Neuronal differentiation and synapse formation in the space-time with temporal fractal dimension

An improvement of the Waliszewski and Konarski approach ([2002] Synapse 43:252-258) to determine the temporal fractal dimension b(t) and scaling factor a(t) for the process of neuronal differentiation and synapse formation in the fractal space-time is presented. In particular the analytical formulae describing the time-dependence of b(t)(t) and a(t)(t), which satisfy the appropriate boundary conditions for t-->0 and t-->infinity, are derived. They have been used to determine the temporal fractal dimension and scaling factor from the two-parametric Gompertz function fitted to experimental data obtained by Jones-Villeneuve et al. ([1982] J Cell Biol 94:253-262) for embryonal carcinoma P19 cells treated by retinoic acid. The results of the calculations differ from those obtained previously by making use of the three- and four-parametric Gompertz function as well as other S-shape functions (Chapman, Hill, Logistic, Sigmoid) evaluated by the fitting of the experimental curve. The temporal fractal dimension can be used as a numerical measure of the neuronal complexity emerging in the process of differentiation, which can be related to the morphofunctional cell organization. A hypothesis is formulated that neuronal differentiation and synapse formation have a lot in common with the process of tumorigenesis. They are qualitatively described by the same Gompertz function of growth and take place in the fractal space-time whose mean temporal fractal dimension is lost during progression.