Comparing Physics-Informed and Neural ODE Approaches for Modeling Nonlinear Biological Systems: A Case Study Based on the Morris-Lecar Model

Nikolaos M. Matzakos; Chrisovalantis Sfyrakis

math.DSMar 27, 2026

Comparing Physics-Informed and Neural ODE Approaches for Modeling Nonlinear Biological Systems: A Case Study Based on the Morris-Lecar Model

Authors: Nikolaos M. Matzakos, Chrisovalantis Sfyrakis

Organizations: Department of Education, School of Pedagogical and Technological Education (ASPETE), 141 21 Athens, Greece. · Department of Mechanical Engineering Educators, School of Pedagogical and Technological Education (ASPETE), 151 22 Athens, Greece.

Abstract

Physics-Informed Neural Networks (PINNs) and Neural Ordinary Differential Equations (NODEs) represent two distinct machine learning frameworks for modeling nonlinear neuronal dynamics. This study systematically evaluates their performance on the two-dimensional Morris-Lecar model across three canonical bifurcation regimes: Hopf, Saddle-Node on Limit Cycle, and homoclinic orbit. Synthetic time-series data are generated via numerical integration under controlled conditions, and training is performed using collocation points for PINNs and adaptive solvers for NODEs (Dormand-Prince method). PINNs incorporate the governing differential equations into the loss function using automatic differentiation, which enforces physical consistency during training. In contrast, NODEs learn the system's vector field directly from data, without prior structural assumptions or inductive bias toward physical laws. Model performance is assessed using standard regression metrics, including Mean Squared Error (MSE), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), and the coefficient of determination. Results indicate that PINNs tend to achieve higher accuracy and robustness in scenarios involving stiffness or sensitive bifurcations, owing to their embedded physical structure. NODEs, while more expressive and flexible, operate as black-box approximators without structural constraints, which can lead to reduced interpretability and stability in these regimes. Although advanced variants of NODEs (e.g., ANODEs, latent NODEs) aim to mitigate such limitations, their performance under stiff dynamics remains an open question. These findings emphasize the trade-offs between physics-informed models, which embed structure and interpretability, and purely data-driven approaches, which prioritize flexibility at the cost of physical consistency.

Figures & tables

Criterion	Morris–Lecar Model Characteristics	Relevance to PINNs & NODEs
Known physical equations	Analytically defined ODE system	Enables direct embedding of physics in PINNs; NODEs learn dynamics from data
Nonlinear dynamics	Exhibits excitability and oscillations	Tests the models’ ability to capture bifurcation behavior
Ground truth availability	Numerical integration feasible	Allows quantitative performance evaluation (Subsection 3.7 )
Low-dimensional yet rich	Two-variable system with complex transitions	Suitable for compact yet expressive architectures
Biophysical interpretability	Neurophysiologically meaningful parameters	PINNs offer insight; NODEs assess black-box adaptability

Table 1: Justification for using the Morris–Lecar model as a benchmark in the comparison of PINNs and NODEs.

Parameter	Hopf	SNLC	Homoclinic
$\varphi$	0.04	0.067	0.23
$g_{Ca}$	4.4	4	4
$V_{3}$	2	12	12
$V_{4}$	30	17.4	17.4
$E_{Ca}$	120	120	120
$E_{K}$	-84	-84	-84

Table 2: Parameter values for the Morris–Lecar model under three canonical bifurcation regimes ( Azizi and Mugabi (2020) ).

Panel of Figure 1 — Figure 1: Bifurcation diagrams of the Morris–Lecar system under the three regimes defined in Table 2 , obtained by sweeping $I_{\text{ext}}$ over the range $[0,\ 120]\,\mu\text{A}/\text{cm}^{2}$ .

Panel of Figure 2 — Figure 2: Equilibrium analysis and phase portraits for each bifurcation regime under selected representative currents.

Scenario	Homoclinic	SNLC	Hopf
Equilibrium Point $(V^{},N^{})$	(5.4540, 0.3203)	(4.8622, 0.3057)	(-26.5969, 0.1294)
Jacobian	$\begin{bmatrix}-35.588&-35.782\\ 0.005655&-0.2341\end{bmatrix}$	$\begin{bmatrix}-35.327&-35.545\\ 0.001601&-0.0684\end{bmatrix}$	$\begin{bmatrix}-22.935&-22.961\\ 0.000269&-0.0446\end{bmatrix}$
Eigenvalues	$-35.583,\ -0.240$	$-35.326,\ -0.070$	$-22.935,\ -0.045$
Type	Stable node/focus	Stable node/focus	Stable node/focus

Table 3: Equilibrium properties of the Morris–Lecar system under representative currents.

Aspect	PINN	Neural ODE
Input–Output Structure	$t\rightarrow[V,N]$	$[V_{0},N_{0}]\rightarrow[V(t),N(t)]$ via ODE integration
Architecture	FCNN (3 hidden layers × 128, Tanh)	FCNN (3 hidden layers × 128, Tanh or SiLU (see Remark 3 ))
Physics Integration	Explicit residuals via autograd	None (data-driven only)
Normalization	No normalization	Min–Max normalization with inverse scaling (see Remark 2 )
Optimizer	Adam (lr = $10^{-3}$ )	Adam (lr = $10^{-3}$ )
Numerical Precision	float32	float32

Table 4: Summary of Methodological Differences between PINNs and NODEs

Figure 3: Predicted vs ground truth voltage trajectories and phase portraits for the Hopf regime with

I_{ext}=50\mu A/cm^{2}

, a current value that does not induce bifurcation. Both PINN and NODE models successfully reproduce the stable resting dynamics.

Panel of Figure 4 — Figure 4: PINN predicted vs ground truth trajectories under the Hopf regime for with $I_{ext}=90\mu A/cm^{2}$ across different training epochs.

Panel of Figure 5 — Figure 5: NODE predicted vs ground truth trajectories under the Hopf regime for $I_{ext}=90\mu A/cm^{2}$ across different training epochs.

Panel of Figure 6 — Figure 6: PINN model under the SNLC regime with $I_{ext}=42\mu A/cm^{2}$ .

Panel of Figure 7 — Figure 7: NODE model for SNLC regime with $I_{ext}=42\mu A/cm^{2}$ .

Panel of Figure 8 — Figure 8: PINN-based predictions for the Morris–Lecar model in the homoclinic regime with $I_{ext}=50\mu A/cm^{2}$ across different training epochs.

Panel of Figure 9 — Figure 9: NODE model for the homoclinic regime with $I_{ext}=50\mu A/cm^{2}$ .

Figure 10: NODE model predictions for the homoclinic regime using the SiLU activation function after

10,000

training epochs. The plots show the predicted membrane voltage

V(t)

, gating variable

N(t)

, phase portrait, and training loss. Compared to earlier results with Tanh (see Figure 9 ), the use of SiLU leads to visibly improved dynamics, particularly in the reproduction of the orbit structure. However, significant deviations remain in the later phase and voltage trajectory, especially near the saddle-node region, indicating that the NODE still struggles to fully capture the global geometry of the homoclinic orbit.

Panel of Figure 11 — Figure 11: Total MSE across epochs for NODE and PINN models in the Hopf, SNLC, and Homoclinic regimes. The total MSE is computed as the sum of the MSE values for $V$ and $N$ ; PINN achieves lower error in all cases, especially in the Homoclinic regime.

Scenario	Method	Epochs	Total MSE	RMSE V	RMSE N	Training Time (s)
Hopf	PINN	20000	3.0577e-01	0.5529	0.0069	145.74
Hopf	NODE	10000	8.1303e+00	2.8514	0.0035	2857.54
SNLC	PINN	5000	1.6022e+00	1.2656	0.0221	37.69
SNLC	NODE	20000	1.9270e+01	4.3898	0.0101	6143.12
Homoclinic	PINN	20000	4.6681e-01	0.6832	0.0091	146.94
Homoclinic	NODE	1000	2.3236e+01	4.8199	0.0672	251.52 sec

Table 5: Comparison of Best Performing Models (PINN vs NODE) per Scenario

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Inferring biophysical parameters and hidden state variables from partial and noisy observations is a fundamental challenge in computational neuroscience. This problem is particularly difficult for fast - slow spiking and bursting models, where strong nonlinearities, multiscale dynamics, and limited observational data often lead to severe sensitivity to initial parameter guesses and convergence failure in the methods replying on the traditional numerical forward solvers. In this work, we developed a physics-informed neural network (PINN) framework for the joint reconstruction of unobserved state variables and the estimation of unknown biophysical parameters in neuronal models. We demonstrate the effectiveness of the method on biophysical neuron models, including the Morris-Lecar model across multiple spiking and bursting regimes and a respiratory model neuron. The method requires only partial voltage observations over short observation windows and remains robust even when initialized with non-informative parameter guesses. These results suggest that PINN can deliver robust and accurate parameter inference and state reconstruction, providing a promising alternative for inverse problems in multiscale neuronal dynamics, where traditional techniques often struggle.

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