Recurrent neural network dynamical systems for biological vision

We thank the reviewer for their time and insightful suggestions. For the sake of consistency across reviews, we would first like to clarify a few definitions used in our response:

• We think that the reviewer is referring to "convolutional RNN" as a CNN connected to an RNN (typically LSTM in literature but could be any type of RNN). This is most likely the widely-adopted definition that the reviewer is using. However, in our response, we will refer to these models as CNN-RNN models.
• In contrast, we will refer to convolutional RNNs as RNNs whose recurrence is represented by a convolution, just like in this work. This is also consistent with the terminology used by other reviewers.

Advantages over CNN-RNN

In the first point raised in the weaknesses section, as well as in the question section, the reviewer made a sharp observation that simply using CNN-RNN architectures would be sufficient to reproduce the results in Figure 3, and therefore questioned the advantage of our proposed model. We raise several orthogonal points below that we hope would help the reviewer see our work in a better light.

• Not all results can be reproduced by CNN-RNN. We agree that the results in Figures 3 and (quite likely but we cannot confirm) Figure 5 can be obtained by the CNN-RNN architecture, but the results in Figures 4B and 6 remain exclusive to our continuous-time model. The phenomenon in Figure 4B is attributed to the temporal lag that arises due to every layer being continuous in time and require time to ramp up. Figure 6 compares neural trajectories across time in CNN layers with neural activity in the macaque ventral stream (their biological visual system), but the CNN in a CNN-RNN architecture is static in time.

• Applying results from CNN and RNN neuroscience. CNNs and RNNs have both been proposed as models of the brain (or parts of it). By building a hybrid model, we may integrate results from both fields which may lead to unified results and potentially generate new hypotheses about how the brain works.

• Redefining expectations for this work. If the end goal of this work is to find a model that can perform the feats that we show in the paper, then CNN-RNNs would probably have an edge over our model. But our goal here is to build models of the brain and help improve our understanding of how continuous-time dynamics work in a competent model of vision. This means that we incorporate biological constraints in our models, even if they are detrimental to final performance. Just like how the brain operates in continuous-time, including its visual system, we build a model that also runs continuous in time.

• CNN-RNNs requiring less complex training. We absolutely agree that the computational resources required to train a model with a static CNN is considerably less than the resources required to train CordsNet. We like to view this as a positive aspect of our work, rather than a negative one. We are knowingly building a model that is harder to train, with constraints that are likely to impede training and testing performance (short inference time, continuous-time and recurrent properties offering nothing to image classification performance), so that we can obtain a model that can be used to study the dynamics underlying visual processing in a continuous-time dynamical system. In fact, we consider the success of our training, using our proposed algorithm, to be a key result of this work.

Suitability of ImageNet and other use cases

The reviewer has raised concern that ImageNet, being a static dataset that does not change in time, may not be the most suitable task for showcasing how our model works. We agree with this viewpoint and are impressed by their pursuit for engineering optimality. Fundamentally, the theoretical and experimental sides of neuroscience work closely with each other. New experiments are conducted from proposed theories, and in turn new models are built from experimental results. Similarly, this work is designed based on influences from a variety of past works in neuroscience. There has been decades of results on dynamical models of the brain, which are still being worked on today. In recent years, CNNs have gained traction in vision neuroscience, and one of the main driving forces of this is the BrainScore platform (link here), where CNNs are trained on ImageNet, and their neural activities are then compared to neural data. This also comes with years of results reported by various neuroscience groups. Our goal here is to bridge the gap between results from dynamical system models and results from CNN models for vision. To that end, training and evaluating our model on ImageNet would best facilitate this.

Spontaneous activity penalty

We thank the reviewer for the suggestion. We have performed ablation studies in the general response that would help clarify your doubts on the impact of this term.

We would also like to specifically comment on the reviewer's intuition, which is largely in the correct direction. We cannot make strong claims for non-linear models, but in the linear case of our model, it would be the largest eigenvalue of the recurrent weight matrix that would determine whether the model is monostable or not. This is definitely related to the magnitude of the weights to some extent, just as the reviewer has suggested.

We are heartened by the extremely positive review and high regard the reviewer has for our work. We thank the reviewer for their time and encouragement.

Comparison of CORNet on tasks in Figure 5

We agree with the reviewer that CORNets could potentially arrive at different solutions compared to CordsNets. This would be in line with our efforts to bring analyses that are commonly found in dynamical systems (such as the cognitive tasks in Figure 5, where a dynamical RNN would have to be plugged) to the CNN vision neuroscience community. It is exactly our hope that our analysis of CordsNet as a dynamical system would spark such applications for CORNets and other models in vision neuroscience.

Performance across time of CordsNet

We note that since CordsNet arrives at a steady-state at time of inference, it can indefinitely maintain the accurate classification for as long as the image is still presented, even far beyond the trained duration. In addition, CordsNet is also able to reset back to baseline when an image is removed, and correctly classify another new image when it is presented at a later time. These results can be found in Figure 3.

Self-supervised learning and embodied tasks

We fully agree with the reviewer that the literature of CNNs in neuroscience have progressed far beyond straightforward supervised learning. As a starting point for incorporating dynamical system, we believe that our work will motivate future endeavors on these ideas (including for ourselves).

Citation error

We thank the reviewer for the clarification.

BrainScore clarification

In order to evaluate our models' ability to capture temporal signatures in neural data, we had to extend the BrainScore similarity metric to account for time.

The original formulation performs a partial-least-squares fit between CNN activations, with shape [number of images, number of nodes] with neural data, with shape [number of images, number of neurons]. This is done with time-averaged neural activity.

Here, we make the most minimal extension, so that we are now fitting CNN activations, with shape [timesteps $\times$ number of images, number of nodes] with neural data, with shape [timesteps $\times$ number of images, number of neurons], which means that the time-averaging step is omitted. Without the time averaging step, the BrainScore drops.

We thank the reviewer for the helpful suggestions. In response to these points, we have performed additional analyses and made several changes to our submission, as explained below (or referred to general response).

Important information in appendix

We thank the reviewer for raising this important point. We would like to direct the reviewer to our general response, which provides a detailed plan on how we plan to rebalance the results in the main text and the appendix.

Issue of unsupported claims

We agree with the reviewer that the performance of CordsNets fine-tuned on ImageNet is lower than their CNN counterparts with the same convolutional layers, and that this discrepancy may possibly be large enough to challenge our claim. We have made the necessary changes to the submission to account for this, as detailed below.

• Line 163 updated to read, "We also find that our fine-tuned models perform only slightly poorer than their initial feedforward CNN counterparts, which nonetheless represents a significant improvement over existing continuous-time dynamical models in visual processing."
• Line 387 updated to read, "Our fine-tuned CordsNets are able to attain accuracies that are only slightly poorer than their feedforward CNN counterparts."

We feel that the term "comparable" is modestly subjective. We initially chose this word because of the current inability of continous-time dynamical systems to process visual information, as seen in the third column of Table S5. Dynamical RNNs are barely beating chance level in a classification task with 1000 classes, which is rightfully "incomparable" to CNNs. By being in a very broad performance range (50% to 80%), we feel that we may use the term "comparable" in this very specific context. However, we will still make the changes above for clarity.

At the same time, we wish to also review and defend the other claims that we have made about this work, which we hope that the reviewer would agree with.

• Dynamical expressivity analysis. We claimed that we have rigorously explored the possible dynamical regimes that our model can express, and also made comparisons to other model architectures. We provided instances of different dynamical regimes in Figure 1B. In addition to CordsNet, we also trained low-rank, dense and sparse RNNs (Section B.3, which is in the appendix in the initial submission) on five different cognitive tasks in neuroscience (Section B.4), for three different network sizes (Section B.5). We then compared the activity trajectories of all these trained models (results from Figures S2 and S3).
• New training algorithm. We have provided full details of our training algorithm in Section 3 in the main text. In addition, we explain certain design choices and perform ablation studies for the loss function in the general response.
• Autonomous and robust inference. We show in Figure 3B that our model is able to reset to baseline levels after stimulus presentation, and can then accurately classify a new input image. This supports our claim on autonomous inference. We also provide evidence that our model is robust to noise, due to the "evidence integration" mechanism that is present in dynamical systems. This is explained in lines 190-196, and equation (7).
• Analytical toolkit. We claim to provide a toolkit consisting of Arnoldi and power iteration algorithms and partial SVD specifically for convolutional architectures (i.e. our model). We demonstrate the effectiveness of our implementation of Arnoldi iteration in Figure 4A, and applied partial SVD to a particular scenario in Figure 4C.
• Image-computable models. We have trained CordsNet appended to a fully-connected RNN on four actual tasks in neuroscience literature (Figure 5, left), and provided evidence of our trained models (Figure 5, right). At the same time, we acknowledge that this analysis is brief, which is why we make no predictions or neuroscientific claims based off our results here. An in-depth investigation on this topic would be outside the scope of this work. The intention is to show the potential applications of CordsNet as a multi-area, image-computable model, which we have delivered.
• Prediction of neural activity. We leveraged on a robust and popular benchmark platform known as BrainScore to compute similarity metrics between model activations in CordsNet to actual neural activity recorded in the visual system of the macaque monkey (Figure 6). We show statistical significance in our results.

We hope that with this overview, the reviewer would agree with the stated accomplishments of this work.

Missing references

We thank the reviewer for the citation. and thoroughly comb through relevant literature in our final version.

Motivation for loss function and ablation studies

We thank the reviewer for this suggestion. We have done an extensive ablation study for our proposed loss function, as detailed in the general response.

Model behavior subjected to ambiguous input

The concept of multistable perception has been extensively studied in neuroscience, with many theories proposed based on dynamical systems. Coincidentally, we have shown preliminary results on this phenomenon in Figure 4C, where the activity just goes to some interpolated activity space between the two interpretations, which is currently still a steady-state response.

This reviewer (8dkQ) seems to not understand the field of "modelling the primate visual cortex". I suggest you to read this paper: https://www.nature.com/articles/s41593-019-0520-2 before making a new judgement. The proposed method is definitely novel because there is no dynamical model of CNN that can model visual cortex before. The proposed method definitely contributes to computational neuroscience. The proposed method also shows some interesting properties, e.g., in Fig. 5.

We thank the reviewer for raising these important points in their review. Our general interpretation of the reviewer's concern is that this work failed to provide what seems to be minimally-required benchmarks, which in turn translates to lack of evidence for any performance improvements. The reviewer is also well-read in the field of convolutional, recurrent and fixed-point architectures, and thus was able to pinpoint key literature that overlap with our proposed work. From this point of view, it can be hard to find significance in this work.

As such, we would like to provide a personalized summary of this work that brings the reviewer's aptly-raised concerns into perspective and explains why we believe our work is still a significant contribution despite those concerns.

• Dynamical systems in neuroscience study how neural activity evolves over time and how these changes relate to brain functions. They are prevalent in neuroscience, recently in the form of continuous-time RNN dynamical systems with biological constraints such as realistic neuron time constants.
• Training with a handicap. Such dynamical RNNs are harder to train than vanilla RNNs in machine learning, because of the continuous-time property that inflates the number of time steps when coupled with the fast time constants present in biological neurons. Since vanilla RNNs are already ill-suited for image processing, this makes dynamical RNNs even harder to train on ImageNet, and for this reason no such model has been proposed to date.
• Motivation and potential. Yet, there is a strong need for such a model to be built, so that decades of dynamical systems theory can be applied to a model that can actually process natural images. Such a model would be studied by the theoretical neuroscience community to understand continuous-time dynamics underlying biological visual information processing, which we hope will give rise to new ideas and theories to be tested.
• Defining expectations. To that end, our objective is to train the first dynamical system to classify ImageNet at an accuracy that would be deemed as “comparable to CNNs”. We approach this goal with the understanding that the constraints we impose on our model are likely to be detrimental to performance, and we will most likely not be achieving anything SoTA outside the field of neuroscience at this stage; we therefore omit benchmarks typically found in machine learning literature.
• Significance of our architecture. CNNs and convolutional RNNs have already been proposed and extensively analyzed in vision neuroscience as candidate models of the visual system. The logical approach would therefore be to build a continuous-time dynamical extension of these models. Such a model would not only build on the work on convolutional architectures by the vision neuroscience community, but also incorporate the legacy of dynamical systems from the wider neuroscience community.
• Evaluating our success. The best top-1 accuracy achieved by our models on ImageNet (<60%) is modest compared to what vision models today can do. Understandably, as a result of the aforementioned biological constraints, our models fail to outperform basic CNNs with the same convolutional layers (but we are close). But ultimately, we can reasonably say that the models that we trained are indeed performing some meaningful processing of visual information, and that we have successfully built a dynamical system that can classify natural images.
• Focusing on the objective. Finally, we dedicated the majority of the main text, including figures 3,4,5 and 6, to show how our trained model can be studied as a continuous-time dynamical system in various fields of neuroscience, thereby highlighting its potential applications and how it opens up new avenues of research in these areas.

We hope that this summary is able to help the author gain a different (and hopefully more optimistic) perspective of our work. We now address specific points raised by the reviewer below.

Issue on DEQs

We thank the reviewer for bringing DEQs into the discussion. Due to the character limit, we refer the reviewer to our discussion on DEQs in the general response.

Issue on linearity

CordsNet is nonlinear in the exact same way that RNNs or DEQs are nonlinear. The evolution of activity across time in CordsNet is described by equation (1) in the main text, which includes a ReLU activation function. While training the nonlinear model, we split training into several stages (Figure 2B), and one of the stages involves linear models (lines 142-152), which may be the source of this misunderstanding.

Comparison with convolutional LSTM/GRU

There are two key reasons why comparison with gated RNNs is not necessary.

• Artificial gating mechanisms are not biologically-realistic and do not fall within our objectives.
• We are trying to model the biological visual system, which is (generally) not the part of the brain that stores memories or makes decisions based on long-term dependencies. Our task is classifying ImageNet, where gating mechanisms may not offer much advantage.

Issue on scattered points

We thank the reviewer for the suggestion. Overall, we feel that this work lies comfortably in the intersection of machine learning and neuroscience, and that there would be an audience for this work in this conference. We also acknowledge the reviewer’s concern that the work presents too many points that have not been sufficiently covered. CordsNet is both a dynamical RNN and a CNN, where each architecture, on their own, has been studied by the neuroscience community for years. As such, the narrative of our paper is focused on introducing the model, how it compares with past RNN or CNN models in neuroscience, and briefly demonstrating the many ways in which new research directions can be explored using this hybrid model. It is also why we developed tools to help analyze our model (lines 214-216).

We truly appreciate the reviewer for the encouraging and positive review.

Technical details in the appendix

The reviewer has expressed concern that important technical details that are in the appendix should briefly be mentioned in the main text. We agree with this point and refer the reviewer to the general rebuttal for all the changes we plan to make to our submission.

Random dot motion task and area MT

The reviewer has correctly mentioned experimental literature pertaining to evidence of neural activity tuned to random dot direction and coherence in area MT in the dorsal stream. In general, the RNN that we append to the back of our model represents either a decision-making part of the brain, such as the prefrontal cortex, or a motor region such as the frontal eye fields (which is still prefrontal). While MT contains direction-tuned cells, the eventual motor response (in the form of eye saccades) would likely be similarly tuned. However, this is ultimately an (unsubstantiated) interpretation on our end. A more rigorous analysis is required for any conclusions to be drawn.

Interpretation of results in Figure 5

The main goal of the results in Figure 5 is to illustrate the point that we can now build completely continuous-time models that accept real images as an input, rather than abstract one-hot vectors as found in many past works. We do provide some interpretation of the solutions we found, but to draw any conclusions about the brain from this would require a much more in-depth analysis that is currently outside the scope of this work. We do not have the neural data to compute BrainScore, and we foresee that the dataset would not be big enough (small number of coherence levels) for strong statistical significance.

Missing citation

We thank the reviewer for pointing out this missing citation, we will add this citation into our introduction. We will also thoroughly find and include other relevant literature in our final version.

In this comment, we will address the references.

Existing RCNN papers

These are important references that we will add to our paper as soon as the editing window is up. They will largely go into the introductory text (specifically the top of page 2), where we already have made reference to a few other recurrent CNN papers. These include:

https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2017.01551/full https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1008215 https://www.pnas.org/doi/abs/10.1073/pnas.1905544116 https://www.sciencedirect.com/science/article/pii/S0959438820301768 https://arxiv.org/abs/2209.11737v2

[1] https://proceedings.neurips.cc/paper/2020/hash/766d856ef1a6b02f93d894415e6bfa0e-Abstract.html This work is great and may have potential uses for continuous-time models. We will add that to our conclusion.

[2] https://arxiv.org/abs/2308.12435 As we have claimed, "Analyzing our architecture as a dynamical system is computationally expensive, so we develop a toolkit consisting of iterative methods specifically tailored for convolutional structures." Analysis of an RNN dynamical system typically involves studying the recurrent weight matrix, which in this case is a convolution. Speaking for myself, I do not see any indication of dynamical systems or tailoring towards convolutional structures in the proposed reference. Nonetheless we will also cite this work as it is relevant.

[3] https://www.biorxiv.org/content/10.1101/2024.09.09.612033v2 Regarding applications in general, we are positioning our work as potential alternatives to CNNs (in the context of Figure 5) and RCNNs (in the context of Figure 6). There are many instances of CNNs being applied in neuroscience, as we mentioned in the second sentence of our abstract, and it would be hard to cite them all. Also, this particular reference seems to be quite recent in late 2024 (in light of the fact that the submission deadline for this conference is in May 2024), but we will also cite it as it is clearly relevant to our work.

Hi and thank you for your interest in this work! It is important that our work reaches the neuroscience RCNN community, because we are presenting a new architecture that aims to be alternatives to RCNN for the same exact uses. We will address the references in a separate comment, but we believe that a single point will largely address the key misunderstanding here.

We would like to point out that none of the references cited are related to continuous-time RNN dynamical systems (which is basically the title of our paper). Some early references of such an architecture are from around 30 years ago (these references are all in the paper, numbered accordingly):

[3] 10.1126/science.274.5293.1724

[5] doi.org/10.1073/pnas.93.23.13339

[7] doi.org/10.1523/JNEUROSCI.16-06-02112.1996

A continuous-time RNN dynamical system is typically characterized by:

1. a leaky term (the -r in equation 2)
2. biological neuron time constants (the T in equation 2)
3. continuous-time smooth trajectories (the dr/dt in equation 2)
4. no normalization (although this can easily be explored)

These properties set dynamical RNNs apart from machine learning RNNs. They are also harder to train because having smooth, continuous trajectories inflate the number of time steps, but they also have cool dynamics such as oscillations, chaos, fixed points, mono- and multi-stability (see Figure 1B-C, S1 and S4). These are the dynamics that we want to bake into CNNs. In fact, one of the references that you cited actually made this distinction, citing Dayan and Abbott:

doi.org/10.1016/j.conb.2020.11.009 in a section titled "Continuous-time versus discrete-time dynamics"

In the paper, we referred to what you call RCNNs as discrete-time CNNs (DT-CNNs), because they lack the properties mentioned above, and dedicated almost an entire figure to differentiate ourselves from this architecture in Figure 3. So training for optimal ImageNet performance over 4-5 timesteps is never the point of this work.

Speaking for myself, I hope that you can review Figures 1, 3 and S1,2,3,4 thoroughly and gain a different perspective about our work, and it would particularly address the following question:

"if you modified your algorithm to only unroll the RCNN for a limited # of timesteps and then artificially interpolate between them, would your results be any worse, I wonder"

Finally, the reviewers are aware of the existence of CORNet, arguably one of the most famous works of RCNN in neuroscience, and have agreed that this work is a contribution beyond that work.

Thank you for reaching back. Although I did/do understand that extending RCNNs to the continuous time domain (what many would call "actual" recurrence) is novel, discrete RCNNs are a related class which is well-studied and I don't understand how the insights found in that literature wouldn't speak to the continuous time domain.

"The activity in CORNet-RT varies over time, and spikes briefly at the time step in which it is trained to make an accurate classification of the input image. It is not able to make an accurate prediction at all other times." - Training a RCNN to output the class only at the last timestep is one way to train them. In the other class of models (Spoerer et al. and also the other papers I mentioned) they are trained to output the correct class over all timesteps. This provides interesting abilities to the RCNN wherein even if trained only for N timesteps, it can maintain its classification accuracy ad-infinitum as long as the stimulus is presented (see Fig.7 in https://arxiv.org/abs/2308.12435). It would be interesting to check how these models behave in the settings in Fig. 3. I reckon the profiles would look quite different from CORNet-RT, although they might still differ from your network - but that would be interesting.

Furthermore, there's no reason that dynamics such as oscillations, chaos, fixed points, mono- and multi-stability cannot arise in regular RCNNs given the right objectives. Fixed points / limit cycles are possibly easy - for e.g. Fig. 2 of https://arxiv.org/abs/2308.12435 already shows the change in representation reducing over time. If the other phenomena are essential for the computations required in a task I apriori see no reason why they can't just emerge in RCNNs (non-linear dynamics in discrete spaces do produce all these phenomena).

"So training for optimal ImageNet performance over 4-5 timesteps is never the point of this work." This is fair - I was not trying to say that you need to beat the state-of-art but was trying to hint at scalability - I should've been more precise. This relates to my Q: "if you modified your algorithm to only unroll the RCNN for a limited # of timesteps and then artificially interpolate between them, would your results be any worse, I wonder" - this would be similar to having larger "steps" in solving your differential equation - but I understand that you want to emphasize the dynamics and not worry too much about accuracy for now. This point was triggered by the abstract wherein you stated "Our models preserve the dynamical characteristics typical of RNNs while having comparable performance with their conventional CNN counterparts on benchmarks like ImageNet." which one can also read as - "this will scale similarly to the counterpart CNNs on Imagenet".

In ref [2], the network is a RCNN - it is convolutional and recurrent and what we analyzed are dynamics. Dynamics can exist in systems which are not mathematically designed to follow the usual state transition equations like Eq. 2 in your paper., and analysis of the dynamics can also take on various forms. This is related to what I'm generally trying to communicate with my comments - that there have been interesting, complimentary, developments in the DT-RCNN space that seem to be closely aligned with many of the interests of this paper.

Ref [3] is very new indeed - it was meant as a reading suggestion - actually all the references were. I thought the camera-ready version was already out which is why I posted a comment instead of writing an email to you - so that these references stay attached here as a comment.

I apologize if the initial comment came across as too aggressive - I just feel that research in both continuous and discrete time RCNN share the same goals and that they should proceed in tandem. Please let me know if I am still mischaracterizing something.

Those are great points being made -- we will briefly reply some specific points and end with a final general statement.

• Continuous vs discrete-time: There are many cases where discrete-time models behave just like continuous-time models, and just as many that do not due to discretization errors. For example, simulating dx/dt = sin(x) for too large of a $\Delta t$ would give you a wrong solution.

• RCNNs in Figure 3: Discrete-time models can potentially achieve the same effects in the figure (I acknowledge Figure 7 and I am curious as well). However, this has been known theoretically for decades in dynamical RNNs, see Chapter 7.4 of Dayan and Abbott. This is an example of a long-known result from dynamical RNNs being applied to RCNNs, which is the point of this work.

• ImageNet discussion: Relevant results are in Table 1, in the last two columns, where we compare fine-tuned CordsNets to their static CNN counterparts without recurrence, showing comparable performance. This is the result the abstract is referencing (it scales from R2 to R8 at least).

Overall, we are proposing a hybrid model between continuous-time RNNs and (R)CNNs. Our audience would be split between the two subfields. The RCNN community can ask questions like "isn't discrete-time good enough?", just like how the dynamical RNN community will ask "is the brain actually performing convolutions?" which is arguably more controversial. If this work can generate such discussions, then we believe we have contributed something to the field.

Thanks for engaging in this discussion - I’ll edit the title of the first comment so it doesn’t come across as negative to someone checking out the paper.

“If this work can generate such discussions, then we believe we have contributed something to the field.” I agree. Looking forward to new ideas on this continuous-time - discrete RCNN continuum. I believe the aforementioned references and this discussion will anyway benefit anyone reading your paper.