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Pulsatile release of cortisol from the adrenal glands is governed by pulsatile release of adrenocorticotropic hormone (ACTH) from the anterior pituitary. In return, cortisol has a negative feedback effect on ACTH release. Simultaneous recording of ACTH and cortisol is not typical, and determining the number, timing, and amplitudes of pulsatile events from simultaneously recorded data is challenging because of several factors: 1) stimulator ACTH pulse activity, 2) kinematics of ACTH and cortisol, 3) the sampling interval, and 4) the measurement error. We model ACTH and cortisol secretion simultaneously using a linear differential equations model with Gaussian errors and sparse pulsatile events as inputs to the model. We propose a novel framework for recovering pulses and parameters underlying the interactions between ACTH and cortisol. We recover the timing and amplitudes of pulses using compressed sensing …