Oscillatory Neural Networks (ONNs) Key Terms
In oscillatory neural networks (ONNs), several key terms and parameters define the dynamics and behavior of the system. Here’s a comprehensive list of the most important ones, including ω (omega) and their roles:
1. Core Parameters & Variables
ω (Omega) – Angular Frequency
- Defines the natural oscillation rate of a neuron or oscillator.
- Units: radians per second (rad/s).
- Higher ω → faster oscillations.
θ (Theta) – Phase
- Represents the current position in the oscillation cycle (0 to \(2\pi\)).
- Dynamics: \( \frac{dθ_i}{dt} = \omega_i + \text{coupling terms} \).
A (Amplitude)
- The magnitude of oscillation (e.g., spike height in neural models).
- Sometimes dynamic (e.g., in amplitude death phenomena).
φ (Phi) – Phase Difference
- Relative phase between two oscillators (\( \phi = \theta_j - \theta_i \)).
- Critical for synchronization.
2. Coupling & Interaction Terms
K (Coupling Strength)
- Determines how strongly oscillators influence each other.
- High \( K \) → faster synchronization.
J (Connection Weights)
- Synaptic-like weights between oscillators (e.g., \( J_{ij} \) for neuron \( i \leftarrow j \)).
g(φ) – Phase Response Curve (PRC)
- Describes how an oscillator’s phase shifts due to input.
3. Synchronization & Collective Dynamics
Sync Order Parameter (R)
- Measures global synchronization:
\[
R e^{i\psi} = \frac{1}{N} \sum_{j=1}^{N} e^{i\theta_j}
\] - \( R \approx 1 \): Perfect sync; \( R \approx 0 \): No sync.
Critical Coupling (Kₐ)
- Threshold coupling strength for synchronization (e.g., in Kuramoto model).
Cluster States
- Subgroups of oscillators sync separately (e.g., phase-locked clusters).
4. Noise & Disturbances
D (Noise Intensity)
- Adds stochasticity (e.g., \( \frac{dθ*i}{dt} = \omega_i + \xi_i(t) \), where \( \langle \xi_i(t) \xi_j(t’) \rangle = 2D \delta{ij} \delta(t-t’) \)).
Hysteresis
- Memory-dependent effects under parameter changes.
5. Network Structure Terms
Topology
- Defines connectivity (e.g., all-to-all, small-world, scale-free).
Delay (τ)
- Time lag in interactions (e.g., \( \sin(\theta_j(t-\tau) - \theta_i(t)) \)).
6. Special Models & Extensions
- Kuramoto Model: \( \frac{dθ_i}{dt} = \omega_i + \frac{K}{N} \sum \sin(\theta_j - \theta_i) \).
- Stuart-Landau Oscillators: Complex amplitudes with \( \dot{z} = (\lambda + i\omega)z - |z|^2z \).
- Hopf Oscillators: Used in central pattern generators (CPGs).
7. Biological Analogies
- Gamma/Beta Bands: Frequency ranges (e.g., 30-100 Hz gamma oscillations in cognition).
- Spike-Phase Coding: Info encoded in spike timing relative to oscillation phase.
Summary Table
| Term | Symbol | Role |
|---|---|---|
| Angular Frequency | ω | Sets oscillation rate |
| Phase | θ | Position in cycle |
| Coupling Strength | K | Sync control |
| Order Parameter | R | Sync measure |
| Noise | D | Stochastic effects |
| Connection Weight | J_{ij} | Network influence |
| Phase Response Curve | g(φ) | Phase shift dynamics |