Identifying multi-compartment Hodgkin-Huxley models with high-density extracellular voltage recordings
Authors: Ian Christopher Tanoh
Michael Deistler
Jakob H. Macke
Scott Linderman
Venue: NeurIPS 2025 poster ()
Keywords: Computational neuroscience
State-space models
Extended Kalman Filter
biophysical modeling
Primary area: neuroscience_and_cognitive_science

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 Hodgkyn models with high-density extracellular voltage recordings
Ian Christopher Tanoh Michael Deistler Jakob H. Macke
Scott W. Linderman 1.2
Depaarment Statistics, Stanford University
Wu Tsai Neurosciences Institute. Stanford University
Machine Learning in Science, University of Tübingen TTübingen AI Center, Tübingen
DDepartment of Empirical Inference, Max Planck Institute for Intelligent Systems, Tübingen [ictanch, scott. linderman] @stanforodoooooooooooooooooo edu
Abstract
Multi-compartment Hodgkin-Huxlen models are biophysical models of how electri- cal 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 sur- rounding a neuron, allowing indirect measurement of many compartments of a cell simultaneously. Here, we propose method for inferring the underlying membrane voltage, biophysical parameters, and the neuron's position relative to the probe, using extracellular measurements alone. We use Extended Kalman Filter infer membrane voltage and channel states using efficient, differentiable simulators. Then, we learn the model parameters by maximizing the marginal likelihood using gradient- baseed methods. We demonstrate the performance this approach using simulated data and real neuron morphologies.
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 environmental perturbations [4, 5]. Multi-compartment Hodgkin-Huxlee models provide principled framework for studying biophysical mechanisms of neurons [6. 7]. Multi-compartment models treat neurons connected electrical compartments, allowing simulation of how voltage signals propagate throughout cell as ion channels open and close. A central limitation for building colpi--comartmentt models-and and for understanding biophysical mechanisms in neurons-ae 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 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 very small number of locations within a neuron (typically, only from the soma) [15].
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Extracellular voltage
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Figure I: Our goal: Inferring biophysical properties of single neurons from dense extracellular measurements: We want to identify key biophysical properties- maximum membranc conductances of ion channels and the cell's relative position to the probe (left, blue) from extracellular voltage traces recorded from multiple sites along the probe (right), capturing the neuron's 's activity over time.
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- identilyinn 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 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 cell to triangulate its position and morphology: (2) differentiable simulators like JAXLEY [20] that enable fast, aareeare-accelerated simulation of complex biophysical models and mendeent-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- culti-compartmentt biophysical models from extracellular observations: We extend classical multi- compartment Hodgkin Huxley models to include biophysical model of extracellular potentials. We then use Extended Kalman Filter (EKF) [22] to infer latent neuronal states such 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 's spatial position relative to the probe, we maximize the marginal likelihood using gradient-based optimization.
We first demonstrate that 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 '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 neuron based only on extracellular voltage recordings.
2 Biophysical Modeling of Neurons
2.1 Modeling Action Potentials with Hodgkin-Huxlyy Dynamics
The multi-compartment Hodgkin-Huxley model (HH) describes how ion channels shape the mem- brane potential across branches of neuron. Each compartment represents small component of the cell with its own voltage, capacitance, and set of ion channels. Let (+) denote the membrane

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potential of compartment i, with surface area S and specific membrane capacitance Cm. Each compartment has set of ion channels, and each channel e characterized by maximum conductance gē reversal potential Ec, and a time-varying state X A nonlinear activation function, a.(X ax()ee) E [0, I. specifies the fraction of the maximum current that flow through the channel, as a function its state. The voltage dynamics for compartment are given by,
Cm dv()
I SO
Σ Le ae (VV) Ee) + Σ g (6:5) (V oEC
(I)
dt
fom
where ((0) the external current applied to compartment i, and g(+.)) is the axial conductance coupling compartment i to its neighbor j. The total transmembrane current in compartment is denoted by (() and is the sum of the ionic current /.00 s Ecec ge (x(( (V Eee) and the
capacitive current Lep g(t) Cm dvlo
di [23]. For instance, in the original single compartment HH model with sodium, potassium, and leak channels, the total current is,
mm = (SN S n2V - ENa) + gK (V Ex) + (V EL) + C.
dV di
(2)
Here, for example, the states of the sodium channel are ANa (m, E [0, and the activation function is ONa (ANa) = m²h. The gating variables evolve according to first-order kinetics dA, ae(V) (V))ee where a a.(V) and (V) are voltage-dependent rate functions that control the opening and closing transitions of the ion channel. Specific forms for the and 3 functions are available in Dayan and Abbott [24], e.g. Importantly, note that by eq. (1), the ionic current and the
dt
capacitive current cancel out to give the following expression of the transmembrane current:
7(0) ( = exi
+ S()
(3.i) (VL)
jori
2.2 Modeling Extracellular Voltage Measurements
Extracellular voltages reflect the collective activity of nearby neurons. While the physics of extracel- lular recordings are well understood [25, 26], our work is, to our knowledge, the first leverage this framework for fitting multi multi-compartment HH models to extracellular measurements. Let & denote the extracellular potential measured by fixed site on a recording probe. We treat each compartment as a point source of transmembrane current located at its center and denote r; the distance from the recording site to compartment i of neuron. It follows from Coulomb's 's law that the measured extracellular voltage is,
m
T
In words, the observed potential & is proportional 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 are simply linear functions of the membrane voltages at time l.
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 neuron from noisy extracellular voltage traces. We can achieve both goals by casting the multi-compartment HH model as 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
state space model (SSM) a latent variable model with latent states 21:7, observations YI.7, and parameters 0. In Markovian SSM, latent states evolve according to dynamics distribution,

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p(zz Zę-1: 0), and they give rise to observations via an emission distribution ply: Ze; 0). There are 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 case, the model parameters, 0, 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, 2, [v(i),... V(N, (xe') "eeC...................... ( discretization of the multi-compartment HH differential equations (see eq. (1)),
]oecx)] To model the state dynamics, we use an Ruler
2 (fo((z, (), At Eayn)
(5)
where fo (z(,,)) t) is the deterministic update function derived from the ODEs above. and Edyn is diagonal covariance matrix with entries from {o? Sa J. This Gaussian noise is intended to account for model uncertainty, such as missing ion channels misspecified channel dynamies, which we will consider in the results section.
We assume that the only source of stochasticity in the emission process is measurement noise. Let Yt E RM denote the observed extracellular potential at time across M recording sites. It is modeled as,
Y+ ~ N (ho (2:), 2 I),
where he (Z:) is the predicted extracellular voltage from eq. (4) and o is the observation noise variance. Since he (Z() only depends 2,, 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 po(zi: | y1a). To address this challenge, we use the extended Kalman filter (EKF), which lincarizes 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 po (yı.)) ) for parameter estimation.
Biologically realistic neuron models often contain dozens or even hundreds of compartments, leading to latent state dimensions in the hundreds 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 al. [27], while exploiting the sparsity inherent in multi-compartment HH dynamics. Specifically, we approximate the EKF filtering distribution Pe (z) A N((z; μu, Enp)
with diagonal Gaussian (z) = (2 ( Enj) where Egje and jú, chosen to minimize the divergence KL(g, l| pt). As shown by Chang et al. [27], this minimization admits a closed-fom solution: the optimal diagonal Gaussian approximation ge matches the marginal means and precisions (inverse variances) of pt. This leads to efficient recursive updates that preserve the EKF structure while requiring only diagonal state.
memory cost from (dd) to (d) for a d-dimensional
In practice this approximation requires evaluating only the diagonal elements of two matrices: (i) H, RH, where H. Jac(hh) (H(j-1) is the Jacobian of the emission model and R, o3 I the observation noise covariance; and (ii) the predicted precision ETi-1 - with
Σ 2 F.E, +Q+
where F, = Jac(fo) is the Jacobian of the dynamics and Q: At Edyn is the dynamics covariance (cf. Appendix B).
The diagonal of H. R 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

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through a fixed linear operator (eq. (3) and (4)), H; R, HH is iime-invariant, allowing its diagonal to be computed once and reused throughout filtering. The remaining challenge lies in efficiently computing diag(2 ())(1)) To simplity calculations, we make the additional assumption that E diagonal, reducing the problem to computing diag(Enjt-1)) Here, we exploit the sparsity pattern of Fr, 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 Fuler solver used in JAXLEY [20], gating variables typically depend only their compartment. This sparsity enables efficient computation of diag (iiag(Zii_ 1) In practice, we compute F, using the SPARSEJAC library [28], which constructs Jacobians directly from sparsity patterns without instantiating the full dense matrix.
"H;
previous value and the voltage of their corresponding
When is the diagonal approximation reasonable? We assume that the initial filtered covariance matrix Eo is diagonal. The EKF covariance update is:
Eqe
- [ H. R, H.
The procedure we described is equivalent to performing this update step with both Eqe-1 and H R, "H, diagonal. Is this valid?
In Appendix B, we show that for well-behaved HH dinetics-smoott transition rates, small integration steps, and not too frequent spiking- the dynamics Jacobian exhibits limited cross- cross-coupling for morphologies of moderate complexity. Under these conditions, a diagonal prior covariance and dynamics covariance can yield an approximately diagonal predicted covariance Enj--1-
The tighter constraint arises from the emission model. The túme-invariant matrix H RRR 'H, reflects how strongly electrode signals overlap across compartments. When a neuron is discretized into many compartments and the electrodes are densely packed around it it-as in Neuropixels or similar high-density probes-ii itts off-diagonal entries can become large, and approximating by a diagonal can severely degrade performance (cf. Appendix B for 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, H 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. R, H, is block diagonna,,,,,, 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 I, we assume that both the predicted and filtered covariance matrices take the form
P
E.
E [t-1,t).
ding(o;
where Σ E KK denotes the dense poltage-voltage covariance block, and of 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, is BD(K), then the BD(K) family is closed under the EKF prediction and update: if the initial covariance matrix En 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) x solve for the voltage block-sinee the measurement Jacobian H, has support only on voltages- and (ii) element- wise variance updates for the diagonal gating blocks (cf. proof of Proposition 1). The prediction step similarly preserves structure by propagating " 1|t-1 1 through the voltage-vottage sub- sub-Jacobian and
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Figure 2: Validation on simple morphologies. Panels A-C show results for a single-brand cell. A: Estimated gK over training iterations for three random initializations (mean +/- 1 sd). B: Distribution of inferred 9L under different observation noise levels aobs (colored dashed lines indicate the mean). C: Distribution of inferred INa for dbbs 1 μVV for various inference methods (orange lines indicate the means). Panels D-F show results for a model with twelve compartments. D: A bulti--ranch neuron and two sets of voltage recording electrodes (red and purple). The EKF correctly identifies the true location of the neuron (EKF in blue, true locations as black lines). E: Conductances inferred across 40 datasets simulated with dbes μV via EKF for the 3rd branch of the cell. F: Recovery of membrane voltages for EKF and diagEKF a compartment of the 3rd branch, averaged over the 40 aforementioned datasets.
updating gating variances via their local Jacobians, yielding overall complexity O(K3 + d) instead of (d for d-dimensional state (ef. proof of Proposition 1).
4 Validation on simple morphologies
4.1 Validation on single-branch neuron model
We first validated our method on simple synthetic neuron: a 10-compartment branch with sodium and potassium HH channels (Appendix E. E) 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 jmm from the soma. The goal was infer maximum conductances assuming that the cell's position was known. We simulated 40 recordings corrupted by Gaussian noise (a = 1 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 7.4μ V in saline and approximately 9.4pV in vivo [16]. Moreover, in practice we expect voltage traces to be corrupted by currents from other nearby neurons, which our model treats noise as well. Fixing the cell-probe distance at 5pm, we simulated 40 noisy EAPs at three increasing noise levels, ranging from V to 30μVV. As expected, found that higher noise leads to higher variance in the parameter estimates (fig. 2B). Nevertheless, estimates remained very close to the ground truth even V RMS, which represents challenging, high-noise regime (fig. E.3).
We also examined the effect increasing the distance between the cell and the recording site. Since extracellular voltage scales proportionally with 1/r, the signal-to -ggiie-t-nooiee ratio declined with distance.

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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. IBI.
Finally, we compared multiple inference peroaaches-Standann EKF, diagonal EKF, block-diagonal EKF, and direct neaa-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 um distance, 1 vV observation noise) and fitting the same 40 random traces, we found that the standard EKF, block diigo-ia 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 bbok-diigonnal 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 single neighboring compartment, leading to minimal cross-visibility and making a diagonal covariance approximation efficient.
4.2 Validation 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 '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 geometrie 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 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 LμV 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 Lngle-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 close to only single compartment, and each compartment has very few neighbors. which together limit compstcompttttnnt measurement interactions and explains why the diagonal approximation still yields informative reconstructions (cf. Appendix B).
geometry, compartments are long enough that most electrodes are
4.3 Robustness to model misspecification
In experimental settings, the true underlying biophysical dynamics of 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 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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Figure Robustness to model misspecification. Two types of channel misspecification are presented. A: missing channels (M-type K and CaL). B: Incorrect gating dynamics for present channels. In both panels, recovered membrane voltage traces using MSE minimization and EKF are shown. EKF provides accurate voltage estimates despite misspecified channel models, whereas MSE minimization fails.
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, retained the correct channel types in the inference model but perturbed the rate constants (:V) and 3(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 MSE minimization. These results suggest that our approach can tolerate moderate misspecification in gating dynamics. Together, these experiments demonstrate that EKF-basee inference, combined with stochastic modeling of dynamics, confers robustness to common forms of model misspecification.
Application to real morphologies
Having established the method's efficiency in controlled settings, we next apply it morphologically detailed neuron models derived from real reconstructions, including a retinal ganglion cell (RGC) and CA1 pyramidal neuron.
5.1 Application to 2D retinal ganglion cell
We applied our method to a morphologically detailed model 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 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 total of six parameters to estimate [29]. For details of the model, see Appendix F.
Here, assumed the spatial location of the neuron to be known and constrained to 2D plane, as typical in retinal MEA experiments [30, 31, 29]. Extracellular voltages were recorded using a simulated MEA with 30μm pitch, matching the geometry of the original 512-electrode array. To reduce computation, we used 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- legliekl diverged from its value at the true parameters, indicating severe model

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Figure 4: Inference of retinal ganglion cell (RGC) properties given multi-electrode array (MEA)
recordings. A: RGC cell geometry and electrode layout. Darker shaded MEA sites match the voltage traces shown in panel B. B: Simulated extracellular recordings at ten sites for noise level Jobs 0.5 μV. C: Marginal log-likelihood during optimization for diagonal EKF, D: The heatmap of the voltage block of H H (log scale) shows that the diagonal approximation is expected to fail. E: Marginal log- log-likelihood for block-diagonal EKF, F: Fitted voltage traces in the soma. The block-diagonal EKF infers the voltages accurately while the diagonal EKF fails.
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 conduetances (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 [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 aegion-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 '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 's spatial configuration (fig. 5A, black vs. blue), and accurately reconstructs membrane voltage (fig. 5D). It also recovers most of the maximum conduc- tances, though not all (fig. 5C). This outcome is consistent with a well-documented phenomenon in conductance-bul based neuron models: parameter degeneracy, where multiple combinations of channel conductances can yield virtually indistinguishable voltage dynamics [36, 37J. Despite this, our method reliably recovers the neuron's 's functional dynamics and spatial structure, demonstrating that biophysical and geometric inference from high-density extracellular recordings feasible in realistic, large-scale neuron models.

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Figure 5: Inference of membrane properties and locations of a C10 model of pyramidal cell. A: Pyramidal cell placed near a Neuropixel Ultra probe. Estimated compartment locations closely match true cell position. B: Marginal log-likes likelihood during optimization, converging to the ground truth value. C: Inferred conductances ONa and 9L in selected dendritic compartments. D: Recovered membrane voltage traces in dendrites of the stratum radiatum (left) and mecuosuummmolecule (right).
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-estimatie 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 computa- tional cost: even with diagonal or block -digonnl approximations, remains more expensive than MSE minimization, particularly for large models. Additionally, EKF approximations may bias pa- rameter recovery: while the block-diagonal EKF recovered parameters accurately our evaluations, the diagonal approximation models often exhibit parameter degeneracy, where different conductance combinations yield similar voltage dynamics, limiting identifiability even when predictions fit the data well.
less reliable in certain cell-probe geometries. Moreover, biophysical
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 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, method-based entiable simulation- openi 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.
combination of state-space modeling and differ-
10

---

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 (U19NS1 3201, ROINS131987, ROINSI13118 & (FIMH133778), the NSF/NIH CRCNS Program (R0INS100789) and the Sloan, Simons, and McKnight Foundations. was also supported by the German Research Foundation (DFG) through Germany's 's Excellence Strategy (EXC 2064 Project number 390727645) and the CRC 1233 "Robust Vision", and the European Union (ERC, "DeepCoMechTomee ref. 10108988).
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13

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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: com/ wttppllllooooooo//o///oooooooooe
JAXLEY [20] (version 0.7.0, Apache License 2.0): differentiable simulator for multi- models, supporting GPU acceleration, just-in-time compi- lation, and gradient-based parameter optimization. Available at: https: //gtthub/.//..//g//// com/ jaxleyverse/ jxxley.
compartment
(version 0.1.4, MIT License): library for state-space modeling and inference, used here for implementing the Extended Kalman Filter (EKF) and its (block) diagonal approximation. Available at: https: /ttpt/ggtttt//tte www.cooo/oooooo httmm/poobmm/d/aaa.....
DYNAMAX [21]
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.coom com wwwoommmffeee wtw.sscuuuetsssarrreeeeeeeeeeeee
For experiments large-scale neuron models (e.g., the retinal ganglion cell and C10 model of pyramidal neuron), we used an NVIDIA H100 GPU. All other experiments were run on a MacBook Pro with 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: httpa:///gthhbb wwt.co///caaaa http://annnnnnneaaaaaaeaaea hhh
B Validity of the Diagonal EKF
The true update step of the EKF is:
Σ/ Σー1 + H. R.
(7)
The procedure described by Chang et al. [27] matches the diagonals of the precision matrices:
diag Emji) = diag (Et)())))))))))))) + diag (H, R, H,).
(8)
Since the filtered covariances are assumed to be diagonal, this amounts to:
diag Eqe) (diag - + diag (H R, (e."
Their procedure is therefore equivalent to the standard EKF (7) with diagonal predicted precision matrix 2/(-1 and diagonal measurement information matrix H. R,'H..
In their work, Chang et al. [27] assume that F, = I. This drastically simplifies the computation of diag( (aggi) since Σmj--1 = E1-1 + Q, and both Q: and Σz-1je- are assumed to be diagonal.
This trivial Jacobian assumption not realistic in our case. But to simplify calculations, we assume that Ejj-1 is diagonal, so that diag(E-)) = diag -.
Our procedure becomes equivalent to the standard EKF with diagonal predicted covariance matrix Emj-1 and diagonal measurement information matrix H. R. H.
Hence, the validity of our diagonal EKF method relies jointly on the dynamics model (through Eti--) and the emission model (through H. HH)) We discuss each turn.
14

---

A diagonal approximation of Emj-11 is reasonable for well-behaved HH models
The predicted covariance verifies
11-11=FEE--1--11 +Q.
(10)
where F, - Jac(
and Q is the dynamics covariance. Assume that both 1/-1 and Q2 are diagonal. The question becomes: What is the structure of 112
Structure of F,, multi-compartment Hodgkin-Huxley models, each compartment voltage depends on its own state and 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
Z
Vts S:
g1 (L)
the Jacobian has the Mlock-arrow structure
A
B
B
(2)
B L
C
D
F
((2)
D
D(4)
0
0
Here A, encodes voltage-voltage coupling, B(") and c capture voltage-gate interactions, and D govern gate self-dynamics. In biophysical HH models, c and D are diagonal, and B,( are nearly diagonal (apart from weak solver-induced mixing). We assume that these off-diagonal elements are negligible and consider in the following that Bi()) are diagonal. The main source of non-diagonality lies in A,, through axial coupling between compartments.
Magnitude of cross-derivatives. We examine the relative size of the partial derivatives that make up F,, In the Hodgkin-Huxlee model, these correspond four types of dependencies: (i) gating variable on voltage (0V)), (ii) gating variable on gating variable (0x,41/0z,) (iii) voltage on gating variable (0V+1100z) and (iv) voltage voltage. We analyze each in turn.
(i) Gate with respect to voltage. Consider a generic gating variable x governed by
t = a<V)(1 - x) - 3VV),,
where (vV) and ((V) are voltage-dependent transition rates. The exponential update gives: E1+1 (V)) + (x: I =vo)) e A1/T(V)),
(11)
with
a(V)
1 T(V) a(V) + 3(V)
Trx (V) avV)+ + 3(V)'
Differentiating 3141 with respect to V gives:
0r = - ( + (x, - Zoo (V:)) K' (V (V2).
0v,
where k(V,) - follows that (using the chain rule on k))
x'cc (V:) -VV)))(V)- [a+VV) + 3(V)]
*((V) = -At ((V)) -A(((())0(V)))
Applying the triangle inequality and using that [2, - Ixal < (since [0, 1]) yields the bound
00
A((a+
< (Ia'l + 181)
Ate Atea-t(a+
0V,
+ 3
15

---

For small At. with moderate and smooth transition rates 5, this is of order O(At). confirming that Poltage-to-gate sensitivities are small under typical HH kinetics.
(ii) Gate self-derivative. It follows directly from eq. (11) that:
00++1 (00+(((())00)) E (0,1);
Or
so it O(1) and typically much larger than the cross term 0x when At is small and |a'), |84| moderate.
(ii)) Voltage with respect to gate. Next, consider the effect of gating variable 2, on the voltage update. In practice, JAXLEY uses an implicit solver. Here for the sake of clarity, we assume an explicit Euler step. For fixed compartment, it can be written as:
Al
Text
E Ege ac()) (V: Ee) + Gax Gax((V))
V4+ V2 +
(12)
Cm
S
where (t) is the activation of channel c. For the channel e* corresponding to gate 2t, the only
dependence is through ag- (Zz). giving
OV+1
At Ic de (xx) (Vi E.
Cm
In practice, ag- (2te) Cark with & E (1,...,,4 4) and C [0, 1]. Indeed recall that the activation a product gating variables (between O and 1) to some integer power. Then la's (xx)) S kC < Thus,
VV dx,
- k kt 9.- V E
With At 0.025 ms, Pe* $ 1 S/cm², and V - Fic1 S 100 mV, these terms are the dominant off-diagonal contribution to F, 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..
(iv) Voltage with respect to voltage. In eq. (12), the voltage of compartment is coupled to the voltage of its neighboring compartments via the axial conductance term:
(V.) GV. g. (v,6) V.()).
It follows that
(v
1
- acll) - Ex (HNO] j=i, ((d)
At C.
0v:(0
Al
j * i.
Thus, diagonal entries are ((1) (since the activations [a.(!)] < 1), while off-diagonal ones scale as ((1)).
What conditions support the diagonal approximation?
The preceding derivative analysis shows that, for broad class of well-behaved Hodgkin- models and sufficiently small time steps At, the state transition Jacobian F, exhibits limited cross- coupling between state variables. In particular, most off-diagonal Jacobian entries scale O (AAt). while the diagonal entries remain (1). The main exception is given by the voltage- gate sensitivities /V which may not be negligible during action potentials.
We refer a model as well-behavel when the following conditions hold:
The gating rate functions a<VV),, ((V) and their derivatives remain bounded over the relevant voltage range, ensuring that voltaee and voltage-voltage cross-derivatives scale as ((At).
16

---

m-gate (Na activation): rates
m-gate (Na activation): derivatives
a(V)
dmidv (ms mV
PM
5-
-0.25
(V)
-0.50
-50
0
50
-50
50
V (mV)
(mV)
h-gate (Na inactivation): rates
higate (Na inactivation): derivalives
dhdvv (ms mv m)))
1.0
(V)
(ms
0.5
(V)
a(V)
((V)
0.00
0.0
-50
50
550
50
V (mV)
(mV)
n-gate (K activation): rates
n-gate (K activation): derivatives
1.0
dnidV (ms mV")
0.010
o(V)
rate (ms
B(V)
0.005
(V)
0.000
0.0
-50
50
-50
V (mV)
(mV)
Figure B.1: Rate functions (left) and their derívatives (right) over plausible values of the membrane voltage for the standard HH model [6]. The rate functions and their derivatives are bounded.
Spiking activity is not excessively frequent, that large voltage- tattgge-gte sensitivities do not persist over time.
Under these conditions, all sources of non-diagonalliy in F, are negligible for small At. except for the voltage-gate derivatives. Standard Bodgkin-Huxley parameterizations typically satisfy the first condition: the transition rates and their derivatives are smooth and moderate magnitude, and choosing At sufficiently small ensures that most off-diagonal Jacobian entries remain negligible relative to the diagonal terms (see fig. B.
The dominant deviation from diagonality arises from the voltage-gad gate terms VV /0x During spikes, these derivatives can become large; at lower firing rates they typically smaller, but 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, 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 given by
Edj-1 F,E, +++
with diagonal prior covariance E,-1 diag(si,..... sa). The off-diagonal entries Egjr1 arise from terms of the form F F F,jk for + Assigning small process and initial variances to gating variables directly suppresses these contributions, even when the corresponding Jacobian entries are negligggee 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 the emission model, and more specifically whether diagonal approximation can be made for H, R, H,.
17

---

A
B
0
1.0
50 +
v (mV)
200
0.5
-50
-100
0.0
20
0
200
Time (ms)
50
4
(mV)
200
-50
2
-100
20
0
(ms)
0
200
Figure B.2: Structure of the dynamics Jacobian for a 100-compartment HH cable (At = 0.025 ms). A: Simulated membrane voltages for a low-frequency spiking regime (top) and high-frequency regime (bottom). B: Square-root-scaled heatmaps of the dynamics Jacobian averaged over time in both regimes. Consistent with our analysis, gate-to-voltage (0x++1/0V] and voltage-neighbor
/0v41/0v(0) couplings remain weak relative to the diagonal terms. The main distinction between regimes is that voltage-to-gate sensitivities (0V++1 increase substantially during high-frequeng, spiking, overwhelming the diagonal entries. At lower firing rates, these terms often smaller, yet can remain comparable to the main diagonal terms.
Diagonality of H. R, HH'
H:
In our work, we assumed whitened observation noise, R, = L. so diagonality depends solely on H. H,. Let the first K state dimensions correspond to compartment voltages and the remainder to gating variables. Since observations depend only on voltages,
H [
0],
H. E M
so that
HH) H 0 0
H H,
Crucially, extracellular potentials are linear in membrane voltages (ef. eq. (4)). Because this mapping static, H. H for all t. The diagonality of (H") " therefore depends only on the spatial arrangement of electrodes and compartments.
Let's analyze the structure of H Recall that the transmembrane current in compartment is
7
ext + S(t) Σ (v)) -V(),
jns
and electrode m measures
N
+
K
(3)
m
Tmi
where Tmi is the ceoetrode-oompartmenn distance and depends properties of the extracellular medium. Differentiating yields the "self + neighbors" structure:
H
Σ gG.))+
Σ brng
K
g(0g (1.5)
Tmj
Tmi
knj
18

---

Define the inverse inverse-distancd distance matrix R E MRM with Rmi = KR/mms. a diagonal matrix D with
Dj5= = D )
Σ glih)
kmj
and an axial coupling matrix G with
(
i~ J, otherwise.
Gij
0,
Then
H = R(G - D),
= (G-D)"
where W = R R.
Let C = G - D and denote its entries by cpis the support of column is the local neighborhood
S: = {i} U (sk: k ~ i}. Then
(
H
Σ
Cpi Cuj Wg.
i5
PES; SE8}
The electrode Gram matrix is
W pq
Σ
(13)
Tmp my
Using (13), Wog becomes large when electrodes are simultaneously close to compartments and Therefore,
(HH") H
grows when both neighborhoods. S, and Sj lic within the sensitive region of multiple electrodes. This regime is typical for high-density probes such as Neuropixels or MEAs, where many electrodes lic close many compartments, yielding strong off-diagonal structure in (H") H and limiting the accuracy of the diagonal EKF (cf. Section 5.1).
When is H"T H" diagonal? When electrode is simultancously close two different compartments, W is approximately diagonal and
Wey
Σ
Cpi Cpj Wpp-
PES,NS,
If the overlap Sy n Sy is small, the off-diagonal terms remain strongly attenuated. In this regime, (H²) HD close to diagonal, and a diagonal EKF update accurately captures the dynamics covari- ance.
concrete example a linear branch with N compartments of length L, and electrodes placed so that electrode lies at distance / < L from compartment i. Each electrode then primarily measures one compartment, yielding nearly diagonal W. Since a linear cable has minimal neighborhood overlap (S, 0 S, @ unless entries. This setting aligns with our single-brand branch experiment and explains why the diagonal EKF performs well there (Section 4.1).
i + 1), (H") "H" becomes tridiagonal with dominant diagonal
H
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 compart- ments. As a result, W' remains approximately diagonal, and (H") H" preserves a meir-tridaagona structure, with the only substantial off-diagonal contributions occurring 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 Bodgkin-HHxxler models, the state transition Jacobian F, exhibits limited cross-coupling, with most off-diagonal entries suppressed by Al. The primary exception arises from voltage-gat 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 nntuurly
19

---

justified by their bounded dynamics- aaloossswlloss the predicted covariance to remain approximately diagonal in practice. contrast, the problem geometry may cause H, R, to deviate strongly from diagonality, which can undermine the validity of diagonal filtered covariance approximation.
Limitation. The above argument for the predicted step is heuristic and intended to motivate when diagonal approximation of the predicted covariance is reasonable. In practice, solver-induced coupling between compartment voltages can introduce additional non-diagonality, particularly partiucllly 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 constrained to 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 we assume that both predicted and filtered covariances take the form
E, Σ/
diag(o?1-)]
E (t-1,4)
where 2t E RRKK is dense and of 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 L:
1. The dynamics covariance Q is BD(K).
2. The dynamics Jacobian F; is BD(K).
If Eo is BD(K) then both predicted and filtered covariances remain BD( K) for all subsequent time steps.
Proof of Proposition I. Writing
[vrig:], assume Σ1-1j(-1 is BD(K):
Σ or 1--1
0
Σ, 11-1/t-1-
diag(o? diag(o?-11 ?-
Under Assumptions 1-2, F, blkdiag( diag(d,)) and Q₁ share this structure. The EKF Q then yields
prediction Eqj-1 F,
1 -F,E,119-11
Emj-1
A, EVE
-
||-
A.
Q
0 dig - + 97)
0
For the update step, use the identity:
Eqjt (Σ + H "
Now,
Se Emp-1
[ (E
-1
Σ 1
diag 1
#|2-1
0
diag (مo
Since H, [H 0).
J
0 0
H R. H, =
J=
R, H.
Therefore the sum inside the inverse is block diagonal,
+
J
diag (oo |
D H H R, H
H,
0
20

---

and inverting yields
(
(E
-1
Sqee
diag(o 2
Hence both predicted and filtered covariances remain BD(K). Note that only the voltage block is updated.
Let's 's derive the update for the filtered mean. We use:
Maje = Maje-1 + Ezje H, R2 ly: - (o(eeeee)))
Let the residual be
Ft := Yr (o
Since H, = [H] 0] and
H R.,r r1
H R, re
diag(o i
(
we obtain the block update
_1 z-1
Σ H.
R, re]
Mij-1
+
Egje Hj
R2
rg
Peje
-
Thus only the first K states (the voltages) are updated; the gating variables remain unchanged, which consistent with the observation depending solely on the voltage block.
This structure reduces the EKF complexity from (dd²) time and Odd²) memory to O(K3 + d) and O(K² + d), respectively, a major gain since K < d in typical large neuron models.
C.2 Validity of the block-diagonal approximation
We now discuss the assumptions of Proposition
Assumption (BD(K) process noise). In our setting, process noise is injected independently per compartment and gating variable, so Q₁ is naturally BD(K) since it is diagonal.
Assumption (BD(K) Jacobian). The exact Jacobian is not strictly BD(K) but has the arrow-shaped form
A
B
B
B
L))
G(
D
F,
C
(2)
D (2)
(14)
:
C (F)
0
D (.)
where A, captures voltage-voltage coupling and the other blocks represent voltage-gon gate interactions and self-dynamies. Under the BD(K) approximation, we set
B ( = c((e) 0, A4,
yielding
F; = bllddaag A, D." ..... (T)).
D diagonal.
This corresponds to neglecting cross-sensitivitte between voltages and gates, ie. assuming 100, R0 and 0r.+1/00 /0V, Rs for each gate æ.
As established in Appendix B, the gate-to-voltage terms (0x441/0V) are negligible under well- behaved HH dynamics. Voltage-to-gate sensitivities (0V++100x;) grow primarily during frequent
21

---

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 block-diagonal covariance structure remains reasonable while still support- ing dense callrtcompart voltage coupling through A,,
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.
model represents the neuron as a collection of compartments, each treated as point source of transmembrane current located its center. Hence, estimating neuron's 's spatial configuration reduces to estimating the 3D positions of its compartments. According to eq. (4), the extracellular potential measured given electrode depends only 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.
Our
We first consider the case where all electrodes lie on single line (fig. D. Fix a single compartment some position in 3D space (black). The set of all locations that preserve its distance to given site is a sphere. The intersection of these ceceeeess-oorrspponnieee 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.
to multiple sites aligned on a line- is
Next, suppose the electrodes lie on two parallel lines, forming a planar grid (fig. D.IB). 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 simultancously preserve distances 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.IC 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 (c.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 multi- coalii-ommpartmentt 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, the cell is known to consist of a single linear branch, then the configuration of all compartments restricted lie along a line. In this case, the full morphology is only ambiguous up to a reflection of that line across the recording planc-ssbstanniaally 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-suu that their electrodes do not lie in the same plane-remove the reflection ambiguity and enable unique localization of each compartment in 3D space.
22

---

A
B
z=0
Y
Y
Figure D.1: Unidentifiability of the positions. A: The distance between the compartment (black) and each electrode (blue) is unchanged as moves along the blue dashed circle. follows that any position of the compartment on the circle yields the same observations at all the electrodes. B: Compartment's distance to each blue (resp. green) electrode is unchanged as moves along the blue (resp. green) dashed circle. Hence, the grey location is the only other position that yields the same observations as the black one at all electrodes. C: Panel viewed from above with now three parallel lines containing electrodes (green, blue, orange ones). Since all the electrodes lic in 2D- 2D-plane (z = 0), the only position that yields the same observations as the true location at each electrode is the true location's reflection with respect to the plane 0.
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 em, and the specific
membrane capacitance was fixed to C. = 1 jF /uF/cm² across all compartments, While we assume homogeneous extracellular resistivity, this simplification neglects known spatial variability [25], which could be addressed future work. Extracellular voltages were recorded at a sampling rate of 40 kHz.
voltages and channel states were initialized at their steady-state values at rest, and thus we assumed the initial state be known in the EKF, We set the initial covariance matrix of the EKF to E, = 0.II. Observation noise was also assumed to known, as commercial devices that record extracellular potentials typically provide noise specifications [16, 38].
In Sections 4.1 and 4.2, where the state dynamies were well-specified and deterministic, we fixed the standard deviation of the membrane potentials and channel states to small values: o, 0.0001 mV //ms>/2 and state = 0.00001 ms 112 (same for all the channels) for all com- partments. 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 μm and radius 2 jum equipped with standard Bodgkin-Huxlle (HH) sodium, potassium, and leak channels [6]. The maximum conductances and reversal potentials provided in Table I. A constant current of amplitude 1A was injected for 20 ms.
As the neuron neuron-clectrene electrode distance increased, the signal-to- -ggnal-to-noise ratio (SNR) decreased, resulting in higher bias and variance in inferred conductances (fig. 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.I The diagonal EKF introduces a slightly larger bias, but its voltage reconstructions remain highly accurate (fig. E.2). This is consistent with our analysis
23

---

A
30WW D," 10pM
Frequency
TpV
Frequency
Frequency
0.00025
0.0003 0.00035
0.020
0.025
9. (S/om")
9k (S/om²)
9ha (Slom")
B
d- 50mm0
Frequency
5.0
Frequency
Frequency
5.0
trae value
2.5
2.5
0.0
0.0
0.00025
0.0003 0.00035
0.015
0.020
0.11
0.12
0.13
9, (S/cm²)
(S/cm?
Sha (S/cm")
0.0003
0.122
0.0204
9. (Slom))
9 (Sion')
Shu (Siam)
True value
0.12
0.02
0.000285
Diag
MSE EKF BD Diag
Diag
Figure E.I: Validation on synthetic single-branch. A: Distributions of inferred maximum conduc- tances under different observation noise levels obb (colored dashed lines indicate the means). B: Distributions inferred maximum conductances for different cell- probe distances (colored dashed lines indicate the means). C: Distributions of inferred maximum conductances for various inference methods with aohe μV and 5um (orange lines indicate the means).
Inferred voltage (mean a 250)
True vollage
50
v (mV)
0
(mV)
0
-100
-100
Time (ms)
Time (ms)
Time (ms)
Figure E.2: Recovery of membrane voltage with diagonal EKF in the single-branch model for compartments (left), 5 (center) and 10 (right). The mean and standard deviation are estimated by considering 40 randomly simulated datasets at fixed observation noise Jobs 1v Despite small bias in the estimation of the parameters, the voltage traces are still reconstructed accurately.
Na
K
Leak
Maximum conductance (S/cm?)
0.12
0.02
0.003
Reversal potential (mV)
53.0
107.0
-8.5188
Table I: Biophysical parameters of the single-branch model.
Appendix only the first (soma) compartment lies near the electrode, with all others much farther away, so cross-visibility is minimal and diagonal covariance approximation is sufficient. As observation noise increased, estimation accuracy declined (fig. E. A)), but even at 30 μV RMS challenging noise regime (fig. (333))estimatee, remained close to the ground truth. Notably, the

---

A
B
0.075
0.15
0.075
Vollage (mV)
Vollage (mV)
0.025
0.05
Voltage
0.025
-0.025
-0.025
-0.05
0
10
20
Time (ms)
Time (ms)
Time (ms)
Figure E.3: Extracellular voltage traces simulated for different noise levels the single-branch
model. A: Jbbs L,V, B: obss 10,VV, C: dobs 30pV represents challenging noise regime. The electrode is 5ymn away from the cell.
6,
50
0 10pV (mean 2SD)
30uV (mean 2SD)
V (mV)
True voltage
-100
0.0
2.5
5.0
7.5
10.0
20.0
Time (ms)
Figure E.4: Recovery of membrane voltage for different observation noise levels in the single- branch model. The mean and standard deviation are estimated by considering 40 datasets simulated fixed observation noise. Even in challenging noise regimes (aabs 30p V), the induced variance on the voltage trace reconstruction is negligible.
induced variance on the reconstructed voltage traces small (fig. E.4), indicating robustness even under challenging conditions.
F.2 Validation on multi-branch neuron model
We also evaluated 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 jam, for total of 12 compartments. Each compartment contains standard Hodgkin-Huxled 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
set to 150 Ω cm. A
current of 2 nA was injected in the first compartment of branch (soma) for 20 ms.
With EKF, all the maximum conductances were inferred accurately for all 40 datasets simulated with Johs V (fig. E.5).
With the diagonal EKF, the recovered conductances exhibit more pronounced bias than in the single- branch case, particularly in branches and (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 distinct compartments. Since each compartment only has two neighbors (except at the branching node), this limits off-diagonal measure- ment coupling. As detailed Appendix B, this geometry ensures that most of the off-diagonal entries of (H*)* H remain significantly smaller than terms near the diagonal (fig. F.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.
25

---

Na
K
Leak
Reversal potential (mV)
53.0
- -077.0
-88.5188
Table 2: Reversal potentials used in the multi-branch neuron model.
Branch
9Na
9K
9Leak
Length
1
0.12
0.02
0.0003
24.0
2
0.08
0.03
0.0004
50.0
3
0.10
0.008
0.0001
50.0
Table 3: Maximum conductances (in S/em²) and branch lengths (in yu) in the multi-branch neuron model.
True value
Branch
Branch 2
Branch
0.0002
0.0004
0.02
0.04
0.1
0.2
9L (S/cm') )
9k (S/cm) )
9Na (S/cm )
Figure E.5: Validation of the EKF on synthetic multi-branch neuron. The maximum conduc- tances 9L. gK and INa were inferred accurately for all the branches, across 40 datasets simulated with abbs LvV.
Omitted Ion Channels. To test robustness to missing channels, we simulated data from a four- compartment branch model (total length 96pm, radius 2pm) 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 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 Jobs 0.1 μV. Biophysical parameters for the simulation model are shown in Table 6.
During EKF inference, we fixed the standard deviation of the channel states to Ossate 0.00001 ms 1/2 (same for all the channels) and treated o 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 likelioodoood.
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 Klogkkin-Hxlley kinetics [6]. The neuron was single cylindrical cable (length 24 jum, radius um), discretized into four compartments. A constant current was injected into the first
26

---

True value
Branch
Branch
Branch 3
0.0002
0.0004
0.02
0,04 9k (Slom) )
0.1
0.2
(S/cm 2
9L (S/om²)
9Na
Figure E.6: Inferred parameters using the diagonal EKF on a synthetic multi-branch neuron. We show results across 40 datasets simulated with dhs pV. The recovered conductances exhibit more pronounced bias than in the single-branch case, particularly in branches 2 and 3.
Interred vollage (meen 2SD)
True voltage
50
50
50
v/ ((MV)
Vo (muu
v (mV)
-50
-50
-50
-100
-100
Time (ms)
20
Time (ms)
Time (ms)
4
V VmVV)
V(mV)
V (m/)
-50
-50
-50
-100
-100
Time (ms) 20
0
Time (ms)
20
Time (ms)
20
Figure E.7: Reconstructed voltage for first (top) and last (bottom) compartment of branch 1 (left), 2 (middle) and (right) using the diagonal EKF on synthetic multi-branch neuron. Voltage traces are averaged over 40 randomly simulated datasets with oohs LμvV. Although conductance estimates exhibit noticeable bias, reconstructed voltages remain accurate up small shifts, consistent with a regime in which the diagonal approximation remains adequate.
compartment (soma) for ms. Extracellular voltages were recorded at 10 different locations with noise level oès 0.1 W. Biophysical parameters for the simulation model are shown in Table 7.
During EKF inference, we treated the voltage standard deviation a, and channel noise levels OK, Na. Leak as learnable parameters. While only the sodium and potassium kinetics are misspeci- fied, we allowed separate noise levels for each channel type to provide flexibility in how uncertainty is accounted for. We also used the same de, Øk, Na, Leak across all compartments, though future work could consider compartment-sper specific noise parameters. before, we performed grid search over learning rates and initializations, selecting the result with the highest marginal log- leglliikehhood.
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.
27

---

(Hm)
-
40
Y (um)
0
0
100
X (um)
200
-40
300
grid 1
grid 2
cell
locations (inferred)
Figure E.8: Diagonal EKF accurately recovers compartment center locations. Estimated 3D positions are averaged over 40 randomly simulated datasets with Jbbs 14V. Although conductance estimates exhibit noticeable bias, the reconstructed compartment locations remain accurate, consistent with regime in which the diagonal approximation remains adequate.
-8
10
log(((H)
-12
10
-14
0
5 10
Figure E.9: Heatmap (in log scale) of H H", the voltage block of H H. The largest coefficients occur between adjacent compartments and at branch points, yielding approximately tridiagonal structure. Off-diagonal entries corresponding to distant compartments are substantially smaller, reflecting the conpaammartt morphology, where few electrodes are simultaneously close two non-neighboring compartments. This structure helps explain why the diagonal approximation remains reasonably effective in this setting.
F Retinal Ganglion Cell (RGC) Model
We use a morphologically detailed model of a retinal ganglion cell (RGC), based on reconstruction from Vilkhu al. [29]. The morphology was originally provided as a NEURON .hoc file,' which 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.
28

---

Sodium (Na")
Potassium (K*)
0.32(V + 47)
0.032(V + 45)
4'm
1 --e-(V+17/4
On 1 - e-e-(V+15))
0.28(V 20)
8m
Bn 0.5e (V+50)/40
e(V+20)/3
0.128 -V+443/18
4
8h
(V+20)/5
Table 4: Rate functions for sodium (Nat) and potassium (K*) channels used in the single-branch model for the missing channel experiment.
M-type K* (KM)
L-type Ca (CaL)
+
27
Px0 1 e (33(5)35)
9 = 0.055
e -V
4000
8g 0.94 -V+45-(++7)//1
Tp 3.3 0.05 +e 0.05(V+335)
0, O. 0.000457 e eVV+-333/50
0.0065
B, 1 (V++15)/25
Table 5: Rate functions for M-type potassium (KM) and L-type calcium (CaL) channels used in the single-branch model for the missing channel experiment.
Na
K
Leak
KM
Cal.
Maximum conductance (S/cm?)
0.05
0.005
0.0001
0.000004
0.0001
Reversal potential (mV)
50.0
-90.0
-70.0
-90.0
120
Table 6: Biophysical parameters for the single-brane neuron model used in the missing channel experiment.
Na
Leak
Maximum conductance (S/em?)
0.12
0.02
0.003
Reversal potential (mV)
53.0
- 107.0
-
-88.5188
Table Biophysical parameters for the single-branch neuron model used for the kinetic mismatch experiment.
The ion channel model consists Hodgkin- (HH) conductances: fast sodium, delayed- rectifier potassium, and leak channels. Voltage-dependent rate constants listed in Table 8, and channel reversal potentials in Table Region-specific maximum conductances for soma and dendrites are given in Table 10. All values were taken directly from Vilkhu et al. [29], with minor adjustment to the leak reversal potential to ensure resting membrane potentials near -65 mV across compartments. Note that Vilkhu al. [29] also consider two additional channels (a calcium channel and a calcium- gated potassium channel as Fohlmeister et al. [40]), which we neglected this experiment. Including these channels could be the subject of future work.
The axial resistivity was 143.2 cm, the membrane capacitance was pF/cmm across all compart- ments, and the extracellular resistivity was 1000 Ω cm.
29

---

Compartment
Compartment 1
50
v Vmm
100
True Vellage
20
0
20
Time (ms)
Time (ms)
Compartment
Compartment
50
50
v (mV)
V (mV)
1000
-100
20
20
Time (ms)
Time (ms)
Figure E.10: Robustness to misspecified channel kineties. The EKF smoothed posterior mean voltage recovers the true voltage trace in each of the four compartments, whereas MSE minimization fails.
Sodium (Na*)
Potassium (K*)
2.725(V + 35)
0.09575(V 37)
Om 1-e-(V+35)/1)
On 1- -VV+3777
8m 90.83 -
8n 1.915 e.915e- -V4-47/880
On = 1.817 e (V+52)/20
27.25
3n + (V++222)110
Table 8: Rate functions in the RGC model.
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 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 be known. We set the initial covariance matrix to Σg 0.II. Observation noise was also assumed to be known, 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: o, 0.001 mV /mm and stare 0.0001 ms 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).
30

---

Na
K
Leak
Reversal potential (m V)
60.60
-
-101.34
-67.1469
Table 9: Reversal potential for ion channels in the RGC model.
INa
[Leak
Dendrites
0.06
0.035
0.0001
Soma
0.06
0.035
0.0001
Table Maximum conductances in the RGC model (in S/em²).
Region
gL
gNa
gK
Soma
[0.0001, 0.005]
[0.05, 0.1]
[0.01, 0.05]
Dendrite
[0.0001, 0.005]
[0.05, 0.1]
[0.01, 0.05]
Table 11: Initialization bounds for the maximum conductances in the RGC model (in S/em?).
Dendrites
9L (Siem'
True value
True value
0.0002
0.0001
9r. (Siom)) }
True value
True value
0.04
0.03
9u (Sioml)
True value
True value
0.08
0.06
1000
2000
3000
1000
2000
3000
Iteration
Iteration
Figure F.l: Inference of retinal ganglion cell (RGC) properties given multi-electrode array (MEA) recordings (block-diagonal EKF). Estimation of all the maximum conductances at noise level aots 0.1pV.
G CA1 Pyramidal Neuron (C10 Model)
We use the C10 model of CA1 pyramidal neuron, originally developed by Cutsuridis et al. [32]. The morphology provided as a NEURON hhoc file,", which we converted 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. the neuron divided into 15 anatomical sections.
Following Cutsuridis al. [32]. we applied the compartmentallizttoo heuristic of Carneaale and Hines [7] to determine the number of compartments per section. Specifically, a section of length L. is discretized into
L
0.1 0.1 + 0.9 2
2/
31

---

A
Inferred vollage Initial guess
B
C
True vollage
0
0
v kmmy
V (m)
-25
-25
-25
-50
-50
-50
0
10
10
Time (ms)
E
Time (ms)
F
Time (ms)
0
VmV)
(mm))
v (mV)
-25
-50
-50
-50
10
10
10
Time (ms)
Time (ms)
Time (ms)
Figure F.2: Membrane voltage recovery for the RGC model in six compartments (bloc-diagonal
EKF). A: soma, B: compartment 31, C: compartment 61, D: compartment 91, E: compartment 106, F: compartment 122. The ordering of the compartments is determined by the SWC file.
LM-t
LM-M
LM-T
RadDist
RadMed
RadProx
Soma
OriProx
OriDist
axon
Figure G.1: Morphology of the C10 model. LM-t: lacunosum moleculare (thin), LM-M: lacunosum moleculare (medium), LM-T: lacunosum moleculare (thick), RadDist: radiatum distal, RadMed: radiatum medial, RadProx: radiatum proximal, OriProx: oriens proximal, OriDist: oriens distal.
where the -requencyyddeeendett electrotonic length is defined Xy = 2/ xfR.C.. with d the diameter, R. the axial resistivity, and Cm the specific membrane capacitance. This yields total of 60 compartments.
d
Each compartment contains Boddkin-Huxlldr (HH) conductances: sodium, potassium, and leak. The sodium current, however, uses three-gate formulation with gating vector ANa (m, h) E [0, 1)3. The total transmembrane current released by a compartment is given by:
Im 9Na m hss( -ENa) + gK ni 2V Ek) (V -E.).
(15)
32

---

Location
m
1
1
Mac
V+48
11:0
V+403
Soma/Axon
1 + 6
1
1 + e
0.05 ms
Th = 1.0 ms
Tn = 3.5 ms
1
Mac
V444
1 + Vi:45
1 1+ V-442
Dendrite
1 +
= 0.05 ms
Th = 0.5 ms
Tn = 2.2 ms
Table 12: Steady-state functions and time constants for gates m, h, and n in the C10 model.
Formula
+ Na_att e (660)2
800 (V)
+ e(V+60)
(..8854(V+6)
(V)
0.0003(1 +0.127(++601) 3.0
Table 13: Kinetics of sodium attenuation gate s as a function attenuation factor Na_att in the C10 model.
Section
Diameter (um)
Length (um)
Na att
Soma
10
10
0.8
Axon
1
150
1.0
RadProx
4
100
0.5
RadMed
3
100
0.5
RadDist
2
200
0.5
OriProx
2
100
1.0
OriDist
1.5
200
1.0
LM-T
2
100
0.5
LM-M
1.5
100
0.5
LM-t
50
0.5
Table 14: Detailed morphology of the C10 model and sodium channel attenuation (Na_att) per section.
Moreover, unlike the canonical HH model, the gating kinetics vary across anatomical regions [41]. Gating variables evolve according first-order kinetics:
dA dt
Tx (V)
dac (V) -
The steady-state functions A (V) and time constants (V) for m, h, and n vary by region and listed in Table 12.
The third gating variable models sodium attenuation. Its dynamics consistent across regions except for region-specific attenuation factor Na_att att E [0, 1). The expressions for 8.0 (V) and Tx(V)
fixed Na_att at_ given at temperature T 34 (Table 13).
The attenuation factor Na_att defined per section based on Migliore et al. [35] (Table 14).
Reversal potentials follow Cutsuridis et al. [32], with small adjustment to Eg 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. 150 Ω cm, specific membrane capacitance C.. af/cm?, and extracellular resistivity Pext 300 Ω cm. While we assume homogeneous extracellular resistivity, this simplification neglects known spatial variability [25],
33

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Na
K
Leak
Reversal potential (m V)
50.0
80.0
- 122.00
Table 15: Reversal potential the C10 model.
Region
9L
gNo
OK
Soma
0.0002
0.007
0.0014
Axon
0.000005
0.1
0.02
OriProx
0.000005
0.007
0.000868
OriDist
0.000005
0.007
0.000868
RadProx
0.000005
0,007
0.000868
RadMed
0.000005
0.007
0.000868
RadDist
0.000005
0.007
0.000868
LM
0.000005
0.007
0.000868
Table 16: Maximum conductances in the C10 model (in S/cm²).
Region
9L
JNa
OK
Soma
[1e-4, 4e-4]
[0.003, 0.015)
[0.001, 0.005)
Axon
[le-6, le-5]
[0.08, 0. 12]
[0.015, 0.03]
Dendrite
[1e-6, le-5]
[0.003, 0.015]
[0.0005, 0.002]
Table 17: Initialization bounds for the maximum conductances in the C10 model (in S/em²).
which could be addressed 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 40 4HkHz sampling rate, with noise standard deviation Jobe 0.
The location of the cell was generated by applying a rigid-body transformation-conss 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.
of
We fit this model using the block-diagonal EKF All voltages and channel states were initialized at their steady-state values rest, and thus we assumed the initial state be known. We the initial covariance matrix to Eg = 0.1I. 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: a, = 0.001 m m and staa 0.0001 ms 112 (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, 2Tr, and translations were initialized to place the soma near its true location. We chose this initialization to reduce the risk of converging 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.
refretee solution, which produces identical extracellular signals when
Although the marginal log- lieeliodl converged to its true value after few hundred iterations (fig. G.3F), and both the voltage traces (fig. G.3A-E) and compartment positions (fig. 5A) were
34

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(8amm))
(Bkm7)
true
0.003
6.00000
c.00215
2020
2000
2000
10-6
0.00
0.00
08 0.01
2000
2000
2000
000000
0.0044
2000
2000
2000
(8lon)
0.0129
0.0015
of 0.0010
2000
2000
2000
1e-15
0.0100
0.0015
0.0079
0.0010
2003
2000
2000
9, (80m))
Du. (konr)
0.000
(6bmr)
0.0000
0.0016
2.6
0.004
0.0010
2000
2000
2000
40
(itee)
0.0100
(96mm))
0.0010
value
mp
b/0011
Sivo walko
(Eer))
0.0010
0000
Irua UM
0.000
3200
2000
Figure G.2: Inference of the maximum conductances of the C10 model in all the regions. A: Soma, B: Axon, C: OriProx, D: OriDist, E: RadProx, F: RadMed, G: RadDist, H: LM. While most maximum conductances converged to their true values, many especialyy the leak conductances-werc still fluctuating.
accurately recovered across the cell, the leak conductances did not converge to their true values or even continued fluctuate (fig. G.2) reflecting the well-known degeneracy in conductance- based neuron models, where multiple parameter combinations can produce nearly identical voltage dynamics.
H DiagEKF vs. EKF runtime
We compare the computational efficiency the standard Extended Kalman Filter (EKF) and our sparse diagonal approximation (DiagEKF). To isolate performance differences, consider simpli- fied neuron model: a 24 μm cable equipped with standard Hodgkin Modgkin-Huxley channels [6]. A constant current is injected for 20 ms, and extracellular voltage is recorded at single electrode at 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. Impor- tantly, the gating dynamics in these models are highly local: each variable typically depends only
35

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A
Inferred vellage
B
Inilial guess
50
voltage
50
(mV)
(mV)
v (mV)
-50
-50
-50
Time (ms)
30
Time (ms)
30
Time (ms)
D
E
F
50
50
Marginal loo-lllkeeellle Sgg-ilkellhood
1e6
(mV)
V (mV)
2.5
True value
0.0
-50
-50
-2.5
Time (ms)
30
Time (ms)
30
0
Iteration
2500
Figure G.3: Recovery of the membrane voltage in different regions of the pyramidal cell. A: Soma, B: Axon, C: RadDist, D: OriDist, E: LM. F: Marginal log- eegllikliihood trace during optimization. Despite some maximum conductances failing to converge to their true values, the membrane voltage was recovered accurately across the cell and the marginal log-likelihood converged its true value, reflecting the well-known degeneracy in conductance- neuron models.
A
B
0.8
EKF
Sparsity
Runtime
diagEKF
0.6
0
200 400
600
800
1000
200
400
600
800
1000
State Dimension
State Dimension
Figure H.1: Computational efficiency of our diagonal approximation to the EKF filtering distribution A: Sparsity of the Jacobian of the dynamics function over state dimension. Especially the filtering covariance
large biophysical models sparse, suggesting that sparse approximations could provide accurate but efficient inference. B: Average runtime over 500 updates for the EKF and sparse diagEKF versus state dimension for a 20ms time horizon. The EKF with diagonal approximation is twice as fast as EKF with a dense covariance large models.
its own previous value and the voltage of its corresponding compartment. As a result, the Jacobian of dynamics function quickly becomes very sparse when the number states increases (fig. H.
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.I.. DiagEKF achieves approximately 2x speedup over the dense EKF in high-dimensional settings. This highlights the computational advantage of our diagonal approximation in large biophysical neuron models.
36

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Reviews:
[{"review_id": "grhZDfYh4E", "reviewer": "Reviewer_co5q", "summary": "This 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.", "questions": "- 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?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\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).", "was_revised": true, "final_justification": "This work introduces an interesting framework for estimating biophysical neuronal models from extracellular voltage recordings. The limitations pointed out by myself and other reviewers are well acknowledged by the authors, and the authors' proposed changes will improve the clarity and framing of the paper. I continue to think this work is a valuable and interesting contribution, and will maintain my score.", "rebuttal": "Firstly, we would like to thank the reviewer for taking the time to review our submission and for their positive comments! Below, we address each of the four questions raised and hope our responses provide the necessary clarifications:\n\n- The reviewer asks for a clearer explanation of how the stated latent space dimensionality and the number of parameters being estimated arise from the model and simulation setup. We agree with the reviewer that the dimensions of the latent and parameter spaces should be made clearer, and we will revise the paper accordingly. In general, the latent space at time $t$ includes the membrane voltage in each compartment as well as the full set of gating variables defined for all the ion channels. If the model has $ N $ compartments and compartment $i$ has $n_i$ gating variables, the latent state at time $t$ has dimension $N + \\sum_{i=1}^N n_i$. As for the parameters being estimated, we include the maximum conductances of the ion channels and, when specified, additional geometric parameters related to the neuron's spatial position relative to the probe.\n\n- The reviewer asks why parameter estimates become biased when the observation noise is large. When measurement noise is high, the EKF places less weight on the observations and relies more heavily on its internal predictions of the system’s state, which are based on the current parameter estimates. Since these parameters are still being learned, it means that the filter adapts more slowly, and the estimates tend to stay closer to their initial values. Another way to understand this effect is through the marginal likelihood: as measurement noise increases, the gradient of this objective shrinks in magnitude, flattening the landscape. A flatter surface makes it easier for optimization to stall in regions that are close to—but not exactly at—the true parameter values. \n\n- The reviewer asks when bias should be considered significant from a modeling perspective. Bias becomes a concern when it leads to inaccurate reconstruction of the membrane voltage or other latent states. In our setting, the primary goal is to estimate biophysical parameters that enable accurate recovery of the membrane and extracellular voltages. If the inferred parameters achieve this—despite being biased relative to ground truth— while being consistent with typical values found in the literature, we consider the result acceptable. In Fig. 4, for instance, we observe some bias in the estimated conductances, but the reconstructed voltages remain highly accurate, so the bias is not problematic in practice.\n\n- Finally, the reviewer asks why the voltage reconstructions in the high-noise regime remain accurate for the retinal ganglion cell, even when the conductance estimates are biased, and whether this relates to the identifiability issues discussed later. We believe this is an insightful observation and agree that the accurate voltage reconstructions, despite parameter bias, reflect the degeneracy discussed later on. Specifically, multiple configurations of conductances can produce nearly indistinguishable voltage dynamics. We acknowledge that this connection should have been made earlier in the paper, and we will revise the relevant sections to clarify the role of parameter non-identifiability."}, {"review_id": "M5259wApmJ", "reviewer": "Reviewer_p6UN", "summary": "This 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.", "questions": "1. 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.", "limitations": "Yes.", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strengths:\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).", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for taking the time to review our submission and for their positive comments! We were particularly pleased by the comments on the novelty and clarity of our work. Below, we address each of the three suggestions made by the reviewer in the \"Questions\" section:\n\n- The reviewer suggests including a brief analysis—either visual or quantitative—comparing the performance of the full EKF, diagonal EKF, and MSE-based optimization. We thank the reviewer for this helpful suggestion. We agree that such a comparison would strengthen the paper by providing concrete evidence of the relative performance of each method. We will include a quantitative and visual analysis comparing full EKF, diagonal EKF, and MSE minimization in the camera-ready version.\n\n- The reviewer notes that applying our method to in vivo recordings will likely require additional modeling layers. We fully agree and will revise our conclusion to emphasize this point more strongly.\n\n- We fully agree with the reviewer on the necessity of providing efficient, well-documented code—including benchmarks and usage instructions for both small- and large-scale models. This is part of our plan, and it will be made available in a public repository by the camera-ready deadline."}, {"review_id": "n1zNPIvuzr", "reviewer": "Reviewer_ntGU", "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.", "questions": "See above weaknesses.", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- This is a novel application of the SSM methods to reconstruct complex HH parameters. \n- 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).\n- The simulations also included results on real neuron morphologies, namely a 2D retinal ganglion cell and CA1 hippocampal cell.\n- The formulation of the SSM and scaling of the EKF, as well as the use of SPARSEJAC, are novel here.\n\nWeaknesses:\n- Is the diagonal Gaussian assumption in the EKF a reasonable approximation in this context? Detail how valid this is, if possible quantitatively.\n- The application of the methods on real Neuropixels or MEA data would strengthen the manuscript.\n-  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?\n- It would be good to compare the methods to alternative inference methods apart from MSE minimization.", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for taking the time to review our submission and for their positive comments! We were particularly pleased by the comments on the novelty of our work. Below, we address each of the four points made in the \"weaknesses\" section of the review and hope to provide the necessary clarifications:\n\n- The reviewer raises an important question about the validity of the diagonal approximation used in the EKF. We adopted the diagonal approximation to the EKF covariance primarily to reduce memory and computational cost. While this simplification is not biophysically exact, it is guided by observed structure in the filtering covariance matrices obtained during full EKF runs. In multi-compartment models, voltage updates depend on neighboring compartments, and channel states are tightly coupled to local voltage. This suggests that latent variables within the same compartment—and to a lesser extent, across adjacent compartments—should be more correlated, leading to filtering covariances with an approximate banded structure.\nTo investigate this, we ran the full EKF on synthetic data (with known parameters) and examined the resulting filtering covariance matrices over time. We observed that, in many time steps, these covariances exhibited an approximate banded structure, with the largest values often concentrated on the main diagonal. This observation motivated us to start with a diagonal approximation as a computationally efficient baseline that captures the dominant structure.\nEmpirically, we found that the diagonal EKF performs very well in Hodgkin–Huxley models, including large multi-compartment neurons. In particular, we validated it on the retinal ganglion cell (RGC) model with over 600 latent dimensions, where it produced accurate voltage reconstructions and reliable conductance estimates. We also tested it on synthetic morphologies such as the 3-branch, 12-compartment neuron. Although the full EKF was used in the paper for this experiment (since it was tractable at that scale), we verified that the diagonal approximation gave nearly identical results: accurate recovery of membrane voltage traces and compartment positions, and only minor bias in conductance estimates.\nThese results suggest that the diagonal approximation is effective in practice for the models and regimes considered in this work. However, we acknowledge its limitations and view it as a first step. The approximation may become insufficient when incorporating more complex channel dynamics—such as calcium-gated conductances—that may introduce stronger intra- and inter-compartment dependencies. A promising direction for future work is to explore more structured covariance approximations, such as block-diagonal (e.g., one block per compartment) or block-diagonal plus low-rank structures, which may better capture the biophysical correlations present in multi-compartment neuron models, while preserving computational efficiency.\n\n- The reviewer suggests applying the method to real Neuropixels or MEA data. We agree, and this is a key direction for future work. Our current study focuses on establishing a principled, robust inference framework in controlled settings with known ground truth. Real recordings introduce additional complexities. We address these in the conclusion and plan to extend our approach to real datasets such as electrical images obtained from retinal ganglion cells.\n\n- The reviewer asks for a deeper discussion of parameter identifiability and whether the identifiable subspace can be characterized for complex models. We thank the reviewer for raising this important point. As discussed in the computational neuroscience literature (e.g., Sterratt et al., 2011), increasing the number of unknown parameters—particularly when morphological properties such as compartment radii or axial resistivity are also unknown—significantly increases the risk of parameter degeneracy. To mitigate this, our experiments make simplifying assumptions: we treat the radii and lengths of compartments as known and fixed. This reduces the effective parameter space and improves identifiability.\nIn our smaller simulated models (e.g. 12-compartment neurons), we did not observe signs of degeneracy: across multiple random initializations, the optimizer consistently converged to the same parameter values and marginal likelihood (matching the simulated ground truth), suggesting identifiability.\nFor the larger CA1 pyramidal neuron, however, we observe a different behavior. While the marginal likelihood converges reliably to the known optimum, the inferred conductance values continue to drift during optimization. Despite these changes, the membrane voltage remains accurately reconstructed. This suggests the presence of a non-identifiable subspace in which multiple parameter combinations produce indistinguishable outputs.\nIdentifying the full identifiable subspace analytically is, to our knowledge, intractable due to the nonlinear and high-dimensional structure of Hodgkin–Huxley-type models. However, several practical strategies can help reduce parameter degeneracy even for larger models like the pyramidal cell. One example, proposed by Migliore and Shepherd (2002), is to constrain conductance densities via spatial parameterizations (e.g., uniform or linearly varying gradients along dendrites). These strategies can reduce the number of free parameters, but must be tailored to the specific cell type.\nWe did not implement such parameterizations in this work, as we aimed to develop a general method for recovering conductance parameters that explain observed extracellular voltages in a general setting, without relying on cell-type-specific assumptions. However, these spatial conductance distribution models can be readily incorporated into our method via simple reparameterizations, and we agree they should be included in future work focused on specific neuron types. We will update the manuscript to clarify this point.\n\n- The reviewer recommends evaluating alternative inference approaches. We agree with the reviewer and appreciate the suggestion. Although not included in the paper, we did experiment with the Unscented Kalman Filter (UKF), which showed performance comparable to the EKF when the number of compartments was low. However, UKF is known to scale poorly with latent dimensionality, which limits its applicability to the larger multi-compartment models considered in this work. For these reasons, we focused on EKF (with diagonal approximation) as a scalable and effective inference method."}, {"review_id": "Fm9tPSMB4J", "reviewer": "Reviewer_aSRG", "summary": "This 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.", "questions": "1. 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?", "limitations": "1. 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.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "The 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.", "was_revised": true, "final_justification": "I think the authors have not been able address the critical concern regarding the question on scalability. Even a single neuron are sometimes decomposed into multiple compartments and they are modeled and analyzed. My question on scalability was on such multi-compartment models, not necessarily multiple neurons. Other assumptions on gating variables and their numbers are significantly limiting the applicability of the method. Hence, I do not intend to change my rating.", "rebuttal": "We thank the reviewer for taking the time to review our submission and highlighting some interesting clarification points. Below we address each of the five questions raised and hope our responses provide the necessary clarifications: \n\n- The reviewer asks about the motivation for choosing the Hodgkin–Huxley framework and its ability to capture neuron behaviors flexibly. We chose the Hodgkin–Huxley (HH) framework over an abstract model because our aim extends beyond reproducing voltage traces—we want to recover biophysically meaningful parameters such as maximal conductances and the neuron’s spatial relationship to the probe, quantities of interest to neuroscientists. HH models are also highly extensible: virtually any ion channel found in the literature can be incorporated by specifying its kinetics. The real difficulty lies in fitting such richly parameterized models, not in their expressiveness. We acknowledge, however, that HH models presuppose some prior knowledge of which channels are present, and that uncertainty can limit identifiability; our method, therefore, includes mechanisms (process noise) to remain robust when the true channel set is only partially known.\n\n- The reviewer raises a concern about how the method generalizes to neurons with unconventional ion channels or spiking behaviors. Our framework starts from the canonical view of a neuron as a discretized cable: each compartment is an RC circuit equipped with voltage‑gated or calcium-gated conductances. This multi‑compartment Hodgkin–Huxley (HH) formulation has been at the center of cellular neurophysiology for decades precisely because it is mechanistic, extensible, and cell‑type agnostic—you obtain different electrophysiological phenotypes by choosing the appropriate set of channels and kinetics. In practice, the HH framework comfortably accommodates the diversity of spiking behaviors: spike‑frequency adaptation (A-current, Ca-activated K+), bursting/chattering (calcium channels combined with Ca-dependent K+ currents), fast spiking (Kv3-family K+ channels).\nOur current experiments focused on the standard Na/K/Leak HH channels to make the methodological contributions clear, but the inference machinery is unchanged when richer channel repertoires are introduced: adding a channel simply augments the state vector and parameter set, and the EKF (or its diagonal variant) and gradient-based learning proceed identically. In short, the limitation is not the HH formalism but which channels you choose to include—a choice that can be guided by prior anatomical/physiological knowledge for each neuron class.\n\n- The reviewer asks about how the method addresses uncertainty or misspecification in the set of gating variables and the implications of fixing the model’s structure in advance. While the neuron’s type and location provide useful prior knowledge about which ion channels are present, we agree with the reviewer that misspecification is almost inevitable because (1) ion channels might have been neglected, (2) gating kinetics might be approximate. Indeed, the standard Hodgkin-Huxley rate functions are empirical fits from voltage-clamp data and may not capture all the biophysical subtleties. Real gating variables often deviate from these idealized kinetics due to modulatory factors such as temperature.\nWe show that naively fitting HH models with mean-squared error (MSE) struggles under these mismatches—it fails to recover the true membrane voltage and may even miss the correct spike count. To address this, we model both voltage and gating variables as stochastic processes, adding small white-noise terms at each time step. This approach, combined with an Extended Kalman Filter (EKF), allows the model to tolerate structural errors and still accurately track the true voltage, even in cases of channel misspecification (cf. Fig. 3).\n\n- The reviewer seeks clarification on the impact of EKF linearization. We demonstrated that MSE minimization and the EKF yield similar, unbiased parameter distributions (cf. Fig. 2c). A similar lack of bias was also observed in the multicompartment setting (e.g., the 3-branch model). This supports the idea that linearization provides a robust approximation of the marginal likelihood and introduces only negligible bias in parameter estimation. Although previous work has shown that the EKF can become unstable at low sampling rates (Lankarany et al., 2014), we did not encounter this issue as we focused on high-resolution recordings such as those from Neuropixels or MEAs.\n\n- The reviewer asks about the scalability of the method to larger neuronal structures or networks, and how the model handles inter-compartmental interactions. While our study focused on single neurons, leaving full network inference for future work, we are already able to handle large single cells efficiently by two design choices: (i) diagonal-covariance EKF—we keep only the diagonal of the filtering covariance, and (ii) sparse Jacobians—each gating variable depends only on its own previous value and the local voltage, so the dynamics’ Jacobian is sparse. These choices make every EKF update run in time proportional to the state dimension, allowing us to fit morphologies as large as the 122-compartment retinal ganglion cell. Inter-compartment interactions enter solely through axial currents (set by the cable’s axial resistivity), and those currents feed directly into the Hodgkin–Huxley equations, so their effect on the transmembrane currents—and hence on the probe’s extracellular signal—is captured automatically within the same formalism."}, {"review_id": "myyEHEWq0Z", "reviewer": "Reviewer_tGNX", "summary": "The 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).", "questions": "This 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)", "limitations": "Limitations 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.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths: 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)?", "was_revised": true, "final_justification": "The paper proposes a cool novel model for HH model parameter estimation in high density electrodes, but in my opinion the evaluation is a bit limited (few cells + lack of comparison to other methods or justification of design choices). Thus, I recomend weak accept", "rebuttal": "We thank the reviewer for taking the time to review our submission and for their thoughtful and constructive feedback. We were especially encouraged by the reviewer’s positive comments on the novelty of our work. Below, we address each of the six points made by the reviewer and hope to provide the necessary clarifications:\n\n- The reviewer suggests clarifying earlier that the method is not applied to in vivo data. We fully agree and thank the reviewer for this suggestion. Although extracellular voltage recordings are well-suited for in vivo acquisition, the current study does not yet include any in vivo experiments. We will clarify this explicitly in the introduction.\n\n- The reviewer suggests clarifying how our method relates to spike sorting. We thank you for raising this important point. Our intention in lines 34–36 was simply to note that extracellular recordings are typically used to obtain spike times and putative neuron labels via spike sorting, but the spike waveform contains rich information that can be used for fitting more detailed biophysical models as well. However, we agree that the current wording could be misleading since our method also relies on spike sorting to identify EAPs. We will revise the introduction to make this distinction clearer.\n\n- The reviewer asks for clarification on what is meant by \"channel states\". In Hodgkin–Huxley models, each ion channel $c\\in\\mathcal{C}$ (where $\\mathcal{C}$ is the set of all ion‑channel types included in our model) is governed by one or more gating variables whose values lie in [0,1]. For example, the fast Na+ current has activation and inactivation gates $m(t)$ and $h(t)$, whereas the delayed‑rectifier K+ current has a single activation gate $ n(t) $. We collect these variables in a vector $\\lambda_c(t)  =  (m_c(t),h_c(t),… )$, which represents the probability that the molecular gates of channel $ c $ are open at time $t$. We define the channel states at time $t$ as the concatenation of all such vectors over channel types $c\\in\\mathcal{C}$ and also compartments. We will revise the manuscript to make this definition—and its connection to the underlying biophysics—more explicit.\n\n- The reviewer asks whether the Hodgkin–Huxley parameters are constant or time-varying, and whether the model can handle dynamic changes in parameters. The Hodgkin-Huxley parameters we infer are the maximum conductances $g_c$​ of each channel type $c$. These biophysical constants are assumed to remain fixed throughout the length of the recordings in each of our experiments. What varies in time are the effective conductances defined as the maximum conductance times the proportion of ion channels opened at time $t$ (a number between 0 and 1). \nAlthough not considered in our current work, one possible source of time-varying conductances is short-term plasticity or activity-dependent modulation. Such effects could be incorporated by introducing additional differential equations into the dynamics model—governing, for instance, how the maximum conductances $g_c$ evolve over time. These added components would introduce new latent states and parameters, which could also be inferred within our framework by extending the EKF and continuing to maximize the marginal likelihood.\n\n- The reviewer suggests it would be more appropriate to emphasize applicability to a broader range of probes, including NP1 and NP2. We fully agree and appreciate this suggestion. Although the Neuropixels Ultra study initially motivated our focus on dense extracellular recordings, we agree that our method should also apply to NP1 and NP2. Indeed, NP1 and NP2 have pitches of 20 µm and 15 µm, respectively, while the MEA used in our RGC experiments has a 30 µm spacing. We will revise our framing to avoid focusing so specifically on NP Ultra and instead emphasize that the method is compatible with a range of high-density probes. We have not yet run experiments on NP1/NP2 data, but this would be a natural direction for future work.\n\n- The reviewer notes that a more recent version of the Neuropixels Ultra paper is now available and should be cited. We thank the reviewer for catching this mistake. We will replace it with the correct reference to Ye et al., 2024, in the camera‑ready version."}]

Author rebuttal:

Meta-review: This paper received largely positive reviews (barring one reviewer). The application is a novel and important one, and the approach is well-executed, making this a good applied neuroscience paper, even if some of the ML is a bit incremental/off the shelf. The biggest weakness concerned the absence of experiments with real-world recordings, though the reviewers were largely of the opinion that this is not fatal. Please go over the discussion with the reviewers when preparing the camera-ready version.
Decision: Accept (poster)