Summary: A joint research initiative has introduced Centered Daydreaming. By combining simultaneous daytime learning and nocturnal pruning, and shifting the network’s focus to local sequence deviations rather than absolute values, the team has successfully pushed the network’s storage capacity to its absolute theoretical limit (100%), even when processing highly biased, realistic datasets.
Key Facts
- The 13% Storage Bottleneck Broken: A standard, classical Hopfield network suffers from severe storage limitations, safely holding a number of memories equal to only about 13% of its total neuron count. Attempting to force more data into the system triggers a cascade of “spurious attractors”, false memories and hallucinations that overwrite actual data.
- Achieving 100% Theoretical Capacity: The “Daydreaming” framework introduced in 2025 solved this storage ceiling. By executing daytime learning and nighttime sleep-pruning simultaneously, the algorithm completely prevented catastrophic forgetting, safely scaling memory capacity to the theoretical absolute maximum: one complete memory for every single neuron (100% capacity).
- The Real-World Data Challenge: Despite achieving a 100% capacity rating on perfectly balanced laboratory data (where black and white pixels are distributed 50/50), the original algorithm failed when confronted with realistic, highly skewed data, such as overexposed, blindingly white images or deeply shadowed dark photographs where identical pixel states dominate the grid.
- Biologically Plausible Local Learning: Traditional computer science patches for data bias rely on global optimizations across the entire network, which are biologically impossible. Human neurons operate purely on local information, only communicating with immediate neighbors. The newly engineered Centered Daydreaming preserves this local framework, making it highly energy-efficient and biologically accurate.
- The Power of Averages and Differences: Instead of evaluating the absolute state of incoming data pixels, the Centered Daydreaming algorithm recalculates data based purely on its variation from the moving average (similar to human facial recognition focusing entirely on distinct structural features that deviate from a standard “average face”).
- Sustained Efficacy Under Strain: Under rigorous testing against heavily biased real-world data distributions, the new Centered Daydreaming algorithm kept the network’s pattern-retrieval fidelity intact. This development paves the way for a new class of clear, easily interpretable, and low-power AI systems modeled directly on statistical physics.
Source: SISSA
During the day, our brain acquires new memories; at night, during sleep, it consolidates the important ones and eliminates the useless ones. A similar principle has been applied to Hopfield networks, one of the classic models of artificial intelligence inspired by the workings of the brain.
In 2025, Federico Ricci-Tersenghi and colleagues developed Daydreaming, an algorithm that combines the learning of new memories with the elimination of spurious ones, drastically improving the network’s capacity.
One limitation remained, however. These networks lose effectiveness when they work with real-world data, which are rarely perfectly balanced — for example, very bright or very dark images, in which white or black pixels overwhelmingly dominate.
In a new study published in the Journal of Statistical Mechanics: Theory and Experiment (JSTAT), Ricci-Tersenghi, together with Japanese colleagues, now presents a new version of the algorithm, capable of effectively handling realistic, strongly biased data.
A “classical” neural network
The networks proposed by John Hopfield in 1982 — work that would earn him the Nobel Prize in 2024 — consist of artificial neurons connected to one another and are among the simplest models of associative memory.
“Whenever we see any tree, our brain recalls the concept of a tree. This ability to associate many different representations with the same concept is what we call associative memory,” explains Federico Ricci-Tersenghi, professor of Theoretical Physics at Sapienza University of Rome and one of the authors of the new study.
The network does something similar: if it is trained, for example, with images of trees, dogs and apples, then when it “sees” a new image of a tree, a dog or an apple, even if partially degraded, it is able to connect it to the correct concept.
The simplest form of the Hopfield network can store a number of memories equal to only about 13% of its number of neurons. A network with one hundred neurons, therefore, can store only 13 memories. The rest of its memory is occupied by “false memories, attractors of the dynamics that do not correspond to any real memory,” Ricci-Tersenghi explains. These spurious memories are configurations that mix elements of real memories — a kind of hallucination — and, besides taking up space in the network’s memory, can also lead it into error.
Daydreaming
To address this problem, algorithms known as “dreaming” have been proposed, inspired by the role of sleep in biological brains. After the learning phase, the network is left to “dream”: starting from random configurations, it explores its own memory and tries to clean it of spurious memories. But if this “cleaning” process goes on for too long, the network ends up erasing correct memories as well, a phenomenon known as catastrophic forgetting.
In 2025, Ricci-Tersenghi and colleagues proposed the Daydreaming algorithm, which carries out learning and cleaning at the same time: the network keeps strengthening correct memories while eliminating spurious ones. “We combined daytime learning with the cleaning and consolidation phase of sleep, as if we were also dreaming during the day,” the researcher explains. Thanks to this strategy, the network’s capacity increased up to the theoretical limit of 100%, meaning one memory for every neuron.
Another problem remained, however — one that the original Daydreaming algorithm did not solve. Hopfield networks work very well when they are trained on perfectly balanced data. In the case of black-and-white images, for example, this means that the number of white pixels and black pixels is roughly the same.
Real-world data, however, are rarely so orderly. Think of heavily overexposed photographs, in which almost all pixels are white, or of very dark images. In these cases, images become very similar to one another, and the network struggles to understand which features really matter for distinguishing one memory from another.
Focusing on differences
The solutions proposed so far required global operations across the entire network, which are not very plausible from a biological point of view. “It is much more realistic for each decision to be made locally,” Ricci-Tersenghi explains. Biological neurons, in fact, are connected to a limited number of other neurons and never communicate with the whole brain.
In the new work, the researchers propose a local modification of the Daydreaming algorithm based on differences.
The example of face recognition helps to understand the idea. If all photographs are close-ups with a similar background, many pixels will be practically identical in every image. The shared information risks dominating the learning process. “If, instead, we work only on what changes relative to the average face, the differences emerge clearly,” Ricci-Tersenghi explains.
The new version of the algorithm, called Centered Daydreaming, no longer compares the absolute values of pixels, but their differences from the average. In the study, Centered Daydreaming kept the network’s ability to retrieve memories almost unchanged even with strongly biased data. The result extends the algorithm to conditions much closer to those of the real world, without giving up local learning rules, which are considered more biologically plausible.
Understanding how simple, brain-inspired models learn to distinguish what matters from what is irrelevant, Ricci-Tersenghi concludes, could in the future contribute to the development of artificial intelligence systems that are easier to understand and more energy-efficient.
Key Questions Answered:
A: Imagine a memory network trained to recognize pictures of apples, dogs, and trees. A spurious memory is an accidental mathematical “glitch” where the network blends elements of those images together, creating a distorted hybrid, like a dog with leaves or an apple with fur. These false attractors act like structural traps. When the network is shown a real, slightly blurry picture of a dog, it can get pulled into one of these glitches instead, confidently returning an erroneous hallucination because its internal energy landscape is cluttered with mathematical junk.
A: In a traditional network, if you try to clean out false memories after training is complete, the cleaning process eventually overcorrects and starts erasing valid, true memories—a breakdown known as catastrophic forgetting. The “Daydreaming” algorithm solves this by running the learning phase and the cleaning phase at the exact same time. It mimics a brain that is continuously dreaming during the day, reinforcing correct structural connections while instantly dissolving false, spurious attractors before they can take root, pushing storage efficiency to the 100% mark.
A: Standard Hopfield networks are built assuming perfect balance, like a black-and-white photo with an equal number of black and white pixels. Real-world data is rarely so neat. In a very dark photograph, nearly every pixel is black. Because the images share the exact same background data, the network gets overwhelmed by the shared information and fails to see the minor details that distinguish a dog from a tree. By shifting to “Centered Daydreaming,” the algorithm subtracts the average background noise, forcing the artificial neurons to focus exclusively on the subtle differences that make each memory unique.
Editorial Notes:
- This article was edited by a Neuroscience News editor.
- Journal paper reviewed in full.
- Additional context added by our staff.
About this AI memory research news
Author: Federica Sgorbissa
Source: SISSA
Contact: Federica Sgorbissa – SISSA
Image: The image is credited to Neuroscience News
Original Research: Open access.
“Daydreaming algorithm for Biased Patterns” by Mikiya Doi, Masayuki Ohzeki and Federico Ricci-Tersenghi. Journal of Statistical Mechanics Theory and Experiment
DOI:10.1088/1742-5468/ae8249
Abstract
Daydreaming algorithm for Biased Patterns
The Daydreaming algorithm has been recently proposed in Serricchio et al (2025 Neural Netw. 186 107216) as a learning rule that simultaneously reinforces stored patterns and suppresses spurious attractors to improve the storage capacity of the Hopfield model. Its effectiveness has been reported for both uncorrelated and correlated data.
However, the existing formulation has mainly assumed unbiased patterns, and the formulation for biased patterns has not yet been sufficiently established. Biased patterns are known to be much more problematic for models of associative memories. In this study, we reformulate Daydreaming for biased patterns motivated by the underlying rationale of the pseudo-inverse rule.
Specifically, we introduce the retrieval dynamics and an energy function based on the centered representation, and we derive a corresponding update rule for centered Daydreaming. We compare the centered pseudo-inverse rule with centered Daydreaming for biased patterns and examine the retrieval maps and eigenvalue distributions of the coupling matrices.
As a result, the centered Daydreaming yields larger basins of attraction than the centered pseudo-inverse rule, and such a beneficial property seems to be due to the broadness of the eigenvalue spectrum of the coupling matrix. To better understand this connection, we construct modified coupling matrices whose spectra interpolate between a pseudo-inverse-like spectrum and the Daydreaming one.
The results clearly indicate that the broader spectrum generated by Daydreaming contributes to the enlarged basin of attraction.
