Comment on "Self-similarity of diffusion-limited aggregates and electrodeposition clusters"

A universal dimensionality function for the fractal dimensions of Laplacian growth

Laplacian growth, associated to the diffusion-limited aggregation (DLA) model or the more general dielectric-breakdown model (DBM), is a fundamental out-of-equilibrium process that generates structures with characteristic fractal/non-fractal morphologies. However, despite diverse numerical and theoretical attempts, a data-consistent description of the fractal dimensions of the mass-distributions of these structures has been missing. Here, an analytical model of the fractal dimensions of the DBM and DLA is provided by means of a recently introduced dimensionality equation for the scaling of clusters undergoing a continuous morphological transition. Particularly, this equation relies on an effective information-function dependent on the Euclidean dimension of the embedding-space and the control parameter of the system. Numerical and theoretical approaches are used in order to determine this information-function for both DLA and DBM. In the latter, a connection to the Rényi entropies and generalized dimensions of the cluster is made, showing that DLA could be considered as the point of maximum information-entropy production along the DBM transition. The results are in good agreement with previous theoretical and numerical estimates for two- and three-dimensional DBM, and high-dimensional DLA. Notably, the DBM dimensions conform to a universal description independently of the initial cluster-configuration and the embedding-space.

Multifractal analysis of the branch structure of diffusion-limited aggregates

We examine the branch structure of radial diffusion-limited aggregation (DLA) clusters for evidence of multifractality. The lacunarity of DLA clusters is measured and the generalized dimensions D(q) of their mass distribution is estimated using the sandbox method. We find that the global n-fold symmetry of the aggregates can induce anomalous scaling behavior into these measurements. However, negating the effects of this symmetry, standard scaling is recovered.

Determination of fractal dimension of physiologically characterized neurons in two and three dimensions

Although there is a growing interest in the application of fractal analysis in neurobiology, questions about the methodology have restricted its wider application. In this report we discuss some of the underlying principles for fractal analysis, we propose the cumulative-mass method as a standard method and we extend the applicability of fractal analysis to both 2 and 3 dimensions. We have examined the relationship between the method of log-log Sholl analysis and fractal analysis and have found that they correlate well. Measurements of physiologically characterized retinal ganglion cells indicate that different cell types can have significantly different fractal dimensions. Such differences may allow the correlation of the physiological type of a neuron with its morphological fractal dimension.