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Aging, the process of growing old or maturing, is one of the most widely seen natural phenomena in the world. For the stochastic processes, sometimes the influence of aging cannot be ignored. For example, in this paper, by analyzing the functional distribution of the trajectories of aging particles performing anomalous diffusion, we reveal that for the fraction of the occupation time of strong aging particles, with coefficient , having no relation with the aging time ta and α and being completely different from the case of weak (none) aging. In fact, we first build the models governing the corresponding functional distributions, i.e. the aging forward and backward Feynman–Kac equations; the above result is one of the applications of the models. Another application of the models is to solve the asymptotic behaviors of the distribution of the first passage time, . The striking discovery is that for weakly aging systems, , while for strongly aging systems, behaves as . [ABSTRACT FROM AUTHOR]

We study the large deviations of additive quantities, such as energy or current, in stochastic processes with intermittent reset. Via a mapping from a discrete-time reset process to the Poland–Scheraga model for DNA denaturation, we derive conditions for observing first-order or continuous dynamical phase transitions in the fluctuations of such quantities and confirm these conditions on simple random walk examples. These results apply to reset Markov processes, but also show more generally that subleading terms in generating functions can lead to non-analyticities in large deviation functions of ‘compound processes’ or ‘random evolutions’ switching stochastically between two or more subprocesses. [ABSTRACT FROM AUTHOR]

Quantum probability (QP) theory can be seen as a type of vector symbolic architecture (VSA): mental states are vectors storing structured information and manipulated using algebraic operations. Furthermore, the operations needed by QP match those in other VSAs. This allows existing biologically realistic neural models to be adapted to provide a mechanistic explanation of the cognitive phenomena described in the target article by Pothos & Busemeyer (P&B). [ABSTRACT FROM PUBLISHER]

Quantum probability models may supersede existing probabilistic models because they account for behaviour inconsistent with classical probability theory that are attributable to normal limitations of cognition. This intriguing position, however, may overstate weaknesses in classical probability theory by underestimating the role of current knowledge states and may under-employ available knowledge about the limitations of cognitive processes. In addition, flexibility in model specification has risks for the use of quantum probability. [ABSTRACT FROM PUBLISHER]

Pothos & Busemeyer (P&B) argue that classical probability (CP) fails to describe human decision processes accurately and should be supplanted by quantum probability. We accept the premise, but reject P&B's conclusion. CP is a prescriptive framework that has inspired a great deal of valuable research. Also, because CP is used across the sciences, it is a cornerstone of interdisciplinary collaboration. [ABSTRACT FROM PUBLISHER]

Although the quantum probability (QP) can be useful to model the context effect, it is not relevant to the order effect, conjunction fallacy, and other related biases. Although the issue of potentiality, which is the intuition behind QP, is involved in the context effect, it is not involved in the other biases. [ABSTRACT FROM PUBLISHER]

Classical probability models of incentive response are inadequate in “large worlds,” where the dimensions of relative risk and the dimensions of similarity in outcome comparisons typically differ. Quantum probability models for choice in large worlds may be motivated pragmatically – there is no third theory – or metaphysically: statistical processing in the brain adapts to the true scale-relative structure of the universe. [ABSTRACT FROM PUBLISHER]

I argue that the overly simplistic scenarios discussed by Pothos & Busemeyer (P&B) establish at best that quantum probability theory (QPT) is a logical possibility allowing distinct predictions from classical probability theory (CPT). The article fails, however, to provide convincing evidence for the proposal that QPT offers unique insights regarding cognition and the nature of human rationality. [ABSTRACT FROM AUTHOR]

Quantum probability (QP) provides a new perspective for cognitive science. However, one must be clear about the outcome the QP model is predicting. We discuss this concern in reference to modeling the subjective probabilities given by people as opposed to modeling the choice proportions of people. These two models would appear to have different cognitive assumptions. [ABSTRACT FROM PUBLISHER]

Pothos & Busemeyer (P&B) argue that quantum probability (QP) provides a descriptive model of behavior and can also provide a rational analysis of a task. We discuss QP models using Marr's levels of analysis, arguing that they make most sense as algorithmic level theories. We also highlight the importance of having clear interpretations for basic mechanisms such as interference. [ABSTRACT FROM AUTHOR]

Pothos & Busemeyer (P&B) argue convincingly that quantum probability offers an improvement over classical Bayesian probability in modeling the empirical data of cognitive science. However, a weakness related to restrictions on the dimensionality of incompatible physical observables flows from the authors' “agnosticism” regarding quantum processes in neural substrates underlying cognition. Addressing this problem will require either future research findings validating quantum neurophysics or theoretical expansion of the uncertainty principle as a new, neurocognitively contextualized, “local” symmetry. [ABSTRACT FROM AUTHOR]

A lot of research in cognition and decision making suffers from a lack of formalism. The quantum probability program could help to improve this situation, but we wonder whether it would provide even more added value if its presumed focus on outcome models were complemented by process models that are, ideally, informed by ecological analyses and integrated into cognitive architectures. [ABSTRACT FROM AUTHOR]

Faced with probabilistic relationships between causes and effects, quantum theory assumes that deterministic causes do not exist, and that only incomplete probabilistic expressions of knowledge are possible. As in its application to physics, this fundamental epistemological stance severely limits the ability of quantum theory to provide insight and understanding in human cognition. [ABSTRACT FROM AUTHOR]

According to the foundations of economic theory, agents have stable and coherent “global” preferences that guide their choices among alternatives. However, people are constrained by information-processing and memory limitations and hence have a propensity to avoid cognitive load. We propose that this in turn will encourage them to respond to “local” preferences and goals influenced by context and memory representations. [ABSTRACT FROM AUTHOR]

We highlight methodological and theoretical limitations of the authors' Dirac formalism and suggest the von Neumann open systems approach as a resolution. The open systems framework is a generalization of classical probability and we hope it will allow cognitive scientists to extend quantum probability from perception, categorization, memory, decision making, and similarity judgments to phenomena in learning and development. [ABSTRACT FROM AUTHOR]

We concentrate on two aspects of the article by Pothos & Busemeyer (P&B): the relationship between classical and quantum probability and quantum probability as a basis for rational decisions. We argue that the mathematical relationship between classical and quantum probability is not quite what the authors claim. Furthermore, it might be premature to regard quantum probability as the best practical rational scheme for decision making. [ABSTRACT FROM AUTHOR]

I consider the general problem of category conjunctions in the light of Pothos & Busemeyer (P&B)'s quantum probability (QP) account of the conjunction fallacy. I argue that their account as presented cannot capture the “guppy effect” – the case in which a class is a better member of a conjunction A^B than it is of either A or B alone. [ABSTRACT FROM AUTHOR]

Quantum probability (QP) theory provides an alternative account of empirical phenomena in decision making that classical probability (CP) theory cannot explain. Cognitive architectures combine probabilistic mechanisms with symbolic knowledge-based representations (e.g., heuristics) to address effects that motivate QP. They provide simple and natural explanations of these phenomena based on general cognitive processes such as memory retrieval, similarity-based partial matching, and associative learning. [ABSTRACT FROM AUTHOR]

Quantum probability theory offers a viable alternative to classical probability, although there are some ambiguities inherent in transferring the quantum formalism to a less determined realm. A number of physicists are now looking at the applicability of quantum ideas to the assessment of physics learning, an area particularly suited to quantum probability ideas. [ABSTRACT FROM AUTHOR]

There is another meeting place for quantum physics and psychology, both within and outside of cognitive modeling. In physics it is known as the issue of classical (probabilistic) determinism, and in psychology it is known as the issue of selective influences. The formalisms independently developed in the two areas for dealing with these issues turn out to be identical, opening ways for mutually beneficial interactions. [ABSTRACT FROM AUTHOR]

Pothos & Busemeyer's (P&B's) query about whether quantum probability can provide a foundation for the cognitive modeling embodies so many underlying implications that the subject is far from exhausted. In this brief commentary, however, I suggest that the conceptual thresholds of the meaningful learning give rise to a typical Boltzmann's weighting measure, which indicates a statistical verisimilitude of quantum behavior in the human cognitive ensemble. [ABSTRACT FROM AUTHOR]

Typically, human decision making is emotionally “hot” and does not conform to “cold” classical probability (CP) theory. As quantum probability (QP) theory emphasises order, context, superimposition states, and nonlinear dynamic effects, one of its major strengths may be its power to unify formal modeling and realistic psychological theory (e.g., information uncertainty, anxiety, and indecision, as seen in the Prisoner's Dilemma). [ABSTRACT FROM AUTHOR]

We propose a way in which Pothos & Busemeyer (P&B) could strengthen their position. Taking a dynamic stance, we consider cognitive tests as functions that transfer a given input state into the state after testing. Under very general conditions, it can be shown that testable properties in cognition form an orthomodular lattice. Gleason's theorem then yields the conceptual necessity of quantum probabilities (QP). [ABSTRACT FROM AUTHOR]

The Hilbert space formalism is a powerful language to express many cognitive phenomena. Here, relevant concepts from signal detection theory are recast in that language, allowing an empirically testable extension of the quantum probability formalism to psychophysical measures, such as detectability and discriminability. [ABSTRACT FROM AUTHOR]

Pothos & Busemeyer (P&B) argue how key concepts of quantum probability, for example, order/context, interference, superposition, and entanglement, can be used in cognitive modeling. Here, we suggest that these concepts can be extended to analyze neurophysiological measurements of cognitive tasks in humans, especially in functional neuroimaging investigations of large-scale brain networks. [ABSTRACT FROM AUTHOR]

One among many misleading quotations about the alleged mysteries of quantum theory is from Feynman (1965): “I think I can safely say that nobody understands quantum mechanics.” Today we know that quantum theory describes many aspects of our world in a fully intelligible fashion. Pothos & Busemeyer (P&B) propose ways in which this may include psychology and cognitive science. [ABSTRACT FROM AUTHOR]

Use of quantum probability as a top-down model of cognition will be enhanced by consideration of the underlying complex-valued wave function, which allows a better account of interference effects and of the structure of learned and ad hoc question operators. Furthermore, the treatment of incompatible questions can be made more quantitative by analyzing them as non-commutative operators. [ABSTRACT FROM AUTHOR]

We construct an equivalent probability description of linear multi-delay Langevin equations subject to additive Gaussian white noise. By exploiting the time-convolutionless transform and a time variable transformation we are able to write a Fokker–Planck equation (FPE) for the 1-time and for the 2-time probability distributions valid irrespective of the regime of stability of the Langevin equations. We solve exactly the derived FPEs and analyze the aging dynamics by studying analytically the conditional probability distribution. We discuss explicitly why the initially conditioned distribution is not sufficient to describe fully out a non-Markov process as both preparation and observation times have bearing on its dynamics. As our analytic procedure can also be applied to linear Langevin equations with memory kernels, we compare the non-Markov dynamics of a one-delay system with that of a generalized Langevin equation with an exponential as well as a power law memory. Application to a generalization of the Green–Kubo formula is also presented. [ABSTRACT FROM AUTHOR]

In one-dimensional systems, the dynamics of a Brownian particle are governed by the force derived from a potential as well as by diffusion properties. In this work, we obtain the first-passage-time statistics of a Brownian particle driven by an arbitrary potential with an exponential temporally decaying superimposed field up to a prescribed threshold. The general system analyzed here describes the sub-threshold signal integration of integrate-and-fire neuron models, of any kind, supplemented by an adaptation-like current, whereas the first-passage-time corresponds to the declaration of a spike. Following our previous studies, we base our analysis on the backward Fokker–Planck equation and study the survival probability and the first-passage-time density function in the space of the initial condition. By proposing a series solution we obtain a system of recurrence equations, which given the specific structure of the exponential time-dependent drift, easily admit a simpler Laplace representation. Naturally, the present general derivation agrees with the explicit solution we found previously for the Wiener process in (2012 J. Phys. A: Math. Theor.45 185001). However, to demonstrate the generality of the approach, we further explicitly evaluate the first-passage-time statistics of the underlying Ornstein–Uhlenbeck process. To test the validity of the series solution, we extensively compare theoretical expressions with the data obtained from numerical simulations in different regimes. As shown, agreement is precise whenever the series is truncated at an appropriate order. Beyond the fact that both the Wiener and Ornstein–Uhlenbeck processes have a direct interpretation in the context of neuronal models, given their ubiquity in different fields, our present results will be of interest in other settings where an additive state-independent temporal relaxation process is being developed as the particle diffuses. [ABSTRACT FROM AUTHOR]

A mathematical model of grain size development during annealing in a polycrystalline metallic system is proposed. The model is based on the simple assumption that at any point during recrystallization the increment in grain size is a random proportion of the grain size at that instant. The model predicts a lognormal distribution in grain sizes under a wide variety of conditions. The width of this distribution is found to broaden with time. The model is compared with experiments and computational predictions. It is argued that this model may represent materials undergoing primary recrystallization in some cases. [ABSTRACT FROM AUTHOR]

Opinion. Focuses on electroencephalogram (EEG) rhythmicity. Formation of the mechanism for disparate EEG phenomena; Significance of the EEG rhythmicity; Interpretation of the mechanism of EEG generation; Effects of the changes in the cortical excitation level to EEG frequencies and amplitude.

Opinion. Proposes a scheme for the study of the chaotic behavior of the brain system. Information provided by the dimension of electroencephalogram; Theoretical approach to brain function based on Hebb's concept of cell assemblies; Relevance of the differences between pointwise dimension.

Opinion. Develops a proposal incorporating a developmental perspective to the microscopic and macroscopic models of brain behavior. Advantages of the developmental perspective to the constrain models; Consideration of the meaning of the neural signals that are stimulated; Factors concerning the developmental processes.

Opinion. Comments on J.J. Wright and D.T.J. Liley's article related to the dynamics of the brain at global and microscopic scales. Consideration of multiscale dynamics; Use of linear approximations; Influence of global boundary conditions on dynamics; Experimental verification; Global influences on local cell assemblies.

Opinion. Points out the major errors in the empirical data used by J.J. Wright and D.T.J. Liley in their article related to the dynamics of the brain at global and microscopic scales. Comparison of simulation results with empirical data; Quantitative differences between the simulation and the real cortex.

Opinion. Recognizes the role of neuromodulation in changing the dynamical state of cortical or neural networks. Presentation of the simulation results of an olfactory cortex model; Discussion on how the system can attain global stability; Necessity of balancing the stability of the system with flexibility and sensitivity.

Opinion. Discusses the validity of using nonlinear models for the cerebral cortex. Exhibition of linearity and nonlinearity on the electroencephalogram (EEG); Argument about the strength of the linear character of EEG; Criticism on the approach to the scale problem; Requirement for nonlinear mechanisms of synchronization in information transfer.

Opinion. Provides an understanding of the overall dynamic properties of the brain. Description of the functional diversity architecture and related intrinsic global control mechanisms; Effect of rhythmic control on the system; Significance of rhythms as control mechanisms to reflect brain dynamics.

Opinion. Comments on J.J. Wright and D.T.J. Liley's article related to the dynamics of the brain at global and microscopic scales. Evaluation of the importance of the approach; Irrelevance of chaos to macroscopic neocortical interactions; Use of quantum mechanics to explain the linearity of macroscopic information processing.

Opinion. Comments on J.J. Wright and D.T.J. Liley's article related to the dynamics of the brain at global and microscopic scales. Contrast of two hypotheses concerning nonlinear elements in complex systems; Utility of linear stability analysis; Analysis of the stability properties of the biosystem.

Opinion. Suggests the validation of the multiscale modeling of the brain in more detail against the biological data. Summary of the assumptions made by authors J.J. Wright and D.T.J. Liley; Reactivity between the inhibitory gamma-aminobutyric acid interneurons and excitatory primary neurons via metabotropic glutamate receptors.

Opinion. Analyzes the some dichotomies related to modeling electrocortical activities in the brain. Discussion of attractor neural networks versus biologically motivated models; Reflection on the stochastic and chaotic patterns; Comparison of local/microscopic and global/macroscopic scales.

Opinion. Comments on J.J. Wright and D.T.J. Liley's article on issues related to the dynamics of the brain at global and microscopic scales. Clarification of the concepts relating to learned, structured functioning of local modules in the neocortex; Correction of the misreadings about the attractor neural network; Perplexity of the article.

We revise the classical problem of characterizing first exit times of a harmonically trapped particle whose motion is described by a one- or multidimensional Ornstein–Uhlenbeck process. We start by recalling the main derivation steps of a propagator using Langevin and Fokker–Planck equations. The mean exit time, the moment-generating function and the survival probability are then expressed through confluent hypergeometric functions and thoroughly analyzed. We also present a rapidly converging series representation of confluent hypergeometric functions that is particularly well suited for numerical computation of eigenvalues and eigenfunctions of the governing Fokker–Planck operator. We discuss several applications of first exit times, such as the detection of time intervals during which motor proteins exert a constant force onto a tracer in optical tweezers single-particle tracking experiments; adhesion bond dissociation under mechanical stress; characterization of active periods of trend-following and mean-reverting strategies in algorithmic trading on stock markets; relation to the distribution of first crossing times of a moving boundary by Brownian motion. Some extensions are described, including diffusion under quadratic double-well potential and anomalous diffusion. [ABSTRACT FROM AUTHOR]

In a discrete model of a neural network, from a limitation on the rank of the matrix of 'coupling coefficients', an upper bound for the maximum length of the transients is derived, generalizing the analogue result of Caianiello (1966a) for the case of rank one. A counterexample shows that the limitation found is the best possible involving only the rank of the matrix. [ABSTRACT FROM AUTHOR]

Pothos & Busemeyer (P&B) provide a compelling case that quantum probability (QP) theory is a better match to human judgment than is classical probability (CP) theory. However, any theory (QP, CP, or other) phrased solely at the computational level runs the risk of being underconstrained. One suggestion is to ground QP accounts in mechanism, to leverage a wide range of process-level data. [ABSTRACT FROM PUBLISHER]