Bursting synchronization dynamics of pancreatic β-cells with electrical and chemical coupling

Abstract

Based on bifurcation analysis, the synchronization behaviors of two identical pancreatic β-cells connected by electrical and chemical coupling are investigated, respectively. Various firing patterns are produced in coupled cells when a single cell exhibits tonic spiking or square-wave bursting individually, irrespectively of what the cells are connected by electrical or chemical coupling. On the one hand, cells can burst synchronously for both weak electrical and chemical coupling when an isolated cell exhibits tonic spiking itself. In particular, for electrically coupled cells, under the variation of the coupling strength there exist complex transition processes of synchronous firing patterns such as “fold/limit cycle” type of bursting, then anti-phase continuous spiking, followed by the “fold/torus” type of bursting, and finally in-phase tonic spiking. On the other hand, it is shown that when the individual cell exhibits square-wave bursting, suitable coupling strength can make the electrically coupled system generate “fold/Hopf” bursting via “fold/fold” hysteresis loop; whereas, the chemically coupled cells generate “fold/subHopf” bursting. Especially, chemically coupled bursters can exhibit inverse period-adding bursting sequence. Fast–slow dynamics analysis is applied to explore the generation mechanism of these bursting oscillations. The above analysis of bursting types and the transition may provide us with better insight into understanding the role of coupling in the dynamic behaviors of pancreatic β-cells.

Appendix

See Figs. 14 and 15.

Fig. 14

a Time series of the single model with g _s  = 2. b Bifurcation diagram of the fast subsystem of [Eqs. (8), (9)]. H refers to the supHopf bifurcation, F1 and F2 refer to the fold bifurcations, HC refers to the homoclinic bifurcation. The C shaped curve represents the maximum and minimum values of periodic solutions originated from H. The phase plot of the whole system is also superposed on the (v, s) plane. The red line is the s-nullcline. (Color figure online)

Fig. 15

a Time series of the single model with g _s  = 4. b Bifurcation diagram of the fast subsystem. The description of the lines and points is the same as that in Fig. 14b

The model for a single pancreatic β-cell is described by the following equations:

$$ \tau \frac{dv}{dt} = - I_{Ca} (v) - I_{k} (v,n) - I_{s} (v,s) - I_{K(ATP)} (v,p) $$

(7)

$$ \tau \frac{dn}{dt} = \lambda [n_{\infty } (v) - n] $$

(8)

$$ \tau_{s} \frac{ds}{dt} = s_{\infty } (v) - s $$

(9)

Three variables in this system are: the membrane potential (v), the delayed rectifier activation (n), and a very slow variable s which has been postulated to be the intracellular Ca²⁺ (Sherman et al. 1988). The s equation is a Ca²⁺-balance equation.

Equation (7) is the current balance equation that determines the membrane potential v and is composed of four ionic currents, which are defined by

$$ \begin{aligned} I_{Ca} (v) = g_{Ca} m_{\infty } (v)(v - v_{Ca} ) \\ I_{K} (v,n) = g_{K} n(v - v_{K} ) \\ I{}_{s}(v,s) = g_{s} s(v - v_{K} ) \\ I_{K(ATP)} (v,p) = g_{K(ATP)} p(v - v_{K} ) \\ \end{aligned} $$

with the constant conductance g _Ca , g _K , g _s and g _K(ATP) and the reversal potentials v _Ca , v _K for Ca²⁺ and K⁺ ions, respectively. I _Ca is assumed to respond instantaneously to a change in membrane potential with the steady state function $m_{\infty } (v)$, while the dynamics for I _K is governed by the gating variable n via Eq. (2), where τ is the activation time constant for the delayed rectifier K⁺ channel, and $n_{\infty } (v)$ is the equilibrium function for the activation variable n. In the expression of I _K(ATP), p is the fraction of the open K(ATP) channels. The dynamics for I _s is similar to that of I _K . But comparing with τ, the time constant τ _s becomes much bigger.

The equilibrium functions $m_{\infty } (v)$, $n_{\infty } (v)$ and $s_{\infty } (v)$ have the general expression given by $x_{\infty } (v)$ as follows:

$$ x_{\infty } (v) = [1 + e^{{(v_{x} - v)/\theta_{x} }} ]^{ - 1} ,\quad x = m, \, n, \, s $$

Since τ _s  >> τ, s responds on a much slower time scale than v and n. Thus Eqs. (7)–(9) construct a system with both slow and fast scales. The fast subsystem is composed of Eqs. (7) and (8), in which the slow variable s is considered as the bifurcation parameter. The trajectory of the whole system is then superimposed on the bifurcation diagram of the fast subsystem to explain the dynamical behavior of the model; this is the fast–slow dynamics analysis (Rinzel 1987).

All the parameters are listed in Table 1. Depending on the value of the parameter g _s , the system either undergoes a spiking solution or generates bursting oscillations. Both of which can be explained by using fast–slow dynamics analysis. The details of the two typical cases are explained as follows.

When g _s  = 2, the single cell gives continuous spiking (see Fig. 14a). The bifurcation diagram of the fast subsystem with respect to the slow variable s is plotted in Fig. 14b. The equilibrium points form a Z shaped curve in (v, s) plane. The Z shaped curve is made up of the upper, middle and the lower branch. With the increase of s, the stable foci on the upper branch become unstable via a supHopf bifurcation H at s = −0.235. A branch of stable limit cycle oscillations emerges from the H point, and it terminates at homoclinic bifurcation HC. The middle and lower branches are composed of saddles and stable nodes, respectively. The s-nullcline is also superimposed onto the bifurcation diagram. The dynamics of the full system can be understood by using the Z shaped curve and the s-nullcline together. It is seen that the s-nullcline intersects the periodic branch and the intersection sufficiently deeps into the periodic branch, so that the system produces continuous spiking (Terman 1992).

When g _s  = 4, the single cell exhibits a typical bursting oscillation (see Fig. 15a). Fast–slow dynamics analysis is illustrated in Fig. 15b. It is noticed that g _s is not involved in the slow subsystem. Hence, it has no effect on the s-nullcline. Compared with Fig. 14b, it is seen that the s-nullcline still intersects with the periodic branch, and at this time the intersection is near the end of the periodic branch. If the variable s varied very slowly, this would result in continuous spiking. However, s does not vary such slow actually in this case so that the bursting can be produced (Terman 1992). Moreover, for values of s between the left knee of the Z shaped curve and the homoclinic bifurcation, there is bistablility between the stable node state on the lower branch, which corresponds to the silent phase of bursting, and the stable limit cycle oscillation corresponding to the active phase of bursting. The silent phase disappears via the fold bifurcation F1, and the active phase ends with the homoclinic bifurcation HC. Since this type of bursting depends on passing through both fold and homoclinic bifurcations, it is called as “fold/homoclinic” bursting (Perc and Marhl 2003). It is also known as “square-wave” bursting, due to the fact that the active phase spikes rise from a plateau (Rinzel 1987).