Parametric Scarring: Memory in Reaction-Diffusion Systems

Teaching patterns to physics: How substrate modification enables memory formation in continuous dynamical systems

This repository contains code for experiments demonstrating parametric scarring—a memory mechanism where learned patterns are encoded as permanent modifications to the physical parameters governing a reaction-diffusion substrate.

🔗 Blog Post: Teaching Patterns to Physics 📄 Paper: Coming soon to arXiv 👤 Author: Barzin Lotfabadi

---

Key Results

• 93.3% recall accuracy from 15% perturbation strength
• 2.86× improvement over baseline (no scarring)
• 3 distinct patterns stored simultaneously with >80% recall quality
• Zero catastrophic forgetting across sequential training

---

What is Parametric Scarring?

Instead of storing memories as weights in a neural network, we modify the laws of physics themselves. When a pattern is learned successfully, we adjust the reaction-diffusion parameters (f and k) locally, creating "scars" that make that pattern easier to recreate in the future.

Key insight: Memory as substrate modification, not just weight adjustment.

---

Quick Start

Installation

# Clone repository
git clone https://github.com/yourusername/parametric-scarring.git
cd parametric-scarring

# Create virtual environment (recommended, double recommend you use `uv` of course)
uv venv
source .venv/bin/activate  # On Windows: .\venv\Scripts\activate on CMD or .\venv\Scripts\Activate.ps1 via PowerShell

# Install dependencies
pip install -r requirements.txt

Run Experiments

Full validation experiment (Phases 1-3, <5 minutes on a cheap GPU):

python scarring_validation.py

Results saved to scarring_results/:

• Training/recall visualizations (PNG)
• Parameter scar maps (PNG)
• Quantitative metrics (JSON)
• Summary plots (PNG)

---

Project Structure

parametric-scarring/
├── scarring_validation.py    # Main experiment script
├── requirements.txt           # Python dependencies
├── LICENSE                    # MIT License
├── README.md                  # This file
├── scarring_results/          # Output directory (generated)
│   ├── *.png                  # Visualizations
│   └── results.json           # Metrics
└── docs/                      # Additional documentation
    └── methodology.md         # Detailed methods

---

How It Works

1. Substrate: Gray-Scott Reaction-Diffusion

Two chemicals (U and V) diffuse and react across a 2D grid:

∂U/∂t = Du∇²U - UV² + f(1-U)
∂V/∂t = Dv∇²V + UV² - (k+f)V

Parameters f and k control pattern formation (spots, stripes, mazes, etc.).

2. Scarring Algorithm

def apply_scarring(V, target, F, K, strength=0.003):
    """Modify F/K where current state matches target"""
    match_map = (V > 0.3) & (target > 0.3)
    F[match_map] += strength          # Increase feed rate
    K[match_map] -= strength * 0.5    # Decrease kill rate
    F.clamp_(0.01, 0.08)
    K.clamp_(0.03, 0.08)

3. Memory Recall

From weak perturbation (15% of target), the scarred substrate reconstructs the full pattern with 93% accuracy.

---

Experiments

Phase 1: Single Pattern Training

• Train pattern with/without scarring
• Control collapses to mazes (11.8% similarity)
• Scarred maintains pattern (51.5% similarity)

Phase 2: Memory Recall

• Weak (15%) perturbation → full reconstruction
• Scarred: 93.3% recall quality
• Control: 32.6% recall quality

Phase 3: Multiple Patterns

• Train 3 patterns sequentially
• All recalled above 80% quality
• Minimal interference between patterns

---

Requirements

• Python 3.8+
• PyTorch 2.0+ (GPU recommended)
• NumPy
• Matplotlib
• OpenCV (for video generation)
• SciPy

See requirements.txt for exact versions.

---

Citation

If you use this code or build upon this work, please cite:

@article{lotfabadi2025parametric,
  title={Parametric Scarring: Memory Formation Through Physics Modification in Reaction-Diffusion Systems},
  author={Lotfabadi, Barzin},
  journal={arXiv preprint},
  year={2025},
  url={https://github.com/BarzinL/parametric-scarring}
}

Blog post:

Lotfabadi, B. (2025). "Teaching Patterns to Physics: Memory Formation in
Reaction-Diffusion Systems." Substack. [URL]

---

Limitations

This is early-stage proof-of-concept research. Known limitations:

• ⚠️ Pattern complexity: Only tested on simple geometric patterns
• ⚠️ Capacity: Unknown how many patterns can be stored before interference
• ⚠️ Temporal sequences: Cannot learn A→B→C transitions yet
• ⚠️ Real-world data: Audio/visual features untested
• ⚠️ Scalability: 256×256 grid has limited representational capacity

See blog post for detailed discussion.

---

Roadmap

Future experiments (contributions welcome!):

• Capacity testing (10+ patterns)
• Temporal sequence learning
• Audio spectrogram encoding
• 3D substrate implementation
• Hierarchical pattern composition
• Comparison with Neural Cellular Automata
• Hardware implementation (analog circuits)

---

Contributing

Contributions are welcome! Areas of interest:

• Experiments: Test new patterns, parameters, or substrates
• Optimization: Improve computational efficiency
• Analysis: Characterize capacity limits, interference patterns
• Extensions: Temporal learning, sensory grounding, 3D systems

Please open an issue before starting major work.

---

License

MIT License - see LICENSE file for details.

---

Related Work

Reaction-Diffusion Computing:

• Turing, A. M. (1952). "The Chemical Basis of Morphogenesis"
• Pearson, J. E. (1993). "Complex Patterns in a Simple System"

Continual Learning:

• Kirkpatrick et al. (2017). "Overcoming catastrophic forgetting" (EWC)
• Zenke et al. (2017). "Continual Learning Through Synaptic Intelligence"

Neural Cellular Automata:

• Mordvintsev et al. (2020). "Growing Neural Cellular Automata"

---

Contact

Barzin Lotfabadi

• 🌐 Website: sanctus.ca
• 💼 LinkedIn: [/in/barzin-lotfabadi]
• 📧 Email: [email protected]

---

"Memory carved into the fabric of simulated physics"