Mathematics of behavior and intelligence Lab
Applied mathematics · AI · Biology

Applied mathematics
and AI

We work at the intersection of applied mathematics, machine learning, and biology, combining basic theory with applications.

We focus on problems where quantitative approaches can drive progress in biology. Current themes include the rules of collective animal behavior, basic principles of hippocampal circuits, genomic signal modeling, and computational creativity in science.

A second strand of our work develops a new mathematical foundation to learning based on abstract algebra, which we are now beginning to apply to biological problems.

For each problem, we typically build tools that support the research and share them with the community.

View publications Explore open tools
Champalimaud Foundation Google Scholar Contact
Members of the Polavieja Lab

Research

Collective behavior Algebraic Machine Learning AI in Biology Scientific discovery

Collective behavior

GroupsDecision rules AI Tracking

We study how animals, including humans, interact in groups. We have explained group decision-making experiments using the idea that each agent uses the behavior of others to estimate where to go.

We have also approached this problem using modular neural networks, which combine high predictive performance with interpretability, thanks to their low-dimensional modules. These methods reveal simple local interactions, including attraction to a small number of nearby, fast-moving neighbors.

References
  1. F. J. Heras, F. Romero-Ferrero, R. Hinz, and G. G. de Polavieja. Deep attention networks reveal the rules of collective motion in zebrafish. PLoS Computational Biology 15.9, e1007354 (2019).
  2. R. C. Hinz, and G. G. de Polavieja. Ontogeny of collective behavior reveals a simple attraction rule. Proc Natl Acad Sci U S A (2017).
  3. S. Arganda, A. Pérez-Escudero, and G. G. de Polavieja. A common rule for decision making in animal collectives across species. PNAS 109, 20508–20513 (2012).

Algebraic Machine Learning

AlgebraGeneralizationTabular dataSmall data

A common strategy for analyzing complex mathematical structures is to decompose them into simpler components. We use Birkhoff’s subdirect representation theorem as the decomposition principle.

η : A → B1 × B2 × ··· × Bn

A learning task is encoded in the algebra A, a subdirect representation is computed, and learning consists of selecting the components Bi that capture the rule underlying the data.

This algebraic does not require hyperparameter tuning or a validation dataset, and can combine data with formal specifications. We are currently applying it to medical datasets.

References
  1. F. Martin-Maroto, N. Abderrahaman, D. Mendez, and G. G. de Polavieja. Algebraic Machine Learning: Learning as computing an algebraic decomposition of a task. arXiv:2502.19944 (2025).
  2. D. Mendez, F. Martin-Maroto, and G. G. de Polavieja. Algebraic Machine Learning for Small-to-Medium Datasets Is Competitive against Strong Standard Baselines. arXiv:2605.22155 (2026).

AI in Biology

Neural systemsGenomicsBiological data

We study biological problems using hypothesis-driven theory and AI methods designed to distill biological knowledge. Past work includes neuronal coding, wiring economy in the brain, and deviations from optimality in biological systems.

We have also modeled systems related to body ownership in humans, bursty behavior, aggression, and species diversity.

Current work applies machine learning to genomic signal modeling, dolphin communication, and hippocampal circuits.

References
  1. F. Nisantzis, C. Gonçalves and G. G. de Polavieja. C.La.P.: Enhancing transformer-based genomic signal modeling by integrating DNA sequences and chromatin accessibility data (2025).
  2. A. Perez-Escudero, M. Rivera-Alba, and G. G. de Polavieja. Structure of deviations from optimality in biological systems. PNAS 106, 20544–20549 (2009).
  3. M. Rivera-Alba et al. Wiring economy and volume exclusion determine neuronal placement in the Drosophila brain. Current Biology 21, 2000 (2011).

Agentic science and scientific discovery

CreativityAI agentsCo-creation

We are developing agentic workflow tools to support the more routine, time-consuming aspects of scientific research.

We are also building a new research line that draws on cognitive science to better understand and enhance creativity in scientific discovery.

Our goal is to create AI agents for co-creation and learning that help researchers generate, test, and refine scientific ideas. A central concern is to avoid the deskilling of scientists: these tools should preserve and strengthen scientific judgment, expertise, and the capacity to learn.

Recent literature in this field (not from us)
  1. J. Gottweis, W.-H. Weng, A. Daryin, et al. Accelerating scientific discovery with Co-Scientist. Nature (2026). doi:10.1038/s41586-026-10644-y.
  2. A. E. Ghareeb, B. Chang, L. Mitchener, et al. A multi-agent system for automating scientific discovery. Nature (2026). doi:10.1038/s41586-026-10652-y.
  3. E. Aygün, A. Belyaeva, G. Comanici, et al. An AI system to help scientists write expert-level empirical software. Nature (2026). doi:10.1038/s41586-026-10658-6.

Publications

Selected publications

Latest publications

Complete publication record: Google Scholar profile.

Media

The author file

Nature Methods profile connected to idtracker.ai and automated animal tracking.

Read in Nature Methods

Deep networks for behavior

Coverage of AI methods used to identify how collective behaviors emerge.

Read at Phys.org

Tracking revolution

News feature on the transformation of animal tracking through machine learning.

Read in Nature

Open tools

idtracker.ai interface and validation workflow

idtracker.ai

Open-source multi-animal tracking based on individual identification, designed for small and large collectives of unmarked animals. The latest version is much faster, more accurate and more robust.

Visit idtracker.ai

References
  1. J. Torrents, T. Costa, G. G. de Polavieja. New idtracker.ai rethinks multi-animal tracking as a representation learning problem to increase accuracy and reduce tracking time. eLife 14:RP107602 (2026).
  2. F. Romero-Ferrero, M. G. Bergomi, R. C. Hinz, F. J. H. Heras, and G. G. de Polavieja. idtracker.ai: tracking all individuals in small or large collectives of unmarked animals. Nature Methods 16, 179–182 (2019).
  3. A. Perez-Escudero, J. Vicente-Page, R. C. Hinz, S. Arganda and G. G. de Polavieja. idTracker: tracking individuals in a group by automatic identification of unmarked animals. Nature Methods, 11, 743-748 (2014)

idmatcher.ai

Match animal identities across different videos.

Open repository

Trajectorytools

Data-analysis tools for trajectories generated by tracking systems.

Open repository

CLaP

Chromatin Language Processing predicts protein binding events.

Open repository

Engine for Algebraic Machine Learning

Open-source engine for algebraic machine-learning experiments.

This is the engine we use for the notebooks, and has additional ready to use examples.

Open repository

Team

Contact

Contact

For research, collaboration, student projects or academic enquiries, contact Gonzalo G. de Polavieja.