NOBLE - Neural Operator with Biologically-informed Latent Embeddings to Capture Experimental Variability in Biological Neuron Models

We thank the reviewer for their thoughtful summary and for recognising the technical quality and relevance of our work within the context of neuroscience research. We are pleased that the reviewer found the use of a latent embedded space for generating voltage traces to be a compelling aspect of our approach, and appreciate the recognition of NOBLE’s ability to produce biophysically plausible outputs efficiently. We have addressed all raised concerns below and hope our response satisfactorily resolves the reviewer’s points. We are happy to provide further elaboration or modifications during the discussion period.

W1: PDE benchmarks are biologically grounded and suitable for learning

We understand the concern about limited comparisons to other baselines. Our primary goal is to evaluate whether the neural operator framework can capture the biological variability seen in real-world experiments. To this end, we used data generated from PDE-based models which were selected via evolutionary multi-objective optimization as the best available approximations of experimental recordings, and exhibit the desired variability. Now, other existing ML approaches [28-33;37-40] in neuroscience are not equipped to capture variability, and can therefore not be used for a comparison. While other models could be used within NOBLE and trained on the same data, we believe this would not offer additional insight. Our focus is on learning representations that reliably capture key e-features across neuron dynamics and parameter regimes, and NOBLE already demonstrates strong performance on this benchmark. Since the PDE solvers generating the dataset themselves are approximations of biological reality with non-negligible errors, further reducing prediction errors would lead to further overfitting to the PDE solver simulations but probably not to substantially better-captured dynamics and e-features when compared to true real-life recordings. For this reason, we tuned hyperparameters up to the level of error seen between the PDE models and experimental data, as marginal error improvements on the PDE data are unlikely to translate meaningfully when compared to real experimental data. We included a note on this in Section 4.1 of the revised paper.

W2: Input Currents Selection

We agree with the reviewer that richer time-varying stimuli would allow for broader validation. We focused on DC step currents because they are widely used in electrophysiological experiments and represent a common protocol for characterising neuron behaviour. More importantly, they offer a readily available source of experimental data to validate NOBLE’s predictions. That said, extending NOBLE to handle arbitrary time-varying inputs is a natural next step that we intend to explore in future work, and added a comment about this at the end of the paper.

Q1: Improving biological understanding

The biologically interpretable embeddings of NOBLE provide a testing ground towards understanding the relationship between neuron properties (electrophysiology, morphology, transcriptomics) and computation. In this first version of NOBLE, we only included 2 prominent e-features in the embedding that is distinct between inhibitory neo-cortical cell-types, but this embedding could incorporate any number of arbitrary features, including morphological features and ion channel gene expression. Through perturbations (most of which are experimentally infeasible and also not possible with biophysical PDE modeling), one could then directly test how these features relate to a neuron’s electrical properties, informing us of how location dependent variations of cellular composition in the human brain might relate to their computations.

Another promising approach involves sampling the embedding space, running NOBLE on each point, and computing e-features to generate heatmaps. This allows to visualise how features vary across the latent space (thereby informing about the relation between the features visualized and those defining the latent space), helps highlight the geometry of the learned representations while also offering new insights into the underlying structure, variability, and diversity of neuronal behaviour. We added an example of such a heatmap and highlighted this as an exciting direction to explore further in future work

While NOBLE does not extend to large scale circuit simulations yet, it addresses current bottlenecks in such modeling:

1. The scalable generation of diverse bio-realistic single neuron models
2. The computational scalability of running these simulations on large populations of neurons

Q2: The interpretability lies in the feature-based latent space

We thank the reviewer for pointing this out. Here, the term “interpretable” refers to the e-features used to embed the models. These features are biophysically meaningful and allow us to condition NOBLE on a feature space that is directly linked to neuronal function (e.g. sthr and Ithr). We clarified this terminology in the revised version.

The voltage traces themselves are not inherently interpretable in the sense of revealing explicit biophysical parameters. However, one could in principle attempt to recover the underlying model parameters (e.g. conductances or membrane properties) using inverse modelling or system identification approaches (e.g. PDE-SINDy)

Q3: Clarification on the HoF models

The HoF models are not constructed through combination or linear mixing. They are obtained via a multi-objective evolutionary optimization over a large space of candidate biophysical models (~600,000 PDEs), which aims to identify parameter configurations that best match a set of e-features extracted from experimental recordings. The final HoF set consists of multiple distinct models, each offering a biophysically plausible fit to the experimental data, but differing in specific parameters (e.g. ion channel distributions, membrane properties). These models reflect the trial-to-trial variation of biological systems. We clarified this process in the paper to avoid confusion with methods involving basis function combinations or linear mixtures of models.

Q4: “Intrinsic variability” refers to repeated responses from the same neuron, not inter-neuron heterogeneity

We thank the reviewer for bringing this source of confusion to our attention. Here, the term “intrinsic variability” specifically refers to the trial-to-trial variability observed in repeated recordings of a single neuron in response to the same current injection. This type of variability can arise from channel noise, spontaneous fluctuations in cellular state, or small experimental perturbations. We revised the paper to make this distinction clearer and avoid ambiguity with neuron-to-neuron heterogeneity which refers to differences across cells.

Q5/C4: Sinusoidal embedding

Equation (2) is not the embedding at a single time point, but rather a “stack of trigonometric time-series”, since t is a vector of discretised time points of length L. The equation defines a matrix of shape K x L, where K is the number of sinusoids (i.e., frequency components), and each row corresponds to one sine or cosine function evaluated across time. This follows the positional encoding convention introduced in “NeRF: Representing Scenes as Neural Radiance Fields for View Synthesis” by Mildenhall et al. (2020), where the entire set of time samples is embedded using fixed-frequency trigonometric functions. The first Fourier transform of the FNO will act on each of these trigonometric time-series separately, and none of the resulting Fourier transforms are trivial by construction. We clarified the embedding formula in Section 3.3.

Q6: Clarification on “Cellular Models representing multiple data modalities” Our biophysical PDE neuron models are constructed considering 3 different types/modalities of experimental measurements including the neuron morphology, electrical properties (electrophysiology), and ion channel gene expression (transcriptomics).

C1-2-3: Clarification on FNO notation and Figure 1

The operator $R_{\phi}$ is a learnable Fourier multiplier, which can be interpreted as the Fourier transform of a periodic function, consistent with the terminology used in the original FNO paper by Li et al. (2021). Similarly, the expression $\sigma (W + K + b)$ follows the same convention and is explained in detail in our appendix and in the original FNO paper.

The diagram in Fig 1A presents a high-level overview of NOBLE, in which the Neural Operator block internally contains the full architecture of a neural operator. In our study, we use an FNO as described in Li et al. (2021). The parameters referred to in Eq. 1 are all encapsulated within the Neural Operator block in our diagram. Since our focus in Fig 1A was to convey the overall structure of the framework (including the embedding space and conditioning mechanism), we chose to abstract away the internal layers of the neural operator. However, these follow the standard FNO implementation, and a more detailed representation of the FNO can be found in Fig 8.

C5: Superimposing PDE and NOBLE

We updated the figure by superimposing the PDE and NOBLE traces, which indeed makes the comparison more direct. Additionally, we have enlarged the figure to improve clarity.

Thank you for the clarifications.

W1: "Now, other existing ML approaches [28-33;37-40] in neuroscience are not equipped to capture variability" I find difficult to believe that intrinsic variability is not captured by previous ML approaches. For example the early work of Paninski et al. 2004 (Neural Computation 16, 2533–2561) estimated the distribution of the membrane potential (intrinsic variability as described by the authors) of an integrate and fire neuron model. Their approach being based on generalized learn models can extend to more complicated neuron models.

We thank the reviewer for their thoughtful summary and generous assessment. We are particularly encouraged by their recognition of NOBLE’s ability to generate biophysically plausible dynamics through latent space interpolation, and of its strong validation against both simulation and experimental benchmarks. We also appreciate the reviewer’s acknowledgment of our efforts to go beyond trace accuracy by evaluating a broad set of e-features, which we believe is essential for meaningful neuroscientific application. We have addressed all raised concerns below and hope our response satisfactorily resolves the reviewer’s points. We are happy to provide further elaboration or modifications during the discussion period.

W1: NeuPRINT and NOBLE are fundamentally different, but complementary

NeuPRINT is a powerful framework, but deals with a very different dataset. It generates robust representations of optical physiology recordings from labeled neurons in living rodents. While our study also leverages neural data from a labeled cell type (in our case using patch-seq labeling), the dynamics are vastly different. In vivo optical recordings are typically slow (resolution ~ tens of ms) and do not capture single action potentials. The in vitro recordings we train on have sub-ms resolution and capture the full complexity of a neuron’s electrophysiological behavior (e.g. spike inflection point, rise time). Thus, NeuPRINT and NOBLE capture opposite parts of the spectrum: the former focuses on slower fluorescence signals recorded in vivo, the latter on fast intracellular membrane dynamics recorded in vitro. Their fundamentally different dynamics make basic comparisons difficult. While Ca-based fluorescence physiology is viable in rodents, these experiments cannot be pursued in humans as they involve genetic manipulation. An alternative would be to develop NOBLE models for rodent neurons to compare the 2 frameworks but, even then, a direct comparison is difficult between the 2 datasets (intracellular patch-clamp voltage recordings with a sampling rate of 20kHz vs. transmembrane fluorescence of Ca-indicators with a sampling rate of >30 Hz). The most direct comparison would be for NeuPRINT to train on human in vitro intracellular physiology data or for NOBLE to train on rodent in vivo optical physiology data. In an ideal scenario, NeuPRINT and NOBLE will inform each other. We discuss NeuPRINT in our updated paper and suggest ways for them to interact.

W2: We demonstrate that the same low-dimensional, biologically interpretable embedding space generalises effectively across neuron types, including VIP neurons

To demonstrate that NOBLE generalises beyond a single cell type, we trained a new model on a vasoactive intestinal peptide (VIP) neuron using the same embedding construction and e-features. After 450 epochs (using the same architecture and optimizer), NOBLE achieves errors:

• Relative L2: 2.5%
• Spikecount: 9.2%
• AP1 width: 22%
• Mean AP amplitude: 10%

These results are consistent with those obtained on PVALB neurons, supporting the robustness and generalisability of our embedding strategy. We included this new experiment in the revised paper.

W3/Q4: NOBLE amortises this cost of costly HoF generation and enables fast, scalable synthesis of biorealistic responses

Training NOBLE on 84,000 samples was completed in ~4 days using a 64GB NVIDIA Tesla P100 GPU (300 epochs). The significant advantage arises post-training: NOBLE enables the instantaneous generation of an infinite number of new HoF model responses within the embedding space's convex hull (Fig. 3). This is a dramatic efficiency gain compared to the 600k CPU hours required to produce the 60 HoF models from which our training data is derived using the traditional evolutionary optimization.

We added an ablation study by varying the number of HoF models used to generate the training data. The analysis below highlights the minimal information NOBLE needs to learn robust underlying representations, allowing for the generation of novel biorealistic HoF responses.

Although the relative L2 error on voltage traces remains largely stable, a clear degradation is observed in spike-related features, particularly spike count and AP1 width, as training diversity decreases. This ablation has been added to the paper since it provides an important perspective to the discussion, illustrating that model diversity, not just quantity, is crucial for learning robust representations of neural dynamics.

Q1/2: Feature selection for embedding

There is no universally optimal strategy. This requires a supervised decision about which features best capture the behaviour or phenotype of interest. Our goal was to test whether NOBLE could succeed using a minimal yet meaningful embedding space. We thus selected 2 widely used e-features derived from each HoF model’s frequency–current (F–I) curve: the threshold current and the slope at threshold. These features are standard in the neuroscience literature, providing a natural 2D space to represent both firing and non-firing dynamics, and offering strong biological interpretability. We plan to explore more complex or multimodal embeddings in future work. More generally, to find features that are both distinct and not redundant, we can quantify the variation of a large set of features for data across/within cell-types as well as feature correlations to determine which subset of features might be suitable for constructing an interpretable embedding space. We note also that within the NOBLE framework the suitability of features (like adaptation and bursting) depends on the level of sophistication of the PDE-based models. NOBLE can account for various complex properties of neuronal membrane dynamics as long as these are represented in the original HoF models NOBLE is trained on.

To quantify the contribution of our feature embeddings, we conducted a comprehensive ablation study systematically varying the number of embedded features. We evaluated five distinct embedding configurations:

• No embedded features
• Only $I_{thr}$
• Only $s_{thr}$
• $s_{thr}$ and $I_{thr}$
• $s_{thr}$, $I_{thr}$ and AHP depth

AHP depth was chosen as it showed large variations across samples and low correlation to sthr and Ith. The results of this ablation study, detailing the quantitative effects on NOBLE's predictive performance, are presented below.

These results show that without the use of embeddings, NOBLE is unable to predict any firing response successfully. Embedding either slope or intercept enables comparable L2 accuracy, with intercept yielding better AP1 width prediction. The best overall performance is achieved by embedding both, with slight additional gains from including AHP depth. We agree that such an ablation adds value to the use of embeddings and has been added to the paper.

Q3: Extrapolating to models outside the training distribution

We agree with the reviewer that NOBLE “demonstrates excellent performance with interpolation” and realize that the paper may not sufficiently highlight the aspect of “extrapolating to models outside the training distribution”. The rightmost plots in Fig. 6 directly addresses this concern, demonstrating NOBLE's robust generalization to unseen, out-of-distribution models. Specifically, these plots show the predicted voltage responses, for different input currents, with 200 randomly sampled unseen HoF models from the convex hull of the embedding space (see Fig 3).

Q5: Subsampling strategy and fast neuronal dynamics Our subsampling rate was carefully chosen so it can resolve the highest frequency neuronal dynamics (such as action potential halfwidths (~0.05ms)) in response to input currents in the considered ranges, for the PVALB neuron selected. As detailed in Section 3.1 and Figs 9-10, we evaluated multiple subsampling strategies and quantified their effects on e-features accuracy. This analysis allowed us to identify a conservative upper bound beyond which subsampling would degrade signal and e-features fidelity. The chosen subsampling factor balances computational efficiency with the need to capture very fast neuronal dynamics

We thank the reviewer for their insightful comments and their recognition of NOBLE’s core contribution: learning a unified neural operator capturing variability across biophysical models. We also appreciate their perspective on foundation models. We have addressed all raised concerns below and hope our response satisfactorily resolves the reviewer’s points. We are happy to provide further elaboration or modifications during the discussion period.

NOBLE’s purpose (W1/Q1)

The goal of NOBLE is not to outperform individual HoF models in matching experimental data. Each HoF model is already optimized to closely fit experimental recordings. Any instance where NOBLE achieves a closer fit would be accidental. Instead, NOBLE is designed to generalize across a biologically realistic population of neuron models, interpolate within a latent feature space to generate new biorealistic models, and dramatically reduce computational costs

The rightmost plots in Fig 6 illustrate NOBLE’s predictive performance on held-out HoF models from the test set, by showing predicted voltage responses at different input currents for 200 randomly sampled models. These models were not seen during training, yet NOBLE's predictions closely match the solver outputs. This strong alignment across a wide set of out-of-distribution models highlights NOBLE’s ability to generate biorealistic responses even in previously unseen regions of the latent space. Also note that during testing, NOBLE is evaluated on random input current values that were not experienced during training

Interpretability (W1/W2)

We appreciate the reviewer’s insight and are glad they anticipate directions we are pursuing. One promising approach involves sampling the embedding space, running NOBLE on each point, and computing e-features to generate heatmaps. This allows to visualise how features vary across the latent space (thereby informing about the relation between the features visualized and those defining the latent space), helps highlight the geometry of the learned representations while also offering new insights into the underlying structure, variability, and diversity of neuronal behaviour. We added an example of such a heatmap and highlighted this as an exciting direction to explore further in future work

Regarding the use of patch-seq data, the 4 inhibitory cell-types presented in Fig 2B exhibit distinct gene expression profiles and ion channel expression. By training a large-scale version of NOBLE that captures neurons from multiple cell-types with patch-seq data included in the embedding space, one then has an efficient tool to probe the effect of ion channel expression on neuronal responses across different cell-types. For example, one can embed and perturb the ionic conductances or gene expression concentrations and analyze how these variables affect a neuron’s electrical properties (Fig 5 illustrates how parameters in the embedding space can be perturbed via interpolation using NOBLE, but not with PDE models)

Subsampling (Q2/Q3)

The entire numerical experiments are conducted on downsampled voltage traces. Our downsampling analysis confirmed that the chosen resolution preserves all relevant e-features (see Section 3.1). In particular, NOBLE is trained on the downsampled traces. The voltage and e-feature errors and the computational times are also computed on the downsampled traces. Now, NOBLE can be queried and remains consistent at higher resolutions (it essentially spectrally interpolates). We tested inference at the original resolution and observed no noticeable difference in computational time compared to predictions at the 3x downsampled resolution.

Training loss (Q2)

The use of L4 for training is motivated by an overfitting study conducted before training on the full dataset where we trained the FNO on a small number of samples using different loss functions to assess their ability to fit voltage traces and preserve key e-features. L4 consistently preserved spike-related features more effectively than L2, particularly in capturing spike amplitudes and widths. Based on these results, we selected L4 as the training loss. We added the details of this overfitting study and the formula for the relative Lp error to the appendix. We only use the relative L2 error for reporting as it is a more standard metric.

“PDE-based” Terminology (C3)

While the neuron models are based on PDE dynamics (specifically the cable equation), their simulation in NEURON relies on a spatial discretization that transforms the system into coupled ODEs. We use the term “PDE-based biophysical models” to refer to models that follow the mathematical formulation of the cable equation. We added sentences to clarify this terminology in the revised paper

Local sampling around HoF model (W1)

This analysis is addressed in Figs 4B and 5B. Fig 4B shows time series responses for a HoF model excluded from training. Fig 5B presents the F-I curves for 50 NOBLE samples drawn from a small ball around the excluded HoF model, along with the voltage trace of one representative sample. A new Appendix figure shows the time series for all 50 sampled models, complementing Fig 6. This confirms that the predicted dynamics remain close to the behaviour of the original HoF model, further highlighting NOBLE’s local continuity in the embedding space

Figure-based modifications (W3/Q4/C1/C2)

• Fig 2: We increased line thickness to enhance readability
• Fig 3: We agree that superimposing the traces improves interpretability and revised the paper accordingly
• Fig 3,4,5,6: We clarified the captions. If interested, the revised versions can be found in the "Figures and Captions" section of our response to Reviewer 5ihX
• Fig 5: The same models were used in both panels, but with different stimulus durations: 1s in 5A (to match experimental data) and 0.4s in 5B (used for NOBLE). The firing rate is not uniform over time, which explains the differences observed in the F–I curves. We do not compare time series between NOBLE and experimental data directly, due to the mismatch in stimulus duration. Instead, we compare summary features in Fig 7, as these are mostly not affected by stimulus length. For spikecount, we approximate a 1s value by linearly scaling the 0.4s response, although we recognise this is only an approximation.
• Fig 7: We agree that reporting mean and std could misleadingly suggest the possibility of negative spike counts. We now report the median and interquartile range which more appropriately reflect the distribution of e-features. Fig 7 is used for ensemble performance visualisation and shows distributions of 4 key e-features compared to experimental data. This complements the feature-level analysis by illustrating the biophysical realism of the generated time series. Following the reviewer’s suggestion, we added a full comparison of NOBLE’s predictions to experimental recordings across the remaining e-features in the appendix

1. Limitations

We extended the discussion relating to the limitations of NOBLE, and created a new section dedicated to that discussion

Training Budget (L1) This is a great question. While training NOBLE involves an upfront cost, it offers significant long-term efficiency and flexibility compared to traditional modelling approaches. Once trained, NOBLE enables the unlimited generation of new, biophysically realistic responses through interpolation (not possible with the original PDE models), and fast inference of voltage traces for arbitrary models in the embedding space

Training NOBLE on 84,000 samples for 300 epochs took ~4 days on a 64 GB NVIDIA Tesla P100 GPU. This is small compared to the 600,000 CPU hours required to generate each HoF model via evolutionary optimization. We added this information to the appendix. Note that we trained until the optimization fully converges, but could have stopped earlier and trained on a smaller number of samples while retaining satisfactory performance

We added an ablation study where we varied the number of HoF models used to generate the training data while keeping the number of training samples fixed, and evaluated performance on the original HoF test set

Although the relative L2 error on voltage traces remains largely stable, a clear degradation is observed in spike-related features, particularly spike count and AP1 width, as training diversity decreases. We added this ablation to the paper to illustrate that model diversity, not just quantity, is crucial for learning robust representations of neural dynamics

2. FNO Limitations (L2) We first emphasize that the NOBLE framework is general and not restricted to the use of the FNO as the neural operator A general limitation of data-driven approaches is that they inherently reflect the assumptions built into the underlying PDE model and numerical solver. Our ensemble approach helps mitigate this limitation by combining predictions from multiple PDE models, each grounded in different assumptions

We selected the FNO as the most suitable neural operator for our setting, given its strong performance and computational efficiency. It offers key advantages over standard neural networks and is faster than many alternative neural operators, by leveraging FFTs to accelerate internal computations. While the FNO requires inputs and outputs on equidistant grids, this aligns perfectly with our data, as both experimental recordings and PDE simulations are sampled at constant timesteps. We added a justification for this choice in Section 4 and discussed alternative neural operators that may be better suited for scenarios where uniform time sampling is not available

First of all I beg the authors pardon for the late response and to keep them waiting. I thank them for their thorough rebuttal and for the detailed answers to my questions. Let me try to respond point by point:

• NOBLE’s purpose: As I understand it NOBLE tries to better and more flexibly capture biological variability, compared to a set of HoF models. When I referred to data, what I meant is to demonstrate that the NOBLE model correctly captures the biological variability (in summary feature space of say PVALB neurons) and does so in a continuous fashion. I feel that this point is not sufficiently shown. In my view Fig. 7 shows this only for a set of 5 inputs and 4 summary statistics, while Fig 6. just shows that NOBLE does not overfit to the training set. The authors could for example show how the summary stats of the NOBLE surrogate vary for a continuous interpolation of the latent space and how that compares to the data. Similar to Fig. 5, but in feature space. This could be added to Fig. 5 for example.

• Interpretability: I appreciate the authors additional inclusion of feature distributions across the latent space. I think this is a valuable analysis that I'd be curious to see the results of.

• Subsampling:

We tested inference at the original resolution and observed no noticeable difference in computational time compared to predictions at the 3x downsampled resolution.

The authors could mention this in their manuscript as it is a potential advantage of the NOBLE framework.

• Training Loss:

L4 consistently preserved spike-related features more effectively than L2, particularly in capturing spike amplitudes and widths.

I did not know this! This is a super interesting and useful nugget of information that I would love the author's included in their revised manuscript.

• PDE-based terminology:

We use the term “PDE-based biophysical models” to refer to models that follow the mathematical formulation of the cable equation.

While biophysical models indeed derive from the cable equation (PDE), the actual model being solved is in fact an ODE. I would argue most practitioners are probably more familiar with the term ODE in this context. I can see that this might be a question of taste though. I appreciate that the authors added a clarifying statement about this to the text.

• Local sampling around HoF model: Thank you for adding the additional analysis. I still feel that showing features would be more intuitive than showing the traces though (also the NOBLE's purpose above).

• Figure based modifications: Why not use spike rate as a feature? This is independent of duration. Thank you for adding the other features to the appendix!

• Training budget: I appreciate the additional ablations! The insights here could be important for considering when a potential practitioner should choose NOBLE over lets say just a few HoF models.

• Limitations: I appreciate that the authors added an additional section to discuss limitations and to justify their choice to use FNOs.

I thank the authors again for their extensive revision of the paper in light of my criticisms. The authors have addressed most of my concerns in their rebuttal. I am in general willing to raise my score, if the authors could respond to the following two points:

Firstly, in my review I raised the point that:

In the introduction the authors allude to patch-seq datasets and how one can use them to study how genes might relate to circuit or brain function, but this is not picked up anywhere in the discussion about NOBLE (or I have missed it). If you could point me to this in the text or elaborate more on how NOBLE can help with this, that would be appreciated.

While the authors mention in their rebuttal that this is illustrated by Fig. 2B and Fig. 5, I maintain that it is not sufficiently discussed (or at least not clearly enough in my opinion), despite being a motivation for NOBLE in the first place. If the authors could make this more obvious / add this to the discussion of the results, I think this would be helpful.

Secondly, if the authors could take another shot at responding to my concerns about NOBLE's purpose (see above), I would greatly appreciate this.

Thank you to the authors for replying on such short notice.

We thank the reviewer for the thoughtful and positive evaluation of our work. We appreciate the recognition of the model’s technical soundness, its ability to capture across-trial variability, its generalization to unseen stimuli, and its substantial improvements in computational efficiency. We are pleased that the reviewer regards our approach as a valuable tool for studying neural responses across neuron types. We have addressed all raised concerns below and hope our response satisfactorily resolves the reviewer’s points. We are happy to provide further elaboration or modifications during the discussion period.

Construction of the Embedding Space

This is indeed an important aspect of NOBLE design. The general idea is to form a low-dimensional embedding space that captures biologically relevant neuronal features (e.g., spikecount, AHP width). Training NOBLE with such features enables ensemble response generation by sampling the embedding space during voltage prediction.

Specifically, we define the embedding space by two electrophysiological features (e-features) derived from each HoF model’s frequency-current curve: the threshold current ($I_{thr}$) and the local slope at threshold ($s_{thr}$). These features are normalised to the range [0.5, 3.5] to account for their differing magnitudes. Ithr values are on the order of $10^{-10}$ and $s_{thr}$ values are on the order of 100. This normalisation ensures that the frequency-modulated sinusoidal embeddings remain within the Nyquist limit while preserving periodicity. The construction of the embedding space is detailed in Section 3.3.

Necessity of the Embedding

We agree that clearer quantitative metrics are required to demonstrate the advantage of incorporating feature embeddings. To address this, we conducted an ablation study comparing NOBLE's performance as the embedding dimension varies from 0 to 3.

These results show that without the use of embeddings, NOBLE is unable to predict any firing response successfully. Embedding either slope or intercept yields similar L2 accuracy, with intercept better capturing AP1 width. Best performance is achieved by embedding both, with minor gains from adding AHP depth. We agree this ablation adds value and included it in the paper

Setting Considered

1. Low firing rates. It is indeed more challenging to learn neural activity across a variety of firing regimes. This is actually the setting we consider. We consider neural activity across a realistic range of current inputs from −0.11 to 0.28 nA, which elicits both non-spiking and spiking behaviors (Fig 4). Specifically, Fig 5B demonstrates NOBLE's ability to accurately capture spike frequencies from 0 Hz up to 60 Hz. While our figures primarily highlight the non-spiking and high-firing extremes, we acknowledge that the transition region near the current threshold, with its lower firing rates and higher variability, presents a greater challenge, as seen in the frequency-current curves in Fig 5B. To further address this, we added an explicit example demonstrating NOBLE's robust performance in this challenging, low-spike count regime. We also emphasize that the same model captures the neuronal responses in the different firing regimes.

2. Time-varying current injections. We agree with the reviewer that richer time-varying stimuli would allow for broader validation. We focused on DC step currents because they are widely used in electrophysiological experiments and represent a common protocol for characterising neuron behaviour. More importantly, they offer a readily available source of experimental data to validate NOBLE’s predictions. That said, extending NOBLE to handle arbitrary time-varying inputs is a natural next step that we intend to explore in future work, and added a comment about this at the end of the paper.

3. Excitatory Neurons. We appreciate the great observation regarding our focus on inhibitory neurons. Although the NOBLE framework can be applied to both excitatory and inhibitory neurons, we concentrated on inhibitory neurons here due to their significant diversity in morphology, gene expression, and electrophysiology within the human cortex. This diversity presents a particularly rich testbed for evaluating the generalization capabilities of our approach. We already have biophysical PDE-based models for both excitatory and inhibitory neurons, and plan to extend NOBLE to broader populations of neuron types in future work, possibly including excitatory neurons.

Clarity of the Exposition

We appreciate the reviewer’s suggestions for improving the clarity and accessibility of our paper.

Limitations section

We thank the reviewer for the suggestion. In response, we added a dedicated Limitations section to the manuscript. This addition improves the clarity of the paper and provides a more complete view of the current constraints of the study. We summarise the main limitations below:

1. Training budget: While NOBLE enables efficient inference post-training, generating the training dataset requires significant resources to generate the HoF models. We added an ablation study to the paper which shows that NOBLE still performs well when trained on a dataset generated from a smaller number of HoF models
2. Single-cell scope: We focus on modelling somatic voltage responses from individual neurons. However, the framework could be extended to multiple neurons by increasing the number of embedded features to capture inter-neuronal variability
3. Setting considered: As discussed earlier, the current study does not consider time-varying current injections and excitatory neurons. We emphasize once again that this is not a limitation of NOBLE as it extends naturally to these settings, but rather a more general setting we intend to pursue in future work

Acronyms

As suggested, we clarified the text by removing certain acronyms. Specifically, we removed FHN (FitzHugh-Nagumo), HH (Hodgkin-Huxley), FFT (Fast Fourier Transform), PINN (physics-informed neural network), and NeRF (Neural Radiance Field).

Figures and Captions

We appreciate the feedback and updated the captions, to enhance clarity. Please see the revised versions below.

Caption 3 Latent representations of neuron models in the normalised ($I_{thr},s_{thr}$)-space. Black dots indicate the 50 training HoF models ({$\mathrm{HoF}^{train}$}), and red crosses represent the 10 test HoF models ({$\mathrm{HoF}^{test}$}) that were excluded during training.

Caption 4 NOBLE predictions for current injections of 0.1 nA (top row) and −0.11 nA (bottom row). A) Predicted somatic voltage response for a HoF model in $\mathrm{HoF}^{train}$ previously seen during training, representing an in-distribution case. B) Predicted somatic voltage response for a HoF model in $\mathrm{HoF}^{test}$ not experienced during training, representing an out-of-distribution case.

Caption 6 Distributions of somatic voltage traces across HoF models for current injections of 0.1 nA (top row) and −0.11 nA (bottom row). From left to right: (1) Ground truth voltage responses from the numerical solver for the 50 training HoF models, $\mathrm{HoF}^{train}$, experienced in training. (2) NOBLE predictions on the same HoFtrain models. (3) NOBLE predictions for 200 randomly sampled models from the convex hull $CH^{train}$.

Line 96

To clarify Line 96 which describes NOBLE as a flexible framework capable of incorporating a range of ‘interpretable neuron characteristics’, we revised the paragraph beginning at Line 99 to explicitly state that, in the experiments presented here, the latent space is defined by two key e-features: the current at the spiking threshold and its slope

Utility of the Fourier Neural Operator (FNO)

The FNO provides a principled and efficient framework for modeling neuronal dynamics by learning mappings from input currents to voltage responses across a broad family of neuron models and current injections. Unlike conventional neural networks that operate on fixed-size vector inputs and outputs, the FNO learns operators, meaning mappings between functions. By operating in the frequency domain, the FNO efficiently captures global, nonlinear, and high-frequency components of voltage responses. Importantly, it generalizes across varying temporal resolutions, different input currents, and different neuron models, enabling accurate simulation of unseen configurations without retraining. This combination of scalability, generalization, and temporal fidelity are typically beyond the reach of standard neural networks. We added a short paragraph at the end of Section 2.2 to emphasize these advantages of the FNO.

We sincerely thank the reviewer for their thoughtful and constructive feedback. We're especially grateful for the recognition of our method’s technical strength, wide applicability, and clear presentation. We have addressed all raised concerns below and hope our response satisfactorily resolves the reviewer’s points. We are happy to provide further elaboration or modifications during the discussion period

W1: Validation against real data

Unlike other existing ML papers [28-33;37-40] in neuroscience focusing exclusively on synthetic neuron dynamics simulations, we validate our model against real in vitro data in Fig 7. We compare NOBLE's predicted e-features to those extracted from real biological human neuron recordings. We report relative errors for both the biophysical models and NOBLE against experimental data for the most important e-features, and show that NOBLE exhibits good performance in predicting key e-features of real experiments when compared to the relative error from fully detailed bio-realistic models

W2: Benchmarks

We understand the concern about limited comparisons to baselines. Our primary goal is to evaluate whether the neural operator framework can capture the biological variability seen in real-world experiments. To this end, we used data generated from PDE-based models which were selected via evolutionary multi-objective optimization as the best available approximations of experimental recordings, and exhibit the desired variability. Now, other existing ML approaches [28-33;37-40] in neuroscience are not equipped to capture variability, and can not be used for a comparison. While other models could be used within NOBLE and trained on the same data, we believe this would not offer additional insight

Our focus is on learning representations that reliably capture key e-features across neuron dynamics and parameter regimes, and NOBLE demonstrates strong performance on this. Since the PDE solvers generating the dataset themselves are approximations of biological reality with non-negligible errors, further reducing prediction errors would lead to further overfitting to the PDE solver simulations but probably not to substantially better-captured dynamics and e-features when compared to true real-life recordings. For this reason, we tuned hyperparameters up to the level of error seen between the PDE models and experimental data, as marginal error improvements on the PDE data are unlikely to translate meaningfully when compared to real experimental data. We included a note on this in Section 4.1 of the revised paper

W3: NOBLE generalizes beyond its training distribution

Fig 6 (rightmost plot) shows that NOBLE generalizes well across the entire convex hull of HoF models, not just near HoF models experienced during training. This highlights NOBLE's significance as the first approach capable of generating voltage responses for unseen HoF models that remain within the distribution of valid outputs produced by the PDE solver To further address concerns about broader applicability, we evaluated NOBLE on a different neuron type: the vasoactive intestinal peptide (VIP) interneuron. We tested two settings:

• Cross-type generalisation, applying the PVALB-trained NOBLE directly to a VIP neuron. Achieves errors: Relative L2: 14%, Spike count: 8.0%, AP1 width: 62%, Mean AP amplitude: 29%
• Training a new NOBLE model on VIP neurons. Achieves errors: Relative L2: 2.5%, Spike count: 9.2%, AP1 width: 23%, Mean AP amplitude: 10%

These results show that NOBLE generalizes modestly across cell types but performs best when trained within a specific type. Its strong performance on both PVALB and VIP neurons highlights the generalizability of the embedding framework. Full results are provided in the updated appendix

Q1: L4 loss was chosen based on empirical performance in early experiments

Before training on the full dataset, we conducted (added to the appendix) an ablation study to assess the effectiveness of various loss functions in capturing voltage traces and e-features. We performed a grid search by training the model on a small subset of samples, providing an estimate of each loss function’s best possible performance under ideal conditions. L4 consistently preserved spike-related features more effectively than L2, so we selected L4 as the training loss

Q2: Figure 5A

The black ‘x’ markers represent experimental firing rates measured at discrete current amplitudes. Unlike the PDE simulations, which cover a finely spaced input range, experimental recordings were limited to a small number of amplitudes due to the temporal constraints of in vitro electrophysiology (one can only keep a neural recording stable during an in vitro experiment for ~20 minutes, see e.g. [13], limiting the number of experiments that can be recorded with repetition). We chose not to connect these points with a line, as the data is too sparse, but clarified this choice in the caption

Q3: Fluctuations in NOBLE’s predictions

We first point out that the oscillations are low in amplitude and do not significantly affect the extracted e-features. Moreover, similar fluctuations can be observed in real experimental voltage recordings, supporting their biological plausibility

Now, capturing high-frequency spiking activity requires the FNO to leverage its higher frequency modes. In contrast, non-firing regimes involve slower dynamics, where these high-frequency components are unnecessary. In attempting to reconcile both regimes within a single model, the FNO relies on interference between higher frequency modes to cancel out their contributions during non-firing activity. However, this cancellation is imperfect, resulting in small residual oscillations. Training a model specifically for non-firing responses would reduce or eliminate these artifacts, but this comes at the cost of generalization. We chose to prioritize a unified model capable of handling the full range of neuronal behaviors, accepting minor imperfections in favor of broader applicability.

Q4: Ablation Studies

We agree with the reviewer’s view on the importance of including ablation studies. As discussed in our response to W2, we tuned hyperparameters up to the level of error seen between PDE models and experimental data. At this level, marginal error differences on the PDE data are unlikely to translate meaningfully when compared to real experimental data, limiting the significance of ablation studies on certain model and training hyperparameters. However, we added to the revised version two additional ablation studies on key components of NOBLE.

1. Feature embedding ablation

To evaluate the benefits of using e-feature embeddings, we vary the embedded features used in 5 different ways:

• No embedded features
• $I_{thr}$
• $s_{thr}$
• $s_{thr}$, $I_{thr}$
• $s_{thr}$, $I_{thr}$, AHP depth

These results show that without the use of embeddings, NOBLE is unable to predict any firing response successfully. Embedding either slope or intercept yields similar L2 accuracy, with intercept better capturing AP1 width. Best performance is achieved by embedding both, with minor gains from adding AHP depth. We agree this ablation adds value and included it in the paper

2. HoF Population Ablation

We vary the number of HoF models used to generate the training data to investigate the importance of variability and diversity of dynamics. Specifically, we constructed training sets using different numbers of HoF models, ensuring that the total number of samples used in training remained the same. This enables an evaluation of the initial cost associated with ensuring a sufficient number of available HoF models

Although the relative L2 error on traces remains stable, there is a degradation in spike-related features, particularly spike count and AP1 width, as training diversity decreases. This ablation has been added to the paper as it highlights the importance of model diversity, not just quantity, for learning robust representations of neural dynamics

Q5: Trial-to-trial variability arises from the diversity across HoF models

We thank the reviewer for raising this important question.The trial-to-trial variability arises from the structured diversity among the biophysical models used for training. The HoF models are generated using an evolutionary optimization procedure that searches for PDE parameters whose simulations, through BMTK, closely match experimental recordings. Since the experimental data naturally exhibit trial-to-trial variability and the HoF models are obtained using random subsets of the entire collection of experimental recordings, the resulting population of optimized PDEs captures a range of biologically plausible dynamics. Simulating across this population reproduces the variability observed in repeated biological trials.

NOBLE can emulate the HoF models, and thus reproduce the structured variability present in the training distribution. We emphasize that our embedding is the key to capture variability, and that prior ML approaches [28-33;37-40] in neuroscience are not equipped to capture variability. At inference time, conditioning NOBLE on different HoF embeddings results in output variations that reflect different realistic biological responses. We updated Section 3.3 to clarify how the trial-to-trial variability is achieved within NOBLE