Applied Topology in Neuroscience - Spring 2018

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​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
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​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​​​