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

Abstract

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 'demon' rectifying the thermal fluctuations to drive unidirectional movements.

Functional dynamics. I: Articulation process

“…However, semantic structure formed through the cognitive process has a dynamical aspect, and hence a fuzzy 'logic' is not sufficient to deal with it. To consider cognitive and semantic processes, a dynamical model of fuzzy logic [13] and a dynamical network of proposition in a fuzzy logic as a functional form [14] have been proposed. However, to study the duality between rule and code directly, we must avoid such a logic-based approach.…”

Internal logic viewed from observation space: Theory and a case study

“…As stated in a previous paper [17], the dynamics of a description can be formulated using a certain type of functional map (see also Refs. [8], [15], [16]). The use of such a map may provide a direction for the development of the present framework.…”

“…(16). Let us denote the statement that a hypothesis H itself is true by the same symbol H, and its negation (i.e., the statement that H is false) byH.…”

Abstract heterarchy: Time/state-scale re-entrant form

“…A chaotic liar reserves his own description, and that evokes the property of self-reference. Tsuda and Tadaki (1997) also enhance the aspect of the liar's self-understanding regarding self-reference. They use a simultaneous equation for a chaotic liar, such as x t+1 = f(x t , y t ) and y t+1 = g(x t , y t ), and examine the significance for which only x t+1 = f(x t , y t ) and y t+1 = g(x t+1 , y t ) show chaotic behavior.…”