Differences in the evolution of network components across groups as the threshold 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. 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. 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. 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 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. 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. Further, as the requisite subsets of a simplex represent regions in the geometric boundary of the simplex Fig.

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 algebraic topology Carlsson ; Lum et al. 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. 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.

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. 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. 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. 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.

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. 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.

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. 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.

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 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.

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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. 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 ; Porter et al. 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. 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. 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 ; Kozlov 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 constructs. 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.

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. 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.

### 1991 Mathematics Subject Classification

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 framework also provides new and powerful ways to measure biological systems. 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. 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. 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.

In many settings, a powerful summary statistic is simply a count of the number of inequivalent cycles of each dimension appearing in the complex.

## Differential cohomology in a cohesive infinity-topos - INSPIRE-HEP

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. 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. 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. One of the most common features of network data is a notion of strength or weight of connections between nodes. However it is difficult to make a principled choice of threshold Ginestet et al.

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 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. 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.

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. 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.

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. Similarly, advanced neurotechnologies including microstimulation, transcranial magnetic stimulation, and neurofeedback enable effective control over larger swaths of cortex Krug et al. 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. 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.