Threshold sensing yields optimal path formation in Physarum polycephalum

The model organism Physarum polycephalum is known to perform decentralised problem solving despite absence of nervous system. Experimental evidence and modelling studies have linked these abilities, and in particular maze-solving, to some sort of memory and adaptation. However, despite compelling hypotheses, it is still not clear whether the tasks are solved optimally, and which key dynamical mechanisms enable Physarum’s impressive abilities. Here, we employ a circuital network model for the foraging behaviour of Physarum polycephalum to prove that threshold sensing yields the emergence of unique and optimal paths that connect food sources and solve mazes. We also prove which conditions lead to alternative paths, thus elucidating how the organism achieves flexibility and adaptation in a self-organised manner. These findings are aligned with experimental evidences and provide insight into the evolution of primitive intelligence. Our results can also inspire the development of threshold-based algorithms for computing applications.

💡 Research Summary

The paper investigates how the acellular slime mold Physarum polycephalum solves mazes and connects food sources by forming optimal transport networks, using a circuit‑theoretic model that abstracts the organism’s tubular network as a graph of memristive elements. The authors first map a maze onto a graph G(N, L) where nodes represent junctions (including the entry and exit points) and edges correspond to potential protoplasmic tubes. Each edge is modeled as a circuit component consisting of a nonlinear memristor in parallel with a capacitor, capturing the pressure‑driven shuttle‑streaming dynamics of the endoplasm and the threshold‑dependent gel‑to‑sol transition observed experimentally.

The memristor’s current‑voltage relationship is defined by a threshold function: when the voltage (analogous to pressure difference) exceeds a critical value V_T, the resistance drops sharply, allowing a large current (fluid flow). Two functional forms are considered—a piecewise‑linear function with parameters α and β, and a smooth hyperbolic‑tangent approximation with steepness j—both reproducing the abrupt current increase reported in Physarum measurements. Kirchhoff’s current‑law together with the memristor‑capacitor dynamics yields the differential equation i = M(v)·v + C·dv/dt, where v denotes the vector of node voltage differences.

To analyse stability and optimality, the authors construct a Lyapunov function V(v) = ½ vᵀ B M(v) Bᵀ v (B being the incidence matrix). They prove V̇ ≤ 0 for all admissible states, guaranteeing convergence to an asymptotically stable equilibrium where all voltages fall below the threshold. At equilibrium, only edges with M(v) > 0 (i.e., those whose voltage exceeds V_T) sustain non‑zero current; all other edges become effectively “pruned.” The set of surviving edges is shown to coincide with the minimal‑resistance subgraph that connects the source and sink, which mathematically corresponds to the shortest path (or, more generally, a minimum‑cost spanning subgraph) on the original graph. Hence, the model demonstrates that local threshold‑based decisions are sufficient for the global emergence of an optimal network.

The paper also derives conditions under which multiple viable paths persist. If the threshold V_T is set too low or the memristor’s gain β is insufficiently large, several edges may simultaneously satisfy M(v) > 0, leading to a network with alternative (longer) branches. This regime captures the experimentally observed flexibility of Physarum, where redundant tubes can be retained to increase fault tolerance or to adapt to changing environments.

The authors discuss the biological plausibility of the model, noting that while the memristor parameters do not map directly onto measurable cellular properties (e.g., viscosity, contractile force), the qualitative agreement with observed current‑voltage curves supports the abstraction. They highlight the model’s advantages: (1) it reduces complex mechano‑chemical interactions to analytically tractable equations, (2) it provides a rigorous proof of optimality for a broad class of maze topologies (any connected graph), and (3) it offers a blueprint for designing threshold‑based algorithms and hardware that mimic Physarum’s problem‑solving abilities. Limitations include the lack of a quantitative calibration between memristor dynamics and real physiological variables, and the omission of stochastic fluctuations that may play a role in biological exploration.

In conclusion, the study establishes that Physarum’s ability to form optimal paths can be explained by a simple threshold‑sensing mechanism embedded in a memristive circuit model. This insight bridges biological self‑organization with electrical network theory, validates the hypothesis that primitive intelligence can arise from local nonlinear feedback, and opens avenues for bio‑inspired computing architectures that exploit threshold‑driven adaptation.