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Networks of coupled phase oscillators are one of the most studied dynamical systems with numerous applications in physics, chemistry, biology, and engineering. Their behaviour is often characterized by the emergence of various partially…

We examine a previouly introduced attractor neural network model that explains the persistent activities of neurons in the anterior ventral temporal cortex of the brain. In this model, the coexistence of several attractors including…

Hopfield neural networks are a possible basis for modelling associative memory in living organisms. After summarising previous studies in the field, we take a new look at learning rules, exhibiting them as descent-type algorithms for…

Recurrently coupled oscillators that are sufficiently heterogeneous and/or randomly coupled can show an asynchronous activity in which there are no significant correlations among the units of the network. The asynchronous state can…

Memory is a complex phenomenon that involves several distinct mechanisms. These mechanisms operate at different spatial and temporal levels. This chapter focuses on the theoretical framework and the mathematical models that have been…

We investigated the properties of mixed states in a sparsely encoded associative memory model with a structural learning method. When mixed states are made of s memory patterns, s types of mixed states, which become equilibrium states of…

A network of propagating nonlinear oscillatory modes (waves) in the human brain is shown to generate collectively synchronized spiking activity (hypersynchronous spiking) when both amplitude and phase coupling between modes are taken into…

Quantum neural networks form one pillar of the emergent field of quantum machine learning. Here, quantum generalisations of classical networks realizing associative memories - capable of retrieving patterns, or memories, from corrupted…

This paper examines the memory capacity of generalized neural networks. Hopfield networks trained with a variety of learning techniques are investigated for their capacity both for binary and non-binary alphabets. It is shown that the…

Motivated by biological considerations, we study sparse neural maps from an input layer to a target layer with sparse activity, and specifically the problem of storing $K$ input-target associations $(x,y)$, or memories, when the target…

A fundamental problem in neuroscience is understanding how working memory -- the ability to store information at intermediate timescales, like 10s of seconds -- is implemented in realistic neuronal networks. The most likely candidate…

To gain a deeper understanding of the behavior and learning dynamics of (deep) artificial neural networks, it is valuable to employ mathematical abstractions and models. These tools provide a simplified perspective on network performance…

This paper studies the capability of a recurrent neural network model to memorize random dynamical firing patterns by a simple local learning rule. Two modes of learning/memorization are considered: The first mode is strictly online, with a…

We test four fast mean field type algorithms on Hopfield networks as an inverse Ising problem. The equilibrium behavior of Hopfield networks is simulated through Glauber dynamics. In the low temperature regime, the simulated annealing…

Recurrently connected neuron populations play key roles in sensory perception and memory storage across various brain regions. While these populations are often assumed to encode information through firing rates, this method becomes…

Associative memory models are content-addressable memory systems fundamental to biological intelligence and are notable for their high interpretability. However, existing models evaluate the quality of retrieval based on proximity, which…

Generalized Hopfield models with higher-order or exponential interaction terms are known to have substantially larger storage capacities than the classical quadratic model. On the other hand, associative memories for sparse patterns, such…

The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as…

We present an algorithm to store binary memories in a Hopfield neural network using minimum probability flow, a recent technique to fit parameters in energy-based probabilistic models. In the case of memories without noise, our algorithm…

Phase oscillators are a common starting point for the reduced description of many single neuron models that exhibit a strongly attracting limit cycle. The framework for analysing such models in response to weak perturbations is now…