PCOMPBIOL-D-25-01766

pyhgf: A neural network library for predictive coding

PLOS Computational Biology

Dear Dr. Legrand,

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Academic Editor

PLOS Computational Biology

Hugues Berry

Section Editor

PLOS Computational Biology

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Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Authors:

Please note here if the review is uploaded as an attachment.

Reviewer #1: This manuscript introduces pyhgf, a Python-based library for implementing predictive-coding neural networks in JAX. The authors position the software as a flexible framework for developing dynamic neural networks that embody predictive-coding principles, and they provide extensive discussion of biological motivation, model structuring, and Bayesian inference workflows.

The work addresses a clear gap in the computational neuroscience and computational psychiatry tool ecosystem, as I am not aware of any similar library: differentiable, and highly modular, and also able to merge with contemporary probabilistic programming tools. A small caveat: there are two predictive coding library in JAX already, that allow to implement a different formulation of the algorithm, and have a different goal: that is, to simulate Rao and Ballard’s formulation of predictive coding (and more general ones) for machine learning and control tasks. The first one is PCX, more suited for neural network training and deep learning, that allows to implement “Equilibrium propagation-like” predictive coding models [1], the second is NGCLearn, more suited for (copy-paste from the GitHub repo): building, simulating, and analyzing biophysical / neurobiological systems, spiking neuronal networks, predictive coding circuitry, and biomimetic (NeuroAI) agents that learn in a biologically-plausible manner [2].

Minor:

I do not think there is any overlap in scope with the library you are proposing, so the existence of these libraries does not influence the novelty of yours. However, I’d like to see a paragraph explaining differences and similarities. In the same way, I’d like to see a small discussion on how the algorithms your library is able to implement relate/differ from the ones often seen in machine learning conferences (a survey can be found in [3], and many references in the already cited [1]).

The manuscript provides a thorough conceptual framing of predictive coding and its computational motivations: The software design principles are clearly articulated, particularly the emphasis on modularity, transparency of internal states, and compatibility with inference engines. A very minor comment is that, while the theoretical exposition is clear, some readers less familiar with predictive-coding frameworks may benefit from a more concise conceptual example early in the manuscript. This also relates to what is probably my main concern in terms of exposition: the lack of code examples. The manuscript describes the software conceptually but does not show code directly in the text. Given that the contribution is a software package, including a very small runnable example would strengthen accessibility for readers.

Minor:

Would it be possible to come up with 2/4 simple code snippets, that can be added in the main body of the manuscript? It can be a running example that gets increasingly complex the more topics are introduced, or anything else that would allow the reader to grasp the simplicity of the library by simply reading the paper.

[1] Pinchetti, Luca, et al. "Benchmarking Predictive Coding Networks--Made Simple." ICLR 2025.

[2] https://github.com/NACLab/ngc-learn

[3] Salvatori, Tommaso, et al. "A survey on neuro-mimetic deep learning via predictive coding." Neural Networks (2025): 108161.

Reviewer #2: Overview:

The manuscript introduces pyhgf, a computational neuroscience/computational psychiatry codebase that focuses on building dynamic neural networks, especially those based on predictive coding networks. pyhgf codebase is entirely written in Python and, as of version 0.2.0, it makes use of JAX, an XLA tensor computing library that is robust in leveraging GPUs and TPUs in order to maximize the computational speed of the neural network implementation. The codebase provides specific examples of building generative models, forward fitting and parameter inference as well as Bayesian multi-level modelling, parameter recovery, and model comparison. The framework natively supports generalized hgf (which is typically used in the inference of parameters) and can be integrated into existing libraries such as PyMC.

Strengths:

+ The paper identifies real limitations in existing tools for predictive coding (PC) research. The development of neural networks in typical tensor computing libraries is an obstacle to biologically-inspired learning principles. Therefore, the introduction of pyhgf aims to tackle/address this gap.

+ This library can be integrated into PyMC, a popular framework for Bayesian statistical modeling and variational inference (VI) algorithms

+ The paper is well-written overall, with effective use of figures. Clear presentation!

+ The manuscript contains comprehensive examples. Section 3 provides detailed demonstrations with figures, including forward fitting, parameter inference through MCMC, and so on.

+ The library is implemented in modules such that it can be extended. Overall, there is good implementation design.

Weaknesses:

- The codebase and examples rely heavily on small/toy simulated data. It would be much better/appropriate to also demonstrate the library’s efficacy on more complex tasks, such those based on MNIST and natural image/video data (CIFAR, etc.), to demonstrate the applicability and robustness of the library.

- Missing performance benchmark comparison with existing, longer-standing/historical implementations (and related work) such as PyMC (Oriol et al., 2023), NGC-Learn (Ororbia & Kifer, 2022; this is also a JAX-based library) and MATLAB HGF toolbox (Frässle et al., 2021). The performance metrics used can include: computational speed, fitting time, memory usage, scalability when the model size increases, and so on.

- The manuscript does not address computational limitations. For example, how large can the network grow/be? What is the relative computational runtime when a particular kind of network grows in parameters? There should be significant discussion in the manuscript and perhaps accompanying analysis of what the library cannot do (to guide future users and inform what the library is not as well as what it is).

- Some missing references (related, PC-backing literature for the communities that this article is targeting); missing two key reviews (2nd is published/long-established) for PC background for the modeling tasks (and computational neuroscience side of this article/library) that are related to this work (and third is another missing JAX-based PC library not compared/mentioned as well):

** van Zwol, Björn, Ro Jefferson, and Egon L. Broek. "Predictive coding networks and inference learning: Tutorial and survey." arXiv preprint arXiv:2407.04117 (2024).

** Salvatori, Tommaso, et al. "A survey on neuro-mimetic deep learning via predictive coding." Neural Networks (2025): 108161.

** Innocenti, Francesco, et al. "Jpc: Flexible inference for predictive coding networks in jax." arXiv preprint arXiv:2412.03676 (2024).

As a result of the points I raised above, I recommend (major) revision for the manuscript as some work needs to be done before it is ready for publication/public dissemination.

Reviewer #3: **Summary**

In this article, the authors present a JAX- and Rust-based library for implementing dynamic networks that perform predictive coding (PC) updates, with a particular focus on hierarchical Gaussian filtering (hGF) models. The manuscript first introduces the theoretical foundations, followed by demonstrations using a generalized hGF model.

Overall, the proposed toolbox represents a valuable addition to the set of computational tools available to researchers working with hGF and predictive coding models. While the core ideas are clearly articulated, the manuscript would benefit from further streamlining (see Point 4 below) that enhances its accessibility and impact. Below, I outline several comments that I hope will help improve the manuscript.

**Major Comments :**

1. Improved description of the library and API components

Although detailed API documentation is understandably delegated to external docs, a methods-oriented paper would benefit from introducing some of the core library components directly in the manuscript. For example, the authors could include pseudocode or a NumPy-like schematic, as is common in machine learning publications, to illustrate the main computational flow. This would enable readers to more easily connect the theoretical framework to its implementation. Additionally, the authors could consider adding and highlighting high-level APIs that allow users—particularly those without a programming background—to instantiate and run an hGF model with minimal code, broadening the library’s accessibility.

2. Stronger integration with existing literature

a. There exist other graph-based implementations of predictive coding that are not discussed. For example, the authors could contrast their network with those of [1].

b. Several neural-network-based predictive coding libraries are also absent from the discussion. The authors are encouraged to acknowledge and situate their contribution relative to this work, including but not limited to PredNets, Predify, pypc, NGC-Learn, and PrediRep. Doing so would clarify the novelty and situate the present toolbox within the broader ecosystem of PC libraries.

3. Lack of a central conceptual contribution

A key concern with the current manuscript is the lack of a clearly articulated overarching message beyond the presentation of the toolbox itself. While the software is undoubtedly a useful resource, many of the principal conceptual insights appear to have been addressed in the authors’ prior publications. Including additional empirical or methodological insights would help strengthen the significance of the work. Given the use of JAX and Rust, maybe the authors could emphasize and quantify the computational advantages that arise from this design choice?

4. Speculative discussion and focus

The discussion surrounding topics such as self-organization, causality, and others is largely speculative and not directly supported by empirical evidence in the manuscript. I would recommend keeping the main text focused on concrete methodological contributions and moving these more speculative considerations to a concise section of the Discussion. Alternatively, providing additional results that directly support these claims would also help address the concern raised in Point 3.

References

1 - Salvatori, Tommaso, Luca Pinchetti, Beren Millidge, et al. 2022. “Learning on Arbitrary Graph Topologies via Predictive Coding.” arXiv:2201.13180. Preprint, arXiv, October 12. https://doi.org/10.48550/arXiv.2201.13180.

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Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: None

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Reviewer #1: No

Reviewer #2: No

Reviewer #3: No

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Dear Mr. Legrand,

We are pleased to inform you that your manuscript 'pyhgf: A neural network library for predictive coding' has been provisionally accepted for publication in PLOS Computational Biology.

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Academic Editor

PLOS Computational Biology

Hugues Berry

Section Editor

PLOS Computational Biology

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Reviewer's Responses to Questions

Comments to the Authors:

Please note here if the review is uploaded as an attachment.

Reviewer #1: Dear authors,

I have reviewed the submitted changes, and evaluated them in line with my review and the one provided by the other reviewers. I'm happy with the re-submitted version, and hence recommend acceptance.

Reviewer #3: I would like to thank the authors for their detailed responses. I think the manuscript reads well now and seems fit for publication.

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Have the authors made all data and (if applicable) computational code underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data and code underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data and code should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data or code —e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #3: Yes

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Reviewer #1: No

Reviewer #3: No