The FMM Project:
An approach for the analysis of oscillatory signals
Introduction
Oscillatory signals such as circadian rhythms, electrocardiograms (ECG), and electroencephalograms (EEG) are central to understanding biological processes. Key questions include identifying genes activated by circadian cycles or detecting cardiac and neurological disorders through automated signal interpretation. The growing need for precise and interpretable analysis has made signal processing a rapidly advancing field, supported by developments in mathematics and statistics.
A major challenge in this area is defining a reduced, interpretable set of features and developing efficient algorithms to extract them reliably. These features often carry valuable physiological information critical for diagnosis and monitoring.
The Frequency Modulated Möbius (FMM) model has emerged as a powerful alternative to classical Fourier and wavelet methods. It offers a physiologically meaningful framework with strong statistical and computational properties. Unlike black-box models, FMM balances interpretability with predictive accuracy, making it particularly suitable for applications in chronobiology, neuroscience, cardiology, and other fields. In fact, the FMM approach has already been successfully applied to the analysis of circadian rhythms, and a variety of bioelectrical signals, among others. It has also proven useful in machine learning pipelines, further supporting its versatility and potential for integration in interdisciplinary research.
Emerging extensions
New domains are opening for FMM applications. In spectroscopy, it shows promise for deconvolution, peak separation, and noise reduction. In demography, FMM models with random effects could help predict fertility patterns and health trends in small populations.
A particularly promising extension is the development of a fully two-dimensional FMM framework, where both variables represent spatial coordinates. This would allow for analysis of spatial diversity and anisotropy in complex systems such as biological tissues or turbulent plasmas, requiring new approaches for modeling and computation.
To support broader use, we are finalizing a Python implementation of the FMM model, enabling scalable applications and integration with machine learning tools. Future work will focus on optimizing efficiency and improving usability.
Contributions to Medicine
Perhaps the most exciting prospect for FMM is its potential to address major medical challenges. These include early detection of cardiac conditions, identifying biomarkers for neurological diseases such as Parkinson’s and visual dysfunctions, and investigating links between biological rhythms and diseases like cancer. Open questions in chronobiology — particularly those related to the temporal organization of gene expression — also stand to benefit from FMM-based analysis. A particularly promising direction lies in the personalized modelling of circadian gene expression, which could inform optimal timing for drug administration (chronotherapy) and enhance precision medicine strategies.
Research People
Itziar Fernández, UVa
itziar.fernandez@uva.es
Yolanda Larriba, UVa
yolanda.larriba@uva.es
Christian Canedo, UVa
christian.canedo@uva.es
Adolfo Fenández-Santamómica, UVa
adolfo.fernandez.santamonica@estudiantes.uva.es
Colaborators
Dra. Rocío Carratalá Sáez, University of Valladolid, Valladolid, Spain.
Dr. Alberto Pérez-Castellanos, Son Espases University Hospital, Mallorca, Spain.
Dra. María Dolores Ugarte, Public University of Navarre, Navarre, Navarre, Spain.
Dr. Shyamal D. Peddada, National Institute of Child Health and Human, NIH, USA.
Dra. Rosa María Coco Martín, Institute of Applied Ophthalmo-Biology (IOBA) of University of Valladolid, Valladolid, Spain.
Dr. Rubén Cuadrado Asensio, Institute of Applied Ophthalmo-Biology (IOBA) of University of Valladolid, Valladolid, Spain.
Dr. Frank Scheer, Division of Sleep Medicine, Harvard Medical School, Boston, USA.
Dr. Richa Saxena, Harvard Medical School. Boston. USA.
Dr. Enrique Hortal. Department of Advanced Computing Sciences, Maastricht University . The Netherlands.
Publications
Rueda, C., Larriba, Y., & Peddada, S. D. (2019). Frequency Modulated Möbius Model Accurately Predicts Rhythmic Signals in Biological and Physical Sciences. Scientific reports, 9(1), 18701. https://doi.org/10.1038/s41598-019-54569-1
Rueda, C., Larriba, Y. & Lamela, A. (2021). The Hidden Wave in the ECG Uncovered Revealing a Sound Automated Interpretation Method. Scientific Reports, 11, 3724. https://doi.org/10.1038/s41598-021-82520-w
Rueda, C., Rodríguez-Collado, A. & Larriba, Y. (2021). A Novel Wave Decomposition for Oscillatory Signals. IEEE Transactions on Signal Processing, 69, 960-972. https://doi.org/10.1109/TSP.2021.3051428
Rodríguez-Collado, A. & Rueda, C. (2021). Simple Parametric Representation of the Hodgkin-Huxley Model. PLOS ONE, 16(7), e0254152. https://doi.org/10.1371/journal.pone.0254152
Rodríguez-Collado, A. & Rueda, C. (2021). Electrophysiological and Transcriptomic Features Reveal a Circular Taxonomy of Cortical Neurons. Frontiers in Human Neuroscience, 15, 410. https://doi.org/10.3389/fnhum.2021.684950
Rueda, C., Fernández, I., Larriba, Y. & Rodríguez-Collado, A. The FMM Approach to Analyze Biomedical Signals: Theory, Software, Applications and Future. (2021). Mathematics, 9, 1145. https://doi.org/10.3390/math9101145
Fernández, I., Rodríguez-Collado, A., Larriba, Y., Lamela, A., Canedo, C. & Rueda, C. FMM: An R Package for Modeling Rhythmic Patterns in Oscillatory Systems. (2022). The R Journal, 14(1), 361-380. https://doi.org/10.32614/RJ-2022-015
Rueda, C., Fernández, I., Larriba, Y., Rodríguez-Collado, A. & Canedo, C. Compelling new electrocardiographic markers for automatic diagnosis. (2022). Computer Methods and Programs in Biomedicine, 221, 106807. https://doi.org/10.1016/j.cmpb.2022.106807
Larriba, Y., Rodríguez-Collado, A. & Rueda, C. (2022). Circular Ordering Methods for Timing and Visualization of Oscillatory Signals. In: García-Escudero, et al. Building Bridges between Soft and Statistical Methodologies for Data Science. SMPS 2022. Advances in Intelligent Systems and Computing, 1433. Springer, Cham. https://doi.org/10.1007/978-3-031-15509-3_34
Rueda, C., Rodríguez-Collado, A., Fernández, I., Canedo, C., Uguarte, M.D. & Larriba, Y. A Unique Cardiac Electrocardiographic 3D Model. Towards Interpretable AI Diagnosis. (2022). iScience. https://www.sciencedirect.com/science/article/pii/S2589004222018892
Jin, M., Watkins, S., Larriba, Y. & Wenzel, S. E. (2021). Real-time imaging of asthmatic epithelial cells identifies migratory deficiencies under type-2 conditions. Journal of Allergy and Clinical Immunology.
Larriba, Y. & Rueda, C. (2023). Modelling the Circadian Variation of Electrocardiographic Parameters with Frequency Modulated Models. In: Larriba, Y. (eds) Statistical Methods at the Forefront of Biomedical Advances. Springer, Cham. https://doi.org/10.1007/978-3-031-32729-2_10
Canedo, C, Fernández, I, Coco, R. M, Cuadrado, R. & Rueda, C (2023). Novel Modeling Proposals for the Analysis of Pattern Electroretinogram Signals. In: Larriba, Y. (eds) Statistical Methods at the Forefront of Biomedical Advances. Springer, Cham. https://doi.org/10.1007/978-3-031-32729-2_10 In: Statistical Methods at the Forefront of Biomedical Advances.
Rueda, C. & Rodríguez-Collado, A. (2023). Functional Clustering of Neuronal Signals with FMM Mixture Models. Helyion, Oct 10;9(10). doi: 10.1016/j.heliyon.2023.e20639
Larriba, Y., Mason, I.C., Scheer, F., Saxena, R., & Rueda, C. (2023). CIRCUST: a novel methodology for reconstruction of temporal order of molecular rhythms; validation and application towards a human circadian gene expression atlas. PLoS Comput Biol. Sep 19(9).doi: 10.1371/journal.pcbi.1011510
Canedo, C., Fernández-Santamónica, A., Larriba, Y., Fernández, I., & Rueda, C. (2023, October). Heart Attack Outcome Predictions Using FMM Models. In 2023 Computing in Cardiology (CinC) (Vol. 50, pp. 1-4). IEEE.
Fernández, I., Cuadrado Asensio, R., Larriba, Y., Rueda, C. & Coco-Martin, R. M. (2024). A Comprehensive Dataset of Pattern Electroretinograms for Ocular Electrophysiology Research: The PERG-IOBA Dataset (version 1.0.0). PhysioNet. https://doi.org/10.13026/d24m-w054
Fernández-Santamónica, A., Catalarrá ,R. Larriba,Y., Pérez-Castellanos, A., and Rueda, C (2024).ECGMiner: A novel Software for Precisely Digitizing Multiple Paper-Based ECG Simultaneously. Computer Methods and Programs in Biomedicine.
Fernández, I., Cuadrado Asensio, R., Larriba, Y., Rueda, C., & Coco-Martin, R. M. (2024) A comprehensive dataset of pattern electroretinograms for ocular electrophysiology research. Scientific Data, 11(1).
Fernández, I., Larriba, Y. Canedo, C., and Rueda, C. (2025). Functional data analysis with Möbius waves: Applications to biomedical oscillatory signals. Annals of Applied Statistics 19(2), 1514-1532.
Canedo, C., Carratalá, R. & Rueda, C. (2025). Combining Adaptive Fourier and Frequency Modulated Möbius Decompositions for Enhanced Analysis of Oscillatory Signals. Submitted.
Fernández-Santamónica, A., Larriba, Y., Horta, E. & Rueda, C. (2025). Explainable AI for Automatic Heart Diagnosis Using 3DFMM Features: A Novel ECG-Based Approach. Submitted.
Rueda, C., Fernández, I., Canedo, C. & Larriba, Y. (2025) Precise Peak Width Estimation: Solving Key Challenges in Biosignal and Spectral Analysis. Submitted.
Rueda, C. (2025). S-FMM: A Spatial Frequency Modulation Framework for Field Modeling on the Torus. Submitted.
Rueda, C. (2025). Frequency Modulated Möbius Model: From Foundations to Frontiers. RSME Springer Series. To appear.
Software
The stable version of the FMM models, implemented in the R programming language, is available on CRAN. Detailed documentation on the package’s usage can be found in its manual and in the paper by Fernández, I., et al. FMM: An R Package for Modeling Rhythmic Patterns in Oscillatory Systems.
For the latest software developments, user documentation, and references, please visit the group’s GitHub repository: FMMGroupVa. It contains the following repositories:
- <<FMM-Extensions>> Implementation of the 3DFMM model for multicomponent and multichannel signals. It includes functions for model fitting, visualization, and parameter inference.
Fernández, I., Larriba, Y. Canedo, C., and Rueda, C. (2025). Functional data analysis with Möbius waves: Applications to biomedical oscillatory signals. Annals of Applied Statistics 19(2), 1514-1532.
- <<s-FMM>> code for modeling structured two-dimensional oscillatory fields. It extends the FMM framework to two spatial dimensions by incorporating Möbius phase functions within a quaternionic representation. It is designed to model spatially distributed signals such as geophysical fields, neural activity, or physiological measurements across structured domains.
Rueda, C. (2025). S-FMM: A Spatial Frequency Modulation Framework for Field Modeling on the Torus. Submitted.
- <<FMM-Applications-3DECG>> Application of the 3DFMM model to multilead ECG signals. Provides preprocessing scripts, ECG wave delineation, and practical examples using datasets like PTB-XL.
Rueda, C., Rodríguez-Collado, A., Fernández, I., Canedo, C., Uguarte, M.D. & Larriba, Y. A Unique Cardiac Electrocardiographic 3D Model. Towards Interpretable AI Diagnosis. (2022). iScience.
- <<FMM-Applications-CIRCUST>> Routines for applying CIRCUST, a robust methodology for temporal order estimation in gene expression data specifically designed for noisy, single-timepoint transcriptomic datasets.
Larriba, Y., Mason, I.C., Scheer, F., Saxena, R., & Rueda, C. (2023). CIRCUST: a novel methodology for reconstruction of temporal order of molecular rhythms; validation and application towards a human circadian gene expression atlas. PLoS Comput Biol.
- <<FMM-Applications-HRV>> Scripts and tools for analyzing heart rate variability using FMM-based rhythm models. Includes beat-to-beat interval preprocessing and temporal pattern modeling.
Larriba, Y. & Rueda, C. (2023). Modelling the Circadian Variation of Electrocardiographic Parameters with Frequency Modulated Models. In: Larriba, Y. (eds) Statistical Methods at the Forefront of Biomedical Advances. Springer, Cham
News
https://massivesci.com/articles/ekg-math-model-waves-diagnosis/
Scientific Dissemination
https://www.youtube.com/watch?v=0szhmvPGKng&t=1s