🧠 neuroscience

Predictive learning with local plasticity in excitatory-inhibitory networks

This paper demonstrates that recurrent excitatory-inhibitory networks with purely local plasticity, particularly BCM-like rules, can achieve predictive inference and learn sparse spatiotemporal features by maintaining weights on a consistency manifold, thereby linking predictive coding to biologically plausible circuit mechanisms without requiring explicit error representations.

Original authors: Reis Aguiar, H., Hennig, M. H.

Published 2026-09-04

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Reis Aguiar, H., Hennig, M. H.

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

The human brain is a vast network of billions of neurons, constantly firing and connecting to make sense of the world. At the heart of this activity are two opposing forces: excitatory signals that encourage neurons to fire, and inhibitory signals that calm them down. For decades, scientists have wondered how these simple local interactions allow the brain to do something as complex as predicting what will happen next. One leading idea, called predictive coding, suggests that the brain is always guessing the causes of its sensory inputs and only pays attention to the surprises. However, a major puzzle has remained: how can a biological network, which can only adjust its connections based on the immediate activity of nearby cells, perform this sophisticated guessing game without a central computer to calculate errors?

A new study from the University of Edinburgh offers a compelling answer by showing how a specific type of neural circuit can learn to predict the future using only local rules. The researchers built a computer model of a network where excitatory neurons send signals forward, while inhibitory neurons provide feedback to keep the system in check. They discovered that for this network to learn effectively, the connections between neurons must stay in a very specific balance. If the excitatory connections drift too far from this balance, the network fails to learn the true patterns in the data. The team found that a particular learning rule, similar to one observed in real animal brains, naturally keeps the network in this sweet spot. This rule allows the network to learn from static images, like edges and shapes, and also from moving videos, where it can anticipate the direction of motion and even fill in missing parts of a sequence when the input disappears.

The researchers started by testing their model on simple, static images, such as crossed bars or patches from natural photographs. They wanted to see if the network could learn to break these complex images down into their basic building blocks, like lines or edges, without being told what to look for. They compared several different ways the network could adjust its connections. Some methods, which relied on simple coincidence between neurons, caused the network to drift away from the optimal balance, leading to messy and inaccurate learning. However, when they used a rule that adjusts connections based on how well the current activity matches a prediction, the network stayed perfectly balanced. This specific rule, which the authors call the "predictive rule," ensured that the inhibitory feedback always matched what was needed to cancel out the noise in the input. As a result, the network learned to represent the images using a sparse set of features, meaning only a few neurons fired at a time to describe the picture, much like how the visual cortex of mammals works.

The study went further by adding a layer of time to the model. Real life is not just a series of still pictures; it is a continuous stream of events. To mimic this, the researchers added a mechanism where the network's current state could influence its future state, creating a form of memory. When they trained this updated network on videos of moving bars, it learned to detect not just the shape of the object, but also the direction it was traveling. Remarkably, the network developed the ability to complete sequences. If the video of a moving bar was suddenly blanked out, the network continued to "see" the bar moving in the same direction, filling in the gap based on what it had learned. This suggests that the network had internalized the rules of motion and could run them forward in its own mind, a capability that is crucial for navigating the world.

To make the model even more realistic, the team removed the artificial pauses between learning steps and let the network learn from a continuous stream of data, just like a living brain does. In this continuous version, the network developed smooth, curved patterns of activity rather than sharp, rigid switches. When the researchers stimulated this network with random noise after it had finished learning, the network spontaneously replayed the sequences it had seen before. It would cycle through the same patterns of activity, as if remembering a past event. These patterns formed a ring-like structure in the network's activity space, a shape that allows the system to represent a single value, like a direction, very precisely. This behavior mirrors what happens in the hippocampus of the brain, where animals replay their paths during rest, suggesting that the same local learning rules could underlie both prediction and memory.

The findings challenge the long-held view that the brain needs special "error neurons" to calculate the difference between what it expects and what it sees. Instead, the study suggests that the brain might simply use the mismatch between its own internal predictions and the incoming signals to adjust its connections directly. The researchers showed that by keeping the excitatory and inhibitory parts of the circuit in a tight, consistent relationship, the network can perform complex inference without needing a separate error signal. This implies that the plasticity, or the ability to change, of the excitatory connections is the key to prediction. The study provides a strong theoretical link between the local, biological rules of learning and the global, intelligent behavior of prediction, suggesting that the brain's ability to anticipate the future is a natural consequence of how its circuits stay balanced.

Technical Summary: Predictive Learning with Local Plasticity in Excitatory-Inhibitory Networks

Problem Statement

Predictive coding offers a normative framework for understanding cortical computation, positing that neural circuits infer latent causes of sensory inputs by reconciling data with an internal generative model. However, a significant gap remains in explaining how biologically plausible networks, specifically recurrent excitatory-inhibitory (E-I) circuits with local plasticity, can support predictive inference and representation learning. Classical predictive coding implementations often rely on non-biological mechanisms, such as symmetric feedback weights, explicit error neurons, or non-local updates. While previous work has shown that Hebbian/anti-Hebbian networks can learn factorized representations, an exact analytic link between these plausible local mechanisms and the principles of predictive coding theory has been lacking. This work addresses the question of whether E-I networks with purely local plasticity can perform predictive inference without explicit error representations.

Methodology

The authors propose a recurrent E-I circuit where learning is driven by local plasticity rules derived from a predictive coding objective.

Theoretical Framework

The model assumes sensory data $x$ is generated from latent causes $y$ via a linear generative model $x = Dy + \epsilon$ with a factorized prior $p(y)$ . The learning objective is to minimize the joint energy function (negative log-likelihood):
$E(y, x; D) = \frac{1}{2}\|x - Dy\|^2_2 + \phi(y)$
where $\phi(y)$ represents the prior energy.

The authors introduce a Hebbian/anti-Hebbian network architecture with:

Feed-forward excitatory weights ( $W$ ): Approximating $D^\top$ .
Recurrent inhibitory weights ( $M$ ): Approximating the second moment of the latent code $E[yy^\top]$ .

The Consistency Manifold

A central theoretical contribution is the identification of a consistency manifold. For the network dynamics to be equivalent to the predictive coding objective, the recurrent inhibition must match the inhibition required by the objective. This requires the condition $M \approx WW^\top$ .
The authors derive that for the network to learn optimally, the weights must remain on a manifold where the feed-forward weights predict the inhibitory dynamics ( $Wx = y$ ). If this condition is violated, the latent code is inferred under a perturbed energy landscape, leading to suboptimal learning.

Plasticity Rules

The paper compares several local plasticity rules for the excitatory weights $W$ :

Predictive Rule: Derived to maintain the consistency condition exactly under a Gaussian prior. The update is proportional to the consistency error $e = y - Wx$ :
$W \leftarrow W + \eta(y - Wx)x^\top$
This rule vanishes when the consistency constraint is satisfied, keeping weights on the manifold.
Hebbian Rule (with weight decay): $W \leftarrow W + \eta(yx^\top - W)$ . The authors show this rule pushes weights off the consistency manifold because the correction term includes a component $W(xx^\top - I)$ that is non-zero for single samples.
BCM-like Rule: A Bienenstock-Cooper-Munro rule with an activity-dependent threshold.
Oja and Földiák Rules: Standard alternatives for normalization and decorrelation.

Experimental Setup

The authors tested these rules on:

Linear Gaussian Case: Non-whitened 2D Gaussian data and crossed-bars datasets to verify whitening and consistency.
Non-Gaussian Case: Sparse coding tasks using crossed-bars, MNIST, and ImageNet patches with exponential priors (requiring rectification).
Temporal Dynamics: Introduction of delayed recurrent excitation ( $P$ ) to model sequential data (moving bars, moving MNIST, CATCAM natural video).
Continuous Model: A time-continuous E-I network with separate excitatory and inhibitory populations, implementing asynchronous updates and Dale's principle, using a BCM-like rule.

Key Results

1. Analytic Link and the Predictive Rule

The authors establish that learning in E-I circuits requires weights to remain on the consistency manifold. The predictive rule is analytically shown to maintain this condition exactly for Gaussian priors. In contrast, the classic Hebbian rule, while optimal for adapting inhibitory weights, pushes excitatory weights away from the manifold, leading to drift and suboptimal solutions.

2. Learning Under Non-Gaussian Priors

Under non-Gaussian (sparse) priors, the consistency condition cannot be satisfied exactly due to non-linearity and weight constraints. However, empirical results show:

The predictive rule and BCM-like rule keep weights sufficiently close to the consistency manifold to learn optimal sparse, factorized features (e.g., edge detectors).
These rules result in lower reconstruction errors and better decorrelation of latent codes compared to Hebbian, Oja, or Földiák rules.
The BCM rule, which depends on the difference between post-synaptic activity and a sliding threshold, approximates the predictive rule's behavior effectively.

3. Spatiotemporal Learning and Sequence Completion

By adding recurrent excitation ( $P$ ) to model temporal dependencies:

Networks learn spatiotemporal factors, developing direction selectivity from natural video data (CATCAM).
The networks acquire the ability to complete partially observed sequences. When inputs are blanked, the recurrent dynamics continue the learned movement patterns.
Both predictive and BCM rules outperform standard Hebbian rules in sequence completion tasks and direction selectivity indices.

4. Continuous Dynamics and Attractor Structures

In the continuous, time-continuous model with asynchronous updates:

The network develops low-dimensional attractor-like structures.
Stimulating the trained network with noise induces replay of learned spatiotemporal sequences.
Activity trajectories form ring-like structures in low-dimensional space, consistent with observed neural population dynamics (e.g., in the hippocampus and visual cortex).
The model develops smooth tuning curves rather than discrete factor detectors, resembling biological neural responses.

Significance and Claims

The paper claims to bridge the gap between normative predictive coding theories and biologically plausible local circuit mechanisms. Its primary contributions are:

No Explicit Error Representation: The work demonstrates that predictive inference does not require explicit "error neurons" or symmetric feedback. Instead, prediction errors are absorbed into the local synaptic update rule as the discrepancy between recurrently inferred activity and feed-forward predictions.
Normative Role for BCM: The results suggest a normative role for BCM-like plasticity in excitatory synapses, showing it effectively approximates the optimal predictive rule required to maintain the consistency manifold.
Local Circuit Optimization: It provides a mechanism by which local plasticity rules can self-organize E-I circuits to perform probabilistic inference and representation learning, linking synaptic dynamics directly to the optimization of a generative model.
Experimental Predictions: The authors propose that excitatory plasticity should depend on the mismatch between recurrently inferred and feed-forward predicted activity. They suggest that manipulating recurrent inhibition should alter the magnitude or sign of excitatory plasticity and the sparseness of learned representations, offering testable hypotheses for sensory systems.

The authors remain modest regarding limitations, noting that the point-estimate approximation (MAP) may struggle with ambiguous inputs and that exact equivalence to the predictive coding energy requires idealized conditions (whitened inputs, centered data) that are only approximately met in non-Gaussian or continuous settings. Nevertheless, the findings offer a unified view of how local plasticity supports cortical computation.