Cell Model

We consider a simplified version of the model that describes the mechanism of excitability in somatotroph cells. The mathematical model has been originally derived in the research by Tsaneva-Atanasova et. al., based on experimental data. The modified version of the model that we used comprises a set of three coupled non-linear ordinary differential equations (ODE), formed by three variables; the cell membrane potential V, the concentration Ca of intracellular Ca²⁺ ions, and the fraction n_dr of open potassium channels in the cells.

Figure 1 The global periodic attractor of the system.

We use the integrators in Matlab and XPP to solve the ODE numerically. The solution in the state space is portrayed by the orbit in Figure 1. This orbit, in fact, is the global attracting periodic attractor that exists in the system. Any arbitrary initial condition to solve the ODE always gives a solution that settles on the same closed curved in the state space. The variables are also plotted independently to produce the time series as in Figure 2.

Figure 2 The time series for V and Ca over a time interval of 30s. The variation of n_dr is not shown in the picture since it varies in a similar manner as V, scaled between 0 to 1.

The periodic behaviour in the system's solution can be divided into two phases, a silent (resting) and an active (bursting) phase. Note that the variation of Ca always happens rather slowly compared with the other two variables during the active phase. This indicates the presence of two different time scales. We reconstruct the model into the fast subsystem formed by the variables V and n_dr, and consider Ca as a parameter. The local behaviour of the global attractor is then described using the bifurcation diagram of this underlying fast subsystem. Such analysis can be adapted to show the influence of the slow Ca-dynamics in alternating the system between the silent and the active phase.