Topological Analysis of Macaca mulatta's Cortical Structures Through the Lens of Poincare Duality

Poincare duality from algebraic topology describes how shapes and spaces interrelate across different dimensions, linking structures of one scale to complementary structures at another. This suggests that densely packed neurons in certain brain areas may correspond to broader connectivity patterns, balancing local processing with global communication. Applying this framework to cortical histological images of Macaca mulatta, we analysed neuronal clustering, connectivity graphs and intensity distributions to identify self-dual structural patterns and reveal the contribution of local organization to large-scale network activity. Using image processing techniques such as contrast enhancement, edge detection and graph-theoretic modeling, we examined how dense neuron clusters correspond to functionally sparse regions and vice versa. We found that high-density neuronal zones form closed-loop topological structures that correspond to homology cycles, while sparser areas function as large-scale integrators, aligning with cohomology properties. Local connectivity hubs in neuron-dense regions support regional specialization, while large-scale, sparser areas, though less connected, facilitate global communication by acting as pathways for long-range integration. Graph-theoretic analysis of connectivity patterns confirmed a reciprocal relationship between clustering coefficients and global centrality. Statistical analysis using Kolmogorov-Smirnov tests revealed conserved topological distributions across different cortical regions, supporting the hypothesis that cortical connectivity maintains structural invariance under local perturbations. These findings provide insights into the mathematical principles governing brain architecture, suggesting that topological methods can enhance our understanding of cortical function. Future research may extend these approaches to higher-dimensional embeddings, network theory in primate brains, functional neuroimaging, human disease modeling and artificial intelligence.