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We study the retrieval accuracy and capacity of modern Hopfield networks of with two-state (Ising) spins interacting via modified Hebbian $n$-spin interactions. In particular, we consider systems where the interactions deviate from the Hebb…

Disordered Systems and Neural Networks · Physics 2025-03-04 Sharba Bhattacharjee , Ivar Martin

The Hopfield associative memory model stores random patterns in synaptic couplings according to Hebb's rule and retrieves them through gradient descent on an energy function. This conventional setting, where neurons are assumed to have…

Disordered Systems and Neural Networks · Physics 2026-01-23 Yoshiyuki Kabashima , Kazushi Mimura

Neural networks are supposed to recognise blurred images (or patterns) of $N$ pixels (bits) each. Application of the network to an initial blurred version of one of $P$ pre-assigned patterns should converge to the correct pattern. In the…

Statistical Mechanics · Physics 2009-11-07 Dietrich Stauffer , Amnon Aharony , Luciano da Fontoura Costa , Joan Adler

We investigate the computational limits of the memory retrieval dynamics of modern Hopfield models from the fine-grained complexity analysis. Our key contribution is the characterization of a phase transition behavior in the efficiency of…

Machine Learning · Computer Science 2024-06-04 Jerry Yao-Chieh Hu , Thomas Lin , Zhao Song , Han Liu

We introduce a three-dimensional vectorial extension of the Hopfield associative-memory model in which each neuron is a unit vector on $S^2$ and synaptic couplings are $3\times 3$ blocks generated through a vectorial Hebbian rule. The…

Disordered Systems and Neural Networks · Physics 2025-12-10 F. Gallavotti , A. Zaccone

Using the Hopfield model as a benchmark case, the present work focuses on the investigation of partially annealed associative neural networks, wherein neural dynamics is coupled to slowly evolving patterns within the…

Disordered Systems and Neural Networks · Physics 2026-05-12 Linda Albanese , Andrea Alessandrelli , Adriano Barra , Silvio Franz , Federico Ricci-Tersenghi

We study the associative-memory network of Kuramoto-type oscillators that stores a set of memorized patterns (memories). In [Phys. Rev. Lett., 92 (2004), 108101], Nishikawa, Lai and Hoppensteadt showed that the capacity of this system for…

Dynamical Systems · Mathematics 2025-05-20 Zhuchun Li , Xiaoxue Zhao , Xiang Zhou

We present extensive simulations of a quantum version of the Hopfield Neural Network to explore its emergent behavior. The system is a network of $N$ qubits oscillating at a given $\Omega$ frequency and which are coupled via Lindblad jump…

Quantum Physics · Physics 2024-10-21 Joaquín J. Torres , Daniel Manzano

A Hopfield network is an auto-associative, distributive model of neural memory storage and retrieval. A form of error-correcting code, the Hopfield network can learn a set of patterns as stable points of the network dynamic, and retrieve…

Neurons and Cognition · Quantitative Biology 2014-07-24 Ila Fiete , David J. Schwab , Ngoc M. Tran

To explore the relation between network structure and function, we studied the computational performance of Hopfield-type attractor neural nets with regular lattice, random, small-world and scale-free topologies. The random net is the most…

Disordered Systems and Neural Networks · Physics 2009-11-10 Patrick N. Mcgraw , Michael Menzinger

We consider $L$-directional associative memories, composed of $L$ Hopfield networks, displaying imitative Hebbian intra-network interactions and anti-imitative Hebbian inter-network interactions, where couplings are built over a set of…

Disordered Systems and Neural Networks · Physics 2025-04-11 Elena Agliari , Andrea Alessandrelli , Paulo Duarte Mourao , Alberto Fachechi

It has been recently shown that a learning transition happens when a Hopfield Network stores examples generated as superpositions of random features, where new attractors corresponding to such features appear in the model. In this work we…

Disordered Systems and Neural Networks · Physics 2024-07-09 Silvio Kalaj , Clarissa Lauditi , Gabriele Perugini , Carlo Lucibello , Enrico M. Malatesta , Matteo Negri

An associative memory has been discussed of neural networks consisting of spiking N (=100) Hodgkin-Huxley (HH) neurons with time-delayed couplings, which memorize P patterns in their synaptic weights. In addition to excitatory synapses…

Disordered Systems and Neural Networks · Physics 2009-10-31 Hideo Hasegawa

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…

Probability · Mathematics 2026-03-30 Matthias Löwe , Franck Vermet

Unsupervised learning in a generalized Hopfield associative-memory network is investigated in this work. First, we prove that the (generalized) Hopfield model is equivalent to a semi-restricted Boltzmann machine with a layer of visible…

Neural and Evolutionary Computing · Computer Science 2017-07-26 Huiling Zhen , Shang-Nan Wang , Hai-Jun Zhou

Real-world physical signals are continuous and high-dimensional, yet the statistical-mechanics machinery of associative memory operates on discrete Ising spins. We bridge this divide through a multilayer Ising framework that couples a…

Disordered Systems and Neural Networks · Physics 2026-05-15 Andrea Ladiana

In neuroscience, classical Hopfield networks are the standard biologically plausible model of long-term memory, relying on Hebbian plasticity for storage and attractor dynamics for recall. In contrast, memory-augmented neural networks in…

Neurons and Cognition · Quantitative Biology 2021-10-28 Danil Tyulmankov , Ching Fang , Annapurna Vadaparty , Guangyu Robert Yang

Storing memory for molecular recognition is an efficient strategy for responding to external stimuli. Biological processes use different strategies to store memory. In the olfactory cortex, synaptic connections form when stimulated by an…

Biological Physics · Physics 2021-06-07 Oskar H Schnaack , Luca Peliti , Armita Nourmohammad

We study a model of spiking neurons, with recurrent connections that result from learning a set of spatio-temporal patterns with a spike-timing dependent plasticity rule and a global inhibition. We investigate the ability of the network to…

Neurons and Cognition · Quantitative Biology 2020-04-22 S. Scarpetta , A. de Candia

As a mathematical model of associative memories, the Hopfield model was now well-established and a lot of studies to reveal the pattern-recalling process have been done from various different approaches. As well-known, a single neuron is…

Disordered Systems and Neural Networks · Physics 2015-05-27 Jun-ichi Inoue