Minimal Cycle Representatives in Persistent Homology Using Linear Programming: An Empirical Study With User's Guide.

Cycle representatives of persistent homology classes can be used to provide descriptions of topological features in data. However, the non-uniqueness of these representatives creates ambiguity and can lead to many different interpretations of the same set of classes. One approach to solving this problem is to optimize the choice of representative against some measure that is meaningful in the context of the data. In this work, we provide a study of the effectiveness and computational cost of several ℓ 1 minimization optimization procedures for constructing homological cycle bases for persistent homology with rational coefficients in dimension one, including uniform-weighted and length-weighted edge-loss algorithms as well as uniform-weighted and area-weighted triangle-loss algorithms. We conduct these optimizations via standard linear programming methods, applying general-purpose solvers to optimize over column bases of simplicial boundary matrices. Our key findings are: 1) optimization is effective in reducing the size of cycle representatives, though the extent of the reduction varies according to the dimension and distribution of the underlying data, 2) the computational cost of optimizing a basis of cycle representatives exceeds the cost of computing such a basis, in most data sets we consider, 3) the choice of linear solvers matters a lot to the computation time of optimizing cycles, 4) the computation time of solving an integer program is not significantly longer than the computation time of solving a linear program for most of the cycle representatives, using the Gurobi linear solver, 5) strikingly, whether requiring integer solutions or not, we almost always obtain a solution with the same cost and almost all solutions found have entries in { - 1,0,1 } and therefore, are also solutions to a restricted ℓ 0 optimization problem, and 6) we obtain qualitatively different results for generators in Erdos-Rényi random clique complexes than in real-world and synthetic point cloud data.

On open and closed convex codes.

Neural codes serve as a language for neurons in the brain. Open (or closed) convex codes, which arise from the pattern of intersections of collections of open (or closed) convex sets in Euclidean space, are of particular relevance to neuroscience. Not every code is open or closed convex, however, and the combinatorial properties of a code that determine its realization by such sets are still poorly understood. Here we find that a code that can be realized by a collection of open convex sets may or may not be realizable by closed convex sets, and vice versa, establishing that open convex and closed convex codes are distinct classes. We establish a non-degeneracy condition that guarantees that the corresponding code is both open convex and closed convex. We also prove that max intersection-complete codes (i.e. codes that contain all intersections of maximal codewords) are both open convex and closed convex, and provide an upper bound for their minimal embedding dimension. Finally, we show that the addition of non-maximal codewords to an open convex code preserves convexity.

Effective learning is accompanied by high-dimensional and efficient representations of neural activity.

A fundamental cognitive process is to map value and identity onto the objects we learn about. However, what space best embeds this mapping is not completely understood. Here we develop tools to quantify the space and organization of such a mapping in neural responses as reflected in functional MRI, to show that quick learners have a higher dimensional representation than slow learners, and hence more easily distinguishable whole-brain responses to objects of different value. Furthermore, we find that quick learners display more compact embedding of their neural responses, and hence have higher ratios of their stimuli dimension to their embedding dimension, which is consistent with greater efficiency of cognitive coding. Lastly, we investigate the neurophysiological drivers at smaller scales and study the complementary distinguishability of whole-brain responses. Our results demonstrate a spatial organization of neural responses characteristic of learning and offer geometric measures applicable to identifying efficient coding in higher-order cognitive processes.

Knowledge gaps in the early growth of semantic feature networks.

Understanding language learning, and more general knowledge acquisition, requires characterization of inherently qualitative structures. Recent work has applied network science to this task by creating semantic feature networks, in which words correspond to nodes and connections to shared features, then characterizing the structure of strongly inter-related groups of words. However, the importance of sparse portions of the semantic network - knowledge gaps - remains unexplored. Using applied topology we query the prevalence of knowledge gaps, which we propose manifest as cavities within the growing semantic feature network of toddlers. We detect topological cavities of multiple dimensions and find that despite word order variation, global organization remains similar. We also show that nodal network measures correlate with filling cavities better than basic lexical properties. Finally, we discuss the importance of semantic feature network topology in language learning and speculate that the progression through knowledge gaps may be a robust feature of knowledge acquisition.

Cliques and cavities in the human connectome.

Encoding brain regions and their connections as a network of nodes and edges captures many of the possible paths along which information can be transmitted as humans process and perform complex behaviors. Because cognitive processes involve large, distributed networks of brain areas, principled examinations of multi-node routes within larger connection patterns can offer fundamental insights into the complexities of brain function. Here, we investigate both densely connected groups of nodes that could perform local computations as well as larger patterns of interactions that would allow for parallel processing. Finding such structures necessitates that we move from considering exclusively pairwise interactions to capturing higher order relations, concepts naturally expressed in the language of algebraic topology. These tools can be used to study mesoscale network structures that arise from the arrangement of densely connected substructures called cliques in otherwise sparsely connected brain networks. We detect cliques (all-to-all connected sets of brain regions) in the average structural connectomes of 8 healthy adults scanned in triplicate and discover the presence of more large cliques than expected in null networks constructed via wiring minimization, providing architecture through which brain network can perform rapid, local processing. We then locate topological cavities of different dimensions, around which information may flow in either diverging or converging patterns. These cavities exist consistently across subjects, differ from those observed in null model networks, and - importantly - link regions of early and late evolutionary origin in long loops, underscoring their unique role in controlling brain function. These results offer a first demonstration that techniques from algebraic topology offer a novel perspective on structural connectomics, highlighting loop-like paths as crucial features in the human brain's structural architecture.

Developmental increases in white matter network controllability support a growing diversity of brain dynamics.

As the human brain develops, it increasingly supports coordinated control of neural activity. The mechanism by which white matter evolves to support this coordination is not well understood. Here we use a network representation of diffusion imaging data from 882 youth ages 8-22 to show that white matter connectivity becomes increasingly optimized for a diverse range of predicted dynamics in development. Notably, stable controllers in subcortical areas are negatively related to cognitive performance. Investigating structural mechanisms supporting these changes, we simulate network evolution with a set of growth rules. We find that all brain networks are structured in a manner highly optimized for network control, with distinct control mechanisms predicted in child vs. older youth. We demonstrate that our results cannot be explained by changes in network modularity. This work reveals a possible mechanism of human brain development that preferentially optimizes dynamic network control over static network architecture.

Topological and geometric measurements of force-chain structure.

Developing quantitative methods for characterizing structural properties of force chains in densely packed granular media is an important step toward understanding or predicting large-scale physical properties of a packing. A promising framework in which to develop such methods is network science, which can be used to translate particle locations and force contacts into a graph in which particles are represented by nodes and forces between particles are represented by weighted edges. Recent work applying network-based community-detection techniques to extract force chains opens the door to developing statistics of force-chain structure, with the goal of identifying geometric and topological differences across packings, and providing a foundation on which to build predictions of bulk material properties from mesoscale network features. Here we discuss a trio of related but fundamentally distinct measurements of the mesoscale structure of force chains in two-dimensional (2D) packings, including a statistic derived using tools from algebraic topology, which together provide a tool set for the analysis of force chain architecture. We demonstrate the utility of this tool set by detecting variations in force-chain architecture with pressure. Collectively, these techniques can be generalized to 3D packings, and to the assessment of continuous deformations of packings under stress or strain.

Choosing Wavelet Methods, Filters, and Lengths for Functional Brain Network Construction.

Wavelet methods are widely used to decompose fMRI, EEG, or MEG signals into time series representing neurophysiological activity in fixed frequency bands. Using these time series, one can estimate frequency-band specific functional connectivity between sensors or regions of interest, and thereby construct functional brain networks that can be examined from a graph theoretic perspective. Despite their common use, however, practical guidelines for the choice of wavelet method, filter, and length have remained largely undelineated. Here, we explicitly explore the effects of wavelet method (MODWT vs. DWT), wavelet filter (Daubechies Extremal Phase, Daubechies Least Asymmetric, and Coiflet families), and wavelet length (2 to 24)-each essential parameters in wavelet-based methods-on the estimated values of graph metrics and in their sensitivity to alterations in psychiatric disease. We observe that the MODWT method produces less variable estimates than the DWT method. We also observe that the length of the wavelet filter chosen has a greater impact on the estimated values of graph metrics than the type of wavelet chosen. Furthermore, wavelet length impacts the sensitivity of the method to detect differences between health and disease and tunes classification accuracy. Collectively, our results suggest that the choice of wavelet method and length significantly alters the reliability and sensitivity of these methods in estimating values of metrics drawn from graph theory. They furthermore demonstrate the importance of reporting the choices utilized in neuroimaging studies and support the utility of exploring wavelet parameters to maximize classification accuracy in the development of biomarkers of psychiatric disease and neurological disorders.

Two's company, three (or more) is a simplex : Algebraic-topological tools for understanding higher-order structure in neural data.

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.

Clique topology reveals intrinsic geometric structure in neural correlations.

Detecting meaningful structure in neural activity and connectivity data is challenging in the presence of hidden nonlinearities, where traditional eigenvalue-based methods may be misleading. We introduce a novel approach to matrix analysis, called clique topology, that extracts features of the data invariant under nonlinear monotone transformations. These features can be used to detect both random and geometric structure, and depend only on the relative ordering of matrix entries. We then analyzed the activity of pyramidal neurons in rat hippocampus, recorded while the animal was exploring a 2D environment, and confirmed that our method is able to detect geometric organization using only the intrinsic pattern of neural correlations. Remarkably, we found similar results during nonspatial behaviors such as wheel running and rapid eye movement (REM) sleep. This suggests that the geometric structure of correlations is shaped by the underlying hippocampal circuits and is not merely a consequence of position coding. We propose that clique topology is a powerful new tool for matrix analysis in biological settings, where the relationship of observed quantities to more meaningful variables is often nonlinear and unknown.

A no-go theorem for one-layer feedforward networks.

It is often hypothesized that a crucial role for recurrent connections in the brain is to constrain the set of possible response patterns, thereby shaping the neural code. This implies the existence of neural codes that cannot arise solely from feedforward processing. We set out to find such codes in the context of one-layer feedforward networks and identified a large class of combinatorial codes that indeed cannot be shaped by the feedforward architecture alone. However, these codes are difficult to distinguish from codes that share the same sets of maximal activity patterns in the presence of subtractive noise. When we coarsened the notion of combinatorial neural code to keep track of only maximal patterns, we found the surprising result that all such codes can in fact be realized by one-layer feedforward networks. This suggests that recurrent or many-layer feedforward architectures are not necessary for shaping the (coarse) combinatorial features of neural codes. In particular, it is not possible to infer a computational role for recurrent connections from the combinatorics of neural response patterns alone. Our proofs use mathematical tools from classical combinatorial topology, such as the nerve lemma and the existence of an inverse nerve. An unexpected corollary of our main result is that any prescribed (finite) homotopy type can be realized by a subset of the form [Formula: see text], where Ρ is a polyhedron.