Beyond single neurons: population response geometry in digital twins of mouse visual cortex
Authors: Dario Liscai
Emanuele Luconi
Alessandro Marin Vargas
Alessandro Sanzeni
Venue: ICLR 2025 Poster (2025)
Keywords: neuroscience
representation learning
population responses
cortical hierarchy
computational biology
visual perception
Primary area: applications to neuroscience & cognitive science

Abstract: Hierarchical visual processing  is essential for cognitive functions like object recognition and spatial localization. Traditional studies of the neural basis of these computations have focused on single-neuron activity, but recent advances in large-scale neural recordings emphasize the growing need to understand computations at the population level. Digital twins-computational models trained on neural data-have successfully replicated single-neuron behavior, but their effectiveness in capturing the joint activity of neurons remains unclear. In this study, we investigate how well digital twins describe population responses in  mouse visual cortex. We show that these models fail to accurately represent the geometry of  population activity, particularly its differentiability and how this geometry evolves across the visual hierarchy. To address this, we explore how dataset, network architecture, loss function, and training method affect the ability of digital twins to recapitulate population properties. We demonstrate that improving model alignment with experiments requires training strategies that enhance robustness and generalization, reflecting principles observed in biological systems. These findings underscore the need to evaluate digital twins from multiple perspectives, identify key areas for refinement, and establish a foundation for using these models to explore neural computations at the population level.

Published as a conference paper at ICLR 2025
BEYOND SINGLE NEURONS: POPULATION RESPONSE GEOMETRY IN DIGITAL TWINS OF MOUSE VISUAL COR- TEX
Dario Liscai".'
Emanuule Luconi
Alessandro Marin Vargas", Alessandro Sanzeni'.
Bocconi University, Italy
"éooe Polytechnique Fédérale de Lausanne, Switzerland
"These authors contributed equally to this work, ceessanddo.ssamzznnn unt.occonininiinnnmmmm
ABSTRACT
Hierarchical visual processing is essential for cognitive functions like object recognition and spatial localization. Traditional studies of the neural basis of these computations have focused on single-neuron activity, but recent advances in large- scale neural recordings emphasize the growing need understand computations at the population level. Digital twins-computational models trained on neural data-have successfully replicated single-neuron behavior, but their effectiveness in capturing the joint activity of neurons remains unclear. In this study, we investi- gate how well digital twins describe population responses in mouse visual cortex. We show that these models fail to accurately represent the geometry of population activity, particularly its differentiability and how this geometry evolves across the visual hierarchy. To address this, explore how dataset, network architecture, loss function, and training method affect the ability of digital twins to recapitulate population properties. We demonstrate that improving model alignment with ex- periments requires training strategies that enhance robustness and generalization. reflecting principles observed in biological systems. These findings underscore the need to evaluate digital twins from multiple perspectives, identify key areas for refinement, and establish foundation for using these models explore neu- ral computations at the population level.
INTRODUCTION
Our ability to perform complex tasks, such object recognition and spatial localization. relies hicrarchical processing of visual information in the brain. Studies using single-neuron recordings have found progressive transformation of visual representations: from simple feature detection in early sensory areas to complex object representations higher visual areas (Felleman & Van Essen 1991 Hubel Wiesel, 1968). Advances in large-scale recordings have shifted the focus from single-neuron activity to population-level responses. These provide a more comprehensive view of the neural processes supporting cognitive functions and have been suggested to the essential unit of computation (Chung & Abboti, 2021 Vaa et al. 2020 Saxena & Cunningham 2019)
major challenge in studying the neural basis of complex computations, such object classifica- tion. is understanding how the joint activity of neurons represent stimuli and how these representa- tions are transformed across brain areas. Limitations in consistently recording from large numbers of neurons under diverse conditions constitute significant barrier to addressing this challenge. Ar- tificial neural networks trained on large-scale neural recordings (Fig. [TA), also referred as digital twins, have recently emerged powerful tools for bridging this gap (Lurz et al. 2020 Safarani et al. 2021 Wang et al. 2023). By learning to predict neural responses, these models offer the opportunity for extensive silico experiments, which would be impractical in live animals, such as identifying the most excitatory stimuli (Walker et al., 2019 Bashivan et al.) 2019 [Pierzchlewicz
et al. 2024) or mapping the landscape of relevant features across neuronal populations (Tong al.;

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2023 Despite their success at the single-neure neuron level, the potential of digital twins for studying
population- neural responses remains underexplored.
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Figure 1: Investigating population geometry of mouse visual cortex with digital twins (A) Schematic of digital twin used predict single neuron responses. We study how well these models capture the neural response geometry and variation across cortical areas. (B) Stringer exposed to thousands of natural images and parametric stimuli, grouped into ensembles of varying dimensions d. while V1 neuronal activity was recorded simultaneously. The collective neuronal activity corresponding each stimulus type formed manifold (left), whose strueture was analyzed using the eigenspectrum of the covariance matrix neural activity, which exhibited power-law decay (center; lighter and darker blue lines represent individual sessions and their average, respectively). The decay exponent a depended on and slightly exceeded + 2/d (right), suggesting that the neural representations are as high-dimensional as possible while still being differentiable. (C) In Froudarakis al. (Froudarakis et al. 2020), mice were presented with objects undergoing presseving transformations (rotation, scaling, translation, and variations in lighting) while simultaneous recordings were made from thousands of neurons (left). The experiment revealed hierarchy of linearly decodable object information across different visual areas (center and right).
(Stringer al. [2019a), mice were
In the visual cortex. neural population responses visual stimuli form manifolds, whose structure determines how efficiently downstream neural circuits can differentiate between stimuli, generalize across appearance variations, and maintain robustness against noise (DiCarlo et al.) 2012, Chung et al. 2018) Chhng & Abbott 2021 Li et al. 2024). Accurately capturing the structure of these population-level responses, both within individual brain areas and as evolves across the visual hierarchy, is important for digital twin models aiming to provide insights into neural computations. In this study, we evaluate the ability digital twins to replicate two landmark findings in the mouse visual cortex hierarchy. First, investigate the differentiability of population responses in primary visual cortex (V1). Stringer et al. (Stringer et al. 2019a) demonstrated that V1 representations ex- hibit high-dimensional geometry that remains differentiable (Fig TB), enabling robustness noise
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and small variations in stimuli. They achieved this by exposing mice to ensembles of visual stimuli (e.g., natural images and parametric stimuli) grouped by dimension d and analyzing the resulting neural activity manifolds using the eigenspectrum of the covariance matrix. The spectrum followed a power-law decay, with an exponent a dependent on d and slightly exceeding + 2/d. Stringer et al. (Stringer et al. (209a)) emphasized that a spectrum with too slow decay is pathological, as it requires an impractically large number of neurons to reconstruct stimuli and makes neural responses extremely sensitive to small stimulus changes-both of which hinder learning to generalize. Math- ematically, they showed through functional analysis that the decay exponent a reflects the geometry of neural representations. Specifically, the dimensionality of representations increases with a, and their differentiability depends on key thresholds: representations with a < 1 are discontinuous, those with 1 S a < 1 + 2/d are continuous but not differentiable, and those with a > + 2/d are differentiable. These results are based on the relationship between the eigenspectrum and the similarity of arbitrary neural response vectors. If the spectrum decays too slowly, the trace of the covariance matrix (linked to vector similarity) diverges, causing instability in neural representations.
Second, we investigate how neural representations transform across cortical areas. Experiments have revealed signatures of hierarchical information processing in mouse visual cortex, at both the anatomical Wang et al. [0022 D'Souza et al., (2022) and functional (Siegle et al. [021 Froudarakis
al. 2020 Hoeller et al. 2024) level. We focus the experiment by Froudarakis al. (Froudarakis let al. 2020), which demonstrated that neural representations evolve to support invariant object recognition (Fig. [CC. They achieved this by recording neural activity in various areas while pre- senting objects undergoing
transformations, such as rotation, scaling, and trans- lation, and evaluating the discriminability of these representations across different areas.
We find that ofate-of-the-art digital twins struggle to replicate these experimental results. To tackle this issue, we examine the impact of different dataset compositions, network architectures, loss func- tions, and regularization techniques on the models' capacity to capture population-level responses. We show that adding regularization during training improves the model's ability to learn neural rep- resentations that match the experimental population response. Our findings underscore the need for digital twin models to move beyond single-neuron predictions and push toward modelling the com- plexities of neural population activity and its hierarchical transformations within the visual system.
2 RELATED WORKS
Encoding models of the visual cortex. Encoding models of the visual system aim to characterize how sensory stimuli are transformed into neural responses. These models have evolved from linear- nonlinear Poisson (LNP) models (Simoncelli et al. (004) to deep artificial neural networks (ANNs), which are now considered digital twins of the brain's visual processing pathway. Two primary approaches dominate: task-driven models, which optimize ANNs to solve specific computational tasks like object recognition (Yamins et 20144 Bakhtiari et al., 2021 Kubilius et al.) 2019) Zhuang let al., 2021 Pogoncheff et al. 2003), , and data-driven models, which are trained directly neural data to replicate observed responses (Klindt et al.] (2017] [Cadena et al. 2019: Lurz et al., 2020) Safarani et al.) 2021 Willeke et al., 2022 Li et al. [003]] Wang et al. 2023). Recently, new models have been developed to address the complexity of dynamic stimuli, like videos. From the best models of the Dynamic Sensorium Competition (Turishcheva et al. (024) to multi-modal models that combine visual images with behavioral variables (Xu et al. 2024). these models have achieved great prediction accuracy. Despite these models' success in capturing single-neuron responses, they struggle to develop hierarchical representations (St- Yves et al.
2023 Dyballa et al. (024)) and we show they also fail to represent the complex structure of neural population responses.
Single-neuron and population-level analysis. Applications of neural network models in visual neuroscience have largely focused on single-neuron properties, investigating how individual neu- rons respond to stimuli. Previous works have examined feature selectivity for object recognition and texture perception (Walker et al. 22119), the columnar organization of selectivity in V4 (Willeke) et al. 2023), and the encoding of phase-invariant fixed spatial patterns in V1 (Ding et al. 2023 Ustyuzbaninov et al.) 2022). While these analyses have provided valuable insights, they fall short in capturing the population structure of neural responses. Recent work has started to address this limitation. Pogoncheff et al. (Pogoncheff et al. (023) explored the relationship between V1 pre- dictability and population-level properties. Similarly, Margalit et al. (Margalit al.) 2024) examined
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population- topological properties across early (V1) and higher-order (VTC) visual areas. We continue this shift towards population- level analysis by using digital twins to investigate the geomet- rical properties of neural population responses, with focus on differentiability and discriminability.
Regularization strategies. To bridge the gap between observed neural activity and artificial models, various studies have introduced regularization techniques, often drawing inspiration from principles like the efficient coding hypothesis (Barlow et al.) 1961. Sparsity constraints, for example, helped neural networks to develop Gabor-like receptive lields, mimicking the tuning properties of early visual neurons (Olshausen & Field 1996. Bell & Sejnowski 1997). These regularizations can be applied during model training, e.g. adversarial training to improve robustness (Guo al., 2002) or through image augmentation techniques that enhance model generalization (Shorten & Khoshgof- taar 2019). More recent approaches apply architectural constraints, e.g. functional or topological constraints, to better align artificial models with biological data, enhancing their interpretability and functional similarity (Pogoncheff et al. 2023; Margalit et al. 2024). Our study builds on these ap- proaches by demonstrating that introducing regularization strategies significantly enhance the mod- els' ability to learn representations aligning with those observed in biological systems.
3 DIGITAL TWIN OF THE MOUSE VISUAL CORTEX
To evaluate the capability of digital twins of the mouse visual cortex in capturing both single-neuron and population-level response properties, we used the baseline model of the SENSORIUM com- petition (Turishcheva et al. 2023), whose architecture is structured into three components: core module, a readout module, and a shifter module. The core is composed of 4 layers of 3D Convolu- tional Neural Network (CNN), inspired by (Höfling et al. 2022), which encodes the spatiotemporal features of visual stimuli and behaviors into a latent representation (see Supp. A. The readout, taken from (Lurz et al.] (020), transforms the core representation into individual neuronal activity. Its functioning is divided into two components: a spatial mask, which determines the neurons' re- ceptive field positions (i.e., where they "look" in the image), and feature weight, which linearly combines features from the core's final layer to generate neural responses shifter takes the pupil center coordinates as input and produces a "gaze-shift" that is linearly ap- plied to the spatial positioning of all neurons" receptive fields. The output is then passed through an activation function (ELU +1) to generate the final neuronal responses. We introduced a few mi- nor modifications to the original architecture (see list in A.1), which we found to facilitate training without modifying performance (see Table D..
The
We trained the model using the MICIONS dataset (Consortium et al. 2021), which includes more than 75,000 neurons spanning multiple areas of the mouse visual cortex (VI, AL, RL) collected trained to minimize the Poisson loss in 14 recording sessions for single mice. The network Throughout this paper, we refer to this between predicted and recorded responses (see Supp. A. trained model as the benchmark model. The model demonstrated strong performance in single- neuron predictions, achieving a median normalized correlation coefficient (CCnorm) of 0.47-0.58 across sessions. This result is comparable to the performance reported by (Wang et al., 2023) (see Table IT. which, to the best of our knowledge, is the only other model trained end-to-ent to- on the MICrONS dataset available in literature. Performance slightly varied across visual areas (Supp. Fig. 1), with the highest prediction accuracy observed in V1 (mean corr-to-averge 0.305), followed by LM (0.28), RL (0.255) and AL (0.253). These performance differences across areas may reflect a higher number of neurons in certain regions, which could significantly impact the loss function (83,222 units in V1, 14,817 in LM, 12,599 in RL and 4,734 in AL across all sessions); alternatively, differences in single-neuron response reliability (6.77% in VI, 6.37% in LM, 5.08% and 5.03% in AL calculated as Fraction of Explainable Variance over repeated stimuli) might bias the model to learn one area better than another, with both factors potentially coexisting.
4 DIGITAL TWINS FAIL TO CAPTURE THE STRUCTURE OF POPULATION RESPONSES IN MOUSE VISUAL AREAS
Traditional metrics for training and evaluating digital twins, such Poisson loss and single-neuron correlation, focus on individual neuron predictions and overlook population-level structures aris- ing from neural interactions. This gap raises questions about whether these models can capture
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Figure 2: Population geometry in 3D convolution model (A) Eigenspectrum of the covariance matrix obtained from our benchmark model, simulating the experiment by Stringer et al. (Stringer et al. (019a) with natural images (lighter green lines individual sessions; the darker green line is the average). The eigen- spectrum exhibits power-law decay (best-fit exponent a 0.82) significantly slower than that observed experimental data from Stringer et al. (Stringer et al.] (019a) (blue, exponent a = 1.04). (B) Power-law exponents for the model (green) and experimental data from Stringer et al. (Stringer et al., 2019a) (blue), calculated across different types of visual stimuli with varying dimensionality. The digital twin's 's representa- tion aligns with experimental observations, showing higher dimensionality with increased stimulus complexity. However, it exhibits slower decay for natural, whitened, spatially localized images and gratings, indicating (C) Hicrarchy of neural representations obtained from the benchmark model when simulating the experiment of Froudarakis et al. (Froudarakis et al. 2020). We evaluated the ability of linear classifier to predict object identity representations from different areas generated in model. Results represent the aggregation of three indepen- dent model runs to ensure robustness and consistency. (D) Relative discriminability across areas computed in the model, in Fig. CC The model fails to capture the differences in discriminability observed experimentally across visual areas. Here and in what follows: error bars represent the standard error of the mean (s.c.m)). ) stars indicate significance levels from Mann Whitney test (x, *4, indicate p-valk. 0.05, 0.01, 0.001).
that the learned representation of these stimuli is not differentiable.
conditions using neural
key properties of population response structure observed experimentally in visual cortex (Stringer) et al. [2119]] Froudarakis et al. 2020). To address this, we replicated these experiments with our benchmark model, testing its ability to capture the observed population-level properties.
To analyze if digital twins capture the geometry of the neural representation of V1, we simulated the experiment of Stringer et (Stringer al. (2019a). We presented ensembles images with varying dimensionalities to the model, extracted predicted responses from neurons in layer 2/3 of V1, and measured the covariance between neural responses across images, mimicking the experimental pro- cedure (Fig. IB and Supp. A.2). Despite the visual stimuli from Stringer al. being substantially different from those used in training, the model did not exhibit any pathological response (Supp. Fig. 1), suggesting that it can generalize to these experimental protocols. Consistent with the ex- perimental data, the spectrum of the covariance matrix between units in the model demonstrated a power-law behavior for natural images (Fig. A and for other types of visual stimuli investigated (Supp. Fig. 1). Furthermore, the model recapitulates the qualitative dependency on stimulus dimen- sionality observed experimentally, showing smaller power-law exponent a for higher-dimensionall stimuli (Fig. 2B). Nevertheless, we found significant discrepancies between the model and the ex- perimental results. Stringer al. showed that the power law exponent of the eigenspectrum
Table 1: Comparison of model single-neuron prediction performance (definitions in Supp. A.1) on the MICrONS (Consortium et al. 2021) and SENSORIUM (Turishcheva et al. 2023) datasets. For SENSORIUM, results for (Turishcheva et al. 2023) are relative to the live test set, whereas VIVIT and ours refer to the validation set.
Dataset (cortical areas)
CC
Corr
Corr-to tove-ao-average
Wang et al. (2023)
MICIONS VII.M.
0.48-0.65
Our benchmark model
MICIONS (VL.L.)
0.47-0.58
0.168
0.294
Turishcheva et al. (002)
SENSORIUM (V1)
0.229
0.416
ViVIT V (Turishcheva et al. 2024
SENSORIUM (V1)
0.261
0.478
Our benchmark model
SENSORIUM (V1)
0.247
0.452
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Figure 3: Non-differentiable geometry of V1 population responses across network models. Eigenspectrum of the covariance matrix for natural stimuli in: (i) model trained on reliable neurons, (ii) model trained the SENSORIUM dataset, (iii) ViVIT, (iv) model trained on correlations, and (v) experimentally observed power-law decay. Eigen- spectra for other stimuli are shown in Supp. Fig. 2. Despite providing better descriptions of either single neuron responses or the correlations between them compared to our benchmark model, none of these models exhibited differentiable representation.
Models ( 0.75
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encapsulates key features of the geometry of neural representations. Smaller indicates higher- dimensional representations; a < means the representation is not continuous, and a < 1 2/d means it is not differentiable. In mice, was found to be slightly above = 1 + 2/d, suggesting a nearly maximal dimensionality while maintaining differentiability. This allows the network to represent a wide range of stimuli smoothly, aiding downstream decoders in generalizing from small stimulus changes. In our benchmark model, for different stimuli (natural images, spatially localized stimuli, whitened stimuli, and gratings). we found a power law exponent a below the critical line 1 + 2/d (Fig. BB.. These findings indicate that, unlike the representation in the mouse VI, the rep- resentation learned by the digital twin is not differentiable and, for natural, whitened and spatially localized images, is not continuous.
To assess whether digital twins can account for hierarchy in discriminabilitty observed across vi- sual areas, we simulated the experiment by Froudarakis et al. (Froudarakis et al.) (020). In their study, mice were shown movies of objects undergoing identity-preserving transformations, includ- ing rotation, scaling, translation, and variations in lighting. We recreated these movies, fed them into our benchmark model and extracted the predicted neural responses from different visual areas (See Supp. A.3). Despite the significant differences between these stimuli and the training data, the model produced plausible responses (Supp. Fig. 1). We then used these extracted responses to train cross-validated logistic regression models for object identification (Fig- DC.D and Supp. A.3). The model showed notable quantitative discrepancies, with discriminability values higher than those re- ported experimentally and less pronounced differences between visual areas (Fig. 2C). While these discrepancies are expected due to the absence of spiking noise, reduced variability, and differences in visual stimuli, changes in discriminability across areas (Fig. 2D) revealed mistepresentation of the hierarchy observed in (Froudarakis et al. (2020), with uniform object identity discriminability across the four brain areas. This qualitative discrepancy indicates that the model fails to capture the hierarchy in discriminability observed across visual areas of the mouse cortex.
These results show that while our model captures single-neuron responses, it fails represent the population-level structure and hierarchical transformations. Capturing the hierarchy of discrim- inability across areas likely requires first understanding the representation geometry within an area, as this determines decoder performance (Chung al.] 2018 Chung & Abbott 2021 Li et al. 2024). Thus, in what follows, we will first modify the model to capture the geometry of population re- sponses in V1, then examine how these changes affect the hierarchy of discriminability across areas.
TOWARDS CAPTURING VI REPRESENTATION GEOMETRY
To develop digital twins of mouse V1 that reflect neural response geometry, we first assessed the robustness of our findings, by evaluating how variations in the dataset, training procedure or net- work architectures may impact the ability to capture the geometry of V1 responses. After finding these methods inadequate, we show that specific regularization techniques push the model toward capturing representations consistent with those of mouse V1.
5.1 DIGITAL TWINS CONSISTENTLY FAIL TO REPLICATE DIFFERENTIABILITY OF NEURAL REPRESENTATIONS
We first examined whether improving model performance could lead improvements in capturing the geometry of population responses. Experimental studies show that spontaneous activity overlaps with visual responses, causing variability in neuron responses repeated stimuli (Niell & Stryker

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2010 Stringer al. 2019b Avitan & Stringer 2022). Although the model partially accounts for this variability by including running and pupil size as inputs, it remains insufficient due to: (1) the influence of additional behavioral variables on spontaneous activity (Stringer et al. 2019b), and (2) the fact that not all spontaneous activity is behavior-driven (Stringer et al. 2019b). These limitations constrain the model's accuracy in predicting single-neuron responses. We trained our model to replicate exclusively the activity of neurons with high reliability to repeated vidcos (see Supp. A.4). As expected, model performance improved for both single trials and trial-averaged responses, with correlation increasing by 19.0% and 36.7%, respectively. Nevertheless, the spectral properties remained unchanged (Fig. 3 and Supp. Fig. 2), suggesting that while individual response predictions improved, the underlying population response geometry was not affected.
We then considered whether the characteristies of the training dataset influenced our findings. To investigate this, we trained our model using the SENSORIUM competition dataset (Turishcheval al. 2023) (see Supp. A.4). The model trained on SENSORIUM significantly outperformed the MiCroNS-trainee model, showing improvements in correlations between single-neuron responses and model predictions by a factor of 47.0% in single trials and 53.7% in trial averages. Despite this improvement, the model's 's ability to capture the power-law exponents observed in experimental data did not improve, either for natural images (Fig. (]i or other tested stimuli (Supp. Fig. 2). As result, the representation remained non-differentiable thereby discarding the training dataset possible factor to influence the population geometry.
We then examined whether the model's 's architecture affected results. CNNs have historically been prominent in computer vision, but recent advances have shown that transformers are now achieving competitive and sometimes superior performance in various tasks (Khan et al., 2022 Az- abou et al. [024; Antoniades et al. 2023). Motivated by this, we employed VIVIT, a transformer- based digital twin trained on the SENSORIUM dataset (Turishcheaa et al.] 2024). to replicate the experiment from Stringer et al. (Sringerr et al.] 2019a). Our results show that VIVIT, like previ-
ous models, fails to capture the geometric properties of mouse VI, with representations remaining non-differentiable despite improved accuracy (Fig. Fii and Supp. Fig. 2).
Finally, we explored whether modifying the target of the loss function could improve the model's ability to capture the covariance between neurons. Previous models were optimized to predict individual neuron responses, which prioritized single-neuron accuracy over population structure. To address this. we developed new training scheme where the loss function is based uniquely on the correlations between neurons. The trained network significantly outperformed our bench- mark model in describing correlations between neurons (root mean squared error correlations decreased by 41.76%). Unsurprisingly. this improvement came at the expense of the model's ability to capture single-neuron responses, which was significantly decreased. We replicated the experi- ment by Stringer et al. (Stringer et al.) 2019a) and analyzed the covariance matrix spectrum (Fig 33y and Supp. Fig. 2). Unlike previous approaches, this model displayed power-law behavior with significantly larger a than the benchmark model (0.93 for natural stimuli compared to 0.82 of the original model). However, the decay was still slower than observed experimentally, and the result- ing representation remained non-differentiable. We expanded this approach with multi-objective optimization to balance single-neuron responses and population correlations (Supp. Fig. 8). We found no evidence of synergistic effect: gradually increasing the relative weight of the Poisson loss led to models better capturing single-neuron responses but at the expense of the diffcrentiability of population representations. Additionally, we explored whether task-driven modeling could yield more differentiable representations. Specifically, we fine-tuned a linear readout to predict neural responses after each of the first three layer block of ResNet50 with frozen weights trained on object recognition. Consistent with the data-driven models, the task-driven model also produced a non-differentiable representation (Supp. Fig. 6), suggesting that the chosen objective functions lack the inductive bias necessary for developing differentiable representations.
5.2 TRAINING REGULARIZATION FACILITATES HIGH-DIMENSIONAL, DIFFERENTIABLE REPRESENTATIONS IN DIGITAL TWINS
Our analyses reveal that digital twins are prone to learning neural representations that are not differ- entiable. We hypothesize that introducing strategies enhance robustness in neural responses could improve differentiability. Specifically, we reasoned that training the model with pairs of images, where neural responses are kept fixed while the images are slightly perturbed, would encourage the
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digital twins simulating responses to natural images (from (Stringer et al.] (019a)) with respect to the amplitude of image transformation used during training (see Supp. A. (B) Power-law exponent of natural images with respect the dropout rate. Digital twins with a dropout rate greater than 0.4 exhibit exponent larger than 1, indicating differentiable representation of natural images. (C) Eigenspectrum of the covariance matrix in response natural images with a dropout rate of 0.4. The model exhibits a power-law decay of a = 1.06, closely matching the experimental data of a = 1.04. (D) Power-law exponent as a function of stimulus dimensionality for dropout rate of 0.4. For all types of visual stimuli, except gratings, the model exhibits a differentiable geometry of the neural representation, consistent with experimental data.
model to learn representations in which small changes in the input produce minor changes in the neural responses. This is effectively a way to ensure that the representations are differentiable.
To test our hypothesis, we trained the model using stimulus augmentation, applying random trans- formations (e.g., translation, rotation, rescaling) to the input images. These transformations were chosen for their relevance to the differentiability neural representations, simulating the way ani- mals perceive stimuli from various angles and distances during movement. The model exhibited modest decline in performance when predicting single-ncuron responses as transformation ampli- tude increased (Supp. Fig. 3), while the covariance matrix retained power-law spectrum (Supp. Fig. 3). Larger transformations during training led to more robust neural representations, with the power-law exponent of the spectrum approaching that of differentiable representations as the trans- formation amplitude increased (Fig. AA).
Another potential approach to enhancing the robustness of neural representations is introduce perturbations directly within the network itself. One widely used machine learning technique that adopts this strategy dropout (Srivastava et al., 2014). By randomly disabling units during train- ing, dropout effectively reduces overfitting and encourages the model to generalize better across unseen data. We hypothesized that this method could also improve the model's ability capture the underlying structure of neural responses. To test this, we applied dropout across layers of the 3D CNN during training and analyzed its impact on the model's representation of neural activity. While dropout reduced performance the dropout rate increased (Supp. Fig. 4), the model still accurately predicted single-neuron responses and maintained a power-law spectrum for the covari- ance matrix (Supp. Fig. 4). Gradually increasing the dropout rate significantly modified the power spectrum across all stimulus types, achieving differentiable representations of natural images for dropout rate 0.4 (Fig. BB Differentiable representations, consistent with those observed ex- perimentally, emerged for all visual stimuli investigated, except for grating stimuli (Fig. 4 and Supp. Fig. 4); we will discuss possible explanations for this exception in the discussion section.
6 IMPACT OF REPRESENTATION GEOMETRY ON THE HIERARCHY LEARNED BY DIGITAL TWINS
The analysis of Fig. CC.D shows that digital twin trained to reproduce single-neuron responses fails to capture the hierarchy of object discriminability observed experimentally (Froudarakis et al.) 2020 Given that population response geometry is critical for determining discriminability (Chung et al. 2018 Chung & Abbott 2021 Li et al. 2024), we investigated whether changes in geometry induced by dropout would influence the hierarchical structure learned by the model.
To explore this, we simulated the experiment Froudarakis et al. (Fruudaraiis et al. 2020 using models trained on the MICrONS dataset with varying dropout rates. We then assessed how well each
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Figure Hierarchy discriminability across visual areas in digital twins with different representation geometry (A) Discriminability
A
B
changes relative to the mean across different ar- eas, computed for the network models with differentiable representations. (B) Same as in A, but for models with differentiable represen- tations. These models more effectively capture the hierarchical relationships between areas ob- served experimentally, accurately reflecting the relationships among LM, VI, and RL. However, the model incorrectly places AL at the bottom of the hierarchy. Results in panels A and B were ob- tained by aggregating simulations from the mod- shown Fig. with a < panel and a > for panel B.
Non-differentiabee
Differentiable
(relative deviation)
0.02
Discriminablitg
0.01
0.00
-0.01
71
41
RL-43
AL
LM V1
RL
LM vi RL
Areas
Areas
model could discriminate between stimuli across different visual areas. Increasing the dropout rate consistently reduced the ability to discriminate stimuli across all areas (Supp. Fig. 5). This finding aligns with prior work, which demonstrates that higher dimensionality, reflected by lower power- law exponent o, facilitates pattern discrimination (Babadi & Sompolinsky observed decrease in discriminability may stem from the model's 's reduced performance in predicting single-neuron responses (Supp. Fig. 5), potentially leading to lower information content in the model's responses to visual stimuli.
Additionally, the
Interestingly, dropout also influenced the hierarchy of discriminability between areas (Fig-5B). In models with non-differentiable representations, object identity was more easily discriminated in RL compared to LM and VI, with AL showing the lowest performance. This is the opposite of what is observed experimentally (Froudarakis et al.] 2020) (Fig. TTC) However, when the model exhibited differentiable representations, the hierarchy aligned more closely with experimental data, positioning LM above VI, which was above RI. at the bottom (Froudarakis al., (2020) (Fig. (B)). While these qualitative results are promising, significant discrepancies remain. The model incor- rectly places AL at the bottom of the hierarchy, whereas was found experimentally to at the top. Additionally, the discriminability values were consistently higher than those observed experimen- tally, and the differences between visual areas were less pronounced (Supp. Fig. 5). We comment on potential causes for these discrepancies in the discussion section.
7 DISCUSSION AND LIMITATIONS
Digital twins have shown promise in modeling single-neuro responses, but their ability to cap- ture population-level response structures remains largely unexplored. In this study, we evaluated their performance in replicating population responses within the mouse visual cortex. Our findings demonstrate that current models struggle to reproduce the differentiability of neural representations and the hierarchy in object discriminability across areas observed experimentally (Stringer et al., 2019a Froudarakis et al. 2020). We observed this limitation across datasets, model architectures, and loss functions.
Our results show that regularization during training enabled models to replicate the observed differ- entiability of population responses. This effect likely arises because introducing stochasticity into the relationship between inputs and neural responses fosters a more distributed and robust coding scheme. our models, stochasticity was introduced through dropout, which randomly deactivates units during training, and data augmentation, which adds variability by shifting and scaling input images. In biological systems, spiking variability and stochastic synaptic transmission effectively introduce an analogous stochastic relationship between inputs and neural representations. Our find- ings suggest that such biological noise could promote differentiable and robust representations.
The only stimuli that maintained non-differentiable representation gratings. This may be be- cause models trained on one domain (e.g., natural images) have limited transferability to different stimulus types, previous studies have indicated (Sinz al. 2018). Consistent with this inter-

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pretation, found that the tuning curves predicted by our model resemble those observed experi- mentally for the oriented stimuli in the MICrONS dataset used for training but not for the gratings used in the experiments by Stringer et al. (2019)) (Supp. Fig. 10). Though recent advancements in foundational models of mouse V1 have improved domain transfer performance (Wang et al. 2023). challenges remain. To properly assess differentiabilityy it is not enough predict simple features like orientation preference; the entire tuning curve shape must be considered. This raises important questions about how well models capture the geometric properties of population responses across stimuli.
We demonstrated that models with differentiable representations better captured the hierarchy in object discriminabilityy across areas observed experimentally. Specifically, the discriminability of stimuli increased from RI to V1, and from V1 to L.M, consistent with the experimental hierarchy. However, our models incorrectly placed AL at the bottom of the hierarchy and consistently overes- timated discriminability and underestimated its variation across areas when compared with precious experimental results (Froudarakis al. 2020). These discrepancies likely stem from differences between artificial and biological networks, including the absence of spiking noise and reduced trial- to-trial variability in the models. Additionally, differences in training stimuli- suuulll-uchhh object move- ment speed or complexity- may contribute to the observed gaps between model predictions and experimental data.
We found that regularization during training improved the ability to capture population-level re- sponses but reduced single-neuron performance (Supp. Fig. 9), revealing a consistent trade-off across models and regularization schemes. This occurs because state-of-the-art models capture only finite fraction of the explainable variance for single neurons. Accurately predicting single-neuron responses reflects an approximation of marginal distributions but does not inherently constrain the model learn the higher-order dependencies that govern population structure and shape the covari- ance matrix. Furthermore, fitting both single-neuron and population-leve responses on one dataset does not ensure generalization, as shown by discrepancies on datasets from Stringer et al. (Stringer) et al. 2019a) and Froudarakis et al. (Froudarakis et al.) 2020), likely due to overfitting to dataset- specific statistics rather than capturing biologically relevant structures. These results highlight key limitations of current approaches and the need for models that balance this trade-off while general- izing effectively across diverse datasets.
Future studies could benefit from using architectures that differentiate between neurons in distinct visual areas, allowing for a more accurate replication of hierarchical transformations. Indeed, a limitation of our architecture is that the core is shared across all visual areas thereby constraining the model's expressivity. In biological systems, receptive fields expand along the visual hierar- chy (Siegle al., 2022). property model does not capture due to its uniform receptive field size. To explore this limitation, we examined more flexible digital twins that allow different areas to rely on distinct internal representations. First, we employed a task-driven modeling approach by finetuning a linear readout from layers of model trained on object recognition and found that it did not capture the discriminability hierarchy observed experimentally (Supp. Fig. 6). Additionally, we trained a 3D CNN model independently on data from each area to capture area-specific differences. While variations in object discriminability across areas were larger than in our benchmark model, the hierarchy of discriminability was still incorrect (Supp. Fig. 7)-
To further improve digital twins, it will be important to move beyond measures of covariance and discrimination and consider more detailed manifold structures that underlie population responses, such as radius and dimensionality (Chung et al. 2018 Froudarakis et al.] (020)). These factors,
which significantly impact discriminability (Chung al. 2018 Chung & Abbott 2021 Li et al.
2024). show clear trends along the visual hierarchy (Froudarakis et al. 2020).
Understanding how complex computations such object recognition and localization emerge from hierarchical neural transformations is a fundamental challenge. Digital twins provide a promising tool for exploring these questions, they allow for detailed control over stimuli and response vari- ables. In this work, we have demonstrated that digital twins can capture key aspects of population response geometry and qualitatively reproduce the hierarchy in object discriminability across areas observed experimentally. However, further research is needed to fully validate these models and to explore how joint activity within neuronal populations contributes to computations across different visual areas.
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REFERENCES
Antonis Antoniades, Yiyi Yu, Joseph Canzano, William Wang, and Spencer Laver Smith. Neu- roformer: Multimodal and multitask generative pretraining for brain data. arXiv preprint arXiv:2311.00136, 2023.
Lilach Avitan and Carsen Stringer. Not so spontaneous: Multi-dimensional representations of be- haviors and context in sensory areas. Neuron, 110(19))3064-3000 2022.
Mehdi Azabou, Vinam Arora, Venkataramana Ganesh, Ximeng Mao, Santosh Nachimuthu, Michael Mendelson, Blake Richards, Matthew Perich, Guillaume Lajoie, and Eva Dyer. A unified, scalable framework for neural population decoding. Advances Neural Information Processing Systems, 36, 2024.
Baktash Babadi and Haim Sompolinsky. Sparseness and expansion in sensory representations. Neu- ron, 83(5):1213-1222 2014.
Shahab Bakhtiari, Patrick Mineault, Timothy Lillicrap. Christopher Pack, and Blake Richards. The functional specialization of visual cortex emerges from training parallel pathways with self- supervised predictive learning. Advances Neural Information Processing Systems, 34:25164- 25178, 2021.
Horace B Barlow et al. Possible principles underlying the transformation of sensory messages. Sensory communication, 1(01):217-233, 1961.
Pouya Bashivan, Kohitij Kar, and James DiCarlo. Neural population control via deep image synthesis. Science, 364(6439))aaav4436 2019.
Anthony J Bell and Terrence J Sejnowski. The "independent components" of natural scenes are edge filters. Vision research, 37(23):3327-3338 1997.
Blender Development Team. Blender. URL.
Santiago Cadena, George H Denfield, Edgar Y Walker, Leon A Gatys, Andreas S Tolias, Matthias Bethge, and Alexander S Ecker. Deep convolutional models improve predictions of macaque vl responses to natural images. PLoS computational biology, 15(4)=e100689, 2019.
Yeo Chung and Larry Abbott. Neural population geometry: An approach for understanding biological and artificial neural networks. Current opinion in neurobiology, 70:137-14: 2021.
YuoYeon Chung, Daniel D Lee, and Haim Sompolinsky. Classification and geometry of general perceptual manifolds. Physical Review X, 8(3):031003, 2018.
MICrONS Consortium, Alexander Bae, Mahaly Baptiste, Caitlyn Bishop, Agnes Bodor, Der- rick Brittain, JoAnn Buchanan, Daniel J Bumbarger, Manuel A Castro, Brendan Celii, et al. Functional connectomics spanning multiple areas of mouse visual cortex. BioRxiv, pp. 2021-07, 2021.
James J DiCarlo, Davide Zoccolan, and Nicole C Rust. How does the brain solve visual object recognition? Neuron, 73(3):415-434 2012.
Zhiwei Ding, Dat Tran, Kayla Ponder, Erick Cobos, Zhuokun Ding, Paul G Fahey, Eric Wang, Taliah Muhammad, Jiakun Fu, Santiago A Cadena, et al. Bipartite invariance in mouse primary visual cortex. bioRxiv, 2023.
Luciano Dyballa, Andra M Rudzite, Mahmood S Hoscini, Mishek Thapa, Michael P Stryker, Greg D Field, and Steven W Zucker. Population encoding of stimulus features along the visual hierarchy. Proceedings of the National Academy of Sciences, 1217712231777311,, 2024.
Rinaldo D D'Szu,, Quanxin Wang, Weiqing Ji, Andrew M Meier, Henry Kennedy, Kenneth Knoblauch, and Andreas Burkhalter. Hierarchical and nonhierarchical features of the mouse visual cortical network. Nature communications, 13(1):503, 2022.
Daniel Felleman and David C Van Essen. Distributed hierarchical processing in the primate cere- bral cortex. Cerebral cortex (New York, NY: 1991), 1(1::1447, 1991.
11

---

Published as a conference paper at ICLR 2025
Katrin Franke, Konstantin F Willeke, Kayla Ponder, Mario Galdamez, Na Zhou, Taliah Muhammad, Saumil Patel, Emmanouil Froudarakis, Jacob Reimer, Fabian H Sinz, et al. -ate-deeenddent pupil dilation rapidly shifts visual feature selectivity. Nature, 120(7930):128-42 134, 2022.
Emmanouil Froudarakis, Uri Cohen, Maria Diamantaki, Edgar Y Walker, Jacob Reimer, Philipp Berens, Haim Sompolinsky, and Andreas S Tolias. Object manifold geometry across the mouse cortical visual hierarchy. BioRxiv, pp. 2020-08, 2020.
Chong Guo, Michael J Lee, Guillaume Leclere, Joel Dapello, Yug Rao, Aleksander Madry, and James J DiCarlo. Adversarially trained neural representations may already be as robust as corre- sponding biological neural representations. arXiv preprint arXiv:2206. 1228, 2022.
Judith Hoeller, Lin Zhong, Marius Pachitariu, and Sandro Romani. Bridging tuning and invariance with equivariant neuronal representations. bioRxiv, pp. 2024-08, 08,4408,5 2024.
Larissa Höfling, Klaudia P Szatko, Christian Behrens, Yuyao Deng. Yongrong Qiu, David A Klindt, Zachary Jessen, Gregory Schwartz, Matthias Bethge, Philipp Berens, et al. A chromatic feature detector in the retina signals visual context changes. bioRxiv, pp. 2,22-11,0022.
David H Hubel and Torsten N Wiesel. Receptive fields and functional architecture of monkey striate cortex. The Journal of physiology. 195(1):215-243, 1968.
Salman Khan, Muzammal Naseer, Munawar Hayat, Syed Waqas Zamir, Fahad Shahbaz Khan, and Mubarak Shah. Transformers in vision: A survey. ACM computing surveys (CSUR), 54(10s):
2022.
David Klindt, Alexander S Ecker, Thomas Euler, and Matthias Bethge. Neural system identification for large populations separating "what" and "where". Advances neural information processing systems, 30, 2017.
Jonas Kubilius, Martin Schrimpf, Kohitij Kar, Rishi Rajalingham, Ha Hong, Najib Majaj. Elias Issa, Pouya Bashivan, Jonathan Prescott-Roy Kailyn Schmidt, et al. Brain-like object recognition with high-performing shallow recurrent anns. Advances in neural information processing systems, 32, 2019.
Bryan Li, Isabel Cornacchia, Nathalie Rochefort, and Amn Onken. Vlt: large-scale mouse vI response prediction using vision transformer. arXiv preprint arXiv:2302.03023. 2023.
Qianyi Li, Ben Sorscher, and Haim Sompolinsky. Representations and generalization in artificial and brain neural networks. Proceedings of the National Academy of Sciences, (2(27)):2311105522 2024.
Konstantin Klemnnns Lurz, Mohammad Bashiri, Konstantin Willeke, Akshay K Jagadish, Eric Wang, Edgar Walker, Santiago A Cadena, Taliah Muhammad, Erick Cobos, Andreas Tolias, et al. Generalization in data-driven models of primary visual cortex. BioRxiv, pp. 2020-10, 2020.
Eshed Margalit, Hyodong Lee, Dawn Finzi, James J DiCarlo, Kalanit Grill-Spector, and Daniel LK Yamins. A unifying framework for functional organization in early and higher ventral visual cortex. Neuron, 2024.
Cristopher M Niell and Michael Stryker. Modulation of visual responses by behavioral state in mouse visual cortex. Neuron, 65(4):472-479, 2010.
Bruno A Olshausen and David J Field. Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature, 381 (6583)))6076666 1996.
Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al. Pytorch: An imperative style, high- performance deep learning library. Advances in neural information processing systems, 32, 2019.
Pawel Pierzchlewicz. Konstantin Willeke, Are Nix, Pavithra Elumalai, Kelli Restivo, Tori Shinn, Cate Nealley, Gabrielle Rodriguez, Saumil Patel, Katrin Franke, et al. Energy guided diffusion for generating neurally exciting images. Advances in Neural Information Processing Systems, 36, 2024.
12

---

Published as a conference paper at ICLR 2025
Galen Pogoncheff, Jacob Granley, and Michael Beyeler. Explaining v1 properties with biologically constrained deep learning architecture. Advances in Neural Information Processing Systems, 36: 11908-13930, 2023.
Shahd Safarani, Arne Nix, Konstantin Willeke, Santiago Cadena, Kelli Restivo, George Denfield, Andreas Tolias, and Fabian Sinz. Towards robust vision by multi-task learning on monkey visual cortex. Advances in Neural Information Processing Systems, 34:739-751, 2021.
Shreya Saxena and John P Cunningham. Towards the neural population doctrine. Current opinion in neurobiology, 55:103-111, 2019.
Oliver Schoppe, Nicol S Harper, Ben DB Willmore, Andrew J King, and Jan WH Schnupp. Mea- suring the performance of neural models. Frontiers in computational neuroscience, 10:10, 2016.
Connor Shorten and Taghi M Khoshgoftaar, A survey on image data augmentation for deep learning. Journal of big data, 6(1):1-48 2019.
Joshua H Siegle, Xiaoxuan Jia, Séverine Durand, Sam Gale, Corbett Bennett, Nile Graddis, Greg- gory Heller, Tamina K Ramirez, Hannah Choi, Jennifer Luviano, et al. Survey of spiking the mouse visual system reveals functional hierarchy. Nature, 592(7852):86 92088866-22--0 2021.
Ecro P Simoncelli, Liam Paninski, Jonathan Pillow. Odelia Schwartz, al. Characterization of neural responses with stochastic stimuli. The cognitive neurosciences, 3(327-338):1, 2004.
Fabian Sinz, Alexander Ecker, Paul Fahey, Edgar Walker, Erick Cobos, Emmanouil Froudarakis, Dimitri Yatsenko, Zachary Pitkow, Jacob Reimer, and Andreas Tolias. Stimulus domain transfer in recurrent models for large scale cortical population prediction on video. Advances in neural information processing systems, 31, 2018.
Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: simple way prevent neural networks from overfitting. The journal of machine learning research, 1919:1992911988 2014.
Ghislain Yee, Emily J Allen, Yihan Wu, Kendrick Kay, and Thomas Naselaris. Brain Brain-optimized deep neural network models of human visual areas learn non-hierarchical representations. Nature communications, 14(1):3329, 2023.
Carsen Stringer, Marius Pachitariu, Nicholas Steinmetz, Matteo Carandini, and Kenneth D Harris. High-dimensional geometry of population responses in visual cortex. Nature, 571(7765):361- 365, 2019a.
Carsen Stringer, Marius Pachitariu, Nicholas Steinmetz, Charu Bai Reddy, Matteo Carandini, and Kenneth D Harris. Spontaneous behaviors drive multidimensional, brainwide activity. Science, 364+(447)eeaav7893, 2019b.
Rudi Tong, Ronan da Silva, Dongyan Lin, Arna Ghosh, James Wilsenach, Erica Cianfarano, Pouya Bashivan, Blake Richards, and Stuart Trenholm. The feature landscape of visual cortex. bioRxiv. pp. 2023-11, 2023.
Polina Turishcheva, Paul G Fahey, Laura Hansel, Rachel Froebe, Kayla Ponder, Michaela Vystrčilová, Konstantin F Willeke, Mohammad Bashiri, Eric Wang, Zhiwei Ding, et al. The dy- namic sensorium competition for predicting large-scale mouse visual cortex activity from videos. ArXiv, 2023.
Polina Turishcheva, Paul G Fahey, Michaela Vystrčilová, Laura Hansel, Rachel Froebe, Kayla Pon- der, Yongrong Qiu, Konstantin F Willeke, Mohammad Bashiri, Ruslan Baikulov, al. Retro- spective for the dynamic sensorium competition for predicting large-scale mouse primary visual cortex activity from videos. arXiv preprint arXiv:2407.09 090000 2024.
Ivan Ustyuzhaninov, Max F Burg, Santiago Cadena, Jiakun Fu, Taliah Muhammad, Kayla Ponder, Emmanouil Froudarakis, Zhiwei Ding, Matthias Bethge, Andreas S Tolias, et al. Digital twin veals combinatorial code of non- non-linear computations in the mouse primary visual cortex. bioRxiv, pp. 2022-02, 2022.
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Published as a conference paper at ICLR 2025
Saurabh Vyas, Matthew D Golub, David Sussillo, and Krishna V Shenoy. Computation through neural population dynamics. Annual review of neuroscience, (3(1):249-273 2020.
Edgar Y Walker, Fabian H Sinz, Erick Cobos, Taliah Muhammad, Emmanouil Froudarakis, Paul G
Fahey, Alexander Beker, Jacob Reimer, Xaq Pitkow, and Andreas S Tolias. Inception loops discover what excites neurons most using deep predictive models. Nature neuroscience, 22(12);
2060-2065, 2019.
Eric Y Wang. Paul G Fahey, Kayla Ponder, Zhuokun Ding, Andersen Chang, Taliah Muhammad, Saumil Patel, Zhiwei Ding, Dat Tran, Jiakun Fu, et al. Towards foundation model of the mouse visual cortex. bioRxiv, 2023.
Quanxin Wang, Olaf Sporns, and Andreas Burkhalter. Network analysis corticocortical connec- tions reveals ventral and dorsal processing streams in mouse visual cortex. Journal of Neuro- science, 32(13):438-4439, 2012.
Konstantin F Willeke, Paul G Fahey, Mohammad Bashiri, Laura Pede, Max F Burg. Christoph Bless- ing, Santiago A Cadena, Zhiwei Ding, Konstantin- Messtntti-KKlmmes Lurz, Kayla Ponder, al. The sen- sorium competition on predicting large-scale mouse primary visual cortex activity. arXiv preprint arXiv:2206.08666 08666, 2022.
Konstantin Willeke, Kelli Restivo, Katrin Franke, Arne F Nix, Santiago A Cadena, Tori Shinn, Cate Nealley, Gabrielle Rodriguez, Saumil Patel, Alexander S Ecker, al. Deep learning-driven characterization of single cell tuning in primate visual area v4 unveils topological organization. bioRxiv, pp. 2023-05, 2023.
Aiwen Xu, Yuchen Hou, Cristopher Niell, and Michael Beyeler. Multimodal deep learning model unveils behavioral dynamics of v1 activity in freely moving mice. Advances in Neural Information Processing Systems, 36, 2024.
Daniel LK Yamins, Ha Hong, Charles F Cadicu, Ethan Solomon, Darren Seibert, and James DiCarlo. Performance-optimize. hierarchical models predict neural responses in higher visual cortex. Proceedings of the national academy of sciences, 1 111(23):8619-8622 2014.
Chengxu Zhuang, Siming Yan, Aran Nayebi, Martin Schrimpf, Michael C Frank, James DiCarlo, and Daniel LK Yamins. Unsupervised neural network models of the ventral visual stream. Pro- ceedings of the National Academy of Sciences, 21803):c014499 18, 2021.
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A
APPENDIX
CONTENTS
A.I
Neural network model description
15
A.I.I
Network architecture
15
1.2
Model training
16
A.1.3
Model evaluation
16
A.2
Details of simulating the experiment of Stringer al. (Stringer al. 2019a)
17
A.3
Details of simulating the experiment of Froudorakis et al. (Froudarakis et al. 20200
18
Details on variations the model discussed Section 5.1
18
A.1 NEURAL NETWORK MODEL DESCRIPTION
This section provides detailed description of the components that make up our benchmark model. We begin by outlining the model architecture, including the key components and how they interact. Specifically, describe the core, readout, and shifter components, along with the modifications made to the original baseline architecture to optimize performance. Next, we discuss the procedures used train the model, highlighting the strategies implemented to ensure robustness and accuracy, followed by a description the metrics we used to evaluate its performance.
A.1.1 NETWORK ARCHITECTURE
The model architecture is composed of three main components: core, shifter, and readout. The model takes videos and behavioral variables as inputs, which are fed to the "core" component and then the "readout" component. The readout component linearly combines the representations learned in the last layer of the core (feature weights) and determines the receptive field associated with each neuron (spatial position). The spatial position is further adjusted by the "shifter" which takes as input the pupil center's s coordinates. A graphical visualization of the model is provided in Fig. TA More details about each element are provided below.
Inputs. The input of the neural network model are: (I) Videos and behavior (Tensor V). This is a 5-d tensor with shape (Batch, Channels, Frames, Height, Width) where the first channel represents the gray screen video shown to the mouse and the last channels the behavior (running speed and pupil size). The behavior tensor was expanded concatenated to the videos tensor in the channel dimension. (2) Pupil center (Tensor P). It is 3-d tensor with shape (Batch, Frames, 2), with data pupil center coordinates frame-by-frame.
match the Height and Width of the videos and
Core. The core, taken from (Höfling et al. 2022). is composed of 4 layers of 3D Convolutional Network (16, 32, 64, 128 channels respectively), factorized in spatial and temporal components. The spatial kernels are 11 x 11 in the 1st layer and 5 x 5 in all of the subsequent layers. Similarly, the temporal kernels are 11 in the 1st layer and 5 x afterwards. After each layer, batch normalization and final non non-linearity (ELU + 1) were applied.
Readout. To compute single responses along stimulus presentation, we adopted the readout pre-
sented by (Lurz et al. (020).. For each neuron, learnable spatial position pv. E [-1, 1] is used to apply a bilinear interpolation feature vector which is then linearly combined with learnable feature weight w. E R and
specified spatial positions the core 's output. This method extracts
bias to predict the activity individual neurons frame-by-frane An (ELU 1) non-linearity was applied to final responses to keep them non-negative.
Shifter. This component a multilayer perceptron (MLP) with three layers having five units each that maps pupil center's coordinates into shift linearly applied to all neurons' positions Pn- At the end of each layer, a Tanh nonlincarity is applied to ensure the output remains within the range 1|1
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Modifications to original architecture. We slightly modified the original baseline architecture proposed in (Turishcheva et al.] 2023) with the following changes: we enforced temporal causality in the temporal convolutions such that representations generated up to certain time were influenced by previous frames only; we included an additional batch normalization layer hetween the two convolutions
found it helped gradient stability and model convergence; lastly, we disabled the Position Network in the readout component, an MLP that maps neurons' anatomical coordinates extracted during recordings to corresponding readout positions pr,
A.1.2 MODEL TRAINING
Loss function and training details. All videos were isotropically downsampled to resolution of 36 x 64 px (H x W) per frame. Functional and behavioral signals were resampled to 30 Hz by linear spline interpolation to match the frequency of visual stimuli. Visual stimuli, behavioral data and pupil center coordinates were normalized by subtracting the mean and dividing by the standard deviation of the corresponding session. Neural responses were standardized to maintain their non- negativity. Behavioral variables (running speed, pupil size) were added as separate channels to the input images (Franke al., (2022) to incorporate the modulatory effects of behavior neural activ- ity. Batches consisted of 150 consecutive frames randomly sampled from the corresponding trials (each consisting of 300 frames). We trained our networks with a batch size of 9, For parameter op- timization, AdamW was used with momentum of 0.1, to minimize the negative likelihood Poisson loss:
Poosson
The (r.llo((r..)
t=1
between the recorded responses and predicted responses r. where n, is the number of frames in one batch and nm is the number of neurons for session s. Gradient updates were performed after a full pass across all sessions, method that has been shown to increase performance (Li et al.] 2023). We employed a learning rate scheduler with linear warm-up over the first 10 epochs. Then, after each epoch, we calculated the correlation to average between predicted and true responses on the validation set and averaged it across all neurons. If the correlation failed to increase for eight consecutive epochs, we stopped the training and restored the model to its state after the best performing epoch. Then, either decreased the learning rate by a factor of 0.3 stopped training altogether, if the number of learning-rate decay steps was reached (n=4 decay steps).
Regularizarion strategies. When we used data augmentation, we applied random affine transfor- mations (rotation, scaling and translation) to input videos. The transformations were parametrized by transformation factor (here called ('afiie) determining the amplitude of the transformation ap- plied as follows:
degrees (-Olffine www.tttttiioooo Qafiine rotat instation m
translate = (-Oaftine t.ransl at i Oaffine trranssale onmax)
scale = (1 Qafline scalemax. + O scalemax)
0.1 and scale 0.2. The parameters translate, scale and degrees were used as inputs of the corresponding arguments of the PyTorch class RandomAffine (Paszke et al. 2019).
with rotation 20, translation
When we used dropout, we applied it at the end of each layer of the core, so that representations generated were more robust by preventing overfitting and ensuring that the model did not rely too heavily on any specific neurons.
A.1.3 MODEL EVALUATION
Model evaluation employed three distinct metrics: single trial correlation, correlation to average on repeated trials and normalized correlation coefficient (COnoem). Single trial correlation calculated as follows:
Eis(ris - (os33 - o)
trial com 0)
v Ees - F)2 En,(or, - -0)2
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where ō and F the average recorded and predicted responses across all trials. average metric is calculated as follows:
The correlation to
avg. com - - () -
(3)
FF -6) 6)²
where F; = E Mij is the average response across repeats.
The normalized correlation coefficient CC is another measure evaluated repeated responses introduced by (Schoppe et al., 2016) to account for the limited number of repetitions and the high variability of neural responses. is described as follows:
CC
CCuorm
Ca
where
Cov(f.
CC
V Var(FF Var(o)
and
NVar(ō) Var(o)
CCmax
(N - Var(()
where r is the predicted response and o is the observed response to N repeated stimuli. C' is normalization factor and it represents the theoretical maximum correlation coefficient between the observed responses o and the model's predicted responses r. In this way, it is possible to evaluate neural models in the presence of noise and variability by accounting for the reliability of individual neurons.
A.2 DETAILS OF SIMULATING THE EXPERIMENT OF STRINGER ET AL. (STRINGER ET AL.. 2019A)
Stimuli composition. Natural stimuli (natural) were composed by a total of 2,800 stimuli selected from the ImageNet database20, from ethologically relevant categories: 'birds', 'cat', 'flowers', 'hamster', 'holes', 'insects', 'mice', 'mushrooms', 'nests', 'pellets'. 'snakes' and 'wildcat'. Spa- tially whitened (white) versions of the 2,800 natural images, were obtained by adjusting their fre- quency content using Fourier transforms, scaling them back to the pixel domain, and normalizing the intensity match the original images' mean and standard deviation. Spatially localized stim- uli (small) were obtained by windowing images over the receptive field of the recorded population. Sparse noise stimuli (sparse) consisted of white or black squares on grey background. The eight- and nour-dimensional stimuli (8D, 4D) were constructed using a reduced-rant regression model by projecting the natural images onto a of basis functions (eight or four depending on the stimuli constructed). Drifting gratings (ori32) consisted of 32 one-dimensional directions, evenly spaced at 11.25%.
Experiment replication. The original experiment consisted in the following procedure: each set of images was presented twice to account for trial-to-trial variability. Images were presented for 0.5 s, alternating with grey-screen inter-stimulus interval lasting a random time between 0.3 and 1.1 Collected responses were averaged across image presentation time to obtain a single response asso- ciated to the corresponding image. Ongoing activity dimensions were removed by subtracting the responses' projection onto the top 32 spontaneous dimension. Then, cvPCA was used to estimate the eigenspectrum from the set of responses collected over two presentations. Power law fits were calculated between the 1 1h and 500th eigenvalues, except for gratings (where the range was 1 11-0).. Since our model was trained on movies, we created videos by repeating images over time (15 frames, corresponding to 0.5 s). Additionally, 50 frames of gray-screen were added before image presen- tation. Since the model takes as input also behavioral data, these were set to 0 (mean value post normalization) across all stimulus presentation. In this way, we ensured that responses dynamics were determined by visual stimulation only. We then collected the output the digital twin re- sponse to the image presentation and computed its mean across the 15 frames, obtaining a single response for each image. In line with the original experiment, we focused on responses of neurons coming from VI at recorded depths 100-350 um from cortical surface (corresponding to layer 2/3). Analyses were conducted only on neurons that were well captured by the model, defined as those selected units with correlation to the average metric greater than 15%. Starting from dataset
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containing responses from all neurons to all stimuli, calculated the covariance matrix and applied PCA to obtain the eigenspectrum for each session. From this, we obtained the power law fits using the same methods of the original experiment.
In Fig. 4A and Fig. 4 we report the average power-law exponent over three models trained with different seeds. Panels in Supp. Fig. 3B and in Supp. Fig. 4B show examples for individual models instead.
A.3 ET AL.. 2020
DETAILS OF SIMULATING THE EXPERIMENT OF FROUDORAKIS ET AL.. (FROUDARAKIS
To replicate the experiment from Froudarakis et. al. (Froudarakis et al. 2020 with our model, used Blender (Blender Development Team.) render four three-dimensional objects (Fig. IP) replicating those used in the original study. Each object was subjected to a set of continuous, random variations in position (X, Y, 2), scaling, tilt/rotation, and environmental lighting (intensity, position), ensuring smooth and coherent movements consistent across all objects. All renders were done over a uniform gray background. The resulting movies were split into 500 clips of 10 seconds each, rendered at 30 fps, and fed our benchmark model trained the MICrONS dataset (Consortium ict al., 2021I. We extracted the predicted neural responses from layer 2/3 for different visual arcas, focusing on neurons well deseribed by the model in the training phase (correlation to average greater than 0.15). We used these extracted responses to construct N-dimensional vectors of neural activity aggregated in 500ms intervals. The vectors models to classify object's identity for different values of neurons count N. In all cases, we used a 10 fold tross-validation approach and reported the average performances for the 10% of the data that were held out from the training of the decoder. We repeated the procedure 50 times with bootstrap sampling of the neurons each count step to evaluate decoding accuracy across visual areas. To measure the discriminability we used the mutual information measured in bits by computing
then used train one versus-uusll all logistic regression
(I(c, ΣΣ Pij logg
Pij
P.Pp
where P.j is the probability of observing the true class i and predicted class j and p; and pj are the marginal probabilities.
The experiment of Froudarakis et. al. (Froudarakis et al., 2020 characterized the object discrim- inability across 10 visual areas; in our analyses, we focus on the four (V1, LM, RL, AL) included in the MICrONS dataset (Consortium et al. 2021). which we used to train our model.
A.4 DETAILS ON VARIATIONS OF THE MODEL DISCUSSED IN SECTION 5.1
Model trained on neurons with high reliability to repeated videos. To select the set of neurons whose response were reliable across repeated stimuli, we used the Fraction of Explained Variance (FEV) metric to quantify neurons' reliability. The metric calculated follows:
FEV(o) Varional (o) Var
Var (o)
where Varioaa (o) is the variance of responses evaluated across trials, repetitions and frames and Vargoise is the variance evaluated on repeated trials, averaged across trials and frames. For each session, we selected the 50% most reliable neurons.
Model trained on SENSORIUM. The SENSORIUM dataset was collected using 5 different mice, all focusing on neurons located in V1. In total, over 38,000 neurons' responses were recorded in response to around 1800 natural scene videos (each lasting 10 s). Behavioral variables (running speed, pupil size and pupil center's coordinates) were recorded along with videos presentation. We repeated the modeling procedure described in Section 3 of the main text on this dataset.
Transformer-basee digital twin. The only difference of the VIVIT model compared to the bench- mark model described in the main text is in the core part. The ViVIT core contains three main components: (1) tokenizer that concatenates the video and behaviour variables
the channel dimensions and extracts overlapping tubelet patches along the temporal and spatial dimensions, fol- lowed by factorized positional embedding which learns the spatiotemporal location of each patch;
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(2) spatial Transformer which receives the tubelet embeddings and learns the spatial relationship over the patches within each frame; (3) temporal Transformer receives the spatial embedding and learns joint spatiotemporal representation of the video and behavioural information. The readout and shifter modules are equal to the model we used.
Model with loss function the correlations between neuron. We developed new loss func- tion that focused on capturing correlations between neurons rather than accurately predicting single neuron activity. This defined
MSE
R - Ozg)²
(4)
where Rig is the correlation between predicted responses for neurons i and j. while Oig is the correlation between their observed responses.
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A
Fraction of neurons
Rizudankki
10
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1d
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Predicted response value
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Vertance
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dimension
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Supplementary Figure 1: Controls and additional measurements for simulations shown in Fig. (A) Distri- bution of correlation to the average measured in the benchmark model for neurons in the four visual areas under investigation. (B) Distribution of responses across neurons measured in the benchmark model when simulating the experiments of Stringer et al. (Stringer al. (019a) and Froudarakis et al. (Froudarakis al. 2020 Dis- tributions are shown in linear (left) and logarithmic (right) scale. Despite the visual stimuli being significantly different from those used to train the model, does not exhibit pathological behavior and generates responses similar to those produced in response to stimuli from the MICrÓNS dataset. (C) Eigenspectrum of the covari- ance matrix obtained from the digital twin (lighter and darker green lines represent individual sessions and their average, respectively), simulating the experiment by Stringer al. for all types of stimuli used (columns). In each panel, the dashed line indicates the decay found the experiment, while the solid gray line represents best fit of a power law to the model results, with the exponent reported as text.
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natural
white
Stal
spanse
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o132
A
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16-
dimensiinn
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#
+
f
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di 1.81
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dimeesion
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cimension
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12
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Supplementary Figure 2: Geometry of V1 population responses across stimuli for the network models shown in Fig.3] Eigenspectrum of the covariance matrix obtained when simulating the experiments of Stringer al. (Stringer al.) 2019a) with: (A) model trained reliable neurons, (B) model trained the SENSO- RIUM dataset, (C) VIVIT, and (D) model trained correlations. Columns correspond to different types of stimuli used (titles). each row, colored lines represent quantities measured in the digital twins (lighter and darker lines indicate individual sessions and their average, respectively). The dashed lines show the decay ob- served in the experiment, while the solid gray lines represent the best fit of a power law to the model results, with the exponent reported text.
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Conglation to avarage
0.52
6.0
03
03 03
54 33 da 0.7
Verssormation factor
natural
whiGe
small
0n13
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Df 0.96
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serrance
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02
dimension
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dimensinn
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Veronce
Vergnce
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Varincc
Varincce
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ttr brg
10
de
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a= 1.04
1.49
== 1.93
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PC dimenson
PC dimenson
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Supplementary Figure 3: Characterization of responses in simulations underlying the results shown in
Fig. 4A (A) Mean correlation average measured across single neuron responses in models trained with dif- ferent transformation amplitudes for each analyzed area. (B) Eigenspectrum of covariance matrix obtained from simulating the experiments of Stringer et al. (Stringer et al. 2019a). Different rows correspond individ- ual models trained with varying transformation amplitudes (text on the left, only models trained with amplitude O are shown).
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A
Correlation to average
0.30
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RL
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Dropout
Ratto
spanse
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171
un
301
dimeicion
dimonsion
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PC dimrision
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PC dimersion
PC 6merston
2- 0.92
= 0.92
d 1.77
d= 174
D 1.62
2 1.98
8 222
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arc
17
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dimession
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RC dimension
PC dimension
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da 134
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dimmnco
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Supplementary Figure Characterization of responses in simulations underlying the results shown in Fig. 4B,D (A) Mean correlation average measured across single neuron responses models trained with different dropout rate for each analyzed area. (B) Eigenspectrum of the covariance matrix obtained from simu- lating the experiments of Stringer et al. (Stinger et al. (019a). Different rows correspond to individual models trained with varying dropout rate (text on the left, only models trained with dropout rate > 0 are shown).
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Non-differentiable models
Differentiable models
0.3
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of neurons
Supplementary Figure 5: Discriminability in simulations shown in Fig. Discriminability as a function of the number of neurons and cortical areas, computed from the network models shown in Fig
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A
B
Layer
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a= 0.57
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0.55
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(relattion deviation)
Variance
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VEAAL
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RL-VI 6.5.
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LM VI RL
PC dimension
Areas
Supplementary Figure 6: Population geometry in task-driven model. We finetuned a ResNet50 model on the MICrÓNS datasct training linear readout after each the ResNet50 weights frozen. The same testing procedure used for the data-driven models was applied to the predicted responses. Each row in the figure presents the eigenspectrum of the covariance matrix (left, as in Fig. (2A) and the relative discriminability across areas (right, as in Fig. 2D) based on the linear readouts from each layer. The results consistent with those from benchmark model: the learned representation not differentiable, and model failed capture hierarchy discriminability observed experimentally.
first three layer block while keeping
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A
B
0.3
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a
0.81
Discriminabiitty (relative deviation)
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Supplementary Figure 7: Population geometry in 3D convolution models trained independently for each area. (A) Eigenspectrum of the covariance matrix, as shown in Fig. A but for 3D CNN model trained exclusively on recordings from The representation learned by the model was not differentiable. (B) Relative discriminability across areas, as shown in Fig. 2D. but computed in four 3D CNN models, each trained using data exclusively from one area. Object discriminability across areas varied significantly more than in our benchmark model. While the model correctly positioned VI above RL in the discriminability hicrarchy, it incorrectly placed VI the top and AL the bottom, contradicting experimental observations.
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A
Fraction of neurons
10-1
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Supplementary Figure 8: Multi-objective optimization of single-neuron responses and population corre- lations. Response properties in network models trained using multi-objective optimization framework bal- ancing single-neuron responses population correlations.
trained three models with a combined Poisson single-ncuron loss and population correlation loss. Pancls show the distribution of correlations and eigen- spectrum of the covariance matrix obtained in models with: (A) larger weight on the Poisson loss; (B) equal contributions from Poisson and population correlation losses; (C) larger weight on the population correlation loss. The results showed evidence of synergistic effect: gradually increasing the relative weight of the Poisson loss improved models' ability to capture single-neuron responses but reduced differentiability of population representations.
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1.6
Power-law exponent a
1.4
1.2
1.0
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Correlation to average
Supplementary Figure 9: Trade-off between single-neuron responses and population geometry. This figure shows the power-law exponent of the eigenspectrum of the covariance matrix as function of the single-neuron correlation to average across all models trained with the Poisson loss. Results reveal negative correlation between the ability capture population geometry and the accuracy single-neuron responses.
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Average shifted responses (MICrONS)
Average shifted responses
(Stringer 2019)
2.5
Hean Zacare
1.0
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Stimulus orientation (reiative preferred)
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Single neuron tuning curves (Stringer 2019)
Sorted neurrs
400
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800
-006
270.0
0.0
riretion
Stimulus oriantation
Supplementary Figure 10: Analysis of model responses to oriented stimuli. (A) Example Monet stimuli from the MICrONS dataset (Consortium et al,
(B) Example used for model training were used test the trained models. (C) Average tuning curve of model neurons in response to Monet stimuli. (D) Average tuning curve of model neurons in response to gratings.
202I]. which were used for training the model.
gratings from the experiment Stringer et al. (Stringer al., 2019ah. These stimuli
(E) Tuning curves all model neurons sorted by their preferred orientation for Monet stimuli. (F) Tuning curves of all model neurons sorted by their preferred orientation for grating stimuli. Responses Monet stimuli, which were used during training, exhibit tuning curves similar those commonly observed in real neurons. In contrast, responses gratings show pathological sighffreyeency oscillations, likely due to the substantial difference between grating stimuli and the Monet stimuli used during training.
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Reviews:
[{"review_id": "DrTGnsCKKa", "reviewer": "Reviewer_YdfR", "summary": "The authors demonstrate that a reasonably successful model of visual cortical neurons does not capture population metrics.  They show that measures of the population geometry and visual hierarchy fail on this model.  The eigenspectrum of the covariance matrix does not show scaling consistent with differentiability and the relative discriminability across areas does not match what is observed empirically.  The authors investigated various aspects of the model and of training to see what effect they had on these measurements.  These include selecting for reliability, choices of dataset, dataset augmentation, model architecture, and using a covariance-based loss function.  All of those failed to show different results for the population metrics.  The approach that did produce differentiable representations was adding dropout during training, which showed an exponent consistent with differentiability for sufficiently high dropout rates.  They claim that the models that show differentiability also show the appropriate hierarchical relationships for discriminability.", "questions": "Does this result hold for task-trained models as well?\n\nCan you test other forms of regularization?  Do they have the same effect, or is dropout special?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "This paper is reasonably clear and well-written.  The question is a very important one as population level characteristics of neural activity are undoubtedly important for neural computation and models need to be able to capture it.  It is extremely important to point out that our current state-of-the-art models are not meeting this criteria and likewise important to address what aspects of training will correct this.  The authors tested a broad set of training procedures.", "weaknesses": "The authors did an excellent job of testing an array of procedures, but the results are still unfortunately limited by the number of models and data sets available for testing.  This is not terribly actionable, sadly, in a practical sense.\n\nThe authors wish to make a general claim about regularization, but only test dropout.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "nnyYmRhpB7", "reviewer": "Reviewer_sa6o", "summary": "The study examines how well digital twins, computational models trained on neural data, capture both single-neuron and population-level responses in the mouse visual cortex. While these models predict single-neuron activity accurately, they fail to replicate the complex geometry and hierarchical organization of population responses seen in biological data. Factors like dataset composition and network architecture didn't resolve these issues. Introducing regularization techniques like dropout improved the models' ability to create more accurate population representations but reduced single-neuron prediction accuracy. The study highlights the need to refine digital twins to better understand neural computations at a population level.", "questions": "Have the authors considered multi-objective optimization to balance single-neuron accuracy and population-level representation? Could adding biological features like spiking noise, synaptic dynamics, or neural variability improve the models' population properties? Have they explored area-specific modules or varied receptive field sizes to better capture hierarchical processing? Can they comment on the models' generalizability to different visual stimuli, given the issues with non-differentiable representations for gratings? Would using spiking neural network models help capture population response geometry and address trial-to-trial variability?", "limitations": "", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "The paper tackles an important gap in computational neuroscience by focusing on the geometry of population-level neural responses in digital twins, not just single-neuron predictions. The study meticulously replicates key experimental findings and explores how factors like dataset variation, architectural changes, and training objectives impact model performance. Regularization techniques, especially dropout, are shown to improve the models' alignment with biological data. The paper is well-explained and accessible, providing valuable insights for both neuroscience and machine learning. By identifying the limits of current models, the research sets the stage for developing more accurate and biologically realistic models, emphasizing the need for principles like robustness and generalization in neural computation replication.", "weaknesses": "Despite improvements with regularization, the models still show quantitative differences from experimental data, especially in discriminability measures and visual area hierarchy, warranting a deeper analysis to uncover the causes. There's a trade-off where enhancing population-level representation reduces single-neuron prediction accuracy, raising practical concerns. The paper's focus on a shared-core architecture limits insights; exploring more varied architectures could better reflect the visual cortex's hierarchical nature. Additionally, the study relies on computational simulations, and direct physiological validation of these models would further substantiate the findings.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "jUEKsr8vld", "reviewer": "Reviewer_A1NT", "summary": "This paper argues that some digital twins (i.e., ANN models that predict single neuron activities well) fail to capture population level properties (specifically having to do with the representational geometry) of neural responses that have been previously observed in the mouse visual cortex. The primary issue is the fact that the population representations in the digital twins have a geometry that is non-differentiable, which suggests that small changes in inputs can lead to very large changes in the network's representation. The paper then shows that dropout regularization can go some way towards mitigating these kinds of discrepancies between the ANN's responses and those of the actual mouse brain.", "questions": "I was a bit confused at a high level as to how it is even possible for a digital twin to do a good job at predicting single neuron responses and yet still not capture population structure. Doesn't the population structure depend on the single neuron responses? How can you capture each individual neuron's activity with a high degree of accuracy and not get the population structure right?\n\nQuestions about the model:\n- I was very surprised to read in the paper that the model does not have increasing receptive field sizes. Isn't that just a standard feature of a CNN?\n- What is the intuition behind the formula for CC_max (where does this formula come from)?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "Understanding how highly predictive ANN models of the brain can fail to capture certain properties of neural population responses that are functionally important is an important issue to address. This paper exposes some shortcomings of existing models, and gives a fairly straightforward way of mitigating some of those shortcomings, using dropout in the network. This is useful because it may lead to models that do a better job of accurately simulating brain responses.", "weaknesses": "The paper could be clearer in a lot of places, especially for readers who are not familiar with the model being used, as well as the background literature which the paper heavily draws upon. \n- Although the model is described in the Appendix, it wasn't very clear (at least, not to me). There are a lot of different parts, and it was hard to see how they fit together. It might help to have a diagram illustrating the architecture and how everything fits together.\n\nAlthough a lot of the background theory relating to the differentiability of neural representational geometry is presented in Stringer 2019, it would help readers if this paper were a bit more self contained, e.g. some intuitive explanation of why the cutoff for differentiability is 1 + 2/d would have been helpful to the uninitiated reader. \n\nTraining the model only on correlations between neurons resulted in a loss of single neuron predictivity, presumably because single neuron predictivity is no longer part of the optimization objective. I would have thought a natural thing to try would be to train the model on a combined objective that optimizes for both objectives at once, and see if that changes the results in any way.\n\nThe claim that the digital twins fail to capture the hierarchical relationships between areas seemed like an overstatement, since what is really shown here is that one aspect of hierarchical relationships was not captured, not that it doesn't capture any relevant aspects of the visual hierarchy.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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:

Decision: Accept (Poster)