# Constrained Predictive Coding as a Biologically Plausible Model of the Cortical Hierarchy

arXiv: https://arxiv.org/abs/2210.15752 · HTML: https://mlsanity.com/paper/2210.15752/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Oct 2022.

## Built on by (most cited first, 18 of 18)

- [Sequential Memory with Temporal Predictive Coding](https://mlsanity.com/paper/2305.11982/) (2023)
- [Predictive Coding Networks and Inference Learning: Tutorial and Survey](https://mlsanity.com/paper/2407.04117/) (2024)
- [Normative framework for deriving neural networks with multi-compartmental neurons and non-Hebbian plasticity](https://mlsanity.com/paper/2302.10051/) (2023)
- [Correlative Information Maximization: A Biologically Plausible Approach to Supervised Deep Neural Networks without Weight Symmetry](https://mlsanity.com/paper/2306.04810/) (2023)
- [Adaptive whitening with fast gain modulation and slow synaptic plasticity](https://mlsanity.com/paper/2308.13633/) (2023)
- [Divide-and-Conquer Predictive Coding: a structured Bayesian inference algorithm](https://mlsanity.com/paper/2408.05834/) (2024)
- [Transformers are Universal Predictors](https://mlsanity.com/paper/2307.07843/) (2023)
- [Meta predictive learning model of languages in neural circuits](https://mlsanity.com/paper/2309.04106/) (2023)
- [Duality Principle and Biologically Plausible Learning: Connecting the Representer Theorem and Hebbian Learning](https://mlsanity.com/paper/2309.16687/) (2023)
- [Curl Descent: Non-Gradient Learning Dynamics with Sign-Diverse Plasticity](https://mlsanity.com/paper/2510.02765/) (2025)
- [Predictive Coding Graphs are a Superset of Feedforward Neural Networks](https://mlsanity.com/paper/2603.06142/) (2026)
- [Context selectivity with dynamic availability enables lifelong continual learning](https://mlsanity.com/paper/2306.01690/) (2023)
- [Biologically-Motivated Learning Model for Instructed Visual Processing](https://mlsanity.com/paper/2306.02415/) (2023)
- [Unlocking the Potential of Similarity Matching: Scalability, Supervision and Pre-training](https://mlsanity.com/paper/2308.02427/) (2023)
- [Score Broadcast and Decorrelation: A General Framework for Broadcast-Based Credit Assignment](https://mlsanity.com/paper/2605.30638/) (2026)
- [Learning in Deep Networks under Dale's Constraint](https://mlsanity.com/paper/2608.06963/) (2026)
- [One-Step Next-Latent Prediction Is Not a World Model](https://mlsanity.com/paper/2609.36227/) (2026)
- [Least Squares for Time Series Forecasting](https://mlsanity.com/paper/2610.03812/) (2026)

## Abstract (arXiv)

Predictive coding has emerged as an influential normative model of neural computation, with numerous extensions and applications. As such, much effort has been put into mapping PC faithfully onto the cortex, but there are issues that remain unresolved or controversial. In particular, current implementations often involve separate value and error neurons and require symmetric forward and backward weights across different brain regions. These features have not been experimentally confirmed. In this work, we show that the PC framework in the linear regime can be modified to map faithfully onto the cortical hierarchy in a manner compatible with empirical observations. By employing a disentangling-inspired constraint on hidden-layer neural activities, we derive an upper bound for the PC objective. Optimization of this upper bound leads to an algorithm that shows the same performance as the original objective and maps onto a biologically plausible network. The units of this network can be interpreted as multi-compartmental neurons with non-Hebbian learning rules, with a remarkable resemblance to recent experimental findings. There exist prior models which also capture these features, but they are phenomenological, while our work is a normative derivation. The network we derive does not involve one-to-one connectivity or signal multiplexing, which the phenomenological models required, indicating that these features are not necessary for learning in the cortex. The normative nature of our algorithm in the simplified linear case allows us to prove interesting properties of the framework and analytically understand the computational role of our network's components. The parameters of our network have natural interpretations as physiological quantities in a multi-compartmental model of pyramidal neurons, providing a concrete link between PC and experimental measurements carried out in the cortex.