Learning and aligning single-neuron invariance manifolds in visual cortex

Thank you for your insightful review and feedback. We believe we can address most of your concerns, but we have one quick clarifying question to ensure we fully understand your concern. In particular, regarding comparisons with Klindt et al. (2017) and Ustyuzhaninov et al. (2019):

If we understood correctly, you’re suggesting adding a comparison showing explicitly that our method is able to identify clusters despite differences in RF position (as in Klindt et al. 2017) and orientation (as in Ustyuzhaninov et al. 2019). We believe our results already show this: we simulated neurons with similar invariances but spatially transformed RFs using different affine transformations, including differences not only in shift and rotation but additionally size and shear (Fig 2A bottom row). Qualitatively (Fig 2B) and quantitatively (Fig 2D,E), our method captures both the RF pattern and its transformations along the manifold, as well as the underlying affine transformations, showing robustness in accounting for spatial variations beyond those in Klindt et al. (2017) and Ustyuzhaninov et al. (2019).

However, it could be that we did not understand the exact issue you are raising. We would therefore be grateful if you could clarify what you felt is missing from these analyses. Thank you again for your input!

We would like to thank you for your detailed and thoughtful review. We are glad that you appreciated the clarity of our work, the quality of its figures, and our focus on continuous transformations. We also appreciate your insightful suggestions and address each of your points below.

Re: Discussion of nullspace and is there a way to game the objective function?

Neurons are generally insensitive to large portions of the visual field (in our case, the input image) outside of their receptive field (RF). These areas could be considered as belonging to the “nullspace” of the neuron. If we understand your question correctly, you’re suggesting that maximally exciting images (MEIs) could potentially be modified in these regions without affecting the neural response, thereby potentially gaming the objective function. Our method incorporates two mechanisms to specifically prevent this. First, the combination of a norm constraint (see section 3.1 Template Learning under Objective function) with the activity maximization objective ensures that features (i.e. pixel variations) are concentrated only in regions relevant to the neuron. Because neurons “like” contrast, this constraint makes the method put contrast only where it gets activation from the neuron. It is important that the selected norm value is low enough to prevent pixel saturation within the RF and to avoid the appearance of random artifacts outside the RF. This mechanism alone should be sufficient to avoid features to appear outside of the RF of the neuron. Second, in addition, we apply the contrastive regularization terms only within the RF of the neuron using a mask computed from the MEI which as well satisfies the norm constraint (see appendix B, Template Learning Details). This ensures that the contrastive regularization term does not encourage features to appear in the neuron’s nullspace. Taken together these mechanisms ensure that the focus remains on regions of the input space that actually drive neurons' response.

Re: Comparison with Klindt at al. (2017) and Ustyuzhaninov et al. (2019)

Klindt at al. 2017 and Ustyuzhaninov et al. 2019 propose to cluster neurons based on the feature weights of the readout of each neuron. Klindt at al. 2017 proposes a method that divides neurons in functional clusters that are obtained up to differences in RF position, Ustyuzhaninov et al. 2019 extend this to rotations. Our method finds clusters that extend this property to the remaining affine transformations. Few additional differences are present between our method and theirs: First, our method is model agnostic, whereas their method is model dependent, meaning that while our method can be applied to any accurate response predicting model of neurons their method necessitates to obtain an accurate response predicting model using a specific type of model architecture (e.g. Ustyuzhaninov needs a rotation equivariant core and could not straightforwardly be applied to a transformer). Second, our method aims at clustering neurons specifically comparing their invariances, whereas their method aims at clustering neurons according to the whole response function. Whether this is to be preferred or not, might depend on the questions that are being asked. It is worth pointing out that in a situation where two neuron clusters differ in invariance, our method will clearly show that the differentiating factor is the invariance and how it is different, whereas the Ustyuzhaninov et al. 2019 method will just indicate that the two clusters of neurons are different but not directly offer insights on how. Third, in our case the metric used for template alignment and neural clustering is neural activity, which has an intuitive interpretation. In contrast, the readout weight vectors used by their method to perform clustering are not immediately interpretable in terms of functional properties, mainly because distances in the space of readout feature weights might not be representative of distances in terms of neural functional properties. This is strongly related to the issue of feature redundancies mentioned in the related work section for which you ask for clarification (see reply to the next point).

Re: 117-119 please clarify:

In Ustyuzhaninov et al. 2019, clustering relies on a model with a core+readout architecture. This architecture incorporates equivariance to shifts and rotations, with the readout combining features from different channels at specific spatial location and orientation using a vector of feature weights for each neuron. Clustering is performed based on these feature weight vectors, which can be considered a fingerprint of the neuron’s properties. The issue of redundancy arises as the same neuronal response function can be realized by different combinations of channels of the core. Consequently, two neurons with very similar properties might present very different fingerprints (weight vectors) and could therefore be grouped into different clusters. The severity of this issue is closely tied to the channel dimensionality of the core’s output (i.e. the model capacity), and while it might be mitigated by some regularization approach or pruning, it still remains a non-trivial issue to address. Since our clustering method is model-agnostic and based on neuron responses rather than readout weights, it is not subject to this issue. In the updated version of the manuscript, we revised the text to make the explanation of this point clearer.

Re: Empirical comparison of our method to Klindt at al. (2017) and Ustyuzhaninov et al. (2019): Regarding the analysis you requested, we are working on it and will update you once the results are ready.

Re: Invariance is a broader concept than just invariance around the MEI

Thank you for this remark. A similar point was raised by another reviewer (5DFt), and we agree that it would be beneficial to expand on the justification behind our choice of focusing on invariances over maximally exciting stimuli only. It would indeed be interesting to explore a neuron’s invariances at different response levels, as this could provide a more comprehensive characterization of the entire response function of the selected neuron. However, capturing invariances at lower-than-optimal response levels presents challenges that go beyond the scope of this study. For instance, images that both satisfy the norm constraint and achieve the target activation level could be generated in two ways: 1) by learning patterns only within the RF, which is the desired outcome, or 2) keeping the pattern within the RF similar to the case of peak activation but with a lower norm and “cheat” the norm constraint by introducing additional patterns outside the neuron’s RF. In the latter case, the features outside the RF represent uninformative contrast allocations unrelated to the neuron’s function and are merely artifacts of the procedure. Resolving this issue would require the development of additional mechanisms. Furthermore, the invariance manifolds are likely to become very high-dimensional and exhibit a complex topology, significantly increasing the difficulty of capturing them effectively. Due to these challenges, exploring invariances at suboptimal activation levels extends beyond the scope of this work. Our focus on invariances at (or close to) the peak activation is motivated by three factors. First, this approach aligns with how invariances are often described in neuroscientific research. For instance, when one says that a complex cell present phase invariance, it is assumed that it is selective to Gabor stimuli with specific parameters and it is invariant to changes in phase of Gabor stimuli with specifically these parameters (i.e. the stimuli for it is particularly selective to). Second, investigating stimuli (and invariances) at peak activation is fundamental to understanding neural coding, as it reveals the features to which neurons are most selective and, therefore, offers critical insights into their functional role. Third, invariance manifolds at peak activation are likely to be lower dimensional, which makes their identification a more constrained yet meaningful problem, that enables us to focus on the alignment aspect of our approach for population-wide comparisons of single-neuron invariance manifolds. In the updated version of the manuscript, we ensured to specify that our method focuses specifically on the invariances at peak activation level, justifying the reasons behind our choice.

Re: Why do you use periodic 1D?

Given the focus of our work on V1, we chose a 1D latent space as we expected to mainly identify neurons with no invariances (simple cells) or a one-dimensional phase invariance (complex cells). Since phase invariance is periodic, we designed the latent space of our INR to be periodic as well. Exploring the dimensionality of neurons belonging to the non-gabor clusters with higher dimensional latent space (as suggested in Baroni et al. 2023) is a natural extension but was not the focus of our current work.

We are happy to let you know that we managed to perform the analysis you requested, comparing our method to that of Klindt et al. (2017) and Ustyuzhaninov et al. (2019), and have included it in the new revision of our manuscript (Appendix H). The results empirically demonstrate the flexibility of our approach in accounting for affine transformations (similar to the results in Figure 2 of the initial submission), while highlighting that the other two methods can only account for shifts (Klindt et al. 2017) and rotations (Ustyuzhaninov et al. 2019). Importantly, we applied our approach on both architectures from Klindt et al. (2017) and Ustyuzhaninov et al. (2019), and show that on both models our method yields consistent clusters, highlighting the model-agnostic nature of our approach.

Thank you again for your valuable input, and we look forward to hearing your feedback on both this new analysis and our earlier clarifications.

We appreciate your positive feedback, recognizing the potential of our approach in classifying neural invariances and uncovering new insights into V1 neurons, and we are thankful for your constructive suggestions to improve clarity and accessibility of our submission.

Re: Clearer introduction of the term “invariance manifold”:

Thank you for pointing out the need for a clearer introduction of this concept. In the revised manuscript, we ensured to define the term “invariance manifold” explicitly before using it and included a concrete example to provide additional context and clarity for readers unfamiliar with the concept.

Re: Affine transformations are reasonable in V1, what would be a suitable choice in high-order visual areas?

This is an intriguing question. In V1, it is well established that neurons perform similar computations across varying locations, orientations, and scales—differences that can be described by affine transformations. This understanding directly motivates our approach, which seeks to ‘filter out’ these established functional variations, allowing for a comparison of neuron invariance properties beyond these aspects. While functional organizational principles such as retinotopic mapping and eccentricity-related scaling of receptive field (RF) size are thought to persist after V1, it remains an open question which specific stimuli these neurons are selective to and what invariance transformations connect these stimuli. One option would be to proceed iteratively, first investigating the code with methods such as ours, identifying the most significant transformations along which the code varies between neurons, incorporating these back into the methods in order to ‘filter them out’ and hence discovering further transformations, and so on. But as you already indicate, these broader questions are beyond the scope of the current work.

Re: This paper relies on existing methods which are only briefly explained

To address this concern, we expanded the related work section and improved the method section ensuring that it is now self-contained and accessible and that it does not rely on familiarity of the reader on past work.

Re: Definition of $\mathcal{Z}+ and \mathcal{Z}-$ and discussion of their importance

Thank you for raising this question. It seems that how near and distant points are defined in the latent space requires further clarification, as this was also noted by another reviewer. In the appendix, we stated that we chose a neighboring radius spanning 10% of the latent space in each direction. This means that if a grid of 20 points is sampled from the latent space during training, 2 points in each direction are considered “near,” while the rest are classified as “distant.” However, the method is robust to the exact criteria used for classifying near and distant points, as long as the classification is reasonable. In the revised version of the manuscript, we improved the corresponding paragraph (Appendix B) to provide greater clarity of the definitions of ‘near’ and ‘distant’ points. Additionally, we included an analysis (Appendix G) to demonstrate the method’s robustness to the specific details of their definition.

Thank you for your insightful feedback and kind words - we are very happy to see that our effort to create clear and informative figures is appreciated.

Re: Exploring invariance manifolds at lower levels of activation:

We agree that modifications of the objective function could enable exploration of broader questions about visual neurons coding properties. The objective you propose, incorporating a target (relative) activation level using the parameter $\lambda \in [0, 1]$ like $\left|\frac{\alpha_i}{\alpha_{\text{MEI}}} - \lambda\right|$, could indeed be used in our approach exactly as you suggested. Such an objective could allow us to investigate more extensively the complex response profiles of neurons, effectively exploring the level-sets of the high-dimensional response function. However, while this would be an interesting and insightful direction to explore, it presents some challenges, as invariances at $\lambda > 0$ can span large portions of the input space. For instance, at $\lambda > 0$, images that both satisfy the norm constraint and achieve the target activation level could be generated in two ways: 1) by learning patterns only within the RF, which is the desired outcome, or 2) keeping the pattern within the RF similar to the case of $\lambda = 0$ but with a lower norm and “cheat” the norm constraint by introducing additional patterns outside the neuron’s RF. In the latter case, the features outside the RF represent uninformative contrast allocations unrelated to the neuron’s function and are merely artifacts of the procedure. Resolving this issue would require the development of additional mechanisms. Furthermore, at $\lambda > 0$ , the invariance manifolds are likely to become very high-dimensional and exhibit a complex topology, significantly increasing the difficulty of capturing them effectively. Due to these challenges, exploring invariances at $\lambda > 0$ extends beyond the scope of this work. Our focus on invariances at (or close to) the peak activation is motivated by three factors. First, this approach aligns with how invariances are often described in neuroscientific research. For instance, when one says that a complex cell present phase invariance, it is assumed that it is selective to Gabor stimuli with specific parameters and it is invariant to changes in phase of Gabor stimuli with specifically these parameters (i.e. the stimuli for it is particularly selective to). Second, investigating stimuli (and invariances) at peak activation is fundamental to understanding neural coding, as it reveals the features to which neurons are most selective and, therefore, offers critical insights into their functional role. Third, invariance manifolds at peak activation are likely to be lower dimensional, which makes their identification a more constrained yet meaningful problem, that enables us to focus on the alignment aspect of our approach for population-wide comparisons of single-neuron invariance manifolds.

Re: Related work using GANs:

Thank you for bringing these two works to our attention. Despite using a different approach, we agree that they are related to our work, as they also investigate neural invariances and response properties using generative models. We have included these works in our revised manuscript as part of the broader context of methods for studying neural invariances.

Re: Redundancy of the numerator of contrastive learning objective:

While we agree with your intuition that the INR's parametrization partially encourages smoothness, due to its nonlinear nature it can learn discontinuities (i.e. big changes in image space corresponding to small changes in the latent space). An objective function with only a diversity enforcing term cannot distinguish between solutions with and without discontinuities. This problem is resolved by the addition of an explicit term enforcing similarity between nearby images (the numerator) which discourages such discontinuities resulting in a smooth transformation along the learned invariance manifold.

Thank you for taking the time to give us valuable feedback on many aspects of our submission. We are confident we can address all of your concerns, which we believe would make this a stronger submission and ultimately more understandable.

Response to the main concerns:

Re: Novelty of the Continuous Invariance Manifold (Abstract Clarification):

You are right that learning a continuous invariance manifold has already been introduced by Baroni et al (2023), and in this paper we build on top of their approach to align and compare invariance manifolds across a population of neurons, uncovering shared invariance properties and allowing for clustering of neurons based on their invariances. It was not our intention to claim that method, and we have explicitly mentioned this in the main text multiple times (lines 64-66, 99, and 149). We agree that the abstract must be improved to make this point clear. We have therefore refined the abstract accordingly.

Re: The phrase “biological neurons from macaque V1”:

You are, of course, right that we do not derive the properties from real neurons, but from a deep learning model trained on real neurons. We fixed the text to be more precise and make sure it is clear that we are applying our method on a response-predicting model of macaque V1 neurons. That said, the type of response-predicting model we use has been used in other works in monkey [1,2] and other species [3,4,5] where its results have been verified with biological neurons, including specifically that MEIs the model generates activate highly corresponding neurons in closed loop experiments. In addition, some of the results in our paper (i.e. simple and complex cell clusters) are well known. Thus, while we agree that any biological statement can only derive from experimentally tested results, these models have proven to be quite accurate in the past.

[1] Willeke, Konstantin F., et al. "Deep learning-driven characterization of single cell tuning in primate visual area V4 unveils topological organization." bioRxiv (2023): 2023-05.

[2] Bashivan, Pouya, Kohitij Kar, and James J. DiCarlo. "Neural population control via deep image synthesis." Science 364.6439 (2019): eaav9436.

[3] Walker, Edgar Y., et al. "Inception loops discover what excites neurons most using deep predictive models." Nature neuroscience 22.12 (2019): 2060-2065.

[4] Franke, Katrin, et al. "State-dependent pupil dilation rapidly shifts visual feature selectivity." Nature 610.7930 (2022): 128-134.

[5] Fu, Jiakun, et al. "Pattern completion and disruption characterize contextual modulation in the visual cortex." bioRxiv (2023): 2023-03.

Re: Using “trivial” to refer to known properties of RF attributes:

We completely agree that these are fundamental properties of receptive fields. What we meant by the word ‘trivial’ was that they are already very well known – in the sense of “obvious”. Therefore, we want to study additional less understood characteristics of sensory coding (invariance) properties across neurons irrespective of these well known properties. To avoid that the use of the word “trivial” might be read as a condescending statement (which we do not intend), we changed “trivial” to “nuisance” to better express what we intend to say, and accordingly rephrased the second bullet point.

Re: Improving the method section and missing information in Figure 1:

The parameters of the response-predicting model $\theta$ are indeed fixed, which is now explicitly noted in the revised version. The INR is not trained on any dataset of images; instead, it serves as a generative model that maps pixel coordinates to pixel values (e.g., grayscale intensities), producing images as continuous functions of pixel coordinates (Lines 158–161, 203-207). In this work, following the approach of Baroni et al. (2023), the INR is trained to generate diverse images along the invariance manifold of a neuron. We have clarified the purpose of using the INR early in the method section so that it is easier to understand for readers unfamiliar with Baroni et al (2023). Additionally, the use of random Fourier projection on pixel coordinates $(x,y)$ to control spatial frequency, as introduced by Tancik et al. (2020) and employed by Baroni et al. (2023), supports accurate representation of the invariance manifold. While not the primary focus of this paper, we adopted a setup inspired by Baroni et al. (2023), with some refinements, to capture the invariance manifold of a single neuron effectively. We have refined the revised manuscript to ensure the method section is self-contained, accessible, and highlights these refinements clearly.

Thank you for your response, your kind words, and for raising your score!

Re: sharp change in the stimuli for 0.40 and 0.45: This is an interesting point. Our intuition is that while having all images be the same will maximize the numerator of the contrastive objective, it results in the worst possible denominator. In contrast, a solution introducing jumps at periodicity equal to half of the latent space (similar to the figure) ensures that the denominator is optimal (since for every $z_i$ its “distant” point is exactly located at half latent space distance and therefore has opposite polarity), but does not result in the optimal numerator. Then the question is: why is the second solution (i.e., having jumps) preferred over the first one (i.e., no jump with all images the same)? Let us consider the argument inside the $\log$ of the contrastive learning objective, which in the above mentioned cases is simple to compute:

• First solution where all images look the same: Both numerator and denominator are $\exp(1/\tau)$, resulting in a final value of $1$.
• Second solution with jumps: Half of the “near” images will have a cosine similarity of $1$ and half of them $-1$, resulting in a numerator of $\left(\exp(1/\tau) + \exp(-1/\tau)\right)/2$, and a denominator of $\exp(-1/\tau)$, resulting in $\left(\exp(2/\tau) + 1\right)/2 > 1$ for temperature $\tau \gt 0$.

Therefore, in terms of optimizing the contrastive objective the second solution is preferrable.

Re: "metric" used for measuring similarity by Williams et al. and Procrustes analysis: At a high level, we believe, these methods can be viewed as optimizing transformations to align representations with the goal of evaluating a distance metric: $\min_{T_1, T_2} D(T_1(x_1), T_2(x_2))$, where $x_1$ and $x_2$ represent shapes, representations, or (in our case) invariance manifolds. Procrustes analysis constrains $T$ to translation, rotation, and scaling with $D$ as a geometric distance in Euclidean space. Williams et al. generalize this framework by defining transformations $T_1$ and $T_2$ that map neural representations into a shared feature space, ensuring that the resulting metrics satisfy properties like symmetry, equivalence, and the triangle inequality, thereby enabling robust comparisons between representations. Our template alignment step fits into this broader view but operates in functional activation spaces, where $D$ is a neuron-specific metric embedded in the functional (activation) space, and we optimize for: $\min_{T_{2,1}} D(F_1(x_1), F_1(T_{2,1}(x_2))) \equiv \min_{T_{2,1}} D_1(x_1, T_{2,1}(x_2)) \equiv \max_{T_{2,1}} F_1(T_{2,1}(x_2))$ where $F_1$ is the response-predicting model of neuron $1$ and $T_{2,1}$ is an affine transformation acting on the invariance manifold of neuron $2$ with the objective of maximizing the response $F_1(T_{2,1}(x_2))$ of neuron 1 (or equivalently minimizing its distance to the response $F_1(x_1)$ of neuron $1$ to its own maximally exciting manifold $x_1$). Overall, in contrast to the other two approaches, which align representations or shapes to evaluate their similarity, our approach aligns single-neuron invariance manifolds to gain insights into functional similarities across neurons, rather than measuring similarities between the manifolds themselves.

Thanks again for your thoughtful comments!

We are happy to let you know that we managed to perform the analysis that was requested by reviewer aC4g (point 6 above), comparing our method to that of Klindt et al. (2017) and Ustyuzhaninov et al. (2019), and have included it in the new revision of our manuscript (Appendix H). The results empirically demonstrate the flexibility of our approach in accounting for affine transformations (similar to the results in Figure 2 of the initial submission), while highlighting that the other two methods can only account for shifts (Klindt et al. 2017) and rotations (Ustyuzhaninov et al. 2019). Importantly, we applied our approach on both architectures from Klindt et al. (2017) and Ustyuzhaninov et al. (2019), and show that on both models our method yields consistent clusters, highlighting the model-agnostic nature of our approach.

Finally, we would like to emphasize that accounting for the full spectrum of affine transformations is not the only advantage of our method over Klindt et al. (2017) and Ustyuzhaninov et al. (2019). Unlike our method, the previous methods are model-based (i.e., not model-agnostic), do not explicitly disentangle invariances from other computational properties (e.g., tuning), and can incorrectly group similar neurons into different clusters due to potential redundancies in the features learned by the model. Please refer to our previous response to reviewer aC4g for more details.

With this analysis, we believe, all issues raised by all the reviewers are addressed. We would like to thank all the reviewers again for their thoughtful feedback and valuable suggestions. Please let us know if you need any further clarification.