## Topological analysis of fNIRS neuroimages

Felipe Orihuela-Espina, INAOE, México

Duration: 180 min

Capacity: 15

Level: Advanced, Coding Experience

Analysis Method

Requirements: Laptop; Matlab and Pyton pre-installed

**Synopsis:** Topology is the branch of mathematics that study deformations of spaces. It underpins many of the operations that we commonly carry out during reconstruction, processing and analysis of fNIRS images regardless of the high-level approach that we may favour. For instance, many transformations of our signals, establishing signals similarity by some metric, comparing statistical distributions for hypothesis testing, or partitioning spaces for modern machine learning solutions are just corollary manifestations of the nearness of elements in geometrical sets. Closer expressions also include dimensionality reduction approaches or changes in coordinate basis for data visualization and exploration or graph theory in connectivity analysis. This course aims to provide an overview of how topology can be used for analysis and understanding of fNIRS neuroimages in research. We shall review the basics of topology and escalate these concepts to answer specific functional neuroimaging research questions, both segregational and integrational. We shall not assume prior knowledge on topology, but the course is not intended to be an introductory course on theoretical topology. The course includes both theory and practical aspects. For theory, we shall aim to strike a compromise between mathematical formal aspects and conceptual intuition. For practice, we shall do exercises on toy datasets including real fNIRS neuroimages.

**Rationale:** As the fNIRS community has matured, our datasets are larger and our analysis approaches have also become more sophisticated. Interpreting and explaining these datasets is becoming increasingly demanding. Traditionally hailed for its virtues in dimensionality reduction, topology can do much more than unfolding local features for data exploration or filtering. It can mimic many existing analysis (or more precisely, it often underpins other forms of analysis), and thus it can address many research questions *within a single mathematical framework*, but further, it can potentially answer questions that would otherwise be difficult with traditional higher level methods. For instance, manifolds can encode any experimental design matrix as vector fields, or infinite many graphs at once which can be useful to separate different coactive circuits, allow for precise comparison of distributions or signal similarity, which in practice is equivalent to solving regression models, or provide exceptional clues of important patterns in the form of topological invariants, etc.Because of its proximity to the foundations of mathematics, topology uses highly abstract mathematical objects, and consequently is capable of high expressivity, translating into a *versatile processing and analysis tool*. The price to pay, which perhaps has withheld widespread acceptance, is that solutions are encoded in unfamiliar mathematical objects, so manipulating and interpreting them is often more demanding than with more ad-hoc solutions. This mini-course intends to instruct the attendants with both, a hint of the capacities of this branch of mathematics when used for fNIRS analysis, and a variety of tricks to encode and interpret the solutions.

**Learning objectives: **

- Getting acquainted with some mathematical objects such as sets, relations, orders, spaces, structures, algebras, charts and atlases, manifolds, geometries and distance functions among others.
- Basic manipulation of the space geometry to support encoding of some information of interest.
- A common flow for manifold-based analysis and how to interpret outcomes.

**Requirements:** You are expected to bring a laptop with the required software installed and have some rudimentary knowledge of coding. Required software is to be announced but may include Matlab (min. R2017a) and/or Python.