Brain-computer interface (BCI) actively translates the brain signals into executable actions by establishing direct communication between the human brain and external devices. Recording brain activity through electroencephalography (EEG) is generally contaminated with both physiological and nonphysiological artifacts, which significantly hinders the BCI performance. Artifact subspace reconstruction (ASR) is a well-known statistical technique that automatically removes artifact components by determining the rejection threshold based on the initial reference EEG segment in multichannel EEG recordings. In real-world applications, the fixed threshold may limit the efficacy of the artifact correction, especially when the quality of the reference data is poor. This study proposes an adaptive online ASR technique by integrating the Hebbian/anti-Hebbian neural networks into the ASR algorithm, namely, principle subspace projection ASR (PSP-ASR) and principal subspace whitening ASR (PSW-ASR) that segmentwise self-organize the artifact subspace by updating the synaptic weights according to the Hebbian and anti-Hebbian learning rules. The effectiveness of the proposed algorithm is compared to the conventional ASR approaches on benchmark EEG dataset and three BCI frameworks, including steady-state visual evoked potential (SSVEP), rapid serial visual presentation (RSVP), and motor imagery (MI) by evaluating the root-mean-square error (RMSE), the signal-to-noise ratio (SNR), the Pearson correlation, and classification accuracy. The results demonstrated that the PSW-ASR algorithm effectively removed the EEG artifacts and retained the activity-specific brain signals compared to the PSP-ASR, standard ASR (Init-ASR), and moving-window ASR (MW-ASR) methods, thereby enhancing the SSVEP, RSVP, and MI BCI performances. Finally, our empirical results from the PSW-ASR algorithm suggested the choice of an aggressive cutoff range of c = 1-10 for activity-specific BCI applications and a moderate range of for the benchmark dataset and general BCI applications.