Physical mechanisms underlying neurite outgrowth: A quantitative analysis of neuronal shape

Abstract

We show that-over a range of length scales r-the shapes of quasi-two-dimensional retinal neurons are fractal objects, and hence may be quantitatively characterized in part by their fractal dimension d/.We analyze the shapes of numerous retinal neurons, both in vivo and in vitro. The neurons in vivo are found to have a fractal dimension df of 1.68 ±0.15. We also propose an explanation of certain stages of neuronal shape development in terms of a diffusion-limited-aggregation model, which predicts <//-1.70±0.1.

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

“…Image acquisition was performed using an Applescanner (resolution of 637 X 825 pixels) to digitize camera lucida line drawings of horseradish peroxidase (HRP) injected retinal ganglion cells. This improved on our earlier work (Caserta et al, 1990) which used a video camera with a resolution of 256 x 256 pixels. Images of neurons (Fig.…”

“…In a previous report (Caserta et al, 1990), our algorithm for fractal dimension calculation was called box counting. However, box counting usually refers to covering an object with a minimal number of boxes, rather than counting the number of pixels in a disk, as in our previous and present algorithm.…”

“…Fractal analysis has also been used to describe how completely a neuron fills its dendritic field or the straightness of individual dendrites (Montague and Friedlander,199 1). However, previous studies have used the following different methods for calculating the fractal dimension (d,) of a neuron: cumulative-mass (Caserta et al, 1990), box counting (Morigiwa et al, 1989) and coastline measurement (Smith et al, 1989). The use of different methods of fractal analysis has made comparison of results difficult because each method of determination can give slightly different results when used to analyze the same structure.…”

“…First, we review the method of Shall analysis which has long been used for quantitative morphological studies of the dendritic arborizations of neurons (Shall, 1953). Second, we compare Shall analysis with fractal analysis (Caserta et al, 1990) for the characterization of neuronal arborizations. To do this, we compare the fractal dimension obtained using the cumulative-mass method, with the results of both log-log and semi-log Shall analysis (Shall, 1953) for the same neurons.…”

“…We did this by analyzing several neurons in both 2 and 3 dimensions using the cumulative-mass method, and comparing the results. Finally, previous fractal analysis studies have largely been performed on neurons whose physiological A properties have not been directly assessed Friedlander, 1989, 1991;Morigiwa et al, 1989;Smith et al, 1989;Caserta et al, 1990). This left open the question of whether there is a correlation between the fractal dimension and physiology of a given cell.…”