Authors

Behnam Babaeian, Marius E. Yamakou

Abstract

We study chaotic synchronization in a 5D Hindmarsh–Rose neuron model augmented with electromagnetic induction and a switchable memristive autapse. For two diffusively coupled identical neurons, we derive the transverse error dynamical system and analyze local synchronization via the linearized error system around the synchronization manifold. A quadratic Lyapunov function yields explicit sufficient conditions for (i) asymptotic stability when the memristive switching remains dissipative and (ii) practical stability with an explicit ultimate bound under non-dissipative switching. We complement this with a Hamiltonian-based viewpoint: a Helmholtz decomposition of the linearized error vector field provides a closed-form synchronization Hamiltonian and its rate identity. Numerical simulations corroborate convergence or ultimate boundedness of the synchronization errors and an overall decay of the synchronization Hamiltonian and its instantaneous rate toward zero after transients, and show consistent trends between Lyapunov- and Hamiltonian-based diagnostics across parameters. Finally, we propose the first port-Hamiltonian physics-informed neural network (pH-PINN) that learns this synchronization Hamiltonian and its rate from data while preserving conservative/dissipative structure, achieving close agreement with the analytical expressions.

Keywords

chaos, hamiltonian function, lyapunov function, memristive neurons, physics-informed neural networks, port-hamiltonian neural networks, synchronization

Citation

  • Journal: Applied Mathematics and Computation
  • Year: 2026
  • Volume: 530
  • Issue:
  • Pages: 130157
  • Publisher: Elsevier BV
  • DOI: 10.1016/j.amc.2026.130157

BibTeX

@article{Babaeian_2026,
  title={{A Lyapunov stability proof and a port-Hamiltonian physics-informed neural network for chaotic synchronization in memristive neurons}},
  volume={530},
  ISSN={0096-3003},
  DOI={10.1016/j.amc.2026.130157},
  journal={Applied Mathematics and Computation},
  publisher={Elsevier BV},
  author={Babaeian, Behnam and Yamakou, Marius E.},
  year={2026},
  pages={130157}
}

Download the bib file

References

  • Neustadter, EEG and MEG probes of schizophrenia pathophysiology. (2016)
  • Lehnertz K, Bialonski S, Horstmann M-T, Krug D, Rothkegel A, Staniek M, Wagner T (2009) Synchronization phenomena in human epileptic brain networks. Journal of Neuroscience Methods 183(1):42–48. https://doi.org/10.1016/j.jneumeth.2009.05.01 – 10.1016/j.jneumeth.2009.05.015
  • Borges FS, Gabrick EC, Protachevicz PR, Higa GSV, Lameu EL, Rodriguez PXR, Ferraz MSA, Szezech JD Jr, Batista AM, Kihara AH (2023) Intermittency properties in a temporal lobe epilepsy model. Epilepsy & Behavior 139:109072. https://doi.org/10.1016/j.yebeh.2022.10907 – 10.1016/j.yebeh.2022.109072
  • Protachevicz PR, Borges FS, Lameu EL, Ji P, Iarosz KC, Kihara AH, Caldas IL, Szezech JD Jr, Baptista MS, Macau EEN, Antonopoulos CG, Batista AM, Kurths J (2019) Bistable Firing Pattern in a Neural Network Model. Front Comput Neurosci 13. https://doi.org/10.3389/fncom.2019.0001 – 10.3389/fncom.2019.00019
  • BOCCALETTI S, LATORA V, MORENO Y, CHAVEZ M, HWANG D (2006) Complex networks: Structure and dynamics. Physics Reports 424(4–5):175–308. https://doi.org/10.1016/j.physrep.2005.10.00 – 10.1016/j.physrep.2005.10.009
  • Osipov GV, Kurths J, Zhou C (2007) Synchronization in Oscillatory Networks. Springer Berlin Heidelber – 10.1007/978-3-540-71269-5
  • Rosenblum MG, Pikovsky AS, Kurths J (1996) Phase Synchronization of Chaotic Oscillators. Phys Rev Lett 76(11):1804–1807. https://doi.org/10.1103/physrevlett.76.180 – 10.1103/physrevlett.76.1804
  • Pikovsky AS, Rosenblum MG, Kurths J (1996) Synchronization in a population of globally coupled chaotic oscillators. Europhys Lett 34(3):165–170. https://doi.org/10.1209/epl/i1996-00433- – 10.1209/epl/i1996-00433-3
  • Parlitz U, Junge L, Lauterborn W, Kocarev L (1996) Experimental observation of phase synchronization. Phys Rev E 54(2):2115–2117. https://doi.org/10.1103/physreve.54.211 – 10.1103/physreve.54.2115
  • Pietras B, Daffertshofer A (2019) Network dynamics of coupled oscillators and phase reduction techniques. Physics Reports 819:1–105. https://doi.org/10.1016/j.physrep.2019.06.00 – 10.1016/j.physrep.2019.06.001
  • Fell J, Axmacher N (2011) The role of phase synchronization in memory processes. Nat Rev Neurosci 12(2):105–118. https://doi.org/10.1038/nrn297 – 10.1038/nrn2979
  • Dahms T, Lehnert J, Schöll E (2012) Cluster and group synchronization in delay-coupled networks. Phys Rev E 86(1). https://doi.org/10.1103/physreve.86.01620 – 10.1103/physreve.86.016202
  • Jalan S, Amritkar RE (2003) Self-Organized and Driven Phase Synchronization in Coupled Maps. Phys Rev Lett 90(1). https://doi.org/10.1103/physrevlett.90.01410 – 10.1103/physrevlett.90.014101
  • Amritkar RE, Jalan S (2003) Self-organized and driven phase synchronization in coupled map networks. Physica A: Statistical Mechanics and its Applications 321(1–2):220–225. https://doi.org/10.1016/s0378-4371(02)01750- – 10.1016/s0378-4371(02)01750-8
  • Abarbanel HDI, Rulkov NF, Sushchik MM (1996) Generalized synchronization of chaos: The auxiliary system approach. Phys Rev E 53(5):4528–4535. https://doi.org/10.1103/physreve.53.452 – 10.1103/physreve.53.4528
  • Zheng Z, Hu G (2000) Generalized synchronization versus phase synchronization. Phys Rev E 62(6):7882–7885. https://doi.org/10.1103/physreve.62.788 – 10.1103/physreve.62.7882
  • Femat R, Solís-Perales G (2002) Synchronization of chaotic systems with different order. Phys Rev E 65(3). https://doi.org/10.1103/physreve.65.03622 – 10.1103/physreve.65.036226
  • Bowong S, McClintock PVE (2006) Adaptive synchronization between chaotic dynamical systems of different order. Physics Letters A 358(2):134–141. https://doi.org/10.1016/j.physleta.2006.05.00 – 10.1016/j.physleta.2006.05.006
  • Bowong S (2004) Stability analysis for the synchronization of chaotic systems with different order: application to secure communications. Physics Letters A 326(1–2):102–113. https://doi.org/10.1016/j.physleta.2004.04.00 – 10.1016/j.physleta.2004.04.004
  • Kobiolka J, Habermann J, Yamakou ME (2024) Reduced-order adaptive synchronization in a chaotic neural network with parameter mismatch: a dynamical system versus machine learning approach. Nonlinear Dyn 113(10):10989–11008. https://doi.org/10.1007/s11071-024-10821- – 10.1007/s11071-024-10821-6
  • Miao, Increasing-order projective synchronization of chaotic systems with time delay. Chin. Phys. Lett. (2009)
  • Al-sawalha MM, Noorani MSM (2011) Adaptive Increasing-Order Synchronization and Anti-Synchronization of Chaotic Systems with Uncertain Parameters. Chinese Phys Lett 28(11):110507. https://doi.org/10.1088/0256-307x/28/11/11050 – 10.1088/0256-307x/28/11/110507
  • Pecora LM, Carroll TL (2015) Synchronization of chaotic systems. Chaos: An Interdisciplinary Journal of Nonlinear Science 25(9). https://doi.org/10.1063/1.491738 – 10.1063/1.4917383
  • Yamakou ME, Desroches M, Rodrigues S (2023) Synchronization in STDP-driven memristive neural networks with time-varying topology. J Biol Phys 49(4):483–507. https://doi.org/10.1007/s10867-023-09642- – 10.1007/s10867-023-09642-2
  • Fujisaka H, Yamada T (1983) Stability Theory of Synchronized Motion in Coupled-Oscillator Systems. Progress of Theoretical Physics 69(1):32–47. https://doi.org/10.1143/ptp.69.3 – 10.1143/ptp.69.32
  • Yamakou EM, Inack EM, Moukam Kakmeni FM (2015) Ratcheting and energetic aspects of synchronization in coupled bursting neurons. Nonlinear Dyn 83(1–2):541–554. https://doi.org/10.1007/s11071-015-2346- – 10.1007/s11071-015-2346-0
  • Tang Y, Qian F, Gao H, Kurths J (2014) Synchronization in complex networks and its application – A survey of recent advances and challenges. Annual Reviews in Control 38(2):184–198. https://doi.org/10.1016/j.arcontrol.2014.09.00 – 10.1016/j.arcontrol.2014.09.003
  • Boccaletti S, Kurths J, Osipov G, Valladares DL, Zhou CS (2002) The synchronization of chaotic systems. Physics Reports 366(1–2):1–101. https://doi.org/10.1016/s0370-1573(02)00137- – 10.1016/s0370-1573(02)00137-0
  • Arenas A, Díaz-Guilera A, Kurths J, Moreno Y, Zhou C (2008) Synchronization in complex networks. Physics Reports 469(3):93–153. https://doi.org/10.1016/j.physrep.2008.09.00 – 10.1016/j.physrep.2008.09.002
  • Protachevicz PR, Hansen M, Iarosz KC, Caldas IL, Batista AM, Kurths J (2021) Emergence of Neuronal Synchronisation in Coupled Areas. Front Comput Neurosci 15. https://doi.org/10.3389/fncom.2021.66340 – 10.3389/fncom.2021.663408
  • Borges FS, Protachevicz PR, Lameu EL, Bonetti RC, Iarosz KC, Caldas IL, Baptista MS, Batista AM (2017) Synchronised firing patterns in a random network of adaptive exponential integrate-and-fire neuron model. Neural Networks 90:1–7. https://doi.org/10.1016/j.neunet.2017.03.00 – 10.1016/j.neunet.2017.03.005
  • Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378:686–707. https://doi.org/10.1016/j.jcp.2018.10.04 – 10.1016/j.jcp.2018.10.045
  • Cuomo S, Di Cola VS, Giampaolo F, Rozza G, Raissi M, Piccialli F (2022) Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next. J Sci Comput 92(3). https://doi.org/10.1007/s10915-022-01939- – 10.1007/s10915-022-01939-z
  • Savaliya D, Yamakou ME (2026) Self-induced stochastic resonance: A physics-informed machine learning approach. Chaos, Solitons & Fractals 207:117998. https://doi.org/10.1016/j.chaos.2026.11799 – 10.1016/j.chaos.2026.117998
  • Greydanus, Hamiltonian neural networks. (2019)
  • Finzi, Simplifying hamiltonian and lagrangian neural networks via explicit constraints. (2020)
  • Chen, Neural symplectic form: learning hamiltonian equations on general coordinate systems. (2021)
  • van der Schaft, Port-Hamiltonian Systems Theory: An Introductory Overview. (2014)
  • Desai SA, Mattheakis M, Sondak D, Protopapas P, Roberts SJ (2021) Port-Hamiltonian neural networks for learning explicit time-dependent dynamical systems. Phys Rev E 104(3). https://doi.org/10.1103/physreve.104.03431 – 10.1103/physreve.104.034312
  • Neary, Compositional learning of dynamical system models using port-hamiltonian neural networks. (2023)
  • Moradi S, Beintema GI, Jaensson NO, Tóth R, Schoukens M (2026) Port-Hamiltonian neural networks with output error noise models. Automatica 187:112892. https://doi.org/10.1016/j.automatica.2026.11289 – 10.1016/j.automatica.2026.112892
  • Persio LD, Ehrhardt M, Outaleb Y, Rizzotto S (2025) Port-Hamiltonian Neural Networks: From Theory to Simulation of Interconnected Stochastic System – 10.21203/rs.3.rs-7572939/v1
  • Ortega, Learnability of linear port-Hamiltonian systems. J. Mach. Learn. Res. (2024)
  • Hindmarsh, A model of neuronal bursting using three coupled first order differential equations. Proc. R. Soc. Lond. Ser. B Biol. Sci. (1984)
  • Selverston AI, Rabinovich MI, Abarbanel HDI, Elson R, Szücs A, Pinto RD, Huerta R, Varona P (2000) Reliable circuits from irregular neurons: A dynamical approach to understanding central pattern generators. Journal of Physiology-Paris 94(5–6):357–374. https://doi.org/10.1016/s0928-4257(00)01101- – 10.1016/s0928-4257(00)01101-3
  • Lv M, Wang C, Ren G, Ma J, Song X (2016) Model of electrical activity in a neuron under magnetic flow effect. Nonlinear Dyn 85(3):1479–1490. https://doi.org/10.1007/s11071-016-2773- – 10.1007/s11071-016-2773-6
  • Yamakou ME (2020) Chaotic synchronization of memristive neurons: Lyapunov function versus Hamilton function. Nonlinear Dyn 101(1):487–500. https://doi.org/10.1007/s11071-020-05715- – 10.1007/s11071-020-05715-2
  • Zhang J, Li Z (2024) Switchable memristor-based Hindmarsh-Rose neuron under electromagnetic radiation. Nonlinear Dyn 112(8):6647–6662. https://doi.org/10.1007/s11071-024-09399- – 10.1007/s11071-024-09399-w
  • Ma J, Wang Y, Wang C, Xu Y, Ren G (2017) Mode selection in electrical activities of myocardial cell exposed to electromagnetic radiation. Chaos, Solitons & Fractals 99:219–225. https://doi.org/10.1016/j.chaos.2017.04.01 – 10.1016/j.chaos.2017.04.016
  • Tsitouras Ch (2011) Runge–Kutta pairs of order 5(4) satisfying only the first column simplifying assumption. Computers & Mathematics with Applications 62(2):770–775. https://doi.org/10.1016/j.camwa.2011.06.00 – 10.1016/j.camwa.2011.06.002
  • Krasovskii, (1963)
  • Kobe DH (1986) Helmholtz’s theorem revisited. American Journal of Physics 54(6):552–554. https://doi.org/10.1119/1.1456 – 10.1119/1.14562
  • Wang, Calculation of Hamilton energy function of dynamical system by using Helmholtz theorem. Acta Phys. Sin. (2016)
  • Ma J, Wu F, Jin W, Zhou P, Hayat T (2017) Calculation of Hamilton energy and control of dynamical systems with different types of attractors. Chaos: An Interdisciplinary Journal of Nonlinear Science 27(5). https://doi.org/10.1063/1.498346 – 10.1063/1.4983469
  • Loshchilov, Decoupled weight decay regularization. (2019)
  • Akiba, Optuna: a next-generation hyperparameter optimization framework. (2019)