Heterogeneity and Delayed Activation as Hallmarks of Self-Organization and Criticality in Excitable Tissue

Abstract

Self-organized critical dynamics is assumed to be an attractive mode of functioning for several real-life systems and entails an emergent activity in which the extent of observables follows a power-law distribution. The hallmarks of criticality have recently been observed in a plethora of biological systems, including beta cell populations within pancreatic islets of Langerhans. In the present study, we systematically explored the mechanisms that drive the critical and supercritical behavior in networks of coupled beta cells under different circumstances by means of experimental and computational approaches. Experimentally, we employed high-speed functional multicellular calcium imaging of fluorescently labeled acute mouse pancreas tissue slices to record calcium signals in a large number of beta cells simultaneously, and with a high spatiotemporal resolution. Our experimental results revealed that the cellular responses to stimulation with glucose are biphasic and glucose-dependent. Under physiological as well as under supraphysiological levels of stimulation, an initial activation phase was followed by a supercritical plateau phase with a high number of global intercellular calcium waves. However, the activation phase displayed fingerprints of critical behavior under lower stimulation levels, with a progressive recruitment of cells and a power-law distribution of calcium wave sizes. On the other hand, the activation phase provoked by pathophysiologically high glucose concentrations, differed considerably and was more rapid, less continuous, and supercritical. To gain a deeper insight into the experimentally observed complex dynamical patterns, we built up a phenomenological model of coupled excitable cells and explored empirically the model's necessities that ensured a good overlap between computational and experimental results. It turned out that such a good agreement between experimental and computational findings was attained when both heterogeneous and stimulus-dependent time lags, variability in excitability levels, as well as a heterogeneous cell-cell coupling were included into the model. Most importantly, since our phenomenological approach involved only a few parameters, it naturally lends itself not only for determining key mechanisms of self-organized criticality at the tissue level, but also points out various features for comprehensive and realistic modeling of different excitable systems in nature.

Keywords: activation delay; beta cells; calcium imaging; cellular heterogeneity; computational model; excitable cells; self-organized criticality.

FIGURE 1

Dynamical features of the Rulkov map. (A) Real and imaginary eigenvalues λ₁ and λ₂ for different values of the control parameter α. (B) Bifurcation diagram of the fast variable with (blue) and without (green) added noise. (C) Traces of the fast variable with (blue) and without (green) noise for different values of excitability levels α.

FIGURE 2

Features of the phenomenological model of beta cell population. (A) A typical simulated beta cell network architecture. Red dots denote individual cells and the arrows depict intercellular electrical coupling. (B) Simulated time course of beta cell excitability rate after switching from substimulatory to stimulatory (Δα = 0.08, A = 0.45, B = 0.0004, $T_{\mathrm{m},i}^{(8)}=35000$, t_s = 10000) and suprastimulatory (Δα = 0.09, A = 0.70, B = 0.0008, $T_{\mathrm{m},i}^{(12)}=20000$, t_s = 10000) conditions.

FIGURE 3

A schematic representation of all three types of heterogeneities in the model. (A) electrical excitability Δα_i, (B) beta cell metabolism T_m,i, (C) electrical coupling g_i. Widths of connections reflect the coupling strength.

FIGURE 4

Experimentally measured beta cell responses after stimulation with 8 mM glucose. (A) Three characteristic Ca²⁺ traces and the raster plot of binarized Ca²⁺ activity of all cells in the islet. The orange dotted line indicates the fraction of active cells within the given time-window that was slid throughout the recording. (B,D) 3D raster plots showing the Ca²⁺ activity waveforms for selected intervals for the activation (B) and plateau (D) phase. Colors denote specific Ca²⁺ events. Gray dots on the y, z plane stand for coordinates of cells. (C,E) The distributions of Ca²⁺ wave sizes for the activation (C) and plateau (E) phase. The gray dashed line indicates the power-law fit. The slope in the critical-like activation phase is –1.69.

FIGURE 5

Experimentally measured beta cell responses after stimulation with 12 mM glucose. (A) Three characteristic Ca²⁺ traces and the raster plot of binarized signals of Ca²⁺ oscillations in all cells in the islet. The orange dotted line indicates the fraction of active cells within the given time-window that was slid throughout the recording. (B,D) 3D raster plots showing the Ca²⁺ waves for selected intervals for the activation (B) and plateau (D) phase. Colors denote specific Ca²⁺ events. Gray dots on the y, z plane stand for coordinates of cells. (C,E) The distributions of Ca²⁺ wave sizes for the activation (C) and plateau (E) phase. The gray dashed line indicates the power-law fit.

FIGURE 6

Simulated beta cell responses after switching from a substimulatory to stimulatory levels of stimulation, i.e., from 6 to 8 mM glucose. (A) Three characteristic traces of simulated cellular dynamics and the raster plot of binarized cellular activity. The orange dotted line indicates the fraction of active cells within the given time-window that was slid throughout the simulation. (B,D) 3D raster plots showing the excitation waves for selected intervals for the activation (B) and plateau (D) phase. Colors denote individual waves. (C,E) The distributions of excitation wave sizes for the activation (C) and plateau (E) phase. The gray dashed line indicates the power-law fit. The slope in the critical-like activation phase is –1.64.

FIGURE 7

Simulated beta cell responses after switching from a substimulatory to a suprastimulatory level of stimulation, i.e., from 6 to 12 mM glucose. (A) Three characteristic traces of simulated cellular dynamics and the raster plot of binarized cellular activity. The orange dotted line indicates the fraction of active cells within the given time-window that was slid throughout the simulation. (B,D) 3D raster plots showing the excitation waves for selected intervals for the activation (B) and plateau (D) phase. Colors denote individual waves. (C,E) The distributions of excitation wave sizes for the activation (C) and plateau (E) phase. The gray dashed line indicates the power-law fit.

FIGURE 8

Simulated beta cell responses without particular types of cellular heterogeneities. Behavior without heterogeneity in the intrinsic excitability level (A,B), without heterogeneity in the delayed responses to stimulation (C,D), and without heterogeneity in intercellular coupling (E,F), for both physiological (A,C,E) and supraphysiological (B,D,F) stimulation levels.

FIGURE 9

Simulated beta cell responses for different degrees of heterogeneities. Simulating the behavior without (A,B), with 10% (C,D), and 20% (E,F) of the used degree of heterogeneities for physiological (A,C,E) and supraphysiological (B,D,F) stimulation levels.