How and why do neurons generate complex rhythms with various frequencies?

Some neurons generate endogenous rhythms with a period of a few hundred milliseconds,while others generate rhythms with a period of a few tens of seconds. Sometimes rhythms appear chaotic. Explaining how these neurons can generate various modes of oscillation with a widely ranging frequency is a challenge. In the first part of this review, we illustrate that such rhythms can be generated from simple yet elegant mathematical models. Chaos embedded in rhythmic activity has interesting characteristics that are not seen in other physical systems. Understanding of how these neurons utilizes endogenous rhythms to communicate with each other is important in elucidating where the brain gets various rhythms and why it can pervert into abnormal rhythms under diseased conditions. Using the islet of Langerhans in pancreas as an example, in the second part of this review, we illustrate how insulin secreting β-cells communicate with glucagon secreting α-cells to achieve an optimal insulin release

Neuronal modeling with intracellular calcium signaling

【中文摘要】 细胞溶质内的游离钙离子在许多细胞活动中发挥着重要的作用。对于神经元,细胞膜上的神经电信号和胞内钙离子化学信号之间有着复杂的相互作用,每个神经元都可看作为一个含有细胞膜和内质网膜的双膜系统,而神经细胞的内质网则可视为神经元内的神经元。本综述探讨了神经元膜上神经电信号与内质网钙通道释放的胞内钙信号相耦合的动力学模型。我们认为,计算神经动力学的一个研究方向是考虑包含胞内钙动力学的神经元模拟,而且该研究前沿转向考虑胞内钙波扩散运动的空间神经元模型。包含内质网钙动力学的神经模型,尤其是考虑胞体和树突内钙扩散的空间神经元模型,将加深我们对神经动力学的认识。 【英文摘要】 Cytosolic Ca2+ ions play an important role in the regulation of numerous aspects of cellular activity in virtually all cell types. There is a complex interaction between the neuronal electrical signals on plasma membrane and the chemical signals of intracellular calcium. Each neuron can be considered as a binary membrane system with plasma membrane and endoplasmic reticulum membrane, and the neuronal endoplasmic reticulum can be regarded as a neuron-within-a-neuron. This review explores the simulation model...supported by the National Natural Science Foundation of China (No. 30970970

Ionic mechanisms and Ca2+ dynamics underlying the glucose response of pancreatic β cells: a simulation study

To clarify the mechanisms underlying the pancreatic β-cell response to varying glucose concentrations ([G]), electrophysiological findings were integrated into a mathematical cell model. The Ca2+ dynamics of the endoplasmic reticulum (ER) were also improved. The model was validated by demonstrating quiescent potential, burst–interburst electrical events accompanied by Ca2+ transients, and continuous firing of action potentials over [G] ranges of 0–6, 7–18, and >19 mM, respectively. These responses to glucose were completely reversible. The action potential, input impedance, and Ca2+ transients were in good agreement with experimental measurements. The ionic mechanisms underlying the burst–interburst rhythm were investigated by lead potential analysis, which quantified the contributions of individual current components. This analysis demonstrated that slow potential changes during the interburst period were attributable to modifications of ion channels or transporters by intracellular ions and/or metabolites to different degrees depending on [G]. The predominant role of adenosine triphosphate–sensitive K+ current in switching on and off the repetitive firing of action potentials at 8 mM [G] was taken over at a higher [G] by Ca2+- or Na+-dependent currents, which were generated by the plasma membrane Ca2+ pump, Na+/K+ pump, Na+/Ca2+ exchanger, and TRPM channel. Accumulation and release of Ca2+ by the ER also had a strong influence on the slow electrical rhythm. We conclude that the present mathematical model is useful for quantifying the role of individual functional components in the whole cell responses based on experimental findings

The electrophysiology of the betacell based on single transmembrane protein characteristics

The electrophysiology of betacells is at the origin of insulin secretion. Betacells exhibit a complex behaviour upon stimulation with glucose including repeated and uninterrupted bursting. Mathematical modelling is most suitable to improve knowledge about the function of various transmembrane currents provided the model is based on reliable data. This is the first attempt to build a mathematical model for the betacell-electrophysiology in a bottom-up approach which relies on single protein conductivity data. The results of previous whole-cell-based models are reconsidered. The full simulation including all prominent transmembrane proteins in betacells is used to provide a functional interpretation of their role in betacell-bursting and an updated vantage point of betacell-electrophysiology. As a result of a number of in silico knock-out- and block-experiments the novel model makes some unexpected predictions: Single-channel conductivity data imply that calcium-gated potassium currents are rather small. Thus, their role in burst interruption has to be revisited. An alternative role in high calcium level oscillations is proposed and an alternative burst interruption model is presented. It also turns out that sodium currents are more relevant than expected so far. Experiments are proposed to verify these predictions.Comment: 28 pages, 5 figures, 54 references, 14 pages supplementary materia

Time-dependent changes in membrane excitability during glucose-induced bursting activity in pancreatic β cells

In our companion paper, the physiological functions of pancreatic β cells were analyzed with a new β-cell model by time-based integration of a set of differential equations that describe individual reaction steps or functional components based on experimental studies. In this study, we calculate steady-state solutions of these differential equations to obtain the limit cycles (LCs) as well as the equilibrium points (EPs) to make all of the time derivatives equal to zero. The sequential transitions from quiescence to burst–interburst oscillations and then to continuous firing with an increasing glucose concentration were defined objectively by the EPs or LCs for the whole set of equations. We also demonstrated that membrane excitability changed between the extremes of a single action potential mode and a stable firing mode during one cycle of bursting rhythm. Membrane excitability was determined by the EPs or LCs of the membrane subsystem, with the slow variables fixed at each time point. Details of the mode changes were expressed as functions of slowly changing variables, such as intracellular [ATP], [Ca2+], and [Na+]. In conclusion, using our model, we could suggest quantitatively the mutual interactions among multiple membrane and cytosolic factors occurring in pancreatic β cells

Pacemaking Neurons in the study of Parkinson’s Disease

Parkinson’s disease is the second most common neurodegenerative disorder with a signiVcant social cost. The disease that develops over years results in signiVcant movement related problems for the aUected. The pathogenesis however is partially understood. Computational approaches are signiVcant in the analysis of events that are multi-factorial. Parkinson’s Disease results from a system failure that leads to severe degeneration in the substantia nigra , a locus in the mid-brain. Traditional approaches tend to focus on isolated sub-components of the pathogenic pathways. However, such an approach may be inadequate to describe the pathogenesis. Substantia nigra neurons function on an expensive energy budget, due to a high level of arborisation and pacemaking activity. Spontaneous oscillations of these neurons are an important feature of motor control. Pacemaking involves the L-type calcium channel, and could impose long-term accumulation of calcium within its organelles. Modelling of this activity is an important part of developing an understanding of the pathogenic process. We develop a mathematical paradigm to describe this activity with a single compartment approach. To develop the mathematical framework we initially identify the components that contribute to the process and investigate an appropriate mathematical representation for the respective components. In the next part, we bring together such representation to develop a model that can reproduce measured data. Global optimisation strategies are adopted to tune important parameters. The model explicitly describes the dynamics of the transmembrane potential with changes in the levels of important cations. The model is veriVed for two major observations in literature regarding its response in the presence of channel blockers. The model is analysed for parameter bifurcation and stability of oscillations. Finally a framework is proposed to extend the model to include aspects of calcium homeostasis