State-Dependent Effects of Na Channel Noise on Neuronal Burst Generation

We explore the effects of stochastic sodium (Na) channel activation on the variability and dynamics of spiking and bursting in a model neuron. The complete model segregates Hodgin-Huxley-type currents into two compartments, and undergoes applied current-dependent bifurcations between regimes of periodic bursting, chaotic bursting, and tonic spiking. Noise is added to simulate variable, finite sizes of the population of Na channels in the fast spiking compartment.During tonic firing, Na channel noise causes variability in interspike intervals (ISIs). The variance, as well as the sensitivity to noise, depend on the model's biophysical complexity. They are smallest in an isolated spiking compartment; increase significantly upon coupling to a passive compartment; and increase again when the second compartment also includes slow-acting currents. In this full model, sufficient noise can convert tonic firing into bursting.During bursting, the actions of Na channel noise are state-dependent. The higher the noise level, the greater the jitter in spike timing within bursts. The noise makes the burst durations of periodic regimes variable, while decreasing burst length duration and variance in a chaotic regime. Na channel noise blurs the sharp transitions of spike time and burst length seen at the bifurcations of the noise-free model. Close to such a bifurcation, the burst behaviors of previously periodic and chaotic regimes become essentially indistinguishable.We discuss biophysical mechanisms, dynamical interpretations and physiological implications. We suggest that noise associated with finite populations of Na channels could evoke very different effects on the intrinsic variability of spiking and bursting discharges, depending on a biological neuron's complexity and applied current-dependent state. We find that simulated channel noise in the model neuron qualitatively replicates the observed variability in burst length and interburst interval in an isolated biological bursting neuron.     

Model for synchronization of pancreatic beta-cells by gap junction coupling.

Pancreatic beta-cells coupled by gap junctions in sufficiently large clusters exhibit regular electrical bursting activity, which is described by the Chay-Keizer model and its variants. According to most reports, however, isolated cells exhibit disorganized spiking. We have previously (Sherman, A. J. Rinzel, and J. Keizer, 1988. Biophys. J. 54:411-425) modeled these behaviors by hypothesizing that stochastic channel fluctuations disrupt the bursts. We showed that when cells are coupled by infinite conductance gap junctions, so that the cluster is isopotential and may be viewed as a single "supercell," the fluctuations are shared over a larger membrane area and hence dampened. Bursting emerges when there are more than approximately 50 cells in the cluster. In the model the temporal organization of spikes into bursts increases the amplitude of intracellular calcium oscillations, which may be relevant for insulin secretion. We now extend the previous work by considering the case of a true "multicell" model with finite gap junctional conductance. Whereas the previous study assumed that the cells were synchronized, we can now study the process of synchronization itself. We show that, for sufficiently large clusters, the cells both synchronize and begin to burst with moderate, physiologically reasonable gap junctional conductance. An unexpected finding is that the burst period is longer, and calcium amplitude greater, than when coupling is infinitely strong, with an optimum in the range of 150-250 pS. Our model is in good agreement with recent experimental data of Perez-Armendariz, M., D. C. Spray, and M. V. L. Bennett. (1991. Biophys. J. 59:76-92) showing extensive gap junctions in beta-cell pairs with mean interfacial conductance of 213 +/- 113 pS. The optimality property of our model is noteworthy because simple slow-wave models without spikes do not show the same behavior.     

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.     

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.     

Theoretical studies on the electrical activity of pancreatic beta-cells as a function of glucose.

The electrical activity of pancreatic beta-cells, which has been closely correlated both with intracellular Ca2+ concentration and insulin release, is characterized by a biphasic response to glucose and bursts of spiking action potentials. Recent voltage clamp and single channel patch clamp experiments have identified several transmembrane ionic channels that may play key roles in the electrophysiological behavior of beta-cells. There is a hypothesis that Ca2+-activated K+ channels are responsible for both the resting potential during low glucose concentration and the silent phase during bursting. The discovery of the ATP-inactivated K+ channel raises the possibility that the current for this latter K+ channel may dominate the resting potential, while the Ca2+-activated K+ current dominates the silent phase potential between bursts. The recent discovery that Ca2+-activated K+ channels are pH sensitive raises an interesting possibility for the biphasic electrical response. In this paper, numerical methods are presented for evaluating these hypotheses against experimental evidence.     

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.     

Phantom bursting is highly sensitive to noise and unlikely to account for slow bursting in beta-cells: considerations in favor of metabolically driven oscillations

Pancreatic beta-cells show bursting electrical activity with a wide range of burst periods ranging from a few seconds, often seen in isolated cells, over tens of seconds (medium bursting), usually observed in intact islets, to several minutes. The phantom burster model [Bertram, R., Previte, J., Sherman, A., Kinard, T.A., Satin, L.S., 2000. The phantom burster model for pancreatic beta-cells. Biophys. J. 79, 2880-2892] provided a framework, which covered this span, and gave an explanation of how to obtain medium bursting combining two processes operating on different time scales. However, single cells are subjected to stochastic fluctuations in plasma membrane currents, which are likely to disturb the bursting mechanism and transform medium bursters into spikers or very fast bursters. We present a polynomial, minimal, phantom burster model and show that noise modifies the plateau fraction and lowers the burst period dramatically in phantom bursters. It is therefore unlikely that slow bursting in single cells is driven by the slow phantom bursting mechanism, but could instead be driven by oscillations in glycolysis, which we show are stable to random ion channel fluctuations. Moreover, so-called compound bursting can be converted to apparent slow bursting by noise, which could explain why compound bursting and mixed Ca(2+) oscillations are seen mainly in intact islets.     

神经起步点自发放电节律及节律转化的分岔规律

在神经起步点的实验中观察到了复杂多样的神经放电([Ca^2 ]o)节律模式,如周期簇放电、周期峰放电、混沌簇放电、混沌峰放电以及随机放电节律等。随着细胞外钙离子浓度的降低,神经放电节律从周期l簇放电,经过复杂的分岔过程(包括经倍周期分岔到混沌簇放电、混沌簇放电经激变到混沌峰放电、以及混沌峰放电经逆倍周期分岔到周期峰放电)转化为周期l峰放电。在神经放电理论模型——Chay模型中,调节与实验相关的参数(Ca^2 平衡电位),可以获得与实验相似的神经放电节律和节律转换规律。这表明复杂的神经放电节律之间存在着一定的分岔规律,它们是理解神经元信息编码的基础。     

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.     

Hamiltonian energy and coexistence of hidden firing patterns from bidirectional coupling between two different neurons

In this paper, bidirectional-coupled neurons through an asymmetric electrical synapse are investigated. These coupled neurons involve 2D Hindmarsh–Rose (HR) and 2D FitzHugh–Nagumo (FN) neurons. The equilibria of the coupled neurons model are investigated, and their stabilities have revealed that, for some values of the electrical synaptic weight, the model under consideration can display either self-excited or hidden firing patterns. In addition, the hidden coexistence of chaotic bursting with periodic spiking, chaotic spiking with period spiking, chaotic bursting with a resting pattern, and the coexistence of chaotic spiking with a resting pattern are also found for some sets of electrical synaptic coupling. For all the investigated phenomena, the Hamiltonian energy of the model is computed. It enables the estimation of the amount of energy released during the transition between the various electrical activities. Pspice simulations are carried out based on the analog circuit of the coupled neurons to support our numerical results. Finally, an STM32F407ZE microcontroller development board is exploited for the digital implementation of the proposed coupled neurons model.     

Channel sharing in pancreatic beta -cells revisited: enhancement of emergent bursting by noise

Secretion of insulin by electrically coupled populations of pancreatic beta -cells is governed by bursting electrical activity. Isolated beta -cells, however, exhibit atypical bursting or continuous spike activity. We study bursting as an emergent property of the population, focussing on interactions among the subclass of spiking cells. These are modelled by equipping the fast subsystem with a saddle-node-loop bifurcation, which makes it monostable. Such cells can only spike tonically or remain silent when isolated, but can be induced to burst with weak diffusive coupling. With stronger coupling, the cells revert to tonic spiking. We demonstrate that the addition of noise dramatically increases, via a phenomenon like stochastic resonance, the coupling range over which bursting is seen.     

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.     

神经放电节律加周期分岔序列中的簇放电到峰放电的转迁

含快慢子系统的神经元数学模型仿真预期,神经放电节律经历加周期分岔序列,可以进一步表现激变,并通过逆倍周期分岔级联进入周期1峰放电。实验调节胞外钙离子浓度,观察到从周期1簇放电开始的带有随机节律的加周期分岔到簇内有多个峰的簇放电,再经激变转迁到峰放电节律的分岔序列,提供了这种分岔序列模式实验证据。实验所见之激变表现为簇放电节律的休止期消失,放电节律变为混沌峰放电和周期峰放电。作者利用随机Chay模型更加逼真地仿真再现了实验所见的分岔序列。该实验结果验证了以前的确定性数学模型的理论预期,并利用随机理论模型仿真了其在现实神经系统的表现;揭示了一类完整的神经放电节律的转换规律。     

How noise and coupling induce bursting action potentials in pancreatic {beta}-cells

Unlike isolated beta-cells, which usually produce continuous spikes or fast and irregular bursts, electrically coupled beta-cells are apt to exhibit robust bursting action potentials. We consider the noise induced by thermal fluctuations as well as that by channel-gating stochasticity and examine its effects on the action potential behavior of the beta-cell model. It is observed numerically that such noise in general helps single cells to produce a variety of electrical activities. In addition, we also probe coupling via gap junctions between neighboring cells, with heterogeneity induced by noise, to find that it enhances regular bursts.     

Variability of bursting patterns in a neuron model in the presence of noise

Spiking and bursting patterns of neurons are characterized by a high degree of variability. A single neuron can demonstrate endogenously various bursting patterns, changing in response to external disturbances due to synapses, or to intrinsic factors such as channel noise. We argue that in a model of the leech heart interneuron existing variations of bursting patterns are significantly enhanced by a small noise. In the absence of noise this model shows periodic bursting with fixed numbers of interspikes for most parameter values. As the parameter of activation kinetics of a slow potassium current is shifted to more hyperpolarized values of the membrane potential, the model undergoes a sequence of incremental spike adding transitions accumulating towards a periodic tonic spiking activity. Within a narrow parameter window around every spike adding transition, spike alteration of bursting is deterministically chaotic due to homoclinic bifurcations of a saddle periodic orbit. We have found that near these transitions the interneuron model becomes extremely sensitive to small random perturbations that cause a wide expansion and overlapping of the chaotic windows. The chaotic behavior is characterized by positive values of the largest Lyapunov exponent, and of the Shannon entropy of probability distribution of spike numbers per burst. The windows of chaotic dynamics resemble the Arnold tongues being plotted in the parameter plane, where the noise intensity serves as a second control parameter. We determine the critical noise intensities above which the interneuron model generates only irregular bursting within the overlapped windows.     

Effects of both glucose and IP<Subscript>3</Subscript> concentrations on action potentials in pancreatic β-cells

Abstarct Considering the ATP-driven (SERCA) pump flux as function of glucose concentration and the calcium flux from the endoplasmic reticulum (ER) through the IP₃R channel, the calcium-based phantom bursting model (PBM) of β-cells (Bertram and Sherman in Bull Math Biol 66:1313, 2004) is theoretically extended to discuss the effects of glucose and inositol 1,4,5-trisphosphate (IP₃) concentration on the membrane potential activities. When IP₃ concentration is fixed, it is found that there is a critical glucose concentration at which electrical bursting oscillations transfer into spiking, and the critical concentration of glucose is increased with the increasing of IP₃ concentration. To get the bursting oscillations in β-cells, our theoretical results show that the stimulatory glucose concentration should be more than 10 mM, which is consistent with the normal physiological IP₃ level. When the stochastic opening and closing of IP₃R channels are considered, it is shown that the membrane potential oscillation transfers from spiking to bursting with the channel number decreasing, and the average cytosolic free Ca²⁺ concentration is increased with the increase of glucose concentration.     

Bursting synchronization dynamics of pancreatic β-cells with electrical and chemical coupling

Based on bifurcation analysis, the synchronization behaviors of two identical pancreatic β-cells connected by electrical and chemical coupling are investigated, respectively. Various firing patterns are produced in coupled cells when a single cell exhibits tonic spiking or square-wave bursting individually, irrespectively of what the cells are connected by electrical or chemical coupling. On the one hand, cells can burst synchronously for both weak electrical and chemical coupling when an isolated cell exhibits tonic spiking itself. In particular, for electrically coupled cells, under the variation of the coupling strength there exist complex transition processes of synchronous firing patterns such as “fold/limit cycle” type of bursting, then anti-phase continuous spiking, followed by the “fold/torus” type of bursting, and finally in-phase tonic spiking. On the other hand, it is shown that when the individual cell exhibits square-wave bursting, suitable coupling strength can make the electrically coupled system generate “fold/Hopf” bursting via “fold/fold” hysteresis loop; whereas, the chemically coupled cells generate “fold/subHopf” bursting. Especially, chemically coupled bursters can exhibit inverse period-adding bursting sequence. Fast–slow dynamics analysis is applied to explore the generation mechanism of these bursting oscillations. The above analysis of bursting types and the transition may provide us with better insight into understanding the role of coupling in the dynamic behaviors of pancreatic β-cells.     

Effect of channel block on the spiking activity of excitable membranes in a stochastic Hodgkin-Huxley model

The influence of intrinsic channel noise on the spontaneous spiking activity of poisoned excitable membrane patches is studied by use of a stochastic generalization of the Hodgkin-Huxley model. Internal noise stemming from the stochastic dynamics of individual ion channels is known to affect the collective properties of the whole ion channel cluster. For example, there exists an optimal size of the membrane patch for which the internal noise alone causes a regular spontaneous generation of action potentials. In addition to varying the size of ion channel clusters, living organisms may adapt the densities of ion channels in order to optimally regulate the spontaneous spiking activity. The influence of a channel block on the excitability of a membrane patch of a certain size is twofold: first, a variation of ion channel densities primarily yields a change of the conductance level; second, a down-regulation of working ion channels always increases the channel noise. While the former effect dominates in the case of sodium channel block resulting in a reduced spiking activity, the latter enhances the generation of spontaneous action potentials in the case of a tailored potassium channel blocking. Moreover, by blocking some portion of either potassium or sodium ion channels, it is possible to either increase or decrease the regularity of the spike train.     

Beating irregularity of single pacemaker cells isolated from the rabbit sinoatrial node.

Single pacemaker heart cells discharge irregularly. Data on fluctuations in interbeat interval of single pacemaker cells isolated from the rabbit sinoatrial node are presented. The coefficient of variation of the interbeat interval is quite small, approximately 2%, even though the coefficient of variation of diastolic depolarization rate is approximately 15%. It has been hypothesized that random fluctuations in interbeat interval arise from the stochastic behavior of the membrane ionic channels. To test this hypothesis, we constructed a single channel model of a single pacemaker cell isolated from the rabbit sinoatrial node, i.e., a model into which the stochastic open-close kinetics of the individual membrane ionic channels are incorporated. Single channel conductances as well as single channel open and closed lifetimes are based on experimental data from whole cell and single channel experiments that have been published in the past decade. Fluctuations in action potential parameters of the model cell are compared with those observed experimentally. It is concluded that fluctuations in interbeat interval of single sinoatrial node pacemaker cells indeed are due to the stochastic open-close kinetics of the membrane ionic channels.     

Output-mode transitions are controlled by prolonged inactivation of sodium channels in pyramidal neurons of subiculum

Transitions between different behavioral states, such as sleep or wakefulness, quiescence or attentiveness, occur in part through transitions from action potential bursting to single spiking. Cortical activity, for example, is determined in large part by the spike output mode from the thalamus, which is controlled by the gating of low-voltage–activated calcium channels. In the subiculum—the major output of the hippocampus—transitions occur from bursting in the delta-frequency band to single spiking in the theta-frequency band. We show here that these transitions are influenced strongly by the inactivation kinetics of voltage-gated sodium channels. Prolonged inactivation of sodium channels is responsible for an activity-dependent switch from bursting to single spiking, constituting a novel mechanism through which network dynamics are controlled by ion channel gating.