- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Brain-inspired, interpretable, resonant recurrent neural networks

Mark A. Kramer*

  • Department of Mathematics and Statistics and Center for Systems Neuroscience, Boston University, Boston, Massachusetts 02215, USA and Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, Maryland 21218, USA

  • *Contact author: mkramer@jhu.edu
PDF

Phys. Rev. Research 8, 023237 – Published 1 June, 2026

DOI: https://doi.org/10.1103/phlp-rhny

Abstract

Traditional artificial neural networks consist of nodes with nonoscillatory dynamics. Biological neural networks, on the other hand, consist of oscillatory components embedded in an oscillatory environment. Motivated by this feature of biological neurons, we describe a neural network framework with explicit damped, oscillatory node dynamics. We express the oscillatory dynamics using two history-dependent terms to connect these dynamics with standard recurrent neural network formulations, and apply physical constraints from observed brain dynamics to choose the oscillatory frequencies and stationary constraints to reduce the number of free parameters. We then optimize and illustrate network performance by classifying hand-written digits and simulated neuronal spike train activity and show that these oscillatory network elements support accurate classification with few trainable parameters. Choosing oscillator frequencies according to a proposed theory for brain rhythms improves classification accuracy compared to alternative frequency configurations and compared to standard recurrent neural network frameworks with comparable numbers of parameters. Compared to existing approaches, the proposed resonant recurrent network (RRN) utilizes oscillatory dynamics expressed as a straightforward extension of standard recurrent neural networks, produces interpretable features for classification, and performs well with few parameters when oscillator frequencies follow a configuration observed in vivo. We propose that RRNs may serve as efficient, biologically inspired building blocks to achieve complex goals in biological and artificial neural networks.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (72)

  1. A. B. L. Tort, J. Brankačk, and A. Draguhn, Respiration-entrained brain rhythms are global but often overlooked, Trends Neurosci. 41, 186 (2018).
  2. E. Marder and D. Bucher, Central pattern generators and the control of rhythmic movements, Curr. Biol. 11, R986 (2001).
  3. E. Marder and D. Bucher, Understanding circuit dynamics using the stomatogastric nervous system of lobsters and crabs, Annu. Rev. Physiol. 69, 291 (2007).
  4. G. Buzsaki, Rhythms of the Brain (Oxford University Press, New York, NY, USA, 2011).
  5. B. P. Staresina, Coupled sleep rhythms for memory consolidation, Trends Cogn. Sci. 28, 339 (2024).
  6. M. Lundqvist, E. K. Miller, J. Nordmark, J. Liljefors, and P. Herman, Beta: Bursts of cognition, Trends Cogn. Sci. 28, 662 (2024).
  7. G. Buzsáki and E. I. Moser, Memory, navigation and theta rhythm in the hippocampal-entorhinal system, Nat. Neurosci. 16, 130 (2013).
  8. W. McCulloch and W. Pitts, A logical calculus of the ideas immanent in nervous activity, Bull. Math. Biol. 5, 115 (1943).
  9. J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. USA 79, 2554 (1982).
  10. F. Rosenblatt, The perceptron: A probabilistic model for information storage and organization in the brain, Psychol. Rev. 65, 386 (1958).
  11. D. H. Hubel and T. N. Wiesel, Receptive fields of single neurones in the cat’s striate cortex, J. Physiol. 148, 574 (1959).
  12. Y. LeCun and Y. Bengio, Convolutional networks for images, speech, and time series, in The Handbook of Brain Theory and Neural Networks, 2nd ed., edited by M. A. Arbib (MIT Press, Cambridge, USA, 1998), p. 255.
  13. W. Singer, The cerebral cortex: A delay-coupled recurrent oscillator network?, in Reservoir Computing: Theory, Physical Implementations, and Applications, edited by K. Nakajima and I. Fischer, (Springer Nature Singapore, Singapore, 2021), pp. 3–28.
  14. G. Hahn, A. Ponce-Alvarez, G. Deco, A. Aertsen, and A. Kumar, Portraits of communication in neuronal networks, Nat. Rev. Neurosci. 20, 117 (2019).
  15. R. F. Helfrich, A. Breska, and R. T. Knight, Neural entrainment and network resonance in support of top-down guided attention, Curr. Opin. Psychol. 29, 82 (2019).
  16. P. Fries, A mechanism for cognitive dynamics: Neuronal communication through neuronal coherence, Trends Cogn. Sci. 9, 474 (2005).
  17. P. Fries, Rhythms for cognition: Communication through coherence, Neuron 88, 220 (2015).
  18. R. T. Canolty and R. T. Knight, The functional role of cross-frequency coupling, Trends Cogn. Sci. 14, 506 (2010).
  19. A. Hyafil, A.-L. Giraud, L. Fontolan, and B. Gutkin, Neural cross-frequency coupling: Connecting architectures, mechanisms, and functions, Trends Neurosci. 38, 725 (2015).
  20. K. Duecker, M. Idiart, M. van Gerven, and O. Jensen, Oscillations in an artificial neural network convert competing inputs into a temporal code, PLoS Comput. Biol. 20, e1012429 (2024).
  21. T. K. Rusch and S. Mishra, Coupled Oscillatory Recurrent Neural Network (coRNN): An accurate and (gradient) stable architecture for learning long time dependencies, International Conference on Learning Representations (2021).
  22. F. Effenberger, P. Carvalho, I. Dubinin, and W. Singer, The functional role of oscillatory dynamics in neocortical circuits: A computational perspective, Proc. Natl. Acad. Sci. USA 122, e2412830122 (2025).
  23. X.-J. Wang, Neurophysiological and computational principles of cortical rhythms in cognition, Physiol. Rev. 90, 1195 (2010).
  24. J. L. Elman, Finding structure in time, Cogn. Sci. 14, 179 (1990).
  25. K. Funahashi and Y. Nakamura, Approximation of dynamical systems by continuous time recurrent neural networks, Neural Networks 6, 801 (1993).
  26. K. Rajan, L. F. Abbott, and H. Sompolinsky, Stimulus-dependent suppression of chaos in recurrent neural networks, Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 82, 011903 (2010).
  27. G. Holzmann and H. Hauser, Echo state networks with filter neurons and a delay&sum readout, Neural Networks 23, 244 (2010).
  28. F. Wyffels, B. Schrauwen, D. Verstraeten, and D. Stroobandt, Band-pass reservoir computing, in 2008 IEEE International Joint Conference on Neural Networks (IEEE World Congress on Computational Intelligence), Hong Kong, China (IEEE, Piscataway, NJ, USA, 2008), pp. 3204–3209.
  29. L. H. B. Liboni et al., Image segmentation with traveling waves in an exactly solvable recurrent neural network, Proc. Natl. Acad. Sci. USA 122, e2321319121 (2025).
  30. A. Ceni et al., Random oscillators network for time series processing, in Proceedings of the 27th International Conference on Artificial Intelligence and Statistics (PMLR, 2024), pp. 4807–4815.
  31. S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D 143, 1 (2000).
  32. M. A. Kramer, Golden rhythms as a theoretical framework for cross-frequency organization, Neuron Behav. Data Anal. Theory 1 (2022).
  33. T. Akam and D. M. Kullmann, Oscillatory multiplexing of population codes for selective communication in the mammalian brain, Nat. Rev. Neurosci. 15, 111 (2014).
  34. K. A. Ferguson and J. A. Cardin, Mechanisms underlying gain modulation in the cortex, Nat. Rev. Neurosci. 21, 80 (2020).
  35. S. Hochreiter and J. Schmidhuber, Long short-term memory, Neural Comput. 9, 1735 (1997).
  36. K. Cho, B. van Merriënboer, C. Gulcehre, D. Bahdanau, F. Bougares, H. Schwenk, and Y. Bengio, Learning phrase representations using RNN encoder-decoder for statistical machine translation, Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP), edited by A. Moschitti, B. Pang, and W. Daelemans (Association for Computational Linguistics, Doha, Qatar, 2014).
  37. D. Percival and A. Walden, Spectral analysis for physical applications (Cambridge University Press, Cambridge, UK, 1993)
  38. G. Buzsaki and A. Draguhn, Neuronal oscillations in cortical networks, Science 304, 1926 (2004).
  39. A. K. Roopun et al., Temporal interactions between cortical rhythms, Front. Neurosci. 2, 145 (2008).
  40. M. A. Kramer, A. K. Roopun, L. M. Carracedo, R. D. Traub, M. A. Whittington, and N. J. Kopell, Rhythm generation through period concatenation in rat somatosensory cortex, PLoS Comput. Biol. 4, e1000169 (2008).
  41. M. Penttonen and G. Buzsáki, Natural logarithmic relationship between brain oscillators, Thalamus Relat. Syst. 2, 145 (2003).
  42. W. Klimesch, An algorithm for the EEG frequency architecture of consciousness and brain body coupling, Front. Hum. Neurosci. 7, 1 (2013).
  43. H. Jaeger, The ‘echo state’ approach to analysing and training recurrent neural networks—With an Erratum note, GMD Rep. 148, 1 (2001).
  44. W. Maass, T. Natschläger, and H. Markram, Real-time computing without stable states: A new framework for neural computation based on perturbations, Neural Comput. 14, 2531 (2002).
  45. H. Jaeger, M. Lukoševičius, D. Popovici, and U. Siewert, Optimization and applications of echo state networks with leaky-integrator neurons, Neural Networks 20, 335 (2007).
  46. G. Tanaka, T. Matsumori, H. Yoshida, and K. Aihara, Reservoir computing with diverse timescales for prediction of multiscale dynamics, Phys. Rev. Res. 4, L032014 (2022).
  47. C. Gallicchio, A. Micheli, and L. Pedrelli, Design of deep echo state networks, Neural Networks 108, 33 (2018).
  48. E. M. Izhikevich and F. C. Hoppensteadt, Weakly Connected Neural Networks, 1st ed. (Springer, New York, NY, USA, 1997).
  49. T. Aonishi, K. Kurata, and M. Okada, Statistical mechanics of an oscillator associative memory with scattered natural frequencies, Phys. Rev. Lett. 82, 2800 (1999).
  50. A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol. 16, 15 (1967).
  51. Y. Kuramoto, Collective synchronization of pulse-coupled oscillators and excitable units, Physica D 50, 15 (1991).
  52. E. M. Izhikevich, Weakly connected quasi-periodic oscillators, fm interactions, and multiplexing in the brain, SIAM J. Appl. Math. 59, 2193 (1999).
  53. E. M. Izhikevich, Weakly-pulse-coupled oscillators, FM interactions, synchronization, and oscillatory associative memory, IEEE Trans. Neural Networks 10, 508 (1999).
  54. F. C. Hoppensteadt and E. M. Izhikevich, Thalamo-cortical interactions modeled by weakly connected oscillators: Could the brain use FM radio principles? Biosystems 48, 85 (1998).
  55. A. K. Roopun et al., Period concatenation underlies interactions between gamma and beta rhythms in neocortex, Front. Cell. Neurosci. 2, (2008)
  56. T. Natschläger, W. Maass, and H. Markram, The ‘liquid computer’: A novel strategy for real-time computing on time series, Telematik 8, 39 (2002).
  57. C. Fernando and S. Sojakka, Pattern recognition in a bucket, in Advances in Artificial Life, edited by W. Banzhaf, J. Ziegler, T. Christaller, P. Dittrich, and J. T. Kim (Springer, Berlin, 2003), pp. 588–597.
  58. Y. Zhong et al., A memristor-based analogue reservoir computing system for real-time and power-efficient signal processing, Nat. Electron. 5, 672 (2022).
  59. J. Torrejon et al., Neuromorphic computing with nanoscale spintronic oscillators, Nature (London) 547, 428 (2017).
  60. H. Cai et al., Brain organoid reservoir computing for artificial intelligence, Nat. Electron. 6, 1032 (2023).
  61. T. A. Keller, L. Muller, T. Sejnowski, and M. Welling, Traveling waves encode the recent past and enhance sequence learning, The Twelfth International Conference on Learning Representations (2024).
  62. L. Muller, P. S. Churchland, and T. J. Sejnowski, Transformers and cortical waves: Encoders for pulling in context across time, Trends Neurosci. 47, 788 (2024).
  63. F. Damicelli, C. C. Hilgetag, and A. Goulas, Brain connectivity meets reservoir computing, PLoS Comput. Biol. 18, e1010639 (2022).
  64. L. E. Suárez et al., Connectome-based reservoir computing with the conn2res toolbox, Nat. Commun. 15, 656 (2024).
  65. G. Buzsáki and M. Vöröslakos, Brain rhythms have come of age, Neuron 111, 922 (2023).
  66. N. J. Kopell, H. J. Gritton, M. A. Whittington, and M. A. Kramer, Beyond the connectome: The dynome, Neuron 83, 1319 (2014).
  67. K. M. Stiefel and G. B. Ermentrout, Neurons as oscillators, J. Neurophysiol. 116, 2950 (2016).
  68. M. Whittington, R. Traub, N. Kopell, B. Ermentrout, and E. Buhl, Inhibition-based rhythms: Experimental and mathematical observations on network dynamics, Int. J. Psychophysiol. 38, 315 (2000).
  69. D. Kototajlo, S. Alexander, T. Larsen, E. Lifland, and R. Dean, AI 2027, AI Futures Project, 2025.
  70. Y. Lecun, L. Bottou, Y. Bengio, and P. Haffner, Gradient-based learning applied to document recognition, Proc. IEEE 86, 2278 (1998).
  71. A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Köpf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai, and S. Chintala, PyTorch: An imperative style, high-performance deep learning library, Advances in Neural Information Processing Systems, edited by H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett (Curran Associates, Inc., 2019), Vol. 32.
  72. M. Kramer, Mark-Kramer/Resonant-Recurrent-Network, Jupyter Notebook, https://github.com/Mark-Kramer/Resonant-Recurrent-Network, accessed 11 May 2026.

Outline

Information

Phys. Rev. Research 8, 023237– Published 1 June, 2026
  • Received 23 June 2025
  • Accepted 9 May 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation