This paper concerns a novel application of Neural Networks to Magnetic Resonance Imaging (MRI) by considering regularized Neural Network models for the problem of image reconstruction from sparsely sampled k-space. Effective solutions to this problem are indispensable especially when dealing with MRI of dynamic phenomena since then, rapid sampling in k-space is required. The goal in such a case is to reduce the measurement time by omitting as many scanning trajectories as possible. This approach, however, entails underdetermined equations and leads to poor image reconstruction. It is proposed here that significant improvements could be achieved concerning image reconstruction if a procedure, based on neural network function approximation methodology and involving regularization techniques, for estimating the missing samples of complex k-space were introduced. To this end, the viability of involving Neural Network algorithms with/without regularization for such a problem is considered and it is found that their image reconstruction results are very favorably compared to the ones obtained by the trivial zero-filled k-space approach or traditional more sophisticated interpolation approaches. Moreover, it is found that regularized Multilayer Perceptrons outperform the ones not involving regularization during their training.