On this page, you will find some interactive simulations of some of the models presented in this preprint. Here we are simulating the system $$ \begin{cases} \dot{\phi} = 2\pi f(x) + k \sin(\theta - \phi) \\ \dot{\theta} = 2 \pi \omega \\ \dot{x} = -\alpha \left(x - \frac{\dot{\phi}}{2\pi}\right) \end{cases} $$ Although this model could potentially be applied to other systems, it was initially motivated by work done on the entrainment of the segmentation clock. More specifically, it was motivated by the results from this paper. In this context, the variables represent:
- \(\phi\): The phase of the segmentation clock.
- \(\theta \): The phase of the external oscillatory perturbation entraining the segmentation clock
- \( x \): A phenomenological dynamical "memory" variable corresponding to the frequency of the segmentation clock.
The motivating biological system corresponds to the oscillations in the Notch signalling pathway inside of the mouse embryo tailbud. These oscillations are part of a plethora of interacting oscillations collectively forming the segmentation clock. After applying periodic pulses of DAPT (a Notch inhibitor) in this system through a microfluidics chip, we observe the Notch oscillations being entrained to the external perturbation. Experimental evidence suggests that the segmentation clock adapts its intrinsic frequency to the one of the entraining oscillator.