# Computational Topology in

@inproceedings{Stolz2014ComputationalTI, title={Computational Topology in}, author={Bernadette J. Stolz}, year={2014} }

Computational topology is a set of algorithmic methods developed to understand topological invariants such as loops and holes in high-dimensional data sets. In particular, a method know as persistent homology has been used to understand such shapes and their persistence in point clouds and networks. It has only been applied to neuronal networks in recent years. While most tools from network science focus solely on local properties based on pairwise connections, the topological tools reveal more… Expand

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#### 3 Citations

The importance of the whole: Topological data analysis for the network neuroscientist

- Computer Science, Biology
- Network Neuroscience
- 2019

An introduction to persistent homology, a fundamental method from applied topology that builds a global descriptor of system structure by chronicling the evolution of cavities as the authors move through a combinatorial object such as a weighted network. Expand

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

- Computer Science, Medicine
- Journal of Computational Neuroscience
- 2016

It is proposed that 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, has the potential to eclipse graph theory in unraveling the fundamental mysteries of cognition. Expand

Topological Data Analysis for Systems of Coupled Oscillators

- Physics
- 2016

Coupled oscillators, such as groups of fireflies or clusters of neurons, are found throughout nature and are frequently modeled in the applied mathematics literature. Earlier work by Kuramoto,… Expand

#### References

SHOWING 1-10 OF 29 REFERENCES

jHoles: A Tool for Understanding Biological Complex Networks via Clique Weight Rank Persistent Homology

- Computer Science, Mathematics
- Electron. Notes Theor. Comput. Sci.
- 2014

Complex networks equipped with topological data analysis are one of the promising tools in the study of biological systems (e.g. evolution dynamics, brain correlation, breast cancer diagnosis,… Expand

Persistent Homology — a Survey

Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multi-scale organization we frequently observe in nature into a mathematical… Expand

Discriminative persistent homology of brain networks

- Computer Science
- 2011 IEEE International Symposium on Biomedical Imaging: From Nano to Macro
- 2011

This paper applies the Rips filtration to construct the FDG-PET based functional brain networks out of 24 attention deficit hyperactivity disorder (ADHD) children, 26 autism spectrum disorder (ASD) children and 11 pediatric control subjects and visually shows the topological evolution of the brain networks using the barcode and performs statistical inference on the group differences. Expand

Barcodes: The persistent topology of data

- Mathematics
- 2007

This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The… Expand

javaPlex: A Research Software Package for Persistent (Co)Homology

- Mathematics, Computer Science
- ICMS
- 2014

A new software package for topological computation, replacing previous jPlex and Plex, enables researchers to access state of the art algorithms for persistent homology, cohomology, hom complexes, filtered simplicial complex, filtered cell complexes, witness complex constructions, and many more essential components of computational topology. Expand

Networks: An Introduction

- Computer Science
- 2010

This book brings together for the first time the most important breakthroughs in each of these fields and presents them in a coherent fashion, highlighting the strong interconnections between work in different areas. Expand

Topology and data

- Computer Science
- 2009

This paper will discuss how geometry and topology can be applied to make useful contributions to the analysis of various kinds of data, particularly high throughput data from microarray or other sources. Expand

Weighted Functional Brain Network Modeling via Network Filtration

- Computer Science
- 2012

The proposed network filtration framework can discriminate the local and global differences of the brain networks of 24 attention deficit hyperactivity disorder (ADHD), 26 autism spectrum disorder (ASD) and 11 pediatric control (PedCon) children obtained through the FDG-PET data. Expand

A Topological Paradigm for Hippocampal Spatial Map Formation Using Persistent Homology

- Computer Science, Medicine
- PLoS Comput. Biol.
- 2012

Using a computational algorithm based on recently developed tools from Persistent Homology theory in the field of algebraic topology, it is found that the patterns of neuronal co-firing can, in fact, convey topological information about the environment in a biologically realistic length of time. Expand

Topological persistence and simplification

- Mathematics, Computer Science
- Proceedings 41st Annual Symposium on Foundations of Computer Science
- 2000

A notion of topological simplification is formalized within the framework of a filtration, which is the history of a growing complex, and a topological change that happens during growth is classified as either a feature or noise, depending on its life-time or persistence within the filTration. Expand