A mathematical model to predict network growth in <i>Physarum polycephalum</i> as a function of extracellular matrix viscosity, measured by a novel viscometer

<i>Physarum polycephalum</i> is a slime mould that forms complex networks, making it an ideal model organism for studying network formation and adaptation. We introduce a novel viscometer capable of accurately measuring extracellular slime matrix (ECM) viscosity in small biological samples, overcoming the limitations of conventional instruments. Using this device, we measured the relative kinematic viscosity and developed continuous models to predict network size over time as a function of ECM viscosity. Our results show that increased ECM viscosity, driven by higher salt (MgCl₂·6H₂O) concentrations, significantly slows network expansion but does not affect the final network complexity. Fractal dimension analysis revealed that network complexity converged to a similar value across all viscosity conditions during the equilibrium state. The models demonstrated strong predictive power, with a mean squared error below 0.4%, closely aligning with experimental data. These findings highlight the critical role of ECM viscosity in influencing network expansion while demonstrating that complexity remains stable across varying conditions. This study advances our understanding of the physical parameters shaping <i>P. polycephalum</i> networks and provides a foundation for exploring network dynamics in other adaptive systems. These insights offer new tools for research in biological systems where sample material is limited.