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.     

Modulation of the bursting properties of single mouse pancreatic beta-cells by artificial conductances

Glucose triggers bursting activity in pancreatic islets, which mediates the Ca2+ uptake that triggers insulin secretion. Aside from the channel mechanism responsible for bursting, which remains unsettled, it is not clear whether bursting is an endogenous property of individual beta-cells or requires an electrically coupled islet. While many workers report stochastic firing or quasibursting in single cells, a few reports describe single-cell bursts much longer (minutes) than those of islets (15-60 s). We studied the behavior of single cells systematically to help resolve this issue. Perforated patch recordings were made from single mouse beta-cells or hamster insulinoma tumor cells in current clamp at 30-35 degrees C, using standard K+-rich pipette solution and external solutions containing 11.1 mM glucose. Dynamic clamp was used to apply artificial KATP and Ca2+ channel conductances to cells in current clamp to assess the role of Ca2+ and KATP channels in single cell firing. The electrical activity we observed in mouse beta-cells was heterogeneous, with three basic patterns encountered: 1) repetitive fast spiking; 2) fast spikes superimposed on brief (<5 s) plateaus; or 3) periodic plateaus of longer duration (10-20 s) with small spikes. Pattern 2 was most similar to islet bursting but was significantly faster. Burst plateaus lasting on the order of minutes were only observed when recordings were made from cell clusters. Adding gCa to cells increased the depolarizing drive of bursting and lengthened the plateaus, whereas adding gKATP hyperpolarized the cells and lengthened the silent phases. Adding gCa and gKATP together did not cancel out their individual effects but could induce robust bursts that resembled those of islets, and with increased period. These added currents had no slow components, indicating that the mechanisms of physiological bursting are likely to be endogenous to single beta-cells. It is unlikely that the fast bursting (class 2) was due to oscillations in gKATP because it persisted in 100 microM tolbutamide. The ability of small exogenous currents to modify beta-cell firing patterns supports the hypothesis that single cells contain the necessary mechanisms for bursting but often fail to exhibit this behavior because of heterogeneity of cell parameters.     

A Hodgkin-Huxley model exhibiting bursting oscillations

We investigate bursting behaviour generated in an electrophysiological model of pituitary corticotrophs. The active and silent phases of this mode of bursting are generated by moving between two stable oscillatory solutions. The bursting is indirectly driven by slow modulation of the endoplasmic reticulum Ca²⁺ concentration. The model exhibits different modes of bursting, and we investigate mode transitions and similar modes of bursting in other Hodgkin-Huxley models. Bifurcation analysis and the use of null-surfaces facilitate a geometric interpretation of the model bursting modes and action potential generation, respectively.     

Differential control of active and silent phases in relaxation models of neuronal rhythms

Rhythmic bursting activity, found in many biological systems, serves a variety of important functions. Such activity is composed of episodes, or bursts (the active phase, AP) that are separated by quiescent periods (the silent phase, SP). Here, we use mean field, firing rate models of excitatory neural network activity to study how AP and SP durations depend on two critical network parameters that control network connectivity and cellular excitability. In these models, the AP and SP correspond to the network's underlying bistability on a fast time scale due to rapid recurrent excitatory connectivity. Activity switches between the AP and SP because of two types of slow negative feedback: synaptic depression—which has a divisive effect on the network input/output function, or cellular adaptation—a subtractive effect on the input/output function. We show that if a model incorporates the divisive process (regardless of the presence of the subtractive process), then increasing cellular excitability will speed up the activity, mostly by decreasing the silent phase. Reciprocally, if the subtractive process is present, increasing the excitatory connectivity will slow down the activity, mostly by lengthening the active phase. We also show that the model incorporating both slow processes is less sensitive to parameter variations than the models with only one process. Finally, we note that these network models are formally analogous to a type of cellular pacemaker and thus similar results apply to these cellular pacemakers. Action Editor: Misha Tsodyks     

Classification of bursting patterns: A tale of two ducks

Bursting is one of the fundamental rhythms that excitable cells can generate either in response to incoming stimuli or intrinsically. It has been a topic of intense research in computational biology for several decades. The classification of bursting oscillations in excitable systems has been the subject of active research since the early 1980s and is still ongoing. As a by-product, it establishes analytical and numerical foundations for studying complex temporal behaviors in multiple timescale models of cellular activity. In this review, we first present the seminal works of Rinzel and Izhikevich in classifying bursting patterns of excitable systems. We recall a complementary mathematical classification approach by Bertram and colleagues, and then by Golubitsky and colleagues, which, together with the Rinzel-Izhikevich proposals, provide the state-of-the-art foundations to these classifications. Beyond classical approaches, we review a recent bursting example that falls outside the previous classification systems. Generalizing this example leads us to propose an extended classification, which requires the analysis of both fast and slow subsystems of an underlying slow-fast model and allows the dissection of a larger class of bursters. Namely, we provide a general framework for bursting systems with both subthreshold and superthreshold oscillations. A new class of bursters with at least 2 slow variables is then added, which we denote folded-node bursters, to convey the idea that the bursts are initiated or annihilated via a folded-node singularity. Key to this mechanism are so-called canard or duck orbits, organizing the underpinning excitability structure. We describe the 2 main families of folded-node bursters, depending upon the phase (active/spiking or silent/nonspiking) of the bursting cycle during which folded-node dynamics occurs. We classify both families and give examples of minimal systems displaying these novel bursting patterns. Finally, we provide a biophysical example by reinterpreting a generic conductance-based episodic burster as a folded-node burster, showing that the associated framework can explain its subthreshold oscillations over a larger parameter region than the fast subsystem approach.     

Synchronous and asynchronous bursting states: role of intrinsic neural dynamics

Brain signals such as local field potentials often display gamma-band oscillations (30-70 Hz) in a variety of cognitive tasks. These oscillatory activities possibly reflect synchronization of cell assemblies that are engaged in a cognitive function. A type of pyramidal neurons, i.e., chattering neurons, show fast rhythmic bursting (FRB) in the gamma frequency range, and may play an active role in generating the gamma-band oscillations in the cerebral cortex. Our previous phase response analyses have revealed that the synchronization between the coupled bursting neurons significantly depends on the bursting mode that is defined as the number of spikes in each burst. Namely, a network of neurons bursting through a Ca(2+)-dependent mechanism exhibited sharp transitions between synchronous and asynchronous firing states when the neurons exchanged the bursting mode between singlet, doublet and so on. However, whether a broad class of bursting neuron models commonly show such a network behavior remains unclear. Here, we analyze the mechanism underlying this network behavior using a mathematically tractable neuron model. Then we extend our results to a multi-compartment version of the NaP current-based neuron model and prove a similar tight relationship between the bursting mode changes and the network state changes in this model. Thus, the synchronization behavior couples tightly to the bursting mode in a wide class of networks of bursting neurons.     

Dissection of a model for neuronal parabolic bursting

We have obtained new insight into the mechanisms for bursting in a class of theoretical models. We study Plant's model [24] for Aplysia R-15 to illustrate our view of these so-called parabolic bursters, which are characterized by low spike frequency at the beginning and end of a burst. By identifying and analyzing the fast and slow processes we show how they interact mutually to generate spike activity and the slow wave which underlies the burst pattern. Our treatment is essentially the first step of a singular perturbation approach presented from a geometrical viewpoint and carried out numerically with AUTO [12]. We determine the solution sets (steady state and oscillatory) of the fast subsystem with the slow variables treated as parameters. These solutions form the slow manifold over which the slow dynamics then define a burst trajectory. During the silent phase of a burst, the solution trajectory lies approximately on the steady state branch of the slow manifold and during the active phase of spiking, the trajectory sweeps through the oscillation branch. The parabolic nature of bursting arises from the (degenerate) homoclinic transition between the oscillatory branch and the steady state branch. We show that, for some parameter values, the trajectory remains strictly on the steady state branch (to produce a resting steady state or a pure slow wave without spike activity) or strictly in the oscillatory branch (continuous spike activity without silent phases). Plant's model has two slow variables: a calcium conductance and the intracellular free calcium concentration, which activates a potassium conductance. We also show how bursting arises from an alternative mechanism in which calcium inactivates the calcium current and the potassium conductance is insensitive to calcium. These and other biophysical interpretations are discussed.     

Anti-phase,asymmetric and aperiodic oscillations in excitable cells—I. Coupled bursters

I seek to explain phenomena observed in simulations of populations of gap junction-coupled bursting cells by studying the dynamics of identical pairs. I use a simplified model for pancreatic β-cells and decompose the system into fast (spike-generating) and slow subsystems to show how bifurcations of the fast subsystem affect bursting behavior. When coupling is weak, the spikes are not in phase but rather are anti-phase, asymmetric or quasi-periodic. These solutions all support bursting with smaller amplitude spikes than the in-phase case, leading to increased burst period. A key geometrical feature underlying this is that the in-phase periodic solution branch terminates in a homoclinic orbit. The same mechanism also provides a model for bursting as an emergent property of populations; cells which are not intrinsic bursters can burst when coupled. This phenomenon is enhanced when symmetry is broken by making the cells differ in a parameter.     

描述肝细胞中两类不同特性钙离子浓度振荡

细胞内第二信使钙离子通常以浓度振荡的方式转导多种生理学信息,影响细胞分化、成熟和凋亡等各种生理过程。肝细胞实验中看到在一定浓度范围的激动剂刺激下,细胞质钙离子浓度的变化可呈现出很不相同的图像。例如顺脱羟肾上腺素刺激下,可出现简单的周期振荡,振荡频率随激动剂浓度的不同有变化;在腺苷三磷酸刺激下,随激动剂浓度从低到高,胞质钙离子浓度的变化可以从开始出现简单振荡,到形成复杂的多峰间歇振荡。肝细胞中钙离子浓度振荡的峰值多在500nmol/L到800nmol/L范围。给出一个四为量数学模型的改进形式,可以模拟肝细胞中钙离子浓度从简单振荡向复杂振荡的变化。数值计算给出与实验结果比较一致的振荡波形和振幅。     

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.     

Phase reset and dynamic stability during human gait

The human walking movement shows transient changes in response to single short-lived external perturbations, termed "stumbling reactions." During the stumbling reactions, the walking phase is reset. It has been considered that the reactions contribute to stabilizing the motion, but less evidence bridging between the rhythm reset and the dynamic stability of the gait has been provided. The present study tries to establish the relationship between them. To this end, we construct a simple dynamical system model of the human musculo-skeletal system interacting with the ground, whose joint kinematics during walking is constrained by a given periodic joint-angles-profile. We show first that the model can exhibit a stable limit cycle corresponding to the steady walking with no perturbations. The responses of the limit cycle oscillation are examined by applying a type of perturbations at various timings with various intensities, elucidating the stability of the model's walking when no phase reset is performed. We then observe that modifications of the periodic joint-angles-profile within a short time interval in response to the perturbation can alter the responses of the limit cycle oscillation and induce phase reset of the model's walking. It is shown that appropriate amounts of the phase reset can prevent the model from falling, even for the perturbation that induces falling in the case without the phase reset. This suggests that those phase resets can improve the dynamic stability of the gait. Moreover, the appropriate phase resets predicted by the model are compared with the experimentally observed phase resets during human stumbling reaction to show they share similar characteristics.     

Intrinsic bursting enhances the robustness of a neural network model of sequence generation by avian brain area HVC

Avian brain area HVC is known to be important for the production of birdsong. In zebra finches, each RA-projecting neuron in HVC emits a single burst of spikes during a song motif. The population of neurons is activated in a precisely timed, stereotyped sequence. We propose a model of these burst sequences that relies on two hypotheses. First, we hypothesize that the sequential order of bursting is reflected in the excitatory synaptic connections between neurons. Second, we propose that the neurons are intrinsically bursting, so that burst duration is set by cellular properties. Our model generates burst sequences similar to those observed in HVC. If intrinsic bursting is removed from the model, burst sequences can also be produced. However, they require more fine-tuning of synaptic strengths, and are therefore less robust. In our model, intrinsic bursting is caused by dendritic calcium spikes, and strong spike frequency adaptation in the soma contributes to burst termination.     

Physiological characteristics of the tympanic organ in noctuoid moths

Summary We show the variations in the spike activity of both auditory receptors inSpodoptera frugiperda, Mocis latipes, Ascalapha odorata (Noctuidae),Maenas jussiae andEmpyreuma pugione (Arctiidae) immediately after 45 ms and 5 s acoustic stimuli at different intensities. The frequency of the applied stimuli was 34 kHz forE. pugione and 20 kHz for the other species. The electrical activity of the auditory receptors was recorded at the tympanic nerve with a stainless steel hook electrode. When the 45 ms pulses cease there is an afterdischarge from both auditory receptors in all the species. The number of spikes in the afterdischarge activity of both receptor cells (A₁ and A₂) shows a linear relation with stimulus intensity (Table 1). This number increases monotonically with increments in stimulus intensity, except for the A₁ cell activity inE. pugione, which decreases at intensities higher than 55 dB (Fig. 1). There are significant species-specific differences in the slope values of the number of spikes in the afterdischarge of both auditory receptors. After a 5 s stimulusM. latipes andM. jussiae show a rapid recovery of the standard spontaneous A₁-cell discharge level. Poststimulus A₁-cell spike activity inS. frugiperda shows a silent period, the duration of which increases with stimulus intensity (Fig. 3).E. pugione andA. odorata show such a silent period after low and moderately intense stimuli, but at high intensities the post-stimulus activity exceeds the pre-stimulus spontaneous discharge (Fig. 3). We demonstrate statistically that these variations cannot be explained by the random fluctuations of the standard spontaneous discharge. They are thus considered a silent and a rebound period respectively (Fig. 5). The presence and duration of either type of period seem to depend on the magnitude of the response to the acoustic stimulus. They thus seem related to the adaptation rate and the previously suggested existence of peripheral inhibitory interaction between the auditory receptors.     

Parabolic bursting revisited

 Many excitable membrane systems display bursting oscillations, in which the membrane potential switches periodically between an active phase of rapid spiking and a silent phase of slow, quasi steady-state behavior. A burster is called parabolic when the spike frequency is lower both at the beginning and end of the active phase. We show that classes of voltage-gated conductance equations can be reduced to the mathematical mechanism previously analyzed by Ermentrout and Kopell in [7]. The reduction uses a series of coordinate changes and shows that the mechanism in [7] applies more generally than previously believed. The key hypothesis for the more general theory is that a certain slow periodic orbit must stay close to a curve of degenerate homoclinic points for the fast system, at least during the active phase. We do not require that the slow system have a periodic orbit when the voltage is held constant. Received 28 March 1995; received in revised form 20 October 1995     

Rhythmic activities of hypothalamic magnocellular neurons: Autocontrol mechanisms

Electrophysiological recordings in lactating rats show that oxytocin (OT) and vasopressin (AVP) neurons exhibit specific patterns of activities in relation to peripheral stimuli: periodic bursting firing for OT neurons during suckling, phasic firing for AVP neurons during hyperosmolarity (systemic injection of hypertonic saline). These activities are autocontrolled by OT and AVP released somato-dentritically within the hypothalamic magnocellular nuclei. In vivo, OT enhances the amplitude and frequency of bursts, an effect accompanied with an increase in basal firing rate. However, the characteristics of firing change as facilitation proceeds: the spike patterns become very irregular with clusters of spikes spaced by long silences; the firing rate is highly variable and clearly oscillates before facilitated bursts. This unstable behaviour dramatically decreases during intense tonic activation which temporarily interrupts bursting, and could therefore be a prerequisite for bursting. In vivo, the effects of AVP depend on the initial firing pattern of AVP neurons: AVP excites weakly active neurons (increasing duration of active periods and decreasing silences), inhibits highly active neurons, and does not affect neurons with intermediate phasic activity. AVP brings the entire population of AVP neurons to discharge with a medium phasic activity characterised by periods of firing and silence lasting 20–40 s, a pattern shown to optimise the release of AVP from the neurohypophysis. Each of the peptides (OT or AVP) induces an increase in intracellular Ca²⁺ concentration, specifically in the neurons containing either OT or AVP respectively. OT evokes the release of Ca²⁺ from IP3-sensitive intracellular stores. AVP induces an influx of Ca²⁺ through voltage-dependent Ca²⁺ channels of T-, L- and N-types. We postulate that the facilitatory autocontrol of OT and AVP neurons could be mediated by Ca²⁺ known to play a key role in the control of the patterns of phasic neurons.     

From simple to complex oscillatory behaviour: analysis of bursting in a multiply regulated biochemical system

We analyze the transition from simple to complex oscillatory behaviour in a three-variable biochemical system that consists of the coupling in series of two autocatalytic enzyme reactions. Complex periodic behaviour occurs in the form of bursting in which clusters of spikes are separated by phases of relative quiescence. The generation of such temporal patterns is investigated by a series of complementary approaches. The dynamics of the system is first cast into two different time-scales, and one of the variables is taken as a slowly-varying parameter influencing the behaviour of the two remaining variables. This analysis shows how complex oscillations develop from simple periodic behaviour and accounts for the existence of various modes of bursting as well as for the dependence of the number of spikes per period on key parameters of the model. We further reduce the number of variables by analyzing bursting by means of one-dimensional return maps obtained from the time evolution of the three-dimensional system. The analysis of a related piecewise linear map allows for a detailed understanding of the complex sequence leading from a bursting pattern with p spikes to a pattern with p + 1 spikes per period. We show that this transition possesses properties of self-similarity associated with the occurrence of more and more complex patterns of bursting. In addition to bursting, period-doubling bifurcations leading to chaos are observed, as in the differential system, when the piecewise-linear map becomes nonlinear.     

A unified model for two modes of bursting in GnRH neurons

Gonadotropin-releasing hormone (GnRH) neurons exhibit at least two intrinsic modes of action potential burst firing, referred to as parabolic and irregular bursting. Parabolic bursting is characterized by a slow wave in membrane potential that can underlie periodic clusters of action potentials with increased interspike interval at the beginning and at the end of each cluster. Irregular bursting is characterized by clusters of action potentials that are separated by varying durations of interburst intervals and a relatively stable baseline potential. Based on recent studies of isolated ionic currents, a stochastic Hodgkin-Huxley (HH)-like model for the GnRH neuron is developed to reproduce each mode of burst firing with an appropriate set of conductances. Model outcomes for bursting are in agreement with the experimental recordings in terms of interburst interval, interspike interval, active phase duration, and other quantitative properties specific to each mode of bursting. The model also shows similar outcomes in membrane potential to those seen experimentally when tetrodotoxin (TTX) is used to block action potentials during bursting, and when estradiol transitions cells exhibiting slow oscillations to irregular bursting mode in vitro. Based on the parameter values used to reproduce each mode of bursting, the model suggests that GnRH neurons can switch between the two through changes in the maximum conductance of certain ionic currents, notably the slow inward Ca²⁺ current I _s, and the Ca²⁺ -activated K⁺ current I _KCa. Bifurcation analysis of the model shows that both modes of bursting are similar from a dynamical systems perspective despite differences in burst characteristics.     

Complex intracellular calcium oscillations. A theoretical exploration of possible mechanisms

Intracellular Ca(2+) oscillations are commonly observed in a large number of cell types in response to stimulation by an extracellular agonist. In most cell types the mechanism of regular spiking is well understood and models based on Ca(2+)-induced Ca(2+) release (CICR) can account for many experimental observations. However, cells do not always exhibit simple Ca(2+) oscillations. In response to given agonists, some cells show more complex behaviour in the form of bursting, i.e. trains of Ca(2+) spikes separated by silent phases. Here we develop several theoretical models, based on physiologically plausible assumptions, that could account for complex intracellular Ca(2+) oscillations. The models are all based on one- or two-pool models based on CICR. We extend these models by (i) considering the inhibition of the Ca(2+)-release channel on a unique intracellular store at high cytosolic Ca(2+) concentrations, (ii) taking into account the Ca(2+)-activated degradation of inositol 1,4,5-trisphosphate (IP(3)), or (iii) considering explicity the evolution of the Ca(2+) concentration in two different pools, one sensitive and the other one insensitive to IP(3). Besides simple periodic oscillations, these three models can all account for more complex oscillatory behaviour in the form of bursting. Moreover, the model that takes the kinetics of IP(3) into account shows chaotic behaviour.     

Analysis of the electrosensory pyramidal cell bursting model for weakly electric fish: model prediction under low levels of dendritic potassium conductance

Pyramidal cells in the electrosensory lateral line lobe (ELL) of weakly electric fish produce burst discharge. A Hodgkin-Huxley-type model, called ghostburster, consisting of two compartments (soma and dendrite) reproduces ELL pyramidal cell bursting observed in vitro. A previous study analyzed the ghostburster by treating Is and gDr,d as bifurcation parameters (Is: current injected into the somatic compartment and gDr,d: maximal conductance of the delayed rectifying potassium current in the dendritic compartment) and indicated that when both Is and gDr,d are set at particular values, the ghostburster shows a codimension-two bifurcation at which both saddle-node bifurcation of fixed points and saddle-node bifurcation of limit cycles occur simultaneously. In the present study, the ghostburster was investigated to clarify the bursting that occurred at gDr,d values smaller than that at the codimension-two bifurcation. Based on the number of spikes per burst, various burst patterns were observed depending on the (Is, gDr,d) values. Depending on the (Is, gDr,d) values, the burst trajectory in a phase space of the ghostburster showed either a high or a low degree of periodicity. Compared to the previous study, the present findings contribute to a more detailed understanding of ghostburster bursting.     

Complex evolution of spike patterns during burst propagation through feed-forward networks

Stable signal transmission is crucial for information processing by the brain. Synfire-chains, defined as feed-forward networks of spiking neurons, are a well-studied class of circuit structure that can propagate a packet of single spikes while maintaining a fixed packet profile. Here, we studied the stable propagation of spike bursts, rather than single spike activities, in a feed-forward network of a general class of excitable bursting neurons. In contrast to single spikes, bursts can propagate stably without converging to any fixed profiles. Spike timings of bursts continue to change cyclically or irregularly during propagation depending on intrinsic properties of the neurons and the coupling strength of the network. To find the conditions under which bursts lose fixed profiles, we propose an analysis based on timing shifts of burst spikes similar to the phase response analysis of limit-cycle oscillators.