Title: NOBLE - Neural Operator with Biologically-informed Latent Embeddings to Capture Experimental Variability in Biological Neuron Models
Venue: NeurIPS 2025
Accepted: yes
Keywords: Computational Neuroscience

Neural Dynamics

Neural Operators

Ensemble Surrogate Modeling

Interpretable Latent Embeddings
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-informed Latent Embeddings to Capture Experimental Variability in Biological Neuron Models

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 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, 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. 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. NOBLE is the first scaled-up deep learning framework that validates its generalization with real experimental data. To this end, 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.

1 Introduction

Hundreds of distinct neuronal cell types co-exist and compute within neural circuits, yet how they shape cognitive functions remains essentially unanswered [1-5]. This is 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].

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., [11-13]). Recently, evolutionary multiobjective optimization algorithms [14] have been used to generate and validate bio-realistic models of neurons in the form of 3D multi-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 to 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 electrophysiological features of parts of an experiment, the ensemble of deterministic HoF models is used as a 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 single-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 as 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 finite-dimensional vectors, which can limit their ability to model physical phenomena that are better described using functions in infinite-dimensional spaces and functional relationships between them [25]. As a result, neural networks can overfit to the training discretization and suffer from limited out-of-distribution capabilities. Neural operators [25-27] are a principled way to generalize neural networks to 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) [29, 30].

Machine learning approaches have been applied to model single-cell electrophysiology, primarily for point-neuron systems such as FitzHugh-Nagumo [31, 32] and Hodgkin-Huxley [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-Huxley systems [43], though the study was limited to simplified data, without capturing biological variability. Related works like NeuPRINT [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 neuroAI.

Contributions. We introduce NOBLE (Neural Operator with Biologically-informed Latent Embeddings), a neural operator framework for learning the nonlinear somatic dynamics across a 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 a 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 an 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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As an example application, we train and evaluate NOBLE on a parvalbumin-positive (PVALB) neuron dataset generated using 50 HoF PVALB models (Figure 2). We show that a single NOBLE model accurately captures both subthreshold and spiking dynamics across all 50 HoF models (in-distribution) as well as 10 unseen HoF models (out-of-distribution) while achieving a significant speedup of 4200 × 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 to 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 a larger set of models (Figure 5). The results showcase NOBLE '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 a 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 biologically-informed 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 a 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 electrophysiological feature of interest.

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2 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 a framework [45] built on the NEURON simulation environment [46], which uses a spatial discretization to simulate the models as a 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 a 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-current curve of each neuron. After 250 generations of evolutionary optimization, the models that best minimize the mean z-score error between simulated and actual experimental electrophysiological features are selected as HoF models (Figure 2B). More details about the electrophysiological features of interest are provided in Appendix B.

To illustrate the proposed approach, we consider a randomly selected PV ALB human cortical neuron, for which we created 60 HoF models. PVALB neurons are fast-spiking inhibitory interneurons regulating high-frequency gamma oscillations (30-80Hz) [48] and their dysfunction has been associated with cognitive impairments such as 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].

3 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 capturing 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-pass 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.1) reveals that 3× 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 515ms with a timestep of 0 . 02ms . For such signals, the subsampling reduces the sequence length from 25 , 750 to 8 , 583 , substantially decreasing the computational load without compromising biological realism.

3.2 Current Amplitude Sampling

Weconsider square DC step current inputs, which are widely used in electrophysiological experiments and common for characterizing neuron behavior. We sample the current amplitudes from a skewnormal distribution whose support matches the experimentally validated range of the HoF models, I ∈ [ -0 . 11 , 0 . 28]nA . 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 as 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 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 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-current (F-I) curve is a useful electrophysiological descriptor that summarizes cellular excitability by relating injected current amplitude to the neuron's firing rate [54]. Differences between HoF parameterizations manifest as shifts in key features of this curve: the threshold current I thr (the minimum amplitude that elicits spiking) and the local slope s thr at I thr (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 I thr and s thr can represent the trial-to-trial intrinsic variability observed when a single neuron is repeatedly recorded under the same current injection.

Using this observation, we propose representing a given neuron model by its threshold current I thr and local slope s thr , that is, using ( I thr , s thr ) . We propose to use this representation as part of a NeRF-style (Neural Radiance Field) embedding [55], where input features are encoded using sine and cosine functions. More precisely, a feature p is encoded as a stack of trigonometric time-series

for some integer K > 0 , where the frequencies are modulated by the feature p . Here t denotes the discretized time coordinates and ⊙ indicates element-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. NeRF-style encodings lead to a representation of the input features that aligns naturally with the spectral approach of FNOs, enhancing their ability to learn high-frequency dynamics. In this context, the sinusoidal embeddings can be thought of as a form of spectral lifting, translating low-dimensional inputs into a richer representation in the frequency domain that FNOs can more efficiently process.

We encode separately the model features I thr and s thr , and the amplitude of the current injection, and stack the resulting embeddings as input channels. To compress the large range of I thr and s thr values into a manageable scale for embedding, we normalize them to [0 . 5 , 3 . 5] 2 , supporting more distinct feature space representations of HoF models. Figure 10 displays the latent representations in normalized ( I thr , s thr ) -space of the 60 HoF PVALB models used in our numerical experiments.

3.5 NOBLE : Neural Operator with Biologically-informed Latent Embeddings

We introduce the Neural Operator with Biologically-informed Latent Embeddings ( NOBLE ), for modeling neuronal voltage dynamics in response to current injections. NOBLE offers a scalable alternative to computationally intensive numerical solvers for biophysically detailed, PDE-based neuron models. It learns a direct mapping from input currents and a continuous, interpretable latent space of neuron features, to the resulting voltage traces (Figure 1A). A key feature of NOBLE is its use of biologically-informed embeddings, which enables interpretability and generalization across biological neuron models. At its core, NOBLE is based on a neural operator, whose discretization invariance allows NOBLE to learn on low-resolution data and infer somatic voltage dynamics at 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.

The proposed NOBLE framework offers key advantages that set it apart from existing approaches:

NOBLE provides a unified framework that learns ensemble dynamics directly, enabling it to generate diverse, biophysically plausible membrane potentials for the same input. Conditioned on a particular electrophysiological 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 a 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, biorealistic 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-informed latent space of NOBLE , combined with its ability to generate unlimited realistic neuron models, enables fine-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).

4 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. In the PVALB setting, NOBLE receives as input the applied current injection I , together with stacked embeddings of I (with K = 9 different frequencies) and of the normalized model features ( I thr , s thr ) associated to a neuron model HoF ℓ (with K = 1 frequency). NOBLE then outputs a 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 train } , and the remaining 10 HoF models { HoF test } are used for testing. Figure 10 displays the latent representations in normalized ( I thr , s thr ) -space of these HoF models. For more details on the data generation, see Appendix D.1.

We use a 1D 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 . 8 Mtrainable parameters is trained in PyTorch for 300 epochs using the Adam optimizer with learning rate 0 . 004 , and the ReduceLROnPlateau scheduler with factor 0 . 4 and patience 4 . The training minimizes the relative L4 error, while the performance metrics are reported using the relative L2 error for interpretability.

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To better evaluate physiologically meaningful behavior, we also report errors on four key electrophysiological features: spikecount , AP1_width , mean_AP_amplitude , and steady_state_voltage (see Appendix B for definitions of the features). For benchmarking, we compare NOBLE predictions against numerical simulations obtained from HoF models and experimental data since the HoF models were optimized to produce the closest approximations to 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.

The PyTorch codes used for our implementation of NOBLE and the numerical experiments are based on the NeuralOperator library [56], and are made available at github.com/neuraloperator/noble.

4.2 Testing on HoF Models Included in the Training Set

We first validate that the trained NOBLE can accurately reproduce the somatic voltage responses of the training { HoF train } models when tested on current injections not seen during training. Figure 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 . 18% with the { HoF train } 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 AP1_width , 8 . 54% for mean_AP_amplitude , and 0 . 83% for steady_state_voltage . To further assess NOBLE '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).

The relatively higher error on AP1_width arises from how the feature is computed: it measures the width of the first spike at half amplitude, where the half level is defined between the spike peak and the subsequent after-hyperpolarization minimum. If the predicted peak or minimum is slightly misaligned relative to the ground truth, the half-voltage reference shifts, and the measured width corresponds to a 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 train } 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 . 1nA .

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4.3 Interpolating Between Models

We now test NOBLE 's ability to interpolate between HoF models within the convex hull CH train of the { HoF train } models used for training. Consider a PDE model HoF test k excluded during training.

We first randomly sample 50 unseen synthetic models from a local neighborhood in the model latent space near the defining ( I thr , s thr ) features of HoF test k (Figure 11 illustrates this neighborhood). Figure 4C shows that NOBLE accurately captures neuronal dynamics when interpolating within the latent space, as it produces a distribution of voltage responses consistent with those of the previously unseen HoF test k . 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-objective evolutionary algorithms lack a consistent structure that would enable meaningful interpolation to 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 test k deviate markedly from experimental data.

Interpolating within the latent embedding space enables NOBLE to efficiently generate novel neuron models at 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 is truly needed for robust generalization. To examine this, we varied the number of HoF models included in { HoF train } 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 { HoF train } models to produce 50 somatic responses. In Figure 5A (left, middle), we compare these 50 predictions with the corresponding 50 numerical solver simulations from { HoF train } . We see that the distribution of curves is very similar.

Since NOBLE enables interpolation between the HoF models used for training, it can generate novel bio-realistic neuron models and produce voltage responses for any neuron model whose latent space representation lies within the convex hull CH train of the training set { HoF train } . We demonstrate this by querying 200 novel models whose features are sampled randomly within CH train . Figure 5A (right) shows that the distribution of curves remains very similar, but the additional samples provide a 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 test } , where predicted and ground-truth responses again show close agreement (Figure 12). Second, we examine local perturbations in 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 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 electrophysiological features relative to experimental data. Here, we report the mean_frequency , representing the average firing rate, to account for the difference in stimulus activation durations between the experimental recordings and NOBLE , since spikecount is a 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. However, experimental recordings for this neuron are limited to a single trace across nine amplitudes. Consequently, our comparison is against a single realization rather than a distribution. In 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-Informed Latent Embeddings

Feature embeddings are central to NOBLE as they enable generalization across and in-between biological neuron models. Here, we have chosen I thr and s thr for their strong biological interpretability, as a natural 2D representation of firing and non-firing dynamics. To quantify the importance of 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 I thr and s thr . Results are reported in Table 1 in terms of the relative L2 error of predicted voltage traces and the four key electrophysiological features on the test set with { HoF train } models.

Without embeddings, NOBLE fails to predict any firing responses, indicating that embeddings are necessary to capture spiking behavior. With a single embedded feature, either s thr or I thr , NOBLE achieves similar accuracy on voltage traces and steady_state_voltage , but errors on spike-related features remain large, with I thr providing better accuracy for AP1_width . As discussed earlier, this feature is particularly sensitive to small misalignments in spike peak and after-hyperpolarization minima, which makes its relative error less robust as a metric. Embedding both s thr and I thr yields the best overall performance across all features as well as the voltage trace. Extending the embedding space with AHP_depth slightly improves AP1_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-Informed Fine-Tuning of NOBLE

To preserve overall neural dynamics while improving accuracy on a single specific feature, NOBLE can be fine-tuned with a weighted composite loss L ( λ ) = L data + λ L F , where L F penalizes deviations in feature F , and λ controls its influence. We illustrate this by fine-tuning NOBLE to improve sag_amplitude accuracy, a feature reflecting the hyperpolarization-activated cation channel (Ih). In the human cortex, Ih expression varies with cortical depth, making sag_amplitude a relevant physiological marker. We fine-tune on 19,600 subthreshold stimuli with negative amplitudes. Even without a feature-specific loss, fine-tuning reduces the L2 feature error from 70% to 19.2%, and incorporating the feature loss L F 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.

5 Conclusion

We introduced NOBLE , a neural operator framework for learning the nonlinear somatic dynamics across a population of HoF models for a 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 4200× speedup over traditional solvers, while maintaining accuracy across key electrophysiological features. Importantly, our work is among the first to 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.

Funding

A.A. is supported by the Bren Endowed Chair, ONR (MURI grant N00014-23-1-2654), and the AI2050 Senior Fellow program at Schmidt Sciences. C.A.A. is supported by the National Institutes of Health R01 - NS120300 and R01 - NS130126. P.H.W. is supported by the National Institutes of Health R01 - NS130126.

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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 HodgkinHuxley [33]. Fully connected and convolutional neural networks were trained to reproduce FitzHughNagumo dynamics [34], while ResNet-based multilayer perceptrons showed promise in learning Hodgkin-Huxley dynamics [35, 36]. Physics-informed 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-Nagumo dynamics [40] and to learn the HodgkinHuxley 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 FitzHugh-Nagumo 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 of electrophysiological behavior, particularly the highly nonlinear onset of firing, remains largely untested. Moreover, as function approximators, they necessitate retraining for each new input stimulus, significantly limiting their practical utility.

To address some of these limitations, Centofanti et al. [43] explored using operator learning approaches for forward simulations of the Hodgkin-Huxley 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 a limited scope: (1) it relies on relatively simple simulated data from a point-neuron model, (2) it 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 in 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 electrophysiological feature extraction and metrics, we use code from the Electrophys Feature Extraction Library (eFEL) available at

The formulas, codes, and more details about each electrophysiological feature can be found at

We list below 16 important electrophysiological features and metrics of interest when constructing neuron models (where AP denotes action potential and AHP denotes after-hyperpolarization):

AHP_depth : Relative voltage values at the first AHP

AHP_time_from_peak : Time between AP peaks and first AHP depth

AHP1_depth_from_peak : Voltage difference between the first AP peak and first AHP depth

AP1_peak : The peak voltage of the first AP

AP1_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

decay_time_constant_after_stim : The decay time constant of the voltage right after the stimulus

depol_block : Check for a depolarization block. Returns 1 if there is a depolarization block or a hyperpolarization block, and returns 0 otherwise.

inv_first_ISI : 1.0 over first interspike interval; returns 0 when no interspike interval

mean_AP_amplitude : The mean of all of the AP amplitudes

mean_frequency : The mean frequency of the firing rate

sag_amplitude : 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

steady_state_voltage : The average voltage after the stimulus

steady_state_voltage_stimend : The average voltage during the last 10% of the stimulus duration.

time_to_first_spike : Time from the start of the stimulus to the maximum of the first peak

voltage_base : 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 in Figures 6 and 7 for the low-pass filtering followed by decimation in 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, we identified the minimum relative error across all HoF models, and then aggregated these minima to compute the mean and standard deviation. These serve as a reference for the inherent worst-case discrepancy between simulations and experimental recordings in the absence of any subsampling. We visualize the mean as a solid black line and the standard deviation as dotted black lines.

Next, we repeated a similar analysis to quantify the additional relative errors introduced by subsampling. For each subsampling factor, we 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 electrophysiological 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 we remain within the bounds of the intrinsic simulation-experiment discrepancy, we adopt a conservative downsampling factor of 3 × , which maintains fidelity while reducing computational load.

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C.3 The Fourier Neural Operator Architecture

Neural operators [25-27] are a principled way to generalize neural networks to learn operators mapping functions to functions, with a universal operator approximation property [28]. Neural operators compose linear integral operators K with pointwise nonlinear activation functions σ to approximate highly nonlinear operators. More precisely, we define the neural operator

where P , Q are the pointwise neural networks that encode the lower dimension function into a higher-dimensional space and vice versa. The model stacks L layers of σ ( W l + K l + b l ) where W l are pointwise linear operators (matrices), K l are integral kernel operators, b l are bias terms, and σ are fixed activation functions. The parameters θ consists of all the parameters in P , Q , W l , K l and b l . Kossaifi et al. [56] maintain a 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].

Avariety of neural operators have been proposed, such as the Fourier Neural Operator (FNO) [29, 30], and successfully applied to a wide range of problems [61-63]. A FNO is a neural operator using Fourier integral operator layers, which are defined via

where R ϕ is the Fourier transform of a periodic function κ parameterized by ϕ . On a uniform mesh, the Fourier transform F can be implemented using the fast Fourier transform (FFT).

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D Experiments

D.1 Dataset

Wehave access to 60 HoF models { HoF } obtained using a multi-objective evolutionary optimization strategy. We use 50 of them during training, { HoF train } , and keep the remaining 10 { HoF test } for testing.

Figure 10 displays the latent representations in normalized ( I thr , s thr ) -space of these HoF models.

The training dataset is composed of 75,600 samples, where the current injections are sampled as described in Section 3.2, each of which is associated randomly to one of { HoF train } . The samples are generated using a numerical solver [45] built on the NEURON simulation environment [46].

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D.2 Evaluation and Evaluation Metrics

NOBLE is trained using the relative L4 error, computed via

The relative L4 error was selected as the training loss after a preliminary training study on a 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 electrophysiological details most relevant to our setting, NOBLE is compatible with any other loss function that may be more suitable in different contexts.

Although NOBLE is trained to minimize the relative L4 error, we report results using the relative L2 error, as it provides a more common and interpretable measure of accuracy. We first evaluate performance on voltage traces. Note that even small temporal shifts between predicted and groundtruth voltage responses can result in 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_width : width of the first spike at half amplitude

mean_AP_amplitude : mean amplitude of all action potentials

steady_state_voltage : 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 as the ones discussed in the introduction, are not 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-negligible error, driving prediction error below the solver-experiment gap risks overfitting to the solver rather than improving alignment with real recordings. Therefore, we tuned hyperparameters only up to the solver-experiment error level, as marginal gains on PDE data are unlikely to yield meaningful improvements relative to experimental recordings.

D.3 Testing on VIP Neuron HoF Models Included in the Training Set

To assess NOBLE 's ability to perform well on different neuron models, we trained it on a VIP neuron using the same architecture as 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 ReduceLROnPlateau scheduler with factor 0.8 and patience 4, minimizing the relative L4 error. For the embeddings, the neural operator in NOBLE takes the stacked embeddings of the normalized model features I thr and s thr associated with HoF ℓ (with K = 1 frequency) and I (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 AP1_width , 10% for mean_AP_amplitude , and 0.99% for steady_state_voltage .

These results are 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

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D.6 On the HoF Models Availability Requirement

Training NOBLE on 75,600 samples required approximately 4 days on a 64GB NVIDIA Tesla P100 GPU (300 epochs). Once trained, for a given input current, NOBLE can synthesize arbitrarily many voltage responses of synthetic HoF models almost instantaneously by interpolating within CH train (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 train } 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 { HoF test } models.

The relative L2 error on voltage traces is largely insensitive to HoF diversity, whereas spike-related features, particularly spikecount and AP1_width , degrade substantially as diversity decreases. As discussed earlier, AP1_width is particularly sensitive to small misalignments in spike peak and afterhyperpolarization minima, which can shift the half-voltage reference and lead to large relative errors even when the underlying traces are close. This makes AP1_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 solver-experiment error gap, the number of HoF models required for training can be adjusted accordingly. Moreover, for any given electrophysiological 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-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-truth signal and ˆ F the corresponding feature computed from NOBLE 's output. A feature-specific loss can then be defined as L F = ∥ F -ˆ F ∥ , which directly penalizes deviations in the feature of interest.

To illustrate this, we fine-tune the previously trained NOBLE to enhance sag_amplitude predictive performance. This feature is particularly relevant as it reflects the presence of the hyperpolarizationactivated cation channel (Ih). In human cortex, the expression of Ih varies with cortical depth, making sag_amplitude 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 a pretrained NOBLE model and further optimize the weights of the neural operator using the feature-specific loss. Relying solely on a feature-specific loss, such as L sag, risks causing NOBLE to overfit to sag_amplitude , 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 L sag 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 electrophysiological features.

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_amplitude :

To evaluate this approach, we construct a smaller dataset of 19,600 stimulus waveforms with negative non-zero amplitudes, since this regime elicits non-firing responses for which the sag_amplitude can be reliably computed. We then fine-tune the pretrained NOBLE model by minimizing the relative L4 error for L data and the relative L2 error for L sag, using the Adam optimizer with learning rate of 0 . 0005 , and the ReduceLROnPlateau scheduler with factor 0 . 4 and patience 12 . The results, reported in Table 3, summarize the relative L2 error of predicted voltage traces and sag_amplitude on the test set of { HoF train } .

Even without an additional sag_amplitude 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 of approximately 10% on sag_amplitude , while maintaining overall signal fidelity. Note that both settings converged after 300 epochs, ensuring that the comparison is fair.

These results demonstrate that prioritizing a 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 to generate 10,000 predictions on a workstation equipped with a single NVIDIA RTX 4090 GPU (24GB VRAM), an AMD Ryzen 9 7900X CPU, and 64GB of system RAM. The numerical solver only generates a single prediction at a time and takes roughly 36,200 seconds to generate 10,000 predictions. When doing one inference at a time with NOBLE (i.e. batch size of 1), we generate 10,000 predictions in 157 seconds, i.e. a speedup of approximately 230× compared to the solver.

In addition, NOBLE can easily be accelerated on a single GPU by generating multiple predictions at the same time. In particular, with a batch size of 1000, NOBLE generates 10,000 predictions in 8.59 seconds, i.e. a speedup of approximately 4200× compared to the solver.

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-neuron configurations with time-varying stimuli, enabling applications such as neuron classification and predicting multi-neuron dynamics. The embedding space used in our experiments is low-dimensional, constructed from two interpretable features derived from a biological neuron model's F-I curve. Extending this to a learnable, higherdimensional continuous embedding space represents a promising direction for future work. Beyond this, NOBLE could also integrate additional modalities such as gene expression or morphology to build a unified latent space linking molecular, electrophysiological, and structural characteristics, as discussed in the main text.

Although NOBLE demonstrates strong performance in modeling nonlinear neuronal dynamics, our study makes a 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 as these are widely used in electrophysiological experiments and represent a 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 can be removed or adapted (e.g., embedding a 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 approximately four days on a 64GB NVIDIA Tesla P100 GPU, which is small compared to the ∼ 600,000 CPU hours required to generate the HoF models via evolutionary optimization. However, once trained, NOBLE enables fast inference and the instantaneous generation of infinitely many biorealistic voltage traces through latent space interpolation, capabilities not possible with the 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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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\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.\n\nQUESTIONS:\n- 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)\n\nLIMITATIONS:\n- The limitation to generate voltage traces from a step function is not sufficiently addressed. PDE based approaches would not have this limitation.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-04"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\nThe 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\n\nQUESTIONS:\nThe 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.\n\nLIMITATIONS:\nThe 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.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 1, "DATE": "2025-06-05"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS AND WEAKNESSES:\n## 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.\n\nQUESTIONS:\n- 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-\n\nLIMITATIONS:\nI 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?", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-25"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\n_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.\n\nQUESTIONS:\n1) 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?\n\nLIMITATIONS:\nLimitations 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.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\n**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.\n\nQUESTIONS:\n1. 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)?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-06-30"}