Topological Sensitivity Connectome Constraints

---
name: topological-sensitivity-connectome-constraints
description: Topological sensitivity analysis of connectome-constrained neural networks. Studies how network topology affects dynamical behavior and sensitivity to perturbations in brain connectome models. Applicable to robust brain dynamics analysis and lesion studies.
version: 1.0.0
author: Research Synthesis
license: MIT
metadata:
  hermes:
    tags: [brain-network, topology, connectome, sensitivity-analysis, neural-dynamics]
---

# Topological Sensitivity Connectome Constraints

## Overview
Methodology for analyzing how brain network topology constrains neural dynamics and determines sensitivity to perturbations. Combines topological data analysis (TDA) with connectome-constrained neural modeling to understand structure-function relationships.

## Core Concepts

### Topological Constraints
- **Connectome topology**: Structural wiring patterns constrain possible dynamics
- **Persistent homology**: Topological features at multiple scales
- **Simplicial complexes**: Higher-order interactions beyond pairwise connectivity
- **Topological invariants**: Features preserved under continuous deformation

### Sensitivity Analysis
- **Structural perturbation**: How changes in connectivity affect dynamics
- **Functional sensitivity**: How topology determines response to stimulation
- **Robustness analysis**: Identifying critical vs. redundant connections
- **Lesion simulation**: Virtual lesion studies on connectome models

### Key Metrics
- **Betti numbers**: Count of topological holes at each dimension
- **Persistence diagrams**: Birth-death of topological features across scales
- **Euler characteristic**: Alternating sum of Betti numbers
- **Topological similarity**: Distance between connectivity patterns

## Implementation

```python
import numpy as np
from scipy.spatial.distance import pdist, squareform

def compute_weighted_clique_complex(adjacency, threshold):
    """Build weighted clique complex from adjacency matrix."""
    n = adjacency.shape[0]
    cliques = {0: list(range(n)), 1: [], 2: [], 3: []}
    
    # 0-simplices (nodes)
    # 1-simplices (edges)
    for i in range(n):
        for j in range(i+1, n):
            if adjacency[i, j] > threshold:
                cliques[1].append((i, j))
    
    # 2-simplices (triangles)
    for edge1 in cliques[1]:
        for edge2 in cliques[1]:
            if edge1[0] != edge2[0] and edge1[1] != edge2[1]:
                triangle = tuple(sorted(set(edge1) | set(edge2)))
                if len(triangle) == 3:
                    if all(adjacency[triangle[i], triangle[j]] > threshold 
                           for i in range(3) for j in range(i+1, 3)):
                        if triangle not in cliques[2]:
                            cliques[2].append(triangle)
    
    return cliques

def betti_numbers_from_cliques(cliques):
    """Compute Betti numbers from clique counts."""
    counts = [len(cliques[k]) for k in sorted(cliques.keys())]
    
    # Euler characteristic
    euler = sum((-1)**k * counts[k] for k in range(len(counts)))
    
    # Betti numbers (simplified)
    b0 = 1  # Connected components
    b1 = counts[1] - counts[0] + b0  # Loops
    b2 = counts[2] - counts[1] + b0 - b1  # Voids
    
    return {'b0': b0, 'b1': max(0, b1), 'b2': max(0, b2)}

def topological_sensitivity(adjacency, perturbation_strength=0.1):
    """Measure topological sensitivity to structural perturbations."""
    n = adjacency.shape[0]
    
    # Original topology
    original_cliques = compute_weighted_clique_complex(adjacency, np.mean(adjacency))
    original_betti = betti_numbers_from_cliques(original_cliques)
    
    # Perturbed topology
    noise = np.random.randn(n, n) * perturbation_strength
    perturbed = adjacency + noise
    perturbed_cliques = compute_weighted_clique_complex(perturbed, np.mean(perturbed))
    perturbed_betti = betti_numbers_from_cliques(perturbed_cliques)
    
    # Sensitivity = change in Betti numbers
    sensitivity = {}
    for k in ['b0', 'b1', 'b2']:
        sensitivity[k] = abs(perturbed_betti[k] - original_betti[k])
    
    return sensitivity

def persistence_diagram(adjacency, max_scale=1.0, num_scales=50):
    """Compute approximate persistence diagram."""
    thresholds = np.linspace(0, max_scale, num_scales)
    betti_history = []
    
    for t in thresholds:
        cliques = compute_weighted_clique_complex(adjacency, t)
        betti = betti_numbers_from_cliques(cliques)
        betti_history.append(betti)
    
    return betti_history
```

## Applications
- **Lesion studies**: Predict effects of structural damage on brain function
- **Brain stimulation**: Identifying robust vs. sensitive stimulation targets
- **Neurodegenerative diseases**: Understanding topology-driven vulnerability
- **Developmental disorders**: Topological differences in atypical connectomes

## References
- Petri, G. et al. (2014). Homological scaffolds of brain functional networks. Journal of The Royal Society Interface.
- Sizemore, A. et al. (2019). Cliques and cavities in the human connectome. Journal of Computational Neuroscience.

## Related
- [[brain-network-topology]]
- [[brain-higher-order-structures]]
- [[tda-neuroscience]]
- [[motif-based-filtrations-persistent-homology-framework-graph]]