Electroencephalogram (EEG) is used for the analysis of brain signals obtained from various electrodes placed across the scalp at specific positions. The collected signals from brain are often contaminated with Ocular Artifacts (OAs), EKG   and EMG artifacts. In this paper a novel technique is used for the removal of ocular artifacts using FastICA algorithm which decomposes the EEG signals into independent components then an LMS (Least Mean Squares) based adaptive algorithm is applied to the independent components so as to get the original EEG signals. In the first step, independent basis functions attributed to OA are computed using FastICA algorithm. In the second step we arrive ocular artifact free EEG signal efficiently comparative to FastICA. In this paper, based on some parameters like Root Mean Square Deviation (RMSD) and Root Mean Square Variance (RMSV) we can say that the EEG signal obtained after second step is better than after the first.