Threshold sensing yields optimal path formation in Physarum polycephalum

TL;DR

This paper investigates how Physarum polycephalum solves mazes without a nervous system by modeling its foraging as flow on a circuital network of memristor-like links. It proves that threshold sensing yields the emergence of a unique, optimal path by casting the problem as a constrained minimisation of $J(i)=\sum_{k=1}^m f_k(i_k)$ with $f_k'(i_k)=g_k(i_k)=M_k^{-1}(i_k)$ under $B i=\bar d$, and shows asymptotic convergence to a steady state. The analysis reveals that an ideal threshold $M_k^{th}$ reduces the solution to the geometrically shortest path, while nonlinear thresholds are required to obtain path selectivity and possibly multiple branches in non-ideal settings. These results bridge foraging dynamics, optimization, and memristor-inspired computing, suggesting new threshold-based algorithms for solving network design and routing problems.

Abstract

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.

Figures (5)

• Figure 1: Circuital model of P. polycephalum to map a maze into a circuit. a) Schematic representation of the circuit component $k$, with capacitance $C_k$ and memristive element $M_k$, connecting nodes $h$ and $r$ of the network. b) Example of maze with walls (grey) and with a source and a sink (red circles) at its entry and exit, respectively representing Physarum and a food source to be reached. The maze is mapped into a network of circuit components, where white circles are associated with nodes and circuit components with links. Encoding the maze topology means having open switches (hence, disconnection) in correspondence to maze walls. In green, the shortest path connecting source and sink; in blue, the additional portion of an alternative (longer) path.
• Figure 2: A network graph (with nodes labelled in black and links labelled in blue) and its corresponding incidence matrix.
• Figure 3: In red, the ideal threshold function with threshold potential $V_{T_k} = 9$. In blue, two examples of threshold-like characteristic functions $i_k = M_k(v_k)$: a) the piecewise linear function in \ref{['eq:char_func1']} with $\alpha=0.05$ and $\beta=5$; b) the smooth function in \ref{['eq:char_func2']} with $j=10$.
• Figure 4: Example of time evolution in scenario (b). Colour encodes the relative abundance of cellular gel, as per model \ref{['eq:model_final']}, and thus highlights Physarum's path from top to bottom. 1) Physarum starts spreading from its entry point ($t=100$), 2) extends ($t=1500$) and 3) probes the environment with several branches ($t=2900$) that are eventually pruned, until 4) only those (ideally, one single branch) most efficiently connecting the entry point with the food sources remain ($t=4300$).
• Figure 5: Top row: Steady-state solutions for the four scenarios (a)-(d) described in Sec. \ref{['sec:numerics']} in the respective panels a)-d). Colour encodes the relative abundance of cellular gel, as per model \ref{['eq:model_final']}, and thus highlights Physarum's path from top to bottom. Note, in a) and b), the formation of a single path; in c), the formation of multiple branches; in d), the homogeneous diffusion in the whole environment. Bottom row: characteristic functions used for the simulations; a)-c) are functions of the form \ref{['eq:char_func1']}, progressively more different from an ideal threshold function; d) is a linear function.