Our research
The applied topology group at Oregon State University employs topology to investigate a variety of complex systems. Primarily, we focus on the following areas of research:
Methods in applied topologyWe develop new mathematical frameworks and computational tools for the study of complex systems through the modern language of shape. We are particularly interested in methods that will allow us to address fundamental scientific questions, and in understanding how and why topological structure arises so often in the natural world.
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Topological neuroscienceIn collaboration with experimental neuroscientists, we study the structure and function of *in vivo* and simulated neural systems with the goal of developing fundamental theory about the encoding of information and performance of computations in the brain.
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The Applied Topology Group is grateful to the National Science Foundation and Air Force Office of Scientific Research for their support of our work.
Methods in applied topology
Modern techniques in applied topology are evolving rapidly, and there are many open questions. We are interested in how measures of topology reflect the organization of systems, and why such structures arise in nature. To study these questions, we develop tools for interpreting topological measures *in situ*.
Our work broadly fits roughly into two categories:
Our work broadly fits roughly into two categories:
1. Interpreting Topological Measures:
Due to computational limitations and underdeveloped mathematics, early applications of persistent homology have largely focused on statistics of persistence diagrams. Interpretation in terms of the original systems from which they were computed is usually *ad hoc* and prone to subtle errors. In our view, these difficulties have inhibited wide-spread application of topological methods in the sciences. One of our primary aims is thus developing rigorous, formal workflows that connect quantitative topological measures and the structure of the system they represent. Our recent work includes the Open Applied Topology software package, which provides full access to the algebraic structures underlying persistence; a user's guide for optimizing persistent cycle representatives; and, the method of analogous cycles, which matches persistent homology classes across systems using observed cross-dissimilarities.
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2. Iterated Integrals As Features For Function Spaces
Persistent homology characterizes structure in filtered spaces. However, some data are better described in terms of functions into a state space. Function spaces are a topic of classical study in algebraic topology, and Chen's iterated integral framework for cochains on mapping spaces, recently recast as the Path Signature in the study of SDEs and ML, provides a well-founded and robust framework for studying such data. We have applied this framework to the study of time-varying persistence diagrams; and, to describe the algebraic properties of the signature transform for higher-dimensional domains.
A topological approach to mapping space signatures
Chad Giusti, Darrick Lee, Vidit Nanda, Harald Oberhauser
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Chad Giusti, Darrick Lee, Vidit Nanda, Harald Oberhauser
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Interpreting Signatures in terms of Time Series Analysis https://link-springer-com.oregonstate.idm.oclc.org/chapter/10.1007/978-3-030-43408-3_9
Topological neuroscience
Topology is the mathematics of "local-to-global" inference. That is, topological tools are designed to sew together local observations to describe nonlinear global phenomena. In this light, it is perhaps less surprising that they are uniquely well-suited for describing systems like the brain, in which the simple computations performed by individual neurons are synthesized into our rich internal representations of the state of the world, and their dynamics give rise to planing and execution of intricate behaviors to interact with it. The introduction of modern topological methods to neuroscience is quite recent, and there are more questions than answers. For a brief overview of some of the big ideas we recommend the following surveys, some of which are written for non-mathematicians.
- - Two's Company, https://link-springer-com.oregonstate.idm.oclc.org/article/10.1007/s10827-016-0608-6
- - What can topology tell us about the neural code? https://www.ams.org/journals/bull/2017-54-01/S0273-0979-2016-01554-0/S0273-0979-2016-01554-0.pdf
- - Importance of the whole https://direct.mit.edu/netn/article/3/3/656/2178/The-importance-of-the-whole-Topological-data
- - Topological Adventures in Neuroscience https://link-springer-com.oregonstate.idm.oclc.org/chapter/10.1007/978-3-030-43408-3_11
- - A hands-on tutorial on network and topological neuroscience https://link-springer-com.oregonstate.idm.oclc.org/article/10.1007/s00429-021-02435-0
- Our recent work has been focused on implications for neural computation of the classical nerve theorem of algebraic topology. In context, it effectively tells us that we can use observations of neural activity to reconstruct the topology (and some geometry) of the information encoded by a neural population. This purely mathematical fact is the reason for the appearance of so-called *neural manifolds*, low-dimensional subspaces of the space of population neuron activation profiles, along which activity is constrained. In his postdoctoral work (https://www.pnas.org/doi/pdf/10.1073/pnas.1506407112), the PI demonstrated that topological measurements of neural population activity alone were sufficient to detect or falsify the presence of a neural manifold, without reference to external stimulus or behavior. In the interim, we have been interested in understanding how these objects arise as a result of neural activity, and developing a mathematical framework for understanding the resulting computational model at the level of neural manifolds, rather than individual or collective neuronal activity. In recent work, we have constructed a method for comparing topological features across neural manifolds, or between neural manifolds and observed stimuli and behaviors; and described combinatorial foundations of convex coding, which underlies such receptive field models of neural activity.
- Recent Products:
- - Tracking Topological Structure Across Neural Manifolds https://www.pnas.org/doi/10.1073/pnas.2407997121 /
- https://github.com/irishryoon/analogous_neural
- - Convex Coding in Neural Systems https://link-springer-com.oregonstate.idm.oclc.org/content/pdf/10.1007/s00454-018-00050-1.pdf
