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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.
A Reader’s Comment on: “Hysteresis Model of
Unconscious-Conscious Interconnection: Exploring Dynamics on
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 (), 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  (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 , 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  and .
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 , 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  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.
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 (). 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 () and F. Murtagh ([25–28]).
A READER’S COMMENT 69
to mechanically formalize memory features of implicit memories of neurophysiology ([7, 23]), so the
authors of  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  has been then applied to formalize other
aspects of human psyche () as well as to deduce a p-adic version of the Weber-Fechner law () and
some of its possible applications to economics and sociology ().
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 () with certain aspects of the physical phenomenology discussed ﬁrst
in , and dealing with conscious-unconscious visual recognition, hence reconsidered in  where
the authors have then pointed out the possible analogical identiﬁcation of hysteresis eﬀects in visual
recognition experiments performed in . 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  and ,
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 ,
then theoretically analyzed in  and , 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 .
On the other hand, following , 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 ().
3As authors themselves point out in , 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 .
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 (). 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 , 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
, where the authors have supposed that hysteresis mechanism roles functionally unconscious realm5
and the related consciousness processes coming from it.
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4Indeed, in , it has been ascertained that functional brain networks become more modular during general anesthesia,
in that coordinated and synchronized interactions across the cerebral cortex break down. So, for the aims of large-scale
modeling, authors of  assume that the conscious brain will have, in aggregate, more synchronized interactions with
respect to the unconscious brain.
5This is in agreement with the hypothesis, assumed in , for which a stronger anesthetics (hence, a deeper unconscious
state) induces a larger hysteresis.
p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS Vol. 12 No. 1 2020
A READER’S COMMENT 71
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p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS Vol. 12 No. 1 2020