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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…

The aim of this tutorial survey is to revisit the basic theory of relaxation processes governed by linear differential equations of fractional order. The fractional derivatives are intended both in the Rieamann-Liouville sense and in the…

Data-driven predictive analytics are in use today across a number of industrial applications, but further integration is hindered by the requirement of similarity among model training and test data distributions. This paper addresses the…

We first review traditional approaches to memory storage and formation, drawing on the literature of quantitative neuroscience as well as statistical physics. These have generally focused on the fast dynamics of neurons; however, there is…

In this paper, we investigate a fractional differential equation involving sequential Caputo derivatives, motivated by recent research on fractional models with multiple memory effects. Using techniques inspired by earlier works on…

Continual learning addresses the problem of continuously acquiring and transferring knowledge without catastrophic forgetting of old concepts. While humans achieve continual learning via diverse neurocognitive mechanisms, there is a…

Memories are stored, retained, and recollected through complex, coupled processes operating on multiple timescales. To understand the computational principles behind these intricate networks of interactions we construct a broad class of…

A growing body of research indicates that structural plasticity mechanisms are crucial for learning and memory consolidation. Starting from a simple phenomenological model, we exploit a mean-field approach to develop a theoretical framework…

Attractor networks are an influential theory for memory storage in brain systems. This theory has recently been challenged by the observation of strong temporal variability in neuronal recordings during memory tasks. In this work, we study…

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…

Humans can learn a variety of concepts and skills incrementally over the course of their lives while exhibiting many desirable properties, such as continual learning without forgetting, forward transfer and backward transfer of knowledge,…

Memory effects are a key feature in the description of the dynamical systems governed by the generalized Langevin equation, which presents an exact reformulation of the equation of motion. A simple measure for the estimation of memory…

Accelerators with power-law memory are proposed in the framework of the discrete time approach. To describe discrete accelerators we use the capital stock adjustment principle, which has been suggested by Matthews.The suggested discrete…

Memory and forgetting constitute two sides of the same coin, and although the first has been rigorously investigated, the latter is often overlooked. A number of experiments under the realm of psychology and experimental neuroscience have…

In this paper, we introduce variational semantic memory into meta-learning to acquire long-term knowledge for few-shot learning. The variational semantic memory accrues and stores semantic information for the probabilistic inference of…

In the theoretical research of Hepatitis B virus, mathematical models of its transmission mechanism have been thoroughly investigated, while the dynamics of the immune process in vivo have not. At present, nearly all the existing models are…

Attention mechanisms underpin the computational power of Transformer models, which have achieved remarkable success across diverse domains. Yet understanding and extending the principles underlying self-attention remains a key challenge for…

Neural dynamical systems with stable attractor structures, such as point attractors and continuous attractors, are hypothesized to underlie meaningful temporal behavior that requires working memory. However, working memory may not support…

We find that multifractal scaling is a robust property of a large class of continuous stochastic processes, constructed as exponentials of long-memory processes. The long memory is characterized by a power law kernel with tail exponent…

Uncovering the mechanisms behind long-term memory is one of the most fascinating open problems in neuroscience and artificial intelligence. Artificial associative memory networks have been used to formalize important aspects of biological…