Two’s company, three (or more) is a simplex

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 observa- tion 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 alge- braic topology, one can work backward from observed coac- tivity 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 geom- etry of the environment, one can in principle introduce infor- mation regarding receptive field size (Curto and Itskov 2008). 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) and shown that theta oscillations improve such learning mechanisms (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 neural networks (Dlotko et al. 2016).

One of the earliest applications of algebraic topology to neural data was to the study of activity in the macaque primary visual cortex (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 example, in (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) and in Bendich et al. (2014), the authors use two different types of cycles derived from the geometry of brain artery trees to infer age and gender in human subjects.

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) to classify pediatric ADHD, ASD and control subjects; in (Khalid et al. 2014) to differentiate mouse models of depression from controls; in (Choi et al. 2014) to differentiate epileptic rat models from controls; and in (Kim et al. 2014) to 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), and a similar approach has been applied to the study of functional brain networks during learning (Stolz 2014). More 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. 2015) (Fig. 3).

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 (Boczko et al. 2005), quantification of contagion spread in social networks (Taylor et al. 2015), characterization of structure in networks of coupled oscillators (Stolz 2014), the study of phylogenic trees (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, 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.

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. 2015) (Fig. 3). In contrast to analyses using standard graph-theoretic tools, the pattern of simplices revealed the presence of geometric structure in only the information encoded 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 systems.

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) (Curto and Itskov 2008); 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). 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. 2015). 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 non-zero entries (Dowker 1952). 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).

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 models.

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 (Dowker 1952). Critically, this formulation refocuses attention (and the output of various vertex-based statistical measures) from individual neural units to patterns of coactivity.

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 (Fortunato 2010; 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 well-represented 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 complex (Kozlov 2007), and can be used to study properties of a system’s community structure such as dynamic flexibility (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).

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 larger simplices).

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 system (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 passes.

The most commonly used of these measurements is the (simplicial) homology of the complex, which is actually a sequence of measurements. The n ^th 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 1-simplices, 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 0 ^{t h} 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 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.

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” connections (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. 2011) 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.

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 cortex (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) in 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 fronto-temporal dementia (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.

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 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.