The maximum entropy random walk in a disordered lattice is obtained as a consequence of the principle of maximum entropy for a particular type of prior information without restriction on the number of steps. This novel result demonstrates that transition probabilities defining the random walk represent a general characterization of information on a defective lattice and does not necessarily reflect a physical process. The localization phenomenon is shown to be a consequence of solution of the Laplacian on the lattice—hence it contradicts the previous interpretation as a spherical Lifshitz state—and naturally generalizes to multiple modes, whose order reflects the significance of information. The dynamics of information flow on the microscale is related to the macroscopic structure of the lattice through a Fokker-Planck formalism. This newly derived theoretical framework is opening doors for a wide range of applications in analysis of (information) flow in disordered systems. That includes potentially breakthrough resolution of the outstanding problem of inferring connectivity from discrete imaging (i.e., neural) data.
We have developed a method for the simultaneous estimation of local diffusion and the global fiber tracts based upon the information entropy flow that computes the maximum entropy trajectories between locations and depends upon the global structure of the multi-dimensional and multi-modal diffusion field. Computation of the entropy spectrum pathways requires only solving a simple eigenvector problem for the probability distribution for which efficient numerical routines exist, and a straight forward integration of the probability conservation through ray tracing of the convective modes guided by a global structure of the entropy spectrum coupled with a small scale local diffusion. The intervoxel diffusion is sampled by multi b-shell multi q-angle DWI data expanded in spherical waves. This novel approach to fiber tracking incorporates global information about multiple fiber crossings in every individual voxel and ranks it in the most scientifically rigorous way. This method has potential significance for a wide range of applications, including studies of brain connectivity.
Characterization of complex shapes embedded within volumetric data is an important step in a wide range of applications. Standard approaches to this problem employ surface based methods that require inefficient, time consuming, and error prone steps of surface segmentation and inflation to satisfy the uniqueness or stability of subsequent surface fitting algorithms. Here we present a novel method based on a spherical wave decomposition (SWD) of the data that overcomes several of these limitations by directly analyzing the entire data volume, obviating the segmentation, inflation, and surface fitting steps, significantly reducing the computational time and eliminating topological errors while providing a more detailed quantitative description based upon a more complete theoretical framework of volumetric data. The method is demonstrated and compared to the current state-of-the-art neuroimaging methods for segmentation and characterization of volumetric magnetic resonance imaging data of the human brain.
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