Modified Oregonator: an Approach from the Complex Networks Theory




Systems Biology, BZ Reaction, Complex Networks, Supervised Learning, Gramian Angular Field


Within the framework of Systems Biology, this paper proposes the complex network theory as a fundamental tool for determining the most critical dynamic variables in complex biochemical mechanisms. The Belousov-Zhabotinsky reaction is proposed as a study model and as a complex bipartite network. By determining the structural property authority, the most relevant dynamic variables are specified, and a mathematical model of the Belousov-Zhabotinsky reaction is obtained. The bidirectional coupling of the proposed model was made with other models associated with biological processes, finding synchronization phenomena when varying the coupling parameter. The time series obtained from the numerical solution of the coupled models were used to construct their images using the Gramian Angular Field technique. In the end, a supervised learning tool is proposed for the classification of the type of coupling by analyzing the images, obtaining score percentages above 94%. The hereby proposed methodology could be extended to the experimental field in order to determine anomalies in the coupling and synchronization of different physiological oscillators.


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Draghici, S., Khatri, P., Tarca, A., Amin, K., Done, A., Voichita, C., Georgescu, C. and Romero, R. A systems biology approach for pathway level analysis. Genome Research. 2007;17(10):1537-1545.

Chuang HY, Hofree M, Ideker T. A Decade of Systems Biology. Annual Review of Cell and Developmental Biology. 2010 Oct;26(1):721–44.

Kell DB. Systems biology, metabolic modelling and metabolomics in drug discovery and development. Drug Discovery Today. 2006;11(23-24):1085–92.

Emmert-Streib F, Dehmer M. Networks for systems biology: conceptual connection of data and function. IET Systems Biology. 2011 Jan;5(3):185–207.

Maeda YT, Sano M. Regulatory Dynamics of Synthetic Gene Networks with Positive Feedback. Journal of Molecular Biology. 2006;359(4):1107–24.

Ferrell JE. Self-perpetuating states in signal transduction: positive feedback, double-negative feedback and bistability. Current Opinion in Cell Biology. 2002;14(2):140–8.

Mullur R, Liu Y-Y, Brent G. Thyroid Hormone Regulation of Metabolism. Physiological Reviews. 2014 Apr;94(2):355-382.

Porte D, Baskin DG, Schwartz MW. Leptin and Insulin Action in the Central Nervous System. Nutrition Reviews. 2002;60:S20-S29.

Chen WW, Niepel M, Sorger PK. Classic and contemporary approaches to modeling biochemical reactions. Genes & Development. 2010;24(17):1861–75.

Alves R, Antunes F, Salvador A. Tools for kinetic modeling of biochemical networks. Nature Biotechnology. 2006;24(6):667–672.

Radulescu O, Gorban AN, Zinovyev A, Noel V. Reduction of dynamical biochemical reactions networks in computational biology. Frontiers in Genetics. 2012;3:131.

Zhabotinsky AM. A history of chemical oscillations and waves. Chaos: An Interdisciplinary. Journal of Nonlinear Science. 1991;1(4):379–86.

Sagues F, Epstein IR. Nonlinear Chemical Dynamics. Dalton Transactions. 2003;32(7):1201-1217.

Epstein IR, Pojman JA, Steinbock O. Introduction: Self-organization in nonequilibrium chemical systems. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2006;16(3):037101.

Noyes RM, Field R, Koros E. Oscillations in chemical systems. I. Detailed mechanism in a system showing temporal oscillations. Journal of the American Chemical Society. 1972;94(4):1394–5.

Field RJ, Koros E, Noyes RM. Oscillations in chemical systems. II. Thorough analysis of temporal oscillation in the bromate-cerium-malonic acid system. Journal of the American Chemical Society. 1972;94(25):8649–64.

Shanks, N. Modeling biological systems: the Belousov–Zhabotinsky reaction. Foundations of Chemistry. 2011; 3(1): 33-53.

Field RJ, Noyes RM. Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction. The Journal of Chemical Physics. 1974;60(5):1877–84.

Györgyi L, Turanyi T, Field RJ. Mechanistic details of the oscillatory Belousov-Zhabotinskii reaction. The Journal of Physical Chemistry. 1990;94(18):7162–70.

Sayama H. Introduction to the modeling and analysis of complex systems. New York: Open SUNY Textbooks, Milne Library, State University of New York at Geneseo; 2015. 478p.

Lesne A. Complex Networks: from Graph Theory to Biology. Letters in Mathematical Physics. 2006;78(3):235–62.

Mason O, Verwoerd M. Graph theory and networks in Biology. IET Systems Biology. 2007;1(2):89–119.

Albert R, Barabási A-L. Statistical mechanics of complex networks. Reviews of Modern Physics. 2002;74(1):47–97.

Costa LDF, Rodrigues FA, Hilgetag CC, Kaiser M. Beyond the average: Detecting global singular nodes from local features in complex networks. EPL (Europhysics Letters). 2009;87(1):18008.

Loscalzo J, Barabási Albert-László, Silverman EK. Network medicine: complex systems in human disease and therapeutics. Cambridge, MA: Harvard University Press; 2017. 448p.

Costa LDF, Rodrigues FA, Cristino AS. Complex networks: the key to systems biology. Genetics and Molecular Biology. 2008;31(3):591–601.

Hornberg J, Bruggeman FJ, Westerhoff HV, Lankelma J. Cancer: A Systems Biology disease. Biosystems. 2006;83(2-3):81-90.

Cardon LR, Bell JI. Association study designs for complex diseases. Nature Reviews Genetics. 2001;2(2):91-99.

DeFronzo RA. Insulin Resistance, Hyperinsulinemia, and Coronary Artery Disease: A Complex Metabolic Web. Journal of Cardiovascular Pharmacology. 1992;20 (Suppl 1):S1-S16.

Karalliedde J, Gnudi L. Diabetes mellitus, a complex and heterogeneous disease, and the role of insulin resistance as a determinant of diabetic kidney disease. Nephrology Dialysis Transplantation. 2016:31(2):206-13.

Upadhyay SK. Chemical kinetics and reaction dynamics. New Delhi: Springer Netherlands; 2006. 256p.

Kotsiantis SB, Zaharakis ID, Pintelas PE. Machine learning: a review of classification and combining techniques. Artificial Intelligence Review. 2006;26:159–90.

Berzal, F. Redes neuronales & deep learning. 1ra. ed. Granada, España: Edición Independiente; 2018. 753p.

Bastian M, Heymann S, Jacomy M. Gephi: an open source software for exploring and manipulating networks. International AAAI Conference on Weblogs and Social Media. 2009;361-362.

Moon FC. Chaotic and fractal dynamics: introduction for applied scientists and engineers. 2nd. ed. New York: Wiley; 1992. 528p.

Afraimovich VS, Lin WW, Rulkov NF. Fractal dimension for poincaré recurrences as an indicator of synchronized chaotic regimes. International Journal of Bifurcation and Chaos. 2000;10(10):2323-2337.

Liu Z, Lai Y, Matías M. Universal scaling of Lyapunov exponents in coupled chaotic oscillators. Physical Review E. 2003;67(4): 045203.

Porta A, Baselli G, Lombardi F, Montano N, Malliani A, Cerutti S. Conditional entropy approach for the evaluation of the coupling strength. Biological Cybernetics. 1999;81:119-129.

Eckmann, JP, Kamphorst SO, Ruelle D. Recurrence plots of dynamical systems. Europhysics Letter. 1987; 4(9):973-977.

Hernández Sánchez S, Fernández Pozo R, Hernández Gómez LA. Deep Neural Networks for Driver Identification Using Accelerometer Signals from Smartphones. In: Abramowicz W, Corchuelo R. (eds) Business Information Systems. Lecture Notes in Business Information Processing. Switzerland: Springer, Cham; 2019, p. 206-220.

Bode BW. Clinical Utility of the Continuous Glucose Monitoring System. Diabetes Technology & Therapeutics. 2000;2(Suppl 1):S35-41.

Yoo EH, Lee SY. Glucose Biosensors: An Overview of Use in Clinical Practice. Sensors. 2010;10(5):4558-4576.

Frost MC, Meyerhoff ME. Implantable chemical sensors for real-time clinical monitoring: progress and challenges. Current Opinion in Chemical Biology. 2002;6(5):633-641.

Frost MC, Meyerhoff ME. Real-Time Monitoring of Critical Care Analytes in the Bloodstream with Chemical Sensors: Progress and Challenges. Annual Review of Analytical Chemistry. 2015;8(1):171-192.

Coveney PV, Fowler PW. Modelling biological complexity: a physical scientist's perspective. Journal of The Royal Society Interface. 2005;2(4):267-280.

Kleinberg JM. Authoritative sources in a hyperlinked environment. Journal of the ACM. 1999;46(5): 604-632.

Aton SJ, Herzog ED. Come Together, Right…Now: Synchronization of Rhythms in a Mammalian Circadian Clock. Neuron. 2005;48(4):531–534.

Lorenzo González MN. Influencia del Ruido Gaussiano Correlacionado en la Sincronización de Sistemas Caóticos [Ph.D.'s thesis]. [Santiago de Compostela]:Universidad de Santiago de Compostela, 2000.161p. Spanish.

Lefever R, Nicolis G, Borckmans P. The brusselator: it does oscillate all the same. Journal of the Chemical Society, Faraday Transactions 1: Physical Chemistry in Condensed Phases. 1988;84(4):1013-1023.

Vaidyanathan, S. Anti-synchronization of Brusselator chemical reaction systems via adaptive control. International Journal of ChemTech Research. 2015;8(6):759-68.

Sel'kov EE. Self-Oscillations in glycolysis. 1. A Simple Kinetic Model. European Journal of Biochemistry. 1968;4(1):79–86.

Keener JP, Sneyd J. Mathematical physiology. 2nd. Ed. New York: Springer; 2009. 470p.

Butcher J. Numerical methods for ordinary differential equations in the 20th century. In: Brezinski C, Wuytack, L. (eds). Numerical Analysis: Historical Developments in the 20th Century. Amsterdam: Elsevier; 2001.449–77p.

Faouzi J, Janati H. pyts: A python package for time series classification. Journal of Machine Learning Research. 2020; 21(46):1−6.

Wang Z, Oates T. Imaging time-series to improve classification and imputation. Proceedings of the 24th International Joint Conference on Artificial Intelligence. 2015;3939-45.

Брагин, АB, Bragin A, Спицын ВГ, Spicyn, V.Electroencephalogram Analysis Based on Gramian Angular Field Transformation. Proceedings of the 29th International Conference on Computer Graphics and Vision. 2019;2485:273-75.

Damaševičius R, Maskeliūnas R, Woźniak M, Polap D. Visualization of physiologic signals based on Hjorth parameters and Gramian Angular Fields. 2018 IEEE 16th World Symposium on Applied Machine Intelligence and Informatics (SAMI). 2018; 91-6.

Qin Z, Zhang Y, Meng S, Qin Z, Choo K-KR. Imaging and fusing time series for wearable sensor-based human activity recognition. Information Fusion. 2020;53:80–7.

Demsar J, Curk T, Erjavec A, Gorup C, Hocevar T, Milutinovic M, Mozina M, Polajnar M, Toplak M, Staric A, Stajdohar M, Umek L, Zagar L, Zbontar J, Zitnik M, Zupan B. Orange: Data Mining Toolbox in Python. Journal of Machine Learning Research. 2013; 14(35): 2349−2353.

Demšar J, Zupan B. Orange: Data Mining Fruitful and Fun- A Hystorical Perspective. Informatica 2013;37:55-60.

Godec P, Pančur M, Ilenič N, Čopar A, Stražar M, Erjavec A, Pretnar A, Demšar J, Starič A, Toplak M, Žagar L, Hartman J, Wang H, Bellazzi R, Petrovič U, Garagna S, Zuccotti M, Park D, Shaulsky G, Zupan B. Democratized image analytics by visual programming through integration of deep models and small-scale machine learning. Nature Communications. 2019;10(1):4551.

Weiss K, Khoshgoftaar T, Wang D. A survey of transfer learning. Journal of Big Data. 2016;3(1):1345–1359.

Tan C, Sun F, Kong T, Zhang W, Yang C, Liu C. A survey on deep transfer learning. In Kůrková V, Manolopoulos Y, Hammer B, Iliadis L, Maglogiannis I. International conference on artificial neural networks. Rhodes: Springer International Publishing. Cham; 2018. p. 270-279.

Szegedy C, Vanhoucke V, Ioffe S, Shlens J, Wojna Z. Rethinking the inception architecture for computer vision. 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR); Las Vegas: IEEE; 2016. 2818-2826p.

Iandola FN, Han S, Moskewicz MW, Ashraf K, Dally WJ, Keutzer K. SqueezeNet: AlexNet-level accuracy with 50x fewer parameters and <0.5MB model size [Internet].; 2020. Available from:

Simonyan K, Zisserman A. Very Deep Convolutional Networks for Large-Scale Image Recognition [Internet].; 2020. Available from:

Almagro Armenteros JJ, Sønderby CK, Sønderby SK, Nielsen H, Winther O. DeepLoc: prediction of protein subcellular localization using deep learning. Bioinformatics. 2017;33(21):3387-3395.

Müller AC, Guido S. Introduction to machine learning with Python a guide for data scientists. Beijing: OReilly; 2018. p. 251-303.

Hawkins DM. The Problem of Overfitting. Journal of Chemical Information and Computer Sciences. 2004;44(1):1-12.

Sokolova M, Japkowicz N, Szpakowicz S. Beyond Accuracy, F-Score and ROC: A Family of Discriminant Measures for Performance Evaluation. In Sattar A, Kang B. (eds). AI 2006: Advances in Artificial Intelligence. Berlin: Springer; 2006. 1015–21p.

Pinto CMA, Mendes Lopes A, Tenreiro Machado JA. A review of power laws in real life phenomena. Communications in Nonlinear Science and Numerical Simulation. 2012;17(9):3558–78.

Gisiger T. Scale invariance in biology: coincidence or footprint of a universal mechanism? Biological Reviews of the Cambridge Philosophical Society. 2001;76(2):161–209.

Amemiya T, Kádár S, Kettunen P, Showalter K. Spiral Wave Formation in Three-Dimensional Excitable Media. Physical Review Letters. 1996;77(15):3244–7.

Krug HJ, Pohlmann L, Kuhnert L. Analysis of the modified complete Oregonator accounting for oxygen sensitivity and photosensitivity of Belousov-Zhabotinskii systems. The Journal of Physical Chemistry. 1990;94(12):4862–6.

Ma J, Li F, Huang L, Jin W-Y. Complete synchronization, phase synchronization and parameters estimation in a realistic chaotic system. Communications in Nonlinear Science and Numerical Simulation. 2011;16(9):3770–3785.

Rajesh S, Sinha S, Sinha S. Synchronization in coupled cells with activator-inhibitor pathways. Physical Review E. 2007;75(1): 011906.

Belykh I, Lange ED, Hasler M. Synchronization of Bursting Neurons: What Matters in the Network Topology. Physical Review Letters. 2005;94(18):1881011-1881014.

Belhaq M, Houssni, M. Quasi-periodic oscillations, chaos and suppression of chaos in a nonlinear oscillator driven by parametric and external excitations. Nonlinear Dynamics. 1999;18(1):1-24.

Ng A. Feature selection, L 1 vs. L 2 regularization, and rotational invariance. In Brodley C (ed.). ICML '04: Proceedings of the twenty-first international conference on Machine learning. New York: Association for Computing Machinery; 2004. 78-86p.

Ng A. Preventing" overfitting" of cross-validation data. In Fisher DH (ed.). ICML '97: Proceedings of the Fourteenth International Conference on Machine Learning. San Francisco: Morgan Kaufmann Publishers Inc; 1997. 245-253p.

Mao W, Mu X, Zheng Y, Yan G. Leave-one-out cross-validation-based model selection for multi-input multi-output support vector machine. Neural Computing and Applications. 2012;24(2):441-451.

Grünauer A, Vincze M. Using Dimension Reduction to Improve the Classification of High-dimensional Data [Internet]. OAGM Workshop; 2020. Available from:

Azure Machine Learning. Evitar el sobreajuste y los datos desequilibrados con el aprendizaje automático automatizado. Microsoft [Internet]. 2019. Available from:

Christodoulou E, Ma J, Collins GS, Steyerberg EW, Verbakel JY, Calster BV. A systematic review shows no performance benefit of machine learning over logistic regression for clinical prediction models. Journal of Clinical Epidemiology. 2019;110:12–22.

Rajagopal S, Hareesha KS, Kundapur PP. Performance analysis of binary and multiclass models using azure machine learning. International Journal of Electrical and Computer Engineering (IJECE). 2020;10(1):978-986.

Kohavi R. A study of cross-validation and bootstrap for accuracy estimation and model selection. IJCAI'95: Proceedings of the 14th international joint conference on Artificial intelligence. 1995;14(2):1137-1145

Alducin Castillo J, Yanez Suárez O. Brust Carmona H. Electroencephalographic analysis of the functional conectivity in habituation by graphics theory. Revista Mexicana de Ingeniería Biomédica. 2016;37(3):181-200.

Glass L. Synchronization and rhythmic processes in physiology. Nature. 2001;410(6825):277–84.




How to Cite

Rojas, J. F., Arzola, J. A., & Vidal Robles, E. (2020). Modified Oregonator: an Approach from the Complex Networks Theory. Revista Mexicana De Ingenieria Biomedica, 41(3), 6–27.



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