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

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 
                            i  ()  j  (i+, )):
                         triangle   cliques[]:
                            cliques[].append(triangle)
    
     cliques

 ():
    
    counts = [(cliques[k])  k  (cliques.keys())]
    
    
    euler = ((-)**k * counts[k]  k  ((counts)))
    
    
    b0 =   
    b1 = counts[] - counts[] + b0  
    b2 = counts[] - counts[] + b0 - b1  
    
     {: b0, : (, b1), : (, b2)}

 ():
    
    n = adjacency.shape[]
    
    
    original_cliques = compute_weighted_clique_complex(adjacency, np.mean(adjacency))
    original_betti = betti_numbers_from_cliques(original_cliques)
    
    
    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 = {}
     k  [, , ]:
        sensitivity[k] = (perturbed_betti[k] - original_betti[k])
    
     sensitivity

 ():
    
    thresholds = np.linspace(, max_scale, num_scales)
    betti_history = []
    
     t  thresholds:
        cliques = compute_weighted_clique_complex(adjacency, t)
        betti = betti_numbers_from_cliques(cliques)
        betti_history.append(betti)
    
     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]]