The FMM Project:

An approach for the analysis of oscillatory signals


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

Some of the most researcher 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 signals 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 the 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. On the one hand, from a theoretical perspective, a rigorous inferential approximation of the FMM models will be presented soon. On the other hand, much work remains to be done from an applied research perspective, such as the design of new electrocardiographic diagnostic rules and the integration of the 3DFMMecg model into AI systems. Specifically, we are working on an automatic procedure that will allow us to digitize and analyze the ECG images taken in the clinic in real time. Finally, the topics of supervised and unsupervised classification of waveforms are of great interest in many disciplines. In Neuroscience, the cluster of cells by their waveforms is one of the problems to which more attention is devoted, is called Spike Sorting. Some work is also being carried out along this line.

In addition, many research opportunities have opened up to study biological systems beyond the heart, such as electrical signals from other organs, such as the brain or 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. Furthermore, in addition to biological signals, the FMM approach would be useful to analyze 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. To appear.

Rodriguez-Collado, A. & Rueda, C. Functional Clustering of neuronal Signals with FMM Mixture Models.

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.



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