One of the possible regimes of bursting activity in the Hodgkin-Huxley model

A mathematical model of the Hodgkin--Huxley neuron membrane that comprises a "fast" second-order system and two "slow" equations is considered. The criterion for a self-oscillatory solution similar to the bursting activity in pacemaker neurons is found. The results obtained agree well with the computations carried out for the initial model.     

Topological and phenomenological classification of bursting oscillations

We describe a classification scheme for bursting oscillations which encompasses many of those found in the literature on bursting in excitable media. This is an extension of the scheme of Rinzel (inMathematical Topics in Population Biology, Springer, Berlin, 1987), put in the context of a sequence of horizontal cuts through a two-parameter bifurcation diagram. We use this to describe the phenomenological character of different types of bursting, addressing the issue of how well the bursting can be characterized given the limited amount of information often available in experimental settings.     

Method for constructing the boundary of the bursting oscillations region in the neuron model

We examine the problem of constructing the boundary of bursting oscillations on a parameter plane for the system of equations describing the electrical behaviour of the membrane neuron arising from the interaction of fast oscillations of the cytoplasma membrane potential and slow oscillations of the intracellular calcium concentration. As the boundary point on the parameter plane we consider the values at which the limit cycle of the slow subsystem is tangent to the Hopf bifurcation curve of the fast subsystem. The method suggested for determining the boundary is based on the dissection of the system variables into slow and fast. The strong point of the method is that it requires the integration of the slow subsystem only. An example of the application of the method for the stomatogastric neuron model [Guckenheimer J, Gueron S, Harris-Warrick RM (1993) Philos Trans R Soc Lond B 341: 345–359] is given. Received: 31 May 1999 / Accepted in revised form: 19 November 1999     

Intermittency in the Hodgkin-Huxley model

We show that action potentials in the Hodgkin-Huxley neuron model result from a type I intermittency phenomenon that occurs in the proximity of a saddle-node bifurcation of limit cycles. For the Hodgkin-Huxley spatially extended model, describing propagation of action potential along axons, we show the existence of type I intermittency and a new type of chaotic intermittency, as well as space propagating regular and chaotic diffusion waves. Chaotic intermittency occurs in the transition from a turbulent regime to the resting regime of the transmembrane potential and is characterised by the existence of a sequence of action potential spikes occurring at irregular time intervals.     

A basic biophysical model for bursting neurons

Presented here is a basic biophysical cell model for bursting, an extension of our previous model (Av-Ron et al. 1991) for excitability and oscillations. By changing a limited set of model parameters, one can describe different patterns of bursting behavior in terms of the burst cycle, the durations of oscillation and quiescence, and firing frequency.     

Birhythmicity, trirhythmicity and chaos in bursting calcium oscillations

We have analyzed various types of complex calcium oscillations. The oscillations are explained with a model based on calcium-induced calcium release (CICR). In addition to the endoplasmic reticulum as the main intracellular Ca2+ store, mitochondrial and cytosolic Ca2+ binding proteins are also taken into account. This model was previously proposed for the study of the physiological role of mitochondria and the cytosolic proteins in gene rating complex Ca2+ oscillations [1]. Here, we investigated the occurrence of different types of Ca2+ oscillations obtained by the model, i.e. simple oscillations, bursting, and chaos. In a bifurcation diagram, we have shown that all these various modes of oscillatory behavior are obtained by a change of only one model parameter, which corresponds to the physiological variability of an agonist. Bursting oscillations were studied in more detail because they express birhythmicity, trirhythmicity and chaotic behavior. Two different routes to chaos are observed in the model: in addition to the usual period doubling cascade, we also show intermittency. For the characterization of the chaotic behavior, we made use of return maps and Lyapunov exponents. The potential biological role of chaos in intracellular signaling is discussed.     

Mixed mode oscillations as a mechanism for pseudo-plateau bursting

We combine bifurcation analysis with the theory of canard-induced mixed mode oscillations to investigate the dynamics of a novel form of bursting. This bursting oscillation, which arises from a model of the electrical activity of a pituitary cell, is characterized by small impulses or spikes riding on top of an elevated voltage plateau. Oscillations with these characteristics have been called “pseudo-plateau bursting”. Unlike standard bursting, the subsystem of fast variables does not possess a stable branch of periodic spiking solutions, and in the case studied here the standard fast/slow analysis provides little information about the underlying dynamics. We demonstrate that the bursting is actually a canard-induced mixed mode oscillation, and use canard theory to characterize the dynamics of the oscillation. We also use bifurcation analysis of the full system of equations to extend the results of the singular analysis to the physiological regime. This demonstrates that the combination of these two analysis techniques can be a powerful tool for understanding the pseudo-plateau bursting oscillations that arise in electrically excitable pituitary cells and isolated pancreatic β-cells.     

A novel network of multipolar bursting interneurons generates theta frequency oscillations in neocortex

GABAergic interneurons can phase the output of principal cells, giving rise to oscillatory activity in different frequency bands. Here we describe a new subtype of GABAergic interneuron, the multipolar bursting (MB) cell in the mouse neocortex. MB cells are parvalbumin positive but differ from fast-spiking multipolar (FS) cells in their morphological, neurochemical, and physiological properties. MB cells are reciprocally connected with layer 2/3 pyramidal cells and are coupled with each other by chemical and electrical synapses. MB cells innervate FS cells but not vice versa. MB to MB cell as well as MB to pyramidal cell synapses exhibit paired-pulse facilitation. Carbachol selectively induced synchronized theta frequency oscillations in MB cells. Synchrony required both gap junction coupling and GABAergic chemical transmission, but not excitatory glutamatergic input. Hence, MB cells form a distinct inhibitory network, which upon cholinergic drive can generate rhythmic and synchronous theta frequency activity, providing temporal coordination of pyramidal cell output.     

Mathematical description of a bursting pacemaker neuron by a modification of the Hodgkin-Huxley equations.

Modifications based on experimental results reported in the literature are made to the Hodgkin-Huxley equations to describe the electrophysiological behavior of the Aplysia abdominal ganglion R15 cell. The system is then further modified to describe the effects with the application of the drug tetrodotoxin (TTX) to the cells' bathing medium. Methods of the qualitative theory of differential equations are used to determine the conditions necessary for such a system of equations to have an oscillatory solution. A model satisfying these conditions is shown to preduct many experimental observations of R15 cell behavior. Numerical solutions are obtained for differential equations satisfying the conditions of the model. These solutions are shown to have a form similar to that of the bursting which is characteristic of this cell, and to preduct many results of experiments conducted on this cell. The physiological implications of the model are discussed.     

Diffusion of extracellular K+ can synchronize bursting oscillations in a model islet of Langerhans.

Electrical bursting oscillations of mammalian pancreatic beta-cells are synchronous among cells within an islet. While electrical coupling among cells via gap junctions has been demonstrated, its extent and topology are unclear. The beta-cells also share an extracellular compartment in which oscillations of K+ concentration have been measured (Perez-Armendariz and Atwater, 1985). These oscillations (1-2 mM) are synchronous with the burst pattern, and apparently are caused by the oscillating voltage-dependent membrane currents: Extracellular K+ concentration (Ke) rises during the depolarized active (spiking) phase and falls during the hyperpolarized silent phase. Because raising Ke depolarizes the cell membrane by increasing the potassium reversal potential (VK), any cell in the active phase should recruit nonspiking cells into the active phase. The opposite is predicted for the silent phase. This positive feedback system might couple the cells' electrical activity and synchronize bursting. We have explored this possibility using a theoretical model for bursting of beta-cells (Sherman et al., 1988) and K+ diffusion in the extracellular space of an islet. Computer simulations demonstrate that the bursts synchronize very quickly (within one burst) without gap junctional coupling among the cells. The shape and amplitude of computed Ke oscillations resemble those seen in experiments for certain parameter ranges. The model cells synchronize with exterior cells leading, though incorporating heterogeneous cell properties can allow interior cells to lead. The model islet can also be forced to oscillate at both faster and slower frequencies using periodic pulses of higher K+ in the medium surrounding the islet. Phase plane analysis was used to understand the synchronization mechanism. The results of our model suggest that diffusion of extracellular K+ may contribute to coupling and synchronization of electrical oscillations in beta-cells within an islet.     

A calcium-based phantom bursting model for pancreatic islets

Insulin-secreting β-cells, located within the pancreatic islets of Langerhans, are excitable cells that produce regular bursts of action potentials when stimulated by glucose. This system has been the focus of mathematical investigation for two decades, spawning an array of mathematical models. Recently, a new class of models has been introduced called ‘phantom bursters’ [Bertram et al. (2000) Biophys. J. 79, 2880–2892], which accounts for the wide range of burst frequencies exhibited by islets via the interaction of more than one slow process. Here, we describe one implementation of the phantom bursting mechanism in which intracellular Ca²⁺ controls the oscillations through both direct and indirect negative feedback pathways. We show how the model dynamics can be understood through an extension of the fast/slow analysis that is typically employed for bursting oscillations. From this perspective, the model makes use of multiple degrees of freedom to generate the full range of bursting oscillations exhibited by β-cells. The model also accounts for a wide range of experimental phenomena, including the ubiquitous triphasic response to the step elevation of glucose and responses to perturbations of internal Ca²⁺ stores. Although it is not presently a complete model of all β-cell properties, it demonstrates the design principles that we anticipate will underlie future progress in β-cell modeling.     

A perm-selective model membrane exhibiting oscillatory behavior

An enzymic model membrane capable of simulating such permeability characteristics of chemically excitable membranes as generation of an action potential-like overshoot, selectivity over permeants, and saturation and hysteresis of transmembrane flow is constructed by means of coupling a nonlinear, interfacial flow regulating the attachment of permeants to the surface of the oligomeric membrane with a transmembrane allosteric conversion flow recently formulated by Blumenthal. Periods of sustained oscillation, as well as the predicted values of threshold, and height of an action potential-like overshoot are calculated for different choices of external and internal parameters of the membrane.     

A model of microtubule oscillations

Simulations of microtubule oscillations have been obtained by a kinetic model including nucleation of microtubules, elongation by addition of GTP-loaded tubulin dimers, disassembly into oligomers, and dissolution of oligomers followed by nucleotide exchange at the free dimers. Dynamic instability is described by the on and off rates for dimer association in the growth phase, the rate of rapid shortening, and the transition rates for catastrophe and rescue. The latter are assumed to be completely determined by the current state of the system (short cap hypothesis). Microtubule oscillations and normal polymerizations measured by time-resolved X-ray scattering were used to test the model. The model is able to produce oscillations without further assumptions. However, in order to obtain good fits to the experimental data one requires an additional mechanism which prevents rapid desynchronization of the microtubules. One of several possible mechanisms that will be discussed is the destabilization of microtubules by the products of disassembly.Abbreviations MT(s) microtubule(s) - G-MT/S-MT microtubule in the state of growth/shortening - GTP guanosine 5-triphosphate - GDP guanosine 5-diphosphate - TU · GDP/TU · GTP tubulin dimer with GDP/GTP bound to the exchangeable nucleotide binding site - MAP(s) microtubule-associated protein(s) - PC tubulin phosphocellulose-purified tubulin - PIPES piperazine-1,4-bis(2-ethane sulfonic acid) - DDT dithiothreitol - EGTA ethylene glycol-O,O-bis(2-amino ethyl ether)-N,N,N,N-tetraacetic acid     

Sinusoidal voltage clamp of the Hodgkin-Huxley model.

A voltage clamp consisting of a sinusoidal voltage of amplitude V1 and frequency f, superimposed on a steady voltage level V0, is applied to the Hodgkin-Huxley model of the squid giant axon membrane. The steady-state response is a current composed of sinusoidal components of frequencies O, f, 2f, 3f,... The frequencies greater than f arise from the nonlinearity of the membrane. The total current is described by a power series in V1; each coefficient of this series is composed of current components for one or more frequencies. For different frequencies one can derive higher-order generalized admittances characterizing the nonlinear as well as the linear properties of the membrane. Formulas for the generalized admittances are derived from the Hodgkin-Huxley equations for frequencies up to 3f, using a perturbation technique. Some of the resulting theoretical curves are compared with experimental results, with good qualitative agreement.     

A simple integrative electrophysiological model of bursting GnRH neurons

In this paper a modular model of the GnRH neuron is presented. For the aim of simplicity, the currents corresponding to fast time scales and action potential generation are described by an impulsive system, while the slower currents and calcium dynamics are described by usual ordinary differential equations (ODEs). The model is able to reproduce the depolarizing afterpotentials, afterhyperpolarization, periodic bursting behavior and the corresponding calcium transients observed in the case of GnRH neurons.     

A quantitative approximation scheme for the traveling wave solutions in the Hodgkin-Huxley model

We introduce an approximation scheme for the Hodgkin-Huxley model of nerve conductance that allows calculation of both the speed and shape of the traveling pulses, in quantitative agreement with the solutions of the model. We demonstrate that the reduced problem for the front of the traveling pulse admits a unique solution. We obtain an explicit analytical expression for the speed of the pulses that is valid with good accuracy in a wide range of the parameters.     

Anodal excitation in the Hodgkin-Huxley nerve model.

Computations show that cathodal rheobase increases with temperature from 0 degrees C to 30 degrees C. Anodal rheobase (stimulation at the end of an indefinitely long anodal pulse) also increases with temperature, but goes to infinity at a critical temperature 17.13 degrees C, above which such excitation is impossible. For a stimulus consisting of any step change of current from I0 to I1, a threshold curve of I1 is plotted against I0. As the temperature increases, this curve rises. Its intersection with the horizontal axis, which determines the anodal rheobase, goes to infinity at the critical temperature. This phenomenon is caused by the saturation of the variables m, h, n for strongly hyperpolarized potentials, combined with the relative speeding up of the inhibitory process with increasing temperature. The threshold charge Q in an instantaneous anodal current pulse (of zero duration) goes to infinity at the same temperature, with a similar explanation in terms of threshold curves in the I1 vs. Q plane. The fact that the critical temperature for both cases is the same is generalized by the conjecture that for any anodal current waveform whatever, as its amplitude approaches infinity, the trajectory in the phase space following its cessation approaches the same limiting trajectory. This limiting trajectory changes from suprathreshold to subthreshold at the critical temperature.     

A two-variable model of somatic-dendritic interactions in a bursting neuron

We present a two-variable delay-differential-equation model of a pyramidal cell from the electrosensory lateral line lobe of a weakly electric fish that is capable of burst discharge. It is a simplification of a six-dimensional ordinary differential equation model for such a cell whose bifurcation structure has been analyzed (Doiron et al., J. Comput. Neurosci., 12, 2002). We have modeled the effects of back-propagating action potentials by a delay, and use an integrate-and-fire mechanism for action potential generation. The simplicity of the model presented here allows one to explicitly derive a two-dimensional map for successive interspike intervals, and to analytically investigate the effects of time-dependent forcing on such a model neuron. Some of the effects discussed include ‘burst excitability’, the creation of resonance tongues under periodic forcing, and stochastic resonance. We also investigate the effects of changing the parameters of the model.     

Low dimensional model of bursting neurons

A computationally efficient, biophysically-based model of neuronal behavior is presented; it incorporates ion channel dynamics in its two fast ion channels while preserving simplicity by representing only one slow ion current. The model equations are shown to provide a wide array of physiological dynamics in terms of spiking patterns, bursting, subthreshold oscillations, and chaotic firing. Despite its simplicity, the model is capable of simulating an extensive range of spiking patterns. Several common neuronal behaviors observed in vivo are demonstrated by varying model parameters. These behaviors are classified into dynamical classes using phase diagrams whose boundaries in parameter space prove to be accurately delineated by linear stability analysis. This simple model is suitable for use in large scale simulations involving neural field theory or neuronal networks.     

Understanding bursting oscillations as periodic slow passages through bifurcation and limit points

We consider a biochemical system consisting of two allosteric enzyme reactions coupled in series. The system has been modeled by Decroly and Goldbeter (J. Theor. Biol. 124, 219 (1987)) and is described by three coupled, first-order, nonlinear, differential equations. Bursting oscillations correspond to a succession of alternating active and silent phases. The active phase is characterized by rapid oscillations while the silent phase is a period of quiescence. We propose an asymptotic analysis of the differential equations which is based on the limit of large allosteric constants. This analysis allows us to construct a time-periodic bursting solution. This solution is jumping periodically between a slowly varying steady state and a slowly varying oscillatory state. Each jump follows a slow passage through a bifurcation or limit point which we analyze in detail. Of particular interest is the slow passage through a supercritical Hopf bifurcation. The transition is from an oscillatory solution to a steady state solution. We show that the transition is delayed considerably and characterize this delay by estimating the amplitude of the oscillations at the Hopf bifurcation point.