Analysis of the noise-induced bursting-spiking transition in a pancreatic $\beta$-cell model

A stochastic model of the electrophysiological behavior of the pancreatic $\beta$ cell is studied, as a paradigmatic example of a bursting biological cell embedded in a noisy environment. The analysis is focused on the distortion that a growing noise causes to the basic properties of the membrane potential signals, such as their periodic or chaotic nature, and their bursting or spiking behavior. We present effective computational tools to obtain as much information as possible from these signals, and we suggest that the methods could be applied to real time series. Finally, a universal dependence of the main characteristics of the membrane potential on the size of the considered cell cluster is presented.

Figure 1

(a) Bifurcation diagram of the deterministic system when the parameter $V_{\text{Ca}}$ is varied. Periodic and chaotic behaviors are shown. (b) Zoom in of (a) in the region of low number of spikes per burst, marked with an arrow in (a).

Figure 2

Plots of the membrane potential for $P6$, $Ch$, and $P40$. (a) Deterministic case, (b) $N=1000$, (c) $N=10$. We can see that the noise affects each orbit in a different way. For small values of $N$, only $P40$ still shows a recognizable bursting behavior.

Figure 3

Largest Lyapunov exponent as a function of the parameter $V_{\text{Ca}}$ for the deterministic model, for $N=5000$, $N=1000$, and $N=150$ cells.

Figure 4

Largest Lyapunov exponent as a function of $N$ for the periodic orbits $P6$ and $P40$, as well as for the chaotic orbit $Ch$.

Figure 5

(a) ISI return maps for $P6$ (six crosses) and the chaotic orbit (dots). (b) ISI return map for $P40$. (c) Zoom in of the region marked with an arrow in (b). The return maps for the periodic orbits are composed of three isolated groups of dots, while the chaotic signal gives rise to an apparently continuous curve.

Figure 6

ISI return maps for $P6$, $Ch$, and $P40$. The pictures to the left correspond to $N=5000$ cells, while the pictures to the right correspond to $N=500$ cells. The three groups of dots in the return map of a periodic orbit become more and more spread when the size of the cluster decreases (and therefore when the noise is increased).

Figure 7

ISI histograms for $P6$. (a) Deterministic case, (b) $N=1000$, (c) $N=500$, (d) $N=100$. The quantities $d$, $d_{max}$, $d_{min}$, $h_H$, and $h_{PDF}$ are shown.

Figure 8

ISI histograms for the chaotic orbit $Ch$. (a) Deterministic case, (b) $N=1000$, (c) $N=500$, (d) $N=100$. In a chaotic orbit, even in the absence of noise the ISI histogram follows an apparently continuous curve.

Figure 9

Variation with $N$ of $d$, $d_{max}$, and $d_{min}$, for $P6$, when we apply the Hartigan’s clustering algorithm (circles). The values of $d$ obtained from direct measurement are also shown (triangles). $N_d\approx 300$ for $P6$. The two solid lines represent mathematical approximations of $d$.

Figure 10

ISI histograms for $P40$. (a) $N=150$, (b) $N=25$, (c) $N=10$, (d) $N=1$. The higher number of spikes per burst, the more robust to noise the signals are.

Figure 11

Variation of $h_H$ (circles) and $h_{PDF}$ (triangles) for $P6$ with the number of cells $N$. The solid line is the mathematical approximation of $h_{PDF}$.

Figure 12

Evolution for $P6$ and $P40$ of the number of spikes per burst with the time, during the first 150 bursts. The curves are plotted for the deterministic case (wide horizontal lines in $\text{NSB}=6$ and $\text{NSB}=40$) and $N=1000$ in the stochastic regime. The dashed lines represent the convergence of the mean $\mu$ in both cases.

Figure 13

(a) Deterministic signal for $P6$ in the $V\text{−}\text{Ca}$ plane. (b) Deterministic signal for $P40$ in the $V\text{−}\text{Ca}$ plane. The $V\text{−}n$ and the $\text{Ca}$ nullclines are plotted both in (a) and (b). (c) Unstable points $\text{FP}1$ and $\text{FP}2$, stable point $\text{FP}3$ and their invariant manifolds for $P6$, just after the homoclinic bifurcation.

Figure 14

Stochastic signals for $P6$ (a) and $P40$ (b) in the $V\text{−}\text{Ca}$ plane, when $N=1000$. The $V\text{−}n$ and the $\text{Ca}$ nullclines are plotted.

Figure 15

(a) Variation of the mean value $\mu$ with $N$, for $P6$ (triangles) and the chaotic orbit (circles). They are independent of the number of cells in the cluster. (b) Variation of the standard deviation $\sigma$ for $P6$ (triangles) and the chaotic orbit (circles).

Figure 16

(a) Variation of the mean value $\mu$ with $N$ for $P40$. It decreases when $N$ decreases. The solid line represents the mathematical approximation. (b) Variation of the standard deviation $\sigma$ for $P40$.