Fluctuations in heart rate are intimately tied to changes in the physiological state of the organism. We examine and exploit this relationship by classifying a human subject's wake/sleep status using his instantaneous heart rate (IHR) series. We use a convolutional neural network (CNN) to build features from the IHR series extracted from a whole-night electrocardiogram (ECG) and predict every 30 seconds whether the subject is awake or asleep. Our training database consists of 56 normal subjects, and we consider three different databases for validation; one is private, and two are public with different races and apnea severities. On our private database of 27 subjects, our accuracy, sensitivity, specificity, and AUC values for predicting the wake stage are 83.1%, 52.4%, 89.4%, and 0.83, respectively. Validation performance is similar on our two public databases. When we use the photoplethysmography instead of the ECG to obtain the IHR series, the performance is also comparable. A robustness check is carried out to confirm the obtained performance statistics. This result advocates for an effective and scalable method for recognizing changes in physiological state using non-invasive heart rate monitoring. The CNN model adaptively quantifies IHR fluctuation as well as its location in time and is suitable for differentiating between the wake and sleep stages.