However, translating the complex behavior of living organisms into mathematical equations and then realizing them as engineered physical hardware—whether silicon circuits or photonic devices—has faced a formidable barrier. The strict constraints built into biological slime mold models had severely limited their compatibility with diverse physical systems, including electronic circuits and optical devices.
A research team led by Assistant Professor Yusuke Miyajima (at the time of the research) and Professor Tadashi Mochizuki of Waseda University's Faculty of Science and Engineering fundamentally reexamined the components of this slime-mold-type algorithm and thoroughly stripped away the mathematical restrictions that had been blocking hardware implementation, constructing a new mathematical model. This model, simplified into a recurrence equation that can be directly mapped onto physical phenomena, demonstrated search performance in computer simulations that far exceeded that of previous models.
This research was published online on June 9, 2026, in the peer-reviewed journal Physical Review Research (Vol. 8, Issue 2, Article 023269, DOI: 10.1103/zgvb-cfpg). The paper presents a simplification of the mathematical model and a demonstration of its performance through numerical simulation; it does not involve fabricating and physically measuring an actual device. Even so, it offers a clear design roadmap serving as a theoretical foundation for bridging bio-inspired computing to real physical devices.
The Dead End Created by Protoplasmic Flow Conservation Laws
The traveling salesperson problem (TSP), a representative example of combinatorial optimization problems, asks for the shortest possible route that visits a set of specified cities exactly once each and returns to the starting point. Because the number of possible routes explodes combinatorially as the number of cities increases even slightly, exhaustively searching for solutions on a conventional von Neumann computer causes computation time and power consumption to balloon astronomically.
In 2018, a research group led by Professor Masashi Aono proposed a "slime mold TSP model" that solves the TSP by placing a living true slime mold (Physarum polycephalum) inside a star-shaped chip with radial channels and applying feedback control via light stimulation (bounce-back control) (Royal Society Open Science, DOI: 10.1098/rsos.180396). Since slime mold has an aversion to light, when an inappropriate route is about to be selected, light is locally applied to make the organism avoid that path. As the slime mold's protoplasm competes among the various channels and settles into a stable volume distribution, a near-shortest tour route emerges as a result.
This biological experiment and the accompanying early mathematical model demonstrated a pathway for converting a self-organizing phenomenon found in nature into an optimization algorithm. However, when attempts were made to transplant this mechanism from a living organism into inorganic hardware devices, serious constraints surfaced. The largest obstacle was the existence of a volume conservation law—the principle that "the total volume of the entire cell is always kept constant"—embedded in the biological slime mold.
When a volume conservation law is built into a mathematical model, a change in the state of one element instantaneously affects the sum total of the entire system, requiring the whole system to continuously reconcile itself. Faithfully reproducing this kind of global constraint in a physical circuit restricts the design to specialized analog electrical circuits in which the total charge within a closed loop is conserved. As a result, the range of usable physical phenomena and materials was severely narrowed.
Furthermore, the conventional slime mold TSP model contained numerous operations that made hardware implementation difficult. These included nonlinear sigmoid functions used to express sharp rises in state, conditional "if-then" branching processes based on state values, and counting operations that tally the number of elements satisfying certain conditions. While these can be written in just a few lines of code in a digital computer program, directly implementing them through continuous physical dynamics demands extremely complex control circuitry.
In research published in 2024 (NOLTA, DOI: 10.1587/nolta.2.1101), Assistant Professor Miyajima and colleagues proposed an "improved slime mold TSP model" and theoretically demonstrated that solution search could still function even when the volume conservation law was partially relaxed. The current study pushes that direction to its logical extreme, completely dismantling and reconstructing the five mathematical elements that had been obstructing hardware implementation.
Five Mathematical Modifications That Connect Directly to Physical Phenomena
The technical core of the "slime mold TSP recurrence-equation model" announced by the Waseda team lies in eliminating the global constraints and complex logical operations that arose from biological mimicry, converting the system into a simple recurrence equation that is self-contained through purely local interactions. Specifically, the research team introduced the following five modifications.
First, they completely eliminated the volume conservation law, which had been the biggest constraint. By redesigning the system so that each element updates its state locally, referencing only information from its immediate neighbors, the need to instantaneously monitor and constrain the sum total of the entire system was removed.
Second, they replaced the sigmoid function with a constant or a simple linear function. The sigmoid function is difficult to reproduce with high precision using analog components and had been a factor driving up circuit complexity; simplifying it substantially reduced the burden at the device level.
Third, they completely eliminated conditional branching processing. The if-then structure, which switches processing depending on whether a state value exceeds a threshold, was removed, and the system was unified into a form that can be consistently described by a single mathematical dynamic equation.
Fourth, they removed the counting operation that tallies the number of nodes satisfying a specific condition. By eliminating global aggregation processing, the framework was revised so that state transitions occur through simple addition and subtraction of local couplings alone.
Fifth, they introduced random noise following a Gaussian distribution to prevent the search from stalling. As fluctuations designed to prevent the optimization algorithm from becoming trapped in local minima (local traps), this allows the thermal noise (thermal fluctuation) inherently present in physical devices to be used directly.
Through these modifications, the slime mold's search dynamics were condensed into a lean recurrence equation. The formula includes tunable parameters that adjust its behavioral characteristics, allowing performance to be flexibly optimized according to the size and difficulty of the problem being solved.
Lead author Assistant Professor Miyajima explained the engineering significance of the work: "Our approach eliminates the volume conservation law, which was the biggest constraint, making it possible to implement slime mold computers using a much wider variety of materials and physical phenomena."
| Comparison Item | Original Slime Mold TSP Model (2018) | Improved Slime Mold TSP Model (2024) | Proposed Recurrence-Equation Model (2026) |
|---|---|---|---|
| Maximum number of cities handled (simulation) | 30 cities | 100 cities | 180 cities |
| Volume conservation constraint | Required (strict sum-total constraint) | Relaxed (partially dependent) | Completely eliminated |
| Conditional branching | Numerous | Present | None (described by a single equation) |
| Sigmoid function | Used | Used | Replaced with constants or simple functions |
| Assumed noise source | Uniform random numbers, etc. | Uniform random numbers, etc. | Gaussian random numbers, compatible with thermal fluctuation |
| Candidate hardware for implementation | Analog circuits relying on charge conservation | Analog electronic circuits | Spintronics, photonic circuits, fluidic systems, etc. |
Simulation Experiments Show Faster Convergence and Greater Scale
Simplifying the equations to this extreme degree did more than just improve the feasibility of hardware implementation. In numerical simulations run on a digital computer, the new recurrence-equation model recorded a substantial improvement in search performance itself.
The research team verified the model's search dynamics using standard benchmark problems for the traveling salesperson problem. The results showed that, compared to both the original slime mold TSP model and the 2024 improved model, the new recurrence-equation model dramatically reduced the number of iterations required to reach a valid route (a feasible solution) that visits every city exactly once without duplication.
In the numerical simulation benchmarks, the proposed model reached feasible solutions with far fewer iterations than the conventional models. Furthermore, the scale of problems that can be handled at once (scalability) also expanded. While the original slime mold TSP model could stably search for solutions only up to roughly 30 cities, and the 2024 improved model raised that ceiling to 100 cities, the new recurrence-equation model successfully derived valid solutions for TSP instances involving up to 180 cities.
It is important to note that these figures are not empirical measurements obtained from a fabricated physical dedicated chip, but rather theoretical benchmarks based on numerical simulation on a digital computer. The successful solution search at 180 cities is strictly a simulation result on a specific TSP benchmark and does not guarantee that the model is immediately universally applicable to every combinatorial optimization problem. Nevertheless, the fact that stripping away complex operations from the model still numerically demonstrated superior performance across solution quality, convergence speed, and scalability provides a strong guideline for physical device design.
Mathematical Equivalence to Recurrent Neural Networks
Another theoretical discovery in this research is that, as a result of simplifying the slime mold's search dynamics, the resulting recurrence equation turned out to be mathematically equivalent to a recurrent neural network (RNN) with fixed connection weights.
In this model, each connection weight within the network is predetermined and fixed as a value representing information such as the distance between cities or a penalty term that prevents the same city from being visited more than once. This is not a framework in which weights are learned (trained) using large amounts of training data through backpropagation, as in typical machine learning neural networks. Instead, within a network with fixed weights, nonlinear dynamics and thermal noise interact and evolve over time, self-organizing into a state of lower overall system energy—and as a result, a solution to the optimization problem emerges.
The paper notes that this mathematical equivalence "suggests the possibility that information processing in slime mold and neural networks share a common computational principle." This does not mean it has been biologically proven that living true slime mold possesses a brain-like neural network inside its body. Rather, it is a discovery that, as the bio-inspired algorithm was stripped down into a form more compatible with physical laws, it naturally converged with the mathematical structure of neural networks—a foundational theory in artificial intelligence and statistical physics. This equivalence firmly situates slime-mold-type computing within a theoretically systematized mathematical framework, rather than leaving it as empirical biomimicry alone.
The Path to Physical Implementation and Remaining Challenges
With engineering barriers such as the volume conservation law and the sigmoid function removed from the mathematical model, the range of candidate platforms for physically realizing the slime-mold-type algorithm has expanded significantly.
The most promising implementation form proposed by the research team in the paper is a hardware configuration using "spintronic devices," which utilize the spin (magnetic property) of electrons. Spintronics technology boasts high stability against heat and radiation and, like non-volatile memory, can retain its state even when the power is switched off—raising the possibility of building a dedicated optimization accelerator with minimal standby power and ultra-low overall power consumption. The paper also presents a conceptual example of a concrete implementation architecture using spintronic devices.
Beyond spintronics, the model's reduced constraints also open the door to a variety of other materials—photonic integrated circuits (optical circuits) that perform instantaneous multiply-accumulate operations using light interference and diffraction, microfluidic devices built with microfabrication technology, and arrays of mechanical microoscillators—any system capable of leveraging local physical interactions and thermal fluctuation.
Professor Mochizuki commented: "Our more flexible model will likely accelerate the development of energy-efficient slime mold computers. This distributed style of information processing may prove valuable for AI and large-scale combinatorial optimization tasks—areas where conventional computers currently consume enormous amounts of power."
However, the "slime mold computer" referred to here is not a biocomputer operated using living slime mold, but rather a non-von-Neumann computing architecture inspired by the operating principles of slime mold. And what this research has completed is the stage of proposing the mathematical model and verifying the principle through numerical simulation—an actual functioning physical hardware device has not yet been fabricated.
To integrate spintronic devices or photonic circuits into a practical optimization chip, numerous engineering challenges must still be overcome, including high-density wiring design between devices, resilience against variations in device characteristics arising from the manufacturing process, and the integration of peripheral circuitry for rapidly inputting problem data from outside and reading out the resulting solutions with low latency.
Lead author Miyajima left Waseda University at the end of June 2026 and relocated to a new research environment as a JST CREST researcher at Osaka University. Whether this mathematical model—refined from a bio-inspired concept into a recurrence equation faithful to physical laws—can actually be realized as an ultra-low-power physical processor will be the focus of the next stage of verification.