The reduction of artifacts in neural data is a key element in improving analysis of brain recordings and the development of effective brain-computer interfaces. This complex problem becomes even more difficult as the number of channels in the neural recording is increased. Here, new techniques based on wavelet thresholding and independent component analysis (ICA) are developed for use in high-dimensional neural data. The wavelet technique uses a discrete wavelet transform with a Haar basis function to localize artifacts in both time and frequency before removing them with thresholding. Wavelet decomposition level is automatically selected based on the smoothness of artifactual wavelet approximation coefficients. The ICA method separates the signal into independent components, detects artifactual components by measuring the offset between the mean and median of each component, and then removing the correct number of components based on the aforementioned offset and the power of the reconstructed signal. A quantitative method for evaluating these techniques is also presented. Through this evaluation, the novel adaptation of wavelet thresholding is shown to produce superior reduction of ocular artifacts when compared to regression, principal component analysis, and ICA.