Deterministic Excitable Cellular Automaton

Emergent Wave Dynamics from Local Threshold Rules

This model is a deterministic discrete simulation framework that demonstrates how complex spatiotemporal excitation patterns can emerge from simple local interaction rules on a 2D grid.

Instead of solving global differential equations, the system uses nearest-neighbor deterministic update rules inspired by excitable media dynamics akin to action potential propagation in neurons and excitable tissues.

Built using NumPy + Matplotlib, the engine simulates the time evolution of discrete waves and visualizes emergent behavior arising from neighbor interactions.

---

Key Features

• Discrete Lattice Representation: 100×100 grid with deterministic update rules.
• Three-State Excitable Dynamics: States cycle through Resting, Excited, Refractory.
• Periodic Boundary Conditions: Wrap-around edges create a seamless topology.
• Nearest-Neighbor Interactions: Moore neighborhood (8 directions).
• Visual Animation Output: Produces a simulation GIF of emergent wave patterns.
• Parameter Tunable: You can adjust grid size, initial activity, and threshold rules.

---

The Core Idea: Local Rules → Global Dynamics

Excitable systems can support wave propagation even when each element only “sees” its nearest neighbors.

In this system, each cell $(x, y)$ takes one of the discrete states:

State Code	Meaning
0	Resting (OFF)
1	Excited (ON)
2	Refractory

The evolution of the grid is fully deterministic — no stochastic elements — and patterns emerge solely from the interaction between neighboring cells.

---

Mathematical Framework

Lattice and Neighborhood

Grid size:

Neighborhood used: Moore neighborhood

$$ \mathcal{N}(x, y) = {g(x+dx,,y+dy)\ :\ dx,dy ∈ {-1,0,1},\ (dx,dy)≠(0,0)} $$

Periodic boundary conditions are applied, making the grid behave like a torus.

---

Update Rules

The grid evolves through discrete time steps according to deterministic local rules:

Rule 1 — Excitation Threshold

A resting cell becomes excited if it has exactly two neighboring excited cells:

$$ g_{t+1}(x,y)=1 \quad \text{if} \quad g_t(x,y)=0 ;\land; |{n∈\mathcal{N}(x,y): n=1}| = 2 $$

Rule 2 — Refractory Transition

An excited cell always becomes refractory at the next step:

$$ g_{t+1}(x,y)=2 \quad \text{if} \quad g_t(x,y)=1 $$

Rule 3 — Recovery

A refractory cell always returns to the resting state:

$$ g_{t+1}(x,y)=0 \quad \text{if} \quad g_t(x,y)=2 $$

Despite their simplicity, these rules produce rich spatiotemporal structure.

---

Implementation

The simulation is coded in:

Key packages used:

• numpy — for efficient array operations
• matplotlib — for visualization and animation
• matplotlib.animation.FuncAnimation — for generating GIF output

The simulation initializes the grid with 10% of cells in the ON state and evolves the system for a fixed number of frames while saving the animation as a GIF.

---

Visual Results

Simulation Animation

This animation visualizes the emergent propagation waves over discrete time steps.

---

Created as a project for IDC621 - Modelling Complex Systems.