Retracing Hodgkin and Huxley: State Recovery Does Not Certify Mechanism

Peiyu Zang; Jiayi Hao; Yongqiang Cai

cs.LGSep 28, 2026

Retracing Hodgkin and Huxley: State Recovery Does Not Certify Mechanism

Authors: Peiyu Zang, Jiayi Hao, Yongqiang Cai

Organizations: School of Mathematical Sciences, Beijing Normal University

Abstract

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.

Figures & tables

Method	Diagnostic role	NRMSE $\downarrow$	OOD gate $R^{2}$ $\uparrow$
Structured ( $K=3$ )	Three-dimensional latent model	$0.4546\pm 0.0004$	$0.378\pm 0.065$
Structured ( $K=5$ )	Additional latent capacity	$0.404\pm 0.033$	$0.806\pm 0.079$
S4D ( $K=5$ )	Sequence representation	$\mathbf{0.193\pm 0.013}$	$\mathbf{0.916\pm 0.020}$
GRU ( $K=5$ )	Recurrent representation	$0.299\pm 0.082$	$0.841\pm 0.118$
Latent ODE ( $K=2$ )	Continuous latent dynamics, command input	$0.510\pm 0.060$	$0.223\pm 0.152$

Table 1: Diagnostic comparisons with observable-only model selection (mean

\pm

sample SD; five seeds for structured models, three for baselines). Gate

R^{2}

measures ID-to-OOD chart transfer.

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Protocol family	Response RMSE	Gate RMSE
Current step	$9.60\!\times\!10^{-5}$ mV	$1.74\!\times\!10^{-7}$
Current chirp	$1.52\!\times\!10^{-4}$ mV	$1.48\!\times\!10^{-7}$
Current PRBS	$1.54\!\times\!10^{-4}$ mV	$1.50\!\times\!10^{-7}$
Voltage step	$0.207$ $\mu$ A cm -2	$8.17\!\times\!10^{-5}$
Voltage prepulse	$0.064$ $\mu$ A cm -2	$7.22\!\times\!10^{-5}$
Active 34	$0.478$ $\mu$ A cm -2	$1.75\!\times\!10^{-4}$

Appendix

Table 2: Production RK4 versus independent DOP853. Response is voltage for current clamp and ionic current for voltage clamp.

Family	Mode	Command	Sampled parameters (duration)
In-distribution
Current step	CC	Step from rest	amplitude $3$ – $16$ ; onset $8$ – $20$ ; offset $55$ – $85$ ( $100$ ms)
Current chirp	CC	Linear frequency sweep	amplitude $3$ – $12$ ; offset $2$ – $8$ ; $0.5$ – $5$ to $30$ – $100$ Hz ( $200$ ms)
Current PRBS	CC	Two-level piecewise-random	block $1$ – $8$ ; levels $-4$ and $14$ ( $150$ ms)
Voltage step	VC	Step with tail	holding $-100$ – $-60$ ; step $-80$ – $40$ ; tail $-100$ – $-50$ ( $60$ ms)
Voltage prepulse	VC	Prepulse–test–tail	holding $-100$ – $-70$ ; prepulse $-120$ – $20$ ; test $-20$ – $30$ ( $70$ ms)

Appendix

Table 3: Stimulus families. In-distribution families supply the training, validation, and test trajectories; out-of-distribution families are withheld in their entirety and used only for protocol-transfer evaluation. Current amplitudes are in

\mu

A cm -2 , voltage levels in mV, and each family reports its trajectory duration. Parameter ranges are the sampling intervals used to generate the fixed corpus; CC denotes current clamp and VC voltage clamp.

$K$	mean	CC	VC	exact $\uparrow$	$R^{2}$ $\uparrow$	$\leq 3$
	Response NRMSE $\downarrow$			spike	OOD gate	PCA95
1	$0.632\pm 0.091$	$0.886\pm 0.067$	$0.379\pm 0.128$	$0.270\pm 0.017$	–	–
2	$0.552\pm 0.039$	$0.846\pm 0.033$	$0.259\pm 0.052$	$0.272\pm 0.016$	–	–
3	$0.4546\pm 0.0004$	$0.755\pm 0.004$	$0.154\pm 0.004$	$0.318\pm 0.003$	$0.378\pm 0.065$	5/5
4	$0.443\pm 0.007$	$0.747\pm 0.012$	$0.138\pm 0.008$	$0.366\pm 0.024$	$0.601\pm 0.057$	5/5
5	$0.404\pm 0.033$	$0.664\pm 0.072$	$0.144\pm 0.029$	$0.490\pm 0.106$	$0.806\pm 0.079$	4/5

Appendix

Table 4: Locked dimension scan. Response NRMSE is reported as the mean over clamp modes and separately for current clamp (CC) and voltage clamp (VC), together with the fraction of test windows whose spike count is reproduced exactly. NRMSE and spike exact are mean

\pm

sample SD over five seeds. Protocol transfer is evaluated from

K=3

onward. The PCA column counts, per dimension, the checkpoints whose 95% variance dimension is at most three; the

K=3

row holds by construction, so the informative counts are the

K=4

–

6

rows,

13

15

overcomplete checkpoints.

Unseen protocol families, chart transferred from in-distribution validation
Model	state $m$	field $n$	field $m$	field $h$	$m$ slope
Structured ( $K=3$ )	$\mathbf{0.992\pm 0.000}$	$0.899\pm 0.008$	$0.269\pm 0.006$	$0.743\pm 0.024$	$0.220\pm 0.009$
S4D ( $K=5$ )	$0.993\pm 0.001$	$0.811\pm 0.120$	$\mathbf{0.406\pm 0.045}$	$0.849\pm 0.074$	$0.414\pm 0.080$
GRU ( $K=5$ )	$0.991\pm 0.003$	$0.564\pm 0.219$	$0.205\pm 0.176$	$0.659\pm 0.173$	$0.165\pm 0.136$
Latent ODE ( $K=2$ )	$0.983\pm 0.004$	$0.023\pm 0.335$	$0.060\pm 0.091$	$0.387\pm 0.048$	$0.038\pm 0.054$
Structured ( $K=3$ )	$0.980\pm 0.001$	$0.664\pm 0.039$	$0.058\pm 0.010$	$0.530\pm 0.044$	$0.046\pm 0.006$

Appendix

Table 5: Finite-difference transported-field audit on identical held-out smooth voltage-clamp windows (39,032 consecutive pairs on the shared 0.1-ms evaluation grid, that is, every fifth sample of the native 0.02-ms stream; the unpaired smooth support is 39,616 samples). Each model’s own finite-difference latent velocity over that interval is transported through the same cubic chart and compared with the canonical HH gate field, on held-out in-distribution windows and on unseen protocol families. State and field columns are

R^{2}

(higher is better); “slope” is the regression coefficient of the transported

m

field on the true

m

field, so exact agreement implies a slope of

1

. Pairs do not cross command transitions. Values are mean

\pm

sample SD over five structured seeds or three seeds per baseline. The Latent ODE has no unseen-protocol row here because its transfer scores are reported in Table 13 .

Model	$n$ increment	$m$ increment	$h$ increment	$m$ slope $\to 1$
Structured ( $K=3$ )	$0.911\pm 0.006$	$0.140\pm 0.010$	$0.730\pm 0.019$	$0.106\pm 0.007$
S4D ( $K=5$ )	$0.850\pm 0.098$	$\mathbf{0.335\pm 0.058}$	$0.865\pm 0.054$	$0.267\pm 0.068$
GRU ( $K=5$ )	$0.640\pm 0.177$	$0.174\pm 0.122$	$0.720\pm 0.110$	$0.138\pm 0.077$
Latent ODE ( $K=2$ )	$0.158\pm 0.673$	$-0.034\pm 0.097$	$0.565\pm 0.050$	$0.019\pm 0.031$
Persistence (all models)	$-0.02$ to $0.00$	$-0.07$ to $0.00$	$-0.02$ to $0.00$	–

Appendix

Table 6: Derivative-free one-step flow on the same smooth voltage-clamp samples. Each model’s own next decoded state, one 0.1-ms step later, is compared with the exact HH relaxation step from the identical decoded state over the same interval; no Jacobian or instantaneous derivative is involved. Values are increment (change)

R^{2}

, mean

\pm

sample SD over five structured seeds or three seeds per baseline; “persistence” is the trivial predictor that the gate does not change.

Checkpoint	$w=0$	$w=0.01$	$w=0.1$	$w=1$
$K=5$ , seed 51	$0.909$ / $1.31$	$0.909$ / $1.11$	$0.900$ / $1.00$	$0.869$ / $0.96$
$K=5$ , seed 54	$0.904$ / $3.25$	$0.897$ / $1.21$	$0.879$ / $1.01$	$0.822$ / $0.97$
$K=6$ , seed 52	$0.926$ / $2.51$	$0.925$ / $1.25$	$0.912$ / $0.97$	$0.871$ / $0.93$
$K=6$ , seed 55	$0.889$ / $4.78$	$0.871$ / $1.23$	$0.859$ / $1.01$	$0.807$ / $0.97$

Appendix

Table 7: Dynamics-aware chart sweep on the same validation split and evaluation windows. Cells are decoded gate

R^{2}

/ transported-field relative RMSE; lower field error is better. Increasing the field weight traces an empirical state–field trade-off and never reaches a low-error field.

Gate	$n$	$m$	$h$
State $R^{2}$	0.9955	0.9965	0.9926
Field relative RMSE	0.322	0.713	0.373

Appendix

Table 8: Minimal

K=3

physical audit over five checkpoints.

$\tau_{m}(V)$ quartile	Q1	Q2	Q3	Q4
Range (ms)	$0.008$ – $0.054$	$0.054$ – $0.158$	$0.158$ – $0.201$	$0.201$ – $0.501$
State $R^{2}_{m}$	$0.992$	$0.992$	$0.991$	$0.992$
Field relative RMSE	$1.00$	$1.00$	$0.66$	$0.72$
Field correlation	$0.03$	$0.06$	$0.76$	$0.74$
Field slope	$0.00$	$0.01$	$0.47$	$0.35$

Appendix

Table 9: Transported

m

field by HH time-constant quartile on the shared smooth voltage-clamp samples, mean over the five

K=3

checkpoints.

$K$	$\Delta t$ (ms)	NRMSE $\downarrow$	$m$ state $R^{2}$ $\uparrow$	$\tau_{m}^{\mathrm{eff}}/\tau_{m}^{\mathrm{HH}}$ $\downarrow$	$m$ -field $R^{2}$ $\uparrow$
5	0.02	0.429	0.768	14.44	0.030
5	0.01	0.389	0.882	13.32	0.046
6	0.02	0.418	0.874	10.16	0.054
6	0.01	0.326	0.939	8.98	0.094

Appendix

Table 10: Matched learner sampling-rate control (mean over three paired seeds). The

m

-state

R^{2}

is evaluated over all points in event-balanced ID test windows using the grouped-CV cubic post-lock chart; it is not the far-event subset or dynamics-aware chart score reported in Figure 5 .

$K$	$R_{m}^{2}(V_{\rm cmd})$	$R_{m}^{2}(z)$	$R_{m}^{2}(z,V_{\rm cmd})$	field $R_{m}^{2}$	field median
3	$.9758\pm.0000$	$.650\pm.031$	$.9921\pm.0002$	$.271\pm.006$	.273
5	$.9758\pm.0000$	$.826\pm.092$	$.9944\pm.0012$	$.303\pm.056$	.311
6	$.9758\pm.0000$	$.922\pm.036$	$.9944\pm.0033$	$.109\pm.337$	.218

Appendix

Table 11: Strict same-support VC audit (five-seed mean

\pm

sample SD). All state and field scores use the identical 39,616 held-out samples satisfying

|\dot{V}_{\rm cmd}|\leq 1\,\mathrm{mV/ms}

per checkpoint. Because the command is piecewise constant on the sampling grid, this threshold selects exactly the samples on which the command is constant between level changes, so the transported field there is the chart pushforward with the command frozen.

Chart	$m$ -state $R^{2}$ $\uparrow$	$m$ -field $R^{2}$ $\uparrow$	$n$
Linear, $K=3$	$0.941\pm 0.002$	$0.039\pm 0.004$	5
Cubic, $K=3$	$0.9921\pm 0.0002$	$0.271\pm 0.006$	5
Linear, $K=5$	$0.942\pm 0.003$	$0.157\pm 0.073$	5
Cubic, $K=5$	$0.9944\pm 0.0012$	$0.303\pm 0.056$	5
Known affine HH	$1.0000\pm 0.0000$	$0.995\pm 0.001$	3
Known nonlinear HH	$0.9999\pm 0.0000$	$0.970\pm 0.010$	3

Appendix

Table 12: State and transported-field recovery for different charts, evaluated on the same smooth voltage-clamp support (

|\dot{V}_{\rm cmd}|\leq 1

mV/ms). The learned rows use the shared

39{,}616

-point support, one free rollout per window, on locked checkpoints; the control rows replace the model with the exact HH gates behind a known invertible conjugacy, resample smooth-support windows independently under the same criterion (

59{,}426

points over three replicates), and refit the same cubic chart class. Values are mean

\pm

sample SD over five checkpoints or three control replicates.

Model, split	$R^{2}_{m}(z,V_{\rm cmd})$	field $R^{2}_{m}$	field $R^{2}_{n}$	field $R^{2}_{h}$
Structured, ID test	$0.9921\pm 0.0002$	$0.271\pm 0.006$	$0.896\pm 0.009$	$0.739\pm 0.024$
Latent ODE, ID test	$0.983\pm 0.004$	$0.068\pm 0.089$	$-0.018\pm 0.368$	$0.338\pm 0.084$
Structured, unseen	$0.980\pm 0.001$	$0.057\pm 0.010$	$0.665\pm 0.040$	$0.531\pm 0.045$
Latent ODE, unseen	$0.950\pm 0.017$	$0.010\pm 0.053$	$-3.57\pm 4.91$	$-1.93\pm 2.76$

Appendix

Table 13: Cross-architecture state and transported-field recovery using the same cubic chart class, ridge coefficient, smooth voltage-clamp support, and evaluation metric. The Latent ODE is a continuous-time latent flow with a parameterization unrelated to the structured model and comes from a different training batch; the state and field columns are matched window by window. Values are mean

\pm

sample SD over five structured checkpoints or three Latent ODE seeds.

$B$	Test NRMSE	OOD $n/m/h$ $R^{2}$
1	0.529	0.808/0.555/0.663
2	0.469	0.843/0.670/0.808
3	0.385	0.916/0.696/0.811
4	0.445	0.711/0.678/0.457
5	0.424	0.779/0.778/0.650

Appendix

Table 14: Branch-count control at

K=5

, seed 56.

$B$	$w$	NRMSE	$n$	$m$	$h$	$\tau_{m}^{\mathrm{eff}}$ scale	$m$ -field $R^{2}$
2	1	0.3411	0.927	0.901	0.899	$6.00\times$	0.175
2	2	0.3416	0.929	0.904	0.918	$5.18\times$	0.238
5	1	0.3287	0.933	0.863	0.866	$5.96\times$	0.091
5	2	0.3307	0.909	0.847	0.846	$5.84\times$	0.103

Appendix

Table 15: Five-seed topology–loss operating points. Gate columns are strict OOD

R^{2}

; the scale column is the fitted descriptive

\tau_{m}^{\mathrm{eff}}/\tau_{m}^{\mathrm{HH}}

Design choice	Control	Observation
Optimization budget	Independent starts, longer and stabilized dynamics-aware chart training	The empirical state–field trade-off persists across the tested settings.
Chart capacity	Four-layer, 256-unit, 20,000-step chart	Repeated resampling leaves the empirical state–field trade-off intact.
Chart complexity	Linear through cubic charts and grouped-CV regularization	On the same support a linear chart lowers both state and field agreement (Table 12 ), so the gap is not unique to the cubic chart.
Voltage chain rule	Voltage-conditioned and clamp-specific charts with the full $\partial_{V}r\,\dot{V}$ term	Accounts for part of the early discrepancy but not the voltage-clamp fast- $m$ field error.
Transient emphasis	Jump-stratified sampling and event-loss weights	Moves the selected state–field operating point without removing the fast- $m$ transported-field gap.
History, horizon, and excitation	Longer histories, direct multihorizon rollout, and active voltage protocols	Improves prediction and audit support; the transported-field gap remains under the tested audit.

Appendix

Table 16: Design choices examined for the fast-

m

comparison. Each row varies one aspect of predictor training or post-lock chart fitting and reports the resulting change.

Claim	Experiment families	Independent units	Paper location
Prediction error and seed spread drop sharply at $K=3$	Latent-dimension scan, $K=1$ – $6$	Five training seeds, $K=1$ – $6$	Figure 3 ; Table 4
Three-dimensional covariance core in 13 of 15 overcomplete models	Latent covariance and geometry audits	$K=4$ – $6$ , five seeds per $K$	Figure 3 ; Appendix A.4
Protocol alignment improves through $K=5$ – $6$	Locked in-distribution-to-OOD chart transfer	Five seeds per $K=3$ – $6$ , grouped-CV repeats	Figure 3 ; Table 4
State recovery does not certify the generator	Gate-state and transported-field audits	Gate-wise, cross-seed, analytic and numerical controls	Figures and 5
Redundancy changes reliability, not correctness	Branch-count and transient-loss factorial	Five training seeds and two chart starts	Table 15 ; Figure 6
Gate information transfers across tested architectures	Baseline architecture comparisons	Three seeds per baseline family	Table 1 ; Figure 4

Appendix

Table 17: Summary of the experimental evidence supporting the main findings.

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