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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

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.

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

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.

Paper Structure

This paper contains 22 sections, 1 theorem, 13 equations, 11 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background
  3. The Morris--Lecar Model
  4. Physics-Informed Neural Networks
  5. Neural ODEs
  6. Methodology
  7. Rationale for Model Selection
  8. Dynamical Regimes of the Morris–Lecar Model
  9. Data Generation and Regime Characterization
  10. Implementation of PINN
  11. Implementation of NODE
  12. Model Architecture and Experimental Setup
  13. Evaluation Metrics
  14. Results and Comparison
  15. Qualitative Analysis of Model Predictions
  16. ...and 7 more sections

Key Result

Proposition 1

Consider the Morris--Lecar model as given in equation (1), with a fixed external input current $I \in \mathbb{R}$ and continuous functions $M_\infty(V)$, $N_\infty(V)$, and $\tau_N(V)$. Then, the system admits at least one equilibrium point $(V^*, N^*) \in \mathbb{R}^2$.

Figures (11)

  • Figure 1: Bifurcation diagrams of the Morris–Lecar system under the three regimes defined in Table \ref{['tab:parameter']}, obtained by sweeping $I_{\text{ext}}$ over the range $[0,\ 120]\,\mu\text{A}/\text{cm}^2$.
  • Figure 2: Equilibrium analysis and phase portraits for each bifurcation regime under selected representative currents.
  • 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.
  • 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.
  • Figure 5: NODE predicted vs ground truth trajectories under the Hopf regime for $I_{ext}=90\mu A/cm^2$ across different training epochs.
  • ...and 6 more figures

Theorems & Definitions (5)

  • Proposition 1
  • proof : Sketch of proof
  • Remark 1: Remark on fast–slow structure.
  • Remark 2
  • Remark 3