Fig. 5: Mathematical model of islet α and β cells.

a Two-cell islet model. The state of each cell is described by its phase θ. Cell secretes hormone at phase π. The hormone secreted by α cell stimulates β cell (with strength K_αβ), and the effect of β cell activation inhibits α cell (with strength K_βα). b Illustration of winding number (w) definition. Two examples are shown. Solid lines are the actual trajectories of two solutions of the equations, and dotted lines are straight lines representing the asymptotic limits of the trajectories that define the winding number. c Left panel: Schematic of α and β cells’ interaction strengths. Four cases from the four regions in d are shown. Middle panel: Example traces of θ_α and θ_β in the four cases, respectively. The parameters used in these examples are indicated by the four colored dots in d. Right panel: Corresponding examples of Ca²⁺ traces found experimentally for the first three cases. d The phase diagram of the system. Depending on the two coupling constants K_αβ and K_βα, the oscillatory behavior of the two cells falls into one of the four phase-locked regions, characterized by the winding number w. In region 3, any rational winding number w < 1 has a stable phase-locking region. For clarity, only the phase-locking regions for lower-order rational numbers (1/2, 1/3 and 1/4) are shown. Color bar codes the period of the oscillation. Except for the four colored dots, 15 randomly selected gray dots are also shown. e Scatter plots of Δθ and T_βα versus T. Each dot represents one oscillation cycle. The color of the dot indicates the parameters used in the simulation, which are shown in phase diagram (d) with the same color.