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.

Building on an idea of Fogelson and Peskin [J. Comput. Phys. 79, 50 (1988)] we describe the implementation and verification of a simulation technique for systems of non-Brownian particles in fluids at Reynolds numbers up to about 20 on the particle scale. This direct simulation technique fills a gap between simulations in the viscous regime and high-Reynolds-number modeling. It also combines sufficient computational accuracy with numerical efficiency and allows studies of several thousand, in principle arbitrarily shaped, extended and hydrodynamically interacting particles on regular work stations. We verify the algorithm in two and three dimensions for (i) single falling particles and (ii) a fluid flowing through a bed of fixed spheres. In the context of sedimentation we compute the volume fraction dependence of the mean sedimentation velocity. The results are compared with experimental and other numerical results both in the viscous and inertial regime and we find very satisfactory agreement.

We present analytical and numerical results suggesting that the optimal path in an energy landscape in the strong disorder limit is in the universality class of the shortest path in invasion percolation with trapping. Our results imply that, in contrast to common belief, invasion percolation with trapping and regular percolation in d 3 are in different universality classes. [S0031-9007(97)04589-4]

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We calculate the minimum growth probability for diffusion-limited aggregation (DLA) as a function of the cluster mass M, and find a novel singularity of the form -\np mw (M)~-(\nM) y with y « 2. We interpret this result in terms of a simple model for DLA structure, which is characterized by a hierarchy of self-similar voids separated by channels whose diameter increases slower than the cluster diameter.