Title: Identifying multi-compartment Hodgkin-Huxley models with high-density extracellular voltage recordings
Venue: NeurIPS 2025
Accepted: yes
Keywords: Computational neuroscience

State-space models

Extended Kalman Filter

biophysical modeling
Abstract: Multi-compartment Hodgkin-Huxley models are biophysical models of how electrical signals propagate throughout a neuron, and they form the basis of our knowledge of neural computation at the cellular level. However, these models have many free parameters that must be estimated for each cell, and existing fitting methods rely on intracellular voltage measurements that are highly challenging to obtain in-vivo. Recent advances in neural recording technology with high-density probes and arrays enable dense sampling of extracellular voltage from many sites surrounding a neuron, allowing indirect measurement of many compartments of a cell simultaneously. Here, we propose a method for inferring the underlying membrane voltage, biophysical parameters, and the neuron's position relative to the probe, using extracellular measurements alone. We use an Extended Kalman Filter to infer membrane voltage and channel states using efficient, differentiable simulators. Then, we learn the model parameters by maximizing the marginal likelihood using gradient-based methods. 
  We demonstrate the performance of this approach using simulated data and real neuron morphologies.

Identifying multi-compartment Hodgkin-Huxley models with high-density extracellular voltage recordings

Abstract

Multi-compartment Hodgkin-Huxley models are biophysical models of how electrical signals propagate throughout a neuron, and they form the basis of our knowledge of neural computation at the cellular level. However, these models have many free parameters that must be estimated for each cell, and existing fitting methods rely on intracellular voltage measurements that are highly challenging to obtain in vivo . Recent advances in neural recording technology with high-density probes and arrays enable dense sampling of extracellular voltage from many sites surrounding a neuron, allowing indirect measurement of many compartments of a cell simultaneously. Here, we propose a method for inferring the underlying membrane voltage, biophysical parameters, and the neuron's position relative to the probe, using extracellular measurements alone. We use an Extended Kalman Filter to infer membrane voltage and channel states using efficient, differentiable simulators. Then, we learn the model parameters by maximizing the marginal likelihood using gradient-based methods. We demonstrate the performance of this approach using simulated data and real neuron morphologies.

1 Introduction

Biological neurons are equipped with a wide range of biophysical mechanisms that contribute to information processing [1-3] and to the brain's robustness to environmental perturbations [4, 5]. Multi-compartment Hodgkin-Huxley models provide a principled framework for studying biophysical mechanisms of neurons [6, 7]. Multi-compartment models treat neurons as connected electrical compartments, allowing simulation of how voltage signals propagate throughout a cell as ion channels open and close. A central limitation for building multi-compartment models-and for understanding biophysical mechanisms in neurons-is the lack of experimental techniques to precisely measure the biophysical parameters of these models. As such, these parameters must be inferred from recordings of the activity of neurons [8-11]. Typically, these recordings are made from within a neuron using intracellular measurements such as patch-clamp electrophysiology, which enables precise recordings of voltage dynamics in the soma [12, 13] and, for large neurons, in dendrites [14]. However, intracellular methods are technically challenging, and only provide voltage signals from a very small number of locations within a neuron (typically, only from the soma) [15].

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Recent advances in neural recording technologies have opened new possibilities for studying neuron biophysics in vivo . High-density silicon probes such as the Neuropixel Ultra [16] can record extracellular voltage traces across hundreds of closely spaced electrodes. These signals, which reflect the transmembrane currents generated by nearby neurons, are traditionally used for spike detection and sorting-identifying when and where neurons fire spikes [17, 18].

While effective for tracking population activity, this approach treats spikes as discrete events and ignores the exquisite biophysical details of neurons [19].

In this work, we revisit the extracellular signal, not just as a source of discrete spike times, but as a rich, spatially structured observation from which we can infer biophysical properties of individual neurons. This is made possible by three key developments: (1) high-density probes and 2D arrays that allow us to sample extracellular voltage at many sites around a cell to triangulate its position and morphology; (2) differentiable simulators like JAXLEY [20] that enable fast, hardware-accelerated simulation of complex biophysical models and gradient-based parameter optimization; and (3) scalable state-space inference frameworks such as DYNAMAX [21], which allow efficient state inference and parameter estimation in high-dimensional, nonlinear dynamical systems.

We leverage these developments to provide a statistical approach for inferring the parameters of multi-compartment biophysical models from extracellular observations: We extend classical multicompartment Hodgkin-Huxley models to include a biophysical model of extracellular potentials. We then use an Extended Kalman Filter (EKF) [22] to infer latent neuronal states-such as membrane voltages and gating variables-from observed extracellular signals and we improve the computational efficiency with diagonal and block-diagonal approximations to the filtering covariance. To estimate model parameters, including ion channel densities and the neuron's spatial position relative to the probe, we maximize the marginal likelihood using gradient-based optimization.

We first demonstrate that our method can successfully infer membrane properties and cell locations in simplified neuron models, and that the use of an EKF largely improves the method's robustness to model misspecification. We then apply our method to synthetic extracellular data generated from real neuron morphologies and demonstrate that the method can scale to morphologies with more than a hundred compartments. Overall, our results suggest that the combination of differentiable simulation and state-space modeling makes it possible to estimate biophysical properties across the entire morphology of a neuron based only on extracellular voltage recordings.

2 Biophysical Modeling of Neurons

2.1 Modeling Action Potentials with Hodgkin-Huxley Dynamics

The multi-compartment Hodgkin-Huxley model (HH) describes how ion channels shape the membrane potential across branches of a neuron. Each compartment represents a small component of the cell with its own voltage, capacitance, and set of ion channels. Let V ( i ) denote the membrane

potential of compartment i , with surface area S ( i ) and specific membrane capacitance C m . Each compartment has a set of ion channels, C , and each channel c ∈ C is characterized by a maximum conductance g ( i ) c , reversal potential E c , and a time-varying state λ ( i ) c . A nonlinear activation function, a c ( λ ( i ) c ) ∈ [0 , 1] , specifies the fraction of the maximum current that can flow through the channel, as a function of its state. The voltage dynamics for compartment i are given by,

where I ( i ) ext is the external current applied to compartment i , and g ( i,j ) is the axial conductance coupling compartment i to its neighbor j . The total transmembrane current in compartment i is denoted by I ( i ) m and is the sum of the ionic current I ( i ) ion = S ( i ) ∑ c ∈C g ( i ) c a c ( λ ( i ) c )( V ( i ) -E c ) and the capacitive current I ( i ) cap = S ( i ) C m d V ( i ) d t [23]. For instance, in the original single compartment HH model with sodium, potassium, and leak channels, the total current is,

Here, for example, the states of the sodium channel are λ Na = ( m,h ) ∈ [0 , 1] 2 and the activation function is a Na ( λ Na ) = m 3 h . The gating variables evolve according to first-order kinetics d λ c d t = α c ( V )(1 -λ c ) -β c ( V ) λ c , where α c ( V ) and β c ( V ) are voltage-dependent rate functions that control the opening and closing transitions of the ion channel. Specific forms for the α and β functions are available in Dayan and Abbott [24], e.g. Importantly, note that by eq. (1), the ionic current and the capacitive current cancel out to give the following expression of the transmembrane current:

2.2 Modeling Extracellular Voltage Measurements

Extracellular voltages reflect the collective activity of nearby neurons. While the physics of extracellular recordings are well understood [25, 26], our work is, to our knowledge, the first to leverage this framework for fitting multi-compartment HH models to extracellular measurements. Let Φ denote the extracellular potential measured by a fixed site on a recording probe. We treat each compartment as a point source of transmembrane current located at its center and denote r i the distance from the recording site to compartment i of a neuron. It follows from Coulomb's law that the measured extracellular voltage is,

In words, the observed potential Φ is proportional to the sum of transmembrane currents from nearby compartments, weighted by the inverse of the distance to the compartment. Importantly, it follows from eq. (3) and (4) that the extracellular voltages at time t are simply linear functions of the membrane voltages at time t .

3 State Inference and Parameter Estimation

We aim to infer the membrane potential and channel states and to estimate the biophysical and structural parameters of a neuron from noisy extracellular voltage traces. We can achieve both goals by casting the multi-compartment HH model as a state space model (SSM) with nonlinear dynamics. However, the nature and scale of these models present unique challenges, which we address below.

3.1 State Space Model Formulation

A state space model (SSM) is a latent variable model with latent states z 1: T , observations y 1: T , and parameters θ . In a Markovian SSM, latent states evolve according to a dynamics distribution,

p ( z t | z t -1 ; θ ) , and they give rise to observations via an emission distribution p ( y t | z t ; θ ) . There are a host of well-established techniques for state inference and parameter estimation in such models [22, 21], depending on the nature of the dynamics and emission distributions.

In our case, the model parameters, θ , may include the maximum conductances of various ion channels as well as parameters governing the neuron's spatial configuration relative to the recording probe. The latent states consist of the voltages and channel states for each compartment, z t = [ V (1) , . . . , V ( N ) , { λ (1) c } c ∈C 1 , . . . , { λ ( N ) c } c ∈C N ] . To model the state dynamics, we use an Euler discretization of the multi-compartment HH differential equations (see eq. (1)),

where f θ ( z t , t ) is the deterministic update function derived from the ODEs above, and Σ dyn is a diagonal covariance matrix with entries from { σ 2 v , σ 2 state } . This Gaussian noise is intended to account for model uncertainty, such as missing ion channels or misspecified channel dynamics, which we will consider in the results section.

We assume that the only source of stochasticity in the emission process is measurement noise. Let y t ∈ R M denote the observed extracellular potential at time t across M recording sites. It is modeled as,

where h θ ( z t ) is the predicted extracellular voltage from eq. (4) and σ 2 obs is the observation noise variance. Since h θ ( z t ) only depends on z t , the emission model is conditionally independent of previous states given the current latent state.

3.2 Scaling the Extended Kalman Filter to Large Biophysical Models

Since the dynamics are nonlinear, there is no closed-form solution for the filtering distribution p θ ( z 1: t | y 1: t ) . To address this challenge, we use the extended Kalman filter (EKF), which linearizes the dynamics and emission functions via first-order Taylor expansions [22]. This linear approximation enables recursive estimation of the latent states, and it also provides an estimate of the marginal likelihood p θ ( y 1: T ) for parameter estimation.

Biologically realistic neuron models often contain dozens or even hundreds of compartments, leading to latent state dimensions in the hundreds or thousands. The standard EKF maintains an estimate of the full state covariance matrix, and updating the filtering distribution requires solving a linear system with this matrix. Since this computation scales cubically with the number of states, it becomes computationally infeasible to apply standard EKF formulations to full-scale neuronal models. To overcome this challenge, we develop two scalable EKF variants for biophysical neuron model fitting.

Diagonal approximation Our first scalable variant adopts a diagonal approximation of the posterior covariance, inspired by Chang et al. [27], while exploiting the sparsity inherent in multi-compartment HH dynamics. Specifically, we approximate the EKF filtering distribution p t ( z ) = N ( z ; µ t , Σ t | t ) with a diagonal Gaussian q t ( z ) = N ( z ; ˜ µ t , ˜ Σ t | t ) where ˜ Σ t | t and ˜ µ t are chosen to minimize the divergence KL( q t ∥ p t ) . As shown by Chang et al. [27], this minimization admits a closed-form solution: the optimal diagonal Gaussian approximation q t matches the marginal means and precisions (inverse variances) of p t . This leads to efficient recursive updates that preserve the EKF structure while requiring only diagonal storage-reducing memory cost from O ( d 2 ) to O ( d ) for a d -dimensional state.

In practice, this approximation requires evaluating only the diagonal elements of two matrices: (i) H ⊤ t R -1 t H t , where H t = Jac( h θ )( µ t | t -1 ) is the Jacobian of the emission model and R t = σ 2 obs I is the observation noise covariance; and (ii) the predicted precision Σ -1 t | t -1 , with

where F t = Jac( f θ )( µ t | t -1 ) is the Jacobian of the dynamics and Q t = ∆ t Σ dyn is the dynamics covariance (cf. Appendix B ).

The diagonal of H ⊤ t R -1 t H t can be computed in linear time with respect to the number of states [27]. Furthermore, because the measurement model maps the membrane voltages to extracellular voltages

through a fixed linear operator (eq. (3) and (4)), H ⊤ t R -1 t H t is time-invariant, allowing its diagonal to be computed once and reused throughout filtering. The remaining challenge lies in efficiently computing diag( Σ -1 t | t -1 ) . To simplify calculations, we make the additional assumption that Σ t | t -1 is diagonal, reducing the problem to computing diag( Σ t | t -1 ) . Here, we exploit the sparsity pattern of F t , which follows directly from the local coupling structure of the HH equations. While voltage updates may depend on all other states due to the implicit Euler solver used in JAXLEY [20], gating variables typically depend only on their own previous value and the voltage of their corresponding compartment. This sparsity enables efficient computation of diag( Σ -1 t | t -1 ) . In practice, we compute F t using the SPARSEJAC library [28], which constructs Jacobians directly from sparsity patterns without instantiating the full dense matrix.

When is the diagonal approximation reasonable? We assume that the initial filtered covariance matrix Σ 0 is diagonal. The EKF covariance update is:

The procedure we described is equivalent to performing this update step with both Σ t | t -1 and H ⊤ t R -1 t H t diagonal. Is this valid?

In Appendix B, we show that for well-behaved HH kinetics-smooth transition rates, small integration steps, and not too frequent spiking-the dynamics Jacobian exhibits limited cross-coupling for morphologies of moderate complexity. Under these conditions, a diagonal prior covariance and dynamics covariance can yield an approximately diagonal predicted covariance Σ t | t -1 .

The tighter constraint arises from the emission model. The time-invariant matrix H ⊤ t R -1 t H t reflects how strongly electrode signals overlap across compartments. When a neuron is discretized into many compartments and the electrodes are densely packed around it-as in Neuropixels or similar high-density probes-its off-diagonal entries can become large, and approximating it by a diagonal can severely degrade performance (cf. Appendix B for a complete derivation). Thus, the diagonal EKF appears best suited for electrode configurations in which each sensor predominantly captures activity from only a few nearby compartments.

Block-diagonal approximation The diagonal EKF procedure described above assumes H ⊤ t R -1 t H t to be diagonal. When this assumption fails, the true update step no longer preserves diagonality even for well-behaved HH models; enforcing it by truncation discards key aspects of the geometry of the problem and harms accuracy.

To avoid this loss, we introduce a more structured approximation that remains consistent under EKF updates. Since the emission model depends only on voltages, H ⊤ t R -1 t H t is block-diagonal, with a single nonzero block corresponding to the voltage dimensions. This motivates a block-diagonal covariance structure, where the EKF predicted and filtered covariances consist of a dense voltage block and diagonal blocks for all remaining (gating) states. Concretely, let the first K dimensions of the latent state vector correspond to compartment voltages and the remaining ones to gating variables. At each time step t , we assume that both the predicted and filtered covariance matrices take the form

where Σ vv t | ⋆ ∈ R K × K denotes the dense voltage-voltage covariance block, and σ 2 t | ⋆ collects the gating variances. We refer to this family of matrices as the BD( K ) family (block-diagonal with K compartments).

We can prove that if the dynamics Jacobian F t is BD( K ), then the BD( K ) family is closed under the EKF prediction and update: if the initial covariance matrix Σ 0 is BD( K ), then subsequent predicted and filtered covariances remain BD( K ) (cf. Proposition 1). For the same reasons as in the diagonal case, the BD( K ) approximation of covariance matrices can be effective for well-behaved HH models.

Computationally, the EKF update step with BD( K ) covariances reduces to (i) a K × K solve for the voltage block-since the measurement Jacobian H t has support only on voltages-and (ii) elementwise variance updates for the diagonal gating blocks (cf. proof of Proposition 1). The prediction step similarly preserves structure by propagating Σ vv t -1 | t -1 through the voltage-voltage sub-Jacobian and

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updating gating variances via their local Jacobians, yielding overall complexity O ( K 3 + d ) instead of O ( d 3 ) for a d -dimensional state (cf. proof of Proposition 1).

4 Validation on simple morphologies

4.1 Validation on single-branch neuron model

We first validated our method on a simple synthetic neuron: a 10-compartment branch with sodium and potassium HH channels (Appendix E.1). We initially assumed that the biophysical model was correctly specified and that the dynamics were fully deterministic. We stimulated the cell with constant current and recorded extracellular voltages from a single electrode placed 5 µ m from the soma. The goal was to infer maximum conductances assuming that the cell's position was known. We simulated 40 recordings corrupted by Gaussian noise ( σ = 1 µ V ) and initialized conductances within biologically plausible ranges. EKF-based inference accurately recovered the true parameters across random seeds, with negligible bias and variance (fig. 2A), supporting the validity of our approach.

Next, we assessed the effect of observation noise. Neuropixels Ultra probes exhibit RMS noise levels around 7 . 4 µ V in saline and approximately 9 . 4 µ V in vivo [16]. Moreover, in practice we expect voltage traces to be corrupted by currents from other nearby neurons, which our model treats as noise as well. Fixing the cell-probe distance at 5 µ m , we simulated 40 noisy EAPs at three increasing noise levels, ranging from 1 µV to 30 µV . As expected, we found that higher noise leads to higher variance in the parameter estimates (fig. 2B). Nevertheless, estimates remained very close to the ground truth even at 30 µ V RMS, which represents a challenging, high-noise regime (fig. E.3).

We also examined the effect of increasing the distance between the cell and the recording site. Since extracellular voltage scales proportionally with 1 /r , the signal-to-noise ratio declined with distance.

In line with this intuition, we found that larger distance between cell and recording site leads to higher variance in the estimate of membrane conductances (fig. E.1B).

Finally, we compared multiple inference approaches-standard EKF, diagonal EKF, block-diagonal EKF, and direct mean-squared-error (MSE) minimization as used in the original JAXLEY paper [20]. The MSE baseline replaces the probabilistic objective with deterministic dynamics and directly minimizes the squared error between observed and predicted extracellular potentials. Using the same setup ( 5 µ m distance, 1 µ V observation noise) and fitting the same 40 random traces, we found that the standard EKF, block-diagonal EKF, and MSE approaches achieved nearly identical estimates (fig. 2C). This confirms that the local linearization in the EKF introduces negligible bias and that the block-diagonal approximation provides an excellent surrogate for the full covariance in HH models. The diagonal EKF exhibited slightly higher bias but its voltage reconstructions remain highly accurate (fig. E.2). This is consistent with our analysis in Appendix B: only the first compartment lies near the electrode, and it has a single neighboring compartment, leading to minimal cross-visibility and making a diagonal covariance approximation efficient.

4.2 Validation on a multi-branch neuron model

We next evaluated our method on a more realistic multi-branch setting, assuming a correctly specified biophysical model and fully deterministic dynamics. Using JAXLEY, we simulated a synthetic neuron with three branches, each divided into four compartments (twelve in total), containing standard HH ion channels. The maximum conductances varied independently across branches, yielding nine biophysical parameters to infer. Details of the setup are provided in Appendix E.2. Unlike the single-branch case, we did not assume full knowledge of the cell's spatial configuration: while branch number, connectivity, lengths, and radii were known, the 3D coordinates of branch endpoints were treated as unknown, introducing 18 additional geometric parameters for a total of 27.

A central question is whether these spatial locations are identifiable from extracellular recordings. With electrodes confined to a single plane (e.g., a Neuropixel probe), mirror reflections of the neuron across the plane produce identical extracellular voltages (cf. Appendix D). To account for this unidentifiability, we simulated recordings from two separate probe positions, ensuring that the sets of recording sites did not lie in the same plane (fig. 2D, red and purple). We also ensured that the sites span the entire neuron to capture signal from all branches.

We fixed the observation noise to 1 µ V and generated 40 sets of extracellular voltage recordings. To initialize parameters, we optimized from multiple random starting points on one dataset and retained the best solution (highest marginal log-likelihood), which we then reused across all runs. In this setting, the EKF consistently recovered the 3D compartment locations (fig. 2D, blue vs black), conductances (fig. 2E), and membrane voltages (fig. 2F), demonstrating the robustness and accuracy of biophysical and geometric inference in multi-compartment neuron models with unknown locations and membrane conductances. The diagonal EKF exhibited somewhat larger bias in the recovered conductances compared with the single-branch case (fig. E.6), but the voltage traces remained accurate up to small shifts (fig. 2F), and the compartment positions were inferred correctly (fig. E.8). In this multi-branch geometry, compartments are long enough that most electrodes are close to only a single compartment, and each compartment has very few neighbors, which together limit cross-compartment measurement interactions and explains why the diagonal approximation still yields informative reconstructions (cf. Appendix B).

4.3 Robustness to model misspecification

In experimental settings, the true underlying biophysical dynamics of a neuron are rarely known with certainty. To evaluate the robustness of our approach under such model misspecification, we conducted experiments where the ground truth and inference models differ in their ion channel composition and dynamics. The detailed experimental setup and additional figures can be found in Appendix E.3.

Omitted Ion Channels In our first experiment, we generated synthetic data from a single-branch cell containing both classical HH channels and two additional conductances: an M-type potassium current and a calcium current. During inference, we deliberately used a simplified model containing only the HH channels, omitting the additional ones (fig. 3A).

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Since we allow for dynamics noise in our model with process noise, we were able to accurately recover the underlying membrane voltage, despite this substantial misspecification. In contrast, naive fitting via mean squared error (MSE) minimization failed to capture the dynamics, yielding voltage estimates that diverged substantially from the ground truth. This result highlights the importance of allowing for uncertainty in the state evolution, using our EKF-based approach.

Perturbed Gating Dynamics We next evaluated the effect of incorrect gating kinetics. Here, we retained the correct channel types in the inference model but perturbed the rate constants α ( V ) and β ( V ) used in gating variable dynamics, simulating errors in channel parameter specification (fig. 3B). Once again, EKF-based inference provided significantly more accurate estimates of both membrane voltage and channel states compared to MSE minimization. These results suggest that our approach can tolerate moderate misspecification in gating dynamics. Together, these experiments demonstrate that EKF-based inference, combined with stochastic modeling of dynamics, confers robustness to common forms of model misspecification.

5 Application to real morphologies

Having established the method's efficiency in controlled settings, we next apply it to morphologically detailed neuron models derived from real reconstructions, including a retinal ganglion cell (RGC) and a CA1 pyramidal neuron.

5.1 Application to 2D retinal ganglion cell

We applied our method to a morphologically detailed model of a retinal ganglion cell (RGC), replicating conditions typical of in vitro studies using planar multi-electrode arrays (MEAs) [29]. In these setups, retinal tissue is placed ganglion-side down on the MEA, enabling simultaneous current injection and extracellular voltage recording from hundreds of electrodes (fig. 4A). We adopted a reconstructed RGC morphology from Vilkhu et al. [29], consisting of a soma and a complex dendritic arbor discretized into 122 compartments. The membrane model includes modified HH sodium and potassium channels with non-standard kinetics. We assumed shared maximum conductances across all dendritic compartments, yielding a total of six parameters to estimate [29]. For details of the model, see Appendix F.

Here, we assumed the spatial location of the neuron to be known and constrained to a 2D plane, as is typical in retinal MEA experiments [30, 31, 29]. Extracellular voltages were recorded using a simulated MEA with 30 µ m pitch, matching the geometry of the original 512-electrode array. To reduce computation, we used a subset of 10 electrodes spanning the entire neuron.

Given the high dimensionality of the latent state space ( d = 610 ), we used our scalable EKF variants for efficient inference. Notably, the diagonal EKF performed poorly: assuming diagonal covariances, the marginal log-likelihood diverged from its value at the true parameters, indicating severe model

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mismatch (fig. 4C). The reconstructed voltage traces were also inaccurate, failing to reproduce the true dynamics (fig. 4F). This is precisely the failure mode predicted in Section 3.2: since each electrode records signal from many close compartments, H ⊤ H cannot be diagonal (fig. 4D). In contrast, the block-diagonal EKF converged to the true marginal likelihood (fig. 4E) and accurately reconstructed both the membrane voltage traces (fig. 4F) and the underlying conductances (fig. F.1).

5.2 Application to CA1 hippocampal pyramidal cell

Finally, we applied our approach to a morphologically detailed model of a rat CA1 hippocampal pyramidal neuron, based on the C10 model introduced by Cutsuridis et al. [32]. This model has been used in prior computational studies to examine hippocampal circuit function and memory encoding, capturing the integration and propagation of signals across dendritic and axonal compartments [32, 33]. We recreated the full morphology of the C10 cell, including the soma, axon, basal dendrites, and apical dendrites (fig. 5A). The cell contained region-specific HH-type channels with subcellularly varying conductances and gating kinetics [32, 34, 35], yielding 24 maximum conductances to infer. Extracellular signals were recorded with a simulated Neuropixel Ultra probe replicating the real probe's geometry. To account for unknown spatial position, we added six rigid-body parameters (rotation and translation), for a total of 30 parameters. Further experimental details and figures are provided in Appendix G. Following Carnevale and Hines [7], the cell was discretized into 60 compartments, resulting in a latent state of dimension 360 . We ran the block-diagonal EKF with over 100 restarts and retained the solution with the highest final marginal log-likelihood (fig. 5B).

Our method successfully recovers the neuron's spatial configuration (fig. 5A, black vs. blue), and accurately reconstructs membrane voltage (fig. 5D). It also recovers most of the maximum conductances, though not all (fig. 5C). This outcome is consistent with a well-documented phenomenon in conductance-based neuron models: parameter degeneracy, where multiple combinations of channel conductances can yield virtually indistinguishable voltage dynamics [36, 37]. Despite this, our method reliably recovers the neuron's functional dynamics and spatial structure, demonstrating that biophysical and geometric inference from high-density extracellular recordings is feasible in realistic, large-scale neuron models.

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6 Conclusions, Limitations, and Future Work

Summary We presented an approach for identifying detailed biophysical neuron models from high-density extracellular voltage recordings. Our method recovers not only membrane conductances but also the spatial position of a neuron relative to the recording probe, solely from extracellular data. We further developed a state-inference and parameter-estimation framework based on an Extended Kalman Filter (EKF), which provides robustness to model misspecification and accurate inference even when the assumed dynamics deviate from the true underlying system.

Limitations and Future Work A limitation of the EKF for parameter estimation is its computational cost: even with diagonal or block-diagonal approximations, it remains more expensive than MSE minimization, particularly for large models. Additionally, EKF approximations may bias parameter recovery: while the block-diagonal EKF recovered parameters accurately in our evaluations, the diagonal approximation was less reliable in certain cell-probe geometries. Moreover, biophysical models often exhibit parameter degeneracy, where different conductance combinations yield similar voltage dynamics, limiting identifiability even when predictions fit the data well.

Finally, while our results demonstrate the feasibility of biophysical inference from extracellular recordings at the single-cell level in idealized settings, extending this framework to real in vivo conditions remains a significant challenge. Real recordings involve complex networks of interacting neurons. A natural next step is to move beyond isolated cells and model local neural populations surrounding the probe. Given knowledge of the targeted brain region, one could incorporate priors on network structure and cell types informed by anatomical and physiological studies, enabling more comprehensive modeling of in vivo extracellular signals.

Conclusion Overall, our method-based on a combination of state-space modeling and differentiable simulation-opens up possibilities to estimate properties across the entire morphologies of neurons from extracellular signals alone. This approach will enable new analysis of how cells generate action potentials and open new avenues for studying the biophysical contributions to neural computation.

Acknowledgments

We thank Alisa Levin, Noah Cowan, and other members of the Linderman Lab for helpful feedback throughout this project. We are also grateful to Amrith Lotlikar, Michael Sommeling, and E.J. Chichilnisky for sharing valuable insights into retinal ganglion cell physiology and model fitting approaches. This work was supported by grants from the NSF EFRI BRAID Program (2223827), the NIH BRAIN Initiative (U19NS113201, R01NS131987, R01NS113119, & RF1MH133778), the NSF/NIH CRCNS Program (R01NS130789), and the Sloan, Simons, and McKnight Foundations. It was also supported by the German Research Foundation (DFG) through Germany's Excellence Strategy (EXC 2064 - Project number 390727645) and the CRC 1233 'Robust Vision', and the European Union (ERC, 'DeepCoMechTome', ref.101089288).

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A Resources

All simulations and inference experiments were conducted using four open-source Python libraries:

JAX (version 0.4.31, Apache License 2.0): A high-performance numerical computing library based on XLA, providing composable function transformations and GPU/TPU support. Available at: https://github.com/google/jax .

JAXLEY [20] (version 0.7.0, Apache License 2.0): A differentiable simulator for multicompartment Hodgkin-Huxley models, supporting GPU acceleration, just-in-time compilation, and gradient-based parameter optimization. Available at: https://github.com/ jaxleyverse/jaxley .

DYNAMAX [21] (version 0.1.4, MIT License): A library for state-space modeling and inference, used here for implementing the Extended Kalman Filter (EKF) and its (block) diagonal approximation. Available at: https://github.com/probml/dynamax .

SPARSEJAC [28] (version 0.2.0, MIT License): A library for efficient sparse Jacobian computation using graph coloring and automatic differentiation. Available at: https: //github.com/mfschubert/sparsejac .

For experiments on large-scale neuron models (e.g., the retinal ganglion cell and C10 model of a pyramidal neuron), we used an NVIDIA H100 GPU. All other experiments were run on a MacBook Pro with an Apple M1 Pro (8-core CPU, 16 GB RAM) running macOS 13.4.

All code used for inference, cell and voltage simulation, and the experiments presented in this paper is available at: https://github.com/ianctanoh/eap-fit-hh .

B Validity of the Diagonal EKF

The true update step of the EKF is:

The procedure described by Chang et al. [27] matches the diagonals of the precision matrices:

Since the filtered covariances are assumed to be diagonal, this amounts to:

Their procedure is therefore equivalent to the standard EKF (7) with a diagonal predicted precision matrix Σ -1 t | t -1 and a diagonal measurement information matrix H ⊤ t R -1 t H t .

In their work, Chang et al. [27] assume that F t = I . This drastically simplifies the computation of diag( Σ -1 t | t -1 ) since Σ t | t -1 = Σ t -1 | t -1 + Q t and both Q t and Σ t -1 | t -1 are assumed to be diagonal.

This trivial Jacobian assumption is not realistic in our case. But to simplify calculations, we assume that Σ t | t -1 is diagonal, so that diag( Σ -1 t | t -1 ) = diag( Σ t | t -1 ) -1 .

Our procedure becomes equivalent to the standard EKF with a diagonal predicted covariance matrix Σ t | t -1 and a diagonal measurement information matrix H ⊤ t R -1 t H t .

Hence, the validity of our diagonal EKF method relies jointly on the dynamics model (through Σ t | t -1 ) and the emission model (through H ⊤ t R -1 t H t ). We discuss each in turn.

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A diagonal approximation of Σ t | t -1 is reasonable for well-behaved HHmodels

The predicted covariance verifies with

where F t = Jac( f θ )( µ t -1 | t -1 ) and Q t is the dynamics covariance. Assume that both Σ t -1 | t -1 and Q t are diagonal. The question becomes: What is the structure of F t Σ t -1 | t -1 F ⊤ t ?

Structure of F t . In multi-compartment Hodgkin-Huxley models, each compartment voltage depends on its own state and on voltages of neighboring compartments, while each gating variable depends only on its own previous value and the voltage of its corresponding compartment. Ordering the latent state as

the Jacobian has the block-arrow structure

Here A t encodes voltage-voltage coupling, B ( ℓ ) t and C ( ℓ ) t capture voltage-gate interactions, and D ( ℓ ) t govern gate self-dynamics. In biophysical HH models, C ( ℓ ) t and D ( ℓ ) t are diagonal, and B ( ℓ ) t are nearly diagonal (apart from weak solver-induced mixing). We assume that these off-diagonal elements are negligible and consider in the following that B ( ℓ ) t are diagonal. The main source of non-diagonality lies in A t , through axial coupling between compartments.

Magnitude of cross-derivatives. We examine the relative size of the partial derivatives that make up F t . In the Hodgkin-Huxley model, these correspond to four types of dependencies: (i) gating variable on voltage ( ∂x t +1 /∂V t ), (ii) gating variable on gating variable ( ∂x t +1 /∂x t ), (iii) voltage on gating variable ( ∂V t +1 /∂x t ), and (iv) voltage on voltage. We analyze each in turn.

(i) Gate with respect to voltage. Consider a generic gating variable x governed by

where α ( V ) and β ( V ) are voltage-dependent transition rates. The exponential-Euler update gives:

Differentiating x t +1 with respect to V t gives:

where k ( V t ) = e -∆ t/τ ( V t ) . It follows that (using the chain rule on k ):

Applying the triangle inequality and using that | x t -x ∞ | ≤ 1 (since x ∈ [0 , 1] ) yields the bound

For small ∆ t , with moderate and smooth transition rates α, β , this is of order O (∆ t ) , confirming that voltage-to-gate sensitivities are small under typical HH kinetics.

(ii) Gate self-derivative. It follows directly from eq. (11) that:

so it is O (1) and typically much larger than the cross term ∂x t +1 /∂V t when ∆ t is small and | α ′ | , | β ′ | are moderate.

(iii) Voltage with respect to gate. Next, consider the effect of a gating variable x t on the voltage update. In practice, JAXLEY uses an implicit solver. Here for the sake of clarity, we assume an explicit Euler step. For a fixed compartment, it can be written as:

where a c ( t ) is the activation of channel c . For the channel c ⋆ corresponding to gate x t , the only dependence is through a c ⋆ ( x t ) , giving

In practice, a c ⋆ ( x t ) = Cx k t with k ∈ { 1 , . . . , 4 } and C, x t ∈ [0 , 1] . Indeed recall that the activation is a product of gating variables (between 0 and 1 ) to some integer power. Then | a ′ c ⋆ ( x t ) | ≤ kC ≤ k . Thus,

With ∆ t = 0 . 025 ms, g c ⋆ ≲ 1 S/cm 2 , and | V t -E c ⋆ | ≲ 100 mV, these terms are the dominant off-diagonal contribution to F t becoming large primarily during spikes. Provided that spikes do not occur at high frequency throughout the cell, these contributions remain intermittent and should not overwhelm the diagonal structure of F t .

(iv) Voltage with respect to voltage. In eq. (12), the voltage of compartment i is coupled to the voltage of its neighboring compartments via the axial conductance term:

It follows that

̸

Thus, diagonal entries are O (1) (since the activations | a c ( t ) | ≤ 1 ), while off-diagonal ones scale as O (∆ t ) .

What conditions support the diagonal approximation?

The preceding derivative analysis shows that, for a broad class of well-behaved Hodgkin-Huxley models and sufficiently small time steps ∆ t , the state transition Jacobian F t exhibits limited crosscoupling between state variables. In particular, most off-diagonal Jacobian entries scale as O (∆ t ) , while the diagonal entries remain O (1) . The main exception is given by the voltage-gate sensitivities ∂V t +1 /∂x t , which may not be negligible during action potentials.

We refer to a model as well-behaved when the following conditions hold:

The gating rate functions α ( V ) , β ( V ) and their derivatives remain bounded over the relevant voltage range, ensuring that gate-voltage and voltage-voltage cross-derivatives scale as O (∆ t ) .

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Spiking activity is not excessively frequent, so that large voltage-gate sensitivities do not persist over time.

Under these conditions, all sources of non-diagonality in F t are negligible for small ∆ t , except for the voltage-gate derivatives. Standard Hodgkin-Huxley parameterizations typically satisfy the first condition: the transition rates and their derivatives are smooth and moderate in magnitude, and choosing ∆ t sufficiently small ensures that most off-diagonal Jacobian entries remain negligible relative to the diagonal terms (see fig. B.1).

The dominant deviation from diagonality arises from the voltage-gate terms ∂V t +1 /∂x t . During spikes, these derivatives can become large; at lower firing rates they are typically smaller, but can still remain of the same order of magnitude as the main diagonal terms (fig. B.2). Consequently, while well-behaved dynamics eliminate most sources of cross-coupling, the diagonal approximation cannot be justified from Jacobian structure alone in the presence of spiking.

Nevertheless, even when voltage-gate sensitivities are large, their impact on the predicted covariance can be controlled through the variances assigned to the gating states. In the EKF prediction step, the covariance update is given by

̸

with diagonal prior covariance Σ t -1 | t -1 = diag( s 1 , . . . , s d ) . The off-diagonal entries of Σ t | t -1 arise from terms of the form F t,ik s k F t,jk for i = j . Assigning small process and initial variances to gating variables directly suppresses these contributions, even when the corresponding Jacobian entries are non-negligible. This choice is also physically well motivated: gating variables represent channel states confined to the interval [0 , 1] , so their intrinsic uncertainty is naturally much smaller than that of membrane voltages. This procedure allows to maintain the validity of the diagonal covariance approximation.

The validity of the diagonal EKF now heavily relies on the emission model, and more specifically on whether a diagonal approximation can be made for H ⊤ t R -1 t H t .

so that

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Diagonality of H ⊤ t R -1 t H t

In our work, we assumed whitened observation noise, R t = σ 2 obs I , so diagonality depends solely on H ⊤ t H t . Let the first K state dimensions correspond to compartment voltages and the remainder to gating variables. Since observations depend only on voltages,

Crucially, extracellular potentials are linear in membrane voltages (cf. eq. (4)). Because this mapping is static, H v t = H v for all t . The diagonality of ( H v ) ⊤ H v therefore depends only on the spatial arrangement of electrodes and compartments.

Let's analyze the structure of H v . Recall that the transmembrane current in compartment i is

where r mi is the electrode-compartment distance and κ depends on properties of the extracellular medium. Differentiating yields the 'self + neighbors' structure:

and electrode m measures

Define the inverse-distance matrix R ∈ R M × N with R mi = κ/r mi , a diagonal matrix D with

and an axial coupling matrix G with

Then where W := R ⊤ R .

Let C := G -D and denote its entries by c pi ; the support of column i is the local neighborhood S i = { i } ∪ { k : k ∼ i } . Then

Using (13), W pq becomes large when electrodes are simultaneously close to compartments p and q . Therefore,

grows when both neighborhoods S i and S j lie within the sensitive region of multiple electrodes. This regime is typical for high-density probes such as Neuropixels or MEAs, where many electrodes lie close to many compartments, yielding strong off-diagonal structure in ( H v ) ⊤ H v and limiting the accuracy of the diagonal EKF (cf. Section 5.1).

Whenis H v ⊤ H v diagonal? When no electrode is simultaneously close to two different compartments, W is approximately diagonal and

If the overlap S i ∩ S j is small, the off-diagonal terms remain strongly attenuated. In this regime, ( H v ) ⊤ H v is close to diagonal, and a diagonal EKF update accurately captures the dynamics covariance.

A concrete example is a linear branch with N compartments of length L , and N electrodes placed so that electrode i lies at distance l ≪ L from compartment i . Each electrode then primarily measures one compartment, yielding nearly diagonal W . Since a linear cable has minimal neighborhood overlap ( S i ∩ S j = ∅ unless j = i ± 1 ), ( H v ) ⊤ H v becomes tridiagonal with dominant diagonal entries. This setting aligns with our single-branch experiment and explains why the diagonal EKF performs well there (Section 4.1).

A similar argument applies in the multi-branch example (Section 4.2). The compartments in each branch are sufficiently long that electrodes are rarely simultaneously close to two distinct compartments. As a result, W remains approximately diagonal, and ( H v ) ⊤ H v preserves a near-tridiagonal structure, with the only substantial off-diagonal contributions occurring near the branch point where neighborhood overlap increases. Although the diagonal approximation is less accurate than in the single-branch case, our results show that it still achieves good reconstruction performance in this geometry.

Takeaway. For well-behaved Hodgkin-Huxley models, the state transition Jacobian F t exhibits limited cross-coupling, with most off-diagonal entries suppressed by ∆ t . The primary exception arises from voltage-gate sensitivities, which can become comparable to diagonal terms during spiking. Nevertheless, because off-diagonal contributions to the predicted covariance are weighted by the variances of the corresponding state variables, assigning small variances to gating states-naturally

The electrode Gram matrix is

justified by their bounded dynamics-allows the predicted covariance to remain approximately diagonal in practice. In contrast, the problem geometry may cause H ⊤ t R -1 t H t to deviate strongly from diagonality, which can undermine the validity of a diagonal filtered covariance approximation.

Limitation. The above argument for the predicted step is heuristic and intended to motivate when a diagonal approximation of the predicted covariance is reasonable. In practice, solver-induced coupling between compartment voltages can introduce additional non-diagonality, particularly for morphologies of increasing complexity with many interconnected compartments. While this effect may reduce the accuracy of the diagonal approximation in such settings, it was not observed to be limiting in the experimental regimes considered in this work.

C Mathematical details of the Block-Diagonal EKF

C.1 Definition and closedness

We now describe the EKF when covariances are constrained to a block-diagonal form. We assume that the first K latent dimensions correspond to compartment voltages and the remaining ones to gating variables. At each time step t , we assume that both predicted and filtered covariances take the form

where Σ vv t | ⋆ ∈ R K × K is dense and σ 2 t | ⋆ contains the gating variances. We denote this family of matrices by BD( K ).

Proposition 1 (Closedness of the BD( K ) family) . Assume that for all t :

The dynamics covariance Q t is BD( K ).

The dynamics Jacobian F t is BD( K ).

If Σ 0 is BD( K ), then both predicted and filtered covariances remain BD( K ) for all subsequent time steps.

Proof of Proposition 1. Writing z t = [ v t ; g t ] , assume Σ t -1 | t -1 is BD( K ):

Under Assumptions 1-2, F t = blkdiag( A t , diag( d t )) and Q t share this structure. The EKF prediction Σ t | t -1 = F t Σ t -1 | t -1 F ⊤ t + Q t then yields

For the update step, we use the identity:

Now,

Since H t = [ H v t 0] ,

Therefore the sum inside the inverse is block diagonal,

and inverting yields

Let the residual be

Since H t = [ H v t 0] and

we obtain the block update

Thus only the first K states (the voltages) are updated; the gating variables remain unchanged, which is consistent with the observation depending solely on the voltage block.

This structure reduces the EKF complexity from O ( d 3 ) time and O ( d 2 ) memory to O ( K 3 + d ) and O ( K 2 + d ) , respectively, a major gain since K ≪ d in typical large neuron models.

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C.2 Validity of the block-diagonal approximation

We now discuss the assumptions of Proposition 1.

Assumption 1 (BD( K ) process noise). In our setting, process noise is injected independently per compartment and gating variable, so Q t is naturally BD( K ) since it is diagonal.

Assumption 2 (BD( K ) Jacobian). The exact Jacobian is not strictly BD( K ) but has the arrow-shaped form

where A t captures voltage-voltage coupling and the other blocks represent voltage-gate interactions and self-dynamics. Under the BD( K ) approximation, we set

yielding

This corresponds to neglecting cross-sensitivities between voltages and gates, i.e. assuming ∂V t +1 /∂x t ≈ 0 and ∂x t +1 /∂V t ≈ 0 for each gate x .

As established in Appendix B, the gate-to-voltage terms ( ∂x t +1 /∂V t ) are negligible under wellbehaved HH dynamics. Voltage-to-gate sensitivities ( ∂V t +1 /∂x t ) grow primarily during frequent

Hence both predicted and filtered covariances remain BD( K ). Note that only the voltage block is updated.

Let's derive the update for the filtered mean. We use:

spiking activity and can become significant in those regimes, although they often remain manageable when spiking rates are low. When these terms become problematic, the same mitigation strategy used for the diagonal approximation applies: assigning very small process and initial variances to gating states suppresses their contribution to the predicted covariance.

Under these conditions, a block-diagonal covariance structure remains reasonable while still supporting dense inter-compartmental voltage coupling through A t .

D Unidentifiability of 3D compartment locations

In this section, we show that when inferring a neuron's spatial position from extracellular recordings, the geometry of the recording sites can impose inherent unidentifiability in the recovered compartment locations.

Our multi-compartment model represents the neuron as a collection of compartments, each treated as a point source of transmembrane current located at its center. Hence, estimating a neuron's spatial configuration reduces to estimating the 3D positions of its compartments. According to eq. (4), the extracellular potential measured at a given electrode depends only on the transmembrane currents and the inverse distance to each compartment. Therefore, if the distances between each compartment and all electrodes remain unchanged, the resulting extracellular signals will be identical.

Wefirst consider the case where all electrodes lie on a single line (fig. D.1A). Fix a single compartment at some position in 3D space (black). The set of all locations that preserve its distance to a given site is a sphere. The intersection of these spheres-corresponding to multiple sites aligned on a line-is a circle. Thus, any position on this circle (blue dashed circle) yields the same set of distances (and hence the same extracellular potentials), rendering the compartment's position unidentifiable up to a rotation around the axis defined by the recording sites.

Next, suppose the electrodes lie on two parallel lines, forming a planar grid (fig. D.1B). In this case, for a given compartment (black), the set of equidistant locations from the first line of sites lies on a circle (blue), and similarly for the second line (green). The only positions that simultaneously preserve distances to both sets of sites lie at the intersection of these two circles. This intersection consists of exactly two points: the true location of the compartment and its mirror reflection across the plane defined by the electrodes (grey).

This ambiguity persists even with more than two lines of electrodes, as long as they all lie in the same plane. For example, fig. D.1C shows three parallel lines of recording sites in a planar grid, viewed from above. The mirror-reflected location remains the only position (besides the true one) that produces identical distances to all recording sites.

In general, if all electrodes lie in a 2D plane (e.g., z = 0 ), then for any compartment, the reflection of its true location across this plane yields the same distances to all electrodes. Consequently, the extracellular voltages at all sites are unchanged. This holds independently of the number or spacing of electrodes and applies to all compartments of a multi-compartment model. Therefore, in the absence of additional constraints, the entire neuron morphology is identifiable only up to reflections of its compartments across the recording plane.

This ambiguity can be partially mitigated by incorporating structural constraints or prior knowledge about the neuron. For example, if the cell is known to consist of a single linear branch, then the configuration of all compartments is restricted to lie along a line. In this case, the full morphology is only ambiguous up to a reflection of that line across the recording plane-substantially reducing the space of indistinguishable configurations. Similarly, if the compartmental connectivity is known (i.e., which compartments are connected to each other and in what topology), then only configurations that preserve both the pairwise distances to electrodes and the known connectivity are allowed. While this does not fully resolve the mirror symmetry, it rules out arbitrary spatial rearrangements. Finally, full identifiability can be achieved by breaking the planar symmetry of the recording sites. For example, recordings from two probes placed at different depths or orientations-such that their electrodes do not lie in the same plane-remove the reflection ambiguity and enable unique localization of each compartment in 3D space.

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E Additional Details and Figures for Section 4

This section provides additional experimental details, biophysical parameters, and figures related to the validation experiments presented in Section 4.

The extracellular resistivity was assumed to be homogeneous and equal to 300 Ω · cm , and the specific membrane capacitance was fixed to C m = 1 µ F / cm 2 across all compartments. While we assume homogeneous extracellular resistivity, this simplification neglects known spatial variability [25], which could be addressed in future work. Extracellular voltages were recorded at a sampling rate of 40 kHz.

All voltages and channel states were initialized at their steady-state values at rest, and thus we assumed the initial state to be known in the EKF. We set the initial covariance matrix of the EKF to Σ 0 = 0 . 1 I . Observation noise was also assumed to be known, as commercial devices that record extracellular potentials typically provide noise specifications [16, 38].

In Sections 4.1 and 4.2, where the state dynamics were well-specified and deterministic, we fixed the standard deviation of the membrane potentials and channel states to small values: σ v = 0 . 0001 mV / ms 1 / 2 , and σ state = 0 . 00001 ms -1 / 2 (same for all the channels) for all compartments. These values ensured numerical stability while avoiding degenerate gradients during optimization.

E.1 Validation on single-branch model

We evaluated a single-branch cell model as a baseline for benchmarking. This branch consists of 10 cylindrical compartments of length 24 µ m and radius 2 µ m , equipped with standard Hodgkin-Huxley (HH) sodium, potassium, and leak channels [6]. The maximum conductances and reversal potentials are provided in Table 1. A constant current of amplitude 1 . 5 nA was injected for 20 ms.

As the neuron-electrode distance increased, the signal-to-noise ratio (SNR) decreased, resulting in higher bias and variance in inferred conductances (fig. E.1B). At short distances with high SNR, we observed that MSE minimization, standard EKF and block-diagonal EKF yielded comparable estimates, indicating that EKF's linearization introduced minimal error and confirming the validity of the block-diagonal approximation (fig. E.1C). The diagonal EKF introduces a slightly larger bias, but its voltage reconstructions remain highly accurate (fig. E.2). This is consistent with our analysis

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in Appendix B: only the first (soma) compartment lies near the electrode, with all others much farther away, so cross-visibility is minimal and a diagonal covariance approximation is sufficient. As observation noise increased, estimation accuracy declined (fig. E.1A), but even at 30 µ V RMS-a challenging noise regime (fig. E.3C)-estimates remained close to the ground truth. Notably, the

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induced variance on the reconstructed voltage traces is small (fig. E.4), indicating robustness even under challenging conditions.

E.2 Validation on multi-branch neuron model

We also evaluated a multi-branch neuron model to test conductance and geometry inference. The model consists of three branches, each discretized into four cylindrical compartments of radius 2 µ m , for a total of 12 compartments. Each compartment contains standard Hodgkin-Huxley sodium, potassium, and leak channels [6]. Reversal potentials are shared across all branches (Table 2), while conductances and lengths vary by branch (Table 3). The axial resistivity was set to 150 Ω · cm . A current of 2 nA was injected in the first compartment of branch 1 (soma) for 20 ms.

With EKF, all the maximum conductances were inferred accurately for all 40 datasets simulated with σ obs = 1 µ V (fig. E.5).

With the diagonal EKF, the recovered conductances exhibit a more pronounced bias than in the singlebranch case, particularly in branches 2 and 3 (fig. E.6), although the voltage traces remain accurate up to small shifts in these branches (fig. E.7), and the compartment center locations were inferred correctly (fig. E.8). In this multi-branch configuration, the diagonal approximation retains decent accuracy because, despite the relatively dense electrode layout, the long-compartment geometry reduces instances where electrodes are simultaneously close to distinct compartments. Since each compartment only has two neighbors (except at the branching node), this limits off-diagonal measurement coupling. As detailed in Appendix B, this geometry ensures that most of the off-diagonal entries of ( H v ) ⊤ H v remain significantly smaller than terms near the diagonal (fig. E.9). Consequently, the diagonal approximation introduces some bias but continues to provide useful and stable inference.

E.3 Robustness to Model Misspecification

In the main text, we considered two forms of model misspecification: missing ion channels and incorrect gating kinetics.

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Omitted Ion Channels. To test robustness to missing channels, we simulated data from a fourcompartment branch model (total length 96 µ m , radius 2 µ m ) containing M-type potassium (KM) and L-type calcium (CaL) channels (Table 5) in addition to Hodgkin-Huxley (HH) channels (Table 4). Inference was performed using a model containing only the HH channels. A constant current was injected for 20 ms. Extracellular voltages were recorded at ten electrodes with a noise level of σ obs = 0 . 1 µ V . Biophysical parameters for the simulation model are shown in Table 6.

During EKF inference, we fixed the standard deviation of the channel states to σ state = 0 . 00001 ms -1 / 2 (same for all the channels) and treated σ v as a free parameter to capture voltage uncertainty due to model mismatch. We performed hyperparameter sweeps over learning rates and initial values of the parameters, selecting the configuration with the highest marginal log-likelihood.

For MSE-based optimization, we used Adam [39] and reported the result with the lowest final MSE.

Perturbed Gating Dynamics. To evaluate robustness to incorrect kinetics, we generated data using gating functions from a retinal ganglion cell (RGC) model (Table 8) and performed inference using the original Hodgkin-Huxley kinetics [6]. The neuron was a single cylindrical cable (length 24 µ m , radius 2 µ m ), discretized into four compartments. A constant current was injected into the first

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compartment (soma) for 20 ms. Extracellular voltages were recorded at 10 different locations with noise level σ obs = 0 . 1 µ V . Biophysical parameters for the simulation model are shown in Table 7.

During EKF inference, we treated the voltage standard deviation σ v and channel noise levels σ K , σ Na , σ Leak as learnable parameters. While only the sodium and potassium kinetics are misspecified, we allowed separate noise levels for each channel type to provide flexibility in how uncertainty is accounted for. We also used the same σ v , σ K , σ Na , σ Leak across all compartments, though future work could consider compartment-specific noise parameters. As before, we performed grid search over learning rates and initializations, selecting the result with the highest marginal log-likelihood.

For MSE optimization, we used Adam and selected the result with the lowest MSE.

The EKF smoothed posterior mean of the voltage recovers the true voltage trace in each of the four compartments, whereas MSE minimization fails (fig. E.10).

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F Retinal Ganglion Cell (RGC) Model

We use a morphologically detailed model of a retinal ganglion cell (RGC), based on a reconstruction from Vilkhu et al. [29]. The morphology was originally provided as a NEURON .hoc file, 1 which we converted to SWC format for compatibility with JAXLEY. The model includes the full soma and dendritic arbor, and was discretized into 122 compartments, assigning one compartment per branch.

The ion channel model consists of Hodgkin-Huxley (HH) conductances: fast sodium, delayedrectifier potassium, and leak channels. Voltage-dependent rate constants are listed in Table 8, and channel reversal potentials in Table 9. Region-specific maximum conductances for soma and dendrites are given in Table 10. All values were taken directly from Vilkhu et al. [29], with a minor adjustment to the leak reversal potential to ensure resting membrane potentials near -65 mV across compartments. Note that Vilkhu et al. [29] also consider two additional channels (a calcium channel and a calcium-gated potassium channel as in Fohlmeister et al. [40]), which we neglected in this experiment. Including these channels could be the subject of future work.

The axial resistivity was 143 . 2 Ω · cm , the membrane capacitance was 1 µ F / cm 2 across all compartments, and the extracellular resistivity was 1000 Ω · cm .

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To generate synthetic extracellular recordings, we applied a constant somatic current injection for 10 ms. Voltage recordings were simulated using a synthetic planar multi-electrode array (MEA) with a sampling rate of 40 kHz.

We fit this model using our diagEKF and block-diagEKF implementations. All voltages and channel states were initialized at their steady-state values at rest, and thus we assumed the initial state to be known. We set the initial covariance matrix to Σ 0 = 0 . 1 I . Observation noise was also assumed to be known, as commercial devices that record extracellular potentials typically provide noise specifications [16, 38]. Since the state dynamics were well-specified and deterministic, we fixed the standard deviation of the membrane potentials and channel states to small values: σ v = 0 . 001 mV / ms 1 / 2 , and σ state = 0 . 0001 ms -1 / 2 (same for all the channels) for all compartments. These values ensured numerical stability while avoiding degenerate gradients during optimization.

We optimized conductance parameters using the Adam optimizer. We tried multiple learning rates and initialized values uniformly at random within the bounds shown in Table 11. We ran over 100 random initializations with different seeds and selected the result that achieved the highest marginal log-likelihood.

With the block-diagonal EKF, all maximum conductances were accurately inferred (F.1) and the voltage traces faithfully reconstructed across the cell (F.2).

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G CA1 Pyramidal Neuron (C10 Model)

We use the C10 model of a CA1 pyramidal neuron, originally developed by Cutsuridis et al. [32]. The morphology is provided as a NEURON .hoc file, 2 , which we converted to SWC format for compatibility with JAXLEY. This file specifies the full 3D morphology, including the diameters and lengths of all the branches (Table 14). As shown in fig. G.1, the neuron is divided into 15 anatomical sections.

Following Cutsuridis et al. [32], we applied the compartmentalization heuristic of Carnevale and Hines [7] to determine the number of compartments per section. Specifically, a section of length L is discretized into

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where the frequency-dependent electrotonic length is defined as λ f ( f ) = 1 2 √ d πfR a C m , with d the diameter, R a the axial resistivity, and C m the specific membrane capacitance. This yields a total of 60 compartments.

Each compartment contains Hodgkin-Huxley (HH) conductances: sodium, potassium, and leak. The sodium current, however, uses a three-gate formulation with gating vector λ Na = ( m,h,s ) ∈ [0 , 1] 3 . The total transmembrane current released by a compartment is given by:

Moreover, unlike the canonical HH model, the gating kinetics vary across anatomical regions [41]. Gating variables evolve according to first-order kinetics:

The steady-state functions λ ∞ ( V ) and time constants τ λ ( V ) for m , h , and n vary by region and are listed in Table 12.

The third gating variable s - models sodium attenuation. Its dynamics are consistent across regions except for a region-specific attenuation factor Na _ att ∈ [0 , 1] . The expressions for s ∞ ( V ) and τ s ( V ) at fixed Na _ att are given at temperature T = 34 ◦ C (Table 13).

The attenuation factor Na_att is defined per section based on Migliore et al. [35] (Table 14).

Reversal potentials follow Cutsuridis et al. [32], with a small adjustment to E L to enforce a resting potential near -70 mV across compartments (Table 15).

The maximum conductances for each region were taken directly from Cutsuridis et al. [32] (Table 16).

We used the following physiological constants: axial resistivity R a = 150 Ω · cm , specific membrane capacitance C m = 1 µ F / cm 2 , and extracellular resistivity ρ ext = 300 Ω · cm . While we assume homogeneous extracellular resistivity, this simplification neglects known spatial variability [25],

which could be addressed in future work. Note that Cutsuridis et al. [32] also consider additional channels, which we neglected here. Including them in our model could also be the subject of future experiments.

Synthetic extracellular recordings were generated by injecting a constant current into the soma for 30 ms. Voltage traces were simulated using a synthetic Neuropixel Ultra probe at a 40 kHz sampling rate, with noise standard deviation σ obs = 0 . 1 µ V .

The true location of the cell was generated by applying a rigid-body transformation-consisting of a 3D rotation and translation-to the positions of the compartment centers defined via the SWC file. The geometric parameters thus included three rotation angles (one for each axis), applied to the base positions of the compartment centers obtained from the SWC file, and a 3D translation vector applied to the base position of the soma compartment.

We fit this model using the block-diagonal EKF. All voltages and channel states were initialized at their steady-state values at rest, and thus we assumed the initial state to be known. We set the initial covariance matrix to Σ 0 = 0 . 1 I . Observation noise was also assumed to be known, as commercial devices that record extracellular potentials typically provide noise specifications [16, 38]. Since the state dynamics were well-specified and deterministic, we fixed the standard deviation of the membrane potentials and channel states to small values: σ v = 0 . 001 mV / ms 1 / 2 , and σ state = 0 . 0001 ms -1 / 2 (same for all the channels) for all compartments. These values ensured numerical stability while avoiding degenerate gradients during optimization.

We optimized maximum conductances and geometric parameters using the Adam optimizer. We tried multiple learning rates and initialized the maximum conductances uniformly at random within the bounds shown in Table 17. The rotation angles were initialized uniformly on [0 , 2 π ) , and translations were initialized to place the soma near its true location. We chose this initialization to reduce the risk of converging to a mirror-reflected solution, which produces identical extracellular signals when using a planar probe (see Section D). We ran 100 random initializations with different seeds and selected the result that achieved the highest marginal log-likelihood.

Although the marginal log-likelihood converged to its true value after a few hundred iterations (fig. G.3F), and both the voltage traces (fig. G.3A-E) and compartment positions (fig. 5A) were

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accurately recovered across the cell, the leak conductances did not converge to their true values or even continued to fluctuate (fig. G.2) -reflecting the well-known degeneracy in conductancebased neuron models, where multiple parameter combinations can produce nearly identical voltage dynamics.

H DiagEKF vs. EKF runtime

We compare the computational efficiency of the standard Extended Kalman Filter (EKF) and our sparse diagonal approximation (DiagEKF). To isolate performance differences, we consider a simplified neuron model: a 24 µ m cable equipped with standard Hodgkin-Huxley channels [6]. A constant current is injected for 20 ms, and extracellular voltage is recorded at a single electrode at a 40 kHz sampling rate.

The neuron is discretized into N compartments, with N ∈ { 1 , 10 , 20 , 50 , 100 , 200 } . As N increases, the number of state variables (membrane voltages and gating variables) grows proportionally. Importantly, the gating dynamics in these models are highly local: each variable typically depends only on

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its own previous value and the voltage of its corresponding compartment. As a result, the Jacobian of the dynamics function quickly becomes very sparse when the number of states increases (fig. H.1A).

To evaluate performance, we ran both EKF and DiagEKF for 500 iterations on an NVIDIA H100 GPU for each value of N . As shown in fig. H.1B, DiagEKF achieves approximately 2× speedup over the dense EKF in high-dimensional settings. This highlights the computational advantage of our diagonal approximation in large biophysical neuron models.

NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: The claims in the abstract and introduction clearly describe the contributions: a method to fit multi-compartment Hodgkin-Huxley models using only extracellular voltage data, leveraging differentiable simulation and an Extended Kalman Filter (EKF). These claims are validated through simulated experiments and applications to realistic morphologies.

Guidelines:

The answer NA means that the abstract and introduction do not include the claims made in the paper.

The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.

It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: The limitations of EKF-based inference (e.g., computational cost, approximation biases) and the challenge of applying this framework in vivo are explicitly discussed in the 'Conclusions, Limitations, and Future Work' section.

Guidelines:

The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.

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The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.

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If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.

While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.

3. Theory assumptions and proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [Yes]

Justification: All the approximations made (diagonal, block-diagonal) are rigorously justified in the appendix.

Guidelines:

The answer NA means that the paper does not include theoretical results.

All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced.

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Theorems and Lemmas that the proof relies upon should be properly referenced.

4. Experimental result reproducibility

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [Yes]

Justification: The paper provides detailed descriptions of the neuron models, simulation tools (JAXLEY, DYNAMAX), experimental setups (e.g., number of compartments, observation noise levels), and inference methods, enabling reproduction of results.

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The answer NA means that the paper does not include experiments.

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Justification: The paper uses publicly available morphology datasets and the open-source JAXLEY and DYNAMAX frameworks. We plan to release our code, and experiment instructions are detailed in the supplementary material.

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Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [Yes]

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The answer NA means that the paper does not include experiments.

The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.

The full details can be provided either with the code, in appendix, or as supplemental material.

7. Experiment statistical significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [Yes]

Justification: The paper includes distributions over repeated simulations and multiple optimization runs and reports 1-sigma confidence intervals on inferred posterior voltage traces (see Fig. 3).

Guidelines:

The answer NA means that the paper does not include experiments.

The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.

The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).

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For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).

If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.

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Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [Yes]

Justification: The paper provides information about computational efficiency and scalability of the EKF-based inference, including runtime for large models. Details on compute time and scaling behavior are provided in the appendix.

Guidelines:

The answer NA means that the paper does not include experiments.

The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.

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The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).

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Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines ?

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Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [NA]

Justification: The work is primarily foundational neuroscience research using synthetic data. Potential downstream impacts are not immediate, and the scope of application remains preclinical.

Guidelines:

The answer NA means that there is no societal impact of the work performed.

If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.

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Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [NA]

Justification: No high-risk models or sensitive datasets were released. All simulations were performed using synthetic or publicly available data.

Guidelines:

The answer NA means that the paper poses no such risks.

Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.

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We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.

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Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [Yes]

Justification: We use two open-source codebases: JAXLEY (Apache 2.0 License) and DYNAMAX (MIT License), both of which are properly cited in the paper. Morphological data for the retinal ganglion cell and pyramidal neuron models are taken from previously published studies [29, 32] and are cited accordingly. The licenses and terms of use for all codebases are respected, and we plan to include URLs and license details in the supplementary material.

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The answer NA means that the paper does not use existing assets.

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Answer: [Yes]

Justification: We plan to release scripts, simulation configurations, and experiment data with documentation, including parameter settings and model variants used in the study, as part of the final submission or camera-ready version.

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The answer NA means that the paper does not release new assets.

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At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.

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Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces a novel approach to estimating the HH parameters of neurons in high density arrays. Authors show good performance on simulated datasets + two neurons from real-wolrd datasets (one 2D retinal ganglion cell and one CA1 hippocampal pyramidal cell).\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: It is a very interesting paper with a novel model for HH parameter estimation, and very good results. \n\nWeaknesses:\n\nI find the paper to be very good overall but the main weakness lies in the lack of comparison to existing methods. \nother methods have been proposed to estimate the parameters of a HH model. For example Multi-objective Differential Evolution (DEMO, Lois et al., 2022) and ElectroPhysiomeGAN (Kim et al., 2025). These methods might not have been developed and tested on high-density extracellular voltage recordings but it would be worth adding some comparisons to one of these methods + citing these approaches and explaining the differences with your approach. \n\nThe results could be improved as well by showing results on more than two cells. Why not extend it to many more cells and report statistics (overall performance of true vs reconstructed signals). It would be great to show this to make sure these two cells are not cherry picked examples. Are you ilmited by data availability (as these are in vitro recordings)?\n\nQUESTIONS:\nThis is a very nice paper, but I believe the clarity of the paper could be improved, which would make the paper a bit more accessible. I struggled a bit understanding the following points:  \n - I would mention that the method is only evaluated / used for in vitro recordings in the intro, as this is only clear after reading section 5.\n - Similarly, why contrast your approach with spike sorting if you're using the output of spike sorting (lines 34-36)? Especially as the method is only evaluated on in-vitro data, I believe it could in the future be complementary to spike sorting, and these lines make it seem like you're targetting in vivo recordings (I'm not sure if it's needed to criticize an approach designed to tackle a very different problem, o reven mention spike sorting)\n - What do the \"states\" (channel states) represent? \n - The parameters of the HH evolve in time (or depend on the neural activity, and events such as bursting, or other recording artefacts that might hinder the recorded signal). Are you restricting the approach to well-defined neurons in specific timesteps? Could you show the evolution of the parameters in time? Or maybe they are stable in-vitro? \n - MEA arrays are not as high resolution as NP Ultra. The motivation mentions NP Ultra but if the method works well on MEA, it should work well on NP1/NP2 as well. Could you extend your analysis to these probes as well? (if there's available data)\n - You're also not citing the more up-to-date paper, not sure if you should update it to the more recent citation? (Ultra-high density electrodes improve detection, yield, and cell type identification in neuronal recordings, Ye et al., 2024)\n\nLIMITATIONS:\nLimitations are well adressed but I would clarify the scope of the paper in the intro - to make it clear to the reader that the method is so far only evaluated and used on in vitro data.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-05"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper discusses a new method for estimating the electrical activity of neurons using extracellular voltage measurements. Traditionally, biophysical models like the Hodgkin-Huxley model require intracellular in vitro voltage data to estimate key parameters. With the advent of high-density probes that capture extracellular voltage from multiple sites around a neuron, the authors propose a technique to indirectly measure the neuron’s membrane voltage, biophysical properties, and its position relative to the probe.\n\nThe method leverages an extended Kalman filter (EKF) to infer membrane voltage and ion channel states, using efficient, differentiable simulators for computation. It also employs gradient-based optimization to learn the model parameters by maximizing the marginal likelihood. The approach is validated with both simulated data and real neuron morphologies, showing its potential for more accurate and practical neuron modeling using extracellular recordings alone.\n\nSTRENGTHS AND WEAKNESSES:\nThe method is non-invasive, relying solely on extracellular recordings, which makes it practical for in vivo studies. \n\nIt uses an Extended Kalman Filter (EKF), which allows for real-time estimation of membrane potentials and ion channel states. \n\nValidation with simulated and real neuron morphologies shows its potential for accurate modeling of neurons. \n\nIt also accounts for the complex morphology of neurons, which is crucial for realistic simulations.\n\nThe method depends heavily on the quality of extracellular data, which is often noisy and may not fully capture the necessary details of individual neuron activity, particularly in densely packed regions. \n\nSeveral assumptions are needed for the proposed method to be effective. These are not explicitly stated. \n\nThe method assumes that the Hodgkin-Huxley model is appropriate for all neuron types, which might not be true for more complex or unconventional neuron behaviors. \n\nNumber of channel variables are assumed to be known, this fixes dimensions for the inferred model. In other words, prior knowledge of the number of gating variables should be known. This assumption makes the inference in the form of HH model to be unnecessary. In case this assumption cannot be met, then any abstract dynamic model can be used to do the inference instead of HH model structure. \n\nThe use of EKF and gradient-based optimization also introduces computational complexity, which could be a bottleneck in large-scale systems or real-time applications. The linearization along with the gradient-based updates may introduce suboptimal estimates and may require further analysis to characterize uncertainties in the estimates. \n\nEstimating the neuron's position relative to the probe adds another layer of complexity, and any errors in this estimation could lead to inaccuracies. Validation with simulated data might not capture all the noise and artifacts found in real-world recordings. \n\nThe number of compartments and the interconnection term in the model needs further analysis as they are dependent on the density of the sensor array. \n\nLastly, while the method is suited for single-neuron modeling, its application to larger networks could become more challenging due to the complexities of multi-neuron interactions and signal overlap.\n\nQUESTIONS:\n1. Why was the Hodgkin-Huxley model chosen for this method, especially given the assumptions about the gating variables and model dimensions, and could an abstract non-linear model offer more flexibility or accuracy in capturing neuron behaviors?\n\n2. What strategies or modifications are in place to handle neurons that don't fit the traditional Hodgkin-Huxley model, especially those with unconventional ion channels or non-linear dynamics? For example, neuros with heterogeneous spike patters, like regular and fast spiking, and chattering, adaption, etc.\n\n3. How does the method address uncertainties related to the number of gating variables in the Hodgkin-Huxley model, and what is the impact of fixing these dimensions on the model's flexibility?\n\n4. Can you elaborate on the potential limitations or inaccuracies introduced by the EKF’s linearization process, and how might these affect the quality of parameter estimates, particularly in real-time applications?\n\n5. What steps are being taken to improve the method's scalability to larger networks of neurons, and how are multi-compartment interactions and signal overlap handled in the current framework, particularly with regard to sensor density, the number of compartments, and the interconnection terms between compartments?\n\nLIMITATIONS:\n1. Please include all the assumptions that are required to be satisfied for the algorithm to work. \n2. Motivation to use HH-based model and the assumptions made regarding the inference need stronger argument. Otherwise, the method, while interesting, may not be useful.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-01"}

comments: SUMMARY:
This paper addresses the estimation of the parameters of a multi-compartment Hodgkin Huxley model with extracellular voltage recordings. The authors use an Extended Kalman Filter to infer the membrane potential, channel states, and the biophysical parameters of a neuron with simple and complex morphologies. This is a fairly novel and interesting study for both the methodology and the application.

STRENGTHS AND WEAKNESSES:
Strengths:
- This is a novel application of the SSM methods to reconstruct complex HH parameters. 
- The simulations include simple and complex multi-compartment morphologies, and also contain different noise levels as well as channel misspecification results (channels omitted or gating dynamics altered).
- The simulations also included results on real neuron morphologies, namely a 2D retinal ganglion cell and CA1 hippocampal cell.
- The formulation of the SSM and scaling of the EKF, as well as the use of SPARSEJAC, are novel here.

Weaknesses:
- Is the diagonal Gaussian assumption in the EKF a reasonable approximation in this context? Detail how valid this is, if possible quantitatively.
- The application of the methods on real Neuropixels or MEA data would strengthen the manuscript.
-  Can the authors address the issue of parameter identifiability a bit more in detail, especially in simulation? Would it be possible to obtain the identifiable subspace, for example, for the larger models that are considered?
- It would be good to compare the methods to alternative inference methods apart from MSE minimization.

QUESTIONS:
See above weaknesses.

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 4
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-12

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a method for fitting multi-compartment Hodgkin–Huxley neuron models using only extracellular recordings from high-density probes. By combining a differentiable simulator (in JAX) with an Extended Kalman Filter (EKF), the method estimates hidden states (e.g., membrane potentials, ion channel dynamics) and learns key biophysical parameters through gradient-based optimization. Validated on synthetic data with both simplified and complex morphologies, the approach recovers conductances, membrane voltages, and neuron position. Fitting full biophysical neuron models from extracellular signals is a novel contribution.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. The combination of high-density extracellular recordings with multi-compartment modeling is novel. By casting the neuron and electrode system as a state-space model and using an EKF, the authors can continually update the latent membrane states and use those to optimize parameters.\n\n2. The method should be scalable. The paper avoids black-box training of a neural network, instead it directly optimizes the mechanistic model’s likelihood, which ensures the learned parameters are physically interpretable. Also, the authors implement efficiency improvements, e.g. using a sparse Jacobian computation for the state transition (via the SPARSEJAC library) so that they never form the full dense Jacobian of the ODE system, and adopting a diagonal approximation to the EKF covariance to reduce complexity.\n\n3. The paper demonstrates the approach on realistic neuron morphologies taken from the literature (a 2D retinal ganglion cell and a 3D CA1 pyramidal neuron). The method scaled to over 100 compartments with many parameters. These experiments show that the algorithm can recover key parameters like sodium/potassium conductances and the neuron’s location in space from synthetic extracellular data.\n\n4. The paper is generally well-written and structured. Technical details of the model (Hodgkin–Huxley equations, extracellular potential calculation) are provided, and the state-space formulation is clearly described. The authors also address reproducibility: they use open-source tools (JAXLEY, DYNAMAX) and plan to release code and instructions.\n\nWeaknesses:\n\n1. The approach makes some assumptions that may limit its real-world applicability. First, the neuron’s morphology (branching structure and compartmental layout) is assumed to be known and fixed during fitting. In practice, for an arbitrary neuron recorded extracellularly, the morphological structure would not be directly known. This limitation is understandable, but it should be clearly stated. Similarly, the method currently fits one neuron at a time in isolation, i.e.  it does not account for overlapping signals from multiple neurons.\n\n2. The study is validated only on synthetic data. There are no real-world experiments or noise beyond what the model assumes (Gaussian noise). While this is completely reasonable for a first proof-of-concept, it remains to be seen if an actual extracellular recording from an unknown neuron can be fitted with this approach. However, the authors do discuss the challenges of real data in their conclusion (e.g. electrode drift, unknown ground truth).\n\nQUESTIONS:\n1. The paper should provide a brief quantitative or visual analysis comparing the full EKF, diagonal EKF, and simpler loss functions (e.g., MSE minimization). This would help readers understand the accuracy vs. efficiency trade-off, and under what conditions the approximations begin to significantly affect parameter recovery.\n2. It should emphasize that applying it to in vivo data with overlapping neuronal sources will likely require additional modeling layers or priors (e.g., from anatomy or known cell types).\n3. The authors should provide efficient code for easy reproduction. They should consider including benchmarks and documentation on how to run the model for small vs. large neurons, to support adoption and experimentation.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-07-15"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work develops an approach for fitting multi-compartment Hodgkin-Huxley models using dense extracellular voltage recordings. Previous work developed differentiable simulators to fit biophysical neuronal models from intracellular recordings. This work adds a straightforward biophysical model of extracellular potentials, as well as a scalable Extended Kalman Filter (EKF) framework to improve parameter estimation given noise and modeling uncertainty. The approach is tested on several biophysical models of increasing complexity, from a single-compartment model to models of retinal ganglion cells and hippocampal pyramidal cells. Across these scenarios, the method shows good ability to reconstruct channel conductances, voltage time courses, and the geometry of the cell, and markedly outperforms a version of the method without the EKF component.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nWhile the core methods in this paper seem to be a combination of largely off-the-shelf tools, the work is well executed. In general, I found the paper to be quite readable. The modeling and inference choices were reasonable. The evaluations were comprehensive and followed a logical build-up. Modern ultra-dense extracellular probes like Neuropixels are most commonly used for neuronal population recordings, but the use of these probes to estimate detailed biophysical models is an interesting alternative application.\n\nWeaknesses:\n\nThe novelty is slightly limited. The differentiable simulation, EKF, and intra- and extracellular biophysical modeling techniques used are all well established. But again, their combination was well executed here. There are also some minor areas where clarity could be improved (see Questions, below).\n\nQUESTIONS:\n- In Sections 4 and 5, when the authors state the dimensionality of the latent space, and the number of parameters that need to be estimated, could they spell out more explicitly how these numbers arise from the model definition and simulation setup?\n- In Section 5.1 (and more generally) why is there estimation bias in the high-noise regime?\n- Throughout the experiments, could the authors help the uninitiated reader appreciate whether or not, in Fig. 4 for example, a bias of 4e-5 is significant? At what order of magnitude would the bias present significant issues from a modeling perspective?\n- In Figure 4 D and E, in the high-noise regime, why are the voltage traces so faithfully reconstructed despite bias in the conductance parameter estimates? Does this point relate to unidentifiability/degeneracy discussed in subsequent sections?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-21"}