Data-Driven Fast Estimation of Hodgkin-Huxley Neuron Model
DOI: Pending DOI
Synopsis
The Hodgkin-Huxley model describes neuron dynamics using stiff nonlinear ODEs that are computationally expensive to simulate. Data-driven methods have emerged as a popular alternative for such hard-to-solve dynamical systems for reducing computational cost compared to conventional ODE solvers. Recent advances include Neural Operators, which learn the underlying operator of a family of differential equations. We applied Fourier and Laplace Neural Operator architectures to approximate Hodgkin-Huxley dynamics across a wide range of input currents and assess each architecture’s ability to predict both action-potential timing and subthreshold response. Our results show that Neural Operators can accelerate Hodgkin-Huxley simulation while maintaining accurate prediction of neural responses.
Background
The Hodgkin-Huxley model uses stiff, nonlinear ordinary differential equations (ODEs) to describe action potential initiation and propagation in response to injected current waveforms [1]. Stiff ODEs are computationally expensive: conventional numerical solvers (RK, Rodas, ode15s, etc.) require very fine temporal resolution to capture fast-changing dynamics, and the resulting small time steps make large-scale simulation impractical. Data-driven methods have emerged as a popular alternative, offering reduced computational cost at high temporal or spatial resolutions. Neural Operators are a recent class of such methods that learn an operator approximating an entire family of differential equations, providing the unique combination of discretization invariance and universal operator approximation [2]. Prior work [3] has demonstrated success in applying Neural Operators to Hodgkin-Huxley dynamics under low-amplitude DC pulse stimuli. We extended this work in two ways. First, we applied multiple Neural Operator architectures to learn Hodgkin-Huxley dynamics under three substantially larger input parameter spaces: a DC pulse, an AC pulse (up to 2 kHz), and a combined DC+AC pulse (with the AC component also up to 2 kHz). Second, we evaluated model performance using error metrics that explicitly quantify the prediction of action potentials, an improvement upon prior work that used generic error metrics that fail to distinguish suprathreshold from subthreshold behaviors.
Methods
DATASETS: Each sample evaluated consisted of a 100 ms Hodgkin-Huxley simulation where the input was injected current and the output was transmembrane voltage. We constructed three datasets. The DC Pulse dataset varied pulse width from 0 to 100 ms and pulse amplitude from 0 to 10 μA [3]. The AC Pulse dataset used a sinusoidal stimulus with frequencies spanning 0–2000 Hz and amplitudes spanning 0–1000 μA. The DC+AC Pulse dataset used the sum of a fixed DC pulse and an AC pulse drawn from the AC parameter space. The DC component for the DC+AC dataset was a 10 μA current applied for 80 ms, which produced 6 action potentials when delivered as the sole input (Fig. 1c, top). When combined with an AC component, this stimulus can become inhibitory, decreasing firing relative to the DC pulse alone (region I in Fig. 1b and Fig. 1c, bottom), excitatory (region E), or neutral (region N).
DATA-DRIVEN MODELS: We evaluated two Neural Operator architectures. The Fourier Neural Operator (FNO) parameterizes the integral kernel in Fourier space, a natural choice for extracting spectral information from oscillatory neuron signals [4]. The Laplace Neural Operator (LNO) parameterizes the kernel in Laplace space, accommodating the combination of transient and stiff dynamics present in Hodgkin-Huxley voltage traces [5].
MODEL EVALUATION: We used Rate of Agreement (RoA) [6] to compare alignment between predicted and reference action potential trains, applied to spike timings extracted from each model’s filtered voltage trace. We additionally evaluated performance using the L2 norm.
Results
We trained 12 variations of each architecture to tune hyperparameters finding that a custom 4th-root absolute error loss accelerates convergence relative to MSE.
Our results demonstrate that the FNO accurately and efficiently captures Hodgkin-Huxley dynamics across both subthreshold and suprathreshold behaviors, and that it substantially outperforms the LNO in both accuracy and computational cost.
Use of the FNO resulted in a full order of magnitude computation speed-up compared to MATLAB’s conventional ode15s solver for all three datasets (Fig. 1A, left). By contrast the LNO was roughly an order of magnitude slower than ode15s, rendering it ineffective for the speedup objective. Importantly, the FNO was also the more accurate model, using RoA, across all three datasets (Fig. 1A, right). The FNO also outperformed the LNO by L2 loss.
Fig. 1B shows a heatmap of FNO test-set RoA across the AC+DC parameter space. We partitioned the space into three regions, labeled E, I, and N, representing regions of excitatory, inhibitory, and neutral response, respectively, to the AC component on spiking relative to the baseline DC pulse alone. In region E, the AC drive elicits additional action potentials beyond the DC baseline. In region I, the AC drive suppresses spiking below the baseline count, and region N the AC drive does not change the number of action potentials. The heatmap shows that the FNO performs strongly in the neutral and inhibitory regions but struggles to capture the additional spikes that arise in parts of the excitatory region.
Fig. 1c shows one representative example from each dataset. Each plot overlays the current stimulus (blue), the ground-truth voltage from the HH model (black), and the FNO prediction (red). The DC plot shows the baseline pulse used as the reference in the AC+DC simulations. The AC plot shows the response to a high-frequency stimulus. The combined AC+DC plot shows the same DC pulse with an inhibitory AC component layered on top, producing a strong suppression of spiking. These examples were chosen to illustrate strong FNO performance in the DC and AC regimes and the model’s ability to capture strong inhibitory dynamics. Fig. 1d shows three additional examples, chosen instead to illustrate cases where the FNO struggles. In the DC sample, the FNO underestimates the peak of the fourth action potential. The AC sample shows a stimulus that yields complex dynamics where the FNO struggles to reproduce the action potential train from the ground truth. The combined AC+DC plot shows a weakly inhibited response in which the FNO predicts an extra action potential, misrepresenting the firing pattern.
Discussion
We have demonstrated advances in data-driven modeling of Hodgkin-Huxley dynamics on three fronts. First, we improved model performance on the DC pulse problem studied in prior work [3], demonstrating that the FNO achieves a full order-of-magnitude speedup over conventional stiff solvers while maintaining accurate prediction. Second, we extended Neural Operator modeling beyond the low-amplitude DC regime to substantially larger and more challenging parameter spaces, including AC stimuli (up to 2 kHz) and combined DC+AC drive capable of eliciting a variety of excitatory, inhibitory, and neutral firing responses. Third, we introduced action-potential-aware evaluation using Rate of Agreement, which quantifies spike-train alignment directly rather than relying on generic error norms that conflate suprathreshold and subthreshold behavior. Together, these contributions establish the FNO as an accurate and computationally efficient surrogate for Hodgkin-Huxley dynamics across a far broader stimulus space than previously demonstrated.
Our work has direct applications for computational neuroscientists running single-neuron simulations and for experimentalists who want to narrow their parameter space in silico before committing to expensive electrophysiology experiments. More broadly, in multiscale models of whole-brain dynamics, neural-operator surrogates offer a computationally efficient path to solving at the single-neuron scale, where the inference speedup demonstrated here becomes especially valuable as the number of required simulations scales with network size.
Disclosures
N/A
References
1. Hodgkin AL, Huxley AF. A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol. 1952;117(4):500–544.
2. Kovachki N, Li Z, Liu B, Azizzadenesheli K, Bhattacharya K, Stuart A, Anandkumar A. Neural Operator: Learning Maps Between Function Spaces. arXiv:2108.08481. 2021.
3. Centofanti E, Ghiotto M, Pavarino LF. Learning the Hodgkin–Huxley model with operator learning techniques. Computer Methods in Applied Mechanics and Engineering. 2024;432:117381.
4. Li Z, Kovachki N, Azizzadenesheli K, Liu B, Bhattacharya K, Stuart A, Anandkumar A. Fourier Neural Operator for parametric partial differential equations. arXiv:2010.08895. 2020.
5. Cao Q, Goswami S, Karniadakis GE. Laplace neural operator for solving differential equations. Nature Machine Intelligence. 2024.
6. Thompson CK, Negro F, Johnson MD, Holmes MR, McPherson LM, Powers RK, Farina D, Heckman CJ. Robust and accurate decoding of motoneuron behaviour and prediction of the resulting force output. J Physiol. 2018;596(13):2643–2659.