  The methods of Nuclear Magnetic Resonance belong to the best developed and
often used tools for studying random motion of particles in different systems,
including soft biological tissues. In the long-time limit the current
mathematical description of the experiments allows proper interpretation of
measurements of normal and anomalous diffusion. The shorter-time dynamics is
however correctly considered only in a few works that do not go beyond the
standard memoryless Langevin description of the Brownian motion (BM). In the
present work, the attenuation function S(t) for an ensemble of spin-bearing
particles in a magnetic-field gradient, expressed in a form applicable for any
kind of stationary stochastic dynamics of spins with or without a memory, is
calculated in the frame of the model of fractional BM. The solution of the
model for particles trapped in a harmonic potential is obtained in an
exceedingly simple way and used for the calculation of S(t). In the limit of
free particles coupled to a fractal heat bath, the results compare favorably
with experiments acquired in human neuronal tissues. The effect of the trap is
demonstrated by introducing a simple model for the generalized diffusion
coefficient of the particle.
