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ISSN 2070-0466, p-Adi c Numbers, Ultrametr ic Analysis and Applications, 2020, Vol. 12, No. 1, pp. 68–71. c

Pleiades Publishi ng, Ltd., 2020.

SHORT COMMUNICATIONS

A Reader’s Comment on: “Hysteresis Model of

Unconscious-Conscious Interconnection: Exploring Dynamics on

m-Adic Trees”∗

Giuseppe Iurato** and Andrei Yu. Khrennikov***

International Center for Mathematical Modeling in Physics, Engineering, Economics

and Cognitive Science, Linnaeus University, S-35195, V ¨axj ¨o, Sweden

Received January 12, 2020; in ﬁnal form, January 15, 2020; accepted January 15, 2020

Abstract—This comment is aimed to point out that the recent work due to H. Kim, J-Y. Moon, G.

A. Mashour and U. Lee ([22]), in which the clinical and experiential assessment of a brain network

model suggests that asymmetry of synchronization suppression is the key mechanism of hysteresis

observed during loss and recovery of consciousness in general anesthesia, has indirectly provided

empirical conﬁrmation of the theoretical model outlined in [8] (Iurato and Khrennikov, 2015), based

on a possible implementation of an hysteretic pattern into a formal model of unconscious-conscious

interconnection worked out on the basis of representations of mental entities by p-adic numbers.

One of the main assumptions done by the authors of [22], is that (physical) hysteresis (of their brain

network model took into account) observed during anesthetic state transitions shares the same

underlying mechanism as that observed in non-biological networks. This makes licit to put into

comparative relations [8] and [22].

DOI: 10.1134/S2070046620010070

Key words: hysteresis model, unconscious-conscious, m-adic tree.

First, rigorous attempts to formalize, through p-adic mathematics, the construct pair conscious-

unconscious of psychology have been undertaken by Andrei Yu. Khrennikov since the late 1990s ([6, 12–

20]). This formalization via p-adic analysis was based on the use of concepts, tools and techniques

drawn from dynamical systems theory and this route is very promising. One of the central points of this

theoretical framework, which lays out the basic concepts and notions of psychology and psychoanalysis,

is the use of p-adic dynamical systems1and related theory, thanks to which it has been possible to

take into account the chief elements of Freudian psychoanalysis, among which the crucial relationships

conscious-unconscious, which may be formalized through discrete dynamical system theory, as brieﬂy

recalled in the next section, and represent the nodal points of the whole psychoanalytic framework.

Therefore, the psychological construct pair conscious-unconscious, say C−UC, is the keystone

of every formalization attempt of psychoanalysis. In [8], the authors have simply taken into account

aﬁrst elementary formal model of hysteretic phenomena (regarding physical context), implemented

into the p-adic dynamical model of the C−UC pair. In doing so, the authors of [8] have tried to use

hysteretic phenomena (belonging to physics) to analogically transfer memory retaining eﬀects into the

phenomenology involved in the pair C−UC. Indeed, hysteretic eﬀects have been considered in attempts

∗The text was submitted by the authors in English.

**E-mail: giuseppeiurato9@gmail.com

***E-mail: andrei.khrennikov@lnu.se

1Mathematically, it is fruitful to proceed with the ﬁelds of p-adic numbers, where p>1is a prime number. These ﬁelds play

an important role in theoretical physics, string theory, quantum mechanics and ﬁeld theory, cosmology –see, e.g., [1, 4, 5]

for recent reviews. However, in cognitive and psychological applications there are no reasons to restrict models to prime

number bases. It is more natural to work with the rings of p-adic numbers, where p>1is an arbitrary natural number.

In general, the language of ultrametric spaces covers completely tree-like representations of information in cognitive

studies and psychology ([16]). However, up to now not so much has been done on general ultrametric spaces. Finally,

we also remark that methods of p-adic and more generally ultrametric analysis, have been used in modeling cognition and

unconscious processing of information by R. Lauro-Grotto ([24]) and F. Murtagh ([25–28]).

68

A READER’S COMMENT 69

to mechanically formalize memory features of implicit memories of neurophysiology ([7, 23]), so the

authors of [8] have thought to extend this idea to C−UC pair, trying to shed light upon a formal issue

raised by the m-adic dynamical model. The model outlined in [8] has been then applied to formalize other

aspects of human psyche ([9]) as well as to deduce a p-adic version of the Weber-Fechner law ([10]) and

some of its possible applications to economics and sociology ([11]).

Hysteresis has a large range phenomenology, and may be understood from either the psychological

and the physical standpoint. A possible conception of hysteresis belonging to psychological context may

be drawn from the APA Dictionary of Psychology which deﬁnes hysteresis as an eﬀect in which the

perception of a stimulus is inﬂuenced by any other stimulus immediately preceding such a perception. It

can be detected, for instance, in experiments making successive changes to a certain stimulus which is

varying along some dimension, hence asking to the participant to describe her or his perception. When

such values along the given dimension are steadily increased, then it will be reached a point in which the

participant will begin to place the related percept into a diﬀerent category (e.g., a sound is loud rather

than quiet2), but, when values along the dimension are decreased, then the crossover point will occur

at a diﬀerent point along such a dimension. In particular, in vision, hysteresis may stand out with the

tendency for a perceptual state to persist under gradually changing conditions: this is, for example, the

case when stereoscopic fusion may persist, so producing the appearance of depth even when binocular

disparity (i.e., the slight diﬀerence between the right and left retinal images) between the two images

becomes so great that they would normally not be able to be merged together.

This last phenomenology of hysteresis (to be meant according to psychology) related to vision may

be also correlated analogically ([3]) with certain aspects of the physical phenomenology discussed ﬁrst

in [21], and dealing with conscious-unconscious visual recognition, hence reconsidered in [3] where

the authors have then pointed out the possible analogical identiﬁcation of hysteresis eﬀects in visual

recognition experiments performed in [2]. Indeed,insuchacontext,H.vonHelmholtz unconscious

inferences, which play a crucial role in the passage from sensation to perception, are considered in

relation to a quantum-like pattern of sensation-perception dynamics –quantically treated, in that not

based on classical logics –so providing a concrete model for unconscious and consciousness processing

of information and their interaction. To be precise, in the cognitive modeling worked out in [21] and [3],

if Srepresents the unconscious information processing and Sthe conscious one, then, in the concrete

instance of von Helmholtz’s unconscious inference, Srepresents just the processing of sensation (its

unconscious nature having been emphasized as early by Hermann von Helmholtz) and Srepresents

processing of perception-conscious representation of sensation. The related experiment performed in [2],

then theoretically analyzed in [21] and [3], concerned the bistable perception (of the type S→S)ofthe

rotation of an ambiguous ﬁgure (i.e., the Schr ¨oder stair), which turned out to be diﬀerent, for each of the

three groups of persons chosen to form statistical test samples, due to the diversity of data’s contextuality

(suitably treatable just by quantum formalism) entailing optical illusions aﬀected by memory biases, and

put into relation with hysteresis eﬀects in [3].

On the other hand, following [22], there already existed a wide literature on computational biology

works which, since the late of 1990s and the beginnings of 2000s, have put attention to possible hystere-

sis phenomena (to be meant according to physics and network systems) occurring in a large-scale brain

network modelled with simple oscillatory patterns, in particular during state transitions of consciousness

and unconsciousness (like in general anesthesia and sleep), with hysteresis observed during the loss and

recovery of consciousness mediated by patterns of synchronization meant, according to general network

systems, as a pathway discontinuous transition between incoherent (unconsciousness) and synchro-

nized (consciousness) states of a network3that is, the asymmetry between the synchronization and

desynchronization paths is just the key network mechanism of hysteresis. The decreasing/increasing of

long-range network synchronization is considered as a basic neural mechanism during the loss/recovery

2This just resembles that typical phenomenology involved in sound experiences called into question in explaining Weber-

Fechner law ([10]).

3As authors themselves point out in [22], consciousness and unconsciousness cannot be trivially reduced to, respectively,

synchronized and incoherent networks, as it is temporal coordination, rather than synchrony, to be critical for conscious-

ness. This is also in agreement with (and provides partial empirical evidence to) what we have stated in [9].

p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS Vol. 12 No. 1 2020

70 IURATO, KHRENNIKOV

of consciousness4. Furthermore, network mechanism of hysteresis is not as a regional brain activity

but rather is a globally conceived mechanism ([22]). This is an remarkable outcome as it proves that

(physical) hysteresis is a phenomenon concerning the general psychic mechanisms of human brain. In

particular, in [22], it has been proved that hysteresis occurs above all during state transitions around a

lower lever of consciousness. This justiﬁes the theoretical implementation of a formal model of hysteretic

phenomena (regarding physical context) into the p-adic dynamical model of the C−UC pair,asdonein

[8], where the authors have supposed that hysteresis mechanism roles functionally unconscious realm5

and the related consciousness processes coming from it.

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A READER’S COMMENT 71

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