Role of topology in complex functional networks of beta cells

The activity of pancreatic $\beta$ cells can be described by biological networks of coupled nonlinear oscillators that, via electrochemical synchronization, release insulin in response to augmented glucose levels. In this work, we analyze the emergent behavior of regular and percolated $\beta$-cells clusters through a stochastic mathematical model where “functional” networks arise. We show that the emergence and robustness of the synchronized dynamics depend both on intrinsic and extrinsic parameters. In particular, cellular noise level, glucose concentration, network spatial architecture, and cell-to-cell coupling strength are the key factors for the generation of a rhythmic and robust activity. Their role in the functional network topology associated with $\beta$-cells clusters is analyzed and discussed.

Figure 1

Construction of the percolated topology. Starting from a compact cubic cluster (left) a human-like architecture is obtained via a site-bond percolation (center) and the biggest connected component is extracted (right).

Figure 2

Different network topologies: linear chain (left), percolated architecture (center), compact structure (right).

Figure 3

Membrane potential time series for a representative cell in: linear chain (top), percolated structure (middle), compact cluster (bottom).

Figure 4

Correlation matrix (top row), reconstructed functional network (central row), and space-time plot of membrane voltage (bottom row) for each topology. (a) Left column: linear chain. (b) Central column: percolated topology. (c) Right column: compact structure. Physiological conditions are considered: $g_c=215\text{pS}$ and $\left[G\right]=7.1\text{mM}$. The color code of the percolated functional network identifies the subpopulations in the space-time plot.

Figure 5

Functional networks sequences for increasing coupling strength, (a–d) $g_c=100,215,300,400\text{pS}$, and fixed glucose concentration $\left[G\right]=7.1\text{mM}$. Top row, linear chain; center row, percolated structure; bottom row, compact cluster.

Figure 6

Degree distribution for different functional networks (FN) of the percolated structure varying the coupling strength ($g_c=100,215,300,400\text{pS}$) and keeping fixed the glucose concentration ($\left[G\right]=7.1\text{mM}$). A smoothing kernel was used for the histograms fitting, previously normalizing the occurrences with the maximum value reached in each case. N600, observables computed from simulations performed with 600 stochastic K-Ca channels per cell; N300, observables computed from simulations performed with 300 stochastic K-Ca channels per cell; AVNND, average nearest neighbor degree; $P_{\text{cum}}$, cumulative degree distribution. Dotted line and continuous line in the AVNND plots represent a power law and a log-normal fit of the data (colored square), respectively.

Figure 7

Functional networks sequences for increasing glucose concentration, (a–d) $\left[G\right]=4.7,8.7,12.7,16.6\text{mM}$ from left to right and fixed gap junction conductance $g_c=215\text{pS}$. Top row, linear chain; center row, percolated structure; bottom row, compact cluster.

Figure 8

Degree distribution for different functional networks (FN) of the percolated structure varying the glucose concentration ($\left[G\right]=4.7,8.7,12.7,16.6\text{mM}$) and keeping fixed the coupling conductance ($g_c=215\text{pS}$). As for variable coupling strength, a smoothing kernel was used for the normalized histograms fitting. N600, observables computed from simulations performed with 600 stochastic K-Ca channels per cell; N300, observables computed from simulations performed with 300 stochastic K-Ca channels per cell; AVNND, average nearest-neighbor degree.

Figure 9

Functional networks sequences for increasing coupling strength, (a–d) $g_c=100,215,300,400\text{pS}$ from left to right at fixed glucose, i.e., $\left[G\right]=7.1\text{mM}$. The number of K-Ca channels per cell was set to 300 to increase noise strength. Top row, linear chain; center row, percolated structure; bottom row, compact cluster.

Figure 10

Functional networks sequences for increasing glucose, (a–d) $\left[G\right]=4.7,8.7,12.7,16.6\text{mM}$ from left to right at fixed gap junction conductance, i.e., $g_c=215\text{pS}$. The number of K-Ca channels per cell was set to 300 to increase noise strength. Top row, linear chain; center row, percolated structure; bottom row, compact cluster.