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.
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
Table 1: Justification for using the Morris–Lecar model as a benchmark in the comparison of PINNs and NODEs.
Parameter
Hopf
SNLC
Homoclinic
φ
0.04
0.067
0.23
gCa
4.4
4
4
V3
2
12
12
V4
30
17.4
17.4
ECa
120
120
120
EK
-84
-84
-84
Table 2: Parameter values for the Morris–Lecar model under three canonical bifurcation regimes ( Azizi and Mugabi (2020) ).
Figure 1: Bifurcation diagrams of the Morris–Lecar system under the three regimes defined in Table 2 , obtained by sweeping Iext over the range [0,120]μA/cm2 .
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
[−35.5880.005655−35.782−0.2341]
[−35.3270.001601−35.545−0.0684]
[−22.9350.000269−22.961−0.0446]
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→[V,N]
[V0,N0]→[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 Iext=50μA/cm2 , 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 Iext=90μA/cm2 across different training epochs.
Figure 5: NODE predicted vs ground truth trajectories under the Hopf regime for Iext=90μA/cm2 across different training epochs.
Figure 6: PINN model under the SNLC regime with Iext=42μA/cm2 .
Figure 7: NODE model for SNLC regime with Iext=42μA/cm2 .
Figure 8: PINN-based predictions for the Morris–Lecar model in the homoclinic regime with Iext=50μA/cm2 across different training epochs.
Figure 9: NODE model for the homoclinic regime with Iext=50μA/cm2 .
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.
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
NODE
10000
8.1303e+00
2.8514
0.0035
2857.54
SNLC
PINN
5000
1.6022e+00
1.2656
0.0221
37.69
NODE
20000
1.9270e+01
4.3898
0.0101
6143.12
Homoclinic
PINN
20000
4.6681e-01
0.6832
0.0091
146.94
NODE
1000
2.3236e+01
4.8199
0.0672
251.52 sec
Table 5: Comparison of Best Performing Models (PINN vs NODE) per Scenario
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.
Changliang Wei, Yangyang Wang, Xueyu Zhu
Department of Mathematics, University of Iowa, Iowa City, IA 52246, USA. · Department of Mathematics, Volen National Center for Complex Systems, Brandeis University, Waltham, MA 02453, USA.
Neural ordinary differential equations (Neural ODEs) often fit training trajectories while generalizing poorly to unseen initial conditions and long horizons. We propose MPINeuralODE, which combines a soft physics-informed residual with a Multiple-Initial-Condition (MIC) multiple-shooting curriculum whose ingredients are structurally complementary: the physics term anchors the vector-field magnitude on the support that MIC enlarges. We evaluate along three axes: out-of-sample error, long-horizon stability, and Hamiltonian drift, which together expose whether the learned dynamics recover the underlying vector field. On Lotka-Volterra, MPINeuralODE achieves the lowest out-of-sample and long-horizon MSE among data-driven methods, with a 24% reduction over the baseline Neural ODE, while essentially matching the PINN ablation on Hamiltonian drift.
Lake Yang, Antonio Malpica-Morales, Frank Ioannis Papadakis Wood +1
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, United Kingdom
Solving time-dependent partial differential equations (PDEs) is an important problem in computational science and engineering. Physics-informed neural networks (PINNs) learn PDE solutions from governing equations. However, accurately capturing temporal evolution remains challenging. Recent sequence-model-based approaches parameterize time evolution using general-purpose sequence models, which capture temporal dependencies but do not explicitly encode the structured dynamics of PDE solutions. In addition, their memory requirements can scale unfavorably with sequence length and resolution, limiting applicability in large-scale or high-dimensional settings. This work introduces a PINN approach that incorporates oscillatory state-space dynamics to represent the modal structure of PDE solutions. The proposed method leverages a linear-oscillator-based temporal evolution, together with a PDE-aware spectral basis in space. This design enables closed-form spatial differentiation and consistent enforcement of boundary conditions. The method is evaluated on forward, inverse, and high-dimensional PDE problems, including cases up to 100 spatial dimensions. The results show improved accuracy and reduced memory usage compared to recent sequence-model-based PINN approaches. Overall, this work highlights the benefits of incorporating structured dynamical priors into the temporal evolution of neural PDE solvers and suggests designing more physics-aligned and computationally efficient PINN architectures.
Abhishek Chandra, Taniya Kapoor
KTH Royal Institute of Technology Stockholm, Sweden · Wageningen University & Research Wageningen, The Netherlands