Applied Topology in Neuroscience - Spring 2018
Week 1: Introduction to applied topology and simplicial complexes.
Notes
Homework
Week 2: Chain Groups (and recap of algebraic structures)
Notes
Homework
Week 3: Boundary operator
Notes
Homework
Week 4: Quotients
Notes
Homework
Week 5: Homology
Notes
Homework
Week 6: Filtrations
Notes
Homework
Week 7: Persistent Homology
Notes
Homework
Week 8: Running Eirene
Notes
Code
Week 9: Extracting Generators
Notes
Week 10: Comparing Persistence Diagrams
Notes
Week 11: Frontiers and Extensions
Notes
References:
Arai et al., The Effects of Theta Precession on Spatial Learning and Simplicial Complex Dynamics in a Topological Model of the Hippocampal Spatial Map
Busaryev et al., Tracking a generator by persistence
Carlsson, Topology and Data
Chung, Bubenik, and Kim, Persistence diagrams of cortical surface data
Cohen-Steiner, Edelsbrunner, and Harer, Stability of persistence diagrams
Curto, What can topology tell us about the neural code?
Dey et al., Optimal Homologous Cycles, Total Unimodularity, and Linear Programming
Estrada and Ross, Centralities in Simplicial Complexes
Gameiro et al., A Topological Measurement of Protein Compressibility
Ghrist, Homological Algebra and Data
Ginestet et al., Brain Network Analysis: Separating Cost from Topology Using Cost-Integration
Giusti et al., Two’s company, three (or more) is a simplex: Algebraic-topological tools for understanding higher-order structure in neural data
Giusti et al., Clique topology reveals intrinsic geometric structure in neural correlations
Horak et al., Persistent Homology of Complex Networks
Madan et al., A novel approach to identifying a neuroimaging biomarker for patients with serious mental illness
Munch, Applications of persistent homology to time varying systems
Petri et al. Homological scaffolds of structural brain networks
Petri et al., Topological Strata of Weighted Complex Networks
Palla et al., Uncovering the overlapping community structure of complex networks in nature and society
Reimann et al., Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function
Rybakken et al, Decoding of neural data using cohomological learning
Sizemore et al., Cliques and Cavities in the Human Connectome
Stolz et al., Persistent homology of time-dependent functional networks constructed from coupled time series
Yoo et al., Topological persistence vineyard for dynamic functional brain connectivity during resting and gaming stages