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Mathematical model of the amoeba-inspired combinatorial optimization machine for physical implementation and its equivalence to recurrent neural networks

Yusuke Miyajima* and Masahito Mochizuki†

  • Department of Applied Physics, Waseda University, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan

  • *Contact author: y.m-ichikawa@akane.waseda.jp
  • †Contact author: masa_mochizuki@waseda.jp
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Phys. Rev. Research 8, 023269 – Published 9 June, 2026

DOI: https://doi.org/10.1103/zgvb-cfpg

Abstract

Information-processing algorithms inspired by survival strategies of slime molds have been intensively studied to realize combinatorial optimization machines with high computational efficiency and low power consumption. However, previously proposed mathematical models are not suitable for physical implementation as they involved constraints due to the volume-conservation law of slime molds, multiple conditional branches, and numerous complex functions such as sigmoid functions. Here, we propose a mathematical model, the Amoeba TSP (traveling salesman problem) recurrence formula model, which simplifies the information-processing procedure and enables physical implementation without sacrificing optimization performance. Specifically, we demonstrate using numerical simulations that (1) the conservation-law constraints assumed in previous models can be removed, (2) several sigmoid functions can be eliminated, and (3) Gaussian-distribution random numbers that can be generated from thermal fluctuations can be employed as random fluctuations. These improvements broaden the range of candidate materials and phenomena for physical implementation and reduce the number of device components required. Furthermore, we discover that our mathematical model integrating these modifications is mathematically equivalent to a recurrent neural network with fixed weights that exploits nonlinear dynamics. Remarkably, despite its significant simplification, the proposed model exhibits superior solution-search performance compared with the previous models in terms of solution quality, convergence speed, and scalability. Our results open a practical pathway toward realization of highly efficient and high-performance combinatorial optimization machines based on non-von Neumann-type architectures and provide insights into bioinspired information processing by uncovering a connection between the computational principles of slime molds and recurrent neural networks.

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References (33)

  1. J. L. Hennessy and D. A. Patterson, Computer Architecture: A Quantitative Approach, 6th ed. (Morgan Kaufmann Publishers, Boston, MA, 2017).
  2. T. Kadowaki and H. Nishimori, Quantum annealing in the transverse Ising model, Phys. Rev. E 58, 5355 (1998).
  3. M. W. Johnson, M. H. S. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A. J. Berkley, J. Johansson, P. Bunyk, et al., Quantum annealing with manufactured spins, Nature (London) 473, 194 (2011).
  4. A. Lucas, Ising formulations of many NP problems, Front. Phys. 2, 1 (2014).
  5. M. Yamaoka, C. Yoshimura, M. Hayashi, T. Okuyama, H. Aoki, and H. Mizuno, A 20k-spin Ising chip to solve combinatorial optimization problems with CMOS annealing, IEEE J. Solid-State Circuits 51, 303 (2016).
  6. T. Inagaki, Y. Haribara, K. Igarashi, T. Sonobe, S. Tamate, T. Honjo, A. Marandi, P. L. McMahon, T. Umeki, K. Enbutsu, O. Tadanaga, H. Takenouchi, K. Aihara, K. Kawarabayashi, K. Inoue, S. Utsunomiya, and H. Takesue, A coherent Ising machine for 2000-node optimization problems, Science 354, 603 (2016).
  7. M. Aramon, G. Rosenberg, E. Valiante, T. Miyazawa, H. Tamura, and H. G. Katzgraber, Physics-inspired optimization for quadratic unconstrained problems using a digital annealer, Front. Phys. 7, 48 (2019).
  8. H. Goto, K. Tatsumura, and A. R. Dixon, Combinatorial optimization by simulating adiabatic bifurcations in nonlinear Hamiltonian systems, Sci. Adv. 5, eaav2372 (2019).
  9. D. Kessler, Plasmodial structure and motility, in Cell Biology of Physarum and Didymium, edited by H. C. Aldrich and J. W. Daniel (Academic Press, New York, 1982), Vol. 1, p. 145.
  10. A. Adamatzky (ed.), Advances in Physarum Machines: Sensing and Computing with Slime Mould (Springer, Cham, 2016).
  11. K. Iwayama, L. Zhu, Y. Hirata, M. Aono, M. Hara, and K. Aihara, Decision-making ability of Physarum polycephalum enhanced by its coordinated spatiotemporal oscillatory dynamics, Bioinspiration Biomimetics 11, 036001 (2016).
  12. S. Bajpai, M. Aono, and P. Kurian, Tracking and distinguishing slime mold solutions to the traveling salesman problem through synchronized amplification in the non-equilibrium steady state, arXiv:2504.03492.
  13. T. Nakagaki, H. Yamada, and Á. Tóth, Maze-solving by an amoeboid organism, Nature (London) 407, 470 (2000).
  14. A. Tero, S. Takagi, T. Saigusa, K. Ito, D. P. Bebber, M. D. Fricker, K. Yumiki, R. Kobayashi, and T. Nakagaki, Rules for biologically inspired adaptive network design, Science 327, 439 (2010).
  15. M. Aono, M. Hara, and K. Aihara, Amoeba-based neurocomputing with chaotic dynamics, Commun. ACM 50, 69 (2007).
  16. M. Aono, Y. Hirata, M. Hara, and K. Aihara, Amoeba-based chaotic neurocomputing: Combinatorial optimization by coupled biological oscillators, New Gener. Comput. 27, 129 (2009).
  17. L. Zhu, M. Aono, S-J. Kim, and M. Hara, Amoeba-based computing for traveling salesman problem: Long-term correlations between spatially separated individual cells of Physarum polycephalum, Biosystems 112, 1 (2013).
  18. M. Aono, Amoeba-inspired combinatorial optimization machines, Jpn. J. Appl. Phys. 59, 060502 (2020).
  19. D. L. Applegate, R. E. Bixby, V. Chvatal, and W. J. Cook, The Traveling Salesman Problem: A Computational Study (Princeton University Press, Princeton, NJ, 2006).
  20. M. Aono, S-J. Kim, L. Zhu, M. Naruse, M. Ohtsu, H. Hori, and M. Hara, Amoeba-inspired SAT solver, in Proceedings of the 2012 International Symposium on Nonlinear Theory and its Applications (NOLTA2012) (IEICE, Majorca, 2012), pp. 586–589.
  21. S. Kasai, M. Aono, and M. Naruse, Amoeba-inspired computing architecture implemented using charge dynamics in parallel capacitance network, Appl. Phys. Lett. 103, 163703 (2013).
  22. Y. Hara-Azumi, N. Takeuchi, K. Hara, and M. Aono, Digital bio-inspired satisfiability solver leveraging fluctuations, Jpn. J. Appl. Phys. 59, 040603 (2020).
  23. M. Aono, M. Naruse, S-J. Kim, M. Wakabayashi, H. Hori, M. Ohtsu, and M. Hara, Amoeba-inspired nanoarchitectonic computing: Solving intractable computational problems using nanoscale photoexcitation transfer dynamics, Langmuir 29, 7557 (2013).
  24. N. Takeuchi, M. Aono, and N. Yoshikawa, Superconductor amoeba-inspired problem solvers for combinatorial optimization, Phys. Rev. Appl. 11, 044069 (2019).
  25. M. Aono, S. Kasai, S-J. Kim, M. Wakabayashi, H. Miwa, and M. Naruse, Amoeba-inspired nanoarchitectonic computing implemented using electrical Brownian ratchets, Nanotechnology 26, 234001 (2015).
  26. K. Saito, M. Aono, and S. Kasai, Amoeba-inspired analog electronic computing system integrating resistance crossbar for solving the travelling salesman problem, Sci. Rep. 10, 20772 (2020).
  27. K. Saito and S. Kasai, Effect of feedback delays on solution quality in amoeba-inspired computing system that solves traveling salesman problem, Appl. Phys. Express 13, 114501 (2020).
  28. L. Zhu, S-J. Kim, M. Hara, and M. Aono, Remarkable problem-solving ability of unicellular amoeboid organism and its mechanism, R. Soc. Open Sci. 5, 180396 (2018).
  29. Y. Miyajima and M. Mochizuki, Proposed modified computational model for the amoeba-inspired combinatorial optimization machine, NOLTA, IEICE 15, 824 (2024).
  30. J. J. Hopfield and D. W. Tank, Computing with neural circuits: A model, Science 233, 625 (1986).
  31. H. Jaeger, The “echo state” approach to analysing and training recurrent neural networks-with a erratum note, GMD Technical Report 148, German National Research Center for Information Technology, Bonn, Germany, 2001, p. 34.
  32. K. Nakajima, Physical reservoir computing—An introductory perspective, Jpn. J. Appl. Phys. 59, 060501 (2020).
  33. K. Nakajima and I. Fischer, Reservoir Computing: Theory, Physical Implementations, and Applications (Springer, Berlin, 2021).

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Phys. Rev. Research 8, 023269– Published 9 June, 2026
  • Received 21 October 2025
  • Accepted 8 May 2026
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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.

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