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

Predicting single spikes and spike patterns with the Hindmarsh–Rose model

Most simple neuron models are only able to model traditional spiking behavior. As physiologists discover and classify different electrical phenotypes, computational neuroscientists become interested in using simple phenomenological models that can exhibit these different types of spiking patterns. The Hindmarsh–Rose model is a three-dimensional relaxation oscillator which can show both spiking and bursting patterns and has a chaotic regime. We test the predictive powers of the Hindmarsh–Rose model on two different test databases. We show that the Hindmarsh–Rose model can predict the spiking response of rat layer 5 neocortical pyramidal neurons on a stochastic input signal with a precision comparable to the best known spiking models. We also show that the Hindmarsh–Rose model can capture qualitatively the electrical footprints in a database of different types of neocortical interneurons. When the model parameters are fit from sub-threshold measurements only, the model still captures well the electrical phenotype, which suggests that the sub-threshold signals contain information about the firing patterns of the different neurons

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