Unclocklike Oscillators with Frequency Memory for the Entrainment of Biological Clocks

Entrainment experiments on the vertebrate segmentation clock have revealed that embryonic oscillators actively change their internal frequency to adapt to the driving signal. This is not consistent with either a one-dimensional clock model or a limit-cycle model, but rather it suggests a new “unclocklike” behavior. In this work, we propose simple, biologically realistic descriptions of such internal frequency adaptation, where a phase oscillator activates a memory variable controlling the oscillator's frequency. We study two opposite limits for the control of the memory variable, one with a smooth phase-averaging memory field, and the other with a pulsatile, phase-dependent activation. Both models recapitulate intriguing properties of the entrained segmentation clock, such as very broad Arnold tongues and an entrainment phase plateauing with detuning. We compute analytically multiple properties of such systems, such as entrainment phases and cycle shapes. We further describe new phenomena, including hysteresis in entrainment, bistability in the frequency of the entrained oscillator, and probabilistic entrainment. Our work shows that oscillators with frequency memory can exhibit new classes of unclocklike properties that can be tested through experimental entrainment. Published by the American Physical Society 2024