On the mechanism of spiking and bursting in excitable cells

Neuronal electrical rhythms described by composite mapped clock oscillators

The Mapped Clock Oscillator (MCO) model is a representation of omnipresent transmembrane voltage oscillations in excitable cells. We present a generalized version of the MCO that can model neuronal electrical oscillations, both labile and omnipresent, entirely within the framework of a system of ordinary differential equations. The previously described MCO was a second-order system, whereas the model presented here, which we call the composite MCO (cMCO) is a fourth-order system. Furthermore, we show how this cMCO can also be adapted to describe a pair of cells that forms a functional unit, as illustrated here by a model of the CA3 pyramidal cell and its basket cell interneuron feedback loop. The model was able to reproduce the high frequencies (super gamma) and possibly chaotic dynamics observed in the biological system.

A synaptic input portal for a mapped clock oscillator model of neuronal electrical rhythmic activity

Neuronal electrical oscillations play a central role in a variety of situations, such as epilepsy and learning. The mapped clock oscillator (MCO) model is a general model of transmembrane voltage oscillations in excitable cells. In order to be able to investigate the behaviour of neuronal oscillator populations, we present a neuronal version of the model. The neuronal MCO includes an extra input portal, the synaptic portal, which can reflect the biological relationships in a chemical synapse between the frequency of the presynaptic action potentials and the postsynaptic resting level, which in turn affects the frequency of the postsynaptic potentials. We propose that the synaptic input-output relationship must include a power function in order to be able to reproduce physiological behaviour such as resting level saturation. One linear and two power functions (Butterworth and sigmoidal) are investigated, using the case of an inhibitory synapse. The linear relation was not able to produce physiologically plausible behaviour, whereas both the power function examples were appropriate. The resulting neuronal MCO model can be tailored to a variety of neuronal cell types, and can be used to investigate complex population behaviour, such as the influence of network topology and stochastic resonance.

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.

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 beta-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 (lambda n), 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 lambda n. 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.

Pump and exchanger mechanisms in a model of smooth muscle

A novel approach to modelling pump and exchanger mechanisms is presented. In this approach, new thermodynamic expressions for the calcium pump, sodium-calcium exchanger and sodium-potassium pump are developed using statistical rate theory (SRT). This theory is well-defined and is not derived empirically. This is in contrast to previous thermodynamic pump expressions which used a simple linear relationship or relied on empirical data for their functional form. The functional form of these new expressions does not require assumptions of steady state or particular forms of voltage dependencies in specific steps. Also, the explicit reaction scheme is not required. Instead, assumptions of a rate-limiting step in the scheme and a near-equilibrium ratio of intermediate substrates are required. These expressions are incorporated into an overall model of gastric smooth muscle. This model presents a novel approach whereby thermodynamic representations for calcium pumps, sodium-calcium exchangers and sodium-potassium pumps have been included together in a model of ionic transport mechanisms for smooth muscle. Variations in basal metabolic concentrations are used to explain the observed amplitude variation in the transmembrane voltage of gastric smooth muscle. The interaction of the various mechanisms are used to illustrate the large depolarization obtained in smooth muscle with ouabain as well as the forward and reverse modes of the sodium-calcium exchanger.

Bursting excitable cell models by a slow Ca2+ current

Bursting in excitable cells is a phenomenon that has attracted the interest of many electrophysiologists and non-linear dynamicists. In this paper, we present two models that give rise to bursting in action potentials. The membrane of the first model contains a voltage-activated Ca2+ channel that inactivates very slowly upon depolarization and a delayed K+ channel that is activated by voltage. This model consists of three dynamic variables--the gating variable of K+ channel (n), inactivation gating variable of the Ca2+ channel (f), and membrane potential (V). The membrane of the second model contains a voltage-activated Na+ channel that inactivates rather fast upon depolarization. This model contains altogether five dynamic variables--the Na+ inactivation gating variable (h) and Ca2+ activation variable (d), in addition to the three dynamic variables in the first model. With the first model, we show how various interesting bursting patterns may arise from such a simple three dynamic variable model. We also demonstrate that a slowly inactivating voltage-dependent Ca2+ channel may play the key role in the genesis of bursting. With the second model, we show how the participation of a quickly inactivating fast inward current may lead to a neuronal type of bursting, multi-peaked oscillations, and chaos, as the rates of the gating variables change.

Theory of the effect of extracellular potassium on oscillations in the pancreatic beta-cell

Based on the observation that potassium ions are compartmentalized near the surface of pancreatic beta-cells in mouse islets (Perez-Armendariz, E.M., I. Atwater, and E. Rojas 1985, Biophys. J. 48:741-749), we present a theoretical treatment of the effect of external potassium on oscillations in the pancreatic beta-cell. Our model includes the effects of ionic diffusion, the Ca2+-activated K+ channel, voltage-gated K+ and Ca2+ channels, and some of the effects of glucose. It is described by four ordinary differential equations. Numerical integration of these equations allows us to examine the effect of glucose, external K+, quinine, and tetraethylammonium ion (TEA) on the oscillations in membrane potential, intracellular Ca2+, and compartmentalized K+. The results are in good agreement with experiment.

Impulse responses of automaticity in the Purkinje fiber

We examined the effects of brief current pulses on the pacemaker oscillations of the Purkinje fiber using the model of McAllister , Noble, and Tsien (1975. J. Physiol. [Lond.]. 251:1-57). This model was used to construct phase-response curves for brief electric stimuli to find "black holes," where rhythmic activity of the Purkinje fiber ceases. In our computer simulation, a brief current stimulus of the right magnitude and timing annihilated oscillations in membrane potential. The model also revealed a sequence of alternating periodic and chaotic regimes as the strength of a steady bias current is varied. We compared the results of our computer simulations with experimental work on Purkinje fibers and pointed out the importance of modeling results of this kind for understanding cardiac arrhythmias.