We present a novel gated recurrent neural network to detect when a person is chewing on food. We implemented the neural network as a custom analog integrated circuit in a 0.18 $\mu$m CMOS technology. The neural network was trained on 6.4 hours of data collected from a contact microphone that was mounted on volunteers' mastoid bones. When tested on 1.6 hours of previously-unseen data, the analog neural network identified chewing events at a 24-second time resolution. It achieved a recall of 91% and an F1-score of 94% while consuming $1.1\,\mu$W of power. A system for detecting whole eating episodes—like meals and snacks—that is based on the novel analog neural network consumes an estimated $18.8\,\mu$W of power.