Generation of genetic codes with 2-adic codon algebra and adaptive dynamics.

This is a brief review on modeling genetic codes with the aid of 2-adic dynamical systems. In this model amino acids are encoded by the attractors of such dynamical systems. Each genetic code is coupled to the special class of 2-adic dynamics. We consider the discrete dynamical systems, These are the iterations of a function F:Z2→Z2, where Z2 is the ring of 2-adic numbers (2-adic tree). A genetic code is characterized by the set of attractors of a function belonging to the code generating functional class. The main mathematical problem is to reduce degeneration of dynamic representation and select the optimal generating function. Here optimality can be treated in many ways. One possibility is to consider the Lipschitz functions playing the crucial role in general theory of iterations. Then we minimize the Lip-constant. The main issue is to find the proper biological interpretation of code-functions. One can speculate that the evolution of the genetic codes can be described in information space of the nucleotide-strings endowed with ultrametric (treelike) geometry. A code-function is a fitness function; the solutions of the genetic code optimization problem are attractors of the code-function. We illustrate this approach by generation of the standard nuclear and (vertebrate) mitochondrial genetics codes.

Contextual measurement model and quantum theory.

We develop a contextual measurement model (CMM) that is used for the clarification of the quantum foundations. This model matches Bohr's views on the role of experimental contexts. CMM is based on a contextual probability theory that is connected with generalized probability theory. CMM covers measurements in classical, quantum and semi-classical physics. The CMM formalism is illustrated by a few examples. We consider the CMM framing of classical probability, the von Neumann measurement theory and the quantum instrument theory. CMM can also be applied outside of physics, e.g. in cognition, decision-making and psychology, the so-called quantum-like modelling.

Adaptive Dynamics Simulation of Interference Phenomenon for Physical and Biological Systems.

Biological systems have been shown to have quantum-like behaviors by applying the adaptive dynamics view on their interaction networks. In particular, in the process of lactose-glucose metabolism, cells generate probabilistic interference patterns similarly to photons in the two-slit experiment. Such quantum-like interference patterns can be found in biological data, on all scales, from proteins to cognitive, ecological, and social systems. The adaptive dynamics approach covers both biological and physical phenomena, including the ones which are typically associated with quantum physics. We guess that the adaptive dynamics can be used for the clarification of quantum foundations, and the present paper is the first step in this direction. We suggest the use of an algorithm for the numerical simulation of the behavior of a billiard ball-like particle passing through two slits by explicitly considering the influence of the two-slit environment (experimental context). Our simulation successfully mimics the interference pattern obtained experimentally in quantum physics. The interference of photons or electrons by two slits is known as a typical quantum mechanical effect. We do not claim that the adaptive dynamics can reproduce the whole body of quantum mechanics, but we hope that this numerical simulation example will stimulate further extensive studies in this direction-the representation of quantum physical phenomena in an adaptive dynamical framework.

Bild Conception of Scientific Theory Structuring in Classical and Quantum Physics: From Hertz and Boltzmann to Schrödinger and De Broglie.

We start with a methodological analysis of the notion of scientific theory and its interrelation with reality. This analysis is based on the works of Helmholtz, Hertz, Boltzmann, and Schrödinger (and reviews of D'Agostino). Following Helmholtz, Hertz established the "Bild conception" for scientific theories. Here, "Bild" ("picture") carries the meaning "model" (mathematical). The main aim of natural sciences is construction of the causal theoretical models (CTMs) of natural phenomena. Hertz claimed that a CTM cannot be designed solely on the basis of observational data; it typically contains hidden quantities. Experimental data can be described by an observational model (OM), often based on the price of acausality. CTM-OM interrelation can be tricky. Schrödinger used the Bild concept to create a CTM for quantum mechanics (QM), and QM was treated as OM. We follow him and suggest a special CTM for QM, so-called prequantum classical statistical field theory (PCSFT). QM can be considered as a PCSFT image, but not as straightforward as in Bell's model with hidden variables. The common interpretation of the violation of the Bell inequality is criticized from the perspective of the two-level structuring of scientific theories. Such critical analysis of von Neumann and Bell no-go theorems for hidden variables was performed already by De Broglie (and Lochak) in the 1970s. The Bild approach is applied to the two-level CTM-OM modeling of Brownian motion: the overdamped regime corresponds to OM. In classical mechanics, CTM=OM; on the one hand, this is very convenient; on the other hand, this exceptional coincidence blurred the general CTM-OM structuring of scientific theories. We briefly discuss ontic-epistemic structuring of scientific theories (Primas-Atmanspacher) and its relation to the Bild concept. Interestingly, Atmanspacher as well as Hertz claim that even classical physical theories should be presented on the basic of two-level structuring.

Quantization of events in the event-universe and the emergence of quantum mechanics.

Quantum mechanics (QM) is derived based on a universe composed solely of events, for example, outcomes of observables. Such an event universe is represented by a dendrogram (a finite tree) and in the limit of infinitely many events by the p-adic tree. The trees are endowed with an ultrametric expressing hierarchical relationships between events. All events are coupled through the tree structure. Such a holistic picture of event-processes was formalized within the Dendrographic Hologram Theory (DHT). The present paper is devoted to the emergence of QM from DHT. We used the generalization of the QM-emergence scheme developed by Smolin. Following this scheme, we did not quantize events but rather the differences between them and through analytic derivation arrived at Bohmian mechanics. We remark that, although Bohmian mechanics is not the main stream approach to quantum physics, it describes adequately all quantum experiments. Previously, we were able to embed the basic elements of general relativity (GR) into DHT, and now after Smolin-like quantization of DHT, we can take a step toward quantization of GR. Finally, we remark that DHT is nonlocal in the treelike geometry, but this nonlocality refers to relational nonlocality in the space of events and not Einstein's spatial nonlocality. By shifting from spatial nonlocality to relational we make Bohmian mechanics less exotic.

Rao-Fisher information geometry and dynamics of the event-universe views distributions.

We developed the novel mathematical model for event-universe by representing events as branches of dendrograms (finite trees) expressing the hierarchic relation between events. At the ontic level we operate with infinite trees. Algebraically such mathematical structures are represented as p-adic numbers. We call this kind of event mechanics Dendrogramic Holographic theory (DHT). It can be considered as a fundamental theory generating both GR and QM. In this paper we endower DHT with Rao-Cramer's information geometry. Following Smolin's derivation of QM from the event-universe, we introduce views from one event to others and by using their probability distributions we invent stochastic geometry. The important mathematical result is that all such views' distributions can be parametrized by four real parameters that are a part of the shape complexity measure introduced by Barbour in his particle shape dynamics theory - adapted to DHT. Hence, within DHT all possible event-universes can be embedded in four-dimensional real space. Asin GR, we introduce proper time. This "proper time" depends only on the change between one distribution of an observer to the other. The linkage of time to change is highlighted in the ideology of Rovelli and Barbour's shape dynamics.

Coherent decision making stimulated within the social laser: open quantum systems framework.

We start with a brief review on social laser theory which is newly framed with the notion of an infon-social energy quantum carrying coarse-grained information content. Infons are the excitations of the quantum social-information field. Humans are analogues of atoms-social atoms absorbing and emitting infons. Another new development is coupling of the social laser with the decision making model based on open quantum systems. The role of the environment for social atoms is played by the strong coherent social-information field-the output of social lasing. We analyse a simple quantum master equation generating decision jumps towards the coherent decision directed by the social laser beam. As an illustrative example, we analyse the possibility to create a societal benefit laser, i.e. directed for societal benefit. This article is part of the theme issue 'Thermodynamics 2.0: Bridging the natural and social sciences (Part 1)'.

Open Systems, Quantum Probability, and Logic for Quantum-like Modeling in Biology, Cognition, and Decision-Making.

The aim of this review is to highlight the possibility of applying the mathematical formalism and methodology of quantum theory to model behavior of complex biosystems, from genomes and proteins to animals, humans, and ecological and social systems. Such models are known as quantum-like, and they should be distinguished from genuine quantum physical modeling of biological phenomena. One of the distinguishing features of quantum-like models is their applicability to macroscopic biosystems or, to be more precise, to information processing in them. Quantum-like modeling has its basis in quantum information theory, and it can be considered one of the fruits of the quantum information revolution. Since any isolated biosystem is dead, modeling of biological as well as mental processes should be based on the theory of open systems in its most general form-the theory of open quantum systems. In this review, we explain its applications to biology and cognition, especially theory of quantum instruments and the quantum master equation. We mention the possible interpretations of the basic entities of quantum-like models with special interest given to QBism, as it may be the most useful interpretation.

EEG-based spatio-temporal relation signatures for the diagnosis of depression and schizophrenia.

The diagnosis of psychiatric disorders is currently based on a clinical and psychiatric examination (intake). Ancillary tests are used minimally or only to exclude other disorders. Here, we demonstrate a novel mathematical approach based on the field of p-adic numbers and using electroencephalograms (EEGs) to identify and differentiate patients with schizophrenia and depression from healthy controls. This novel approach examines spatio-temporal relations of single EEG electrode signals and characterizes the topological structure of these relations in the individual patient. Our results indicate that the relational topological structures, characterized by either the personal universal dendrographic hologram (DH) signature (PUDHS) or personal block DH signature (PBDHS), form a unique range for each group of patients, with impressive correspondence to the clinical condition. This newly developed approach results in an individual patient signature calculated from the spatio-temporal relations of EEG electrodes signals and might help the clinician with a new objective tool for the diagnosis of a multitude of psychiatric disorders.

On Applicability of Quantum Formalism to Model Decision Making: Can Cognitive Signaling Be Compatible with Quantum Theory?

This note is devoted to the problem of signaling (marginal inconsistency) in the Bell-type experiments with physical and cognitive systems. It seems that in quantum physics, this problem is still not taken seriously. Only recently have experimenters started to check the signaling hypothesis for their data. For cognitive systems, signaling was statistically significant in all experiments (typically for decision making) performed up to today. Here, one cannot simply ignore this problem. Since signaling contradicts the quantum theory of measurement for compatible observables, its statistical significance in experiments with humans can be considered as an objection for quantum-like modeling-applications of quantum theory to cognition, decision making, psychology, economics and finance, social and political science. In this paper, we point to two possible sources of signaling generation that are consistent with quantum measurement theory. Thus, the signaling objection for quantum-like modeling is not catastrophic. One of these sources is the direct physical signaling about selection of experimental settings, questions or tasks in quantum-like studies. Another possible source is a state modification dependent on experimental settings. The latter was a rather common source of signaling in quantum physics. Since the physical size of the brain is very small comparing with the light velocity, it seems to be impossible to prevent the direct physical signaling (with electromagnetic waves) between the brain's areas processing two questions a and b. However, if, for these questions, not the electromagnetic waves, but electrochemical communication plays the crucial role, the experimenter may hope to make signaling weaker by answering the questions faster. The problem of question-dependent mental state modification seems to be solvable via smarter experimental design. This paper can be useful both for physicists interested in quantum foundations and for researchers working in quantum-like studies, e.g., applying the quantum theory to model decision making or psychological effects. This paper is solely about quantum theory. Thus, we do not consider general contextual probabilistic models.

Ambivalence in decision making: An eye tracking study.

An intuition of ambivalence in cognition is particularly strong for complex decisions, for which the merits and demerits of different options are roughly equal but hard to compare. We examined information search in an experimental paradigm which tasked participants with an ambivalent question, while monitoring attentional dynamics concerning the information relevant to each option in different Areas of Interest (AOIs). We developed two dynamical models for describing eye tracking curves, for each response separately. The models incorporated a drift mechanism towards the various options, as in standard drift diffusion theory. In addition, they included a mechanism for intrinsic oscillation, which competed with the drift process and undermined eventual stabilization of the dynamics. The two models varied in the range of drift processes postulated. Higher support was observed for the simpler model, which only included drifts from an uncertainty state to either of two certainty states. In addition, model parameters could be weakly related to the eventual decision, complementing our knowledge of the way eye tracking structure relates to decision (notably the gaze cascade effect).

Towards Unification of General Relativity and Quantum Theory: Dendrogram Representation of the Event-Universe.

Following Smolin, we proceed to unification of general relativity and quantum theory by operating solely with events, i.e., without appealing to physical systems and space-time. The universe is modelled as a dendrogram (finite tree) expressing the hierarchic relations between events. This is the observational (epistemic) model; the ontic model is based on p-adic numbers (infinite trees). Hence, we use novel mathematics: not only space-time but even real numbers are not in use. Here, the p-adic space (which is zero-dimensional) serves as the base for the holographic image of the universe. In this way our theory is connected with p-adic physics; in particular, p-adic string theory and complex disordered systems (p-adic representation of the Parisi matrix for spin glasses). Our Dendrogramic-Holographic (DH) theory matches perfectly with the Mach's principle and Brans-Dicke theory. We found a surprising informational interrelation between the fundamental constants, h, c, G, and their DH analogues, h(D), c(D), G(D). DH theory is part of Wheeler's project on the information restructuring of physics. It is also a step towards the Unified Field theory. The universal potential V is nonlocal, but this is relational DH nonlocality. V can be coupled to the Bohm quantum potential by moving to the real representation. This coupling enhances the role of the Bohm potential.

Order stability via Fröhlich condensation in bio, eco, and social systems: The quantum-like approach.

Stability of social and behavioural order in biological, ecological, and social systems is modelled within the formalism of the Fröhlich condensation. The latter is a high temperature analogue of the Bose-Einstein condensation and stability is approached via intensive pumping of energy into a system interacting with a bath. We start with the review of this formalism considering nonequilibrium thermodynamic and quantum frameworks. Although Fröhlich applied this formalism to bio-systems and the physical energy flows (electromagnetic, chemical, vibrational), he pointed out on the possibility to apply it to wider class of systems. We realize this program by using quantum-like modelling in combination with the information approach to biological and social systems, by treating them as information processors and introducing the notion of social energy (with its versions, as, e.g., social and behavioural energy). This formalism is applied to modelling of social stability in the modern open society characterized by powerful flows of information and huge information reservoir based on internet, including the variety of social networks. Then, it is applied to modelling of coherent behaviour in herds and flocks with the illustrative example of wolf packs. The essence of the paper is extracting conditions for the Fröhlich condensation and reformulating them in the purely information framework.

Contextuality, Complementarity, Signaling, and Bell Tests.

This is a review devoted to the complementarity-contextuality interplay with connection to the Bell inequalities. Starting the discussion with complementarity, I point to contextuality as its seed. Bohr contextuality is the dependence of an observable's outcome on the experimental context; on the system-apparatus interaction. Probabilistically, complementarity means that the joint probability distribution (JPD) does not exist. Instead of the JPD, one has to operate with contextual probabilities. The Bell inequalities are interpreted as the statistical tests of contextuality, and hence, incompatibility. For context-dependent probabilities, these inequalities may be violated. I stress that contextuality tested by the Bell inequalities is the so-called joint measurement contextuality (JMC), the special case of Bohr's contextuality. Then, I examine the role of signaling (marginal inconsistency). In QM, signaling can be considered as an experimental artifact. However, often, experimental data have signaling patterns. I discuss possible sources of signaling-for example, dependence of the state preparation on measurement settings. In principle, one can extract the measure of "pure contextuality" from data shadowed by signaling. This theory is known as contextuality by default (CbD). It leads to inequalities with an additional term quantifying signaling: Bell-Dzhafarov-Kujala inequalities.

Dendrogramic Representation of Data: CHSH Violation vs. Nonergodicity.

This paper is devoted to the foundational problems of dendrogramic holographic theory (DH theory). We used the ontic-epistemic (implicate-explicate order) methodology. The epistemic counterpart is based on the representation of data by dendrograms constructed with hierarchic clustering algorithms. The ontic universe is described as a p-adic tree; it is zero-dimensional, totally disconnected, disordered, and bounded (in p-adic ultrametric spaces). Classical-quantum interrelations lose their sharpness; generally, simple dendrograms are "more quantum" than complex ones. We used the CHSH inequality as a measure of quantum-likeness. We demonstrate that it can be violated by classical experimental data represented by dendrograms. The seed of this violation is neither nonlocality nor a rejection of realism, but the nonergodicity of dendrogramic time series. Generally, the violation of ergodicity is one of the basic features of DH theory. The dendrogramic representation leads to the local realistic model that violates the CHSH inequality. We also considered DH theory for Minkowski geometry and monitored the dependence of CHSH violation and nonergodicity on geometry, as well as a Lorentz transformation of data.

EEG p-adic quantum potential accurately identifies depression, schizophrenia and cognitive decline.

No diagnostic or predictive instruments to help with early diagnosis and timely therapeutic intervention are available as yet for most neuro-psychiatric disorders. A quantum potential mean and variability score (qpmvs), to identify neuropsychiatric and neurocognitive disorders with high accuracy, based on routine EEG recordings, was developed. Information processing in the brain is assumed to involve integration of neuronal activity in various areas of the brain. Thus, the presumed quantum-like structure allows quantification of connectivity as a function of space and time (locality) as well as of instantaneous quantum-like effects in information space (non-locality). EEG signals reflect the holistic (nonseparable) function of the brain, including the highly ordered hierarchy of the brain, expressed by the quantum potential according to Bohmian mechanics, combined with dendrogram representation of data and p-adic numbers. Participants consisted of 230 participants including 28 with major depression, 42 with schizophrenia, 65 with cognitive impairment, and 95 controls. Routine EEG recordings were used for the calculation of qpmvs based on ultrametric analyses, closely coupled with p-adic numbers and quantum theory. Based on area under the curve, high accuracy was obtained in separating healthy controls from those diagnosed with schizophrenia (p<0.0001), depression (p<0.0001), Alzheimer's disease (AD; p<0.0001), and mild cognitive impairment (MCI; p<0.0001) as well as in differentiating participants with schizophrenia from those with depression (p<0.0001), AD (p<0.0001) or MCI (p<0.0001) and in differentiating people with depression from those with AD (p<0.0001) or MCI (p<0.0001). The novel EEG analytic algorithm (qpmvs) seems to be a useful and sufficiently accurate tool for diagnosis of neuropsychiatric and neurocognitive diseases and may be able to predict disease course and response to treatment.

Quantum-like model for unconscious-conscious interaction and emotional coloring of perceptions and other conscious experiences.

Quantum measurement theory is applied to quantum-like modeling of coherent generation of perceptions and emotions and generally for emotional coloring of conscious experiences. In quantum theory, a system should be separated from an observer. The brain performs self-measurements. To model them, we split the brain into two subsystems, unconsciousness and consciousness. They correspond to a system and an observer. The states of perceptions and emotions are described through the tensor product decomposition of the unconscious state space; similarly, there are two classes of observables, for conscious experiencing of perceptions and emotions, respectively. Emotional coloring is coupled to quantum contextuality: emotional observables determine contexts. Such contextualization reduces degeneration of unconscious states. The quantum-like approach should be distinguished from consideration of the genuine quantum physical processes in the brain (cf. Penrose and Hameroff). In our approach the brain is a macroscopic system which information processing can be described by the formalism of quantum theory. The paper is concluded with experimental test of contextual emotional coloring of conscious experiences based on Bell type inequalities which are treated in the contextual framework.

Ultrametric diffusion equation on energy landscape to model disease spread in hierarchic socially clustered population.

We present a new mathematical model of disease spread reflecting some specialties of the COVID-19 epidemic by elevating the role of hierarchic social clustering of population. The model can be used to explain slower approaching herd immunity, e.g., in Sweden, than it was predicted by a variety of other mathematical models and was expected by epidemiologists; see graphs Fig. 1, 2. The hierarchic structure of social clusters is mathematically modeled with ultrametric spaces having treelike geometry. To simplify mathematics, we consider trees with the constant number p > 1 of branches leaving each vertex. Such trees are endowed with an algebraic structure, these are p -adic number fields. We apply theory of the p -adic diffusion equation to describe a virus spread in hierarchically clustered population. This equation has applications to statistical physics and microbiology for modeling dynamics on energy landscapes. To move from one social cluster (valley) to another, a virus (its carrier) should cross a social barrier between them. The magnitude of a barrier depends on the number of social hierarchy's levels composing this barrier. We consider linearly increasing barriers. A virus spreads rather easily inside a social cluster (say working collective), but jumps to other clusters are constrained by social barriers. This behavior matches with the COVID-19 epidemic, with its cluster spreading structure. Our model differs crucially from the standard mathematical models of spread of disease, such as the SIR-model; in particular, by notion of the probability to be infected (at time t in a social cluster C ). We present socio-medical specialties of the COVID-19 epidemic supporting our model.

Representation of the Universe as a Dendrogramic Hologram Endowed with Relational Interpretation.

A proposal for a fundamental theory is described in which classical and quantum physics as a representation of the universe as a gigantic dendrogram are unified. The latter is the explicate order structure corresponding to the purely number-theoretical implicate order structure given by p-adic numbers. This number field was zero-dimensional, totally disconnected, and disordered. Physical systems (such as electrons, photons) are sub-dendrograms of the universal dendrogram. Measurement process is described as interactions among dendrograms; in particular, quantum measurement problems can be resolved using this process. The theory is realistic, but realism is expressed via the the Leibniz principle of the Identity of Indiscernibles. The classical-quantum interplay is based on the degree of indistinguishability between dendrograms (in which the ergodicity assumption is removed). Depending on this degree, some physical quantities behave more or less in a quantum manner (versus classic manner). Conceptually, our theory is very close to Smolin's dynamics of difference and Rovelli's relational quantum mechanics. The presence of classical behavior in nature implies a finiteness of the Universe-dendrogram. (Infinite Universe is considered to be purely quantum.) Reconstruction of events in a four-dimensional space type is based on the holographic principle. Our model reproduces Bell-type correlations in the dendrogramic framework. By adjusting dendrogram complexity, violation of the Bell inequality can be made larger or smaller.