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The idea that memory behavior relies on a gradually-changing internal state has a long history in mathematical psychology. This chapter traces this line of thought from statistical learning theory in the 1950s, through distributed memory…
Traditional statistical estimation, or statistical inference in general, is static, in the sense that the estimate of the quantity of interest does not change the future evolution of the quantity. In some sequential estimation problems…
This paper discusses the evolution of probability distributions for certain time-dependent dynamical systems. Exponential loss of memory is proved for expanding maps and for one-dimensional piecewise expanding maps with slowly varying…
Consider first a memoryless population model described by the usual branching process with a given mean reproduction matrix on a finite space of types. Motivated by the consequences of atavism in Evolutionary Biology, we are interested in a…
A large class of linear memory differential equations in one dimension, where the evolution depends on the whole history, can be equivalently described as a projection of a Markov process living in a higher dimensional space. Starting with…
This article deals with dynamical systems depending on a slowly varying parameter. We present several physical examples illustrating memory effects, such as metastability and hysteresis, which frequently appear in these systems. A…
Memory is often defined as the mental capacity of retaining information about facts, events, procedures and more generally about any type of previous experience. Memories are remembered as long as they influence our thoughts, feelings, and…
Earlier we proposed the stochastic point process model, which reproduces a variety of self-affine time series exhibiting power spectral density S(f) scaling as power of the frequency f and derived a stochastic differential equation with the…
Using kicked differential equations of motion with derivatives of noninteger orders, we obtain generalizations of the dissipative standard map. The main property of these generalized maps, which are called fractional maps, is long-term…
Episodic control enables sample efficiency in reinforcement learning by recalling past experiences from an episodic memory. We propose a new model-based episodic memory of trajectories addressing current limitations of episodic control. Our…
Derivatives of fractional order with respect to time describe long-term memory effects. Using nonlinear differential equation with Caputo fractional derivative of arbitrary order $\alpha>0$, we obtain discrete maps with power-law memory.…
We examine the effects of memory and different updating paradigms in a game-theoretic model of competitive learning, where agents are influenced in their choice of strategy by both the choices made by, and the consequent success rates of,…
Standard macroeconomic models assume that households are rational in the sense that they are perfect utility maximizers, and explain economic dynamics in terms of shocks that drive the economy away from the stead-state. Here we build on a…
In the paper the memory effect in the system consisting from a trajectory of process and an environment is considered. The environment is presented by scalar potential and noise. The evolution of system is interpreted as process of the…
In this paper we introduce a model of dynamic crack growth in viscoelastic material, where the damping term depends on the history of the deformation. The model is based on a dynamic energy dissipation balance and on a maximal dissipation…
We obtain option pricing formulas for stock price models in which the drift and volatility terms are functionals of a continuous history of the stock prices. That is, the stock dynamics follows a nonlinear stochastic functional differential…
Recent advances at the intersection of control theory, neuroscience, and machine learning have revealed novel mechanisms by which dynamical systems perform computation. These advances encompass a wide range of conceptual, mathematical, and…
Understanding the statistical dynamics of growth and inequality is a fundamental challenge to ecology and society. Recent analyses of wealth and income dynamics in contemporary societies show that economic inequality is very dynamic and…
Evolution equations are derived for the amplitudes of associative memories: heterogeneous states stored in the connectivity of distributed systems with non-local interactions. The resulting coupled amplitude equations describe the…
We consider the evolution of correlation functions in a non-Markov version of the contact model in the continuum. The memory effects are introduced by assuming the fractional evolution equation for the statistical dynamics. This leads to a…