Local dissipation and coupling properties of cellular oscillators: a case study on calcium oscillations

Dynamics of in-phase and anti-phase bursting in the coupled pre-Bötzinger complex cells

Activity of neurons in the pre-Bötzinger complex within the mammalian brain stem has an important role in the generation of respiratory rhythms. Previous experimental results have shown that the dynamics of sodium and calcium within each cell may be responsible for various bursting mechanisms. In this paper, we study the bursting dynamics of the two-coupled pre-Bötzinger complex neurons. Using a combination of fast-slow decomposition and two-parameter bifurcation analysis, we explore the possible forms of dynamics that the model network can produce as well the transitions of in-phase and anti-phase bursting respectively.

The relationship between node degree and dissipation rate in networks of diffusively coupled oscillators and its significance for pancreatic beta cells

Self-sustained oscillatory dynamics is a motion along a stable limit cycle in the phase space, and it arises in a wide variety of mechanical, electrical, and biological systems. Typically, oscillations are due to a balance between energy dissipation and generation. Their stability depends on the properties of the attractor, in particular, its dissipative characteristics, which in turn determine the flexibility of a given dynamical system. In a network of oscillators, the coupling additionally contributes to the dissipation, and hence affects the robustness of the oscillatory solution. Here, we therefore investigate how a heterogeneous network structure affects the dissipation rate of individual oscillators. First, we show that in a network of diffusively coupled oscillators, the dissipation is a linearly decreasing function of the node degree, and we demonstrate this numerically by calculating the average divergence of coupled Hopf oscillators. Subsequently, we use recordings of intracellular calcium dynamics in pancreatic beta cells in mouse acute tissue slices and the corresponding functional connectivity networks for an experimental verification of the presented theory. We use methods of nonlinear time series analysis to reconstruct the phase space and calculate the sum of Lyapunov exponents. Our analysis reveals a clear tendency of cells with a higher degree, that is, more interconnected cells, having more negative values of divergence, thus confirming our theoretical predictions. We discuss these findings in the context of energetic aspects of signaling in beta cells and potential risks for pathological changes in the tissue.

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

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.

Toward the virtual stomach: progress in multiscale modeling of gastric electrophysiology and motility

Experimental progress in investigating normal and disordered gastric motility is increasingly being complimented by sophisticated multiscale modeling studies. Mathematical modeling has become a valuable tool in this effort, as there is an ever-increasing need to gain an integrative and quantitative understanding of how physiological mechanisms achieve coordinated functions across multiple biophysical scales. These interdisciplinary efforts have been particularly notable in the area of gastric electrophysiology, where they are beginning to yield a comprehensive and integrated in silico organ modeling framework, or 'virtual stomach'. At the cellular level, a number of biophysically based mathematical cell models have been developed, and these are now being applied in areas including investigations of gastric electrical pacemaker mechanisms, smooth muscle electrophysiology, and electromechanical coupling. At the tissue level, micro-structural models are being creatively developed and employed to investigate clinically significant questions, such as the functional effects of ICC degradation on gastrointestinal (GI) electrical activation. At the organ level, high-resolution electrical mapping and modeling studies are combined to provide improved insights into normal and dysrhythmic gastric electrical activation. These efforts are also enabling detailed forward and inverse modeling studies at the 'whole body' level, with implications for diagnostic techniques for gastric dysrhythmias. These recent advances, together with several others highlighted in this review, collectively demonstrate a powerful trend toward applying mathematical models to effectively investigate structure-function relationships and overcome multiscale challenges in basic and clinical GI research.

Connexin channel permeability to cytoplasmic molecules

Connexin channels are known to be permeable to a variety of cytoplasmic molecules. The first observation of second messenger junctional permeability, made approximately 30 years ago, sparked broad interest in gap junction channels as mediators of intercellular molecular signaling. Since then, much has been learned about the diversity of connexin channels with regard to isoform diversity, tissue and developmental distribution, modes of channel regulation, assembly, expression, biochemical modification and permeability, all of which appear to be dynamically regulated. This information has expanded the potential roles of connexin channels in development, physiology and disease, and made their elucidation much more complex--30 years ago such an orchestra of junctional dynamics was unanticipated. Only recently, however, have investigators been able to directly address, in this more complex framework, the key issue: what specific biological molecules, second messengers and others, are able to permeate the various types of connexin channels, and how well? An important related issue, given the ever-growing list of connexin-related pathologies, is how these permeabilities are altered by disease-causing connexin mutations. Together, many studies show that a variety of cytoplasmic molecules can permeate the different types of connexin channels. A few studies reveal differences in permeation by different molecules through a particular type of connexin channel, and differences in permeation by a particular molecule through different types of connexin channels. This article describes and evaluates the various methods used to obtain these data, presents an annotated compilation of the results, and discusses the findings in the context of what can be inferred about mechanism of selectivity and potential relevance to signaling. The data strongly suggest that highly specific interactions take place between connexin pores and specific biological molecular permeants, and that those interactions determine which cytoplasmic molecules can permeate and how well. At this time, the nature of those interactions is unclear. One hopes that with more detailed permeability and structural information, the specific molecular mechanisms of the selectivity can be elucidated.

Detecting and controlling unstable periodic orbits that are not part of a chaotic attractor

A method for controlling unstable periodic orbits (UPOs) that have not been controllable before is presented. The method is based on detecting UPOs that are situated outside the skeleton of a chaotic attractor. The main idea is to exploit flexible parts of the attractor, which under weak external perturbations allow variable excursions of the trajectory away from its originally determined path. After the perturbation, the trajectory of the autonomous system seeks its path back to the chaotic attractor and reveals additional UPOs that are otherwise not used by the system. It is shown that these UPOs can be controlled as easily as the UPOs that form the basic chaotic attractor. The effectiveness of the proposed method is demonstrated on two different chaotic systems with very distinct response abilities to external perturbations. Additionally, some applications of the method in the fields of laser technology, information encoding, and biomedical engineering are discussed.