The FMM Project:

An approach for the analysis of oscillatory signals


The biological variables following the circadian rhythm, the electrocardiogram (ECG), or the electroencephalograph (EEG) are examples of oscillatory signals. Questions such as which genes are activated by the circadian rhythms, and how to detect heart rhythm failures or mental disorders, by automatically interpreting the ECG or EEG are only some examples of relevant questions related to oscillatory signal analysis.  The field is in continuous advance, much due to the statistics and mathematics basic research.

Some of the most research questions in data signal analysis are the extraction of features, detection of fiducial marks, generating synthetic data, or denoising signals. In particular, to define a reduced set of interpretable features and an efficient algorithm to extract these features from the recorded signal accurately, are the top requirements of an efficient signal analysis method. In the oscillatory signals, the main features to extract are the number of oscillatory components and the amplitude and peak time of each oscillation. For instance, it is well known for physiological signal that these features contain plenty of information about a person’s health condition.

In general, inferring the dynamical information from an oscillatory signal is challenging. The FMM approach is a universal approach that competes with Fourier or wavelet decompositions. It combines a physically meaningful formulation with good statistical and computational properties. It has been recently presented in a series of papers, listed below, where besides the theory and computer properties, diverse applications in different fields have been shown.

Challenges for the future

There are different lines of work for the future. Firstly, from a theoretical perspective, a rigorous inferential approximation of the FMM models will be presented soon. Secondly, we have created FMM-AIS, 1.0, a new tool for researchers and clinicians to facilitate the use and understanding of the FMM methodology for the automatic analysis of ECG signals. And we have developed an automatic procedure that will allow us to digitize and analyze the ECG images taken in the clinic in real-time. Thirdly, we have started to research different FMM approaches for the analysis of EEG signals useful in the prediction of mental disorders and outcomes of patients after a heart attack.

Furthermore, some work is also being carried out related to supervised and unsupervised classification of oscillatory signals with specific applications in Neuroscience, where the cluster of cells by their waveforms is one of the problems to which more attention is devoted, is called Spike Sorting. Finally, many research opportunities have opened up to study biological systems beyond the heart or the brain, such as electrical signals from other organs, such as the eyes, which can be modeled by adapted 3DFMM models. Other biological signals that can be analyzed with the FMM approach, in addition to electrical ones, are signals associated with the circadian cycle, such as hormone levels and body temperature or food intake in animals. Finally, besides biological signals, the FMM approach would be useful for analyzing oscillatory signals from different disciplines. Indices of refraction and luminance in optics, spectrophotometric curves in experimental chemistry, website traffic, or atmospheric pressure are more examples, to name a few from different disciplines. Also, even in less scientific fields, we can find examples of 24-hour oscillatory signals such as electricity demand profiles or the daytime pattern of water consumption.

Medical advances contributions

The most interesting, by far, that the future of research in this field holds is contributing to solving relevant medical problems. Among the most important ones are the detection of cardiology pathologies, and the determination of factors that influence the course of neurological diseases as Parkinson.  Furthermore, many exciting questions in chronobiology remain open, such as the relation of rhythmicity patterns with diseases as cancer or how the hormone patterns are related to physiological processes.

Research People

Cristina Rueda

Cristina Rueda, Project leader
University of Valladolid (UVa)

Itziar Fernández, UVa

Yolanda Larriba, UVa

Cristian Canedo, UVa

Adolfo Fenández-Santamómica, UVa


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. Javier Gómez Pilar, University of Valladolid, Valladolid, Spain.

Dr. Roberto Hornero Sánchez, University of Valladolid, Valladolid, Spain.

Dr. Alejandro Rodríguez-Collado.


Rueda, C., Larriba, Y., & Peddada, S. D. (2019). Frequency Modulated Möbius Model Accurately Predicts Rhythmic Signals in Biological and Physical Sciences. Scientific reports9(1), 18701.

Rueda, C., Larriba, Y. & Lamela, A. (2021). The Hidden Wave in the ECG Uncovered Revealing a Sound Automated Interpretation Method. Scientific Reports, 11, 3724.

Rueda, C., Rodríguez-Collado, A. & Larriba, Y. (2021). A Novel Wave Decomposition for Oscillatory Signals. IEEE Transactions on Signal Processing, 69, 960-972.

Rodríguez-Collado, A. & Rueda, C. (2021). Simple Parametric Representation of the Hodgkin-Huxley Model. PLOS ONE, 16(7), e0254152.

Rodríguez-Collado, A. & Rueda, C. (2021). Electrophysiological and Transcriptomic Features Reveal a Circular Taxonomy of Cortical Neurons. Frontiers in Human Neuroscience, 15, 410. https://

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.

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.

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.

Larriba, Y., Rodríguez-Collado, A., Rueda, C. (2023). 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.

Rueda, C., Rodríguez-Collado, A., Fernández, I., Canedo, C., Uguarte, M.D. & Larriba, Y. A Unique Cardiac Electrocardiographic 3D Model. Towards Interpretable AI Diagnosis. (2023). iScience.

Larriba, Y., Rueda, C. and Rodríguez-Collado, A (2023).  Circular Ordering Methods for Timing and Visualization of Oscillatory Signals. In: Building Bridges between Soft and Statistical Methodologies for Data Science . SMPS 2022. Advances in Intelligent Systems and Computing, vol 1433. Springer

Rodriguez-Collado, A. & Rueda, C (2023). Functional Clustering of neuronal Signals with FMM Mixture Models. (to appear)

Larriba, Y. and  Rueda, C (2023).   Modelling the circadian variation of electrocardiographic parameters with Frequency Modulated Models. In Statistical Methods at the Forefront of Biomedical Advances. (to appear)

Canedo, C, Fernández, I, Coco, R. M, Cuadrado, R. and Rueda, C (2023). Novel Modeling Proposals for the Analysis of Pattern Electroretinogram Signals. In: Statistical Methods at the Forefront of Biomedical Advances.  (to appear)

Rueda, C., Canedo, C., Larriba, Y. & Fernández, I.FDA with Möbius Waves: Estimation of Signals and their Derivatives with Applications. Preprint.

Larriba, Y.,  Rueda, C., Mason, I., Saxena, R. & Scheer, F.  CIRCUST: a novel methodology for reconstruction of temporal order of molecular rhythms; validation and application towards a human circadian gene expression atlas. Preprint.

Rueda, C, Larriba, Y, Canedo, C., Pérez-Castellanos, A., and Fernández, I.(2023) FMM-AIS 1.0: A new automatic system for ECG interpretation. Preprint.

Fernández-Santamónica, A., Catalarrá,R. Larriba,Y., Pérez-Castellanos, A., and Rueda, C. (2023) ECGMiner: A Software for Precisely Digitizing Paper-Based Electrocardiograms. Preprint.




The stable version of the FMM models implementation on the programming language R is available on CRAN. Documentation about the package’s use can be found on its manual and in Fernández, I., et al. FMM: An R Package for Modeling Rhythmic Patterns in Oscillatory Systems. Furthermore, the package’s latest development version as well as its issue tracker can be found on Github.

Other applications developed using FMM models can be found below:




Scientific Dissemination