Reproducibility of predictive networks for mouse visual cortex
Authors: Polina Turishcheva
Max F Burg
Fabian H. Sinz
Alexander S Ecker
Venue: NeurIPS 2024 spotlight (2024)
Keywords: reproducibility
predictive models for visual cortex
neuroscience
Primary area: neuroscience_and_cognitive_science
TL;DR: Better performing predictive models for V1 might be less consistent that the older generation. Step towards solving it: pruning and adaptive regularisation.

Abstract: Deep predictive models of neuronal activity have recently enabled several new discoveries about the selectivity and invariance of neurons in the visual cortex.
These models learn a shared set of nonlinear basis functions, which are linearly combined via a learned weight vector to represent a neuron's function.
Such weight vectors, which can be thought as embeddings of neuronal function, have been proposed to define functional cell types via unsupervised clustering.
However, as deep models are usually highly overparameterized, the learning problem is unlikely to have a unique solution, which raises the question if such embeddings can be used in a meaningful way for downstream analysis.
In this paper, we investigate how stable neuronal embeddings are with respect to changes in model architecture and initialization. 
We find that $L_1$ regularization to be an important ingredient for structured embeddings and develop an adaptive regularization that adjusts the strength of regularization per neuron.  
This regularization improves both predictive performance and how consistently neuronal embeddings cluster across model fits  compared to uniform regularization.
To overcome overparametrization, we propose an iterative feature pruning strategy which reduces the dimensionality of performance-optimized models by half without loss of performance and improves the consistency of neuronal embeddings with respect to clustering neurons.
Our results suggest that to achieve an objective taxonomy of cell types or a compact representation of the functional landscape, we need novel architectures or learning techniques that improve identifiability. 
The code is available https://github.com/pollytur/readout_reproducibility.

Reproducibility of predictive networks for mouse visual cortex
Polina Turishcheva," 1.f Max F. Burg, Fabian H. Sinz, Alexander Ecker!44
Institute of Computer Science and Campus Institute Data Science, University Göttnngen Germany
International Max Planck Research School for Intelligent Systems, Tübingen, Germany
3 Tübingen AI Center, University of Tübingen, Germany
Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
uni uni-goettingen de, "eckerêcs. uni uni-goettingen.co
Abstract
Deep predictive models of neuronal activity have recently enabled several new discoveries about the selectivity and invariance of neurons in the visual cortex. These models learn a shared set of nonlinear basis functions, which are linearly combined via learned weight vector to represent a neuron's function. Such weight vectors, which can be thought as embeddings of neuronal function, have been proposed to define functional cell types via unsupervised clustering. However, as deep models usually highly overparameterized, the learning problem is unlikely to have a unique solution, which raises the question such embeddings can be used in a meaningful way for downstream analysis. In this paper, we investigate how stable neuronal embeddings are with respect to changes in model architecture and initialization. We find that L1 regularization to be important ingredient for structured embeddings and develop an adaptive regularization that adjusts the strength of regularization per neuron. This regularization improves both predictive performance and how consistently neuronal embeddings cluster across model fits compared uniform regularization. To overcome overparametrization, we propose an iterative feature pruning strategy which reduces the dimensionality performance-optimized models by half without loss performance and improves the consistency of neuronal embeddings with respect to clustering neurons. Our results suggest that achieve objective taxonomy of cell types or a compact representation of the functional landscape, we need novel architectures or learning techniques that improve identifiability. The code is available https: https://github htw.poollyyuurrraddotrr.rrrooooo
//github
1 Introduction
One central idea in neuroscience is that neurons cluster into distinct cell types defined by anatomical, genetic, electrophysiological or functional properties single cells [23 49 unified taxonomy of cell types across these different descriptive properties research. The functional propertics of neurons, i. which computations they implement on the sensory input, constitute important dimension along which cell types should manifest. However, each neuron's 's function is a complex object, mapping from a high-dimensional sensory input to the output of the neuron. Classical work in neuroscience has described neuronal function by few parameters such as tuning to orientation, spatial frequency, or phase invariance [ S 8
69 77). Defining
active area of
13 25 40). However, when considering neurons from higher visual areas, deeper into the brain's processing network, neuronal functions become more complex and not easily described by few
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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manually picked parameters. In addition, recent work in the mouse visual cortex demonstrated that even in early areas, neuronal functional properties are not necessarily well-described by simple properties such as orientation [47. 71J. One possible avenue to address these challenges is to use
functional representations learned by data-driven deep networks [36] 37 44 trained to predict neuronal responses from sensory inputs. The central design element of these networks is split into a core - a common feature representation shared across neurons, and a neuron specific readout typically linear layer with a final nonlinearity (see Fig. D These networks are state-of-the-ar in predicting neuronal responses on arbitrary visual input, and several studies have used these networks as "digital twins" to identify novel functional properties and experimentally verify them in vivo [37 57
47
67 68 Because readout weights determine a compact representation of a neuron's 's imput- output mapping. they can serve embedding of a neuron's 's function. Such embeddings could be used to describe the functional landscape of neurons or obtain cell types via unsupervised clustering [59]. However, because the feature representations provided by the core are likely overcomplete, there will be many readout vectors that represent approximately the same neuronal function. Thus, it not clear how identifiable functional cell types are based on clustering these embeddings. Moreover, since early data-driven networks [22 28 there have been several advances in readout (48) 51 65) and
58
network architecture [33 36 56 67 to decrease the number of per-neuron parameters, incorporate additional signals such as behavior and eye movements, or add specific inductive biases, such as rotation equivariance. Currently, there no study systematically investigating the impact of these architectural choices on the robustness and the consistency of embedding based clustering.
Here address this question by quantifying how consistent the models are across several fits with different seeds. We measure consistency by (1) adjusted rand index (ARI) of clustering partitions across models, (2) correlations of predicted responses across models, and (3) consistency of tuning indexes describing known nonlinear functional properties of visual neurons. Our contributions are:
We show that L1 regularization, used in early models 28 59 structured clustering when using embeddings of newer readout mechanisms [51].
is instrumental in obtaining a
We introduce new adaptive regularization scheme, which improves consistency of learned neuron embeddings while maintaining close to the state-of-the- art predictive performance.
We address the identifiability problem by proposing an iterative feature pruning strategy which models by half without loss of performance
reduces the dimensionality of performance-opimnin and improves the consistency of neuronal embeddings.
We show that even though our innovations improve the consistency of clustering neurons, older readout mechanisms 28 reproduce neuronal tuning properties more consistently across models.
Our results suggest that to achieve an objective taxonomy of cell types compact representa- tion of the functional landscape, we need novel architectures or learning techniques that improve identifiability while preserving neuronal functional properties.
2 Background and related work
Predictive models for visual cortex. Starting with the work of Antolík et al. [2 number of deep predictive models for neuronal population responses to natural images have been proposed. These models capture the stimulus-response function of a population of neurons by learning nonlinear feature space shared by all neurons (Fig. [ the "core"), typically implemented by a convolutional
neural network [26] 28 5T]. sometimes including recurrence 55). The core can be pretrained
and shared across datasets [67]: it forms a set of basis functions spanning the space of all neuron's input-output function. The second part of these model architectures is the "readout": it lincarly combines the core's features using a set of neuron-specific weights.
As the dimensionality of the core's features is quite high (height x width x feature channels), several different ways have been proposed constrain and regularize the readout, accounting to the fact that neurons do not see the whole stimuli but rather focus on a small receptive field (RF). Initially, Klindt al. (28) proposed a factorization across space (RF location) and features (function) (Fig. A "Factorized readout"). This approach required relatively strong L,1 regularization to constrain the RE More recently, Lurz et al. (51) further simplified the problem by learning only an (x. coordinate, representing the mean of a Gaussian for the receptive field location, as well as vector of feature

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A
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Input [x.ycc]
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Figure 1: A. Model architecture: The models used in our study are separated into a core learning non-linear feature representation shared across neurons, and a neuron specific linear readout. In our case, the core is oolation-qquivarinn 360/n degrees, resulting in f n output channels. We apply batch norm on the learnt feature channels only and do not learn scale and bias parameters in the core's 's last layer, as this would interfere with the readout regularization. Two commonly used linear readouts are factorized and Gaussian readout. The factorized readout learns spatial weights accounting for neuron's 's receptive field position and feature weightings the neural embeddings. The Gaussian readout replaced the spatial weights by sampling receptive field positions from a Gaussian and inference time samples the Gaussian's mean. B. Readout alignment: To yield rotation invariant neural embedding clusters, we align the orientations of the embedding vectors by cyclically rotating the elements in the embedding vectors to minimize the sum of pairwise distances between the rotated embeddings [46).C. Our pipeline: For each model architecture and training configuration we train 3 models with different parameter initialization seeds. We orientation-alige their embeddings, cluster them, and eventually compute all model pair-wise/
core, i.c. each learnt feature f is analytically rotated n times by
wise ARIs
weights (Fig. IA:
"Gaussian readout"). This removed the necessity for strong L₁ regularization. Current state-of of-the-ar art models use Gaussian readout with weak or no regularization 66 64 67].
There is a long history of work investigating neuronal functional properties experimentalyy y, beginning with Hubel and Wiesel (1962) [[) who showed bar stimuli cats and found that many neurons in VI are orientation selective. Since then, a number of studies showed that many VI neurons are - in linear model approximation optimally driven by Gabor stimulus of specific orientation, size, spatial frequency, and phase 66 20 32 40 Later, non-linear phenomena were investigated, such as surround suppression and cross-orienten orientation inhibition. Neurons that are surround suppressed can be driven by stimulating their receptive field, while an additional stimulation of their surround reduces neural activity. Interestingly, only stimulating the surround would not elicit a response. Cross-orientation inhibition reduces neural activity of some neurons if neuron's 's optimal Gabor stimulus is linearly combined with a 90 degrees rotated version of that Gabor stimulus, forming a plaid (.55 10 14
Functional properties. Previous work verified these model in vivo 37
58 67 68
[7]. Recently, such experiments were performed in silico with neural predictive models, allowing for large-scale analysis of neuron's 's tuning propertics without experimental limitations [53 59
Network pruning. While modern deep neural networks are typically highly overparameterrize, as
this simplifies optimization (38] 50 52). overparametrization poses a problem for reproducibility of neural embeddings: model training might end in various local optima that leads to different neuronal representations even model outputs and predictive performance are similar. Prior work has shown that learned deep neural networks can be pruned substantially: Frankle and Carbin 34) showed
3

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that up to 80% of weights can be removed without sacrificing model performance. Further work investigated various pruning strategies. For instance Li al. (29) pruned CNN channels based on their 1.1 norm, Luo and Wu 30) used entropy as the pruning criteria, and various other criteria were studied [24 27 3TJ. Interestingly, a later study showed that model performance relatively robust against the specific pruning strategy used (35]
3 Data and model architecture
Data. We used the data from the NeurIPS 2022 Sensorium Competition for our study [60]. This dataset contains responses primary visual cortex (V1) neurons to gray-scale natural images of seven mice, recorded using two-photon calcium imaging. In addition the mice's 's running speed, pupil center positions, pupil dilation and its time-derivative were recorded.
Model architecture. Our model (Fig. D builds upon the baseline model from the NeurIPS 2022
Sensorium competition [60]. Similar as the baseline model, our model includes the shifter sub- network (36] to account for eye movements and correct the resulting receptive field displacement. We
also concatenate the remaining behavioral variables to the stimuli as the input to the model [56] 60]. For the model core, we use
convolutional neural network, roughly following earlier work [33 46 59]. Compared to this prior line of work, our model includes an additional convolutional layer (as the dataset is larger). It consists of four layers with filter sizes 13, 5, 5, and pixels. In all layers, we use sixteen channels and eight rotations for each of the channels, which results 228-dimensional neuronal embeddings. The details of model and training config are provided in Appendix 1. For the readout, we use both the factorized readout and the Gaussian readout (Fig. [ The factorized readout has been used in prior work on functional cell types (46] 59]. As it does not support the shifter network, this version of the model cannot account for eye movements.
4 Methods
Training. Following prior work 228 33 36 47
66], we trained the model minimizing (F(o) (() log f(t) between predicted f(i) and observed r(() spike
53
58
60
64
the Poisson loss L, N Σ>
i=
counts for all i = 1,.... N neurons, as it is oftentimes assumed that neuron's 's firing rates follow Poisson process [15] For factorized readouts add L₁ regularization of the readout mask
and embeddings the loss, i.e L1 Σ (Enget |magl + ΣE emmeedding
(wxl) for spatial mask
m, embedding w, and regularization coefficient y, line with previous work [28
33
39 46
53 59]. As the Gaussian readout selects one spatial position, the L1 penalty term reduces to Li Zi=o? Dembeddiing |wn|- Thus, for both models the total loss becomes Loss Lp L1. For
the factorized readout y is typically chosen to maximize performance (Fig. 6). while the choice of ? for Gaussian readouts is the subject of this paper.
Evaluation model performance. Following earlier work 22 we report model's predictive performance using Pearson correlation between each measured and predicted neural activity pair across images on the test set.
33
36
53
60
64
66)
Evaluation of embedding consistency. Assessing how "consistent" neuronal embeddings across model architectures and initializations is a non-trivial question. They will naturally differ across runs in absolute terms, because core and readout are learned jointly. In the context of identifying cell types, we are interested the relative organization of the embedding space: Will the same sets of neurons consistently cluster in embedding space? To address this question, clustering and evaluate how frequently pairs of neurons end up in the same group by computing the adjusted rand index (ARI, 44) The ARI quantifies the similarity of two cluster assignments X and Y:
Σg (32)
Σ, (3)
ARI
1/
(t)
:
Σ, | Σ (3) 6
Here, a; is the number of data points in cluster of partition X, by the number of data points in cluster j of partition Y, ny is the number of data points in clusters and of partition X and Y.

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respectively, and n is the total number of data points. The ARI is invariant under permutations of the cluster labels. It is one if and only if the two partitions match exactly. is zero when the agreement between the two partitions is as good as expected by chance and, hence, can be negative (61].
For clustering, we follow the notation-invarian clustering procedure (Fig. [ introduced by Ustyuzhaninov et al. (46] and then cluster the aligned embeddings with k means. As we do not know the true underlying number of clusters, we explore a range of i: = 5 up to k = 100 clusters, which
covers the typical amount of cell types suggested in various works and modalities 44 42 45 59
To visualize the neuronal embeddings, Berens [43). We use a perplexity given by 1% of the size of the dataset and set early exaggeration to 15. To avoid visual clutter, we plot a randomly sampled subset of 2,000 neurons from each of the seven animals in the dataset. We use the same neurons across all ARI experiments and t-SNE visualizations. The results do not depend on the specific cluster consistency metric used (see Appendix A.9). We assessed the credibility of t-SNE visualizations in Appendix A.
use t-SNE [16] using the recommendations of Kobak and
Neuronal tuning properties. Unlike the embeddings, which were not yet biologically validated, the predicted activity from our models has been consistently verified in vivo across several studies [7 188 59 68). By focusing on predicted activity, we leverage the stability and biological relevance of these outputs and relate our findings to real-world scenarios. Specifically, we investigated the predicted neural activity to parametrie, artificial stimuli that are known to yield interesting phenomena and were used extensively in biological experiments (see Section 2
To quantify neuron's 's tuning properties for phase or orientation tuning, we use the parameters of the optimal Gabor and change only the phase or orientation to create a set of new Gabors. We feed this set of stimuli through the model and fit sinusoidal functions to the stimulated response curve, namely f(a) = sin(a + x) + B for the phase sweep. For the orientation sweep, rotating the Gabor-based stimuli by 180-degrees will lead the same stimulus, accounted for by the additional factor of two when fitting/ ((a) = A sin 22 + Φ) + B to the stimulated responses. We define the corresponding tuning indices as the ratio of the fitted amplitude A to the mean B of the curve.
For cross-orientation inhibition we added an optimal Gabor and its orthogonal copy and stimulated the tuning curve by varying the contrast of the orthogonal Gabor. Analogically, for surround suppression we varied the size and the contrast of a grating based on the optimal Gabor's parameters. To quantify both indices, we followed [11, 53) and saved the highest model prediction fmaz- Next, we computed the according tuning indices as suupp/ maz. where f supp corresponds to the highest suppression/inhibiton observed. See Appendix Section A.2 for parameter details.
We compute tuning index consistency across three trained models only differing in their parameter initialisation before fitting. Specifically, we compare their similarity by the normalized mean absolute error N M AE((, y) Σ different model fits and T denotes the mean across neurons.
sdeurons ((I;; w/1/((2; - =l)) where x and y are tuning indices for
Optimal Gabor search. All tuning properties investigated in our paper are based on optimal Gabor parameters. Previous work (33] 53 (33,53,, [3] brute-force searched the optimal stimulus. Since their models relied on the factorized readout which could integrate over multiple spatial input positions, they had to show Gabor stimuli of various properties at all spatial input positions. Thus, they needed to iterate through millions of artificially generated Gabor stimuli for each neuron. Here, introduce a technical improvement upon their work significantly speeding up the optimal Gabor search for Gaussian readouts: as this readout type selects only one spatial position per neuron is sufficient to limit the presented stimulus canvas size to the model's receptive field size. This significantly decreased the canvas size from 36 64 2, 304 to 25 x 25 625 input pixels, leading to an approximately 3. 3.7-fold speedup. Additionally, our improved approach reduces the number of floating point operations in the model's forward pass, further benefiting the computation time. Another important detail is that for the optimal Gabor search and tuning curves experiments we completely ignored the learnt neuronal position as there was only one position to choose. For all in silico experiments, we removed models" shifter pupil position prediction network (36), as pupil position only impacted the spacial position selection, which now limited to a single pixel. All other behavior variables we set to their training set median values, following [56] 67).
5

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Figure 2: A: Model performance: correlation between predicted and observed neuronal response. B-F: T-SNE projections of the embeddings from differently regularized models. All models used seed = 42. G: The histogram of weights before alignment from different models. Unregularized weights (7 = 0) are crucially different, while 7 50 and 7 = 100 are close to the factorized readout's 's distribution. "Hogoomm 10 is not as sparse as the factorized one. H: Example of 'adaptive" regularization for the factorized mask. The "mask" a 2D matrix, selecting a "receptive field" from the latent space and the "embedding" is linear vector representing learned neuron function. Both are weights learned the readout. In the top, the mask learnt is much smaller, hence, regularizing a neuronal embedding is more important reduce the L1 penalty, while in the bottom can keep the neuronal embedding less sparse; its impact is smaller as the mask is bigger. I: Adjusted rand index (ARI) for clustering embeddings using k-means. We take embeddings from models trained with different seeds, cluster them and compute ARI between the clusterings. Note that even clustering the same embeddings twice with different clustering initializations will result in ART 1 (see Appendix A.4).
5 Results
From factorized to Gaussian readouts. We start with our observation that the structure of the neuronal embeddings depends quite strongly on the type of readout mechanism employed by the model (Fig. 2 Earlier studies employing the factorized readout reported clusters in the neuronal embeddings space 46 59 and suggested that those could represent functional cell types (Fig. B However, for more recent models with a Gaussian readout [51] 60 this structure is much less clear (Fig. (C). although these models exhibit substantially better predictive performance (Fig. ( This vizualization difference was also evident in quantitative terms: Clustering the neuronal embeddings led significantly more consistent clusters for the factorized readout than for Gaussian readout (Fig.) 2 red vs. blue), independent of the number of clusters used.
We hypothesized that this difference might be caused by the L1 regularization in earlier models. which could reveal the structure in neuronal function better and lead to more consistent embeddings. Indeed, the weight distribution for the factorized model was much sparser compared to the Gaussian readout (Fig.G)) Increasing the L2 regularization indeed resulted in more structured embeddings least qualitatively (Fig. E). and in terms of the distribution of embeddings (Fig. 2G; pink and green). It also came with a drop in predictive performance comparable to that of the factorized readout (Fig. 2 A). However, somewhat surprisingly, did not lead to more consistent cluster assignments (Fig. 2 pink and green). This result suggests that the structure that emerges for the Gaussian readout, arises somewhat trivially by very aggressively forcing weights to zero due the strong I.I penalty, but this does not happen in consistent way across models, therefore not improving the consistency

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0.38
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Figure 3: Selection of adaptive regularization hyperparameters. A: Validation correlation of model with 0.1, 0.5 and different overall regularization strength. For smaller the performance decreases more fast. B: Distribution of values learnt. As expected, for smaller a the distribution is closer Log.N Norm orrm and smaller for bigger o. This also explains, why the performance in panel A decreases slower, as the model is less regularized. C: ARI for different hyperparameters. We that for
regularize both embeddings and 8-s, the learnt B-s
0.5 ARI for 10 and = 30 are mostly equivalent and = 50 is slightly worse. The ARIs trend is more visible on a 0.1. It seems like the better the predictive performance, the better is the ARI curve. As the performances for y = 10, y 15, and y = 20 are identical within std, we choose the least regularized model to introduce less bias.
of embeddings across runs. Note that the factorized readout is always L1 reggullarieed - it does not work without it (28) and very sensitive to the choice of regularization strength (Appendix (A.3). This is also why we show only single gamma value in the table.
Adaptive regularization improves consistency of embeddings while maintaining predictive performance. We next asked what could explain the difference in embedding consistency between factorized and Gaussian readout. One important difference is that the factorized readout jointly regularizes the mask and the neuronal weight. As consequence, the factorized readout could reduce the L1 penalty on the neuron weights by increasing the size of its receptive field mask (Fig. 2).. As consequence, the neuronal weights may not be regularized equally.
Motivated by this observation, we devised a new adaptive regularization strategy for the Gaussian readout, where the strength of the L, regularizer is adjustable for each neuron:
readout
Σ
3 w,ll
Here, 7 controls the overall strength of regularization as before. The variables 3, are learned coefficients for neuron n. which we imposed log-normal hyperprior p(B,) LogNormal(11 achieved by adding the according loss term L, = 1/o² En log(3.).
This adaptive regularization procedure introduces an additional hyperparameter, a, that controls how far each neuron's regularization strength is allowed to deviate from the overall average controlled by 7. We found that for relatively large = 0.5, the model essentially learns to down-regulate all neurons' weights (Fig. B)) and falls back very mildly regularized mode countering the effect of increasing 7 (Fig. 3A; blue). When constraining the relative weights more closely around I (a = 0.1), the model indeed learns redistribute the regularization across neurons. The average weights remain closer to 1 (Fig. 3) and changing the overall regularization strength does have an effect (Fig. (A).
The optimal model chosen by this adaptive regularization strategy (y 10, 0.1) indeed performs well on all dimensions we explored: First, its predictive performance matches the state-of-the-art model with the Gaussian readout (Fig. A). Second, its learned neuronal embeddings more consistent that those both factorized and uniformly regularized Gaussian readout (Fig. 2 purple).
Pruning models post-hoc improves consistency neuronal clusters. We saw that the consistency of neuronal clusters could be improved by adaptive regularization of the readout. However, in absolute terms the consistency is still not very high. One potential reason for this could be that deep models typically highly overparameterized (34]. As a result, several of the feature maps of the core
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Figure 4: Pruning. A: Validation score the pruned models with different regularization. Lines: average across three models. Shaded areas: standard deviation across three models. Colors: model type and regularization (see legend in B). Stars: selected model. Crosses: non-pruned model with channels. B: ART for non-pruned and models selected after pruning (stars from panel B). Pruning consistently improves consistency of clustering measured by ARI. Adaptive regularization readout pruning is shown in A.I1 C: perfformmaee trade-off (20 clusters). Ideally we want high ARI and high performance.
could be redundant, resulting in multiple ways to represent the same neuronal function. In this case, two neurons with the same function could have highly different embeddings. We therefore now investigate whether pruning trained networks could lead to more consistent neuronal embeddings across model fits. We took the following approach to pruning models:
1. Train a model as usual.
2. In the trained model, mask one core output convolutional channel and compute performance.
3. Remove the channel for which masking decreases model performance the least.
4. Finetune the model on the reduced core.
5. Repeat steps 2-4 until only one channel is left.
In agreement with prior work pruning neural nets (34) we observe slight performance improve- ment during first pruning iterations and find that 1/3 to 1/2 of the convolutional channels can be removed without affecting performance (Fig. 4A). We selected the smallest number of channels before the performance starts to decrease. We also observed that for the more regularized models more channels should be kept after pruning (Fig. 4A), which makes sense severe Lı regularization is also feature selection mechanism. Pruning noticeably improved the consistency of clustering neurons for models with the Gaussian readout (Fig. 43,, but interestingly only for the regularized models. However, this improvement in consistency still comes at the cost of predictive performance. Overall, the (non-pruned) model with the adaptive readout achieves the best consistency -erireencce-effomaecee trade-off (Fig. 4C, purple).
Neuronal tuning properties. To explore to how improving reproducibility of neural embeddings impact neurons' actual behavior, we examined the predicted neurons' tuning with respect to four known properties of VI neurons: orientation tuning, phase invariance, surround suppression and cross-orientation inhibition (see Section for details). We chose these properties because they are well-established and widely studied nonlinear phenomena of V1 neurons [12,17] 17 We performed in-silico experiments showing classical grating-based stimuli and computed neuronal tuning curves from the predicted model responses. Note that the neurons have not seen any of these stimuli during the invvivo experiment. Hence, our models have not been explicitly trained these stimuli. Thus, we cannot compare these predictions against ground truth. However, we can evaluate qualitatively whether the predicted tuning curves reproduce what is known from biology and we quantify how consistently these phenomena are expressed across predictive models that just differ in their initialization (in ideal world tuning curves would be identical across seeds).
We found that different readouts gave rise to substantially different tuning properties for the same neurons (Fig. top row; one example neuron's tuning curves). Strong uniform regularization of Gaussian readouts (> and 7 100) made neuron's responses highly biased towards their mean response, while the non-regularized Gaussian model exhibited the biggest responses amplitude,

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Phase invariaacc
Orientation Tuning
Surround Suppression
Cross Orientation Inhibiion
Tuaing Curous nauron
2.
5
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Figure 5: Impact of regularization on neurons' tuning properties. In silico analysis of neural tuning properties Top: Example of tuning curves for neuron well- well-predicted by the model. Bottom: Population distribution of tuning indices across neurons.
followed by the adaptively regularized Gaussian and the factorized model, which both exhibit moderate shrinkage effects.
For each of the four response properties, computed tuning index that measures the degree of modulation. As tuning indices measure the effect size, tuning curves with small amplitude and high baseline lead to high phase invariance, and low orientation tuning, surround suppression, and cross- orientation inhibition tuning indices (distributions in Fig. bottom). Non-regut Non-regularized Gaussian and factorized models exhibited similar distributions. This result suggests that even though regularization leads to sparser and less overfitted embeddings. it also removes important functional properties due to reduced expressive power. The adaptively regularized models stayed in between of the non- regularized and highly regularized models, as it puts different regularization weights across various neurons, effectively combining both modes. Thus, keeps the overall regularization coefficient ? low, offering good combination of both viable tuning properties and preserved consistent clustering.
Discussion
We performed several important reproducibility checks for the commonly used readout mechanisms of predictive models the visual cortex. We found that the older factorized readout led to a more structured embedding space, more consistent neuronal clusters and reproducible in-silico tuning
Readout
7
Response Correlation f
Cross-Ori. Inhibition
Phase Invar. 1
Orientation
Surround
Tuning 1
Suppr. 1
Factorized
0.003
0.81
0.52
0.54
0.50
0.78
0.76
0.72
0.69
0.69
0.85
Gaussian
0,86
1.10
0.92
1.03
1.26
100
0.82
0.91
0.83
0.96
1.04
Gaussian adaptive
10
0.85
0.81
1.02
0.81
0.94
Gaussian pruned
0.85
0.72
0.71
0.79
0.92
0.86
0.91
0.91
0.99
0.95
100
0.83
0.96
0.82
1.02
0.99
Table 1: Consistency of model predictions and tuning properties across model fits. We compared pairwise correlations response predictions across three model fits (1: higher is better) and the normalized mean absolute enror (NMAE; Sec. across tuning indeces (l: lower is better).

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curves than the more recent, performance-ppimizee Gaussian readout. The structure of the embed- ding space was primarily related to the L, regularization of the weights. By equipping the more recent Gaussian readout with a novel, adaptive I1 regularization, we could recover a similarly struc- tured embedding space and achieve more consistent neuronal clusters without sacrificing predictive performance. However, all of the models with regularized Gaussian readout exhibited strongly biased neuronal tuning curves, calling into question how faithfully they can represent neuronal tuning beyond experimentally verified properties such maximally exciting stimuli [37 47 62
63 68) Rigorous, biologically meaningful evaluation remains an open question, but the presence of non-linear phenomena (tuning curves) is crucial. While current methods focus on explained variance/correlation, models with similar scores can differ in explainability (53). To our knowledge, we conducted state-of-the ahr evaluations for biological meaning by testing the tuning curves.
An important goal models is that they should be resistant to small hyperparameters changes and different initial conditions in order to lead to meaningful biological conclusions. The comparably low consisteney of state-of-the-an models across seeds suggests that future research is required to improve the consistency of models, potentially by improving identifiability improved training paradigms that find robust solutions.
One limitation of our work is that we focused on one core architecture crotation-equivriinn CNN), We do not have a reason to believe that other core architectures would behave fundamentally differently. One sanity check with a regular CNN core (Appendix (A.6) yielded similar results. Whether the same holds for Vision Transformer cores 64 remains to be checked.
In conclusion, our work (1) raises the important question how reproducible current neuronal predictive models are; (2) suggests framework to assess model consistency along orthogonal dimensions (clustering of neuronal embeddings, predicted responses and tuning curves); (3) provides technique to improve the consistency of the clusters with keeping the predictive performance high.
Acknowledgements
We would like thank Konstantin F. Willeke, Dmitry Kobak and Ivan Ustyuzhaninov for discussions, as well as Rasmus Steinkamp and the GWDG team for the technical support. MFB thanks the International Max Planck Research School for Intelligent Systems (IMPRS-IS).
This work was supported by the Deutsche Forschungsgemeinschatt (DFG, German Research Foun- dation)- project ID 432680300 (SFB 1456, project B05) and by the European Research Council (ERC) under the European Union's ' Horizon 2020 research and innovation programme (grant agrec- ment number 101041669). Computing time made available the high-performance computers HLRN-IV at GWDG at the NHR Center NHR @ Göttingen. The center is jointly supported by the Federal Ministry of Education and Research and the state governments participating in the NHR htttr--ereinnddenseeeeeeeee
FHS is supported by the German Federal Ministry of Education and Research (BMBF) via the Collaborative Research in Computational Neuroscience (CRCNS) (FKZ O1GQ2107.
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A Appendix
A.1 Model config and Training pipeline
We stayed close to the Willeke et al. [60] benchmark with list of the following changes. For the model, we used rotation equivariant model with 16 hidden_ccammnn channels and 8 rotations per chan- nel, which resulted in the total readout dimensionality of g28. _aaammmaaannatte input_kers was 13 and hidden_kent was 5, to make models closer to Ecker et al. (33) and Ustyuzhani- nov et al. [59]. gamma_ caamaa_inpt,, corresponding to the input Laplace smoothing was set to 500 and gumma shidden applying Laplace smoothing over the first layer of convolutuonal kernels was set to 500,000, depth_ deptth_separable- separable=False For the training, we also used early stopping and learning rate scheduling with steps. The only differences were that used batch size=256 instead of 128 and initial learning rate of 0.005 instead 0.009.
was turned-off,
A.2 Parameters optimal gabor and tuning curves sweep
For the optimal Gabor search, we followed Burg al. [53 and generated the Gabors with contrasts steps and 8 sizes steps. The maximum input value for the stimuli in the dataset was 1.75, which we used for the maximum contrast and the minimal contrast we set 1% of the maximum. The stimuli diameter sizes were between 4 and 25. Minimum spacial frequency was 1.3 1.3 i 0, 1, ...., 10.
For orientation tuning 24 orientations equidistantly partitioning the interval between 0 and used with phases and maximum response acrross phases was taken for each orientation.
For phase invariance created stimuli for twelve equidistant phases between 0 and 2T.
For surround suppression we used the following stimulus creation parameters
p {
'total_size' : 40, mnii_ssizecennter 0.05,
p 'num_sizes_center 15,
'size_ Bisize__enter_inng tetp_ceenttrr_iccmeeetttt 1.23859,
"min_siize_surround 0.05, Pnmmm "aszee.uurroumd_iccreennn'' 1. 1.23859,
"num_sizes_surround" 15,
"min_contraast_center 2,
nnm mmmm_contrasts_center" : 1,
coontrast_center_inrrreet.
1,
e "min_contraat__urrund 2,
, 'num_ gaam__oontratt_surrund. 1,
contrast_ conntrast_suroundddto "narround_incrmment" increment'
}
For cross-orientation inhibition 9 contrasts and phases were used to create the combinations of the optimal Gabor and its orthogonal version.
A.3 Choice of regularization for the factorized readout
For factorized readouts severe L, regularization is design necessity because mask and readout are regularized jointly. The idea of the mask is to learn the receptive field, ideally "pixel' only, so the strong regularization needed and tightly connected with the performance. We select the regularization based on the best performance 7 0.003 (see Fig 6
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0.34
Single-trial S validation correlation
0.32
0.30
0.28
0.26
0.002
0.004
0.006
0.008
0.010
Regularisation coefficient (y)
Figure 6: Regularization strength impact on performance for factorized readouts
A.4 ARI stability model and 3 seeds for k-means
50
100
1.0
0ch
0ch
0.9
9ch
5ch
4ch
0.8
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Figure 7: Here for the original and pruned models we took the model trained in 1 seed (seed = 101) and performed the clustering using 3 different seeds in k-Means and computed ARI across different partitions. As k-Means is initialisation dependent we can see that in case non-clear clusters even the same vectors the ARI values could be very low. As do not know the true amount of clusters, we tried clusters range [5, 90] with step 5.
A.5 Selection of channels during pruning
To select the amount of channels, train same models with different starting seeds and estimate the mean barp and 0p of performance on each pruning stage. accounting for std using the following logic: P Jes
P O where is the amount of channels pruned.
A.6 Motivation for rotation equivariant core and non-equivariant control
The motivation behind the rotation- -quuvvin-qu core is that V1 neurons are orientation selective and neurons with all preferred orientations exist. Thus, any feature is likely to be extracted in multiple rotated versions [33 46 59 To control that our results are not specific to the rotation-equivvariat core, performed a few control experiments with non-rotation-equivrii
CNN core (Fig. A.6):
The general pattern of results the same: the adaptive regularization improves consistency of clustering and so does pruning feature maps.
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0.20
y=0
y-0, 16ch
0.225
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y= 13ch
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y-50, 13ch
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y=
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40
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Number clusters
Figure Non-rotation equivariant CNN with 128 channels ((rotation equívariant core with 16 channels and 8 rotations would have "128" channels). Adaptive regularization outper- forms both regularized Gaussian and factor- ized models in terms of ARI. In both plots red factorised, purple adaptive Gaussian, blue is non-regularized Gaussian, magenta is regularized Gaussian.
Figure 9: Pruned non-rotation equivariant CNN core with 16 channels (we decreased the amount of channels to make the computa- tions feasible). We see that pruning improves ARI and adaptive model is still better then regularized Gaussian and factorized models. Though, changing the dimensionality might require additional research to scale regulariza- tion strength.
A.7 Table with correlations for shifter turned on
Readout
Responses Correlation t
0.65
Gaussian
0.80
100
0.75
Gaussian
0.80
pruned
50
0.81
100
0.77
Table 2: This table same logic as Table here for the Gaussian readouts the shifter used. We can see that with grid mean predictor undefined the shifter actually improves the consistency of responces I while pruning still helps, especially for the non-regularized models. However, turning off the shifter causes some displacements in the receptive fields, probably in different directions. This effect is much more present, when grid mean predictor is can compensate for the shifter displacement even more.
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A.8
Optimal Gabors
Optimal Neuron
non-pruned models
Neuron
Neuron
Neuron
Neuron
Weuron
Neuran
Neuron
Neuron
Neuron
Camins-
508d-42
Carima-0 seed-10 101
Camma-o So0d 7607
Camma- 50
send-42
Gamma- 50
seod seod-10 101
Gamma-50
2ax5-1667
Gammas 1.00
seed-42
Cammss 1000
sond-101
Camms- 1.00
5a8d 7687
Figure 10: Optimal Gabors for non-pruned models. Columns per neuron, per model, for each of three regularization strengths three seeds were trained. We can see that the optimal Gabors selected somewhat consistent, however, ideally same. More regularized models also tend to select Gabors with smaller sizes.
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Optimal Gabors of pruned models 6114
1770
2566
6890
7736
Command
seed-42
Cammoon sead- 101
Camma-D
saed 7687
Comma 50
Fo0dm42
Camina-S0
S00d-101 101
Cammaa-S
saed-760 7607
Camma- 100
2eed-42
Gomma- 100
5080-101 01
Comma-1 100
se6d-7607
Figure 11: Optimal Gabors for pruned models. Columns are neuron, rows are model, for each of three regularization strengths three seeds were trained. Compared Fig. 10 we see that for = 0 the model became much more consistent across seeds, suggesting that the model was originally overparametrised. Moreover, pruned 7 = 0 models now select optimal Gabors, which close to the non-pruned majority choice across models. While this is not true for the more regularized models, they now disagree more between seeds and tend to choose Gabors with higher- frequencies, which is not biologically plausible choice. This is probably happening due to too severe regularization. However, this explains why there is bigger differences between the tuning indexes for regularized models. The indexes computed modifications of the optimal Gabors, and if the Gabors are different the responses and indexes would be different as well.
A.9 Other clustering consistency metrics
ARI is the most popular metric for pairwise cluster comparisons [19]. which we think more appropriate than set- bet-based methods. To ensure are not missing anything, we also considered alternative metrics such as Fowlkes-Mallows, completeness, homogeneity, and v-measure particular case of mutual information) and repeated the analysis. The results show the same ordering of methods A.9 Do the metrics reflect biological significance? We reasoned that if neuronal embeddings determine the response function of neuron in the model, then two neurons that have similar
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embeddings under one instance of a model, should also have similar embeddings under another instance of this model. Therefore, measuring similarity of clustering is a reasonable proxy.
y= Y-
100
Fowikes Mallows
0.4
Ylogoom 10
0, 7ch
0.3
50, 11ch
100, 12ch
0.003
0.30
0.2
0.25
20
40
60
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0.20
Number clusters
20
60 Number clusters
40
100
Figure 12: Fowlkes-Mallows index for es- timating the cluster consistency. It shows same trends as ARI. In both plots- reids-reed red is fac- torised, purple adaptive Gaussian, blue is non-regularized Gaussian, magenta and green regularized Gaussian.
Figure 13: V-measure index for estimating the cluster consistency. The models order is same as ARI, though it is biased towards bigger amount of clusters due to the set-based nature.
A.10 t-SNE reliability
Please note that we used I-SNE only visualization purposes to show the presence of the density modes in the data. While t-SNE plots are indeed not sufficient to make any conclusions about the data, many papers showed that they are a good tool for exploratory data analysis [72] and we additionally checked trustworthiness metrie (18) after performing t-SNE. Trustworthiness shows higher scores for more regularized models with varying numbers neighbors confirming the presence of the density modes (Fig. 14..
A.11
Adaptive regularization pruning
turns out that pruning does not further increase ARI the adaptively regularized model (Fig. Thus, regularizing neurons in non-uniform fashion is already effective enough that additional pruning does not improve We speculate that this is because some noisy neurons become sufficiently regularized without pruning, while before such regularization was only achieved with pruning and regularization combined.
15
A.12 Computational resources
All of the experiments were performed on local infrastructure cluster with 8 NVIDIA RTX A5000 GPUs with 24Gb memory each. 2 models could be trained on single GPU simultaneously hours. Pruning experiment, without parallelization, would take a week. No extensive cpu resources are required. Optimal Gabor search, without parallelization takes around 12hours for Gaussian readout model and days for the factorized readout model. We also pre-saved optimal Gabors in batches to improve the speed, which required 100Gb of memory.
A.13 Broader impact
Our work is important step towards obtaining functional cell types taxonomy for the primary visual cortex neurons. Such classification can significantly improve our understanding of the brain and help on the way to understand the mechanisms and develop the treatment for different neurode- generative disease.
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y=0 16ch
y=0, 10ch
trustvorthinesss
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Magrom 10, 16ch
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y-
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Figure 14: Trustworthiness of dimen- sional t-SNE projections. It shows higher scores for more regularized models, confirm- ing the presence of the density modes. In both plots red is factorised, purple adaptive Gaussian, blue is non-regularized Gaussian, magenta and green are regularized Gaussian.
Figure 15: Pruning for rotation-equivariant modeling with adaptive regularization of Gaussian readout and factorized readouts.
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The abstract and/or introduction should clearly state the claims made, including the
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is fine to include aspirational goals as motivation as long as is clear that these goals are not attained by the paper.
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Justification: We clearly admit that nevertheless the improvement was done ARI, it could have been improved more. The Discussion section discussed the ponetial acpects we have not addressed.
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Justification: In appendix there are 2 sections describing the model and training pipeline we followed and modifications to it. We also report the seeds and neuron ids for the pictures and the data subsampling procedure when applicable and specific vizualization settings.
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Reviews:
[{"review_id": "AjljrieViJ", "reviewer": "Reviewer_RvHn", "summary": "The paper \"Reproducibility of predictive networks for mouse visual cortex,\" explores the reproducibility of neuronal embeddings in the mouse visual cortex using deep predictive models. By introducing adaptive regularization and iterative feature pruning, the authors address key issues related to model overparameterization and provide a robust framework for achieving consistent functional representations of neurons. The work lays the groundwork for future research aimed at developing reliable and interpretable models for understanding neuronal function. \n\nThe primary goal is to investigate the stability and reproducibility of neuronal function embeddings derived from deep predictive models. These models aim to predict neuronal responses to sensory inputs and have been proposed to define functional cell types via unsupervised clustering. The paper addresses the concern that deep models are often highly overparameterized, leading to multiple solutions that can represent the same neuronal function, thereby questioning the reliability of embeddings for downstream analysis.\n\nThe paper demonstrates that L1 regularization, which was used in early models, is crucial for obtaining structured and consistent neuronal embeddings when newer readout mechanisms are used. A novel adaptive regularization scheme is introduced, which adjusts the strength of regularization for each neuron. This method improves the consistency of neuronal embeddings across different model fits while maintaining predictive performance. The paper proposes an iterative feature pruning strategy to reduce the dimensionality of performance-optimized models by half without losing predictive performance. This pruning improves the consistency of neuronal embeddings concerning clustering neurons.  The consistency of neuronal embeddings is evaluated using the Adjusted Rand Index (ARI) of clustering partitions across models, correlations of predicted responses across models, and the consistency of tuning indexes describing known nonlinear functional properties of visual neurons.", "questions": "- Is ARI a metric you developed or did it exist before? If it existed before, it requires referencing. \n\n- Have you considered evaluating other core architectures, such as Vision Transformers or regular CNNs, to see if the findings generalize across different model types? What specific reasons led you to choose a rotation-equivariant CNN core for this study?\n\n- Could you elaborate on the potential biases introduced by the adaptive regularization and pruning strategies? How do you mitigate these biases? Have you explored other regularization techniques or pruning strategies that might offer a better balance between consistency and predictive performance?\n\n- Are there specific aspects of model performance that are most critical for maintaining biological plausibility?\n\n- The adaptive regularization introduces additional hyperparameters. How do you approach hyperparameter tuning to ensure optimal model performance? What are the computational costs associated with this tuning process? \n\n- Why did you choose the Adjusted Rand Index (ARI) and other metrics for evaluating consistency? Are there alternative metrics that might provide additional insights? How do you ensure that the chosen metrics accurately reflect the biological significance of the neuronal embeddings?", "limitations": "NA", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Adaptive Regularization Scheme: The introduction of an adaptive regularization method that adjusts the regularization strength per neuron is a significant novelty. This approach helps achieve better consistency in clustering neuronal embeddings without compromising on predictive accuracy.\n\n- Iterative Feature Pruning: The feature pruning strategy to address overparameterization in deep models is another novel contribution. By systematically reducing the model’s dimensionality, the authors enhance the robustness and consistency of the neuronal embeddings.\n\n- Comprehensive Consistency Evaluation: The paper provides a comprehensive evaluation of model consistency across different dimensions (embedding clustering, predicted responses, and tuning curves). This thorough approach highlights the robustness and reproducibility of the proposed methods.\n\n- The paper is well-written with generally sufficient referencing to the previous methods.", "weaknesses": "- Bias from Regularization and Pruning: While the adaptive regularization and pruning strategies improve consistency, they may introduce biases that affect the biological validity of the neuronal embeddings. Over-regularization, for instance, can reduce the model’s expressive power and lead to less biologically plausible representations.\n\n- The study focuses primarily on models with a rotation-equivariant convolutional neural network (CNN) core. The authors acknowledge that other core architectures, such as regular CNNs or Vision Transformers, were not evaluated. This limitation means the findings may not generalize across different model architectures.\n\n- Despite the improvements in clustering consistency, there is a trade-off between consistency and predictive performance. The pruning and regularization strategies, while improving consistency, sometimes result in a drop in predictive performance, which is not ideal.\n\n- The introduction of adaptive regularization adds another layer of hyperparameters (e.g., the log-normal hyperprior parameter) that need to be carefully tuned. This increases the complexity of the model training process and may require significant computational resources.\n\n- The focus on achieving high consistency in clustering and embeddings may overshadow other important factors, such as the interpretability and biological relevance of the model outputs. Balancing consistency with these factors is crucial for developing useful predictive models in neuroscience.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for thoroughly reviewing our paper and providing valuable feedback. We are happy that you found the adaptive regularization scheme “a significant novelty”, iterative pruning as “another novel contribution” and our consistent evaluation procedure to “highlight the robustness and reproducibility of the proposed methods”.\nWe have addressed all your comments and will incorporate them into our manuscript to strengthen it further:\n* **Is ARI a metric you developed?**\nARI is a standard metric introduced by Hubert and Arabie [1] to measure how consistent two clusterings are. We will explicitly point this out and add the references.\n* **Have you considered evaluating other core architectures?**\nThe motivation behind the rotation-equivariant core is that V1 neurons are orientation selective and neurons with all preferred orientations exist. Thus, any feature is likely to be extracted in multiple rotated versions [2,3]. We do not see a reason why our results should be specific to the rotation-equivariant core, though. To justify this intuition, we performed a few control experiments with a non-rotation-equivariant CNN core (Fig. 2 + 5 in attached PDF): The general pattern of results is the same: the adaptive regularization improves consistency of clustering and so does pruning feature maps. We hope these controls solidify your trust in our results.\n* **Could you elaborate on the potential biases introduced by the adaptive regularization and pruning strategies? How do you mitigate these biases? Have you explored other regularization techniques or pruning strategies that might offer a better balance between consistency and predictive performance?**\nThanks for pointing that out. The tradeoff between cluster consistency and predictive performance was exactly the starting point of our paper, and the adaptive regularizer is a first answer that mitigates it by performing good along both axes. Of course every model has (implicit or explicit) biases. To assess these biases, we performed the in-silico experiments shown in Fig. 5 and Table 1. This analysis shows that regularized Gaussian readout models strongly shrink the responses and do not faithfully predict neuronal tuning w.r.t. orientation and other nonlinear phenomena, whereas factorized readouts produce more biologically plausible predictions. We hypothesized that this might be related to the factorized readout being able to adapt its regularization strength per neuron, which motivated the new adaptive readout. As the in-silico analyses show, this readout indeed produces biologically plausible tuning curves and does not sacrifice the predictive performance, while maintaining consistency.\\\nRegarding your question about alternative regularization techniques: Yes we explored several other ways of adapting the strength per neuron by using L1 or L2 penalties instead of the lognormal prior, but these did not work well as many of the adaptive coefficients collapsed to zero, weakening regularization.\\\nYou mention in the weaknesses that *“The focus on achieving high consistency [...] may overshadow other important factors, such as the interpretability and biological relevance of the model outputs.”* We believe it is a misunderstanding, as biological relevance is also a key goal for us, as discussed above. Figure 5 and Table 1 address exactly this issue and suggest that the adaptive regularization does not only improve consistency, but also maintains biological relevance of the models.\n\n* **Are there specific aspects of model performance that are most critical for maintaining biological plausibility?** \nRigorous, biologically meaningful evaluation remains an open question, but the presence of non-linear phenomena (tuning curves) is crucial. While current methods focus on explained variance/correlation, models with similar scores can differ in explainability [4]. To our knowledge, we conducted state-of-the-art evaluations for biological meaning by testing the tuning curves. \n* **The adaptive regularization introduces additional hyperparameters. How do you approach hyperparameter tuning […]? What are the computational costs […]?**  \nIn fact it introduces only one additional hyperparameter, because the overall strength $\\gamma$ is also a parameter of the non-adaptive regularizer. The only one that needed to be optimized was $\\sigma$, which we chose by a simple line search using five different values. The goal here was to keep the distribution of coefficients mean centered around 1 and clearly non-zero but not too restricted. Thus, the additional computational cost is manageable.\n\n* **Why ARI? Are there alternatives? Do they reflect the biological significance**\nARI is the most popular metric for pairwise cluster comparisons (Section 2 in [5]), which we think is more appropriate than set-based methods. To ensure we are not missing anything, we also considered alternative metrics such as Fowlkes-Mallows, completeness, homogeneity, and v-measure (a particular case of mutual information) and repeated the analysis. The results show the same ordering of methods (e.g. Fig. 4 in attached PDF). \\\n*Do the metrics reflect biological significance?* We reasoned that if neuronal embeddings determine the response function of a neuron in the model, then two neurons that have similar embeddings under one instance of a model, should also have similar embeddings under another instance of this model. Therefore, measuring similarity of clustering is a reasonable proxy.\n\nWe hope these comments shed some light on our motivation, technical and design choiced. We also hope that it would help the reviewer to reconsider some criticism. \n \n[1] Hubert and Arabie (1985) doi: 10.1007/BF01908075  \n[2] Ustyuzhaninov et al. (2022) doi: 10.1101/2022.02.10.479884   \n[3] Ecker et al. (2019) doi: 10.48550/arXiv.1809.10504  \n[4] Burg et al. (2021) doi: 10.1371/journal.pcbi.1009028  \n[5] Vinh et al. (2010) doi: 10.1145/1553374.1553511", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "bq6boVo4h1", "reviewer": "Reviewer_mU1L", "summary": "This work studies models trained to predict the responses of neurons in visual cortex. The model has a shared multilayer network core followed by a final layer which maps the core features into individual neuron responses. The key question that this paper asks is, How reproducible are the individual neuron properties and \"cell-type\" clusterings inferred by such techniques? The main conclusion is that sparsity-inducing regularization is important for finding consistent cell properties. Pruning certain channels in the core is also shown to improve consistency.", "questions": "* Line 35 \"well described\" -> \"well-described\" typo\n* Line 62 \"We address the identifiability problem, by proposing\" typo, remove comma\n* Fig 1 caption \"sampling receptive field positions form a Gaussian\" typo\n* Is the clustering shown in Fig 1B truly \"rotation invariant\" or related to cyclic permutations of the vector indices? It is unclear from the text. The figure does not depict a rotation (i.e. action by an orthogonal matrix), since the orbit is elliptical rather than circular and it is not centered at the origin.\n* Fig 3A has $\\sigma=01$ which should be $\\sigma=0.1$\n* Fig 3B axis \"amount of such weights\" sounds strange", "limitations": "The authors mention that their work is limited to a particular type of core architecture. It's also limited to being applied to just one dataset, whereas open data in other animals (or collected by other research groups, e.g. the Allen Institute's work) are available. I am not suggesting the authors do this in their revision but that they mention it as a limitation.", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "Consistency and reliability of these methods for understanding visual cortical processing is important for the scientific interpretation of these models. The design of the approach seems sound. A number of metrics are used, both scores as well as response properties used in neuroscience.", "weaknesses": "* I found the comic sans font and cartoon diagram style distracting in Figs 1-4.\n* It isn't clear that much is gained from the t-SNE plots in Fig 2. Some evidence of different clusters, yes, but it's known that these kind of plots can be deceiving.\n* How do you pick the optimal regularization parameters? By what metric/cross-validation procedure? (Line 242 gives $\\gamma=10, \\sigma=0.1$) Validation performance is reported but I'm unclear of the splitting procedure and whether there was a train/valid/test 3-way split or not.\n* Minor points are made in \"questions\" section", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks a lot for your time and effort for the review. We are very happy to see your feedback and high evaluation of our work. We will fix the typos, clarify details and address the style comments in the final version. \n\nYou basically raise two main concerns about the t-SNE plot in Fig. 2 and the selection of hyperparameters in regularization:\n\n* **It isn't clear that much is gained from the t-SNE plots in Fig 2. Some evidence of different clusters, yes, but it's known that these kind of plots can be deceiving.** Please note that we used t-SNE only for visualization purposes to show the presence of the density modes in the data. While t-SNE plots are indeed not sufficient to make any conclusions about the data, many papers showed that they are a good tool for exploratory data analysis [1] and we additionally checked `trustworthiness` metric [2] after performing t-SNE. Trustworthiness shows higher scores for more regularized models with varying numbers of neighbors (see Fig. 1 in attached PDF) confirming the presence of the density modes.  \n* **“How do you pick the optimal regularization parameters? By what metric/cross-validation procedure? Validation performance is reported but I'm unclear of the splitting procedure and whether there was a train/valid/test 3-way split or not.”** We do not use a test set but report performance on the validation set. However, that’s not a problem because (a) predictive performance is not our primary target and (b) in all cases where we select a single model from multiple (Fig. 3A, Fig. 4A, Fig. 6) the performance of the models left and right of the one selected are virtually the same (difference < 0.01), so none of the results depends on this particular model. Hence, we use the original train/val splits of the Sensorium 2022 competition as the additional effort of doing another data split is not warranted.\\\nFor Gaussian readouts $\\gamma = 0, 50, 100$ were chosen arbitrarily, as there is no globally “optimum” parameter, as higher regularization leads to better ARI but hurts predictive performance. The factorized readout is regularization chosen to maximize performance on the validation set, however, this model simply does not learn if the regularization is not correctly tuned (see Fig 6 in the appendix or [4]). \n\nAs for the questions:\n* **“Is the clustering shown in Fig 1B truly \"rotation invariant\" or related to cyclic permutations of the vector indices? It is unclear from the text. The figure does not depict a rotation (i.e. action by an orthogonal matrix), since the orbit is elliptical rather than circular and it is not centered at the origin.”** We follow the procedure and visualization from the original paper [3]. The reason the figure shows an elliptic orbit is because it is a projection of a circle from a higher space for 2d, therefore, it does not have to stay a zero-mean circle. The idea is that we have rotation equivariant vectors from the model, which consist of n blocks, where n is the number of rotations. Then these vectors are transformed to be rotation-invariant with the help of the cyclic permutations along rotations.\n\nWe hope these comments shed some light and clarified our motivation, technical and design choices.\n\n[1] Lause et al. (2024) doi: 10.1101/2024.03.26.586728   \n[2] Van Der Maaten (2009) AI-STATS, PMLR 5:384-391, 2009  \n[3] Ustyuzhaninov et al., ICLR 2020  \n[4] Klindt et al. (2017) doi: 10.48550/arXiv.1711.02653", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6Vss27oYNf", "reviewer": "Reviewer_8oHp", "summary": "The authors present a systematic investigation of the use of deep neural network fits to biological neurons as the basis for neuron cell type classification. The authors explore how various factors such as regularization and model pruning can influence both the predictive model fit and the consistency of the neural clustering.", "questions": "- Why is the adaptive model not pruned in the experiments shown in fig 4? I feel like it makes sense to include it in the pruning experiments if possible for consistent comparison across experiments.\n\n- I think more explanation of table 1 is warranted. First, what is the ideal case? It's not clear to me that the optimal computational model has neurons with precise tuning curves for these chosen functional biological properties. If this is obvious to the authors, then perhaps there should be explicit motivation in the text. Why these biological properties and not others? Do biological neurons in HVC demonstrate each of these properties robustly (and not others)? Is it fair to expect that neurons in silica should each individually exhibit these properties? Second, the difference in NMAE of tuning indices between the factorized and the gaussian readouts seems very large. Large enough that I would expect more discussion of it in the text. How big of a difference in score is this really, is there something that can put it in context? If the biological property fits are a big deal, then is this a nail in the coffin for gaussian readout? Is gaussian readout a hack for improving overall fit while sacrificing something more important? Or is the functional biological properties just a cool trick that would be nice to have but not essential.", "limitations": "Limitations are clear.", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "This paper presents a well-motivated, calculated set of experiments for evaluating how well deep neural networks can be used as models of mouse visual cortex. With increasing interest in this area and increasingly bold claims, work like this is a breath of fresh air. The controlled experiments and attention to detail provides compelling evidence that though DNNs may be our best working computational model of visual cortex, the are likely not a global optimum.", "weaknesses": "Overall, the biggest weakness I think is some lack of explanation. As someone who is very familiar with work in computational neuroscience but not a neuroscientist myself, there were a few things that felt under-explained and it wasn't clear if it was because I'm not the correct audience or because the authors did not provide enough context in the text. The authors do a good job of providing clear motivation for some of the fundamentals (L24-53) but then leave out context when things get a little more technical specialized (i.e. comparisons between readout mechanisms, significance of biological properties and tuning indices, etc) as a result it feels unclear what the reader is expected to know before reading.\n\n- The font in the figures is a little unprofessional. Comic sans? In this economy?\n- Figure 2G is hard to parse. I recommend splitting up the histograms\n- The intuitive explanation of the factored vs gaussian vs adaptive readouts is not super clear to me. It might be worthwhile to spend more time on it, especially to motivate your proposed adaptive mechanism.\n- minor typos (\"initialiation\" in L183)\n- L216 is nonsensical. I think some words are out of order\n- Confidence intervals in Table 1 would be nice to gauge significance. Also, the table format makes it a little hard to visualize. Perhaps a graphical representation would be better.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for appreciating our paper stating that “with increasing interest in this area and increasingly bold claims, work like this is a breath of fresh air.” Regarding your questions:\n\n* **“Why is the adaptive model not pruned in the experiments shown in fig 4? I feel like it makes sense to include it in the pruning experiments if possible for consistent comparison across experiments.”** Thank you for bringing this up. As you suggested, we now pruned the adaptively regularized model as well (see Fig. 3 in attached PDF). It turns out that pruning does not further increase ARI for the adaptively regularized model. Thus, regularizing neurons in a non-uniform fashion is already effective enough that additional pruning does not improve it. We speculate that this is because some noisy neurons become sufficiently regularized without pruning, while before such regularization was only achieved with pruning and regularization combined.\n\n* **“More explanation of table 1 is warranted. First, what is the ideal case?”** Ideally, a model would accurately predict the measured neural responses (response correlation across models should be high - ideally one (1) and result in the same biological phenomena. However, we cannot compare biological phenomena against the ground truth, as they are not contained in the dataset. We can, though, quantify how consistently these phenomena are contained across predictive models that just differ in their initialization seed, and would expect a perfect match (low NMAE, ideally 0).\\\n\\\n*“It's not clear to me that the optimal computational model has neurons with precise tuning curves for these chosen functional biological properties. [...] Why these biological properties and not others? Do biological neurons in HVC demonstrate each of these properties robustly (and not others)? Is it fair to expect that neurons in silico should each individually exhibit these properties?”*\\\nNote that we train the models to predict the activity of neurons in the brain and analyze only the output neurons (i.e. the “digital twins” of the real neurons). Thus, each in-silico output neuron should behave as its corresponding real neuron. We chose the biological properties because they are well-established and widely studied nonlinear phenomena of V1 neurons ([1,2]  and many more) and our dataset is from V1 only (i.e. no HVAs). Note also that we do not expect all neurons – neither in-vivo nor in-silico – to exhibit all these properties. The indices we measure merely quantify *to what degree* they exhibit them (Fig 5, original manuscript) – and all model instances should consistently come to the same answer (Table 1, original manuscript).\\\n\\\n*“Second, the difference in NMAE of tuning indices between the factorized and the gaussian readouts seems very large [...] is there something that can put it in context? If the biological property fits are a big deal, then is this a nail in the coffin for gaussian readout? Is gaussian readout a hack for improving overall fit while sacrificing something more important? Or is the functional biological properties just a cool trick that would be nice to have but not essential.”*\\\nNo, it is not a nail in the coffin. It shows that the factorized readout produces more consistent indices across model runs. Whether these indices are more *accurate* is not captured by these numbers. That question can be resolved quantitatively only by an actual experiment where one measures real tuning curves. However, Fig. 5 in the original manuscript suggests qualitatively that the tuning curves by non-regularized Gaussian, factorized and adaptive readouts are more consistent with what we know from biology than regularized Gaussian readouts. \n\n\nIn addition, we will include your other suggestions into the final version of our manuscript substantially improving the presentation of our results.\n\n[1] Cavanaugh et al. (2002) doi: 10.1152/jn.00692.2001  \n[2] Busse et al. (2009) doi: 10.1016/j.neuron.2009.11.004", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "zLtfBxcIgN", "reviewer": "Reviewer_iat5", "summary": "This work demonstrates that overparameterized neural network models, which have many non-unique solutions, can lead to inconsistencies in representing the mouse visual cortex. It suggests a novel approach called \"adaptive regularization,\" where the regularization parameters in the loss term are learnable rather than fixed. The study also examines other approaches, such as normal regularization and pruning, and finds that these methods are beneficial for improving representation consistency. The proposed method significantly enhances consistency. This contribution is significant to the computational neuroscience community.", "questions": "1. Oother models that, despite lacking rotational equivariance, still demonstrate high capability in predicting neural responses measured by electrodes [1]. Why did the authors choose the Rotational Equivariance CNN? What was the motivation behind this choice?\n\n2. Figure 2A shows the models' performances. However, what about the factorized readout with L1 regularization? Does the predictive performance decrease?\n\n3. Why did the authors choose a log-normal prior for the learnable coefficient?\n\n4. Why did the authors not examine adaptive regularization for factorized readout models? The implementation should be straightforward, similar to what was done for Gaussian.\n\n5. Why does the factorized model have one γ in Table 1?\n\n\nReference\n\n[1] https://www.brain-score.org/vision/", "limitations": "This study investigates the consistency of neuronal properties and the prediction performance of regularized models. However, the models used by the authors are trained to predict neural responses. There is another type of computational model that is trained to perform downstream tasks, with neural-like representations emerging from the training. Future work could consider these latter models.", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 2, "contribution": 4, "strengths": "1. This work addresses a critical problem in computational neuroscience: reproducibility. It proposes solutions to mitigate this issue, which have significant impact on the computational neuroscience community. \n\n2. The authors rigorously define the consistency of computational models and thoroughly examine how regularization strength affects neuronal properties.", "weaknesses": "1. The study is limited to only one type of model, the Rotational Equivariance CNN.\n\n2. In the Methods section, the terms \"Embedding\" and \"Mask\" are not defined. Moreover, Lp and L1 are described as functions of model parameters, but no parameters are defined in the text. This lack of clarity poses an issue for understanding. For instance, Lp could be interpreted in two different ways: as a function of the core but not the readout, or as a function of both the core and the readout. The same ambiguity applies to L1.\n\n3. There are two types of computational neuroscience models regarding their outputs. The first type is task-optimized models, whose outputs are task-related, such as object class, object representation, or action. These models are trained to perform downstream tasks, such as supervised object classification, reconstruction, or playing games [1,2,3]. The second type is neural response fitting models, whose outputs are predicted neural responses, and they are trained to predict these responses directly. However, the reasons why the authors chose response fitting models over task-optimized models are not mentioned.\n\nReference\n\n[1] Performance-optimized hierarchical models predict neural responses in higher visual cortex\n\n[2] Unsupervised neural network models of the ventral visual stream\n\n[3] Using deep reinforcement learning to reveal how the brain encodes abstract state-space representations in high-dimensional environments", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for your rigorous review. We highly appreciate your feedback, which helps us to improve our paper.\n\nRegarding the weaknesses you mention:\n* **“The study is limited to only one type of model, the Rotational Equivariance CNN”** \nThe motivation behind the rotation-equivariant core is that V1 neurons are orientation selective and neurons with all preferred orientations exist. Thus, any feature is likely to be extracted in multiple rotated versions. The rotation-invariant framework allows to assess response features independent of the preferred orientation of a neuron [1,2]. Moreover, it does not lead to any sacrifice in performance compared to the classic CNNs reported in Sensorium 2022 competition [3]. We do not see a reason why our results should be specific to the rotation-equivariant core, though. To justify this intuition, we performed a few control experiments with a non-rotation-equivariant CNN core (Fig. 2 + 5 in attached PDF): The general pattern of results is the same: the adaptive regularization improves consistency of clustering and so does pruning feature maps. We hope these controls solidify your trust in our results.\n\n* **“In the Methods section, the terms \"Embedding\" and \"Mask\" are not defined.”** \nWe will improve the clarity for “masks” and “embeddings” definitions. In fact, Fig. 1A and 2H are supposed to show it, but we realize it’s not entirely clear. We will add annotations to the figures and include a sentence in the text. Both are weights learned in the readout. The “mask” is a 2D matrix, selecting a “receptive field” from the latent space and the “embedding” is a linear vector representing a learned neuron function.  \n\n* **“Moreover, Lp and L1 are described as functions of model parameters, but no parameters are defined in the text. This lack of clarity poses an issue for understanding. For instance, Lp could be interpreted in two different ways: as a function of the core but not the readout, or as a function of both the core and the readout. The same ambiguity applies to L1.”**\nWe apologize for the lack of clarity and will improve the exposition. To clarify: The regression loss Lp is the main loss driving model fitting. It penalizes the difference between observed neuronal responses $r$ and model predictions $\\hat r$. As the response predictions $\\hat r$ are the model *outputs*, they are a function of both core and readout and hence guide optimization of *all* parameters. L1, in contrast, is only a function of the readout parameters, which we believe should be clear from its definition in line 133, which sums over mask $m$ and embeddings $w$ – both readout parameters (but admittedly not entirely clear as already discussed in the previous point).\n\n* **“The reasons why the authors chose response fitting models over task-optimized models are not mentioned.”**\nIt’s true, we could also have used a core from a task optimized model and finetune the readout with it. However, (a) this would lose the rotation equivariance, (b) task-driven models are not great representations for mouse V1 [4] and (c) our main question is about the reproducibility of the learned representations, so we would need multiple versions of the task-trained model. This would (i) imply retraining the task-trained models several times, (ii) would raise lots of questions about which ones to use and (iii) how the specifics of the task training affect the results as well as (iv) whether these representations should be fine-tuned when training the readout or not. While all of these are valid and interesting questions, we opted for focusing on data-driven models for this study and deferring an analysis of task-driven models to future work. \n\nAs for the questions:\n\n* **Why did the authors choose the Rotational Equivariance CNN?** Answered above\n* **“Figure 2A shows the models' performances. However, what about the factorized readout with L1 regularization? Does the predictive performance decrease?” and “Why does the factorized model have one γ in Table 1?”**\nNote that the factorized readout is always L1-regularized – it does not work without it [5] and is very sensitive to the choice of regularization strength $\\gamma$ (Appendix, Fig. 6). This is also why we show only a single gamma value in the table.\n\n* **“Why did the authors choose a log-normal prior for the learnable coefficient?”**\nUsing this prior we separate the overall strength of regularization ($\\gamma$) from the weight $\\beta_n$ of each individual neuron. In this parameterization defined in Eq. (2) we want the weights $\\beta_n$ to be non-negative and 1 on average, which is what a lognormal prior achieves. We also experimented with restricting coefficients to be positive and using L1 or L2 penalties, but this did not work well because they pushed many coefficients to zero, effectively avoiding regularization. \n\n* **“Why did the authors not examine adaptive regularization for factorized readout models? The implementation should be straightforward, similar to what was done for Gaussian.”**\nThe factorized readouts are always (implicitly) adaptively regularized. This insight is what inspired us to develop the adaptive regularization approach. As Figure 2H illustrates, the factorization allows moving weight between the mask and the embedding. By changing the size of the mask, one can effectively change the L1 penalty of the embedding vector. As the Gaussian readout always selects exactly one location, it does not have this degree of freedom.\n\nWe hope these comments shed some light and clarified our motivation, technical and design choices.\n\n[1] Ustyuzhaninov et al. (2022) doi: 10.1101/2022.02.10.479884   \n[2] Ecker et al. (2019) doi: 10.48550/arXiv.1809.10504  \n[3] Willeke et al. (2022) doi: 10.48550/arXiv.2206.08666  \n[4] Cadena et al. (2019) NeurIPS 2019 Workshop Neuro AI  \n[5] Klindt et al. (2017) doi: 10.48550/arXiv.1711.02653", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We thank all reviewers for their constructive and positive feedback on our paper, highlighting its novelty and importance. 

One concern was that our analysis might not generalize beyond our architecture choice of a rotation-equivariant neural predictive model. We now have performed additional experiments showing our results generalize to a more generic architecture as well (Fig 2 and 5 in attached pdf). 

Another question was whether our analyses are biologically plausible. We now have thoroughly discussed in our answers to the reviewers that we verify biological validity through a number of experiments (Fig 5 and Table 1 in the original manuscript) and will revise our final paper to describe this more clearly.

We want to thank all reviewers once again for these and their other great comments - comments we were keen to incorporate as they further improve our paper.

Meta-review: Reviewers were impressed by the paper's contributions in the problem of DNN modeling of biological neurons: the novel methods of iterative feature pruning and adaptive regularization, and well-designed and clearly motivated experiments, as well as the focus on addressing the reproducibility of learned embeddings.  Other strengths of the paper included testing of the biological plausibility of the tuning curves which enhanced the evaluation of the method in addition to more standard ML metrics.  The results on the non-rotationally invariant CNN added during the rebuttal further contribute to the potential generalizability of the results of the paper, which were originally shown only for rotationally invariant networks (due to biological considerations).

Some reviewer comments indicate that the paper could potentially benefit by adding more explanations and clarifying the motivation and technical details, which the authors have agreed to attempt in their revision.

This work has demonstrated a high level of technical sophistication and quality, and is likely to have high impact on of neuronal data modeling.  Given the size of the audience that would find this work to be of interest, I am recommending it for a spotlight presentation.
Decision: Accept (spotlight)