$p$-Adic Mathematics in Behavioural Models. The Case of Physarum Polycephalum
The plasmodium of Physarum polycephalum (order Physarales, class Myxomecetes and subclass Myxogastromycetidae) is a single cell, visible with the naked eye, with many diploid nuclei. When it is placed in an environment with distributed sources of nutrients, the plasmodium forms a network of protoplasmic tubes connecting the food sources. The topology of the plasmodium's protoplasmic network optimizes the plasmodium's harvesting of nutrient resource from the scattered sources of nutrients and makes more efficient the transport of intracellular components. In this paper, we propose to use $p$-adic valued fuzziness and probabilities for measuring behaviours which cannot be measured additively. Then we construct a natural deductive system for describing all possible experiments with Physarum polycephalum. This system is $p$-adic many-valued.