Decidable and Undecidable Arithmetic Functions in Actin Filament Networks

The plasmodium of Physarum polycephalum, called slime mould, is very sensitive to its environment and reacts to stimuli by its appropriate motions. The sensitive stage as well as the motor stage of these reactions are explained by actin filament networks. This paper is devoted to actin filament networks as a computation medium. The point is that actin filaments are sensitive to outer cellular stimuli (attractants as well as repellents) and they appear and disappear at different places of the cell to change the cell structure, e.g. its shape. Due to assembling and disassembling actin filaments, Physarum polycephalum as well as other unicellular organisms like Amoeba proteus can move in responses to different stimuli. As a result, these organisms can be considered a simple reversible logic gate, where outer cellular signals are its inputs and the motions are its outputs. In this way, we can implement different logic gate on the amoeboid behaviours. The actin filament networks have the same basic properties as neural networks: lateral inhibition; lateral activation; recurrent inhibition; recurrent excitation; feedforward inhibition; feedforward excitation; convergence/divergence. These networks can embody arithmetic functions defined recursively and corecursively within p-adic valued logic. Furthermore, within these networks we can define the so-called diagonalization for deducing undecidable arithmetic functions.