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Retracing Hodgkin and Huxley: State Recovery Does Not Certify Mechanism

Peiyu Zang, Jiayi Hao, Yongqiang Cai

arXiv:2609.34566v1cs.LG

TL;DR

Prediction alone does not establish whether a learned model has recovered the physical states and dynamics underlying observed neural responses. The paper trains structured latent models from simulated current and voltage without gate information, then tests prediction, state recovery, transfer, and generator agreement. Prediction and state-decoding results do not guarantee transported-field agreement with HH, motivating separate evaluation of states and dynamics across interventions.

  • Problem

    Whether models trained on current and voltage can recover both physical hidden states and their dynamics remains unclear, because accurate response prediction does not reveal what the model has rediscovered.

    Source: §1 · p. 2

  • Method

    The authors train structured latent models on simulated current and voltage histories, withholding gate identities and trajectories from training and model selection, then evaluate prediction, gate decoding, protocol transfer, and transported-field agreement.

    Source: §1 · p. 2, §2.2 · p. 3

  • Results

    Prediction error and cross-seed spread fall sharply at three latent dimensions, while latent-state-plus-command-voltage charts reach m-state R2 > 0.99 but transported fields still agree poorly with HH.

    Source: §6 · p. 9

  • Takeaways & Limitations

    State recovery is only part of the evidence; evaluations should also report chart fitting, chart inputs, and transported-field agreement across interventions.

    Source: §6 · p. 9

    The findings concern simulated HH dynamics with fixed parameters and structured latent models; extension to richer neuronal dynamics and biological recordings remains a next step.

    Source: §5 · p. 9

Abstract

from arXiv · show

Predicting observed dynamics does not establish recovery of the underlying physical mechanism. Can machine learning retrace the hidden-state reasoning behind the Hodgkin-Huxley (HH) model? We train structured latent models on simulated current and voltage, withholding gate identities and trajectories from training and model selection. We then test response prediction, state recovery, protocol transfer, and agreement with HH dynamics. Prediction error and its cross-seed spread both drop sharply at three latent dimensions under the tested protocols, while gate recovery under new protocols improves through five to six coordinates. State recovery depends on which observations the chart uses. Observed voltage improves current-clamp decoding relative to freely predicted voltage. Under voltage clamp, adding latent state to command voltage raises m-state $R^2$ from 0.976 to above 0.99, yet the transported field disagrees with HH on identical smooth samples. Known invertible HH coordinates achieve high fast-m field agreement under the same audit procedure. An exact HH identity decomposes the discrepancy into time-scale-weighted state error and a residual in the transported field; these terms can cancel or reinforce. These findings concern the tested models and charts. They support evaluating state and dynamics recovery separately, including chart inputs and transported-field agreement across interventions.

1 Introduction

The paper asks whether models trained on current and voltage can recover Hodgkin–Huxley (HH) states and the laws governing them. It tests prediction, state decoding, protocol transfer, and transported-field agreement, finding that state recovery and dynamics recovery can diverge.

1 Introduction

  • The study asks whether latent states recover physical gates and whether their evolution follows the governing law, since prediction alone cannot establish what a model has rediscovered.The HH setting provides observable responses, hidden states with physical meaning, and known governing equations.Source: §1 · pp. 1–2
  • Models train on simulated current and voltage histories without gate identities, trajectories, HH powers, or true kinetics entering training or model selection.The structured model assumes bounded latent states, voltage-dependent relaxation, and an additive conductance-like current law.Source: §1 · p. 2
  • At three latent dimensions, prediction error and its cross-seed spread fall sharply, while additional coordinates improve gate recovery under new protocols.The three-dimensional setting also supports accurate gate decoding on held-out trajectories from the training protocol families.Source: §1 · p. 3
  • Under voltage clamp, charts using latent state and command voltage reach m-state R2 > 0.99, yet their transported fields agree poorly with HH on identical smooth samples.The paper also reports that observed voltage improves current-clamp decoding relative to freely predicted voltage.Source: §1 · p. 3
  • An exact HH identity decomposes transported-field error into time-scale-weighted state error and a chart–field residual, which can cancel or reinforce.Consequently, similar state scores can accompany different dynamics errors.Source: §1 · p. 3

2 Related Work

Prior work reconstructs hidden states, infers gates when conductance equations are specified, and learns latent equations with varying structural assumptions. The paper distinguishes those aims from testing whether learned states and their dynamics remain physically meaningful under interventions.

2.1 From known neuron models to learned hidden states

  • Known-conductance methods infer hidden channel states and parameters, while partially learned and hybrid approaches retain specified channel or state structure.State identifiability can depend on observation protocol and parameterization; a voltage step may not distinguish steady-state gate values from maximal conductance.Source: §2.1 · p. 3

2.2 From state reconstruction to mechanism recovery

  • State-reconstruction and equation-learning methods do not by themselves establish recovery of physical variables and their dynamics under new interventions.The study tests whether a chart decodes gates across interventions and whether its expressed dynamics agree with the physical generator, using current- and voltage-clamp protocols.Source: §2.2 · p. 3

3 Hidden-State Discovery in the Hodgkin–Huxley Model

The study uses the known HH system under current- and voltage-clamp protocols, trains structured latent models from observable histories, and evaluates gates only after model selection. Its evaluation separates state decoding and transfer from agreement between the transported dynamics and the HH generator.

3.1 HH dynamics and observation protocols

  • Current clamp commands Iapp(t) and observes V(t), whereas voltage clamp prescribes V(t) and observes ionic clamp current across varied stimulus families.The protocol families include current steps, chirps, piecewise-random currents, voltage steps, prepulse–test–tail sequences, and multistep protocols.Source: §3.1 · p. 4

3.2 Learning from current and voltage

  • Training and model selection use only time, current, voltage, and control mode; simulated gates and separated ionic currents are reserved for evaluation.The learner is not given gate names, trajectories, their number, HH rate functions, or the n4/m3h powers, but it does use bounded latent coordinates, voltage-dependent relaxation, and an additive conductance-like current law.Source: §3.2 · p. 4

3.3 A structured latent model

  • Separate GRU encoders map clamp histories to bounded latent states, which share voltage-dependent relaxation dynamics and a conductance-like current equation.Branch count and latent dimension vary independently; training selects models by validation error across clamp modes without using true gates in the loss.Source: §3.3 · p. 5

3.4 Testing state and dynamics recovery

  • After model selection, a chart decodes gates on held-out training-family trajectories and unseen protocols, while the final test compares its chain-rule transported field with the HH field.Prediction and gate decoding alone do not establish generator agreement; the chart’s voltage input and derivative terms matter to the comparison.Source: §3.4 · p. 5
  • The exact field-error decomposition separates the chart–field residual from the state-error term, which scales as −ex/τx(V).For the same state error, a shorter gate time constant enlarges that term; the total error depends on whether the two terms cancel or reinforce.Source: §3.4 · p. 5

4 Experiments

The experiments test prediction, gate recovery, transfer across protocols and agreement with Hodgkin–Huxley dynamics using models selected without access to gate identities or trajectories. They find that prediction, state recovery and generator agreement provide distinct evidence, with outcomes depending on latent dimension, chart inputs and model architecture.

4.1 Experimental protocol

  • The study uses 2,600 simulated trajectories across three current- and five voltage-clamp protocol families, with a 15-ms history to predict the next 20 ms.Gate identities and trajectories are withheld from training and model selection; charts are fitted after selection, and OOD charts are selected using ID validation data.Source: §4.1 · p. 6

4.2 How many latent dimensions are needed?

  • At K = 3, response NRMSE falls from 0.632±0.091 at K = 1 to 0.4546±0.0004, while OOD gate recovery improves through K = 5–6.The cross-seed sample SD falls to 0.0004 at K = 3; at least 95% of variance lies in three principal directions for 13 of 15 overcomplete checkpoints.Source: Figure 3 · p. 6, §4.2 · p. 6

4.3 Gate information across models and interventions

  • Generic history models recover transferable gate information, but S4D’s strong prediction and decoding do not establish HH-generator agreement.DDAE variants have comparable gate recovery, yet the autonomous variant has negative voltage-clamp velocity R2 while the controlled variant retains positive OOD field R2; Latent ODE and S4D audits also show state–field gaps.Source: §4.3 · p. 7

4.4 State recovery does not certify the generator

  • Charts using observed response voltage improve current-clamp m-state decoding, but much of the gain disappears when observed voltage is replaced by freely predicted voltage.Thus current-clamp chart scores do not establish equally accurate state recovery during free prediction.Source: §4.4 · p. 7
  • On identical smooth voltage-clamp points, m-state R2 reaches 0.992–0.994, while transported-field scores are only 0.271, 0.303 and 0.109, respectively.Affine and nonlinear invertible HH controls reach smooth-support m-field R2 of 0.995 and 0.970 under the same audit; the error identity attributes field discrepancy to time-scale-weighted state error and a chart–field residual that can cancel or reinforce.Source: §4.4 · pp. 7–8, Table 2 · p. 8

4 Experiments

  • The same chart must transfer across interventions, and its transported dynamics must agree with the target generator to support mechanism recovery.The experiments use held-out trajectories and smooth voltage-clamp points to separate gate decoding from generator agreement.Source: Table 2 · p. 8

4.4 State recovery does not certify the generator

  • For two-branch models, 6/10 large-field-error fits have RMSE > 0.5, versus 0/10 for five-branch fits, despite similar response and gate recovery.These comparisons quantify chart variability across checkpoints and chart fits.Source: §4.4 · p. 8

5 Discussion

Under the tested protocols, strong gate-state recovery does not establish agreement with the HH dynamics. The discussion connects this gap to chart inputs, field-error decomposition, and the study’s simulated scope.

5 Discussion

  • Short time constants amplify state error, while state error and the chart–field residual can cancel or reinforce in the transported field.Matched retraining at 0.01-ms resolution preserves the state–field gap despite improved prediction and state recovery.Source: §5 · p. 8
  • Known invertible HH coordinates reach smooth voltage-clamp m-field R² values of 0.995 and 0.970 under the same evaluation procedure.In current clamp, observed voltage improves decoding relative to freely predicted voltage; under voltage clamp, voltage is the imposed command.Source: §5 · p. 8
  • The authors suggest designing protocols to distinguish hidden states and checking transported-field agreement, intervention transfer, and gate continuity at command changes.The decoded m derivative also depends on other latent coordinates, and decoded gates jump at voltage-command changes.Source: §5 · p. 8
  • High gate-state agreement can coexist with low, variable transported-field agreement across five seeds.The figure reports sample standard deviations for the transported-field scores.Source: §5 · p. 8, Figure 4 · p. 9
  • The findings are limited to simulated HH dynamics with fixed parameters and structured latent models; richer neuronal dynamics and biological recordings remain future work.The broader discussion also notes that kinetic identifiability can depend on observation protocols and model assumptions, even when HH equations are specified.Source: §5 · pp. 8–9

6 Conclusion

The conclusion reports separate outcomes for prediction, gate recovery, and dynamics agreement in the tested models and charts. It recommends evaluating state recovery alongside chart inputs and transported-field agreement across interventions.

6 Conclusion

  • Prediction error and its cross-seed spread fall sharply at three latent dimensions, while additional coordinates improve gate recovery under new protocols.Most variance remains in three principal directions.Source: §6 · p. 9
  • Charts using latent state and command voltage achieve m-state R² > 0.99, yet their transported fields agree poorly with HH on identical smooth voltage-clamp samples.The exact error decomposition shows that time-scale-weighted state error and the chart–field residual can cancel or reinforce.Source: §6 · p. 9
  • Scientific self-discovery studies should report chart fitting, chart inputs, and transported-field agreement across interventions, not state recovery alone.Additional current branches reduce variation across fitted charts without establishing dynamics recovery.Source: §6 · p. 9

A.1 Simulator and corpus validation

The simulator and corpus checks compare the production implementation with independent references and test numerical convergence. They support numerical correctness for the fixed-parameter canonical squid-axon HH system.

A.1 Simulator and corpus validation

  • Current-clamp voltage error is below 1.54 × 10^-4 mV, and voltage-clamp gate RMSE is below 1.55 × 10^-3 against the independent DOP853 reference.Table 3 compares production RK4 with independent DOP853 for one held-out realization of every protocol family.Source: Table 3 · p. 12, §A.1 · p. 12
  • At clamps of −100, −65, −20, and 40 mV, numerical gates match the closed form with maximum absolute error 2.51 × 10^-5.The gates are also compared with adaptive references at three fixed-step resolutions; gate error decreases under refinement for every protocol family.Source: §A.1 · p. 12
  • These checks establish numerical correctness for the simulated canonical HH system, a fixed-parameter squid-axon model, not variability across biological preparations.Larger current differences for the most extreme active protocols reflect fast gate error amplified by ionic conductance, rather than a different command or equation.Source: §A.1 · p. 12

A.2 Corpus construction and hidden-label firewall

The corpus uses trajectory-level splits, balanced sampling, and a firewall that keeps gates and separated ionic currents out of training and model selection. Entire stimulus families are reserved for protocol-transfer evaluation.

A.2 Corpus construction and hidden-label firewall

  • The fixed corpus contains 2,600 trajectories split 2,080/260/260, with each trajectory assigned wholly to train, validation, or test.Current-clamp families include steps, chirps, and piecewise-random commands; voltage-clamp families include steps, prepulse–test–tail sequences, and active multistep families.Source: §A.2 · p. 12
  • Training files expose time, command, response, and clamp mode, while gates and separated ionic currents are stored separately and opened only by post-lock audits.Input checks prevent these hidden quantities from entering training or model selection.Source: §A.2 · p. 12
  • Windows use a 750-sample (15 ms) history followed by a 1,000-sample (20 ms) free rollout, and OOD evaluation withholds complete stimulus families.Trajectory-balanced, stratified windows prevent quiet samples from dominating the objective.Source: §A.2 · p. 12
  • Observable-history sensitivity scores ranked candidate voltage commands, and three selected active protocols enlarged excitation support for later audits.The ranking used validation histories without gate values; selected protocols had robust minimum eigenvalues of 0.1648, 0.1617, and 0.1421.Source: §A.2 · pp. 12–13
  • Table 4 organizes stimulus families by distribution split, parameter ranges, and trajectory duration; OOD families are reserved for protocol transfer.The table reports current amplitudes, voltage levels, and clamp mode.Source: Table 4 · p. 13

A.3 Training, selection, and chart protocol

The dimension study and post-lock charts use fixed training and validation procedures, with gate information reserved for evaluation. Protocol-transfer charts are calibrated on whole trajectories and evaluated once on unseen protocol families.

A.3 Training, selection, and chart protocol

  • Five independent seeds define the locked dimension study; later topology–loss experiments use five different seeds and two chart initializations per checkpoint.No checkpoint, dimension, branch count, loss weight, or ridge parameter is selected using test gates.Source: §A.3 · p. 13
  • After model selection, charts map (z, V) to (n, m, h), with regularization chosen by trajectory-grouped cross-validation before one evaluation on unseen protocol families.Repeated windows and initializations are reliability audits, not retrospective selection; absolute transfer scores are comparable only within a fixed setting.Source: §A.3 · p. 13
  • Field relative RMSE normalizes field error by the true-field magnitude on the same samples; far-event samples lie more than 2 ms from command transitions.These conventions define how the field and far-event comparisons are computed.Source: §A.3 · p. 13

A.4 Complete latent-dimension results

The dimension scan separates predictive performance, protocol-invariant gate alignment, and latent geometry. Prediction and seed-to-seed reproducibility improve sharply at three dimensions, while transfer decoding continues to improve through five to six coordinates.

A.4 Complete latent-dimension results

  • Prediction improves sharply at K = 3 and then gradually; protocol-invariant alignment continues through K = 5–6, while 13 of 15 overcomplete checkpoints remain concentrated around a three-dimensional covariance core.Source: §A.4 · p. 13
  • Prediction improves sharply at K = 3 and then gradually; protocol-invariant alignment continues through K = 5–6, while 13 of 15 overcomplete checkpoints have a 95% variance dimension of at most three.Source: §A.4 · p. 13, Table 5 · p. 14
  • Gate information transfers across tested architectures, while autonomous DDAE transfers under current clamp but fails under voltage clamp; controlled DDAE does not show that pattern.Figure 5 reports sample SD across five structured-model seeds and three baseline-family seeds.Source: Figure 5 · p. 14
  • At K = 3, post-lock in-distribution gate R2 is 0.9949 ± 0.0004, but correlated on-manifold gates can already be interpolated at K = 1 and K = 2.The matched-seed mean gate R2 is 0.937/0.983/0.995 for K = 1/2/3, so static regression alone is not a dimension certificate.Source: §A.4 · p. 14

A.5 Baseline implementation and full comparison

Across the audits, accurate gate decoding does not establish agreement with HH gate dynamics. The results test that distinction across architectures, chart choices, interventions, and training controls.

A.5 Baseline implementation and full comparison

  • Autonomous DDAE reaches sparse OOD field performance of 0.786 ± 0.073 in current clamp but −0.005 ± 0.009 in voltage clamp; controlled DDAE restores positive voltage-clamp scores.The controlled extension scores 0.631 ± 0.152 at K = 3 and 0.541 ± 0.058 at K = 5.Source: §A.5 · p. 14
  • Across architectures, decoded gates can be accurate while fast-m dynamics disagree with HH; even S4D, the strongest predictor, reaches only 0.406 m-field R2 and a field-to-truth slope of 0.41.The derivative-free one-step audit likewise finds weak fast-m increment agreement, while n and h increments agree better for several models.Source: §A.6 · p. 15, Table 6 · p. 15, Table 7 · p. 16

A.7 Chart reliability and regularization

  • Whole-trajectory calibration improves test–retest stability, but increasing transported-field weight reduces field error at the cost of decoded gate accuracy.For K = 4, trajectory-calibrated fits have m-field RMSE SD 0.028 versus 0.125 for point calibration; tested dynamics-aware settings do not achieve low errors on both quantities.Source: §A.7 · p. 16, Table 8 · p. 16

A.8 State recovery and the fast-m field gap

  • On smooth voltage-clamp samples, fast-m field error persists despite accurate decoded states; its correlation with HH varies by time constant, and fitted effective relaxation is several-fold slower than HH.Removing command-jump neighborhoods and adding voltage to the chart with the full chain rule do not remove the discrepancy.Source: §A.8 · p. 17, Table 10 · p. 17

A.8.1 Learner temporal-resolution control

  • Finer sampling improves prediction and state recovery but leaves the state–field gap; richer or voltage-conditioned charts also improve some scores without consistently removing it.The same audit finds high smooth-support field agreement for known invertible HH coordinates: affine-coordinate m-field R2 is 0.995 and nonlinear-coordinate m-field R2 is 0.970.Source: §A.8.1 · p. 18, Table 11 · p. 18, §A.8.2 · p. 19

A.8.2 Chart closure, command jumps, and positive control

  • Restoring latent state to (m, V) improves prediction of the transported m derivative by R2 0.15–0.16, supporting additional latent-state dependence beyond those tested regressors.At voltage-command jumps, chart jumps and physical one-step gate changes are distinct quantities, so smooth-support field evidence is reported separately.Source: §A.8.2 · pp. 18–19

A.9 Architecture and transient-loss factorial study

  • For B = 5, w = 1, far-event m-field RMSE is 0.237 ± 0.044 with 0/10 runs above 0.5, but B = 5, w = 2 is less reliable.Figure 7 compares far-event errors and counts of fits above 0.5; transient weighting effects depend on topology, and none removes the fast-m gap.Source: Figure 7 · p. 20, §A.9 · p. 20

A.10 Robustness and additional-coordinate audits

  • Forty-millisecond rollouts remain stable, but errors rise; additional latent coordinates are reconstructible from observable history, consistent with a history-derived role within sampled protocol support.Across design controls, the fast-m gap persists; Table 18 maps claims to the supporting experiment families and independent units.Source: §A.10 · pp. 20–21, §A.11 · p. 21, §A.12 · p. 22

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