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

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:

• 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.

• 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.

• 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.

• 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). Although 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.

• 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.

• 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.

Thank you again for your thoughtful comments on our paper. We wanted to know whether you had a chance to read our follow-up response addressing the “Weaknesses” section, and if it helped address your concerns. We appreciate your time and feedback.

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:

• 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.

• 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). Our 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.

• 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. We 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).

• 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.

• 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.

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:

• 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. To 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. Empirically, 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. These 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.

• 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.

• 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. In 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. For 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. Identifying 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. We 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.

• 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.

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:

• 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.

• 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.

• 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.

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:

• 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.

• 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.

• 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.

• 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.

Thank you very much for the feedback and for advocating the acceptance of this paper.