Multi-input Adaptive Notch Filter and Observer for Circadian Phase Estimation

Accurate estimation of circadian phase is critical to the assessment and treatment of circadian disruption. Direct measurements of circadian rhythm markers such as dim light melatonin onset and core body temperature are inconvenient and acquired at best at low rate. On the other hand, measurements of other circadian rhythm-modulated signals such as actigraphy, heart rate, and body temperature are convenient but are typically masked by many other factors.

In this paper, we present a new multi-input adaptive notch filter algorithm that can be used to extract the periodic components from multiple circadian signals simultaneously. We also prove some stability properties of the proposed filter. Once the periodic components are extracted, the next step is to relate their phases with the circadian phase. For this, we propose a nonlinear observer, which is based on a model of the circadian phase dynamics. The model takes the form of a first-order ODE, incorporating the concept of phase response curve, which is widely used in the study of biological oscillators. We also prove the stability of the observer. We evaluate our algorithms using simulation data generated from a circadian rhythm model for fruit flies (Drosophila melanogaster).