363 results on '"Harrington, Heather"'
Search Results
2. Geometry of rational quasi-independence models as toric fiber products
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Coons, Jane Ivy, Harrington, Heather A., and Paul, Niharika Chakrabarty
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Mathematics - Algebraic Geometry ,Mathematics - Combinatorics ,Mathematics - Statistics Theory ,62R01, 14M25, 62F30, 05C90 - Abstract
We investigate the geometry of a family of log-linear statistical models called quasi-independence models. The toric fiber product is useful for understanding the geometry of parameter inference in these models because the maximum likelihood degree is multiplicative under the TFP. We define the coordinate toric fiber product, or cTFP, and give necessary and sufficient conditions under which a quasi-independence model is a cTFP of lower-order models. We show that the vanishing ideal of every 2-way quasi-independence model with ML-degree 1 can be realized as an iterated toric fiber product of linear ideals. We also classify which Lawrence lifts of 2-way quasi-independence models are cTFPs and give a necessary condition under which a $k$-way model has ML-degree 1 using its facial submodels.
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- 2024
3. Quiver Laplacians and Feature Selection
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Sumray, Otto, Harrington, Heather A., and Nanda, Vidit
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Statistics - Machine Learning ,Computer Science - Machine Learning ,Mathematics - Combinatorics ,Mathematics - Representation Theory ,Mathematics - Statistics Theory ,Quantitative Biology - Quantitative Methods ,16G20, 05C50, 62P05, 62H25 - Abstract
The challenge of selecting the most relevant features of a given dataset arises ubiquitously in data analysis and dimensionality reduction. However, features found to be of high importance for the entire dataset may not be relevant to subsets of interest, and vice versa. Given a feature selector and a fixed decomposition of the data into subsets, we describe a method for identifying selected features which are compatible with the decomposition into subsets. We achieve this by re-framing the problem of finding compatible features to one of finding sections of a suitable quiver representation. In order to approximate such sections, we then introduce a Laplacian operator for quiver representations valued in Hilbert spaces. We provide explicit bounds on how the spectrum of a quiver Laplacian changes when the representation and the underlying quiver are modified in certain natural ways. Finally, we apply this machinery to the study of peak-calling algorithms which measure chromatin accessibility in single-cell data. We demonstrate that eigenvectors of the associated quiver Laplacian yield locally and globally compatible features., Comment: 40 pages, 7 figures
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- 2024
4. Algebraic identifiability of partial differential equation models
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Byrne, Helen, Harrington, Heather, Ovchinnikov, Alexey, Pogudin, Gleb, Rahkooy, Hamid, and Soto, Pedro
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Quantitative Biology - Quantitative Methods ,Computer Science - Symbolic Computation ,Electrical Engineering and Systems Science - Systems and Control ,Mathematics - Analysis of PDEs ,92B05, 12H05, 35R30, 93C20, 93B25, 93B30 - Abstract
Differential equation models are crucial to scientific processes. The values of model parameters are important for analyzing the behaviour of solutions. A parameter is called globally identifiable if its value can be uniquely determined from the input and output functions. To determine if a parameter estimation problem is well-posed for a given model, one must check if the model parameters are globally identifiable. This problem has been intensively studied for ordinary differential equation models, with theory and several efficient algorithms and software packages developed. A comprehensive theory of algebraic identifiability for PDEs has hitherto not been developed due to the complexity of initial and boundary conditions. Here, we provide theory and algorithms, based on differential algebra, for testing identifiability of polynomial PDE models. We showcase this approach on PDE models arising in the sciences.
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- 2024
5. Grounded Persistent Path Homology: A Stable, Topological Descriptor for Weighted Digraphs
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Chaplin, Thomas, Harrington, Heather A., and Tillmann, Ulrike
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- 2024
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6. Absolute concentration robustness: Algebra and geometry
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Puente, Luis David García, Gross, Elizabeth, Harrington, Heather A, Johnston, Matthew, Meshkat, Nicolette, Millán, Mercedes Pérez, and Shiu, Anne
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Mathematics - Algebraic Geometry ,Mathematics - Dynamical Systems ,Quantitative Biology - Molecular Networks ,37N25, 92E20, 12D10, 37C25, 65H14, 14Q20 - Abstract
Motivated by the question of how biological systems maintain homeostasis in changing environments, Shinar and Feinberg introduced in 2010 the concept of absolute concentration robustness (ACR). A biochemical system exhibits ACR in some species if the steady-state value of that species does not depend on initial conditions. Thus, a system with ACR can maintain a constant level of one species even as the environment changes. Despite a great deal of interest in ACR in recent years, the following basic question remains open: How can we determine quickly whether a given biochemical system has ACR? Although various approaches to this problem have been proposed, we show that they are incomplete. Accordingly, we present new methods for deciding ACR, which harness computational algebra. We illustrate our results on several biochemical signaling networks., Comment: 44 pages
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- 2023
7. Kuramoto Oscillators: algebraic and topological aspects
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Harrington, Heather, Schenck, Hal, and Stillman, Mike
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Mathematics - Dynamical Systems ,Mathematics - Algebraic Geometry ,Mathematics - Classical Analysis and ODEs ,90C26, 90C35, 34D06, 35B35 - Abstract
We investigate algebraic and topological signatures of networks of coupled oscillators. Translating dynamics into a system of algebraic equations enables us to identify classes of network topologies that exhibit unexpected behaviors. Many previous studies focus on synchronization of networks having high connectivity, or of a specific type (e.g. circulant networks). We introduce the Kuramoto ideal; an algebraic analysis of this ideal allows us to identify features beyond synchronization, such as positive dimensional components in the set of potential solutions (e.g. curves instead of points). We prove sufficient conditions on the network structure for such solutions to exist. The points lying on a positive dimensional component of the solution set can never correspond to a linearly stable state. We apply this framework to give a complete analysis of linear stability for all networks on at most eight vertices. Furthermore, we describe a construction of networks on an arbitrary number of vertices having linearly stable states that are not twisted stable states., Comment: 22 pages, 6 figures
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- 2023
8. Multiscale topology classifies cells in subcellular spatial transcriptomics
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Benjamin, Katherine, Bhandari, Aneesha, Kepple, Jessica D., Qi, Rui, Shang, Zhouchun, Xing, Yanan, An, Yanru, Zhang, Nannan, Hou, Yong, Crockford, Tanya L., McCallion, Oliver, Issa, Fadi, Hester, Joanna, Tillmann, Ulrike, Harrington, Heather A., and Bull, Katherine R.
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- 2024
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9. Brain chains as topological signatures for Alzheimer’s disease
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Goodbrake, Christian, Beers, David, Thompson, Travis B., Harrington, Heather A., and Goriely, Alain
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- 2024
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10. Active shape control by plants in dynamic environments
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Oliveri, Hadrien, Moulton, Derek E., Harrington, Heather A., and Goriely, Alain
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Physics - Biological Physics ,92-10, 92B99 - Abstract
Plants are a paradigm for active shape control in response to stimuli. For instance, it is well-known that a tilted plant will eventually straighten vertically, demonstrating the influence of both an external stimulus, gravity, and an internal stimulus, proprioception. These effects can be modulated when a potted plant is additionally rotated along the plant's axis, as in a rotating clinostat, leading to intricate shapes. We use a morphoelastic model for the response of growing plants to study the joint effect of both stimuli at all rotation speeds. In the absence of rotation, we identify a universal planar shape towards which all shoots eventually converge. With rotation, we demonstrate the existence of a stable family of three-dimensional dynamic equilibria where the plant axis is fixed in space. Further, the effect of axial growth is to induce steady behaviors, such as solitary waves. Overall, this study offers new insight into the complex out-of-equilibrium dynamics of a plant in three dimensions and further establishes that internal stimuli in active materials are key for robust shape control.
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- 2023
11. Topological fingerprints for audio identification
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Reise, Wojciech, Fernández, Ximena, Dominguez, Maria, Harrington, Heather A., and Beguerisse-Díaz, Mariano
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Computer Science - Sound ,Electrical Engineering and Systems Science - Audio and Speech Processing ,Mathematics - Algebraic Topology ,55N31, 68U10, 62R40 - Abstract
We present a topological audio fingerprinting approach for robustly identifying duplicate audio tracks. Our method applies persistent homology on local spectral decompositions of audio signals, using filtered cubical complexes computed from mel-spectrograms. By encoding the audio content in terms of local Betti curves, our topological audio fingerprints enable accurate detection of time-aligned audio matchings. Experimental results demonstrate the accuracy of our algorithm in the detection of tracks with the same audio content, even when subjected to various obfuscations. Our approach outperforms existing methods in scenarios involving topological distortions, such as time stretching and pitch shifting., Comment: 26 pages
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- 2023
12. Relational persistent homology for multispecies data with application to the tumor microenvironment
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Stolz, Bernadette J., Dhesi, Jagdeep, Bull, Joshua A., Harrington, Heather A., Byrne, Helen M., and Yoon, Iris H. R.
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Mathematics - Algebraic Topology ,Quantitative Biology - Cell Behavior ,Quantitative Biology - Quantitative Methods ,55N31, 92C17 - Abstract
Topological data analysis (TDA) is an active field of mathematics for quantifying shape in complex data. Standard methods in TDA such as persistent homology (PH) are typically focused on the analysis of data consisting of a single entity (e.g., cells or molecular species). However, state-of-the-art data collection techniques now generate exquisitely detailed multispecies data, prompting a need for methods that can examine and quantify the relations among them. Such heterogeneous data types arise in many contexts, ranging from biomedical imaging, geospatial analysis, to species ecology. Here, we propose two methods for encoding spatial relations among different data types that are based on Dowker complexes and Witness complexes. We apply the methods to synthetic multispecies data of a tumor microenvironment and analyze topological features that capture relations between different cell types, e.g., blood vessels, macrophages, tumor cells, and necrotic cells. We demonstrate that relational topological features can extract biological insight, including the dominant immune cell phenotype (an important predictor of patient prognosis) and the parameter regimes of a data-generating model. The methods provide a quantitative perspective on the relational analysis of multispecies spatial data, overcome the limits of traditional PH, and are readily computable.
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- 2023
13. Topological classification of tumour-immune interactions and dynamics
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Yang, Jingjie, Fang, Heidi, Dhesi, Jagdeep, Yoon, Iris H. R., Bull, Joshua A., Byrne, Helen M., Harrington, Heather A., and Grindstaff, Gillian
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Quantitative Biology - Cell Behavior ,Mathematics - Algebraic Topology ,92C17, 55N31 - Abstract
The complex and dynamic crosstalk between tumour and immune cells results in tumours that can exhibit distinct qualitative behaviours - elimination, equilibrium, and escape - and intricate spatial patterns, yet share similar cell configurations in the early stages. We offer a topological approach to analyse time series of spatial data of cell locations (including tumour cells and macrophages) in order to predict malignant behaviour. We propose four topological vectorisations specialised to such cell data: persistence images of Vietoris-Rips and radial filtrations at static time points, and persistence images for zigzag filtrations and persistence vineyards varying in time. To demonstrate the approach, synthetic data are generated from an agent-based model with varying parameters. We compare the performance of topological summaries in predicting - with logistic regression at various time steps - whether tumour niches surrounding blood vessels are present at the end of the simulation, as a proxy for metastasis (i.e., tumour escape). We find that both static and time-dependent methods accurately identify perivascular niche formation, significantly earlier than simpler markers such as the number of tumour cells and the macrophage phenotype ratio. We find additionally that dimension 0 persistence applied to macrophage data, representing multi-scale clusters of the spatial arrangement of macrophages, performs best at this classification task at early time steps, prior to full tumour development, and performs even better when time-dependent data are included; in contrast, topological measures capturing the shape of the tumour, such as tortuosity and punctures in the cell arrangement, perform best at intermediate and later stages. The logistic regression coefficients reveal detailed shape differences between the classes., Comment: 29 pages, 12 figures
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- 2023
14. Detecting Temporal shape changes with the Euler Characteristic Transform
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Marsh, Lewis, Zhou, Felix Y., Qin, Xiao, Lu, Xin, Byrne, Helen M., and Harrington, Heather A.
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Quantitative Biology - Quantitative Methods ,Mathematics - Algebraic Topology ,Quantitative Biology - Tissues and Organs - Abstract
Organoids are multi-cellular structures which are cultured in vitro from stem cells to resemble specific organs (e.g., brain, liver) in their three-dimensional composition. Dynamic changes in the shape and composition of these model systems can be used to understand the effect of mutations and treatments in health and disease. In this paper, we propose a new technique in the field of topological data analysis for DEtecting Temporal shape changes with the Euler Characteristic Transform (DETECT). DETECT is a rotationally invariant signature of dynamically changing shapes. We demonstrate our method on a data set of segmented videos of mouse small intestine organoid experiments and show that it outperforms classical shape descriptors. We verify our method on a synthetic organoid data set and illustrate how it generalises to 3D. We conclude that DETECT offers rigorous quantification of organoids and opens up computationally scalable methods for distinguishing different growth regimes and assessing treatment effects.
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- 2022
15. Topological Data Analysis Detects Percolation Thresholds in Arctic Melt-Pond Evolution
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Offord, Wilfred, Coughlan, Michael, Hewitt, Ian J., Harrington, Heather A., and Grindstaff, Gillian
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Physics - Data Analysis, Statistics and Probability ,Mathematics - Algebraic Topology ,Physics - Atmospheric and Oceanic Physics ,86A40 - Abstract
During the summer melt period, ponds form on the surface of Arctic sea ice as it melts, with important consequences for ice evolution and marine ecology. Due to the ice-albedo feedback, these melt ponds experience uneven heat absorption, and exhibit complex patterns, which has motivated the development of modelling and data analysis to understand their particular dynamics. We provide a multiscale shape analysis using tools from computational algebraic topology, simultaneously capturing convexity, proximity, integrity, and feature size complementing existing single-scale quantification. Of particular interest in modelling the ponds is a percolation threshold at which local pond structure begins merging into macroscopic features. This percolation threshold has previously been observed using fractal dimension techniques. The signed Euclidean distance transform (SEDT) is a topological encoding of heterogeneous shape in binary images, and has been previously applied to porous media for percolation as well as other material behaviours. Here we adapt the SEDT for Arctic melt pond data to give a rich characterization and computation of shape, quantifying overall melt pond development in several complementary ways, and from which classical percolation and dimension results can be extracted. This orientation-invariant topological approach distinguishes different dynamical network models of melt pond evolution of varying complexity., Comment: 13 pages, 13 figures
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- 2022
16. Multiscale topology classifies and quantifies cell types in subcellular spatial transcriptomics
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Benjamin, Katherine, Bhandari, Aneesha, Shang, Zhouchun, Xing, Yanan, An, Yanru, Zhang, Nannan, Hou, Yong, Tillmann, Ulrike, Bull, Katherine R., and Harrington, Heather A.
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Quantitative Biology - Quantitative Methods ,Mathematics - Algebraic Topology ,Quantitative Biology - Genomics ,Statistics - Methodology ,92-08, 55N31, 62R40, 68T09 - Abstract
Spatial transcriptomics has the potential to transform our understanding of RNA expression in tissues. Classical array-based technologies produce multiple-cell-scale measurements requiring deconvolution to recover single cell information. However, rapid advances in subcellular measurement of RNA expression at whole-transcriptome depth necessitate a fundamentally different approach. To integrate single-cell RNA-seq data with nanoscale spatial transcriptomics, we present a topological method for automatic cell type identification (TopACT). Unlike popular decomposition approaches to multicellular resolution data, TopACT is able to pinpoint the spatial locations of individual sparsely dispersed cells without prior knowledge of cell boundaries. Pairing TopACT with multiparameter persistent homology landscapes predicts immune cells forming a peripheral ring structure within kidney glomeruli in a murine model of lupus nephritis, which we experimentally validate with immunofluorescent imaging. The proposed topological data analysis unifies multiple biological scales, from subcellular gene expression to multicellular tissue organization., Comment: Main text: 8 pages, 4 figures. Supplement: 12 pages, 5 figures
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- 2022
17. Algebraic network reconstruction of discrete dynamical systems
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Harrington, Heather A., Stillman, Mike, and Veliz-Cuba, Alan
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Quantitative Biology - Quantitative Methods ,Mathematics - Algebraic Geometry ,Quantitative Biology - Molecular Networks ,13P25, 37N25, 92B05, 05E40, 46N60, 92C42, 68R10, 90B10, 97N70, 62-07 - Abstract
We present a computational algebra solution to reverse engineering the network structure of discrete dynamical systems from data. We use monomial ideals to determine dependencies between variables that encode constraints on the possible wiring diagrams underlying the process generating the discrete-time, continuous-space data. Our work assumes that each variable is either monotone increasing or decreasing. We prove that with enough data, even in the presence of small noise, our method can reconstruct the correct unique wiring diagram., Comment: 19 pages, 5 figures
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- 2022
18. Stability of topological descriptors for neuronal morphology
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Beers, David, Harrington, Heather A., and Goriely, Alain
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Quantitative Biology - Neurons and Cognition ,Mathematics - Algebraic Topology - Abstract
The topological morphology descriptor of a neuron is a multiset of intervals associated to the shape of the neuron represented as a tree. In practice, topological morphology descriptors are vectorized using persistence images, which can help classify and characterize the morphology of broad groups of neurons. We study the stability of topological morphology descriptors under small changes to neuronal morphology. We show that the persistence diagram arising from the topological morphology descriptor of a neuron is stable for the 1-Wasserstein distance against a range of perturbations to the tree. These results guarantee that persistence images of topological morphology descriptors are stable against the same set of perturbations and reliable., Comment: 11 pages, 4 figures
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- 2022
19. Grounded persistent path homology: a stable, topological descriptor for weighted digraphs
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Chaplin, Thomas, Harrington, Heather A., and Tillmann, Ulrike
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Mathematics - Algebraic Topology ,55N31 (Primary), 05C20 (Secondary), 05C22 (Secondary) ,G.2.2 - Abstract
Weighted digraphs are used to model a variety of natural systems and can exhibit interesting structure across a range of scales. In order to understand and compare these systems, we require stable, interpretable, multiscale descriptors. To this end, we propose grounded persistent path homology (GrPPH) - a new, functorial, topological descriptor that describes the structure of an edge-weighted digraph via a persistence barcode. We show there is a choice of circuit basis for the graph which yields geometrically interpretable representatives for the features in the barcode. Moreover, we show the barcode is stable, in bottleneck distance, to both numerical and structural perturbations. GrPPH arises from a flexible framework, parametrised by a choice of digraph chain complex and a choice of filtration; for completeness, we also investigate replacing the path homology complex, used in GrPPH, by the directed flag complex.
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- 2022
20. Hypergraphs for multiscale cycles in structured data
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Barbensi, Agnese, Yoon, Iris H. R., Madsen, Christian Degnbol, Ajayi, Deborah O., Stumpf, Michael P. H., and Harrington, Heather A.
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Mathematics - Algebraic Topology ,Computer Science - Computational Geometry ,Quantitative Biology - Quantitative Methods ,55N31, 62R40, 55P10, 60C05, 92B05, 92-10 - Abstract
Scientific data has been growing in both size and complexity across the modern physical, engineering, life and social sciences. Spatial structure, for example, is a hallmark of many of the most important real-world complex systems, but its analysis is fraught with statistical challenges. Topological data analysis can provide a powerful computational window on complex systems. Here we present a framework to extend and interpret persistent homology summaries to analyse spatial data across multiple scales. We introduce hyperTDA, a topological pipeline that unifies local (e.g. geodesic) and global (e.g. Euclidean) metrics without losing spatial information, even in the presence of noise. Homology generators offer an elegant and flexible description of spatial structures and can capture the information computed by persistent homology in an interpretable way. Here the information computed by persistent homology is transformed into a weighted hypergraph, where hyperedges correspond to homology generators. We consider different choices of generators (e.g. matroid or minimal) and find that centrality and community detection are robust to either choice. We compare hyperTDA to existing geometric measures and validate its robustness to noise. We demonstrate the power of computing higher-order topological structures on spatial curves arising frequently in ecology, biophysics, and biology, but also in high-dimensional financial datasets. We find that hyperTDA can select between synthetic trajectories from the landmark 2020 AnDi challenge and quantifies movements of different animal species, even when data is limited., Comment: 6 Figures, 15 pages and Supplementary Information (including figures) as an Appendix. Associated GitHub repositories: github.com/degnbol/hyperTDA and github.com/irishryoon/minimal_generators_curves
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- 2022
21. Zigzag persistence for coral reef resilience using a stochastic spatial model
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McDonald, Robert A., Neuhausler, Rosanna, Robinson, Martin, Larsen, Laurel G., Harrington, Heather A., and Bruna, Maria
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Quantitative Biology - Quantitative Methods ,Mathematics - Algebraic Topology ,Quantitative Biology - Populations and Evolution - Abstract
A complex interplay between species governs the evolution of spatial patterns in ecology. An open problem in the biological sciences is characterising spatio-temporal data and understanding how changes at the local scale affect global dynamics/behaviour. Here, we extend a well-studied temporal mathematical model of coral reef dynamics to include stochastic and spatial interactions and generate data to study different ecological scenarios. We present descriptors to characterise patterns in heterogeneous spatio-temporal data surpassing spatially averaged measures. We apply these descriptors to simulated coral data and demonstrate the utility of two topological data analysis techniques--persistent homology and zigzag persistence--for characterising mechanisms of reef resilience. We show that the introduction of local competition between species leads to the appearance of coral clusters in the reef. We use our analyses to distinguish temporal dynamics stemming from different initial configurations of coral, showing that the neighbourhood composition of coral sites determines their long-term survival. Using zigzag persistence, we determine which spatial configurations protect coral from extinction in different environments. Finally, we apply this toolkit of multi-scale methods to empirical coral reef data, which distinguish spatio-temporal reef dynamics in different locations, and demonstrate the applicability to a range of datasets.
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- 2022
22. Brain Chains as Topological Signatures for Alzheimer's Disease
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Goodbrake, Christian, Beers, David, Thompson, Travis B., Harrington, Heather A., and Goriely, Alain
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Quantitative Biology - Neurons and Cognition ,Mathematics - Algebraic Topology ,Mathematics - Combinatorics - Abstract
We propose a topological framework to study the evolution of Alzheimer's disease, the most common neurodegenerative disease. The modeling of this disease starts with the representation of the brain connectivity as a graph and the seeding of a toxic protein in a specific region represented by a vertex. Over time, the accumulation of toxic proteins at vertices and their propagation along edges are modeled by a dynamical system on this graph. These dynamics provide an order on the edges of the graph according to the damage created by high concentrations of proteins. This sequence of edges defines a filtration of the graph. We consider different filtrations given by different disease seeding locations. To study this filtration we propose a new combinatorial and topological method. A filtration defines a maximal chain in the partially ordered set of spanning subgraphs ordered by inclusion. To identify similar graphs, and define a topological signature, we quotient this poset by graph homotopy equivalence, which gives maximal chains in a smaller poset. We provide an algorithm to compute this direct quotient without computing all subgraphs and then propose bounds on the total number of graphs up to homotopy equivalence. To compare the maximal chains generated by this method, we extend Kendall's $d_K$ metric for permutations to more general graded posets and establish bounds for this metric. We then demonstrate the utility of this framework on actual brain graphs by studying the dynamics of tau proteins on the structural connectome. {We show that the proposed topological brain chain equivalence classes distinguish different simulated subtypes of Alzheimer's disease., Comment: 33 pages, 13 figures, submitted to Journal of Applied and Computational Topology (APCT)
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- 2022
23. Multiscale methods for signal selection in single-cell data
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Hoekzema, Renee S., Marsh, Lewis, Sumray, Otto, Carroll, Thomas M., Lu, Xin, Byrne, Helen M., and Harrington, Heather A.
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Quantitative Biology - Quantitative Methods ,Computer Science - Social and Information Networks ,Mathematics - Algebraic Topology ,Mathematics - Spectral Theory ,Statistics - Machine Learning - Abstract
Analysis of single-cell transcriptomics often relies on clustering cells and then performing differential gene expression (DGE) to identify genes that vary between these clusters. These discrete analyses successfully determine cell types and markers; however, continuous variation within and between cell types may not be detected. We propose three topologically motivated mathematical methods for unsupervised feature selection that consider discrete and continuous transcriptional patterns on an equal footing across multiple scales simultaneously. Eigenscores ($\text{eig}_i$) rank signals or genes based on their correspondence to low-frequency intrinsic patterning in the data using the spectral decomposition of the Laplacian graph. The multiscale Laplacian score (MLS) is an unsupervised method for locating relevant scales in data and selecting the genes that are coherently expressed at these respective scales. The persistent Rayleigh quotient (PRQ) takes data equipped with a filtration, allowing the separation of genes with different roles in a bifurcation process (e.g., pseudo-time). We demonstrate the utility of these techniques by applying them to published single-cell transcriptomics data sets. The methods validate previously identified genes and detect additional biologically meaningful genes with coherent expression patterns. By studying the interaction between gene signals and the geometry of the underlying space, the three methods give multidimensional rankings of the genes and visualisation of relationships between them., Comment: 32 pages, 15 figures, 1 table. Revised and published in Entropy, special issue Applications of Topological Data Analysis in the Life Sciences
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- 2022
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24. Barcodes distinguish morphology of neuronal tauopathy
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Beers, David, Goniotaki, Despoina, Hanger, Diane P., Goriely, Alain, and Harrington, Heather A.
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Quantitative Biology - Neurons and Cognition ,Mathematics - Algebraic Topology - Abstract
The geometry of neurons is known to be important for their functions. Hence, neurons are often classified by their morphology. Two recent methods, persistent homology and the topological morphology descriptor, assign a morphology descriptor called a barcode to a neuron equipped with a given function, such as the Euclidean distance from the root of the neuron. These barcodes can be converted into matrices called persistence images, which can then be averaged across groups. We show that when the defining function is the path length from the root, both the topological morphology descriptor and persistent homology are equivalent. We further show that persistence images arising from the path length procedure provide an interpretable summary of neuronal morphology. We introduce {topological morphology functions}, a class of functions similar to Sholl functions, that can be recovered from the associated topological morphology descriptor. To demonstrate this topological approach, we compare healthy cortical and hippocampal mouse neurons to those affected by progressive tauopathy. We find a significant difference in the morphology of healthy neurons and those with a tauopathy at a postsymptomatic age. We use persistence images to conclude that the diseased group tends to have neurons with shorter branches as well as fewer branches far from the soma., Comment: 25 pages, 10 figures
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- 2022
25. Homology of homologous knotted proteins
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Benjamin, Katherine, Mukta, Lamisah, Moryoussef, Gabriel, Uren, Christopher, Harrington, Heather A., Tillmann, Ulrike, and Barbensi, Agnese
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Mathematics - Algebraic Topology ,Mathematics - Geometric Topology ,Quantitative Biology - Biomolecules ,62R40, 55N31, 57K10 - Abstract
Quantification and classification of protein structures, such as knotted proteins, often requires noise-free and complete data. Here we develop a mathematical pipeline that systematically analyzes protein structures. We showcase this geometric framework on proteins forming open-ended trefoil knots, and we demonstrate that the mathematical tool, persistent homology, faithfully represents their structural homology. This topological pipeline identifies important geometric features of protein entanglement and clusters the space of trefoil proteins according to their depth. Persistence landscapes quantify the topological difference between a family of knotted and unknotted proteins in the same structural homology class. This difference is localized and interpreted geometrically with recent advancements in systematic computation of homology generators. The topological and geometric quantification we find is robust to noisy input data, which demonstrates the potential of this approach in contexts where standard knot theoretic tools fail., Comment: 3 figures + 2 SI figures
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- 2022
26. Principal Components Along Quiver Representations
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Seigal, Anna, Harrington, Heather A., and Nanda, Vidit
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Algebra ,Mathematics - Abstract
Quiver representations arise naturally in many areas across mathematics. Here we describe an algorithm for calculating the vector space of sections, or compatible assignments of vectors to vertices, of any finite-dimensional representation of a finite quiver. Consequently, we are able to define and compute principal components with respect to quiver representations. These principal components are solutions to constrained optimisation problems defined over the space of sections and are eigenvectors of an associated matrix pencil., Author(s): Anna Seigal [sup.1] [sup.2], Heather A. Harrington [sup.1], Vidit Nanda [sup.1] Author Affiliations: (1) grid.4991.5, 0000 0004 1936 8948, University of Oxford, , Oxford, UK (2) grid.38142.3c, 000000041936754X, Harvard [...]
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- 2023
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27. Algebra, Geometry and Topology of ERK Kinetics
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Marsh, Lewis, Dufresne, Emilie, Byrne, Helen M., and Harrington, Heather A.
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Quantitative Biology - Quantitative Methods ,Mathematics - Algebraic Geometry ,Mathematics - Algebraic Topology ,Statistics - Applications - Abstract
The MEK/ERK signalling pathway is involved in cell division, cell specialisation, survival and cell death. Here we study a polynomial dynamical system describing the dynamics of MEK/ERK proposed by Yeung et al. with their experimental setup, data and known biological information. The experimental dataset is a time-course of ERK measurements in different phosphorylation states following activation of either wild-type MEK or MEK mutations associated with cancer or developmental defects. We demonstrate how methods from computational algebraic geometry, differential algebra, Bayesian statistics and computational algebraic topology can inform the model reduction, identification and parameter inference of MEK variants, respectively. Throughout, we show how this algebraic viewpoint offers a rigorous and systematic analysis of such models.
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- 2021
28. Differential elimination for dynamical models via projections with applications to structural identifiability
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Dong, Ruiwen, Goodbrake, Christian, Harrington, Heather A, and Pogudin, Gleb
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Mathematics - Algebraic Geometry ,Computer Science - Computational Geometry ,Computer Science - Symbolic Computation ,Electrical Engineering and Systems Science - Systems and Control ,Quantitative Biology - Quantitative Methods - Abstract
Elimination of unknowns in a system of differential equations is often required when analysing (possibly nonlinear) dynamical systems models, where only a subset of variables are observable. One such analysis, identifiability, often relies on computing input-output relations via differential algebraic elimination. Determining identifiability, a natural prerequisite for meaningful parameter estimation, is often prohibitively expensive for medium to large systems due to the computationally expensive task of elimination. We propose an algorithm that computes a description of the set of differential-algebraic relations between the input and output variables of a dynamical system model. The resulting algorithm outperforms general-purpose software for differential elimination on a set of benchmark models from literature. We use the designed elimination algorithm to build a new randomized algorithm for assessing structural identifiability of a parameter in a parametric model. A parameter is said to be identifiable if its value can be uniquely determined from input-output data assuming the absence of noise and sufficiently exciting inputs. Our new algorithm allows the identification of models that could not be tackled before. Our implementation is publicly available as a Julia package at https://github.com/SciML/StructuralIdentifiability.jl.
- Published
- 2021
29. Topological Approximate Bayesian Computation for Parameter Inference of an Angiogenesis Model
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Thorne, Thomas, Kirk, Paul D. W., and Harrington, Heather A.
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Quantitative Biology - Quantitative Methods ,Statistics - Methodology - Abstract
Inferring the parameters of models describing biological systems is an important problem in the reverse engineering of the mechanisms underlying these systems. Much work has focused on parameter inference of stochastic and ordinary differential equation models using Approximate Bayesian Computation (ABC). While there is some recent work on inference in spatial models, this remains an open problem. Simultaneously, advances in topological data analysis (TDA), a field of computational mathematics, have enabled spatial patterns in data to be characterised. Here we focus on recent work using topological data analysis to study different regimes of parameter space for a well-studied model of angiogenesis. We propose a method for combining TDA with ABC to infer parameters in the Anderson-Chaplain model of angiogenesis. We demonstrate that this topological approach outperforms ABC approaches that use simpler statistics based on spatial features of the data. This is a first step towards a general framework of spatial parameter inference for biological systems, for which there may be a variety of filtrations, vectorisations, and summary statistics to be considered. All code used to produce our results is available as a Snakemake workflow., Comment: 7 pages, 2 figures. For associated code see: https://github.com/tt104/tabc_angio
- Published
- 2021
30. Principal Components along Quiver Representations
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Seigal, Anna, Harrington, Heather A., and Nanda, Vidit
- Subjects
Mathematics - Representation Theory ,16G20, 05C20, 62H25 - Abstract
Quiver representations arise naturally in many areas across mathematics. Here we describe an algorithm for calculating the vector space of sections, or compatible assignments of vectors to vertices, of any finite-dimensional representation of a finite quiver. Consequently, we are able to define and compute principal components with respect to quiver representations. These principal components are solutions to constrained optimisation problems defined over the space of sections, and are eigenvectors of an associated matrix pencil., Comment: 31 pages; v2: more examples, results added to section 5
- Published
- 2021
31. What are higher-order networks?
- Author
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Bick, Christian, Gross, Elizabeth, Harrington, Heather A., and Schaub, Michael T.
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Computer Science - Social and Information Networks ,Mathematics - Combinatorics ,Mathematics - Dynamical Systems ,Mathematics - Statistics Theory ,Nonlinear Sciences - Adaptation and Self-Organizing Systems ,05C82, 34B45, 68R10, 55U10, 05C65 - Abstract
Network-based modeling of complex systems and data using the language of graphs has become an essential topic across a range of different disciplines. Arguably, this graph-based perspective derives its success from the relative simplicity of graphs: A graph consists of nothing more than a set of vertices and a set of edges, describing relationships between pairs of such vertices. This simple combinatorial structure makes graphs interpretable and flexible modeling tools. The simplicity of graphs as system models, however, has been scrutinized in the literature recently. Specifically, it has been argued from a variety of different angles that there is a need for higher-order networks, which go beyond the paradigm of modeling pairwise relationships, as encapsulated by graphs. In this survey article we take stock of these recent developments. Our goals are to clarify (i) what higher-order networks are, (ii) why these are interesting objects of study, and (iii) how they can be used in applications.
- Published
- 2021
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32. Topological data analysis distinguishes parameter regimes in the Anderson-Chaplain model of angiogenesis
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Nardini, John T., Stolz, Bernadette J., Flores, Kevin B., Harrington, Heather A., and Byrne, Helen M.
- Subjects
Quantitative Biology - Quantitative Methods - Abstract
Angiogenesis is the process by which blood vessels form from pre-existing vessels. It plays a key role in many biological processes, including embryonic development and wound healing, and contributes to many diseases including cancer and rheumatoid arthritis. The structure of the resulting vessel networks determines their ability to deliver nutrients and remove waste products from biological tissues. Here we simulate the Anderson-Chaplain model of angiogenesis at different parameter values and quantify the vessel architectures of the resulting synthetic data. Specifically, we propose a topological data analysis (TDA) pipeline for systematic analysis of the model. TDA is a vibrant and relatively new field of computational mathematics for studying the shape of data. We compute topological and standard descriptors of model simulations generated by different parameter values. We show that TDA of model simulation data stratifies parameter space into regions with similar vessel morphology. The methodologies proposed here are widely applicable to other synthetic and experimental data including wound healing, development, and plant biology.
- Published
- 2021
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33. Continuous Indexing of Fibrosis (CIF): improving the assessment and classification of MPN patients
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Ryou, Hosuk, Sirinukunwattana, Korsuk, Aberdeen, Alan, Grindstaff, Gillian, Stolz, Bernadette J., Byrne, Helen, Harrington, Heather A., Sousos, Nikolaos, Godfrey, Anna L., Harrison, Claire N., Psaila, Bethan, Mead, Adam J., Rees, Gabrielle, Turner, Gareth D. H., Rittscher, Jens, and Royston, Daniel
- Published
- 2023
- Full Text
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34. Multiscale Topology Characterises Dynamic Tumour Vascular Networks
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Stolz, Bernadette J., Kaeppler, Jakob, Markelc, Bostjan, Mech, Franziska, Lipsmeier, Florian, Muschel, Ruth J., Byrne, Helen M., and Harrington, Heather A.
- Subjects
Quantitative Biology - Quantitative Methods ,Mathematics - Algebraic Topology ,Quantitative Biology - Tissues and Organs - Abstract
Advances in imaging techniques enable high resolution 3D visualisation of vascular networks over time and reveal abnormal structural features such as twists and loops, and their quantification is an active area of research. Here we showcase how topological data analysis (TDA), the mathematical field that studies `shape' of data, can characterise the geometric, spatial and temporal organisation of vascular networks. We propose two topological lenses to study vasculature, which capture inherent multi-scale features and vessel connectivity, and surpass the single scale analysis of existing methods. We analyse images collected using intravital and ultramicroscopy modalities and quantify spatio-temporal variation of twists, loops, and avascular regions (voids) in 3D vascular networks. This topological approach validates and quantifies known qualitative trends such as dynamic changes in tortuosity and loops in response to antibodies that modulate vessel sprouting; furthermore, it quantifies the effect of radiotherapy on vessel architecture.
- Published
- 2020
35. Grid diagrams as tools to investigate knot spaces and topoisomerase-mediated simplification of DNA topology
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Barbensi, Agnese, Celoria, Daniele, Harrington, Heather A., Stasiak, Andrzej, and Buck, Dorothy
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Quantitative Biology - Biomolecules - Abstract
Grid diagrams with their relatively simple mathematical formalism provide a convenient way to generate and model projections of various knots. It has been an open question whether these 2D diagrams can be used to model a complex 3D process such as the topoisomerase-mediated preferential unknotting of DNA molecules. We model here topoisomerase-mediated passages of double-stranded DNA segments through each other using the formalism of grid diagrams. We show that this grid diagram-based modelling approach captures the essence of the preferential unknotting mechanism, based on topoisomerase selectivity of hooked DNA juxtapositions as the sites of intersegmental passages. We show that grid diagram-based approach provide an important, new and computationally convenient framework for investigating entanglement in biopolymers., Comment: 22 pages, 9 figures, Supplementary Information at the end
- Published
- 2019
36. Geometric anomaly detection in data
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Stolz, Bernadette J, Tanner, Jared, Harrington, Heather A, and Nanda, Vidit
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Mathematics - Algebraic Topology ,Mathematics - Algebraic Geometry ,57N80 - Abstract
This paper describes the systematic application of local topological methods for detecting interfaces and related anomalies in complicated high-dimensional data. By examining the topology of small regions around each point, one can optimally stratify a given dataset into clusters, each of which is in turn well-approximable by a suitable submanifold of the ambient space. Since these approximating submanifolds might have different dimensions, we are able to detect non-manifold like singular regions in data even when none of the data points have been sampled from those singularities. We showcase this method by identifying the intersection of two surfaces in the 24-dimensional space of cyclo-octane conformations, and by locating all the self-intersections of a Henneberg minimal surface immersed in 3-dimensional space. Due to the local nature of the required topological computations, the algorithmic burden of performing such data stratification is readily distributable across several processors.
- Published
- 2019
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37. Topological Methods for Characterising Spatial Networks: A Case Study in Tumour Vasculature
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Byrne, Helen M, Harrington, Heather A, Muschel, Ruth, Reinert, Gesine, Stolz, Bernadette J, and Tillmann, Ulrike
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Quantitative Biology - Quantitative Methods ,Mathematics - Algebraic Topology ,Mathematics - Dynamical Systems ,Quantitative Biology - Tissues and Organs - Abstract
Understanding how the spatial structure of blood vessel networks relates to their function in healthy and abnormal biological tissues could improve diagnosis and treatment for diseases such as cancer. New imaging techniques can generate multiple, high-resolution images of the same tissue region, and show how vessel networks evolve during disease onset and treatment. Such experimental advances have created an exciting opportunity for discovering new links between vessel structure and disease through the development of mathematical tools that can analyse these rich datasets. Here we explain how topological data analysis (TDA) can be used to study vessel network structures. TDA is a growing field in the mathematical and computational sciences, that consists of algorithmic methods for identifying global and multi-scale structures in high-dimensional data sets that may be noisy and incomplete. TDA has identified the effect of ageing on vessel networks in the brain and more recently proposed to study blood flow and stenosis. Here we present preliminary work which shows how TDA of spatial network structure can be used to characterise tumour vasculature.
- Published
- 2019
38. Double branched covers of knotoids
- Author
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Barbensi, Agnese, Buck, Dorothy, Harrington, Heather A., and Lackenby, Marc
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Mathematics - Geometric Topology ,57M25, 57M27 - Abstract
By using double branched covers, we prove that there is a 1-1 correspondence between the set of knotoids in the 2-sphere, up to orientation reversion and rotation, and knots with a strong inversion, up to conjugacy. This correspondence allows us to study knotoids through tools and invariants coming from knot theory. In particular, concepts from geometrisation generalise to knotoids, allowing us to characterise invertibility and other properties in the hyperbolic case. Moreover, with our construction we are able to detect both the trivial knotoid in the 2-sphere and the trivial planar knotoid., Comment: 39 pages, 40 figures. It incorporates the referees' comments and a small change in Theorem 1.3
- Published
- 2018
39. Coloured Noise from Stochastic Inflows in Reaction-Diffusion Systems
- Author
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Adamer, Michael F, Harrington, Heather A, Gaffney, Eamonn A, and Woolley, Thomas E
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Quantitative Biology - Quantitative Methods ,Physics - Biological Physics ,Quantitative Biology - Molecular Networks ,92C42, 92C15, 60H30, 34C08 - Abstract
In this paper we present a framework for investigating coloured noise in reaction-diffusion systems. We start by considering a deterministic reaction-diffusion equation and show how external forcing can cause temporally correlated or coloured noise. Here, the main source of external noise is considered to be fluctuations in the parameter values representing the inflow of particles to the system. First, we determine which reaction systems, driven by extrinsic noise, can admit only one steady state, so that effects, such as stochastic switching, are precluded from our analysis. To analyse the steady state behaviour of reaction systems, even if the parameter values are changing, necessitates a parameter-free approach, which has been central to algebraic analysis in chemical reaction network theory. To identify suitable models we use tools from real algebraic geometry that link the network structure to its dynamical properties. We then make a connection to internal noise models and show how power spectral methods can be used to predict stochastically driven patterns in systems with coloured noise. In simple cases we show that the power spectrum of the coloured noise process and the power spectrum of the reaction-diffusion system modelled with white noise multiply to give the power spectrum of the coloured noise reaction-diffusion system., Comment: 31 pages, 8 figures
- Published
- 2018
40. On Some Configurations of Oppositely Charged Trapped Vortices in the Plane
- Author
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Dufresne, Emilie, Harrington, Heather A, Kevrekidis, Panayotis G, and Tripoli, Paolo
- Subjects
Nonlinear Sciences - Pattern Formation and Solitons ,Mathematics - Commutative Algebra - Abstract
Our aim in the present work is to identify all the possible standing wave configurations involving few vortices of different charges in an atomic Bose-Einstein condensate (BEC). In this effort, we deploy the use of a computational algebra approach in order to identify stationary multi-vortex states with up to 6 vortices. The use of invariants and symmetries enables deducing a set of equations in elementary symmetric polynomials, which can then be fully solved via computational algebra packages within Maple. We retrieve a number of previously identified configurations, including collinear ones and polygonal (e.g. quadrupolar and hexagonal) ones. However, importantly, we also retrieve a configuration with 4 positive charges and 2 negative ones which is unprecedented, to the best of our knowledge, in BEC studies. We corroborate these predictions via numerical computations in the fully two-dimensional PDE system of the Gross-Pitaevskii type which characterizes the BEC at the mean-field level., Comment: 26 pages, 3 figures
- Published
- 2018
41. Joining and decomposing reaction networks
- Author
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Gross, Elizabeth, Harrington, Heather A, Meshkat, Nicolette, and Shiu, Anne
- Subjects
Mathematics - Dynamical Systems ,Mathematics - Algebraic Geometry ,Quantitative Biology - Molecular Networks - Abstract
In systems and synthetic biology, much research has focused on the behavior and design of single pathways, while, more recently, experimental efforts have focused on how cross-talk (coupling two or more pathways) or inhibiting molecular function (isolating one part of the pathway) affects systems-level behavior. However, the theory for tackling these larger systems in general has lagged behind. Here, we analyze how joining networks (e.g., cross-talk) or decomposing networks (e.g., inhibition or knock-outs) affects three properties that reaction networks may possess---identifiability (recoverability of parameter values from data), steady-state invariants (relationships among species concentrations at steady state, used in model selection), and multistationarity (capacity for multiple steady states, which correspond to multiple cell decisions). Specifically, we prove results that clarify, for a network obtained by joining two smaller networks, how properties of the smaller networks can be inferred from or can imply similar properties of the original network. Our proofs use techniques from computational algebraic geometry, including elimination theory and differential algebra., Comment: 44 pages; extensive revision in response to referee comments
- Published
- 2018
42. Multiparameter persistent homology landscapes identify immune cell spatial patterns in tumors
- Author
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Vipond, Oliver, Bull, Joshua A., Macklin, Philip S., Tillmann, Ulrike, Pugh, Christopher W., Byrnea, Helen M., and Harrington, Heather A.
- Published
- 2021
43. Topological Data Analysis of Task-Based fMRI Data from Experiments on Schizophrenia
- Author
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Stolz, Bernadette J., Emerson, Tegan, Nahkuri, Satu, Porter, Mason A., and Harrington, Heather A.
- Subjects
Quantitative Biology - Quantitative Methods ,Condensed Matter - Disordered Systems and Neural Networks ,Mathematics - Algebraic Topology ,Nonlinear Sciences - Adaptation and Self-Organizing Systems ,Quantitative Biology - Neurons and Cognition - Abstract
We use methods from computational algebraic topology to study functional brain networks, in which nodes represent brain regions and weighted edges encode the similarity of fMRI time series from each region. With these tools, which allow one to characterize topological invariants such as loops in high-dimensional data, we are able to gain understanding into low-dimensional structures in networks in a way that complements traditional approaches that are based on pairwise interactions. In the present paper, we use persistent homology to analyze networks that we construct from task-based fMRI data from schizophrenia patients, healthy controls, and healthy siblings of schizophrenia patients. We thereby explore the persistence of topological structures such as loops at different scales in these networks. We use persistence landscapes and persistence images to create output summaries from our persistent-homology calculations, and we study the persistence landscapes and images using $k$-means clustering and community detection. Based on our analysis of persistence landscapes, we find that the members of the sibling cohort have topological features (specifically, their 1-dimensional loops) that are distinct from the other two cohorts. From the persistence images, we are able to distinguish all three subject groups and to determine the brain regions in the loops (with four or more edges) that allow us to make these distinctions.
- Published
- 2018
44. Linear compartmental models: input-output equations and operations that preserve identifiability
- Author
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Gross, Elizabeth, Harrington, Heather A., Meshkat, Nicolette, and Shiu, Anne
- Subjects
Mathematics - Algebraic Geometry ,Mathematics - Dynamical Systems ,Quantitative Biology - Molecular Networks - Abstract
This work focuses on the question of how identifiability of a mathematical model, that is, whether parameters can be recovered from data, is related to identifiability of its submodels. We look specifically at linear compartmental models and investigate when identifiability is preserved after adding or removing model components. In particular, we examine whether identifiability is preserved when an input, output, edge, or leak is added or deleted. Our approach, via differential algebra, is to analyze specific input-output equations of a model and the Jacobian of the associated coefficient map. We clarify a prior determinantal formula for these equations, and then use it to prove that, under some hypotheses, a model's input-output equations can be understood in terms of certain submodels we call "output-reachable". Our proofs use algebraic and combinatorial techniques., Comment: v2: 26 pages, 3 figures; expanded introduction to identifiability, including updating definitions; deleted two remarks from v1; improved exposition throughout
- Published
- 2018
45. Topological data analysis of continuum percolation with disks
- Author
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Speidel, Leo, Harrington, Heather A., Chapman, S. Jonathan, and Porter, Mason A.
- Subjects
Condensed Matter - Statistical Mechanics ,Condensed Matter - Disordered Systems and Neural Networks ,Mathematics - Algebraic Topology ,Nonlinear Sciences - Adaptation and Self-Organizing Systems ,Physics - Data Analysis, Statistics and Probability - Abstract
We study continuum percolation with disks, a variant of continuum percolation in two-dimensional Euclidean space, by applying tools from topological data analysis. We interpret each realization of continuum percolation with disks as a topological subspace of $[0,1]^2$ and investigate its topological features across many realizations. We apply persistent homology to investigate topological changes as we vary the number and radius of disks. We observe evidence that the longest persisting invariant is born at or near the percolation transition.
- Published
- 2018
- Full Text
- View/download PDF
46. Sampling real algebraic varieties for topological data analysis
- Author
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Dufresne, Emilie, Edwards, Parker B., Harrington, Heather A., and Hauenstein, Jonathan D.
- Subjects
Mathematics - Algebraic Topology ,Mathematics - Algebraic Geometry ,Mathematics - Numerical Analysis - Abstract
Topological data analysis (TDA) provides a growing body of tools for computing geometric and topological information about spaces from a finite sample of points. We present a new adaptive algorithm for finding provably dense samples of points on real algebraic varieties given a set of defining polynomials. The algorithm utilizes methods from numerical algebraic geometry to give formal guarantees about the density of the sampling and it also employs geometric heuristics to reduce the size of the sample. As TDA methods consume significant computational resources that scale poorly in the number of sample points, our sampling minimization makes applying TDA methods more feasible. We provide a software package that implements the algorithm and also demonstrate the implementation with several examples., Comment: Updated section 5, added links to implementation software package, improved figure resolution in section 5
- Published
- 2018
47. Constructing custom-made radiotranscriptomic signatures of vascular inflammation from routine CT angiograms: a prospective outcomes validation study in COVID-19
- Author
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Ahern, David J, Ai, Zhichao, Ainsworth, Mark, Allan, Chris, Allcock, Alice, Angus, Brian, Ansari, M Azim, Arancibia-Cárcamo, Carolina, Aschenbrenner, Dominik, Attar, Moustafa, Baillie, J Kenneth, Barnes, Eleanor, Bashford-Rogers, Rachael, Bashyal, Archana, Beer, Sally, Berridge, Georgina, Beveridge, Amy, Bibi, Sagida, Bicanic, Tihana, Blackwell, Luke, Bowness, Paul, Brent, Andrew, Brown, Andrew, Broxholme, John, Buck, David, Burnham, Katie, Byrne, Helen, Camara, Susana, Ferreira, Ivan Candido, Charles, Philip, Chen, Wentao, Chen, Yi-Ling, Chong, Amanda, Clutterbuck, Elizabeth, Coles, Mark, Conlon, Christopher, Cornall, Richard, Cribbs, Adam, Curion, Fabiola, Davenport, Emma, Davidson, Neil, Davis, Simon, Dendrou, Calliope, Dequaire, Julie, Dib, Lea, Docker, James, Dold, Christina, Dong, Tao, Downes, Damien, Drakesmith, Hal, Dunachie, Susanna, Duncan, David, Eijsbouts, Chris, Esnouf, Robert, Espinosa, Alexis, Etherington, Rachel, Fairfax, Benjamin, Fairhead, Rory, Fang, Hai, Fassih, Shayan, Felle, Sally, Fernandez Mendoza, Maria, Ferreira, Ricardo, Fischer, Roman, Foord, Thomas, Forrow, Aden, Frater, John, Fries, Anastasia, Gallardo Sanchez, Veronica, Garner, Lucy, Geeves, Clementine, Georgiou, Dominique, Godfrey, Leila, Golubchik, Tanya, Gomez Vazquez, Maria, Green, Angie, Harper, Hong, Harrington, Heather, Heilig, Raphael, Hester, Svenja, Hill, Jennifer, Hinds, Charles, Hird, Clare, Ho, Ling-Pei, Hoekzema, Renee, Hollis, Benjamin, Hughes, Jim, Hutton, Paula, Jackson-Wood, Matthew, Jainarayanan, Ashwin, James-Bott, Anna, Jansen, Kathrin, Jeffery, Katie, Jones, Elizabeth, Jostins, Luke, Kerr, Georgina, Kim, David, Klenerman, Paul, Knight, Julian, Kumar, Vinod, Kumar Sharma, Piyush, Kurupati, Prathiba, Kwok, Andrew, Lee, Angela, Linder, Aline, Lockett, Teresa, Lonie, Lorne, Lopopolo, Maria, Lukoseviciute, Martyna, Luo, Jian, Marinou, Spyridoula, Marsden, Brian, Martinez, Jose, Matthews, Philippa, Mazurczyk, Michalina, McGowan, Simon, McKechnie, Stuart, Mead, Adam, Mentzer, Alexander, Mi, Yuxin, Monaco, Claudia, Montadon, Ruddy, Napolitani, Giorgio, Nassiri, Isar, Novak, Alex, O'Brien, Darragh, O'Connor, Daniel, O'Donnell, Denise, Ogg, Graham, Overend, Lauren, Park, Inhye, Pavord, Ian, Peng, Yanchun, Penkava, Frank, Pereira Pinho, Mariana, Perez, Elena, Pollard, Andrew, Powrie, Fiona, Psaila, Bethan, Quan, T Phuong, Repapi, Emmanouela, Revale, Santiago, Silva-Reyes, Laura, Richard, Jean-Baptiste, Rich-Griffin, Charlotte, Ritter, Thomas, Rollier, Christine, Rowland, Matthew, Ruehle, Fabian, Salio, Mariolina, Sansom, Stephen Nicholas, Sanches Peres, Raphael, Santos Delgado, Alberto, Sauka-Spengler, Tatjana, Schwessinger, Ron, Scozzafava, Giuseppe, Screaton, Gavin, Seigal, Anna, Semple, Malcolm, Sergeant, Martin, Simoglou Karali, Christina, Sims, David, Skelly, Donal, Slawinski, Hubert, Sobrinodiaz, Alberto, Sousos, Nikolaos, Stafford, Lizzie, Stockdale, Lisa, Strickland, Marie, Sumray, Otto, Sun, Bo, Taylor, Chelsea, Taylor, Stephen, Taylor, Adan, Thongjuea, Supat, Thraves, Hannah, Todd, John, Tomic, Adriana, Tong, Orion, Trebes, Amy, Trzupek, Dominik, Tucci, Felicia Anna, Turtle, Lance, Udalova, Irina, Uhlig, Holm, van Grinsven, Erinke, Vendrell, Iolanda, Verheul, Marije, Voda, Alexandru, Wang, Guanlin, Wang, Lihui, Wang, Dapeng, Watkinson, Peter, Watson, Robert, Weinberger, Michael, Whalley, Justin, Witty, Lorna, Wray, Katherine, Xue, Luzheng, Yuen Yeung, Hing, Yin, Zixi, Young, Rebecca, Youngs, Jonathan, Zhang, Ping, Zurke, Yasemin-Xiomara, Banning, Adrian, Antonopoulos, Alexios, Bajaj, Amrita, Kelion, Andrew, Deshpande, Aparna, Kardos, Attila, Hudson, Benjamin, Koo, Bon-Kwon, Shirodaria, Cheerag, Xie, Cheng, Kotanidis, Christos, Mahon, Ciara, Berry, Colin, Adlam, David, Newby, David, Connolly, Derek, Scaletta, Diane, Alexander, Donna, Nicol, Ed, McAlindon, Elisa, Oikonomou, Evangelos, Pugliese, Francesca, Pontone, Gianluca, Benedetti, Giulia, He, Guo-Wei, West, Henry, Kondo, Hidekazu, Benedek, Imre, Das, Intrajeet, Deanfield, John, Graby, John, Greenwood, John, Rodrigues, Jonathan, Ge, Junbo, Channon, Keith, Fabritz, Larissa, Fan, Li-Juan, Kingham, Lucy, Guglielmo, Marco, Lyasheva, Maria, Schmitt, Matthias, Beer, Meinrad, Anderson, Michelle, Desai, Milind, Marwan, Mohamed, Takahashi, Naohiko, Mehta, Nehal, Dai, Neng, Screaton, Nicholas, Sabharwal, Nikant, Maurovich-Horvat, Pál, Rao, Praveen, Kotronias, Rafail, Kharbanda, Rajesh, Preston, Rebecca, Wood, Richard, Blankstein, Ron, Rajani, Ronak, Mirsadraee, Saeed, Munir, Shahzad, Thomas, Sheena, Neubauer, Stefan, Klömpken, Steffen, Petersen, Steffen, Achenbach, Stephan, Anthony, Susan, Mak, Sze, Mittal, Tarun, Benedek, Theodora, Sharma, Vinoda, Lin, Wen-Hua, Kotanidis, Christos P, Rodrigues, Jonathan C L, O’Connor, Daniel, Siddique, Muhammad, Lockstone, Helen, Oikonomou, Evangelos K, Badi, Ileana, Lumley, Sheila F, Constantinides, Bede, Sanderson, Nicholas, Rodger, Gillian, Chau, Kevin K, Lodge, Archie, Tsakok, Maria, Gleeson, Fergus, Indrajeet, Das, Hudson, Benjamin J, Srivastava, Vivek, Farid, Shakil, Krasopoulos, George, Sayeed, Rana, Newby, David E, Channon, Keith M, and Antoniades, Charalambos
- Published
- 2022
- Full Text
- View/download PDF
48. Quantification of vascular networks in photoacoustic mesoscopy
- Author
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Brown, Emma L., Lefebvre, Thierry L., Sweeney, Paul W., Stolz, Bernadette J., Gröhl, Janek, Hacker, Lina, Huang, Ziqiang, Couturier, Dominique-Laurent, Harrington, Heather A., Byrne, Helen M., and Bohndiek, Sarah E.
- Published
- 2022
- Full Text
- View/download PDF
49. Algebra, Geometry and Topology of ERK Kinetics
- Author
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Marsh, Lewis, Dufresne, Emilie, Byrne, Helen M., and Harrington, Heather A.
- Published
- 2022
- Full Text
- View/download PDF
50. Stratifying multiparameter persistent homology
- Author
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Harrington, Heather A., Otter, Nina, Schenck, Hal, and Tillmann, Ulrike
- Subjects
Mathematics - Algebraic Topology ,Mathematics - Commutative Algebra ,55B55, 68U05, 68Q17, 13P25 (primary) - Abstract
A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter family of spaces obtained from the data. In applications data often depend on several parameters, and in this case one is interested in studying the persistent homology of a multiparameter family of spaces associated to the data. While the theory of persistent homology for one-parameter families is well-understood, the situation for multiparameter families is more delicate. Following Carlsson and Zomorodian we recast the problem in the setting of multigraded algebra, and we propose multigraded Hilbert series, multigraded associated primes and local cohomology as invariants for studying multiparameter persistent homology. Multigraded associated primes provide a stratification of the region where a multigraded module does not vanish, while multigraded Hilbert series and local cohomology give a measure of the size of components of the module supported on different strata. These invariants generalize in a suitable sense the invariant for the one-parameter case., Comment: Minor improvements throughout. In particular: we extended the introduction, added Table 1, which gives a dictionary between terms used in PH and commutative algebra; we streamlined Section 3; we added Proposition 4.49 about the information captured by the cp-rank; we moved the code from the appendix to github. Final version, to appear in SIAGA
- Published
- 2017
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