Two’s company, three (or more) is a simplex
J Comput Neurosci
Two's company, three (or more) is a simplex
Chad Giusti 0 1 2 3
Robert Ghrist 0 1 2 3
Danielle S. Bassett 0 1 2 3
Action Editor: Bard Ermentrout 0 1 2 3
0 Department of Bioengineering, University of Pennsylvania , Philadelphia, PA 19104 , USA
1 Department of Mathematics, University of Pennsylvania , Philadelphia, PA 19104 , USA
2 Danielle S. Bassett
3 Department of Electrical & Systems Engineering, University of Pennsylvania , Philadelphia, PA 19104 , USA
The language of graph theory, or network science, has proven to be an exceptional tool for addressing myriad problems in neuroscience. Yet, the use of networks is predicated on a critical simplifying assumption: that the quintessential unit of interest in a brain is a dyad - two nodes (neurons or brain regions) connected by an edge. While rarely mentioned, this fundamental assumption inherently limits the types of neural structure and function that graphs can be used to model. Here, we describe a generalization of graphs that overcomes these limitations, thereby offering a broad range of new possibilities in terms of modeling and measuring neural phenomena. Specifically, we explore the use of simplicial complexes: a structure developed in the field of mathematics known as algebraic topology, of increasing applicability to real data due to a rapidly growing computational toolset. We review the underlying mathematical formalism as well as the budding literature applying simplicial complexes to neural data, from electrophysiological recordings in animal models to hemodynamic fluctuations in humans. Based on the exceptional flexibility of the tools and recent ground-breaking insights into neural function, we posit that this framework has the potential to eclipse graph theory in unraveling the fundamental mysteries of cognition.
Networks; Topology; Simplicial complex; Filtration
The recent development of novel imaging techniques and
the acquisition of massive collections of neural data make
finding new approaches to understanding neural structure
a vital undertaking. Network science is rapidly
becoming an ubiquitous tool for understanding the structure of
complex neural systems. Encoding relationships between
objects of interest using graphs (Figs. 1a–b, 4a) enables the
use of a bevy of well-developed tools for structural
characterization as well as inference of dynamic behavior. Over
the last decade, network models have demonstrated broad
utility in uncovering fundamental architectural principles
(Bassett and Bullmore 2006; Bullmore and Bassett 2011)
and their implications for cognition (Medaglia et al. 2015)
. Their use has led to the
development of novel diagnostic biomarkers
conceptual cognitive frameworks
illustrate a paradigm shift in systems, cognitive, and clinical
neuroscience: namely, that brain function and alteration are
inherently networked phenomena.
All graph-based models consist of a choice of vertices,
which represent the objects of study, and a collection of
edges, which encode the existence of a relationship between
pairs of objects (Figs. 1a–b, 4a). However, in many real
systems, such dyadic relationships fail to accurately capture
the rich nature of the system’s organization; indeed, even
when the underlying structure of a system is known to be
Fig. 1 Extensions of network models provide insights into neural
data. a Network models are increasingly common for the study of
whole-brain activity. b Neuron-level networks have been a driving
force in the adoption of network techniques in neuroscience. c Two
potential activity traces for a trio of neural units. (top) Activity for
a “pacemaker”-like circuit, whose elements are pairwise active in all
combinations but never as a triple. (bottom) Activity for units driven
by a common strong stimulus, thus are simultaneously coactive. d A
network representation of the coactivity patterns for either population
in (c). Networks are capable of encoding only dyadic relationships, so
do not capture the difference between these two populations. e A
simplicial complex model is capable of encoding higher order interactions,
thus distinguishing between the top and bottom panels in (c). f A
similarity measure for elements in a large neural population is encoded as
dyadic, its function is often understood to be polyadic. In
large-scale neuroimaging, for example, cognitive functions
appear to be performed by a distributed set of brain regions
and their interactions
(Medaglia et al.
. At a smaller scale, the spatiotemporal patterns of
interactions between a few neurons is thought to underlie
basic information coding
(Szatmary and Izhikevich 2010)
and explain alterations in neural architecture that
(Feldt et al. 2011)
Drawing on techniques from the field of algebraic
topology, we describe a mathematically well-studied
generalization of graphs called simplicial complexes as an alternative,
often preferred method for encoding non-dyadic
relationships (Fig. 4). Different types of complexes can be used
to encode co-firing of neurons
(Curto and Itskov 2008)
co-activation of brain areas
(Crossley et al. 2013)
structural and functional connections between neurons or brain
(Bullmore and Sporns 2009)
(Fig. 5). After
choosing the complex of interest, quantitative and theoretical tools
can be used to describe, compare, and explain the statistical
properties of their structure in a manner analogous to graph
statistics or network diagnostics.
We then turn our attention to a method of using
additional data, such as temporal processes or frequency of
a matrix, thought of as the adjacency matrix for a complete, weighted
network, and binarized using some threshold to simplify quantitative
analysis of the system. In the absence of complete understanding of
a system, it is difficult or impossible to make a principled choice of
threshold value. g A filtration of networks is obtained by thresholding
at every possible entry and arranging the resulting family of networks
along an axis at their threshold values. This structure discards no
information from the original weighted network. g Graphs of the number
of connected components as a function of threshold value for two
networks reveals differences in their structure: (top) homogeneous
network versus (bottom) a modular network. (dotted lines) Thresholding
near these values would suggest inaccurately that these two networks
have similar structure
observations, to decompose a simplicial complex into
constituent pieces, called a filtration of the complex (Fig. 1f–h).
Filtrations reveal more detailed structure in the complex,
and provide tools for understanding how that structure arises
(Fig. 7). They can also be used as an alternative to
thresholding a weighted complex, providing a principled approach
to binarizing which retains all of the data in the original
In what follows, we avoid introducing technical details
beyond those absolutely necessary, as they can be found
(Ghrist 2014; Nanda and Sazdanovic´ 2014;
, though we include boxed mathematical
definitions of the basic terms to provide context for the
interested reader. These ideas are also actively being applied in
the theory of neural coding, and for details we highly
recommend the recent survey (Curto 2016). Finally, although the
field is progressing rapidly, we provide a brief discussion of
the current state of computational methods in the Appendix.
1 Motivating examples
We begin with a pair of simple thought experiments, each of
which motivates one of the tools this article surveys.
1.1 Complexes for relationship
First, imagine a simple neural system consisting of three
brain regions (or neurons) with unknown connectivity. One
possible activity profile for such a population includes some
sort of sequential information processing loop or
“pacemaker” like circuit, where the regions activate in a rotating
order (Fig. 1c, top). A second is for all three of the regions to
be active simultaneously when engaged in certain
computations, and otherwise quiescent or uncorrelated (Fig. 1c,
bottom). In either case, an observer would find the activity of
all three possible pairs of regions to be strongly correlated.
Because a network can only describe dyadic relationships
between population elements, any binary coactivity network
constructed from such observations would necessarily be
identical for both (Fig. 1d). However, a more versatile
language could distinguish the two by explicitly encoding the
triple coactivity pattern in the second example (Fig. 1e).
One possible solution lies in the language of
hypergraphs, which can record any possible collection of
relations. However, this degree of generality leads to a
combinatorial explosion in systems of modest size. In contrast,
the framework of simplicial complexes (Fig. 4b–d) gives
a compact and computable encoding of relations between
arbitrarily large subgroups of the population of interest
while retaining access to a host of quantitative tools for
detecting and analyzing the structure of the systems they
encode. In particular, the homology1 of a simplicial complex
is a collection of topological features called cycles that one
can extract from the complex (Fig. 6b). These cycles
generalize the standard graph-theoretic notions of components
and circuits, providing a mesoscale or global view of the
structure of the system. Together, these methods provide a
quantitative architecture through which to address modern
questions about complex and emergent behavior in neural
1.2 Filtrations for thresholding
Second, consider a much larger neural system, consisting
of several hundred units, whose activity is summarized as
a correlation or coherence matrix (Fig. 1f, top). It is
common practice to binarize such a matrix by thresholding it at
some value, taking entries above that value to be
“significant” connections, and to study the resulting, much sparser
network (Fig. 1f, bottom). Selecting this significance level
is problematic, particularly when the underlying system has
a combination of small-scale features, some of which are
noise artifacts, and some of which are critically important.
1Names of topological objects have a seemingly pathological tendency
to conflict with terms in biology, so long have the two subjects been
separated. Mathematical homology has no a priori relationship to the
usual biological notion of homology.
One method for working around this difficulty is to take
several thresholds and study the results separately.
However, this approach still discards most of the information
contained in the edge weights, much of which can be of
inherent value in understanding the system. We propose
instead the use of filtrations, which record the results of
every possible binarization of the network,2 along with the
associated threshold values (Fig. 1g). Filtrations not only
retain all of the information in the original weighted
networks, but unfold that information into a more accessible
form, allowing one to lift any measure of structure in
networks (or simplicial complexes) to “second order” measures
as functions of edge weight (Fig. 1h). Such functions carry
information, for example, in their rate of change, where
sudden phase transitions in network structure as one varies the
threshold can indicate the presence of modules or rich clubs
in networks (Fig. 1h). The area under such curves was used
(Giusti et al. 2015)
to detect geometric structure in the
activity of hippocampal neural populations (Fig. 3).
Further, even more delicate information can be extracted from
the filtration by tracking the persistence of cycles as the
threshold varies (Fig. 7c).
2 A growing literature
Before we proceed to an explicit discussion of the tools
described above, we pause to provide a broad overview of
how they have already been applied to address questions in
neuroscience. The existing literature can be roughly divided
into two branches:
Describing neural coding and network properties
Because of their inherently non-pairwise nature,
coactivation patterns of neurons or brain regions can be naturally
encoded as simplicial complexes. Such techniques were first
introduced in the context of hippocampal place cells in
(Curto and Itskov 2008)
, where such an encoding was used
to describe how one can reconstruct the shape of an animal’s
environment from neural activity. Using the simple
observation that place fields corresponding to nearby locations will
overlap, the authors conclude that neurons corresponding
to those fields will tend to be co-active (Fig. 5b). Using the
aptly (but coincidentally) named “Nerve Theorem” from
algebraic topology, one can work backward from observed
coactivity patterns to recover the intersection pattern of the
receptive fields, describing a topological map of the animal’s
environment (Fig. 6c). Further, in order to recover the
geometry of the environment, one can in principle introduce
information regarding receptive field size
(Curto and Itskov 2008)
2Or some suitably dense subset of the binarizations, in the case of very
However, it seems plausible that place cells intrinsically
record only these intersection patterns and rely on
downstream mechanisms for interpretation of such geometry.
This hypothesis was tested in the elegant experiment of
(Dabaghian et al. 2014)
, in which place cell activity was
recorded before and after deformation of segments of the
legs of a U-shaped track. A geometric map would have been
badly damaged by such a change in the environment, while
a topological map would remain consistent, and indeed the
activity is shown to be consistent across the trials. Further
theoretical and computational work has explored how such
topological maps might form
(Dabaghian et al. 2012)
shown that theta oscillations improve such learning
(Arai et al. 2014)
, as well as demonstrating how
one might use this understanding to decode maps of the
environment from observed cell activity
(Chen et al. 2014)
Even in the absence of an expected underlying collection
of spatial receptive fields like those found in place cells,
these tools can be employed to explore how network
modules interact. In
(Ellis and Klein 2014)
, the authors study
the frequency of observation of coactivity patterns in fMRI
recordings to extract fundamental computational units. Even
when those regions which are coactive will change
dynamically over time, cohesive functional units will appear more
often than those that happen coincidentally, though a priori
it is impossible to set a threshold for significance of such
observations. Using a filtration, it becomes possible to make
reasonable inferences regarding the underlying
organization. The same approach was used in (Pirino et al. 2014),
to differentiate in vivo cortical cell cultures into functional
sub-networks under various system conditions. Finally, an
extension of these ideas that includes a notion of
directedness of information flow has been used to investigate
the relationship between simulated structural and functional
(Dlotko et al. 2016)
Characterizing brain architecture or state One of the
earliest applications of algebraic topology to neural data
was to the study of activity in the macaque primary visual
(Singh et al. 2008)
, where differences in the cycles
computed from activity patterns were used to distinguish
recordings of spontaneous activity from those obtained
during exposure to natural images.
Cycles provide robust measures of mesoscale structures
in simplicial complexes, and can be used to detect many
different facets of interest in the underlying neural system. For
(Chung et al. 2009)
, the authors compute cycles
that encode regions of thin cortex to differentiate human
ASD subjects from controls; in
(Brown and Gedeon 2012)
cycles built from physical structure in afferent neuron
terminals in crickets are used to understand their organization
(Brown and Gedeon 2012)
Bendich et al. (2014)
authors use two different types of cycles derived from the
geometry of brain artery trees to infer age and gender in
A common theme in neuroscience is the use of
correlation of neuronal population activity as a measure of strength
of the interaction among elements of the population. Such
measures can be used as weightings to construct weighted
simplicial complexes, to which one can apply a threshold
analogously to thresholding in graphs. Using the language
of filtrations, one can compute persistence of cycles,
recording how cycles change as the thresholding parameter varies.
Such measurements provide a much finer discrimination
of structure than cycles at individual thresholds. The
simplest case tracks how the connected components of the
complex evolve; it has been used in
(Lee et al. 2011)
classify pediatric ADHD, ASD and control subjects; in
et al. 2014)
to differentiate mouse models of depression
from controls; in
(Choi et al. 2014)
epileptic rat models from controls; and in
(Kim et al. 2014)
study morphological correlations in adults with hearing loss
(Fig. 2). Studying more complex persistent cycles
computed from fMRI recordings distinguishes subjects under
psilocybin condition from controls
(Petri et al. 2014)
a similar approach has been applied to the study of
functional brain networks during learning
recently, these techniques have been adapted to detect
structure, such as that possessed by a network of hippocampal
place cells, in the information encoded by a neural
population through observations of its activity without reference
to external correlates such as animal behavior
(Giusti et al.
The small, budding field of topological neuroscience
already offers an array of powerful new quantitative
approaches for addressing the unique challenges inherent in
understanding neural systems, with initial, substantial
contributions. In recent years, there have been a number of
innovative collaborations between mathematicians interested in
applying topological methods and researchers in a variety of
biological disciplines. While it is beyond the scope of this
paper to enumerate these new research directions, to
provide some notion of the breadth of such collaborations we
include the following brief list: the discovery of new genetic
markers for breast cancer survival
(Nicolau et al. 2011)
measurement of structure and stability of biomolecules
(Gameiro et al. 2013; Xia et al. 2015)
, new frameworks for
understanding viral evolution (Chan et al. 2013),
characterization of dynamics in gene regulatory networks
et al. 2005)
, quantification of contagion spread in social
(Taylor et al. 2015)
, characterization of structure in
networks of coupled oscillators
, the study of
(Miller et al. 2015)
, and the classification
of dicotyledonous leaves
(Katifori and Magnasco 2012)
This wide-spread interest in developing new research
directions is an untapped resource for empirical neuroscientists,
parameter varies provides insight into differences in structure. It is
unclear how one would select a particular threshold which readily
reveals these differences without a priori knowledge of their presence.
Figure reproduced with permission from
(Kim et al. 2014)
which promises to facilitate both direct applications of
existing techniques and the collaborative construction of novel
tools specific to their needs.
We devote the remainder of this paper to a careful
exposition of these topological techniques, highlighting specific
ways they may be (or have already been) used to address
questions of interest to neuroscientists.
3 Mathematical framework: simplicial complexes
We begin with a short tutorial on simplicial complexes, and
illustrate the similarities and differences with graphs.
Recall that a graph consists of a set of vertices and a
specified collection of pairs of vertices, called edges. A
simplicial complex, similarly, consists of a set of vertices, and
CA1 pyramidal cell
data betti curves
data vs. geometric
Fig. 3 Betti numbers detect the existence of geometric organizing
principles in neural population activity from rat hippocampus. a Mean
cross correlation of N=88 rat CA1 pyramidal cells activity during
spatial navigation. b Betti numbers as a function of graph edge density
(# edges / possible # edges) for the clique complex of the pairwise
correlation network in (a). c Comparison of data Betti numbers (thick
lines) to model random networks with (top) geometric weights given
by decreasing distance between random points in Euclidean space and
(bottom) with no intrinsic structure obtained by shuffling the entries
of the correlation matrix. d Integrals of the curves from panel B show
that the data (thick bars) lie in the geometric regime (g) and that
the unstructured network model (s) is fundamentally different (p <
0.001). Similar geometric organization was observed in non-spatial
behaviors such as REM sleep. Figure reproduced with permission from
(Giusti et al. 2015)
a collection of simplices — finite sets of vertices. Edges
are examples of very small simplices, making every graph a
particularly simple simplicial complex. In general, one must
satisfy the simplex condition, which requires that any subset
of a simplex is also a simplex.
Just as one can represent a graph as a collection of points
and line segments between them, one can represent the
simplices in a simplicial complex as a collection of solid regions
spanning vertices (Fig. 4d). Under this geometric
interpretation, a single vertex is a zero-dimensional point, while
an edge (two vertices) defines a one-dimensional line
segment; three vertices span a two-dimensional triangle, and so
on. Terminology for simplices is derived from this
geometric representation: a simplex on (n + 1) vertices is called
an n-simplex and is viewed as spanning an n-dimensional
region. Further, as the requisite subsets of a simplex
represent regions in the geometric boundary of the simplex
(Fig. 4c), these subsets of a simplex are called its faces.
Because any given simplex is required to “contain all of
its faces”, it suffices to specify only the maximal simplices,
those which do not appear as faces of another simplex
(Fig. 4c). This dramatically reduces the amount of data
necessary to specify a simplicial complex, which helps make
both conceptual work and computations feasible.
In real-world systems, simplicial complexes possess
richly structured patterns that can be detected and
characterized using recently developed computational tools from
(Carlsson 2009; Lum et al. 2013)
as graph theoretic tools can be used to study networks.
Importantly, these tools reveal much deeper properties of
the relationships between vertices than graphs, and many
are constructed not only to see structure in individual
simplicial complexes, but also to help one understand how
two or more simplicial complexes compare or relate to one
another. These capabilities naturally enable the study of
complex dynamic structure in neural systems, and formalize
statistical inference via comparisons to null models.
4 How do we encode neural data?
To demonstrate the broad utility of this framework, we turn
to describing a selection of the many types of simplicial
complexes that can be constructed from data: the clique
complex, the concurrence complex
(Ellis and Klein 2014;
Curto and Itskov 2008; Dowker 1952)
, its Dowker dual
(Dowker 1952), and the independence complex
, as summarized in Table 1. In each case, we describe
the relative utility in representing different types of neural
data – from spike trains measured from individual neurons
to BOLD activations measured from large-scale brain areas.
Clique complex One straightforward method for
constructing simplicial complexes begins with a graph where
vertices represent neural units and edges represent structural
or functional connectivity between those units (Fig. 4a–
b). Given such a graph, one simply replaces every clique
(all-to-all connected subgraph) by a simplex on the
vertices participating in the clique (Fig. 5a). This procedure
produces a clique complex, which encodes the same
information as the underlying graph, but additionally completes
the skeletal network to its fullest possible simplicial
structure. The utility of this additional structure was recently
demonstrated in the analysis of neural activity measured
in rat hippocampal pyramidal cells during both spatial and
non-spatial behavior (including REM sleep)
(Giusti et al.
(Fig. 3). In contrast to analyses using standard
graphtheoretic tools, the pattern of simplices revealed the
presence of geometric structure in only the information encoded
Fig. 4 Simplicial complexes generalize network models. a A graph
encodes elements of a neural system as vertices and dyadic
relations between them as edges. b–c Simplicial complex terminology.
A simplicial complex is made up of vertices and simplices, which
are defined in terms of collections of vertices. b A n-simplex can be
thought of as the convex hull of (n + 1) vertices. c The boundary of
a simplex consists of all possible subsets of its constituent vertices,
called its faces, which are themselves required to be simplices in
the complex. A simplex which is not in the boundary of any other
simplex is called maximal. d A simplicial complex encodes polyadic
relations through its simplices. Here, in addition to the dyadic relations
specified by the edges, the complex specifies one four-vertex
relation and three three-vertex relations. The omission of larger simplices
where all dyadic relations are present, such as the three bottom-left
vertices or the four top-left vertices, encodes structure that cannot be
specified using network models
in neural population activity correlations that – surprisingly
– could be identified and characterized independently from
the animal’s position. This application demonstrates that
simplicial complexes are sensitive to organizational
principles that are hidden to graph statistics, and can be used to
infer parsimonious rules for information encoding in neural
Clique complexes precisely encode the topological
features present in a graph. However, other types of simplicial
complexes can be used to represent information that cannot
be so encoded in a graph.
Concurrence complex Using cofiring, coactivity, or
connectivity as before, let us consider relationships between
two different sets of variables. For example, we can
consider (i) neurons and (ii) times, where the relationship is
given by a neuron firing in a given time (Fig. 5b)
; a similar framing exists for (i) brain regions
and (ii) times, where the relationship is given by a brain
region being active at a given time
(Ellis and Klein 2014)
General framework for encoding dyadic relations
Canonical polyadic extension of existing network models
Relationships between two variables of interest
e.g., time and activity, or activity in two separate regions
Structure where non-membership satisfies the simplex property
e.g., communities in a network
Alternatively, we can consider (i) brain regions in the motor
system and (ii) brain regions in the visual system, where
the relationship is given by a motor region displaying
similar BOLD activity to a visual region (Fig. 5c)
(Bassett et al.
. In each case, we can record the patterns of
relationships between the two sets of variables as a binary matrix,
where the rows represent elements in one of the variables
(e.g., neurons) and the columns the other (e.g., times), with
non-zero entries corresponding to the row-elements in each
column sharing a relation (e.g., firing together at a single
time). The concurrence complex is formed by taking the
rows of such a matrix as vertices and the columns to
represent maximal simplices consisting of those vertices with
. A particularly interesting
feature of this complex is that it remains naive to coactivity
patterns that do not appear, and this naivety plays an
important role in its representational ability; for example, such
a complex can be used to decode the geometry of an
animal’s environment from observed hippocampal cell activity
(Curto and Itskov 2008)
Fig. 5 Simplicial complexes encode diverse neural data modalities. a
Correlation or coherence matrices between regional BOLD time series
can be encoded as a type of simplicial complex called a clique complex,
formed by taking every complete (all-to-all) subgraph in a binarized
functional connectivity matrix to be a simplex. b Coactivity patterns
in neural recordings can be encoded as a type of simplicial complex
called a concurrence complex. Here, we study a binary matrix in which
each row corresponds to a neuron and each column corresponds to a
collection of neurons that is observed to be coactive at the same time
(yellow boxes) – i.e., a simplex. c Thresholded coherence between
the activity patterns of motor regions and visual regions in human
fMRI data during performance of a motor-visual task
(Bassett et al.
. (top) We can construct a concurrence complex whose vertices
are motor regions and whose simplices are families of motor regions
whose activity is strongly coherent with a given visual region. (bottom)
We can also construct a dual complex whose vertices are families of
motor regions. The relationship between these two complexes carries
a great deal of information about the system (Dowker 1952)
Moving to simplicial complex models provides a
dramatically more flexible framework for specifying data
encoding than simply generalizing graph techniques. Here we
describe two related simplicial complex constructions from
neural data which cannot be represented using network
Dowker dual Beginning with observations of coactivity,
connection or cofiring as before, one can choose to represent
neural units as simplices whose constituent vertices
represent patterns of coactivity in which the unit participates.
Expressing such a structure as a network would necessitate
every neural unit participating in precisely two activity
patterns, an unrealistic requirement, but this is straightforward
in the simplicial complex formalism. Mathematically
speaking, one can think of the matrix encoding this complex as the
transpose of the matrix encoding the concurrence complex;
such “dual” complexes are deeply related to one another, as
first observed in
. Critically, this formulation
refocuses attention (and the output of various vertex-based
statistical measures) from individual neural units to patterns
Independence complex It is sometimes the case that an
observed structure does not satisfy the simplicial complex
requirement that subsets of simplices are also simplices,
but its complement does. One example of interest is the
collection of communities in a network
Porter et al. 2009)
: communities are subgraphs of a network
whose vertices are more densely connected to one another
than expected in an appropriate null model. The collection
of vertices in the community is not necessarily a simplex,
because removing densely connected vertices can cause a
community to dissolve. Thus, community structure is
wellrepresented as a hypergraph (Bassett et al. 2014), though
such structures are often less natural and harder to work with
than simplicial complexes. However, in this setting, one can
take a simplex to be all vertices not in a given community.
Such a simplicial complex is again essentially a concurrence
complex: simply negate the binary matrix whose rows are
elements of the network and columns correspond to
community membership. Such a complex is called an independence
, and can be used to study
properties of a system’s community structure such as dynamic
(Bassett et al. 2011, 2013)
Together, these different types of complexes can be used
to encode a wide variety of relationships (or lack thereof)
among neural units or coactivity properties in a simple
matrix that can be subsequently interrogated
mathematically. This is by no means an exhaustive list of complexes of
potential interest to the neuroscience community; for further
examples, we recommend
(Ghrist 2014; Kozlov 2007)
5 How do we measure the structure of simplicial complexes?
Just as with network models, once we have effectively
encoded neural data in a simplicial complex, it is necessary
to find useful quantitative measurements of the resulting
structure to draw conclusions about the neural system of
interest. Because simplicial complexes generalize graphs,
many familiar graph statistics can be extended in interesting
ways to simplicial complexes. However, algebraic
topology also offers a host of novel and very powerful tools that
are native to the class of simplicial complexes, and
cannot be naturally derived from well known graph theoretical
Graph theoretical extensions First, let us consider how
we can generalize familiar graph statistics to the world
of simplicial complexes. The simplest local measure of
structure – the degree of a vertex – naturally becomes a
vector-measurement whose entries are the number of
maximal simplices of each size in which the vertex participates
(Fig. 6a). Although a direct extension of the degree, this
vector is perhaps more intuitively thought of as a
generalization of the clustering coefficient of the vertex: in this setting
we can distinguish empty triangles, which represent three
dyadic relations but no triple-relations, from 2-simplices
which represent clusters of three vertices (and similarly for
Just as we can generalize the degree, we can also
generalize the degree distribution. Here, the simplex distribution
or f-vector is the global count of simplices by size, which
provides a global picture of how tightly connected the
vertices are; the maximal simplex distribution collects the same
data for maximal faces (Fig. 6a). While these two
measurements are related, their difference occurs in the complex
patterns of overlap between simplices and so together they
contain a great deal of structural information about the
simplicial complex. Other local and global statistics such as
efficiency and path length can be generalized by considering
paths through simplices of some fixed size, which provides
a notion of robust connectivity between vertices of the
(Dlotko et al. 2016)
; alternately, a path through general
simplices can be assigned a strength coefficient
depending on the size of the maximal simplices through which it
Algebraic-topological methods Such generalizations of
graph-theoretic measures are possible, and likely of
significant interest to the neuroscience community, however
they are not the fundamental statistics originally developed
to characterize simplicial complexes. In their original
context, simplicial complexes were used to study shapes, using
algebraic topology to measure global structure. Thus, this
Fig. 6 Quantifying the structure of a simplicial complex. a
Generalizations of the degree sequence for a simplicial complex. Each vertex
has a degree vector giving the number of maximal simplices of each
degree to which it is incident. The f-vector gives a list of how many
simplices of each degree are in the complex, and the maximal
simplex distribution records only the number of maximal simplices of
each dimension. b Closed cycles of dimension 1 and 2 in the
complex from panel (a). (left) There are two inequivalent 1-cycles (cyan)
up to deformation through 2-simplices, and (right) a single 2-cycle
(cyan) enclosing a 3-d volume. The Betti number vector β gives an
framework also provides new and powerful ways to measure
The most commonly used of these measurements is the
(simplicial) homology of the complex, which is actually a
sequence of measurements. The nth homology of a
simplicial complex is the collection of (closed) n-cycles, which are
structures formed out of n-simplices (Fig. 6b), up to a notion
of equivalence. While the technical details are subtle, an
n-cycle can be understood informally to be a collection of
n-simplices that are arranged so that they have an empty
geometric boundary (Fig 6b). For example, a path between
a pair of distinct vertices in a graph is a collection of
1simplices, the constituent edges, whose boundary is the pair
of endpoints of the path; thus it is not a 1-cycle. However, a
circuit in the graph is a collection of 1-simplices which lie
end-to-end in a closed loop and thus has empty boundary;
therefore, circuits in graphs are examples of 1-cycles.
Similarly, an icosahedron is a collection of twenty 2-simplices
which form a single closed 2-cycle.
We consider two n-cycles to be equivalent if they form
the boundary of a collection of (n + 1)-simplices. The
simplest example is that the boundary of any (n + 1)-simplex,
while necessarily a cycle, is equivalent to the trivial n-cycle
consisting of no simplices at all because it is “filled in” by
the (n + 1)-simplex (Fig. 4c). Further, the endpoints of any
path in a graph are equivalent 0-cycles in the graph (they
are precisely the boundary of the collection of edges which
make up the path) and so the inequivalent 0-cycles of a
graph (its 0th homology) are precisely its components.
Cycles are an example of global structure arising from
local structure; simplices arrayed across multiple vertices
must coalesce in a particular fashion to encircle a “hole”
not filled in by other simplices, and it is often the case that such
a structure marks feature of interest in the system (Fig. 6c).
In many settings, a powerful summary statistic is simply a
count of the number of inequivalent cycles of each dimension
enumeration of the number of n-cycles in the complex, here with
n = 0, 1 and 2; the single 0-cycle corresponds to the single connected
component of the complex. c Schematic representation of the
reconstruction of the presence of an obstacle in an environment using a
concurrence complex constructed from place cell cofiring
. By choosing an appropriate cofiring threshold, based on
approximate radii of place cell receptive fields, there is a single 1-cycle
(cyan), up to deformation through higher simplices, indicating a large
gap in the receptive field coverage where the obstacle appears
appearing in the complex. These counts are called Betti
numbers, and we collect them as a vector β (Fig. 6b).
In the context of neural data, the presence of multiple
homology cycles indicates potentially interesting structure
whose interpretation depends on the meaning of the
vertices and simplices in the complex. For example, the open
triangle in the complex of Fig. 5b is a 1-cycle
representing pairwise coactivity of all of the constituent neurons but
a lack of triple coactivity; thus, the reconstructed receptive
field model includes no corresponding triple intersection,
indicating a hole or obstacle in the environment. In the
context of regional coactivity in fMRI, such a 1-cycle might
correspond to observation of a distributed computation that
does not involve a central hub. Cycles of higher
dimension are more intricate constructions, and their presence
or absence can be used to detect a variety of other more
complex, higher-order features.
6 Filtrations: a tool to assess hierarchical and temporal structure
In previous sections we have seen how we can construct
simplicial complexes from neural data and interrogate the
structure in these complexes using both extensions of
common graph theoretical notions and completely novel tools
drawn from algebraic topology. We close the
mathematical portion of this exposition by discussing a computational
process that is common in algebraic topology and that
directly addresses two critical needs in the neuroscience
community: (i) the assessment of hierarchical structure in
relational data via a principled thresholding approach, and
(ii) the assessment of temporal properties of stimulation,
neurodegenerative disease, and information transmission.
Filtrations to assess hierarchical structure in weighted
networks One of the most common features of network
data is a notion of strength or weight of connections between
nodes. In some situations, like measurements of
correlation or coherence of activity, the resulting network has
edges between every pair of nodes and it is common to
threshold the network to obtain some sparser, unweighted
network whose edges correspond to “significant”
(Achard et al. 2006)
. However it is difficult to make
a principled choice of threshold
(Ginestet et al. 2011;
Bassett et al. 2012; Garrison et al. 2015; Drakesmith et al.
2015; Sala et al. 2014; Langer et al. 2013)
, and the resulting
network discards a great deal of information. Even in the
case of sparse weighted networks, many metrics of structure
are defined only for the underlying unweighted network, so
in order to apply the metric, the weights are discarded and
this information is again lost
(Rubinov and Bassett 2011)
Here, we describe a technique that is commonly applied in
the study of weighted simplicial complexes which does not
discard any information.
Generalizing weighted graphs, a weighted simplicial
complex is obtained from a standard simplicial complex by
assigning to each simplex (including vertices) a numeric
weight. If we think of each simplex as recording some
relationship between its vertices, then the assigned weight
records the “strength” of that relationship. Recall that we
require that every face of a simplex also appears in a
simplicial complex; that is, every subgroup of a related population
is also related. Analogously, we require that the strength of
the relation in each subgroup be at least as large as that in the
whole population, so the weight assigned to each simplex
must be no larger than that assigned to any of its faces.
Given a weighted simplicial complex, a filtration of
complexes can be constructed by consecutively applying
each of the weights as thresholds in turn, constructing
an unweighted simplicial complex whose simplices are
precisely those whose weight exceeds the threshold, and
labeling each such complex by the weight at which it was
binarized. The resulting sequence of complexes retains all
of the information in the original weighted complex, but one
can apply metrics that are undefined or difficult to compute
for weighted complexes to the entire collection, thinking
of the resulting values as a function parameterized by the
weights of the original complex (Fig. 7d). However, it is also
the case that these unweighted complexes are related to one
another, and more sophisticated measurements of structure,
like homology, can exploit these relations to extract much
finer detail of the evolution of the complexes as the
threshold varies (Fig. 7c). We note that the omni-thresholding
approach utilized in constructing a filtration is a common
theme among other recently developed methods for network
characterization, including cost integration
(Ginestet et al.
and functional data analysis
(Bassett et al. 2012; Ellis
and Klein 2014)
The formalism described above provides a principled
framework to translate a weighted graph or simplicial
complex into a family of unweighted graphs or complexes that
retain all information in the weighting by virtue of their
relationships to one another. However, filtrations are much more
generally useful: for example, they can be used to assess the
dynamics of neural processes.
Filtrations to assess temporal dynamics of neural
processes in health and disease Many of the challenges
faced by cutting edge experimental techniques in the field
of neuroscience are driven by the underlying difficulties
implicit in assessing temporal changes in complex patterns
of relationships. For example, with new optogenetics
capabilities, we can stimulate single neurons or specific groups
of neurons to control their function
(Grosenick et al. 2015)
Similarly, advanced neurotechnologies including
microstimulation, transcranial magnetic stimulation, and
neurofeedback enable effective control over larger swaths of
(Krug et al. 2015; Sulzer et al. 2013)
. With the advent
of these technologies, it becomes imperative to develop
computational tools to quantitatively characterize and assess
the impact of stimulation on system function, and more
broadly, to understand how the structure of a simplicial
complex affects the transmission of information.
To meet this need, one can construct a different type of
filtration, such as that introduced in
(Taylor et al. 2015)
Fig. 7 Filtrations of a weighted simplicial complex measure dynamic
network properties. a A neural system can be stimulated in precise
locations using electrical, magnetic or optogenetic methods and the
resulting activity recorded. b A filtration of simplicial complexes is
built by recording as maximal faces all patterns of coactivity observed
up to a given time. A filtration can be constructed from any weighted
simplicial complex by thresholding at every possible weight to produce
a sequence of standard simplicial complexes, each sitting inside the
next.. c A persistence diagram recording the appearance (“birth”) and
disappearance or merging (“death”) of homology cycles throughout
the context of graphs: construct a sequence of simplicial
complexes with a time parameter, labeling each simplex as
“on” or “off” at each time, and require that once simplices
“turn on” they remain so indefinitely. If the function has the
further requirement that in order for a simplex to be active,
all of its faces must be as well, then a filtration is obtained by
taking all active simplices at each time. Such functions are
quite natural to apply to the study of the pattern of neurons
or neural units that are activated following stimulation.
Interestingly, this type of filtration is also a natural way in
which to probe and reason about models of
neurodegenerative disease such as the recently posited diffusion model of
(Raj et al. 2012; Zhou et al. 2012)
Here, critical network epicenters form points of
vulnerability that are effected early in the disease, and from which
toxic protein species travel via a process of transneuronal
spread. Indeed, these filtrations were first introduced in the
context of contagion models (Taylor et al. 2015), where
a simplex becomes active once sufficiently many nearby
simplices are active.
Measuring the structure of filtrations Assuming we have
encoded our data in an appropriate filtration, guided by
our scientific hypothesis of interest, we might next wish
to quantitatively characterize and measure the structure in
those filtrations. It is important to note that any given
measure of the structure of a simplicial complex can be applied
to each complex in a filtration in turn, producing a function
the filtration in panel (b). Cycles on the top edge of the diagram are
those that do not die. Tracking equivalent cycles through the filtration
provides information about the evolution of structure as the filtration
parameter changes. d Betti curves are the Betti numbers for each
complex in the filtration of panel (b) represented as functions of time.
Such curves can be constructed for any numerical measurement of
the individual unweighted simplicial complexes in the filtration and
provide a more complete description of structure than the individual
measurements taken separately
from the set of weights appearing in the complex to the set
of values the measure can take (Fig. 7d). This function is
a new measure of the structure of the complex which does
not rely on thresholds and can highlight interesting details
that would not be apparent at any fixed threshold (or small
range of thresholds), as well as being more robust to
perturbations in the weights than measurements of any individual
complex in the filtration.
Of particular interest in this setting are those
quantitative measures whose evolution can be explicitly understood
in terms of the relationships between successive complexes
in the filtration, as then we can exploit this framework
to gain a more refined picture of the structure present in
the weighted simplicial complex. Central among these in
terms of current breadth of application and computability is
persistent homology, which extends the homology of each
individual complex in the filtration by tracking how cycles
change as simplices are added when their weight exceeds
the threshold: new cycles can form, and due to the notion
of equivalence, cycles can also merge change shape, and
potentially finally be filled in by larger simplices. Therefore,
the sequence of complexes in the filtration is transformed
by homology into an inter-related family of evolving cycles.
Inside this sequence, cycles have well-defined birth and
death weights, between which very complex interactions are
possible. This information is often encoded in persistence
diagrams for each degree n (Fig. 7c), scatter plots of birth
and death weights for each cycle which give a schematic
overview of how the cycles are born and die.
Understanding these persistence lifetimes of individual cycles in the
system and their statistics can provide critical information
about how the system is arranged.
We are at a uniquely opportune moment, in which a wealth
of tools and computational methods are poised for
principled development directed toward specific critical
neuroscience challenges. With the feverish rise of data being
collected from neural systems across species and spatial
scales, mathematicians and experimental scientists must
necessarily engage in deeper conversation about how
meaning can be drawn from minutia. Such conversations will
inevitably turn to the common understanding that it is not
necessarily the individual objects of study themselves, but
their relations to one another, that provide the real structure
of human and animal thought. Though originally developed
for entirely different purposes, the algebraic topology of
simplicial complexes provides a quantitative methodology
uniquely suited to address these needs.
Acknowledgments RG acknowledges support from the Air Force
Office of Scientific Research (FA9550-12-1-0416 and
FA9550-14-10012) and the Office of Naval Research (NO0014-16-1-2010). DSB
acknowledges support from the John D. and Catherine T. MacArthur
Foundation, the Alfred P. Sloan Foundation, the Army Research
Laboratory and the Army Research Office through contract numbers
W911NF-10-2-0022 and W911NF-14-1-0679, the National Institute
of Child Health and Human Development (1R01HD086888-01), the
National Institute of Mental Health (2-R01-DC-009209-11), the Office
of Naval Research, and the National Science Foundation
(BCS1441502, PHY-1554488 and BCS-1430087).
Open Access This article is distributed under the terms of the
Creative Commons Attribution 4.0 International License (http://
creativecommons.org/licenses/by/4.0/), which permits unrestricted
use, distribution, and reproduction in any medium, provided you give
appropriate credit to the original author(s) and the source, provide a
link to the Creative Commons license, and indicate if changes were
Compliance with Ethical Standards
Conflict of interests
The authors declare that they have no conflict
from the view of computational complexity. The
following facts are well-known
(Dumas et al. 2003)
: the standard
algorithm for computing homology involves computing the
Smith normal form of a matrix (the boundary operator). In
general, the time-complexity is doubly-exponential in the
size of the matrix; it reduces to cubic complexity for simple
(binary) coefficients and quadratic complexity for a sparse
matrix. Worse still, the size of the relevant matrix is the
total number of simplices, which grows exponentially with
the dimension. This makes both space complexity (memory)
and time complexity (runtime) an issue. For example,
computing the persistent homology from correlation data of 100
neurons leads to a simplicial complex with approximately
107 4-simplices, while the same computation for a
population of 200 neurons involves two orders of magnitude more.
However, there are a number of ways to exploit the
structure inherent in simplicial complexes to mitigate this
combinatorial growth in complexity. One effective approach
is to use preprocessing to collapse the size of the complex
– sometimes dramatically – without changing the
homology. This is the basis of various reduction algorithms: see,
(Kaczynski et al. 2004; Mischaikow and Nanda 2013)
Another approach is to use the fact that homology can be
computed locally and then aggregated, allowing for
distributed computation over multiple processors and memory
cores (Bauer et al. 2014). Finally, computing approximate
homology further reduces complexity in difficult cases
while still providing useful statistical information
A comprehensive survey on the state-of-the-art in
homology software with benchmarks as of late 2015 appears in
(Otter et al. 2015)
. Because algorithmic computations of
topological quantities is a relatively recent innovation and
because the field is now being driven by accelerating
interest in the broader scientific community, it is our expectation
that new ideas and better software implementations will
dramatically improve our ability to perform these computations
over the next few years. Further, as is commonly the case,
computations on “organic” data outperform the worst-case
expectations for the algorithms; in practice, the difficulty of
homology computation in a particular dimension tends to
grow linearly in the number of simplices in that dimension.
Thus, we are optimistic that the computational tools
necessary to apply these ideas to neural data will be available
to meet the needs of the neuroscience community as they
The primary computational challenge in the methods here
surveyed is computing the homology of a simplicial
complex (or a filtration thereof). This is a significant challenge,
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