Transcranial doppler (TCD) ultrasound is a non-invasive imaging technique that can be used for continuous monitoring of blood flow in the brain through the major cerebral arteries by calculating the cerebral blood flow velocity (CBFV). Since the brain requires a consistent supply of blood to function properly and meet its metabolic demand, a change in CBVF can be an indication of neurological diseases. Depending on the severity of the disease, the symptoms may appear immediately or may appear weeks later. For the early detection of neurological diseases, a classification model is proposed in this study, with the ability to distinguish healthy subjects from critically ill subjects. The TCD ultrasound database used in this study contains signals from the middle cerebral artery (MCA) of 6 healthy subjects and 12 subjects with known neurocritical diseases. The classification model works based on the maximal blood flow velocity waveforms extracted from the TCD ultrasound. Since the signal quality of the recorded TCD ultrasound is highly dependent on the operator's skillset, a noisy and corrupted signal can exist and can add biases to the classifier. Therefore, a deep learning classifier, trained on a curated and clean biomedical signal can reliably detect neurological diseases. For signal classification, this study proposes a Self-organized Operational Neural Network (Self-ONN)-based deep learning model Self-ResAttentioNet18, which achieves classification accuracy of 96.05% with precision, recall, f1 score, and specificity of 96.06%, 96.05%, 96.06%, and 96.09%, respectively. With an area under the ROC curve of 0.99, the model proves its feasibility to confidently classify middle cerebral artery (MCA) waveforms in near real-time.