UROP Project
Integrate-and-Fire Sampling Meets Optimal Transport
Integration, Sampling, Optimal Transport, Signal Encoding and Reconstruction
Research Mentor: Rocio Diaz Martin,
Department, College, Affiliation: Department of Mathematics, Arts and Sciences
Contact Email: rdiazmartin@fsu.edu
Research Assistant Supervisor (if different from mentor):
Research Assistant Supervisor Email:
Faculty Collaborators:
Faculty Collaborators Email:
Department, College, Affiliation: Department of Mathematics, Arts and Sciences
Contact Email: rdiazmartin@fsu.edu
Research Assistant Supervisor (if different from mentor):
Research Assistant Supervisor Email:
Faculty Collaborators:
Faculty Collaborators Email:
Looking for Research Assistants: Yes
Number of Research Assistants: 1
Relevant Majors: mathematics, computer science, physics, engineering
Project Location: On FSU Main Campus
Research Assistant Transportation Required: Remote or In-person: Partially Remote
Approximate Weekly Hours: 5, Flexible schedule (Combination of business and outside of business. TBD between student and research mentor.)
Roundtable Times and Zoom Link:
Not participating in the roundtable
Number of Research Assistants: 1
Relevant Majors: mathematics, computer science, physics, engineering
Project Location: On FSU Main Campus
Research Assistant Transportation Required: Remote or In-person: Partially Remote
Approximate Weekly Hours: 5, Flexible schedule (Combination of business and outside of business. TBD between student and research mentor.)
Roundtable Times and Zoom Link:
Not participating in the roundtable
Project Description
This project studies the relationship between the classical integrate-and-fire sampling scheme and one-dimensional optimal transport.The integrate-and-fire sampling scheme is a method for recording a signal by tracking when its accumulated value reaches a fixed threshold. Instead of measuring the values of the signal at fixed times, this sampler starts a counter at zero and continuously integrates, or accumulates, the signal until the total mass reaches a prescribed threshold. It then records that time stamp as a spike and resets the counter to zero. This process is repeated to obtain subsequent time stamps.
At its core, this scheme is closely related to optimal transport, since both ideas involve moving from one representation of a signal to another in terms of accumulated mass. The integrate-and-fire sampler records the times at which equal amounts of the signal have been accumulated. In one dimension, optimal transport similarly describes how to move uniformly distributed mass into the shape of a given signal. Therefore, the firing times can be understood as samples of the map that transports uniform mass to the signal.
Although this connection is hinted in the literature, it has not been explicitly stated. The first goal of this project is to make this connection precise. We will then investigate and implement the reconstruction of signals from spike times by interpolating the transport map, and compare density-based error with Wasserstein reconstruction error, that is, the error measured using a distance from optimal transport. The goal is to determine whether optimal transport geometry provides a natural and stable framework for analyzing integrate-and-fire sampling.
Research Tasks: - Literature review.
- Developing and implementing algorithms through programming.
- Formulate and rigorously prove mathematically grounded results.
Skills that research assistant(s) may need: Foundation in Calculus (required) and linear algebra (recommended).
Basic programming skills, preferably in Python (recommended).
Some knowledge of probability theory is preferred (recommended).
Mentoring Philosophy
I consider students as junior colleagues, empowering them to grow as collaborators rather than passive learners. I begin by sharing the theoretical foundations (especially intuitive and proof-based thinking) while suggesting programming as a tool for exploration.As a young researcher, I recognize that mentoring presents a challenge for me, but it is one that excites me deeply. I view this process as a two-way learning experience: while students grow as mathematicians and programmers, I grow as a mentor and teacher. Coming from Argentina, I also bring a perspective shaped by my own educational journey, which helps me relate to students navigating diverse paths and backgrounds.
I set clear, mutual goals and maintain open, respectful communication. By inviting students to co-create the learning path, align expectations and build trust. My mentorship is inclusive: I’m attentive to different learning styles and backgrounds, ensuring all students feel valued and encouraged to share their ideas. Being approachable is essential, and I actively cultivate this quality in myself to create a welcoming and supportive environment for everybody.
I guide them to develop mathematical intuition by asking guiding questions and encouraging reflection, helping them formulate logical results and rigorous proofs in their own words. We periodically assess progress, celebrate small victories, and iterate our process to strengthen understanding and confidence.
Ultimately, I aspire for students to become confident, independent thinkers: capable programmers, insightful mathematicians, and critical collaborators who continue learning beyond our time together.
Additional Information
The goal of this project is to connect the nonlinear sampling scheme of integrate-and-fire with tools and concepts from Optimal Transport theory. Optimal Transport is a central area of my current research, and as a new Assistant Professor in the Department of Mathematics, I am excited to continue studying it, connect it with different areas of mathematics, and share my background with undergraduate students. We will begin by exploring both the fundamental theory and applications of Optimal Transport. For example, the Wasserstein metric, also known as the Earth Mover's Distance, is a key object in Optimal Transport and is widely used in Machine Learning, inspiring developments such as Wasserstein Generative Adversarial Networks, introduced by M. Arjovsky and collaborators in 2017. This theoretical study will also provide an opportunity to explore other important concepts in applied mathematics.We will then focus on making a clear connection with the integrate-and-fire technique. Classical integrate-and-fire sampling is a nonlinear sampling mechanism in which a nonnegative signal is integrated until the accumulated mass reaches a prescribed threshold. At that moment, a spike is recorded and the integrator is reset. We will review the literature on this procedure, including the recent work “Model Agnostic Signal Encoding by Leaky Integrate-and-Fire: Performance and Uncertainty” by Diana Carbajal and José Luis Romero.
The guiding questions for the project are: Can the integrate-and-fire scheme be characterized exactly as sampling a specific optimal transport map? Can one reconstruct a signal, or at least its induced probability measure, from spike times? Can the stability of spike times be described using Wasserstein distances? Can optimal transport geometry provide improved reconstruction guarantees from spike data?
The resulting algorithms will preferably be implemented in Python.
We will work in-person during the Fall semester 2026 and remotely during the Spring semester 2027.