NOBLE - Neural Operator with Biologically-informed Latent Embeddings to Capture Experimental Variability in Biological Neuron Models
Authors: Luca Ghafourpour
Valentin Duruisseaux
Bahareh Tolooshams
Philip H. Wong
Costas A. Anastassiou
Anima Anandkumar
Venue: NeurIPS 2025 poster ()
Keywords: Computational Neuroscience
Neural Dynamics
Neural Operators
Ensemble Surrogate Modeling
Interpretable Latent Embeddings
Primary area: neuroscience_and_cognitive_science
TL;DR: A neural operator framework that maps biologically interpretable embeddings of neuron models to realistic neuronal responses, thereby enabling the fast generation of ensembles of neuron models that capture experimental variability.

Abstract: Characterizing the cellular properties of  neurons is fundamental to understanding their function in the brain. In this quest, the generation of bio-realistic models is central towards integrating multimodal cellular data sets and establishing causal relationships. However, current modeling approaches remain constrained by the limited availability and intrinsic variability of experimental neuronal data. The deterministic formalism of bio-realistic models currently precludes accounting for the natural variability observed experimentally. While deep learning is becoming increasingly relevant in this space, it fails to capture the full biophysical complexity of neurons, their nonlinear voltage dynamics, and variability. To address these shortcomings, we introduce $\texttt{NOBLE}$, a neural operator framework that learns a mapping from a continuous frequency-modulated embedding of interpretable neuron features to the somatic voltage response induced by current injection. Trained on synthetic data generated from bio-realistic neuron models, $\texttt{NOBLE}$ predicts distributions of neural dynamics accounting for the intrinsic experimental variability. Unlike conventional bio-realistic neuron models, interpolating within the embedding space offers models whose dynamics are consistent with experimentally observed responses. $\texttt{NOBLE}$ enables the efficient generation of synthetic neurons that closely resemble experimental data and exhibit trial-to-trial variability, offering a $4200$× speedup over the numerical solver. $\texttt{NOBLE}$ is the first scaled-up deep learning framework that validates its generalization with real experimental data. To this end, $\texttt{NOBLE}$ captures fundamental neural properties in a unique and emergent manner that opens the door to a better understanding of cellular composition and computations, neuromorphic architectures, large-scale brain circuits, and general neuroAI applications.

NOBLE - Neural Operator with Biologically-nformeed Latent Embeddings to Capture Experimental Variability in Biological Neuron Models
Luca Ghatourpour!? 1.2 Valentin Duruisseaux
Bahareh Tolooshams
Philip H. Wong
Costas Anastassion 5.6
Anima Anandkumar
CCalifornia Institute of Technology University of Alberta 4 Alberta Machine Intelligence Institute (Amii)
ETT Zürich
*Cedars-Sinal Medical Center
Archimedes AI, Athena Research Center
Abstract
Characterizing the cellular properties of neurons fundamental understanding their function in the brain. In this quest, the generation of bio-realistic models is central towards integrating multimodal cellular data sets and establishing causal relationships. However, current modeling approaches remain constrained by the limited availability and intrinsic variability of experimental neuronal data. The deterministic formalism of bio-realistic models currently precludes accounting for the natural variability observed experimentally. While deep learning is becoming increasingly relevant in this space, it fails to capture the full biophysical complexity neurons, their nonlinear voltage dynamics, and variability. To address these shortcomings, introduce NOBLE, neural operator framework that learns map- ping from continuous beqquencymmddulane features to the somatic voltage response induced by current injection. Trained on synthetic data generated from bio-realistic neuron models, NOBLE predicts distributions of neural dynamics accounting for the intrinsic experimental vari- ability. Unlike conventional bio-realistic neuron models, interpolating within the embedding space offers models whose dynamics are consistent with experimentally observed responses. NOBLE enables the efficient generation of synthetic neurons that closely resemble experimental data and exhibit trial-to-tree variability, offering a 4200x speedup over the numerical solver. NOBLE is the first scaled-up deep learning framework that validates generalization with real experimental data. To this end, NOBLE captures fundamental neural properties in unique and emergent manner that opens the door better understanding of cellular composition and computations, neuromorphic architectures, large-scale brain circuits, and general neuroAl applications.
embedding of interpretable neuron
Introduction
Hundreds distinct neuronal cell types co-exist and compute within neural circuits, yet how they shape cognitive functions remains essentially unanswered [1-5]. This particularly true in the human brain, where access and monitoring capabilities are severely limited compared to animal models. Over the past decade, multimodal cellular datasets that integrate electrophysiology. morphology, and transcriptomics have emerged for human cell types [6-10].
"These authors contributed equally to this work.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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While integrating across the different modalities remains a challenge, clear differences in gene expression, morphology, and electrophysiology are evident across cell types. However. understanding how these differences impact brain processing is crucial, e.g. to uncover how expression of specific genes relates to neurological diseases.
Cellular models representing multiple data modalities are invaluable as they offer a degree of control and perturbations that are experimentally impossible (e.g.. [1I-13]. Recently, evolutionary multi- objective optimization algorithms [14] have been used to generate and validate bio-realistic models of neurons in the form of 3D nulti-compartment partial differential equation (PDE) models that mirror both their shape and ion channel expression, shaping their electrical properties [12, 13, 15, 16]. Yet, such models are deterministic and fail capture the intrinsic variability observed experimentally, where identical input to the same neuron often results in different electrophysiological responses. One approach is to generate families of models, referred to as "hall-of-fame" (HoF) models [12, 13, 17] to represent a single cell. While each HoF model is distinct and reproduces the electrophysiologica features of parts of an experiment, the ensemble of deterministic HoF models is used as collective representation that captures both the main features as well as their variability in an experiment [12, 13]. Typically, neurons exhibit highly nonlinear behavior, necessitating equally complex models rendering the optimization computationally demanding (i.e. requiring about 600k CPU core hours per negle-neuron model [12, 13). Yet, even tiny perturbations of the model parameters lead to large deviations from experimental data [18]. Other approaches have explored capturing variability through introducing stochasticity in neuron models [19-21]. However, the synthetic injection of white noise is non-mechanistic and introduces perturbations that can lead to unrealistic predictions [22-24]. In summary, capturing the nature and variability of neurons is a challenge with existing computational techniques.
The challenges of scalability and the computational cost associated with traditional numerical modeling approaches, such numerical integrators and evolutionary optimization algorithms, have led the scientific community to explore the use of machine learning to accelerate simulations by learning underlying relationships between variables directly from experimental and synthetic data. While neural networks have been used successfully for many applications, they learn mappings between diite-dimensional vectors, which can limit their ability to model physical phenomena that are better described using functions in dimente-ddieniioalll spaces and functional relationships between them [25]. As result, neural networks can overfit the training discretization and suffer from limited distr--iiitiiittoo capabilities. Neural operators [25-27] are principled way to generalize neural networks learn operators mapping functions to functions, with a universal operator approximation property [28]. A variety of neural operators have been proposed, such as the Fourier Neural Operator (FNO) (0J.
Machine learning approaches have been applied to model single-cell electrophysiology. primarily for point-neuron systems such as FitzHugh-Nagumo [31, 32] and Modgkin--Huxle, [33]. Conventional neural networks [34-36] and physics-informed neural networks [37-42] successfully reproduced their dynamics but remain highly specific to deterministic formulations and require retraining for each new stimulus. More recently, neural operators demonstrated strong potential for learning the governing dynamics of Hodgkin-Huxle,, systems [43], though the study was limited to simplified data, without capturing biological variability. Related works like NeuPRIÑT [44] captured biological neuronal variability, but on slower in vivo 2-photon calcium imaging data and models population-level fluorescence dynamics, rather than fast intracellular voltage dynamics of individual neurons.
We build on these advances and address current limitations to enable deeper insights into brain function and neuroAL.
Contributions. We introduce NOBLE (Neural Operator with Biologically-nfoomme Latent Embed- dings), a neural operator framework for learning the nonlinear somatic dynamics across population of HoF models for a single neuron (Figure 1). NOBLE is the first scaled-up deep learning framework whose performance is validated with experimental human cortex data. Rather than training separate independent surrogate for each HoF model, NOBLE learns a single neural operator that maps from a continuous latent space of user-defined, interpretable neuron characteristics to ensemble of somatic voltage responses induced by current injection. This latent space is constructed using an embedding strategy informed by the specified characteristics of the neuron models.

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Figure I: The Neural Operator with Biologically-infon informed Latent Embeddings (NOBLE) framework. A) In NOBLE, current injection and neuron model features are first encoded using the proposed embedding strategy, before passing through neural operator to produce a prediction for the somatic voltage response. B) NOBLE can be queried in parallel with different model latent representations to produce ensemble predictions. C) The proposed embedding in NOBLE encodes specified neuron features and the imput current stack of trigonometric time-series, as described in Section 3.4.
As an example application, we train and evaluate NOBLE on parvalbumin-positive (PVALB) neuron dataset generated using 50 HoF PVALB models (Figure 2). We show that single NOBLE model accurately captures both subthreshold and spiking dynamics across all 50 HoF models (in-distribution) as well as 10 unseen HoF models -outro--distributiin)) while achieving a significant speedup of 4200x over the numerical solver used to generate the dataset (Figure 3). In addition, the NOBLE predictions across 16 electrophysiological features of interest (including spike count, amplitude, and width) remain within the variability observed in experimental data (Figure 5B). Additional ablation studies confirm that biologically informed embeddings are critical for capturing both firing and non-firing dynamics, and demonstrate that we can enhance performance on targeted features without compromising overall dynamics by introducing a feature-specific fine-tuning approach.
We further show that NOBLE can successfully generate novel bio-realistic neuron models by sampling and interpolating within the latent space of models. The dynamics of novel neuron models generated by NOBLE align both with previously unseen HoF PVALB models and experimentally observed somatic responses. In contrast, direct interpolation between the parameters of bio-realistic PDE-based neuron models fails due to the sensitivity and nonlinearity of the underlying PDEs [45, 46] (Figure 4). We also successfully instantiate and train an additional NOBLE on a vasoactive intestinal peptide (VIP) interneuron to demonstrate the generalizability of our embedding framework. Owing to NOBLE'S ability generate novel bio-realistic neuron models, ensemble predictions are no longer constrained to the original 50 HoF models used for training. We demonstrate that NOBLE can produce somatic voltage responses for an arbitrary number of biologically plausible neurons by predicting responses to input stimuli across larger set of models (Figure 5). The results showcase NOBLE's s ability to accurately capture a broad range of neuron dynamics while enabling dense interpolation across model space. By unlocking the efficient, unlimited generation of diverse yet realistic neurons from continuous embedding, NOBLE offers a scalable alternative to computationally intensive, scale-limited evolutionary approaches, laying the foundation for brain-scale neural circuit modeling.
Finally, the baaogiccally-nformen latent representation of the neuron models together with the capability of NOBLE to generate arbitrarily many new bio-realistic neuron models also offers further insight into the behavior of neural dynamics. We can use NOBLE to obtain the somatic responses to current injections on fine grid in the model latent representation space, and consequently construct heat maps and surface plots to better understand how neuron features used for the model latent representation affect any clectrophysiological feature of interest.
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Figure 2: Creation of Bio-rcalistic PDE-based Neuron Models. A) Evolutionary optimization process for a neuron interest, with voltage responses sampled at different generations (top) and the error history with other experimental neurons overlaid in the background (bottom). B) Sample HoF models of various inhibitory cell-types, showing morphology (top), experimental voltage traces (2nd row), simulated voltage traces (3rd row), and spike waveform and frequency-current curves (bottom).
Background on Bio-Realistic Neuron Modeling
We create bio-realistic PDE-based neuron models (based on the cable equation [47]) using actual reconstructions of neuron morphologies from human cortical data [6. 8, 9] (Figure 2). We instantiate these models using framework [45] built on the NEURON simulation environment [46], which uses a spatial discretization to simulate the models as system of coupled ordinary differential equations. We place ion channels in an "all-active" configuration [12, 13], where active ion channels are distributed along both somatic and dendritic compartments along the neuron morphology. For each experimental neuron, models are generated using a multi-objective evolutionary optimization framework [13] to find ion conductance parameters replicating standard set of electrophysiological features from patch clamp recordings (Figure 2A). We adopt a two-stage optimization strategy, first fitting passive subthreshold responses, followed by capturing the active dynamics above the spiking threshold and the full frequency-curred curve of each neuron. After 250 generations evolutionary optimization, the models that best minimize the mean z-score error between simulated and actual experimental sloloralhy features are selected as HoF models (Figure 2B). More details about the clectrophysiological features of interest provided in Appendix B.
To illustrate the proposed upproach, we consider randomly selected PVALB human cortical neuron, for which we created 60 HoF models. PVALB neurons are fast-spiking inhibitory interneurons regu- lating high-frequency gamma oscillations (30-80Hz) [48] and their dysfunction has been associated with cognitive impairments such schizophrenia and Alzheimer's disease [49, 50]. We also consider the class VIP of inhibitory interneurons, known for its disinhibitory role in cortical circuits [51-53].
Method
3.1 Subsampling
NOBLE utilizes the notable property of neural operators of training on low-resolution data while reserving the capability to generate dynamics at higher resolution. In this regard, we subsample the reference HoF simulations in time. To avoid discarding high-resolution information necessary for cap- turing neuron features of interest, we analyze how these features are affected by different subsampling factors and strategies, in particular via the discrepancy between HoF simulations and experimental data. We consider (i) low-pac filtering followed by decimation in time, (ii) low-pass filtering followed by truncation in the frequency domain, (iii) truncation in the frequency domain, and (iv) decimation in time without filtering. Across neuron features, we observe no consistent differences in performance between these strategies and thus opt for low-pass filtering followed by decimation in time. Our analysis (see Appendix C. reveals that 3x subsampling preserves the fidelity of extracted features, within the bounds of the intrinsic discrepancy between HoF simulations and experimental data, and without inducing notable aliasing. For the HoF simulations, we consider time series of 51 5ms with a timestep of 0.02ms. For such signals, the subsampling reduces the sequence length from 25,750 to 8, 8,583, substantially decreasing the computational load without compromising biological realism.

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3.2 Current Amplitude Sampling
We consider square DC step current inputs, which are widely used in electrophysiological experiments and common for characterizing neuron behavior. We sample the current amplitudes from skew- normal distribution whose support matches the experimentally validated range of the HoF models, E [-0111, 0. n.28 To effectively capture the highly nonlinear dynamics around the spiking threshold (0 to 0.05nA) where neural responses transition abruptly from being non-spiking to spiking, the mode of our sampling distribution is strategically located within this peri-threshold window. To address the greater learning challenge posed by the high-frequency components of depolarizing. spiking responses (characterized by features such as spike width, latency to first spike, and spike count), we deliberately use a heavier positive tail in our sampling distribution. This ensures the model encounters numerous examples of spike onset and complex spiking patterns during training while still covering the full input range. The distribution of square-pulse amplitudes is shown in Figure 8.
3.3 Neural Operators for Neuron Dynamics Simulation
We choose to use neural operators they offer clear advantages for modeling complex dynamics (Appendix C.3). Among neural operators, the Fourier Neural Operator (FNO) [29, 30] is very efficient as it leverages fast Fourier transforms on equidistant grids, which aligns naturally with our setting where both experimental recordings and PDE simulations are sampled at constant timesteps. FNOs provide principled and efficient framework for modeling neuronal dynamics by learning mappings from input currents to voltage responses across broad family of neuron models and current injections. Unlike conventional neural networks that operate on vector inputs and outputs of fixed sizes, the FNO learns operators, that is, mappings between functions. By operating in the frequency domain, the FNO efficiently captures global, nonlinear, and high-frequency components of voltage responses. These properties allow the model to generalize across different temporal resolutions, input currents, and neuron types, enabling the accurate simulation of unseen configurations without retraining.
3.4 Embedding Strategy for Neuron-Model Variability
NOBLE learns a single neural operator that maps from a continuous latent space of user-defined, interpretable neuron characteristics to an ensemble of somatic voltage responses induced by current injection. The frequency-curee current (F-I) curve is a useful electrophysiological descriptor that sum- marizes cellular excitability by relating injected current amplitude to the neuron's 's firing rate [54]. Differences between HoF parameterizations manifest as shifts in key features of this curve: the thresh- old current Teir (the minimum amplitude that elicits spiking) and the local slope S Tahr (the rate of increase in the firing rate upon spiking). Figure 3A displays examples of F-I curves for different HoF PVALB models, illustrating how variability in Lihr and Syhr can represent the trial-to-trie to-trial intrinsic variability observed when a single neuron is repeatedly recorded under the same current injection.
Using this observation, we propose representing given neuron model by its threshold current Lahr and local slope Syhr, that is, using (Tahrs $.hr). We propose to use this representation as part of NeRF-style (Neural Radiance Field) embedding [55], where input features
encoded using sine and cosine functions. More precisely, a feature p is encoded as a stack of trigonometric time-series
(pp t) = (2 TP C t), cos(22 mp @ t),
sin xP O t), cos(2" Tp G t)) (I)
for some integer K O, where the frequencies modulated by the feature p. Here t denotes the dis- cretized time coordinates and O indicates element-wis wise multiplication with appropriate broadcasting.
The use of sine and cosine functions for encoding features is particularly synergistic with FNOs, which operate in the frequency domain to learn mappings between functions. FNOs leverage the Fourier transform to represent and manipulate data as sums of sine and cosine functions, effectively learning complex patterns by capturing interactions among frequency components. NcRF-style encodings lead to representation of the input features that aligns naturally with the spectral approach of FNOs, enhancing their ability to learn high-frequee frequency dynamics. In this context, the sinusoidal embeddings can be thought of as a form of spectral lifting. translating low-dimensional inputs into richer representation in the frequency domain that FNOs can more efficiently process.
We encode separately the model features I,hr and Syhr, and the amplitude of the current injection, and stack the resulting embeddings as input channels. To compress the large range of Iar and Sar values into manageable scale for embedding, we normalize them to [0.5, 3.5)% supporting more distinct feature space representations of HoF models. Figure 10 displays the latent representations in normalized (Lih,, 8th) )space of the 60 HoF PVALB models used in our numerical experiments.
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3.5 NOBLE: Neural Operator with Biologically-infrrmed Latent Embeddings
We introduce the Neural Operator with gologically-informer Latent Embeddings (NOBLE), for modeling neuronal voltage dynamics in response current injections. NOBLE offers a scalable alternative to computationally intensive numerical solvers for biophysically detailed, PDE-based neuron models. It learns direct mapping from input currents and a continuous, interpretable latent space of neuron features, to the resulting voltage traces (Figure IA). A key feature of NOBLE is its use of coologically-infrmmen biological neuron models. At its core, NOBLE is based on a neural operator, whose discretization invariance allows NOBLE learn on low-resolution data and infer somatic voltage dynamics higher resolutions. By combining a neural operator with the proposed continuous interpretable embedding, NOBLE learns a continuous operator over the space of bio-realistic neuron models.
embeddings, which enables interpretability and generalization across
The proposed NOBLE framework offers key advantages that set it apart from existing approaches:
NOBLE provides a unified framework that learns ensemble dynamies directly, enabling it to generate diverse, biophysicall, plausible membrane potentials for the same input. Conditioned on particular clectrophysiological feature, it produces one realization of the intrinsic variability observed in biological neurons. This stands in contrast to previous deep learning approaches, which are inherently deterministic and produce single trace for each input, failing to capture the trial-to-trial variability observed experimentally. To account for different neuronal behaviors, such models must be retrained for each variation, resulting in inefficiency and fragmentation.
Through the latent embedding space of electrophysiological features, NOBLE can interpolate between known HoF models to produce new, bio-realistic neuronal responses. This capability is significant because HoF models are restricted to the finite set discovered by evolutionary optimization, and direct interpolation between their parameters does not result in realistic traces. As shown in Figure 4, interpolation in NOBLE 's latent space consistently produces valid, bio- realistic responses, whereas interpolations between PDE parameters do not.
NOBLE can rapidly generate arbitrarily many distinct neuron models by sampling points within this continuous latent embedding space and producing the corresponding dynamics. This enables a single model to capture both spiking and subthreshold behaviors beyond the finite set of HoF models, providing an effectively infinite ensemble of bio-realistic responses that remain consistent with the variability observed in biological neurons. This is distinct from previous deep-learning methods, which were limited to predicting either spiking or subthreshold regimes in isolation.
The bio- nio-informed latent space of NOBLE, combined with its ability to generate unlimited realistic neuron models, enables fine-grain ene-grained exploration of neural dynamics. By sampling models across this space, NOBLE can produce somatic responses for different latent features, and reveal how they influence electrophysiological behavior via visualizations (e.g. heat maps and surface plots).
Results
4.1 Experimental Setup
For evaluating NOBLE, we focus on the PVALB neuron example introduced in Section 2, and further assess the framework's generality using a VIP neuron. the PVALB setting, NOBLE receives as input the applied current injection I, together with stacked embeddings of / (with K = 9 different frequencies) and of the normalized model features (Jırr, &nw) associated to neuron model HoF, (with K = 1 frequency). NOBLE then outputs corresponding somatic voltage response. The current injections are square DC steps with an activation duration of 400ms, consistent across all stimuli used for training and testing. We have access to 60 HoF models, where 50 are used during training, {HoF"aal)) }. and the remaining 10 HoF models (HoF are used for testing. Figure 10 displays the latent representations in normalized (Lshr, -space of these HoF models. For more details on the data generation, see Appendix D.I.
We use ID FNO (implemented as in the NeuralOperator library [56]) with 12 layers, each with 24 hidden channels and 256 Fourier modes. The resulting NOBLE with 1.8M trainable parameters is trained in PyTorch for 300 epochs using the Adam optimizer with learning rate 0.004, and the Reducel Reducel.RRnmlatte. ROnPlateau scheduler with factor 0.4 and patience 4. The training minimizes the relative 1.4 error, while the performance metrics are reported using the relative L2 error for interpretability.

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Figure 3: A) F-I curves from experimental recordings, PDE simulations, and NOBLE predictions on (HoPtruin) }. For one HoF model, B) compares experimental voltage responses with PDE simulations at current injection of 0. InA (top) and -0.11nA (bottom), and C) compares the corresponding PDE simulations with the NOBLE predictions the same HoF model.
To better evaluate physiologically meaningful behavior, also report errors four key electrophysi- ological features: spikecount, AP1_ _r1_iddth, mean_ Baaaa___ampllliiaaaaaaaaaa amplitude, tude, and steady_state _ss_tt__tteeeee saseeevlttaggee (see Appendix B for definitions of the features). For benchmarking, against numerical simulations obtained from HoF models and experimental data since the HoF models were optimized produce the closest approximations experimental recordings and capture the biological variability required for a meaningful benchmark. Further details on the evaluation and evaluation metrics are provided in Appendix D.2.
compare NOBLE predictions
The Py Torch codes used for our implementation of NOBLE and the numerical experiments are based on the NeuralOperator library [56], and are made available at giitub..om/huumlogrrato/o/no
4.2 Testing on HoF Models Included in the Training Set
We first validate that the trained NOBLE accurately reproduce the somatic voltage responses of the training (HoF* models when tested on current injections not seen during training. Fig- 3C shows that the voltage traces exhibit minimal differences, confirming that NOBLE generalizes well to unseen inputs. This is supported by a relative L2 test error of 2. 2.18% with the (HoF"min) models. Figure 3B also shows that the numerical solver outputs align closely with experimental recordings, and together with Figure 3C, indicates that NOBLE inherits this agreement and captures physiologically meaningful dynamics. Note that the available experimental recordings correspond to stimuli with activation durations of 1s. NOBLE also achieves errors of 3% for spikecount, 32.8% for APi_width, 8.54% for mean_AP aaP_aallittteeeeeaee assess NOBLE's s applicability across neuron types, we trained it on a VIP neuron using the same architecture and observed similarly strong performance (see Appendix D.3 for more details).
and 0. 0.83% for Steady_state stoatttoltttaeeeee..mmmmm To further
The relatively higher error on AP1_width arises from how the feature is computed: it measures the width of the first spike half amplitude, where the half level defined between the spike peak and the subsequent after-hypente hyperpolarization minimum. If the predicted peak or minimum is slightly misaligned relative the ground truth, the half-voltage reference shifts, and the measured width corresponds to different portion of the trace. This sensitivity makes AP1_width less stable to small deviations, so its relative error should be interpreted with caution compared to the other features.
We also generate the F-I curves using the trained NOBLE for (HoF) and compare them with the reference F-I curves produced by the numerical solver for the same HoF models. As shown in Figure 3A, the curves from both methods remain close overall, although for 3 out of the 50 HoF models the NOBLE predictions show larger deviations in firing rate between 0.0 0.InA.

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Figure 4: A) F-I curves from experimental recordings, PDE simulations, and NOBLE predictions on 50 interpolated HoF models. B) Experimental response numerical simulation after interpolating in between PDE parameterizations. C) Numerical simulation HoFF vs. distribution of 50 NOBLE predictions after interpolating within the model latent space near the (Lebor S features of H
4.3 Interpolating Between Models
We now test NOBLE's ability to interpolate between HoF models within the convex hull Mirrai of the (HoF models used for training. Consider a PDE model HoF excluded during training.
We first randomly sample 50 unseen synthetic models from local neighborhood in the model latent space near defining (Ttir; Sihr) Sthr) features HoF (Figure 11 illustrates this neighborhood), Figure 4C shows that NOBLE accurately captures neuronal dynamics when interpolating within the latent space, as produces a distribution of voltage responses consistent with those of the previously unseen H In addition, Figure 4A shows that the F-I curves generated by NOBLE remain biophysically meaningful and closely aligned with experimentally observed neuronal behavior.
On the other hand, the parameterizations of the HoF models obtained using multi objei-objective evolu- tionary algorithms lack consistent structure that would enable meaningful interpolation discover new bio-realistic models. This is illustrated in Figure 4B, where the prediction made by interpolating in between PDE parameterizations deviates significantly from experimental data. This can also be observed in Figure 4A, where the F-I curves obtained by numerically solving PDE models constructed from slightly perturbed parameterizations of HoF deviate markedly from experimental data.
Interpolating within the latent embedding space enables NOBLE to efficiently generate novel neuron models scale, while providing up to 4,200 faster model predictions than the numerical solver (see Appendix E). Yet, constructing the initial set of bio-realistic HoF models remains time-consuming, prompting the question of how much diversity truly needed for robust generalization. To examine this. we varied the number of HoF models included in [HoF*min) while keeping the dataset size fixed. Performance on voltage traces remained largely stable, but spike-related features degraded markedly as model diversity decreased. highlighting the importance of experiencing sufficient biophysical variability during training. Further details of this analysis are provided in Appendix D.6.
4.4 Ensemble Predictions
We now examine how the single trained NOBLE can be used for ensemble predictions. Given a current injection, we run 50 inferences in parallel of NOBLE for the (HoFrmii) models to produce 50 somatic responses. In Figure 5A (left, middle), we compare these 50 predictions with the corresponding 50 numerical solver simulations from (Hoin J. We see that the distribution of curves is very similar.
Since NOBLE enables interpolation between the HoF models used for training, bio-realistic neuron models and produce voltage responses for any neuron model whose latent space representation lies within the convex hull C'H of the training set (HoFr ). We demonstrate this by querying 200 novel models whose features are sampled randomly within C'H Figure 5A (right) shows that the distribution of curves remains very similar, but the additional samples provide denser coverage of the response space while maintaining bio-realism, with no artifacts or implausible predictions. We include additional ensemble analyses in Appendix D.5. There, we first validate NOBLE on the test models (HoF*er), where predicted and ground ground-truth responses again show close agreement (Figure 12). Second, we examine local perturbations the latent embedding space by sampling 50 synthetic models from a small circle around a held-out test model (Figure 11). The resulting responses (Figure 13) resemble small perturbations around the unseen ground-truth trace, demonstrating NOBLE's ability to generalize to unseen models and the smoothness of the latent space.
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Figure 5: A) Distributions somatic voltage traces across HoF and synthetic models for current injections of 0.1nA (top) and -0.11nA (bottom). B) Relative errors of ensemble predictions from PDE simulations and NOBLE models on (HoF"rim) compared to experimental data across key features.
Figure 5B shows that NOBLE's ensemble predictions for the full collection of HoF models achieve comparable accuracy to the HoF models themselves when evaluated on the four key electrophysio- logical features relative to experimental data. Here, we report the mean_ faaanffrruueccy representing the average firing rate, to account for the difference in stimulus activation durations between the experimental recordings and NOBLE, since spikecount is nonlinear function of time and cannot be directly rescaled. These results demonstrate NOBLE's ability to faithfully represent a diverse set of bio-realistic neuron models while enabling dense interpolation across the model space. How- experimental recordings for this neuron are limited to single trace across nine amplitudes. Consequently, our comparison is against a single realization rather than distribution. practice, experimental features exhibit trial-to-trial variability, which NOBLE is designed to capture, but cannot be directly validated here due to limited experimental data availability.
4.5 Choice of Biologically-Inoorme Latent Embeddings
Feature embeddings are central NOBLE as they enable generalization across and in-between biolog- ical neuron models. Here, we have chosen Ier and sthr for their strong biological interpretability, as natural 2D representation of firing and non-firing dynamics. To quantify the importance these embeddings, we conduct an ablation study evaluating NOBLE with lower-dimensional embeddings, as well as without embeddings. We also consider a higher-dimensional embedding that includes AHP depth, selected for its large variation across intracell HoF models and its low correlation with Tuhr and Sthr- Results are reported Table terms of the relative 1.2 error of predicted voltage traces and the four key clectrophysiologic features on the test set with {HoF models.

---

Table 1: Relative L2 error NOBLE on voltage traces and the four key features, when trained with different embedded features. Results evaluated on the test set with (HoF"min) models.
None API width
Features embedded
Voltage
Steady state voltage
Spikecount
Mean AP amplitude
12.1%
1.31%
Never fires
Never fires
Never fires
Sehr
2.83%
1.33%
4.9%
233%
13%
ltee
2.73%
1.20%
4.4%
107%
14%
Sthrs Lahor
1.92%
1.02%
3.1 %
27%
8.9%
Sthrs Lehor AHP_ AHP_depppp _dppth
2.16%
1.04%
3.3% 22%
9.5%
Without embeddings, NOBLE fails to predict any firing responses, indicating that embeddings are necessary to capture spiking behavior. With a single embedded feature, either Sthr or Ithr, NOBLE achieves similar accuracy on voltage traces and steady_staa _state_____ss _voltee, but errors on spike-related features remain large, with Ler providing better accuracy for API_width. As discussed earlier, this feature is particularly sensitive to small misalignments in spike peak and affer-hyperpolaariztionn minima, which makes its relative error less robust as a metric. Embedding both Sthr and Itrr yields the best overall performance across all features as well as the voltage trace. Extending the embedding space with AHP _de_hhepphhhpp slightly improves AP1 APi_width, but reduces accuracy for the other features. These results show that feature embeddings are important for NOBLE to capture both firing and non-firing dynamics, and more broadly for its ability to generalize across diverse neuron models.
While we only considered constructing the latent space from electrophysiological features, NOBLE can readily incorporate additional modalities such as gene expression profiles from patch-sequencing data [57]. Learning joint embeddings across modalities could yield a unified latent space linking gene expression, electrophysiology. and morphology. providing a means to test hypotheses that are infeasible experimentally, such as how genes associated with neurological diseases [50] influence neuronal dynamics. In doing so, NOBLE would pave the way for improved statistical analysis, more reliable uncertainty quantification, and robust predictive modeling of neuronal behavior.
4.6 Feature Specific Physics-Informe Fine- line-Tuning of NOBLE
To preserve overall neural dynamics while improving accuracy on single specific feature, NOBLE can be fine-tuned with weighted composite loss L(A) Lasa + XLE, where LE penalizes deviations in feature F, and À controls its influence. We illustrate this by fine-tuning NOBLE to improve agg_amliitude accuracy, a feature reflecting the peerpollrization-acttiee In the human cortex, Ih expression varies with cortical depth, making sag- _ag_ampl.. i tude a relevant physiological marker. We fine-tune without a feature- specific loss, fine-tuning reduces the L2 feature error from 70% to 19.2%, and incorporating the feature loss LF further lowers it to 9.6% while preserving overall signal fidelity. These results show that NOBLE can be selectively refined to prioritize biophysical features of interest without compromising overall performance. Further details are provided in Appendix D.7.
cation channel (Ih).
19,600 subthreshold stimuli with negative amplitudes. Even
5
Conclusion
We introduced NOBLE, a neural operator framework for learning the nonlinear somatic dynamics across a population of HoF models for single neuron. Rather than training separate surrogates for each bio-realistic model, NOBLE learns a single neural operator that captures the inherent variability observed in experimental neuron recordings by mapping biologically interpretable embeddings to voltage responses from current injections. Demonstrated on PVALB and VIP neurons, NOBLE correctly captured the diverse neuron dynamics with a 4200x speedup over traditional solvers, while maintaining accuracy across key electrophyssologicct features. Importantly, our work is among the first benchmark a deep learning based method for predicting membrane potential responses to intracellular current injections against experimental data from the human cortex. NOBLE also allows for generating novel, bio-realistic neuron models through interpolation in the latent space, which is not feasible with HoF models. This allows for realistic and efficient ensemble predictions beyond the original set of HoF models. NOBLE's interpretable latent space also offers new insights into how neuron characteristics affect neuron dynamics. NOBLE opens a pathway toward modeling larger-scale brain circuits and leveraging multimodal latent spaces to determine relationships between gene expression, electrophysiology, and morphology, as discussed in Appendix F.
10

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Acknowledgements
L.G. was responsible for the complete technical implementation of this work. C.A.A. and A.A. conceptualized this work. L.G., V.D., and B.T. jointly developed the methodology of the novel NOBLE framework. P.H.W. and C.A.A. provided neuroscience-specift domain expertise, contextualizing the relevance and impact of this work within the broader field of neuroscience. P.H. W. supplied the biophysical PDE models and developed the multi-objectivv optimization pipeline. L.G. and P.H. W. produced the figures. L.G., V.D., B.T., P.H. H.W,, and C.A.A. co-wrote the manuscript. C.A.A. and A.A. provided supervision and editorial comments.
Funding
A.A. is supported by the Bren Endowed Chair, ONR (MURI grant N00014-23-1-26544, and the AI2050 Senior Fellow program at Schmidt Sciences. C.A.A. supported by the National Institutes of Health R01 NSIN1330 and RO1 N8130126. P.H.W. is supported by the National Institutes of Health RO1 N8130126.
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A Related Works in Machine Learning for Single-Cell Electrophysiology
Earlier applications of machine learning to single-cell electrophysiology focused on directly learning the dynamics of canonical point-neuron models like FitzHugh-Nagumo [31, 32] and Hodgkin- Huxley [33]. Fully connected and convolutional neural networks were trained to reproduce FitzHugh- Nagumo dynamics [34], while ResNet- based multilayer perceptrons showed promise in learning Bodgkin-Huxllee dynamics [35, 36). Physics-infonee neural networks (PINNs) extend conventional neural networks by introducing prior knowledge about the underlying dynamical system [37-39]. This formulation was used to predict FitzHugh-Nagun dynamics [40] and to learn the Hodgkin- Huxley model ionic conductances from simulated voltage recordings [41]. Further refinements with PINNs incorporated wavelet bases to capture localized multiscale dynamics and compute derivatives analytically, improving both accuracy and convergence speed when training on the HatHHugh-Naguma model [42]. While these methods demonstrate capabilities in capturing the core spiking dynamics of these simplified models, their ability to accurately represent the full spectrum clectrophysiological behavior, particularly the highly nonlinear onset of firing, remains largely untested. Moreover, as function approximators, they necessitate retraining for each new imput stimulus, significantly limiting their practical utilityy.
To address some of these limitations, Centofanti et al. [43] explored using operator learning ap- proaches for forward simulations of the Hodgkin-Huxee model. Among other approaches, FNOs showed promising results by demonstrating a strong capacity for learning the governing operator of this biophysical system. However, this work still exhibits key limitations and limited scope: (I) it relies on relatively simple simulated data from a point-neuron model, (2) does not explicitly attempt to capture the full spectrum of electrophysiological dynamics, particularly the highly nonlinear onset of firing, and (3) its formulation on a single operator inherently lacks the capacity to represent the trial-to-trial variability observed in biological recordings. Related work such as NeuPRINT [44] also leverages deep learning to capture trial-to-trial variability. However, it operates on slower vivo 2-photon calcium imaging data and models population-level fluorescence dynamics, rather than the fast intracellular voltage dynamics of individual neurons.
B Electrophysiological Features
For clectrophysiological feature extraction and metrics, use code from the Electrophys Feature Extraction Library (eFEL) available at
The formulas, codes, and more details about each electrophysiologicaa feature can be found at
httpa//eeoss..eeee//aaattteeerre htmm
We list below 16 important electrophysiologial features and metrics of interest when constructing neuron models (where AP denotes action potential and AHP denotes afferr-hpperpolarizaaion)::
AHP depth: Relative voltage values at the first AHP
s__timm_ffro_peede Time between AP peaks and first AHP depth
AHP1_depth_ feoaatthfrrmmppaatt Voltage difference between the first AP peak and first AHP depth
AP1_peak: The peak voltage of the first AP
AP1 @Pi_width: Width of first spike at half spike amplitude, with the spike amplitude taken as the difference between the minimum between two peaks and the next peak
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- http::/i___oottttttfforrossstaae The decay time constant of the voltage right after the stimulus
depol_blocks Check for depolarization block. Returns if there is depolarization block or hyperpolarization block, and returns otherwise.
inv inv_first_In_ ISI: 1.0 over first interspike interval; returns 0 when no interspike interval
mean mean_AP_amplitune The mean of all of the AP amplitudes
mean freuufrey The mean frequency of the firing rate
19g_mmllitude: The difference between the minimal voltage and the steady state at the end of the stimulus
spikecount: Number of spikes in the trace, including outside of stimulus interval
saaedyy_stat__oottttee The average voltage after the stimulus
steadd_sttate_voltaee..ee stimeme The average voltage during the last 10% the stimulus duration.
time_to to tettoooffstsssieeeeee Time from the start of the stimulus to the maximum of the first peak
bastag_bbase: The average voltage during the last 10% of time before the stimulus
C
Method
C.1
Impact of Subsampling on Neuron Features
We present the results of the analysis conducted to determine the maximum subsampling factor that preserves the fidelity of extracted neuron features, mentioned in Section 3.1. The results are displayed Figures 6 and 7 for the low-pass filtering followed by decimation time subsampling strategy.
We first computed the relative error between the raw, non-subsampled HoF model voltage responses and the experimental data across all amplitudes. For each amplitude, identified the minimum relative error across all HoF models, and then aggregated these minima to compute the mean and standard deviation. These serve reference for the inherent worst-case discrepancy between simulations and experimental recordings in the absence of any subsampling. We visualize the mean as solid black line and the standard deviation dotted black lines.
Next, we repeated a similar analysis to quantify the additional relative errors introduced by subsam- pling. For each subsampling factor, calculated the relative error between the subsampled and original HoF responses for each amplitude. These errors were then averaged across all HoF models, and the distribution of these averages is summarized using the mean (solid line), standard deviation (shaded region), and min/max (error bars).
This study shows that for most clectrophysiological features, subsampling introduces negligible additional error. The most sensitive features were AP1 Width and AP1_Peak, which exhibited noticeable deviations at higher subsampling rates. To ensure remain within the bounds of the intrinsic simulation- experrmmentt discrepancy, we adopt a conservative downsampling factor of 3x, which maintains fidelity while reducing computational load.
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Relative Experimental
Decimation
AHP1 Depth From Peak
AHP Depth
Relative Error (%6
10
20
Relative
2
10
2
6
10
Downsampling Factor
Downsampling Factor
AHP Time From Peak
AP1 Peak
15
Relative Error (%)
150
10
100
Relative
50
0
2
10
2
10
Downsampling Factor
Downsampling Factor
Width
Spikecount
0.050
Relative Error (%)
Relative Error
20
0.000
10
-0.025
-0.050
2
10
2
10
Downsampling Factor
Downsampling Factor
Decay Time Constant After Stim
Depol Block
30
0.050
Relative Error 156)
20
0.000
Relative
10
-0.025
00.550
2
10
2
10
Downsampling Factor
Downsampling Factor
Figure 6: Analysis of the relative errors introduced in neuron feature computation function of subsampling factor, using low-pass filtering followed by decimation in time. The solid and dotted black lines indicate the mean and standard deviation, respectively, the minimum relative error between non -on-subsampled HoF and experimental responses. The solid blue line, shaded region, and error bars represent the mean, standard deviation, and minimum-maximum statistics of the relative enror between son-subsadpled and subsampled HoF responses.
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Relative
Experimental
Decimation
INV First
Mean AP Amplitude
Relative Error (%)
Relative (%6
2
10
Downsampling Factor
Downsampling Factor
Steady State Voltage
Steady State Voltage Stimend
10
Relative Error (%)
Relative Error
2
4
10
2
8
10
Downsampling Factor
Downsampling Factor
Time To First Spike
Voltage Base
15.0
12.5
Relative Error (%6)
Relative Error
10.0
4
7.5
Downsampling Factor
Downsampling Factor
Figure 7: Analysis of the relative errors introduced in neuron feature computation function of subsampling factor, using low-pass filtering followed by decimation in time. The solid and dotted black lines indicate the mean and standard deviation, respectively, of the minimum relative error between non-subsampled HoF and experimental responses. The solid blue line, shaded region, and enror bars represent the mean, standard deviation, and minimum-maximum statistics of the relative error between -on-subsampled and subsampled HoF responses.
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C.2
Input Current Amplitude Distribution
4
Non-spiking
Spiking
3
Density
-0.11
0.05
Current (nA)
0.28
Figure 8: Distribution of square-pulse amplitudes in [-0.11, 0.28) nA considered. There is spiking threshold (between 0 to 0.05nA) where neuron responses transition from being non-spiking to spiking.
C.3 The Fourier Neural Operator Architecture
Neural operators [25-27] are a principled way to generalize neural networks to learn operators mapping functions functions, with universal operator approximation property [28]. Neural operators compose linear integral operators K. with pointwise nonlinear activation functions a to approximate highly nonlinear operators. More precisely, define the neural operator
Go = - o oWW, + K., + br.) ... o(W, + K, + +b) o P
(2)
where P. are the pointwise neural networks that encode the lower dimension function into ligher-dimensional space and vice versa. The model stacks I, layers of o(W; - Kc₃ + by) where Wi are pointwise linear operators (matrices), K., are integral kernel operators, by are bias terms, and are fixed activation functions. The parameters consists of all the parameters in P. Q, W, Kn and by Kossaifi al. [56] maintain comprehensive open-source PyTorch library for learning neural operators, which serves as the foundation for our implementation. Prior knowledge of the relevant physics laws and differential equations can also be incorporated as additional loss terms during training, to supplement or replace reference data, as done with physics-informed neural operators [58-60].
A variety of neural operators have been proposed, such as the Fourier Neural Operator (FNO) [29, 30], and successfully applied to wide range of problems [61-63]. A FNO is neural operator using Fourier integral operator layers, which are defined via
(K))))))())=2)) = =F (Fo))
(Fu))
where Rs is the Fourier transform periodic function parameterized by é. On uniform mesh. the Fourier transform F can be implemented using the fast Fourier transform (FFT).
afx)
Fourier layer
Fourier layer
Fourier layer
u(xx
Fourier layer
vido
W
Figure 9: The Fourier Neural Operator (FNO) architecture (extracted from [29]).
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D Experiments
Chtrriin
D.1 Dataset
We have access to 60 HoF models (HoF) obtained us- ing multi-objective evolutionary optimization strat- egy. We use 50 of them during training, {HoF"mi)) J. and keep the remaining 10 (HoFee) for testing.
thr
Figure 10 displays the latent representations nor- malized (Lehe -space of these HoF models.
The training dataset is composed of 75,600 samples, where the current injections are sampled described Section 3.2, each of which associated randomly to one of (HoF"ai)) }. The samples are generated us- ing numerical solver [45] built on the NEURON simulation environment [46].
Tthr
Figure 10: Latent representations of neuron models in the normalized (T Sthr Sthr Spaace. Black dots indicate the 50 training HoF mod- els (HoF"ain) ), and red crosses the 10 test HoF models (HoF"e)) excluded during training.
D.2 Evaluation and Evaluation Metries
NOBLE is trained using the relative L4 error, computed via
||
y ullp
[2 Yb p)1/P 1 PiPP//P +
Relative Lp error(x,,
(4)
11y
The relative 1.4 error was selected as the training loss after preliminary training study on small dataset, where it consistently preserved spike-related features, especially amplitudes and widths, more effectively than the relative L2 error. While this choice ensured the model captured the clectrophysiological details most relevant our setting, NOBLE is compatible with any other loss function that may more suitable in different contexts.
Although NOBLE is trained to minimize the relative L4 error, we report results using the relative common and interpretable measure of accuracy. We first evaluate
L2 error, it provides performance on voltage traces. Note that even small temporal shifts between predicted and ground- truth voltage responses can result large relative errors. To better evaluate physiologically meaningful behavior, we also report errors on four key electrophysiological features:
spikecount: number of spikes in the trace
AP1_widtha width of the first spike at half amplitude
mean _____AP_aaaa ___mmliiuue: mean amplitude of all action potentials
steady_state stalloy-tttee average voltage after the stimulus
For benchmarking, we compare NOBLE predictions against numerical simulations obtained from HoF models and experimental data. HoF models were optimized to produce the closest approximations to experimental recordings and capture the biological variability required for a meaningful benchmark. In contrast, existing machine learning methods, such the ones discussed in the introduction, are designed to reproduce such variability and are therefore unsuitable as baselines. Furthermore, since the PDE solvers used to generate the dataset are themselves approximations with non nog-negli error, driving prediction error below the sxlver-experiment gap risks overfitting to solver rather than improving alignment with real recordings. Therefore, we tuned hyperparameters only up to solver-experiment error level, as marginal gains on PDE data are unlikely to yield meaningful improvements relative to experimental recordings.
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D.3 Testing on VIP Neuron HoF Models Included in the Training Set
To assess NOBLE is ability to perform well on different neuron models, we trained it on a VIP neuron using the same architecture in the PVALB case with 1.8M trainable parameters. The model was optimized for 450 epochs with the Adam optimizer with learning rate 0.004 and the scheduler with factor 0.8 and patience 4, minimizing the relative 144 error. For the embeddings, the neural operator in NOBLE takes the stacked embeddings of the normalized model features Thrr and associated with HoFz (with K = frequency) and 7 (with K = 11 different frequencies).
The trained model achieves a relative L2 error of 2.5% on voltage traces and relative L2 errors of 9.0% for spikecount, 20% for APi_width, 10% for aean_AP_mmlliteeeee and 0.99% for steady_sttate_vlltage
These results comparable to those obtained for PVALB, indicating that NOBLE, with the same latent embedding space, also performs well when trained on different neuron types.
D.4 Neighborhood Considered in Interpolation Experiments
CH
thr
Tthr
Figure 11: Latent representations in normalized (Thr, -spcce of (HoFfmin (black dots) and (HoF*e") (red crosses) models. The latter lie in the convex hull CH of the [HoFrmin] models. In the interpolation experiment of Section 4.3, construct small neighborhood around given Ho that defines a region of latent space not encountered during training, and sample 50 unseen models from this neighborhood. The boundary of this neighborhood is shown (blue circle) for an example Ho model.
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D.5
Auxiliary Ensemble Prediction Figures
A
PDE
B
NOBLE
20
Voltage (mV)
-80
-70
Voltage (mV)
-90
0
200
400
0
200
400
Time (ms)
Time (ms)
Figure 12: Distribution of somatic voltage traces across HoF models {HoF**) } for current injections of 0.InA (top row) and InA (bottom row). A) Ground truth voltage responses from HoF simulations, B) NOBLE predictions.
NOBLE
PDE
20
Voltage (mV)
-80
-70
Voltage (mV)
-90
200
400
Time (ms)
Figure 13: Distribution of somatic voltage traces across 50 synthetic HoF models sampled from small circle centered a HoF model in (HoFeer) not experienced during training, as shown in Figure 11. Results are shown for current injections of 0.1 (top) and ~0.11nA (bottom). The ground truth voltage response from the HoF simulation not experienced during training is shown in blue, and the synthetic NOBLE predictions in orange.
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D.6 On the HoF Models Availability Requirement
Training NOBLE on 75,600 samples required approximately days 64GB NVIDIA Tesla P100 GPU (300 epochs). Once trained, for given input current, NOBLE can synthesize arbitrarily many voltage responses of synthetic HoF models almost instantaneously by interpolating within Cmraan (Figure 10). Thus, NOBLE amortizes the high upfront cost of bio-realistic neuron model generation and enables scalable response synthesis at negligible inference cost. To investigate how much model diversity is needed during training for effective amortization, we varied the number of HoF models used to construct the training set {HoF"min) j while keeping the total number of samples fixed. Results are summarized in Table 2 in terms of the relative L2 error of predicted voltage traces and the four key electrophysiological features when evaluated on the (HoFles) models.
Table 2: Predictive performance of NOBLE on voltage traces and the four key electrophysiolgicca features using the relative L2 error metric when the training set [HoFuain) } constructed by including varying numbers of models. Results are evaluated on (HoF*y).
#HoF's included
Voltage
Steady state voltage
Spikecount
API width Mean AP amplitude
50
11.7%
2.0%
9.2%
350%
14%
40
10.9%
1.8%
19%
920%
17%
30
10.9%
1.9%
20%
3004%
20%
20
10.6%
1.9%
45%
1698%
20%
The relative 1.2 error on voltage traces is largely insensitive to HoF diversity, whereas spike-related features, particularly spikecount and APi_widtte degrade substantially as diversity decreases. As discussed earlier, APi_width is particularly sensitive to small misalignments in spike peak and after- hyperpolarization minima, which can shift the half-voltage reference and lead to large relative errors even when the underlying traces are close. This makes APi_width less reliable for direct comparison than other features. Overall, these findings indicate that while NOBLE amortizes the cost of HoF generation, effective bio-realistic synthesis still requires sufficient model diversity to learn robust representations of neural dynamics. Depending on the purpose of the study and the error gap, the number of HoF models required for training can be adjusted accordingly. Moreover, for any given electrophysiolgical feature of particular interest, further fine-tuning can be used to refine predictions and improve generalization, as we discuss next.
D.7 Additional Information on Fine- Fine-Tuning
Suppose the objective is to capture overall neural dynamics while placing particular emphasis on one specific electrophysiological feature. In this setting, the loss function can be designed to prioritize the feature of interest. Let F' denote a feature computed on the ground -round-truth signal and F the corresponding feature computed from NOBLE 's output. A feature-specific loss then be defined as C. = F Fl1, which directly penalizes deviations in the feature of interest.
To illustrate this, we fine-tune the previously trained NOBLE to enhance sag_ampl i tude predictive performance. This feature is particularly relevant it reflects the presence of the hyperpolarization- activated cation channel (Ih). In human cortex, the expression of varies with cortical depth, making smg_ampl tude a useful marker. Although the broader role of Ih in shaping neuronal and network properties is not yet fully understood, it is thought to regulate neural excitability and coincidence detection.
We start from pretrained NOBLE model and further optimize the weights of the neural operator using the feature-specific loss. Relying solely on feature-specift loss, such as Luug. risks causing NOBLE to overfit sag_ampll implitude, improving that metric while degrading performance on other features and overall voltage trace fidelity. To mitigate this issue, one possible strategy is to introduce an anchor loss, as proposed in PINO [58], which penalizes deviations from the pretrained operator during fine-tuning, Combining the anchor loss with Lang could constrain optimization so that gains on a single feature do not come at the expense of overall signal fidelity, thereby encouraging balanced gains across all clectrophyssological features.
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Another approach, which we adopt, is to define a composite loss that encourages further accuracy on the voltage traces while prioritizing the feature of interest, sag- amglaiudetue
(()) datta + Saaa
To evaluate this approach, we construct smaller dataset of 19,600 stimulus waveforms with negative non-zero amplitudes, since this regime elicits non-firing responses for which the sag_ amp1 sag.mmmlttuul tude can be reliably computed. We then fine-tune the pretrained NOBLE model by minimizing the relative L4 error for Latta and the relative L2 error for Lsag. using the Adam optimizer with learning rate of 0.0005, and the in Table 3, summarize the relative L2 error of predicted voltage traces and sig_amplitun on the test set of (HoF
scheduler with factor 0.4 and patience 12. The results, reported
Table 3: Predictive performance of NOBLE fine-tuned on aag_amplituue Metries are reported relative L2 errors on voltage traces and on eag.amplitude with the training set (HoF"oin) constrained to samples with negative non-zero amplitudes. Results are evaluated on the test set of
(HoFiruin) j. Here, denotes the weighting factor of the feature- specific loss the composite loss.
Before optimization
Epoch 100
Epoch 200
Epoch 300
A Voltage
Sag amplitude
Voltage
Sag amplitude
Voltage
Sag amplitude
Voltage
Sag amplitude
0
0.14%
69.6% 69.6%
0.064% 0.055 %
22. 13.2%
0.041 %
20.3% 12.2%
0.043%
19.2%
25
0.14%
0.047%
0.035
9.6%
Even without an additional sag-ann aos__mmpiitue loss, fine-tuning on the restricted non-firing regime of the dataset improves both trace prediction and sag_amplitude performance, since in this setting every sample is subthreshold and the feature can be computed consistently.
Including the feature-specific loss provides a further improvement approximately 10% on sag_ arg_ammlitude while maintaining overall signal fidelity. Note that both settings converged after 300 epochs, ensuring that the comparison is fair.
These results demonstrate that prioritizing feature through the loss function can yield targeted improvements without sacrificing global accuracy.
E Comparison of Sample Generation Time
The trained NOBLE generates predictions significantly faster than the reference numerical solver from [45].
We record the time necessary generate 10,000 predictions on workstation equipped with single NVIDIA RTX 4090 GPU (24GB VRAM), an AMD Ryzen 9 7900X CPU, and 64GB of system RAM. The numerical solver only generates single prediction at time and takes roughly 36,200 seconds to generate 10,000 predictions. When doing one inference at time with NOBLE (i.e. batch size of 1), we generate 10,000 predictions in 157 seconds, i.e. speedup of approximately 230x compared to the solver.
In addition, NOBLE easily be accelerated on single GPU by generating multiple predictions the same time. In particular, with a batch size of 1000, NOBLE generates 10,000 predictions 8.59 seconds, i.e. speedup of approximately 4200x compared the solver.
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F
Scope and Future Directions
NOBLE successfully captures the dynamics across HoF models but is currently limited to single-neuron settings. A natural next step is to extend NOBLE to multi- multi-neuron configurations with time-varying stimuli, enabling applications such as neuron classification and predicting multi-neure dynamics, The embedding space used in our experiments is low-dimensional, constructed from two interpretable features derived from biological neuron model's F-I curve. Extending this learnable, higher- dimensional continuous embedding space represents promising direction for future work. Beyond this, NOBLE could also integrate additional modalities such gene expression morphology to build unified latent space linking molecular, electrophysiologial. and structural characteristics, as discussed in the main text.
Although NOBLE demonstrates strong performance in modeling nonlinear neuronal dynamics, our study makes few deliberate scope choices. These do not represent inherent limitations of the framework but rather natural starting points, each of which can be extended with minimal or no modifications.
Input currents: We restricted our attention to square-pulse DC step currents these are widely used in electrophysiological experiments and represent common protocol for characterizing neuron behavior. However, the NOBLE framework is not specific to these types of input currents: time-varying inputs can be incorporated directly by including examples during training. In such cases, the amplitude embedding be removed or adapted (e.g., embedding function of the amplitude, such as a moving-average modulation or the maximum amplitude).
Choice of operator learning architecture: We used the FNO primarily for its computational efficiency and strong generalizability. Moreover, since FNOs use the FFT and thus require inputs and outputs on equidistant grids, they align naturally with our data, where both simulations and experimental recordings are sampled at constant timesteps. For non-uniformly spaced data, alternative neural operators such as geometry-informed neural operators (GINOs) [64, 59] could be used within the same framework.
Training cost vs. efficiency: Training NOBLE on 75,600 samples for 300 epochs took approxi- mately four days on a 64GB NVIDIA Tesla P100 GPU, which is small compared to the ~600,000 CPU hours required generate the HoF models via evolutionary optimization. However, once trained, NOBLE enables fast inference and the instantaneous generation of infinitely many bio- realistic voltage traces through latent space interpolation, capabilities not possible with original PDE models.
Neuron populations considered: We focused on inhibitory neurons (PVALB and VIP), which show strong heterogeneity in morphology, gene expression, and electrophysiology, making them a stringent test for generalization. However, NOBLE is not restricted to inhibitory neurons and can naturally extend to excitatory types. It can also be applied to larger populations or multiple neurons within a family by expanding the latent embedding space with additional electrophysiological features that capture intracellular variability.
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Reviews:
[{"review_id": "1A9EeM6laV", "reviewer": "Reviewer_tkUK", "summary": "This paper proposes NOBLE, a neural operator framework based on Fourier Neural Operators (FNO), designed to efficiently approximate detailed multi-compartment neuron simulations. By training on neuron models, the method learns to predict voltage traces, achieving substantial speed-ups over traditional PDE solvers while maintaining biophysical interpretability. It also enables interpolation across different parameter regimes, making it well-suited for integration into a variety of neuroscience modeling studies.", "questions": "1. Why was the L4 error function chosen as the training objective?\n2. In Figure 5A, why are there no black lines for the 'Experiment' data, and what do the black x markers indicate?\n3. NOBLE produces noisier outputs compared to the PDE solution (e.g, voltage traces in Figure 6). If training were optimal, should these fluctuations be eliminated? Do they reflect imperfect training?\n4. Conducting simple ablation tests that vary training hyperparameters would help assess the robustness of the FNO training.\n5. How does NOBLE have trial-to-trial variability? Where does the variability come from (e.g., noise injection or random sampling)?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n1. **Quality and significance**: This method is technically sound and can be used across a wide range of research.\n2. **Clarity**: The manuscript is well written.\n\n**Weaknesses**\n1. **Lack of real experiment**: This paper provides experiments based on comparing neuron models, without direct validation with real experimental data.\n2. **Narrow set of benchmarks**: Evaluation is limited to comparisons with the PDE solver, without any learning-based models. While interpolation-based comparison may be ambiguous, direct comparison of fitting performance should be possible.\n3. **Generalization to other neuron types**: All experiments focus on PVALB neuron models. The out-of-distribution samples (blue circle in Figure 12) remain close to the point that is used for training. \n\n     However, this may be acceptable, as the authors conducted extensive experiments to demonstrate the method’s reliability within the PVALB neuron domain.\n.", "was_revised": true, "final_justification": "I appreciate the authors’ detailed responses and the additional experiment provided during the discussion. I am keeping my score unchanged, as my initial evaluation was already positive.", "rebuttal": "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\n\n---\n\n**W1: Validation against real data**\n\nUnlike 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\n\n---\n\n**W2: Benchmarks**\n\nWe 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\n\nOur 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\n\n---\n\n**W3: NOBLE generalizes beyond its training distribution**\n\nFig 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\nTo further address concerns about broader applicability, we evaluated NOBLE on a different neuron type: the vasoactive intestinal peptide (VIP) interneuron. We tested two settings:\n* 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%\n* 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%\n\nThese 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\n\n---\n\n**Q1: L4 loss was chosen based on empirical performance in early experiments**\n\nBefore 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\n\n---\n\n**Q2: Figure 5A**\n\nThe 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\n\n---\n\n**Q3: Fluctuations in NOBLE’s predictions** \n\nWe 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\n\nNow, 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.\n\n---\n\n**Q4: Ablation Studies** \n\nWe 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.\n\n**1. Feature embedding ablation**\n\nTo evaluate the benefits of using e-feature embeddings, we vary the embedded features used in 5 different ways:\n\n* No embedded features\n* $I_{thr}$\n* $s_{thr}$\n* $s_{thr}$, $I_{thr}$\n* $s_{thr}$, $I_{thr}$, AHP depth\n\nFeatures embedded|rel L2|Steady State Voltage|Spikecount|AP1 Width|Mean AP Amplitude\n-|-|:-:|:-:|:-:|:-:\nNone|12.1%|1.31% |Never fires|Never fires|Never fires\n$s_{thr}$|2.83% |1.33%|4.9%|233%|13%\n$I_{thr}$|2.73%|1.20% |4.4%|107%|14%\n$s_{thr}$, $I_{thr}$|1.92%|1.02%|3.1%|27%|8.9%\n$s_{thr}$, $I_{thr}$, AHP depth|2.16%|1.04%|3.3%|22%|9.5%\n\nThese 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\n\n**2. HoF Population Ablation**\n\nWe 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\n\n#HoFs excluded|rel L2|Steady State Voltage|Spikecount|AP1 Width|Mean AP Amplitude\n-|-|:-:|:-:|:-:|:-:\n10|11.7%|2.0%|9.2%|350%|14%\n20|10.9%|1.8%|19%|920%|17%\n30|10.9%|1.9%|20%|3004%|20%\n40|10.6%|1.9%|45%|1698%|20%\n\nAlthough 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\n\n---\n\n**Q5: Trial-to-trial variability arises from the diversity across HoF models**\n\nWe 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.\n\nNOBLE 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"}, {"review_id": "yQ5FrcAelB", "reviewer": "Reviewer_5ihX", "summary": "This contribution proposes a neural operator framework for predicting voltage dynamics of single neurons in response to the input current. The model is a deep network and operates on the latent space of user-defined neuron properties. The latent space is constructed using embedding of observed neuron properties. Compared to previous approaches, the proposed framework is significantly faster, captures better across-trial variability and generalizes better to unseen levels of stimuli.", "questions": "1) How is the embedding space constructed?\n \n2) Have the authors tested the performance of the model using current injections that vary over time?\n\n3) Why are Pyramidal (Excitatory) neurons not shown?\n\n4) All tested neurons seem to respond with high firing rates and relatively regular spike trains. Have authors tried to reproduces neural activity with lower firing rates? This might be more challenging because of increased variability of responses in such firing regimes.\n\n5) What is the utility of the Fourier Neural operator?", "limitations": "Limitations are briefly discussed in the section “Conclusion”. I suggest to give more emphasis on discussing limitations and to give such discussion a separate section instead of merging it with Conclusion. I suggest to add to limitations the fact that the current contribution does not test the model on weak and time-varying input currents.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "_Strengths_: \nSubmission is technically sound and proposes an important upgrade to previous approaches on reconstructing activity of single neurons. It also outperforms previous approaches on the ability to capture variability across trials, its generalization capacity to reproduce neural responses to current injections that were not sued during training and strongly outperforms previous approaches in speed of learning. Overall, this paper reads as a useful tool to study neural responses of different neuron types.\n\n_Weaknesses_:\n1.) No clear metrics of performance is used, even though this would be useful to quantitatively compare the performance of NOBLE with HoF models without embeddings.\n2) Along the paper and in particular in the Introduction, a number of acronyms is used, to an extent where this decreases the clarity of the presentation. I advise the authors to limit the number of acronyms to only a handful  (3-4) that are the most well known (besides the acronym of their model). This will ease the reading for readers that are less familiar with the topic.\n3) The writing is somewhat convoluted all along the paper. To make the paper more accessible, I kindly suggest to the authors to revise the text so that it gains in clarity. For example, on several occasions the text is unspecific, which decreases clarity. In line 96, authors mention that the latent space is constructed from “observable neuron characteristics”. Later on (line 257) we learn model features that define the latent space are only two, the current at spiking threshold and its slope. It would be clearer to simply state what those features are when they are first mentioned.\n4) Several figure captions are too short and do not explain what is shown on the figure. For example, it remains ambiguous what is shown on Figure 3. Figure 4 and 6 show two types of voltages, one seems to be with spikes, the other without spikes. This is not clearly explained in the text of the figure caption.", "was_revised": true, "final_justification": "The paper provides a useful, generally applicable and efficient method to study properties of single neurons from neural data. Also, the method can be extended in various ways, e.g., different objective function, different features, etc.", "rebuttal": "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.\n\n---\n\n## Construction of the Embedding Space\n\nThis 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.\n\nSpecifically, 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.\n\n---\n\n## Necessity of the Embedding\n\nWe 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. \n\nFeatures embedded|rel L2|Steady State Voltage|Spikecount|AP1 Width|Mean AP Amplitude\n-|-|:-:|:-:|:-:|:-:\nNone|12.1%|1.31% |Never fires|Never fires|Never fires\n$s_{thr}$|2.83% |1.33%|4.9%|233%|13%\n$I_{thr}$|2.73%|1.20% |4.4%|107%|14%\n$s_{thr}$, $I_{thr}$|1.92%|1.02%|3.1%|27%|8.9%\n$s_{thr}$, $I_{thr}$, AHP depth|2.16%|1.04%|3.3%|22%|9.5%\n\nThese 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\n\n---\n\n## Setting Considered\n\n1. **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.\n\n\n2. **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.\n\n\n3. **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.\n\n---\n\n## Clarity of the Exposition\nWe appreciate the reviewer’s suggestions for improving the clarity and accessibility of our paper. \n\n### Limitations section\n\nWe 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:\n1. **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\n2. **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\n3. **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\n\n### Acronyms\n\nAs 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).\n\n### Figures and Captions\n\nWe appreciate the feedback and updated the captions, to enhance clarity. Please see the revised versions below.\n\n**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. \n\n**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.\n\n**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 $CH^{train}$.\n  \n  \n### Line 96\n\nTo 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\n\n---\n\n## Utility of the Fourier Neural Operator (FNO) \n\nThe 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."}, {"review_id": "d1hTcMnEkR", "reviewer": "Reviewer_VJJQ", "summary": "The manuscript introduces Neural Operator with biologically-informed Latent Embeddings (NOBLE). In this framework a neural operator, or more specifically a Fourier Neural Operator (FNO) is trained to predict the somatic voltage from a f(I) curves for an entire collection of biophysical (Hall of Fame aka. HoF) models. This yields a single representative model for a diverse set of HoF models which collectively represent the dynamics of a single cell. This NOBLE model can then be used to generate biophysically plausible responses at any point within the embedding space, while being much cheaper to evaluate than the original biophysical models.", "questions": "- Does the analysis done in Fig. 6 also work for the test data? It would be nice to see this hold there.\n- In L274 you mention a \"relative L4 error\" and in L290 you say \"L2 error\". If this is not a typo, could the authors explain why they picked 2 different metrics and add a formula for the rel. L4 error to the appendix? Was the error computed on the full or downsampled traces?\n- I am not very familiar with FNOs. Could the authors comment on whether sampling time is affected by the fidelity of the samples? If yes, does the 4200x speedup refer to the downsampled version of the voltage trace?\n- In Fig 7. how does NOBLE compare on the remaining summary statistics? This would be great to have in the appendix. \n-", "limitations": "I feel like technical limitations were insufficiently discussed. For example how does NOBLE compare when the training budget is factored in? What are limitations of FNOs that also apply to NOBLE?", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "## Strengths\n- Using FNOs to capture the variability across a collection of biophysical models in a single emulator by encoding this into a shared latent space is a neat idea. With a more interesting latent space, i.e. including more behavioral features or multimodal features one could imagine training a \"foundation model\" for a certain cell type this way.\n- Another advantage of FNOs is the ability to generate dynamics at higher resolution than what was trained on, which can be especially helpful for generating voltage traces in the hundreds of milliseconds. (This was unfortunately not shown in the paper though).\n\n## Weaknesses\n- The main argument the paper seems to be making, is that the NOBLE surrogate is a _better_ and _more flexible_ model of the data than a family of HoF models, however I feel like that this was not demonstrated sufficiently. While the authors show that NOBLE approximates the HoF models well in different settings, the authors comparison of NOBLE predictions with the original data is limited (only Fig. 7 really does this for a subset of features). To show that the NOBLE surrogate is indeed a potentially _better_ model of the data it would be great to find a latent input (or set of inputs) that match the data better than the HoF models (or come up with another metric which shows NOBLE models are superior to HoF models). In addition the 2nd selling point of a NOBLE model, _more flexibility_, seems underutilized to me. More examples of how the latent space can be used would effectively, i.e. by showing the distribution of summary statistics across the entire latent space  (i.e. via pairplots) and comparing them to the data. An interesting comparison would also be to see the NOBLE predictions in a small ball around the f(I) of the data / for a HoF model.\n- 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.\n- The paper would benefit from more polished figures. The line thickness in Fig 2. is really small (I printed the paper out) and it would be nice to include scale bars or axes. The side by side arrangement of Fig 3. makes it hard to compare HoF and NOBLE predictions. Overlaying them would make this more obvious. Also the in and out of distribution currents appear to be the same. It would also be great to show  distributions of important summary features rather than the full voltages traces in Fig. 6, as it is hard to judge how the NOBLE models compare here.\n\n## Comments\n- In Fig 7. I'd argue that report medians and quantiles makes more sense. Reporting the $\\mu-\\sigma$ suggests existance of negative spike counts. \n- In Fig 5. it would be nice to also see NOBLE compared to the experimental data. In addition the plot could be improved by replacing the lines with error bars or a shaded area. Also why did you use seemingly different models in A and B? This makes it a bit hard to compare.\n- The PDE-based biophysical models are in fact ODE-based. While multicompartment biophysical models could be seen as spatially discretized PDEs, what NEURON simulates is a system of coupled ODEs, **not** PDEs.", "was_revised": true, "final_justification": "The authors have done extensive revisions of the manuscript and addressed all my major concerns during the rebuttal. While the novelty beyond the use of FNOs is limited in my opinion (training of a neural emulator + embedding space), I think the work should be published. I hence err on the site of acceptance.", "rebuttal": "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.\n\n---\n\n**NOBLE’s purpose (W1/Q1)**\n\nThe 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\n\n\nThe 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\n\n---\n\n**Interpretability (W1/W2)**\n\nWe 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\n\nRegarding 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)\n\n---\n\n**Subsampling (Q2/Q3)**\n\nThe 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. \n\n---\n\n**Training loss (Q2)**\n\nThe 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.\n\n---\n\n**“PDE-based” Terminology  (C3)**\n\nWhile 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\n\n---\n\n**Local sampling around HoF model (W1)**\n\nThis 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\n \n---\n \n**Figure-based modifications (W3/Q4/C1/C2)**\n\n* Fig 2: We increased line thickness to enhance readability\n* Fig 3: We agree that superimposing the traces improves interpretability and revised the paper accordingly\n* 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\n* 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.\n* 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\n\n---\n\n**1. Limitations**\n\nWe extended the discussion relating to the limitations of NOBLE, and created a new section dedicated to that discussion\n\n**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\n\nTraining 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 \n\nWe 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\n\n#HoFs excluded|rel L2|Steady State Voltage|Spikecount|AP1 Width|Mean AP Amplitude\n-|-|:-:|:-:|:-:|:-:\n10|11.7%|2.0%|9.2%|350%|14%\n20|10.9%|1.8%|19%|920%|17%\n30|10.9%|1.9%|20%|3004%|20%\n40|10.6%|1.9%|45%|1698%|20%\n\nAlthough 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\n\n**2. FNO Limitations (L2)**\nWe first emphasize that the NOBLE framework is general and not restricted to the use of the FNO as the neural operator\nA 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\n\nWe 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"}, {"review_id": "dhXLBD86o0", "reviewer": "Reviewer_1sYR", "summary": "This paper introduces NOBLE, a neural operator framework with biologically-informed latent embeddings, to address the challenges of modeling neuronal dynamics. It targets the key problems of high computational cost and the failure of existing models to capture the experimental variability inherent in biological neurons. The authors propose a deep learning solution that learns a direct mapping from an interpretable, continuous space of neuron features to the somatic voltage response following a current injection. Trained on data from biophysically realistic simulations, NOBLE can generate distributions of neural dynamics, effectively accounting for variability. A central contribution is its ability to interpolate within this latent embedding space to create novel, yet biophysically plausible, neuron models. This approach yields a significant 4200x speedup over conventional numerical solvers and is validated against real experimental data, demonstrating its potential to advance the scale and realism of brain circuit modeling.", "questions": "The findings presented are compelling, yet several questions remain regarding the framework's broader applicability and robustness. It would be beneficial to understand the methodology for selecting the embedding features. How might one identify a sufficient set of interpretable features for neurons with different properties, such as bursting or adaptation, and how does the model's performance scale with a higher-dimensional embedding? The paper demonstrates excellent performance with interpolation, but its behavior when extrapolating to models outside the training distribution is unknown; clarifying the model's performance degradation in this regime would help define its valid operational domain. From a practical standpoint, the data efficiency of the framework is also a key concern. Understanding how performance varies with the number of initial models used for training would be crucial for assessing the trade-off between the upfront simulation cost and the resulting surrogate's accuracy. Lastly, further discussion on the impact of the chosen time-series subsampling on the ability to capture very fast neuronal dynamics would be valuable.", "limitations": "The authors have commendably provided a forthright discussion of the work's limitations within the paper's conclusion. They correctly identify that the framework is currently developed for single neurons and that the biologically-informed embedding space is low-dimensional and constructed manually from pre-selected features. They appropriately position these limitations as clear and exciting directions for future research. Specifically, they suggest that natural extensions would involve scaling the method to multi-neuron networks and exploring the potential of learnable, higher-dimensional embedding spaces. This transparent handling of the current boundaries of the work is a positive aspect of the paper.", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 1, "strengths_and_weaknesses": "The paper is elegantly shown to succeed where direct interpolation of model parameters fails, highlighting its unique value. Furthermore, the technical quality of the study is high, featuring a rigorous validation process on models derived from human neuron data. The framework's performance is not only evaluated on voltage trace accuracy but is also benchmarked against a wide array of electrophysiological features, ensuring its neuroscientific relevance. \n\nDespite these considerable strengths, the work has weaknesses related to its scope and generalization:\n\nThe core idea for this framework is similar to NeuPRINT [1]. The authors should compare their framework with NeuPRINT.\n\nThe demonstrated embedding space is low-dimensional and hand-engineered for a specific type of neuron, the PVALB interneuron. It remains an open question how this feature selection process would generalize to other neuron types with different dynamic characteristics. \n\nThe validation is also confined to models of a single neuron, and demonstrating the framework's effectiveness on a different cell type would have greatly strengthened its claims of general applicability. \n\nWhile the framework dramatically accelerates inference, it still depends on an initial set of models generated through a computationally expensive process. The method does not eliminate this upfront cost but rather amortizes it, a practical consideration for its adoption. \n\n[1] Learning Time-Invariant Representations for Individual Neurons from Population Dynamics. NeurIPS 2023", "was_revised": true, "final_justification": "The starting point of the work is not novel, ignoring a lot of previous work about single neuron embeddings based on electrophysiological activities. \n\nThe work is entirely based on simulated data, and the generalization of real scenes needs to be considered. \n\nThe exploration of the resulting embeddingsn is not sufficient, and the association at the molecular, structural, and other levels is not fully discussed. Other similar work has relevant content.\n\nI choose to reject it. Thanks!", "rebuttal": "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.\n\n---\n\n**W1: NeuPRINT and NOBLE are fundamentally different, but complementary**\n\nNeuPRINT 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. \nTheir 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.\nIn an ideal scenario, NeuPRINT and NOBLE will inform each other. We discuss NeuPRINT in our updated paper and suggest ways for them to interact.\n\n---\n\n**W2: We demonstrate that the same low-dimensional, biologically interpretable embedding space generalises effectively across neuron types, including VIP neurons**\n\nTo 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:\n* Relative L2: 2.5%\n* Spikecount: 9.2%\n* AP1 width: 22%\n* Mean AP amplitude: 10%\n\nThese 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.\n\n---\n\n\n**W3/Q4: NOBLE amortises this cost of costly HoF generation and enables fast, scalable synthesis of biorealistic responses**\n\nTraining 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.\n\nWe 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.\n\n#HoFs excluded|rel L2|Steady State Voltage|Spikecount|AP1 Width|Mean AP Amplitude\n-|-|:-:|:-:|:-:|:-:\n10|11.7%|2.0%|9.2%|350%|14%\n20|10.9%|1.8%|19%|920%|17%\n30|10.9%|1.9%|20%|3004%|20%\n40|10.6%|1.9%|45%|1698%|20%\n\nAlthough 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.\n\n---\n\n**Q1/2: Feature selection for embedding**\n\nThere 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.\nMore 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.  \n\nTo 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:\n* No embedded features\n* Only $I_{thr}$\n* Only $s_{thr}$\n* $s_{thr}$ and $I_{thr}$\n* $s_{thr}$, $I_{thr}$ and AHP depth\n\nAHP 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.\n\nFeatures embedded|rel L2|Steady State Voltage|Spikecount|AP1 Width|Mean AP Amplitude\n-|-|:-:|:-:|:-:|:-:\nNone|12.1%|1.31% |Never fires|Never fires|Never fires\n$s_{thr}$|2.83% |1.33%|4.9%|233%|13%\n $I_{thr}$|2.73%|1.20% |4.4%|107%|14%\n$s_{thr}$ and $I_{thr}$|1.92%|1.02%|3.1%|27%|8.9%\n$s_{thr}$, $I_{thr}$ and AHP depth|2.16%|1.04%|3.3%|22%|9.5%\n\nThese 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.\n\n---\n\n**Q3: Extrapolating to models outside the training distribution**\n\nWe 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).\n\n---\n\n**Q5: Subsampling strategy and fast neuronal dynamics**\nOur 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"}, {"review_id": "eNbITU21Ld", "reviewer": "Reviewer_d3WE", "summary": "This paper describes a method to generate neural voltage traces starting from easily measurable neuron features (F-I curve). The method allows to sample voltage traces of synthetic neurons that are consistent with electrophysiological data. Using a neural operator approach, the method is insensitive to the discretization/integration method used in numerical solvers. The result is a speedup in generating voltages traces compared to numerical solvers. The technical contribution of the paper is the application of an existing neural operator approach to neural electrophsiological data.", "questions": "- I'm not convinced by how the demonstrated use cases of the method enables a \"better understanding of cellular composition and computations, neuromorphic architectures, large-scale brain circuits, and general neuroAI applications\". The proposed method does apparently not reveal biophysical parameters to identify the neuron types so I'm not sure how it can reveal cellular composition. Its help to inform neuromorphic architectures and large-scale brain circuits is a stretch from the shown results. Figure 7 shows consistency in some electrophysiological features, but this seems somewhat limited towards \"predictive modeling of neuron dynamics\". \n- \"It learns a direct mapping from input currents and an interpretable, continuous latent space of neuron features to voltage traces (Figure 1A).\": How are the traces \"interpretable\"? Is there a way to extract the complete biophysical parameters of the models generating the voltage traces?\n- \"...generate families of models, sometimes referred to as \"hall-of-fame\" (HoF) models to represent a single cell.\" The authors don't clearly describe whether (or how) the models are combined. Does this family form a pool from which the best fit is selected, or are they somehow combined (e.g. a linear combination as with basis functions).\n- \"intrinsic variability\" is mentioned throughout the article. Does this term include neuron heterogeneity or specifically refer to variability in the recording of a given neuron? I think it would be helpful to distinguish the two.\n- The fourier transform of each embedding dimension yields trivial functions. Couldn't the first fourier transform of FNO and the sine/cosine be trivially discarded?\n- what does \"Cellular models representing multiple data modalities\" mean? Does this refer to a model of the measurement (e.g. LFP, multi unit recording etc?)\n\nOther minor comments\n- I'm pretty sure that in \"where Rϕ is the Fourier transform of a periodic function\", you meant F (R seems to be a linear mixing layer). Same in the appendix.\n- The notation \\sigma(W+K+b) is a bit misleading. Please describe the meaning of this notation (mathematically)\n- If possible, it would be helpful to relate the parameters in eq (1) to figure 1.\n- It took me more time that I wanted to understand that eq (2) is not a time series itself, but rather a vector of the time series at a given time point. Perhaps add some notation to clarify this. \n- figure 4: I see no reason to separate the plots PDE and Noble. Please superimpose them to better assess the differences (as in figure4)", "limitations": "- The limitation to generate voltage traces from a step function is not sufficiently addressed. PDE based approaches would not have this limitation.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The generation of voltage traces from an latent embedded space is a neat idea. \n- The technical quality of the paper and the positioning relative to neuroscience research is well explained\n\nWeaknesses:\n- There are no direct comparisons with other state-of-the-art methods (deep learning or other statistical methods) to generate voltage traces. It is hard to assess the relative quality of the fits without such benchmarks.\n- The paper lacks in clarity ( some expressions cannot be deciphered without digging into references), and precision (math is not precisely described.). Comments in Questions below  reflect this lack of clarity.\n- The shown input currents appear to be step functions, capturing an arguably limited extent of a neuron's dynamical features. Assuming there is no direct way to relate the mapped voltage traces to the biophysical parameters of the neuron model, the HoF models could not be used to predict the neuron states to dynamic input currents.", "was_revised": true, "final_justification": "Thanks for addressing my questions. I think this paper is a useful contribution to the area. I raised my score accordingly.", "rebuttal": "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.\n\n---\n\n**W1: PDE benchmarks are biologically grounded and suitable for learning**\n\nWe 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.\n\n---\n\n**W2: Input Currents Selection**\n\nWe 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.\n\n---\n\n**Q1: Improving biological understanding**\n\nThe 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. \n\nAnother 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\n\nWhile NOBLE does not extend to large scale circuit simulations yet, it addresses current bottlenecks in such modeling: \n1. The scalable generation of diverse bio-realistic single neuron models \n2. The computational scalability of running these simulations on large populations of neurons\n\n---\n\n**Q2: The interpretability lies in the feature-based latent space**\n\nWe 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.\n\nThe 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)\n\n---\n\n**Q3: Clarification on the HoF models**\n \nThe 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.\n\n---\n\n**Q4: “Intrinsic variability” refers to repeated responses from the same neuron, not inter-neuron heterogeneity**\n\nWe 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.\n\n---\n\n**Q5/C4: Sinusoidal embedding**\n\nEquation (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.\n\n---\n\n**Q6: Clarification on “Cellular Models representing multiple data modalities”**\nOur 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). \n\n---\n\n**C1-2-3: Clarification on FNO notation and Figure 1**\n\nThe 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.\n\nThe 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.\n\n---\n\n**C5: Superimposing PDE and NOBLE**\n\nWe 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."}]

Author rebuttal:

Meta-review: This paper presents NOBLE, a framework for predicting single neuron voltage responses to injected currents. The method is based on an artificial neural network that learns to approximate the solutions the results of simulations of multi-compartmental models. NOBLE defines a latent space of interpretable neuronal features that enables meaningful interpolation between different parameter regimes and efficient generation of synthetic neurons with realistic behavior.

The reviewers appreciated the relevance of the work to the NeurIPS community and its potential impact for a broad range of computational neuroscience applications, as well as its technical soundness. The main concerns raised by the initial reviews included a lack of more extensive benchmarking against other methods and types of data, along several requests for clarification about the details of the method and its relationship with other approaches. The authors responded by providing additional explanations and analyses, testing their method on an additional neuronal type (as the original manuscript only included one type) and arguing why comparisons with other learning-based methods may not be fair/provide additional insight. A substantial discussion with one of the reviewers focused on the merits and novelty of the proposed method vs a specific prior work (NeuPRINT).

Overall, following also the internal discussion among reviewers, the consensus is that the merits of the paper and its potential usefulness to the community outweigh its limitations. I recommend acceptance.
Decision: Accept (poster)