Teresa Ree Chay

Chaos in a three-variable model of an excitable cell☆

Abstract We describe chaotic behavior in a model that consists of three first-order, non-linear differential equations, which represent ionic events in excitable membranes. For a certain range of conductances, the model generates chaotic action potentials, and the intracellular calcium concentration also varies chaotically. The chaos was characterized by constructing phase portraits and one-variable maps using the membrane potentials and calcium concentrations. This is the first, simple, biophysically realistic model for excitable cells that shows endogenous chaos.

BURSTING, SPIKING, CHAOS, FRACTALS, AND UNIVERSALITY IN BIOLOGICAL RHYTHMS

Biological systems offer many interesting examples of oscillations, chaos, and bifurcations. Oscillations in biology arise because most cellular processes contain feedbacks that are appropriate for generating rhythms. These rhythms are essential for regulating cellular function. In this tutorial review, we treat two interesting nonlinear dynamic processes in biology that give rise to bursting, spiking, chaos, and fractals: endogenous electrical activity of excitable cells and Ca2+ releases from the Ca2+ stores in nonexcitable cells induced by hormones and neurotransmitters. We will first show that each of these complex processes can be described by a simple, yet elegant, mathematical model. We then show how to utilize bifurcation analyses to gain a deeper insight into the mechanisms involved in the neuronal and cellular oscillations. With the bifurcating diagrams, we explain how spiking can be transformed to bursting via a complex type of dynamic structure when the key parameter in the model varies. Understanding how this parameter would affect the bifurcation structure is important in predicting and controlling abnormal biological rhythms. Although we describe two very different dynamic processes in biological rhythms, we will show that there is universality in their bifurcation structures.

Effects of extracellular calcium on electrical bursting and intracellular and luminal calcium oscillations in insulin secreting pancreatic beta-cells.

The extracellular calcium concentration has interesting effects on bursting of pancreatic beta-cells. The mechanism underlying the extracellular Ca2+ effect is not well understood. By incorporating a low-threshold transient inward current to the store-operated bursting model of Chay, this paper elucidates the role of the extracellular Ca2+ concentration in influencing electrical activity, intracellular Ca2+ concentration, and the luminal Ca2+ concentration in the intracellular Ca2+ store. The possibility that this inward current is a carbachol-sensitive and TTX-insensitive Na+ current discovered by others is discussed. In addition, this paper explains how these three variables respond when various pharmacological agents are applied to the store-operated model.

Electrical bursting and luminal calcium oscillation in excitable cell models

Abstract. It is shown in this paper that electrical bursting and the oscillations in the intracellular calcium concentration, [Ca2+]i, observed in excitable cells such as pancreatic β-cells and R-15 cells of the mollusk Aplysia may be driven by a slow oscillation of the calcium concentration in the lumen of the endoplasmic reticulum, [Ca2+]lum. This hypothesis follows from the inclusion of the dynamic changes of [Ca2+]lum in the Chay bursting model. This extended model provides answers to some puzzling phenomena, such as why isolated single pancreatic β-cells burst with a low frequency while intact β-cells in an islet burst with a much higher frequency. Verification of the model prediction that [Ca2+]lum is a primary oscillator which drives electrical bursting and [Ca2+]i oscillations in these cells awaits experimental testing. Experiments using fluorescent dyes such as mag-fura-2-AM or aequorin could provide relevant information.

Generation of periodic and chaotic bursting in an excitable cell model

There are interesting oscillatory phenomena associated with excitable cells that require theoretical insight. Some of these phenomena are: the threshold low amplitude oscillations before bursting in neuronal cells, the damped burst observed in muscle cells, the period-adding bifurcations without chaos in pancreatic β-cells, chaotic bursting and beating in neurons, and inverse period-doubling bifurcation in heart cells. The three variable model formulated by Chay provides a mathematical description of how excitable cells generate bursting action potentials. This model contains a slow dynamic variable which forms a basis for the underlying wave, a fast dynamic variable which causes spiking, and the membrane potential which is a dependent variable. In this paper, we use the Chay model to explain these oscillatory phenomena. The Poincaré return map approach is used to construct bifurcation diagrams with the ‘slow’ conductance (i.e., gK,C) as the bifurcation parameter. These diagrams show that the system makes a transition from repetitive spiking to chaotic bursting as parameter gK,C is varied. Depending on the time kinetic constant of the fast variable (λn), however, the transition between burstings via period-adding bifurcation can occur even without chaos. Damped bursting is present in the Chay model over a certain range of gK,C and λn. In addition, a threshold sinusoidal oscillation was observed at certain values of gK,C before triggering action potentials. Probably this explains why the neuronal cells exhibit low-amplitude oscillations before bursting.

Endogenous Bursting Patterns in Excitable Cells

Abstract The electrical burst pattern of pancreatic β-cells is quite different from that of neurons. Our goal is to find a unified model which gives rises to both types of patterns, and in this paper we consider such a model. The central features of the model are: (1) a voltage-activated fast inward conductance g I combined with a voltage-gate K + conductance g K which creates spikes, (2) a voltage-activated Ca 2+ conductance g s , which brings about a slow underlying wave, and (3) the intracellular Ca 2+ concentration, [Ca 2+ ] i , which inactivates g s . If the g s gate opens very fast (∼ms), the model gives rise to square-wave bursting as in the pancreatic β-cell; however, if the gate opens slowly (∼s), it gives rise to parabolic bursting as in R-15. Furthermore, as the dynamics of both g s and g n gating become slow variables, we find that small oscillations develop during the silent phase and the model may give rise to aperiodic chaos.

Role of single-channel stochastic noise on bursting clusters of pancreatic beta-cells.

To study why pancreatic beta-cells prefer to burst as a multi-cellular complex, we have formulated a stochastic model for bursting clusters of excitable cells. Our model incorporated a delayed rectifier K+ channel, a fast voltage-gated Ca2+ channel, and a slow Cai-blockable Ca2+ channel. The fraction of ATP-sensitive K+ channels that may still be active in the bursting regime was included in the model as a leak current. We then developed an efficient method for simulating an ionic current component of an excitable cell that contains several thousands of channels opening simultaneously under unclamped voltage. Single channel open-close stochastic events were incorporated into the model by use of binomially distributed random numbers. Our simulations revealed that in an isolated beta-cell [Ca2+]i oscillates with a small amplitude about a low [Ca2+]i. However, in a large cluster of tightly coupled cells, stable bursts develop, and [Ca2+]i oscillates with a larger amplitude about a higher [Ca2+]i. This may explain why single beta-cells do not burst and also do not release insulin.

Abnormal discharges and chaos in a neuronal model system

Using the mathematical model of the pacemaker neuron formulated by Chay, we have investigated the conditions in which a neuron can generate chaotic signals in response to variation in temperature, ionic compositions, chemicals, and the strength of applied depolarizing current.

Electrical bursting and intracellular Ca 2+ Oscillations in excitable cell models

In bursting excitable cells such as pancreaticβ-cells and molluscanAplysia neuron cells, intracellular Ca2+ ion plays a central role in various cellular functions. To understand the role of [Ca2+]i (the intracellular Ca2+ concentration) in electrical bursting, we formulate a mathematical model which contains a few functionally important ionic currents in the excitable cells. In this model, inactivation of Ca2+ current takes place by a mixture of voltage and intracellular Ca2+ ions. The model predicts that, although the electrical bursting patterns look the same, the shapes of [Ca2+]i oscillations could be very different depending on how fast [Ca2+]i changes in the cytosolic free space (i.e., how strong the cellular Ca2+ buffering capacity is). If [Ca2+]i changes fast, [Ca2+]i oscillates in bursts in parallel to electrical bursting such that it reaches a maximum at the onset of bursting and a minimum just after the termination of the plateau phase. If the change is slow, then [Ca2+]i oscillates out-of-phase with electrical bursting such that it peaks at a maximum near the termination of the plateau and a minimum just before the onset of the active phase. During the active phase [Ca2+]i gradually increases without spikes. In the intermediate ranges, [Ca2+]i oscillates in such a manner that the peak of [Ca2+]i oscillation lags behind the electrical activity. The model also predicts the existence of multi-peaked oscillations and chaos in certain ranges of the gating variables and the intracellular Ca2+ buffer concentration.

Modeling slowly bursting neurons via calcium store and voltage-independent calcium current

Recent experiments indicate that the calcium store (e.g., endoplasmic reticulum) is involved in electrical bursting and [Ca2+]i oscillation in bursting neuronal cells. In this paper, we formulate a mathematical model for bursting neurons, which includes Ca2+ in the intracellular Ca2+ stores and a voltage-independent calcium channel (VICC). This VICC is activated by a depletion of Ca2+ concentration in the store, [Ca2+]CS. In this model, [Ca2+]CS oscillates slowly, and this slow dynamic in turn gives rise to electrical bursting. The newly formulated model thus is radically different from existing models of bursting excitable cells, whose mechanism owes its origin to the ion channels in the plasma membrane and the [Ca2+]i dynamics. In addition, this model is capable of providing answers to some puzzling phenomena, which the previous models could not (e.g., why cAMP, glucagon, and caffeine have ability to change the burst periodicity). Using mag-fura-2 fluorescent dyes, it would be interesting to verify the prediction of the model that (1) [Ca2+]CS oscillates in bursting neurons such as Aplysia neuron and (2) the neurotransmitters and hormones that affect the adenylate cyclase pathway can influence this oscillation.