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This comment is aimed to point out that the recent work due to Kim, et al. in which the clinical and experiential assessment of a brain network model suggests that asymmetry of synchronization suppression is the key mechanism of hysteresis has coupling with our theoretical hysteresis model of unconscious-conscious interconnection based on dynamics on m-adic trees.

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In this study, we discuss a non-Kolmogorovness of the optical illusion in the human visual perception. We show subjects the ambiguous figure of "Schröeder stair", which has two different meanings [1]. We prepare 11 pictures which are inclined by different angles. The tendency to answer "left side is front" depends on the order of showing those pictures. For a mathematical treatment of such a context dependent phenomena, we propose a non-Kolmogorovian probabilistic model which is based on adaptive dynamics.

Hysteresis, the discrepancy in forward and reverse pathways of state transitions, is observed during changing levels of consciousness. Identifying the underlying mechanism of hysteresis phenomena in the brain will enhance the ability to understand, monitor, and control state transitions related to consciousness. We hypothesized that hysteresis in brain networks shares the same underlying mechanism of hysteresis as other biological and non-biological networks. In particular, we hypothesized that the principle of explosive synchronization, which can mediate abrupt state transitions, would be critical to explaining hysteresis in the brain during conscious state transitions. We analyzed high-density electroencephalogram (EEG) that was acquired in healthy human volunteers during conscious state transitions induced by the general anesthetics sevoflurane or ketamine. We developed a novel method to monitor the temporal evolution of EEG networks in a parameter space, which consists of the strength and topography of EEG-based networks. Furthermore, we studied conditions of explosive synchronization in anatomically informed human brain network models. We identified hysteresis in the trajectory of functional brain networks during state transitions. The model study and empirical data analysis explained various hysteresis phenomena during the loss and recovery of consciousness in a principled way: (1) more potent anesthetics induce a larger hysteresis; (2) a larger range of EEG frequencies facilitates transitions into unconsciousness and impedes the return of consciousness; (3) hysteresis of connectivity is larger than that of EEG power; and (4) the structure and strength of functional brain networks reconfigure differently during the loss vs. recovery of consciousness. We conclude that the hysteresis phenomena observed during the loss and recovery of consciousness are generic network features. Furthermore, the state transitions are grounded in the same principle as state transitions in complex non-biological networks, especially during perturbation. These findings suggest the possibility of predicting and modulating hysteresis of conscious state transitions in large-scale brain networks.

In the framework of p-adic analysis (the simplest version of analysis on trees in which hierarchic structures are presented through ultrametric distance) applied to formalize psychic phenomena, we would like to propose some possible first hypotheses about the origins of human consciousness centered on the basic notion of time symmetry breaking as meant according to quantum field theory of infinite systems. Starting with Freud’s psychophysical (hydraulic) model of unconscious and conscious flows of psychic energy based on the three-orders mental representation, the emotional order, the thing representation order, and the word representation order, we use the p-adic (treelike) mental spaces to model transition from unconsciousness to preconsciousness and then to consciousness. Here we explore theory of hysteresis dynamics: conscious states are generated as the result of integrating of unconscious memories. One of the main mathematical consequences of our model is that trees representing unconscious and consciousmental states have to have different structures of branching and distinct procedures of clustering. The psychophysical model of Freud in combination with the p-adic mathematical representation gives us a possibility to apply (for a moment just formally) the theory of spontaneous symmetry breaking of infinite dimensional field theory, to mental processes and, in particular, to make the first step towards modeling of interrelation between the physical time (at the level of the emotional order) and psychic time at the levels of the thing and word representations. Finally, we also discuss some related topological aspects of the human unconscious, following Jacques Lacan’s psychoanalytic concepts.

We present a quantum-like model of sensation-perception dynamics (originated in Helmholtz theory of unconscious inference) based on the theory of quantum apparatuses and instruments. We illustrate our approach with the model of bistable perception of a particular ambiguous figure, the Schröder stair. This is a concrete model for unconscious and conscious processing of information and their interaction. The starting point of our quantum-like journey was the observation that perception dynamics is essentially contextual which implies impossibility of (straightforward) embedding of experimental statistical data in the classical (Kolmogorov, 1933) framework of probability theory. This motivates application of nonclassical probabilistic schemes. And the quantum formalism provides a variety of the well-approved and mathematically elegant probabilistic schemes to handle results of measurements. The theory of quantum apparatuses and instruments is the most general quantum scheme describing measurements and it is natural to explore it to model the sensation-perception dynamics. In particular, this theory provides the scheme of indirect quantum measurements which we apply to model unconscious inference leading to transition from sensations to perceptions.

This paper illustrates the recontextualization of a formal theory from the domain of cognitive science to psychoanalysis. Starting from an approach driven by information theory, a description of mental space in terms of a peculiar topographical structure is derived. This "ultrametric" structure can be seen to fit the constraints of primary process thinking, as presented by Matte Blanco in his essays on bi-logic. The author's reformulation of the distinction between primary and secondary processes in topological terms leads to a speculative proposal of "quasi-symbols" as possible objects of mental life. The paper also seeks to capture some aspects of the crucial interplay between the phenomenological experience of personal change and the proposed conceptual advance in theorization.

In a companion paper, Murtagh (2012), we discussed how Matte Blanco's work
linked the unrepressed unconscious (in the human) to symmetric logic and
thought processes. We showed how ultrametric topology provides a most useful
representational and computational framework for this. Now we look at the
extent to which we can find ultrametricity in text. We use coherent and
meaningful collections of nearly 1000 texts to show how we can measure inherent
ultrametricity. On the basis of our findings we hypothesize that inherent
ultrametricty is a basis for further exploring unconscious thought processes.

We mathematically model Ignacio Matte Blanco's principles of symmetric and
asymmetric being through use of an ultrametric topology. We use for this the
highly regarded 1975 book of this Chilean psychiatrist and pyschoanalyst (born
1908, died 1995). Such an ultrametric model corresponds to hierarchical
clustering in the empirical data, e.g. text. We show how an ultrametric
topology can be used as a mathematical model for the structure of the logic
that reflects or expresses Matte Blanco's symmetric being, and hence of the
reasoning and thought processes involved in conscious reasoning or in reasoning
that is lacking, perhaps entirely, in consciousness or awareness of itself. In
a companion paper we study how symmetric (in the sense of Matte Blanco's)
reasoning can be demarcated in a context of symmetric and asymmetric reasoning
provided by narrative text.

We develop a model of the process of thinking in the presence of noise (which is produced by the simultaneous action of a huge number of neurons in the brain as well as by external information and internal cognitive processes). Our model is based on Freud's idea on the splitting of cognitive processes into two (closely connected) domains: consciousness and subconsciousness. We represent the process of thinking as a random dynamical process in a space of ideas endowed with a non-Euclidean geometry (which differs extremely from the ordinary Euclidean geometry of spatial location of neurons in the brain). The so-called p-adic geometry on a space of ideas describes the ability of cognitive systems to form associations. We show that random dynamical thinking systems on a p -adic space of ideas still generate only deterministic ideas. We also study positive and negative effects of noise (in particular, creativeness and stress).

We propose a mathematical model of the process of thinking based on dynamical systems over a configuration space of ideas. These dynamical systems are assumed to be located in the human subconscious and are controlled by the human conscious which fixes parameters of the dynamical systems in the subconscious and transmits to the subconscious generating ideas which initiate iterations of the dynamical systems in the subconscious. Thus, we present the mathematical model which is not based on the rule of reason. Mathematically the space of ideas is described by so-called p-adic numbers. In fact, a p-adic metric on the space of ideas corresponds to the following nearness between ideas: two ideas x and y are close if they have sufficiently long common root. Already the simplest p-adic dynamical systems might describe some sides of human psychological and social behaviour.

this paper we develop a p-adic probability formalism based on measure theory of [19]. By probabilistic reasons we use the special case of this measure theory: measures defined on algebras (such measures have some special properties). However, probabilistic applications stimulate also the development of the general theory of non-Archimedean measures defined on rings. We prove the formula of the change of variables for these measures and use this formula for developing the formalism of conditional expectations for

Considering the main aspects of a previous formal model of the relationships unconscious-conscious based on the representation of mental entities by m-adic numbers through hysteresis phenomenology, a pattern which has been then used to work out a possible psychoanalytic model of human consciousness as well as to argue on a simple derivations of p-adic Weber-Fechner laws of psychophysics, we now carry on along this formal analysis putting forward some remarks about the possible applications and consequences of this model of human psyche in regard to central themes of economics and sociology.

From a simple extension of a previous formal pattern of unconscious-conscious interconnection based on the representation of mental entities by m-adic numbers through hysteresis phenomenology, a pattern which has been then used to work out a possible psychoanalytic model of human consciousness, we now argue on related simple derivations of p-adic Weber-Fechner laws of psychophysics.

p$-Adic mathematical physics is a branch of modern mathematical physics based on the application of $p$-adic mathematical methods in modeling physical and related phenomena. It emerged in 1987 as a result of efforts to find a non-Archimedean approach to the spacetime and string dynamics at the Planck scale, but then was extended to many other areas including biology. This paper contains a brief review of main achievements in some selected topics of $p$-adic mathematical physics and its applications, especially in the last decade. Attention is mainly paid to developments with promising future prospects.

We discuss differences in mathematical representations of the physical and mental worlds. Following Aristotle, we present the mental space as discrete, hierarchic, and totally disconnected topological space. One of the basic models of such spaces is given by ultrametric spaces and more specially by m-adic trees. We use dynamical systems in such spaces to model flows of unconscious information at different level of mental representation hierarchy, for “mental points”, categories, and ideas. Our model can be interpreted as an unconventional computational model: non-algorithmic hierarchic “computations” (identified with the process of thinking at the unconscious level).

In this brief note, we focus attention on a possible implementation of a basic hysteretic pattern (the Preisach one), suitably generalized, into a formal model of unconscious-conscious interconnection and based on representation of mental entities by m-adic numbers.

We present a brief information on “The Workshop on p-Adic Methods for Modeling of Complex Systems”, which was held in the Center for Interdisciplinary Research (Zentrum für interdisziplinäre Forshung — ZiF), Bielefeld University, Bielefeld, Germany, April 15–19, 2013.

We summarize some areas of great potential for data analytics based on ultrametric topology. These areas include search and discovery, in a way that is computationally efficient and scalable, as well as with demonstrated effectiveness. Further areas discussed are the analysis and synthesis of narrative. We conclude with a preliminary description of work on emotion that has developed out of our work on narrative.

This review covers an important domain of p-adic mathematical physics — quantum mechanics with p-adic valued wave functions. We start with basic mathematical constructions of this quantum model: Hilbert spaces over quadratic
extensions of the field of p-adic numbers ℚ
p
, operators — symmetric, unitary, isometric, one-parameter groups of unitary isometric operators, the p-adic version of Schrödinger’s quantization, representation of canonical commutation relations in Heisenberg andWeyl forms,
spectral properties of the operator of p-adic coordinate.We also present postulates of p-adic valued quantization. Here observables as well as probabilities take values in ℚ
p
. A physical interpretation of p-adic quantities is provided through approximation by rational numbers.

In this paper we present a model of processing of mental information based on encoding by points of ultrametric space. Basic
mental entities categories are encoded by ultrametric balls. Our model describes processes which take place in subconsciousness.
It seems that ultrametric is a right tool for modeling of unconscious mental processes. Properties of ultrametric balls match
well properties of unconscious representation of information which have been discussed in psychology.

The triangular inequality is a defining property of a metric space, while the
stronger ultrametric inequality is a defining property of an ultrametric space. Ultrametric
distance is defined from p-adic valuation. It is known that ultrametricity is a natural
property of spaces in the sparse limit. The implications of this are discussed in this article.
Experimental results are presented which quantify how ultrametric a given metric space
is. We explore the practical meaningfulness of this property of a space being ultrametric.
In particular, we examine the computational implications of widely prevalent and perhaps
ubiquitous ultrametricity.

Violation of contextual generalization of the Leggett-Gargin equality for recognition of ambiguous figures

- M Asano
- A Yu
- M Khrennikov
- Y Ohya
- I Tanaka
- Yamato

M. Asano, A.Yu. Khrennikov, M. Ohya, Y. Tanaka, I. Yamato,
"Violation of contextual generalization of the Leggett-Gargin equality for recognition of ambiguous figures", Physica Scripta (2014),
https://doi.org/10.1088/00318949/2014/t163/014006

Classical and Quantum Mental Models and Freud's Theory of Unconscious Mind

- A Yu
- Khrennikov

A.Yu. Khrennikov, Classical and Quantum Mental Models and
Freud's Theory of Unconscious Mind (Växjö University Press, Växjö,
SWE, 2002).

Toward theory of p-adic valued probabilities

- A Yu
- Khrennikov

A.Yu. Khrennikov, "Toward theory of p-adic valued probabilities",
Studies in Logic, Grammar and Rethoric, 14 (27), 137-154 (2008).

- M L Latash

M.L. Latash, Neurophysiological Basis of Movement (2nd Edition,
Human Kinetics, Champaign, IL, 2008).