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Top1. Introduction
The p-adic1 and more generally ultrametric approach to mathematical modeling of brain’s functioning was started in the works of one the authors of this paper (Khrennikov, 1997, 1998, 2002, 2004a, 2004b, 2007) and his collaborators (Albeverio et al., 1997; Dubischar, 1999).This approach was initially used merely for modeling of functioning of human subconsciousness and, in particular, for development of a mathematical model for Freud’s psychoanalysis (Freud, 1900, 1923). Recently the ultrametric approach to the mathematical representation of some essential features of human unconscious mind was also explored in the works of other authors (Lauro-Grotto, 2007; Murtagh, 2012a, 2012b; Contreras & Murtagh, 2012). We also remark that our ultrametric model of information processing by the brain matches well the general program of development of Cognitive Informatics and Cognitive Computing (Wang et al., 2010).
Applications of p-adics to cognitive science and psychology were stimulated by its applications to theoretical physics: string and superstring theory, cosmology, quantum mechanics and quantum field theory, spin glasses (Vladimirov et al., 1994; Khrennikov, 1997). As was mentioned in footnote 1, in many biological applications it is not important to restrict encoding of information to the prime number quantization, {0,1,.., p-1}. The use of quantization based on an arbitrary natural number, {0,1,…,m-1}, can be useful as well. In general the problem of a proper selection the prime or natural basis of the model has the contextual nature. It depends on the class of cognitive or biological phenomena. In image analysis the choice of the prime p (or more generally m) may depend on the class of images and it can be justified in the process of learning by using a database of images. The biological brain can represent a huge variety of trees. It may be able to select a tree having the structure matching a problem.
Information processed by complex cognitive systems and especially by human beings (but already a single cell is a complex information processor) is characterized by the presence of various closely connected hierarchic structures. The most natural geometry for the representation of such structures is geometry of trees and the corresponding topology is the ultrametric topology of on these trees. And the p-adic trees provide the simplest model for representation of mental hierarchies (Khrennikov, 1997; Khrennikov & Nilsson, 2004).