Izhikevich Neuron Simulation: Numerical & PINN Solvers

Biomedical Engineering | Numerical Analysis | Physics-Informed Deep Learning

This repository implements and evaluates multiple computational approaches to solving the Izhikevich model, a system of two non-linear Ordinary Differential Equations (ODEs) that simulate spiking neuron behavior. We compare classical numerical integration against Physics-Informed Neural Networks (PINNs) to analyze stability, accuracy, and computational performance.

---

🔬 Mathematical Framework

The model approximates neural excitability through two coupled equations:

$$\frac{dv}{dt} = 0.04v^2 + 5v + 140 - u + I$$ $$\frac{du}{dt} = a(bv - u)$$

Where $v$ represents membrane potential and $u$ represents the recovery variable. Upon reaching the peak threshold ($v \geq 30$ mV), the variables are reset: $v \leftarrow c$ and $u \leftarrow u + d$.

Solvers & Methodology

We evaluated five distinct methods to address the system's non-linearity:

• Midpoint Runge-Kutta (RK2): Our primary benchmark; it provided the best balance of local truncation error and processing speed.
• Explicit & Backward Euler: Used to analyze the trade-off between computational simplicity and numerical stability in "stiff" regions.
• ExpRESS-Euler: An exponential integrator specifically implemented to handle sharp biological transitions.
• PINNs (Physics-Informed Neural Networks): A deep learning approach using PyTorch where the loss function is constrained by the ODE residuals, allowing the model to learn the physics of the neuron without traditional time-stepping.

---

📈 Benchmarking Results

Based on the results detailed in the Technical Report:

• Numerical Stability: While Implicit methods (Backward Euler) showed superior stability, they struggled to capture the sharp, discontinuous resets characteristic of the Izhikevich model.
• PINN Performance: The neural network successfully captured recovery dynamics ($u$), but showed a "smoothing" effect on high-frequency voltage peaks ($v$) compared to RK2.
• Real-time Visualization: An interactive Unity interface was developed to allow for live parameter tuning ($a, b, c, d$), enabling immediate observation of transitions between spiking regimes (e.g., Tonic Spiking vs. Chattering).

---

📁 Repository Structure

Module	Contents
Explicit_Euler_method/	Implementation of baseline explicit integration.
Backward_Euler_method.py	Robust implicit solver for stable integration.
Midpoint_(RK2)_method/	Second-order Runge-Kutta implementation.
adaptive_exponential/	Advanced stiff solver (Rosenbrock method).
DL_PINN_model/	PyTorch implementation of the Physics-Informed model.
unity/	C# scripts and assets for the 3D interactive viewer.

---

🚀 Execution

Requirements

• Python 3.8+
• numpy, torch, matplotlib, scipy
• Unity (for interactive visuals)

Running Solvers

# Clone the repository
git clone [https://github.com/MohamedBadawy19/Dynamic-Neuron-Model-Project.git](https://github.com/MohamedBadawy19/Dynamic-Neuron-Model-Project.git)
cd Dynamic-Neuron-Model-Project

# Run a numerical method (Example: RK2)
cd "Midpoint_(RK2)_method"
python simulate_rk2.py

# Train the PINN model
cd ../DL_PINN_model
python train_pinn.py

👨‍💻 Contributors Developed by Karim Hassan, Bassel Shaheen, Mohamed Badawy, Kareem Taha, Engy Wael, Alaa Essam, Amat Al-Rahman, Ahmed Salem, Omar Gamal, Omar Amein, Ahmed AbdelMoety

📄 License: This project is licensed under the MIT License.