Artificial Kuramoto Oscillatory Neurons

Authors: Takeru Miyato, Sindy Löwe, Andreas Geiger, Max Welling
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
Keywords: oscillatory neurons, feature binding, object-centric learning, reasoning, adversarial robustness

Abstract:
It has long been known in both neuroscience and AI that ``binding'' between neurons leads to a form of competitive learning where representations are compressed in order to represent more abstract concepts in deeper layers of the network. More recently, it was also hypothesized that dynamic (spatiotemporal) representations play an important role in both neuroscience and AI. Building on these ideas, we introduce Artificial Kuramoto Oscillatory Neurons (*AKOrN*) as a dynamical alternative to threshold units, which can be combined with arbitrary connectivity designs such as fully connected, convolutional, or attentive mechanisms. Our generalized Kuramoto updates bind neurons together through their synchronization dynamics. We show that this idea provides performance improvements across a wide spectrum of tasks such as unsupervised object discovery, adversarial robustness, calibrated uncertainty quantification, and reasoning. We believe that these empirical results show the importance of rethinking our assumptions at the most basic neuronal level of neural representation, and in particular show the importance of dynamical representations.

Decision: accept (oral)
Reviewer scores: 8, 8, 10, 10

Reviews and discussion:
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creation date:
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thread_type:
official_review

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creator: reviewer_dawq

<content>
summary:
The paper proposes a novel type of neural network where neurons’ activities are defined on the unit sphere (in N-dimensional space, where N is typically 2 or 4), and are updated based on Kuramoto equations. This can model oscillatory dynamics that can be leveraged for binding activations across neurons. The model is applied to various settings (unsupervised object discovery, image classification adversarial or natural noise robustness, Sudoku problem solving) and compared against relevant baselines (e.g. slot-based models, Transformers), each time demonstrating competitive performance. The model has clear potential to improve the state-of-the-art across domains.

strengths:
* A biologically inspired architecture, well described and motivated by neuroscientific findings
* A thorough set of experiments across diverse machine learning domains
* Meaningful comparisons with baselines revealing competitive performance

weaknesses:
* The dependence of the model on the choice of dimensionality N, although it is explicitly acknowledged in the paper, warrants further investigations (which can be the goal of future studies)
* Although baselines are roughly matched for memory/parameter size and flops, it is possible that GPU optimizations will make them more efficient than the AKOrN versions. There is unfortunately no discussion of this issue nor any report of absolute time required for training or inference. This could be added to the Appendix.

questions:
* on Lines 140-141, you argue that “we found that the use of C is necessary for stable trainings […] as a symmetry-breaking field”. But isn’t C also –and more simply—required for the network to receive inputs, and thus to perform any useful function?
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<thread>
creation date:
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thread_type:
official_review

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creator: reviewer_atlx

<content>
summary:
- This paper developed a feature-rotation-like neuron models (high-dim feature rotation encodes synchrony binding), a promissingly neuron-level building block that can be combined with diverse network architectures, like CNN and transformer. 
- The capability of the new architecture is validated on three independent tasks: object discovery (synthetic & natural image), reasoning on Sudoku task, robustness w.r.t adversarial attack in CI
FAR10. 
- On each task, the proposed model shows SOTA performance and interesting shining points: (1) For object discovery, the model scales to natural image without pretrained DINO model; (2) for reasoning, the model's correctness is naturally implied by the "internal coherence", indicated by the task-independent energy measure; (3) for adversarial attack, robustness seems a built-in property of the dynamics and again, the energy could imply the accuracy.

strengths:
I enjoyed reading this paper, impressed by the achievements on diverse tasks and also the authors's thoughts dispersed over the paper.  In brief, the strength of the paper includes:
- Soft grouping/clustering in neural network is an important question and often missing in main-stream ANNs.
- The proposed neuron model is neuroscientific motivated yet has concise fomula. The physical origin of the model makes it plausible to construct an "energy function", which turns out to be quite informative on the performance.
- While previous work on binding by synchrony mainly focuses on object discovery, the author extends these insights into reasoning tasks, and adversarial attack. And the performance on these new tasks is very successful.
- Lastly, the author provides a different conceptual angle for the binding by synchronization: the synchronization enables competitive learning, and therefore compresses representation to construct a bottle neck, which facilitate abstraction. As far as I know, it is a novel argument not presented in previous works and it also makes sense to me.

weaknesses:
For the weakness:
- It is not clear what is the actual conceptual contribution / nolvelty of the Kura model proposed in this paper compared with Lowe's Rotating Feature (and recent updates). For example, equation (6) is very similiar to the "binding mechanism activation" in  [1] and I have the intuition that this is the essential mechanism for the binding ability of model, similiar to [1]. If I am wrong, please correct me.
- When generalizing the original Kura model to high-dim, it is not clear whether the Proj in eq(2) is an 'equivalent' generalization of $sin(\theta_j-\theta_i)$. So it is not clear how much this term actually contribute to the computation of the network. Note that $sin(\theta_j-\theta_i)$ is a essential mechanism in original Kura model to guarentee the synchrony as a stable state and sin acts among each pair of neurons before the $\sum$, but Proj only acts after the $\sum$. (The first two concerns are mainly about how "Kura" the model is ?)
- Since the activity is constraints on the sphere, it loses the representational ability for encoding feature presence, which is an important conceptual diaviation from synchrony binding idea and is likely to limit its ability to infer in certain cases, e.g. when the feature is uncertain or weakly related to the task.
- On the experimental side, it mostly ignored the Lowe's model for comparison (Fig.3, Fig.4,Fig.5,Tab.1,Tab2, Tab3...) and other synchrony based model, e.g. [2].
- The code is not provided, so it is hard to evaluate the the reproducibility of the results.
- The author misses several essential recent works: e.g. [2] shows how oscillation emerges in spike-based model to bind features; [3] combine Kura model with ANNs and also achieved synchrony binding.
- The first two sentences in Abstract is quite confusing and the first paragraph in Introduction is not very informative.

[1] Löwe S, Lippe P, Locatello F, et al. Rotating features for object discovery[J]. Advances in Neural Information Processing Systems, 2024, 36.
[2] Zheng H, Lin H, Zhao R. GUST: combinatorial generalization by unsupervised grouping with neuronal coherence[J]. Advances in Neural Information Processing Systems, 2024, 36.
[3] Ricci M, Jung M, Zhang Y, et al. KuraNet: systems of coupled oscillators that learn to synchronize[J]. arXiv preprint arXiv:2105.02838, 2021.

questions:
(1) For the claim in the paper: 
> “binding” between neurons leads to a form of competitive learning where representations are compressed  in order to represent more abstract concepts in deeper layers of the network. 

> In fact, neighboring neurons tend to cluster their activities, and clusters tend to compete to explain the input. This “competitive learning” has the advantage that information is compressed as we move through the layers, facilitating the process of abstraction by creating an information bottleneck. Additionally, the competition encourages different higher-level neurons to focus on different aspects of the input (i.e., they specialize). This process is made possible by synchronization: like fireflies in the night, neurons tend to synchronize their activities with their neighbors’, which leads to the compression of their representations.

Can author clarify the reference if such claim is well-documented? If it is the novel claim of the author, the author should modify the tone to avoid overclaim or provide rationale or experiments to make sense of such claims.

(2) (Weakness section). What is the contribution of the "Proj", and will there be synchronization, e.g. given input image, if we replace Proj into general linear sum (ablation of Proj)? And what is the difference between eq(6) and binding mechanism in Lowe's work? Since the binding mechanism itself can generate synchronization, how to disentangle the contribution from (6) to contribution from the "Kura idea".

(3) Can author shows the segmentation results on binary datasets, like Zheng2024. The different color of object may make it eaiser to break symmetry compared with binary images. e.g. in Fig.4 the segmentation is more likely based on color instead of object-instance (the highlight is treated as an object). Besides, what is the actual input dimmension of the image to the model? To what extent it is downscaled?
(4) Why the energy function still make sense even if the J and $\Omega$ is not constraint by symmetry?
(5) What is the rationale of robustness? Is it the general outcome of dynamical models or is it the outcome of this Kura model? I suggest compare with related non-synchrony dynamical ANN models to disentagle the causal-effect.
(6) Why energy function, or the synchrony level in the network, can indicate the task performance? Can the author explain that more solidly?
(7) For the Sudoku task, can the author compare with those Ising-based models? e.g. Hopfield network or modern hopfield network. So as to disentagle the contribution for the "Kura idea" from that generally comes from attractor dynamics.

In general, I would like to vote for the accept of the paper, but I will also revisit the evaluation based on the author's response and other reviewer's comments.
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<thread>
creation date:
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thread_type:
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creator: reviewer_ynsz

<content>
summary:
the authors propose a neural ntwork architecture in which units are high-dimensional Kuramoto oscillatory neurons. This architecture leads synchronization or 'binding' between neurons which enables competitive performance in object discovery, reasoning and robustness.

strengths:
- the model is interesting and well defined
- the quantitative results appear impressive, though I am have some issues with their presentation (see below)
- the many numerical results presented, and comparisons to other models, suggests that overall a good deal of thought/time has been put in the manuscript

weaknesses:
- there are numerous minor issues with the text/presentation (see question below) which impacts the clarity of the paper 
- This not a field I have experience in, but I am not entirley convinced a fair comparison was made to other models for object discovery. Is the idea that AKOrN is a highly competitive model for models 'trained from scratch', and that comparing it to models with some pretrained parameters (e.g. Lowe et al. 2024) is unfair? I think this could be better clarified if so
- there is a notion from the abstract and sections 1/2 that this work is relevant to neuroscience, but the link feels quite tenuous to me. Is it reasonable to think of neurons as high-d points on a sphere?

questions:
- The format of figure references could be improved. I would recommend references as 'Fig. X'. Currently there is either the period missing or there's no space (e.g. 'Fig1' on line 96). Same with Tab X -> Table X. Should also be Figs for multiple figures (not Fig as in line 368)
- line 36: 'we follow a more modern dynamical view of neurons as oscillatory units'. Is this correct? Oscilliations have been studied in the brain for a long time; moreover, I would say that the majority of computational neuroscientists currently model neurons as scalar values.
- For table 3 results are presented with error bars over 5 seeds (which I favour), why is this not the case for any of the other results presented?
- line 121: 'we introduce a multi-dimensional vector version of the model' - this line is confusing as this was already employed by Chandra et al. and Lipton et al. (as cited). Perhaps replace with 'incorporate a mutli-dimensional model...'. On this subject could you clarify to me the exact novelty of the model? Is it that it's not been introduced in deep neural nets before? Or is it about the fact connections between oscillators J is not symetric in AKOrN? Relatedly, line 308 'our..models differ from these approaches in various aspects such as...' should really provide all fundemental differences beyond some examples. In fact, in any case what's the difference between 'asymmetric connections' and 'their symmetry breaking term'? 
- line 123 (and other places): what's C? the size of the neural population? this should be clarified
- line 245: what is 'proper' energy?
- line 323: I don't understand how these convolution/self-attention layers are iterated upon. What are the shared parameters?
- I didn't understand the implementation for the sudoku task. 1. is the entire 9x9 soduku grid fed all at once, or one digit at a time? and 2. how come x_i takes the value c_i when a digit is given, does the model no longer follow equation 2? Or is this just the initial value taken before the Kuramoto steps?
- line 425: 'the energy is a good indication of the solution's correctness'. Why? I can believe that stability is often a desirable thing in tasks but think this line deserves some elaboration
- line 464: would it be possible to give a brief description/overview of what 'common corruptions' are?
- line 506: why is it surprising that confidence and accuracy are linearly correlated? 
- line 527: 'the oscillator...'. I didn't understand the logic of this sentence, could you explain it to me please?

Other comments:

- the notation x_{chw} is confusing, I think x_{c,h,w} is better
- on line 201 the terms should be switched around (the initial state of the next block is the thing being defined). But also, how does this fit in with line 189 where X^(l,0) = X^l?
- the metrics in all of the tables should be clarified. I presume the unit is %, but e.g. in the case of Table 1 I'm not entirely sure what the metric is. % objects correctly identified?
- in the results section the language can be overly strong. For example 'far better' (line 377) and 'vastly better' (line 416) do not appear to me as very scientific language. 
- in the discussion there is talk of small N resulting in a strongly regularized model. Given that the size of N nor regularization was mentioned in the prior text, this comes rather abruptly as is confusing to the reader

typos: 
- line 150: 'set C a silhouette' -> 'set C as a silhouette'
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creation date:
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thread_type:
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creator: reviewer_uqq5

<content>
summary:
In this work, the authors proposed to use Kuramoto oscillatory neurons (AKOrN) to replace the usual thresholding units in deep neural networks. The basic neuron architecture can be applied to layers with different connectivity patterns, including convolutional and self-attention layers. Models with Kuramoto oscillatory neurons show promising performance in visual discovery and reasoning tasks, they are also more robust to adversarial attacks and input corruptions.

strengths:
1. The neuronal architecture draws inspiration from physics and neuroscience and is very novel in deep learning.
2. Model performance is impressive across a range of tasks.
3. Extensive experiments show that the method applies to mainstream architectures including convolutional neural networks and transformers.

weaknesses:
1. While AKOrN shows promising performance on the selected tasks, the model might not work as well on more "classical tasks" such as image classification. 
2. Although the motivation of AKOrN is clear from a neuroscience perspective, how, and why the model shows superior performance in the tested task is not well understood.

questions:
1. Judging from the method, AKOrN seems to be much more computationally costly than the usual thresholding units. I noticed that the authors controlled the number of channels in AKOrN models, but showing the training/inference time would be helpful.
2. It would be helpful to include some explicit discussion on the weakness of the method. i.e., what tasks are probably more suitable for this method and why?
3. As mentioned above, more analysis might help establish the advantage of AKOrN. For example, the paper shows that the AKOrN models show better robustness, but it is not clear from the design that they are more robust. Thus analysis of the network dynamics on some toy tasks might be helpful (e.g., synthesized classification tasks on low-dim vectors).
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creator: authors

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title:
General comments

comment:
We owe great thanks to all reviewers for their helpful comments and suggestions for improving our manuscripts! 
We have revised our manuscript since the initial submission, incorporating reviewers’ suggestions and including updated results for the object discovery tasks.

Let us highlight the changes we made:
1. Added **Model analysis** section in the Appendix, where we provide an extensive comparison of various model designs. This section addresses some of the concerns and questions raised by the reviewers. 
2. Improved the object discovery performance by **up-tiling**. Up-tiling enables us to compute finer object assignments on SSL features, and substantially improves SSL models’ performance including our AKOrN. AKOrN is now better than any other prior works on PascalVOC, and is better on COCO than existing methods except for a recent model SPOT (See Figures 4,5 and Tables 1,2 for the improved results and Section D.1.2 for the methodological details of up-tiling). 

---

We here address some concerns that were commonly raised by different reviewers.

- ***Regarding the computational overhead in AKOrN*** (by reviewers DAWQ and uqq5)

The main computational overheads in AKOrN are the projection and normalization in Eq.(4) and Eq.(5). The computational order of those operations is linear w.r.t the feature sizes, thus the overhead becomes ignorable as the model size increases. The below table shows the training and inference time of AKOrNs and their non-Kuramoto counterpart models in different tasks. Training time is the time taken to complete a single gradient step (excluding data loading). Inference time is the time taken for a single forward pass with a mini-batch size of 100.  The unit is milliseconds.
| Dataset   | ItrConv | AKOrN |
|----------|-----------------:|-----------------:|
| CIFAR10 (Train)  | 30.72         | 52.84             |
| CIFAR10 (Infer)   | 8.84          | 13.58           |

| Dataset   | ItrSA   | AKOrN  |
|----------|-----------------:|-----------------:|
|   Sudoku (Train) | 88.18           | 96.59              |
|   Sudoku (Infer)  |  29.00       | 33.40           |


| Dataset  | ItrSA  | AKOrN |
|----------|------------------:|------------------:|
|   CLEVRTex (Train) | 637.10           | 661.50             |
|  CLEVRTex (Infer)  |   41.40        | 45.50            |

 In CIFAR-10 experiments using relatively small convolutional networks, AKOrN requires approximately 1.7 times the training computation time compared to its non-Kuramoto counterparts. However, in experiments with larger channel sizes, such as the CLEVRTex object discovery and the Sudoku solving, the overhead drops significantly to 1.1x and 1.04x, respectively. For inference, the additional computational cost with AKOrN is slightly higher compared to training but remains modest, at 1.15x for CLEVRTex and 1.09x for Sudoku. We include these runtime comparisons in the updated manuscript as Figure 17.

- ***Regarding the rationale of the robustness by AKOrN*** (by reviewers uqq5, AtLx, and ynsZ) 

One possible explanation for AKOrN’s robustness is the fluctuations occurring in the oscillatory states.  We observe that, especially in the first layer, each Kuramoto update generates _noise-like_ perturbations (Please see the animation file **cifar10_block1.gif** in the Supplementary Material). This _deterministic noise_ would help the model avoid learning features vulnerable to perturbations. Interestingly, simply adding Gaussian noise during ResNet training does not produce models as robust as AKOrN. The table below compares the robustness of ResNets trained with Gaussian noise to that of AKOrN. We tuned the noise strength to match the clean test set accuracy of AKOrN and selected two ResNet models with similar clean accuracy. Across all robustness metrics, AKOrN consistently outperforms the ResNets, particularly excelling in adversarial robustness. The same table is shown in Table 8, and a visualization of the fluctuations is provided in Figure 15.

| Model              | Clean Accuracy ↑ | Adv Accuracy ↑ | CC Accuracy ↑ | ECE CC ↓ |
|--------------------|------------------|----------------|---------------|----------|
| ResNet (σ = 0.2)   | 85.2            | 22.3           | 75.1          | 2.3      |
| ResNet (σ = 0.225) | 83.9            | 25.5           | 73.6          | 2.6      |
| AKOrN (N = 2)      | 84.6            | **64.9**       | **78.3**      | **1.8**  |
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Metareview:
metareview:
This paper takes on a long-standing idea from the computational neuroscience community, namely, that binding of different features in an input can be done using oscillations in unit activity that synchronize to indicate binding. The authors propose Artificial Kuramoto Oscillatory Neurons (AKOrN), which employs oscillatory units based on the Kuramoto (1984) model.

The authors claim that their system is capable of binding together features in an input (e.g. objects in an image). They further claim that this provides performance improvements across various tasks such as unsupervised object discovery, adversarial robustness, calibrated uncertainty quantification, and reasoning. They demonstrate the vailidity of these claims using a range of benchmarks, such as CLEVRTex, PascalVOC, and Sodoku puzzles.

The primary strength of this paper is that it manages to solve a long-standing technical issue (how to bind features) with a brain-inspired solution that no one else has previously been able to get working really well. It also provides strong empirical support for the claims, and illuminating analyses of the model. There were some weaknesses with clarity on the model's computational overhead and critical parts, but those were addressed in rebuttal. Altogether, it is a very strong paper. As such, a decision to accept for an oral was reached.