| name | synaptic-matrix-eigenvalues-analysis |
| description | Spectral analysis of synaptic matrix eigenvalues for stability, transient dynamics, and memory capacity analysis in sparsely connected neural networks |
| authors | Mohd. Gayas Ansari, Pragya Shukla |
| arxiv_id | 2606.00326v1 |
| submitted | 2026-05-29T00:00:00.000Z |
| categories | q-bio.NC, cond-mat.dis-nn |
| keywords | synaptic matrix, eigenvalue analysis, neural network stability, spectral analysis, transient dynamics, memory capacity, synaptic sparsity |
| activation_words | synaptic matrix eigenvalues, neural network stability, spectral analysis, memory capacity, synaptic sparsity, brain dynamics |
On the Synaptic Matrix Eigenvalues of Sparsely Connected Neural Networks
Overview
Spectral analysis framework for synaptic matrices in neural networks, providing mathematical tools to analyze stability, transient dynamics, learning capacity, and effects of different sparsity patterns on brain function.
Core Innovation
Statistical Spectral Analysis
- Problem: Exact synaptic matrix determination technically difficult + meaningless for complex brains
- Solution: Statistical spectral analysis of different sparsity patterns
- Key Insight: Eigenvalue distribution determines network properties
Sparsity-Function Relationship
- Hypothesis: Specific brain functions require specific synaptic sparsity types
- Applications: Pharmacological effects, physiological modulators
- Framework: Statistical approach to transient mechanisms
Key Technical Components
1. Synaptic Matrix Model
S = sparse connectivity matrix
Eigenvalues: λ_i determine dynamics
Eigenvalue distribution: ρ(λ) determines stability
2. Spectral Analysis Methods
- Random Matrix Theory: Analytical predictions
- Density of States: Eigenvalue distribution
- Correlation Functions: Statistical properties
3. Sparsity Effects
| Sparsity Type | Eigenvalue Distribution | Dynamics |
|---|
| Homogeneous | Compact spectrum | Stable |
| Modular | Clustered eigenvalues | Multi-timescale |
| Scale-free | Broad spectrum | Critical dynamics |
Applications
1. Stability Analysis
- Network Dynamics: Eigenvalue stability criterion
- Seizure Dynamics: Spectral signatures
- Homeostasis: Eigenvalue regulation
2. Memory Capacity
- Eigenvalue Density: Information capacity
- Storage Limit: Matrix rank bounds
- Sparsity Optimization: Capacity maximization
