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This paper is concerned with the modeling and analysis of two of the most commonly used recurrent neural network models (i.e., Hopfield neural network and firing-rate neural network) with dynamic recurrent connections undergoing Hebbian…

Optimization and Control · Mathematics 2024-03-25 Veronica Centorrino , Francesco Bullo , Giovanni Russo

We propose and analyze a new variation of the so-called {\em exponential Hopfield model}, a recently introduced family of associative neural networks with unprecedented storage capacity. Our construction is based on a cost function defined…

Disordered Systems and Neural Networks · Physics 2025-09-09 Linda Albanese , Andrea Alessandrelli , Adriano Barra , Peter Sollich

Spin-glass models of associative memories are a cornerstone between statistical physics and theoretical neuroscience. In these networks, stochastic spin-like units interact through a synaptic matrix shaped by local Hebbian learning. In…

Disordered Systems and Neural Networks · Physics 2025-04-08 Gianni V. Vinci , Andrea Galluzzi , Maurizio Mattia

We present a computational model based on the CRISP theory (Content Representation, Intrinsic Sequences, and Pattern completion) of the hippocampus that allows to continuously store pattern sequences online in a one-shot fashion. Rather…

Neural and Evolutionary Computing · Computer Science 2019-05-31 Jan Melchior , Mehdi Bayati , Amir Azizi , Sen Cheng , Laurenz Wiskott

The brain can reproduce memories from partial data; this ability is critical for memory recall. The process of memory recall has been studied using auto-associative networks such as the Hopfield model. This kind of model reliably converges…

Neurons and Cognition · Quantitative Biology 2016-05-18 James P. Roach , Leonard M Sander , Michal R. Zochowski

Associative memory or content addressable memory is an important component function in computer science and information processing and is a key concept in cognitive and computational brain science. Many different neural network…

Neural and Evolutionary Computing · Computer Science 2025-02-19 Anders Lansner , Naresh B Ravichandran , Pawel Herman

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…

Machine Learning · Computer Science 2025-11-26 Shurong Wang , Yuqi Pan , Zhuoyang Shen , Meng Zhang , Hongwei Wang , Guoqi Li

Recent research has established a connection between modern Hopfield networks (HNs) and transformer attention heads, with guarantees of exponential storage capacity. However, these models still face challenges scaling storage efficiently.…

Machine Learning · Computer Science 2025-04-11 Saul Santos , António Farinhas , Daniel C. McNamee , André F. T. Martins

We consider the class of Hopfield models of associative memory with activation function $F$ and state space $\{-1,1\}^N$, where each vertex of the cube describes a configuration of $N$ binary neurons. $M$ randomly chosen configurations,…

Probability · Mathematics 2025-04-08 Véronique Gayrard

Dense Associative Memories or Modern Hopfield Networks have many appealing properties of associative memory. They can do pattern completion, store a large number of memories, and can be described using a recurrent neural network with a…

Neural and Evolutionary Computing · Computer Science 2021-07-29 Dmitry Krotov

Hopfield networks are a variant of associative memory that recall information stored in the couplings of an Ising model. Stored memories are fixed points for the network dynamics that correspond to energetic minima of the spin state. We…

Quantum Physics · Physics 2014-12-15 Hadayat Seddiqi , Travis S. Humble

Dense associative memory, a fundamental instance of modern Hopfield networks, can store a large number of memory patterns as equilibrium states of recurrent networks. While the stationary-state storage capacity has been investigated, its…

Disordered Systems and Neural Networks · Physics 2025-10-29 Kazushi Mimura , Jun'ichi Takeuchi , Yuto Sumikawa , Yoshiyuki Kabashima , Anthony C. C. Coolen

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

We study a simple extended model of oscillator neural networks capable of storing sparsely coded phase patterns, in which information is encoded both in the mean firing rate and in the timing of spikes. Applying the methods of statistical…

Disordered Systems and Neural Networks · Physics 2009-10-31 Masaki Nomura , Toshio Aoyagi

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

Associative memory, traditionally modeled by Hopfield networks, enables the retrieval of previously stored patterns from partial or noisy cues. Yet, the local computational principles which are required to enable this function remain…

Generative models, including diffusion models, are increasingly used as foundation models and adapted through sequential fine-tuning, making continual learning an essential problem setting. However, continual learning in such generative…

Machine Learning · Computer Science 2026-05-29 Ken Takeda , Masafumi Oizumi , Ryo Karakida

Neuromorphic computing with spiking neural networks is promising for energy-efficient artificial intelligence (AI) applications. However, different from humans who continually learn different tasks in a lifetime, neural network models…

Neural and Evolutionary Computing · Computer Science 2024-02-20 Mingqing Xiao , Qingyan Meng , Zongpeng Zhang , Di He , Zhouchen Lin

We generalize the standard Hopfield model to the case when a weight is assigned to each input pattern. The weight can be interpreted as the frequency of the pattern occurrence at the input of the network. In the framework of the statistical…

Disordered Systems and Neural Networks · Physics 2012-05-07 Iakov Karandashev , Boris Kryzhanovsky , Leonid Litinskii

Recent generalizations of the Hopfield model of associative memories are able to store a number $P$ of random patterns that grows exponentially with the number $N$ of neurons, $P=\exp(\alpha N)$. Besides the huge storage capacity, another…

Disordered Systems and Neural Networks · Physics 2024-02-14 Carlo Lucibello , Marc Mézard