Alarm fatigue caused by false alarms and alerts is an extremely important issue for the medical staff in Intensive Care Units. The ability to predict electrocardiogram and arterial blood pressure waveforms can potentially help the staff and hospital systems better classify a patient's waveforms and subsequent alarms. This paper explores the use of Echo State Networks, a specific type of neural network for mining, understanding, and predicting electrocardiogram and arterial blood pressure waveforms. Several network architectures are designed and evaluated. The results show the utility of these echo state networks, particularly ones with larger integrated reservoirs, for predicting electrocardiogram waveforms and the adaptability of such models across individuals. The work presented here offers a unique approach for understanding and predicting a patient's waveforms in order to potentially improve alarm generation. We conclude with a brief discussion of future extensions of this research.