The fractal geometry of the cortex: from synthetic meshes to geometric eigenmodes

Statistical self-similarity in the cortical folding is one of the key structural features of the human brain. It may be considered as the underlying design principle capable of unifying and reconciling apparently contradictory theories of neural dynamics. Traditional connectome-based models treat the cortex as an assembly of discrete nodes linked by white-matter tracts, overlooking continuous wave propagation and geometry-dependent conduction delays. Conversely, Neural Field Theory models the cortex as a continuous excitable medium supporting a brain-wide wave propagation. Accordingly, more recent theories explain experimental fMRI data as the result of excitations of resonant modes of brain geometry, thus implying that cortical geometry constrains its activity. This thesis aims to extend the geometric eigenmode framework of large-scale brain dynamics by integrating the fractal complexity of cortical morphology since it could play a fundamental role in shaping the spectrum and, consequently, functional dynamics. To investigate this, brain-like 2D synthetic meshes were generated with controlled fractal dimension and the associated geometric eigenmodes were computed by solving the Laplace-Beltrami operator. Then, the effect of fractal complexity was analyzed as related to both spectral properties and mesh reconstruction accuracy. Results showed that an increased fractal complexity enhances the geometric eigenmodes spectral distribution providing the foundation for more effective and complex brain functioning.