are more stable.
6. Discussion
In this paper, we first described a variation of persistent hom-
ology that allows us to deal with weighted and signed
networks. We then introduced two new objects, the homolo-
gical scaffolds, to go beyond the picture given by persistent
homology to represent and summarize information about
individual links. The homological scaffolds represent a new
measure of topological importance of edges in the original
system in terms of how frequently they are part of the genera-
tors of the persistent homology groups and how persistent
are the generators to which they belong to. We applied this
method to an fMRI dataset comprising a group of subjects
injected with a placebo and another injected with psilocybin.
By focusing on the second homology group H
1
,wefound
that the stability of mesoscopic association cycles is reduced by
the action of psilocybin, as shown by the difference in the
probability density function of the generators of H
1
(figure 3).
It is here that the importance of the insight given by the
homological scaffolds in the persistent homology procedure
becomes apparent. A simple reading of this result would be
that the effect of psilocybin is to relax the constraints on brain
function, ascribing cognition a more flexible quality, but when
looking at the edge level, the picture becomes more complex.
The analysis of the homological scaffolds reveals the existence
of a set of edges that are predominant in terms of their persist-
ence although they are statistically part of the same number of
cycles in the two conditions (figure 5). In other words, these
functional connections support cycles that are especially stable
and are only present in the psychedelic state. This further
implies that the brain does not simply become a random
system afterpsilocybin injection, but instead retains some organ-
izational features, albeit different from the normal state, as
suggested by the first part of the analysis. Further work is
required to identify the exact functional significance of these
edges. Nonetheless, it is interesting to look at the community
structure of the persistence homological scaffolds in figure 6.
The two pictures are simplified cartoons of the placebo (figure
6a) and psilocybin (figure 6b) scaffolds. In figure 6a,b,the
nodes are organized and coloured accordingto their community
membership in the placebo scaffold (obtained with the Louvain
algorithm for maximal modularity and resolution 1 [50]). This is
done in order to highlight the striking difference in connectivity
structure in the two cases. When considering the edges in the tail
of the distribution, weight greater than or equal to 80, in figure
5a, only 29 of the 374 edges present in the truncated psilocybin
scaffold are shared with the truncated placebo scaffold (165
edges). Of these 374 edges, 217 are between placebo commu-
nities and are observed to mostly connect cortical regions. This
supports our idea that psilocybin disrupts the normal organiz-
ation of the brain with the emergence of strong, topologically
long-range functional connections that are not present in a
normal state.
The two key results of the analysis of the homological scaf-
folds can therefore be summarized as follows (i) there is an
increased integration between cortical regions in the psilocybin
state and (ii) this integration is supported by a persistent scaffold
of a set of edges that support cross modular connectivity prob-
ably as a result of the stimulation of the 5HT2A receptors in
the cortex [51].
We can speculate on the implications of such an organiz-
ation. One possible by-product of this greater communication
across the whole brain is the phenomenon of synaesthesia
which is often reported in conjunction with the psychedelic
state. Synaesthesia is described as an inducer-concurrent
pairing, where the inducer could be a grapheme or a visual
stimulus that generates a secondary sensory output—like a
colour for example. Drug-induced synaesthesia often leads
to chain of associations, pointing to dynamic causes rather
1
10
–1
10
–2
P(w
f
)
10
–3
10
–4
10 10
2
link frequency w
f
10.0 100.0
1.0
0.1
0.01
0.001
0.0001 1000.0
10
3
0
50
100
150
200
250
300
350
(a)(b)
200 400 600
psi
pla
800 1000 1200
pla
p
psi
p
e
e
link persistence w
p
e
link frequency w f
e
P(wp)
e
link persistence wp
e
Figure 5. Statistical features of group homological scaffolds. Panel (a) reports the (log-binned) probability distributions for the edge weights in the persistence
homological scaffolds (main plot) and the frequency homological scaffolds (inset). While the weights in the frequency scaffold are not significantly different, the
weight distributions for the persistence scaffold display clearly a broader tail. Panel (b) shows instead the scatter plot of the edge frequency versus total persistence.
In both cases, there is a clear linear relationship between the two, with a large slope in the psilocybin case. Moreover, the psilocybin scaffold has a larger spread in
the frequency and total persistence of individual edges, hinting to a different local functional structure within the functional network of the drugged brains. (Online
version in colour.)
rsif.royalsocietypublishing.org J. R. Soc. Interface 11: 20140873
7
than fixed structural ones as may be the case for acquired
synaesthesia [52]. Broadly consistent with this, it has been
reported that subjects under the influence of psilocybin
have objectively worse colour perception performance
despite subjectively intensified colour experience [53].
To summarize, we presented a new method to analyse
fully connected, weighted and signed networks and applied
it to a unique fMRI dataset of subjects under the influence
of mushrooms. We find that the psychedelic state is associ-
ated with a less constrained and more intercommunicative
mode of brain function, which is consistent with descriptions
of the nature of consciousness in the psychedelic state.
7. Methods
7.1. Dataset
A pharmacological MRI dataset of 15 healthy controls was used
for a proof-of-principle test of the methodology [54]. Each subject
was scanned on two separate occasions, 14 days apart. Each scan
consisted of a structural MRI image (T1-weighted), followed by a
12 min eyes-close resting-state blood oxygen-level-dependent
(BOLD) fMRI scan which lasted for 12 min. Placebo (10 ml
saline, intravenous injection) was given on one occasion and psi-
locybin (2 mg dissolved in 10 ml saline) on the other. Injections
were given manually by a study doctor situated within the scan-
ning suite. Injections began exactly 6 min after the start of the
12-min scans, and continued for 60 s.
7.1.1. Scanning parameters
The BOLD fMRI data were acquired using standard gradient-echo
EPI sequences, reported in detail in reference [54]. The volume
repetition time was 3000 ms, resulting in a total of 240 volumes
acquired during each 12 min resting-state scan (120 pre- and 120
post-injection of placebo/psilocybin).
7.1.2. Image pre-processing
fMRI images were corrected for subject motion within individual
resting-state acquisitions, by registering all volumes of the
functional data to the middle volume of the acquisition using
the FMRIB linear registration motion correction tool, generating
a six-dimension parameter time course [55]. Recent work demon-
strates that the six parameter motion model is insufficient to
correct for motion-induced artefact within functional data,
instead a Volterra expansion of these parameters to form a 24
parameter model is favoured as a trade-off between artefact cor-
rection and lost degrees of freedom as a result of regressing
motion away from functional time courses [56]. fMRI data
were pre-processed according to standard protocols using a
high-pass filter with a cut-off of 300 s.
Structural MRI images were segmented into n¼194 cortical
and subcortical regions, including white matter cerebrospinal
fluid (CSF) compartments, using FREESURFER (http://surfer.nmr.
mgh.harvard.edu/), according to the Destrieux anatomical atlas
[57]. In order to extract mean-functional time courses from
the BOLD fMRI, segmented T1 images were registered to the
middle volume of the motion-corrected fMRI data, using bound-
ary-based registration [58], once in functional space mean
time-courses were extracted for each of the n¼194 regions in
native fMRI space.
7.1.3. Functional connectivity
For each of the 194 regions, alongside the 24 parameter motion
model time courses, partial correlations were calculated between
all couples of time courses (i,j), non-neural time courses (CSF,
white matter and motion) were discarded from the resulting
functional connectivity matrices, resulting in a 169 region corti-
cal/subcortical functional connectivity corrected for motion
and additional non-neural signals (white matter/CSF).
7.2. Persistent homology computation
For each subject in the two groups, we have a set of persistence
diagrams relative to the persistent homology groups H
n
. In this
paper, we use the H
1
persistence diagrams of each group to
construct the corresponding persistence probability densities
for H
1
cycles.
Filtrations were obtained from the raw partial-correlation
matrices through the PYTHON package Holes and fed to javaplex
[46] via a Jython subroutine in order to extract the persistence
(a)(b)
Figure 6. Simplified visualization of the persistence homological scaffolds. The persistence homological scaffolds Hp
pla (a) and Hp
psi (b) are shown for comparison.
For ease of visualization, only the links heavier than 80 (the weight at which the distributions in figure 5abifurcate) are shown. This value is slightly smaller than
the bifurcation point of the weights distributions in figure 5a. In both networks, colours represent communities obtained by modularity [49] optimization on the
placebo persistence scaffold using the Louvain method [50] and are used to show the departure of the psilocybin connectivity structure from the placebo baseline.
The width of the links is proportional to their weight and the size of the nodes is proportional to their strength. Note that the proportion of heavy links between
communities is much higher (and very different) in the psilocybin group, suggesting greater integration. A labelled version of the two scaffolds is available as GEXF
graph files as the electronic supplementary material. (Online version in colour.)
rsif.royalsocietypublishing.org J. R. Soc. Interface 11: 20140873
8
intervals and the representative cycles. The details of the
implementation can be found in reference [30], and the software
is available at Holes [59].
Funding statement. G.P. and F.V. are supported by the TOPDRIM project
supported by the Future and Emerging Technologies programme of
the European Commission under Contract IST-318121. I.D. P.E. and
F.T. are supported by a PET methodology programme grant from
the Medical Research Council UK (ref no. G1100809/1). The authors
acknowledge support of Amanda Feilding and the Beckley
Foundation and the anonymous referees for their critical and
constructive contribution to this paper.
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... For an introduction to TDA and applications see [3,10,11]. For applications of clique homology to computational neuroscience see [12][13][14]. ...
... The action of the boundary operator is that in (2.6), sending k-cliques to (k − 1)-cliques. Finding k-dimensional holes in clique complexes is an important task in TDA, with many applications including neuroscience [12][13][14] and the study of protein networks. Another simplicial complex which is defined naturally given a graph G is the "independence complex" I(G). ...
... In this section, we will prove that this problem is QMA 1 -hard when the simplicial complex, K, is a clique complex Cl(G) represented by G. As a stepping stone we consider the closely related problem when the complex is an independence complex I(G) represented by G: 12 For readers who have already covered the connection to supersymmetry in Section 5 the form of these tensor product gadgets can be checked very quickly. Clearly if the qubit with the classical projector acting on it is in the kernel of the classical projector then the mediators are forced to be empty by the hard core condition, and the homology is non-trivial regardless of the state of the remaining qubits. ...
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We tackle the long-standing question of the computational complexity of determining homology groups of simplicial complexes, a fundamental task in computational topology, posed by Kaibel and Pfetsch 20 years ago. We show that this decision problem is QMA1-hard. Moreover, we show that a version of the problem satisfying a suitable promise and certain constraints is contained in QMA. This suggests that the seemingly classical problem may in fact be quantum mechanical. In fact, we are able to significantly strengthen this by showing that the problem remains QMA1-hard in the case of clique complexes, a family of simplicial complexes specified by a graph which is relevant to the problem of topological data analysis. The proof combines a number of techniques from Hamiltonian complexity and homological algebra. We discuss potential implications for the problem of quantum advantage in topological data analysis.
... However, the decrease of within-network connectivity is accompanied by increases in between-network connectivity (i.e., deviation of functional connections from network pathways). This pattern of reduced-within and increased-between connectivity-coined as disintegration and desegregation (Carhart-Harris et al., 2016b)-is documented across resting-state networks under psychedelics (Barnett et al., 2019;Varley et al., 2019) and alters the functional integration of brain regions (Petri et al., 2014) (Fig. 5). Cognitive function depends on the dynamic, context-sensitive regulation of functional segregation and integration in the brain and is mediated by neural gain modulation. ...
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... These network models are premised on the hypothesis that the components of a system are connected only through pairwise links, schematized by graphs. However, this pairwise representation is merely a first-order estimation [7,8] in many real-world systems, such as structural [9] and functional [10] brain networks, ecological networks [11,12], collaboration graphs [13,14], protein interaction networks [15], social systems [16,17], and consensus dynamics [18], where manybody interactions between system components are ubiquitous and of immense interest. To schematize these higher-order interactions, one then requires a generalization of classical network theory [19,20]. ...
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... These network models are premised on the hypothesis that the components of a system are connected only through pairwise links, schematized by graphs. However, this pairwise representation is merely a first-order estimation [7,8] in many realworld systems, such as structural [9] and functional [10] brain networks, ecological networks [11,12], collaboration graphs [13,14], protein interaction networks [15], social systems [16,17], and consensus dynamics [18], where many-body interactions between system components are ubiquitous and of immense interest. To schematize these higher-order interactions, one then requires a generalization of classical network theory [19,20]. ...
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... Finally, our analyses assessing high-order functional interactions are based on some specific metrics such as mean values of O-information at each interaction order. Future studies may consider different algebraic or topological properties of the full O-information hypergraph [69][70][71][72][73][74], which may provide complementary insights. It is also worth noting that the reported values of the O-information are not indicative of 'pure' synergy or redundancy, but correspond to the balance between them. ...
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