A logic-based dynamical theory for a genesis of biological threshold

To help the study of constructing a formal neuron by computer, we propose a logic-based dynamical theory for a genesis of the biological threshold which specific proteins, such as ion channel proteins, or their networks can produce. By viewing such a protein or a protein network as a computational machine, the statements concerning the states of reaction chains which eventually activate or inactivate the protein are treated. By introducing dynamical systems, associated with an inference process based on statements with continuous truth values, we investigated invariant characters of such dynamics and we obtained a sigmoidal function for an invariant distribution function of the truth values. The domain of solutions of the functional equations regarded as representing the self-description of proteins or protein networks as a machine indicates the emergence of a threshold, namely the realization of dyadic value, 0 or 1, based on the continuous truth values. The results obtained may highlight the mechanism of neuronal threshold in a framework which differs from population dynamics. The derived dynamical systems may also provide a simple model of ‘demonʹ rectifying the thermal fluctuations to drive unidirectional movements.