On the Gompertzian growth in the fractal space-time.

An analytical approach to determination of time-dependent temporal fractal dimension b(t)(t) and scaling factor a(t)(t) for the Gompertzian growth in the fractal space-time is presented. The derived formulae take into account the proper boundary conditions and permit a calculation of the mean values b(t)(t) and a(t)(t) at any period of time. The formulae derived have been tested on experimental data obtained by Schrek for the Brown-Pearce rabbit's tumor growth.

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.

On time-space of nonlinear phenomena with Gompertzian dynamics.

This paper describes a universal relationship between time and space for a nonlinear process with Gompertzian dynamics, such as growth. Gompertzian dynamics implicates a coupling between time and space. Those two categories are related to each other through a linear function of their logarithms.

Coherent states of Gompertzian growth.

The origin of the Gompertz function G(t)=G(0)e(b/a(1-e(-at))) widely applied to fit the biological and medical data, particularly growth of organisms, organs, and tumors is analyzed. It is shown that this function is a solution of a time-dependent counterpart of the Schrödinger equation for the Morse oscillator with anharmonicity constant equal to 1. The coherent states of the Gompertzian systems, which minimize the time-energy uncertainty relation, have been found.

Neuronal differentiation and synapse formation occur in space and time with fractal dimension.

The analysis of a set of experimental data obtained by an independent team of researchers confirms that neuronal differentiation or synapse formation do occur in time and space with fractal dimension. The interacting cells create first a dynamic system with its own attractor, (i.e.