Multiple modes of a conditional neural oscillator

We present a model for a conditional bursting neuron consisting of five conductances: Hodgkin-Huxley type time- and voltage-dependent Na⁺ and K⁺ conductances, a calcium activated voltage-dependent K⁺ conductance, a calcium-inhibited time- and voltage-dependent Ca⁺⁺ conductance, and a leakage Cl(_– conductance. With an initial set of parameters (versionS), the model shows a hyperpolarized steady-state membrane potential at which the neuron is silent. Increasingg _Na and decreasingg _Cl, whereg _i , is the maximal conductance for speciesi, produces bursts of action potentials (BursterN). Alternatively, an increase ing _Ca produces a different bursting state (BursterC). The two bursting states differ in the periods and amplitudes of their bursting pacemaker potentials. They show different steady-stateI–V curves under simulated voltage-clamp conditions; in simulations that mimic a steady-stateI–V curve taken under experimental conditions only BursterN shows a negative slope resistance region. ModelC continues to burst in the presence of TTX, while bursting in ModelN is suppressed in TTX. Hybrid models show a smooth transition between the two states.     

Mechanisms of Generation of Pacemaker Activity by Identified Helix Neurons

We studied the mechanisms of generation of pacemaker activity in identified neurons of Helix pomatia. For this purpose, we isolated the PPa2 and PPa7 neurons generating spontaneous rhythmic monomodal activity and PPa1 neuron with bursting activity. It was demonstrated that isolated PPa2 and PPa7 cells produce endogenous rhythmic activity that was not considerably modified by external application of 1 mM CdCl₂. Sometimes, only low-amplitude dendritic action potentials (AP) were observed instead of generation of full-amplitude somatic AP. In contrast, isolation of the PPa1 neuron eliminated its bursting activity, but subsequent application of oxytocin on this neuron recovered such activity. This finding shows that the bursting activity of the PPa1 neuron is of an exogenous nature. Application of 1 mM CdCl₂ suppressed this bursting activity, but when Cd²⁺ was applied against the background of superfusion of the neuron with Ringer solution containing a bursting activity-initiating neuropeptide obtained from the molluscan CNS, this blocker was incapable of suppressing the bursting activity. A blocker of the hyperpolarization-activated current (I _h , H current), Cs⁺ (10 mM) exerted no noticeable effect on the activity of the studied neurons. Our findings allow us to conclude that the pacemaker activity is initiated within the dendritic tree of a cell and is then electrotonically spread to the soma, where full-amplitude AP are generated. It seems probable that Ca²⁺ ions and H current are not directly involved in generation of the pacemaker activity in the studied snail neurons.     

Dependence of spontaneous neuronal firing and depolarisation block on astroglial membrane transport mechanisms

Exposed to a sufficiently high extracellular potassium concentration ([K^?+?]_o), the neuron can fire spontaneous discharges or even become inactivated due to membrane depolarisation (??depolarisation block??). Since these phenomena likely are related to the maintenance and propagation of seizure discharges, it is of considerable importance to understand the conditions under which excess [K^?+?]_o causes them. To address the putative effect of glial buffering on neuronal activity under elevated [K^?+?]_o conditions, we combined a recently developed dynamical model of glial membrane ion and water transport with a Hodgkin?CHuxley type neuron model. In this interconnected glia-neuron model we investigated the effects of natural heterogeneity or pathological changes in glial membrane transporter density by considering a large set of models with different, yet empirically plausible, sets of model parameters. We observed both the high [K^?+?]_o-induced duration of spontaneous neuronal firing and the prevalence of depolarisation block to increase when reducing the magnitudes of the glial transport mechanisms. Further, in some parameter regions an oscillatory bursting spiking pattern due to the dynamical coupling of neurons and glia was observed. Bifurcation analyses of the neuron model and of a simplified version of the neuron-glia model revealed further insights about the underlying mechanism behind these phenomena. The above insights emphasise the importance of combining neuron models with detailed astroglial models when addressing phenomena suspected to be influenced by the astroglia-neuron interaction. To facilitate the use of our neuron-glia model, a CellML version of it is made publicly available.     

Intrinsic bursting activity in the pre-Bötzinger Complex: Role of persistent sodium and potassium currents

Computational models of single pacemaker neuron and neural population in the pre-Bötzinger Complex (pBC) were developed based on the previous models by Butera et al. (1999a,b). Our modeling study focused on the conditions that could define endogenous bursting vs. tonic activity in single pacemaker neurons and population bursting vs. asynchronous firing in populations of pacemaker neurons. We show that both bursting activity in single pacemaker neurons and population bursting activity may be released or suppressed depending on the expression of persistent sodium (I_NaP) and delayed-rectifier potassium (I_K) currents. Specifically, a transition from asynchronous firing to population bursting could be induced by a reduction of I_K via a direct suppression of the potassium conductance or through an elevation of extracellular potassium concentration. Similar population bursting activity could be triggered by an augmentation of I_NaP. These findings are discussed in the context of the possible role of population bursting activity in the pBC in the respiratory rhythm generation in vivo vs. in vitro and during normal breathing in vivo vs. gasping.     

Monosynaptic pathway responsible for generation of bursting activity in theHelix pomatia RPal neuron

The connection between a visceral ganglia interneuron initiating bursting pacemaker activity in the RPal neuron and the RPal neuron itself was investigated inHelix pomatia. Stimulating the interneuron either initiated or intensified bursting activity in the RPal neuron, depending on initial electrical activity in this cell. Replacing calcium with magnesium ions in the extracellular fluid and adding CdCl₂ to this fluid reversibly inhibited the effect of interneuronal stimulation on the RPal neuron. The latter effect was unaffected by increasing the concentration of extracellular Ca²⁺ 10 to 70 mM. Intracellular injection of both Cs⁺ and TEA into the interneuron produced an increase in the duration of its action potentials and rendered the link connecting the neurons more effective. It is deduced that a monosynaptic chemical connection exists between the interneuron and the RPal neuron for which a peptide compound serves as transmitter.A. A. Bogomolets Institute of Physiology, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Neirofiziologiya, Vol. 19, No. 1, pp. 20–28, January–February, 1987.     

Electrical bursting and luminal calcium oscillation in excitable cell models

 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. Received: 17 August 1995/Accepted in revised form: 10 July 1996     

Analysis of the effects of modulatory agents on a modeled bursting neuron: Dynamic interactions between voltage and calcium dependent systems

In a computational model of the bursting neuron R15, we have implemented proposed mechanisms for the modulation of two ionic currents (I _R andI _SI) that play key roles in regulating its spontaneous electrical activity. The model was sufficient to simulate a wide range of endogenous activity in the presence of various concentrations of serotonin (5-HT) or dopamine (DA). The model was also sufficient to simulate the responses of the neuron to extrinsic current pulses and the ways in which those responses were altered by 5-HT or DA. The results suggest that the actions of modulatory agents and second messengers on this neuron, and presumably other neurons, cannot be understood on the basis of their direct effects alone. It is also necessary to take into account the indirect effects of these agents on other unmodulated ion channels. These indirect effects occur through the dynamic interactions of voltage-dependent and calcium-dependent processes.     

A computational study of the effects of serotonin on a molluscan burster neuron

A mathematical model of burster neuron R₁₅ from the abdominal ganglion of Aplysia is presented. This is an improvement over earlier models in that the bursting mechanism is more accurately represented. The improved model allows for simulated application of the neurotransmitter serotonin, which has been reported to have profound effects on the voltage waveform produced by R₁₅. Computational analysis indicates that the serotonin-induced modulation of the waveform can be explained in terms of competition between stationary, bursting, and beating attractors. Analysis also indicates that, as a result of this competition, serotonin increases the sensitivity of the neuron to synaptic perturbations. This may have important consequences with regard to water balance in the Aplysia, particularly during egg laying.     

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.     

Mechanisms of membrane potential oscillation in bursting neurons on the snail,Helix pomatia

• 1.1. The mechanism of generation of membrane potential (MP) oscillations was studied in identified bursting neurons from the snail Helix pomatia.
• 2.2. Long-lasting stimulation of an identified peptidergic interneuron produced a persistent bursting activity in a non-active burster.
• 3.3. External application of calcium channel blockers (1 mM Cd²⁺ or 5 mM La²⁺) resulted in a transient increase in the slow-wave amplitude and subsequent prevention of pacemaker activity generation in bursting neurons. Application of these blockers together with endogenous neuropeptide initiating bursting activity generation, increased MP wave amplitude without prevention of bursting activity generation.
• 4.4. Replacement of all NaCl in normal Ringer's solution with isoosmotic CaCl₂, glucose or Tris-HCl produced a reversible block of bursting activity generation. Stationary current-voltage relation (CVR) of bursting neuron membrane has a region of negative resistance (NRR) and does not intersect the potential axis in threshold region for action potential (AP) generation in normal Ringer's solution. In Na-free solution stationary CVR is linear and intersects the potential axis near — 52 mV.
• 5.5. Novel potential- and time-dependent outward (E_rev = − 58 mV) current, I_B, activated by hyperpolarization was found in the bursting neuron membrane. Having achieved a maximal value, this current decayed with a time constant of about 1 sec. Hyperpolarization inactivated maximal conductance, g_B, responsible for I_B, and depolarization abolished inactivation of g_B.
• 6.6. Short-lasting (0.01 sec) hyperpolarization of the bursting neuron membrane by inward current pulse induced the development of prolonged hyperpolarization wave lasting up to 10 sec.
• 7.7. These results suggest that: (a) persistent bursting activity of RPal neuron in the snail Helix pomatia is not endogenous but is due to a constant activation of peptidergic synaptic inputs of these neurons; (b) Ca²⁺ ions do not play a pivotal role in the ionic mechanism of MP oscillations but play a determining role in the process of secretion of a peptide initiating bursting activity by the interneuron presynaptic terminal; (c) depolarizing phase of the MP wave is due to specific properties of stationary CVR and hyperpolarization phase is due to regenerative properties of hyperpolarization-activated outward current I_B. The minimal mathematical version of MP oscillations based on the experimental data is presented.

     

Identifying Crucial Parameter Correlations Maintaining Bursting Activity

Recent experimental and computational studies suggest that linearly correlated sets of parameters (intrinsic and synaptic properties of neurons) allow central pattern-generating networks to produce and maintain their rhythmic activity regardless of changing internal and external conditions. To determine the role of correlated conductances in the robust maintenance of functional bursting activity, we used our existing database of half-center oscillator (HCO) model instances of the leech heartbeat CPG. From the database, we identified functional activity groups of burster (isolated neuron) and half-center oscillator model instances and realistic subgroups of each that showed burst characteristics (principally period and spike frequency) similar to the animal. To find linear correlations among the conductance parameters maintaining functional leech bursting activity, we applied Principal Component Analysis (PCA) to each of these four groups. PCA identified a set of three maximal conductances (leak current, _Leak; a persistent K current, _K2; and of a persistent Na+ current, _P) that correlate linearly for the two groups of burster instances but not for the HCO groups. Visualizations of HCO instances in a reduced space suggested that there might be non-linear relationships between these parameters for these instances. Experimental studies have shown that period is a key attribute influenced by modulatory inputs and temperature variations in heart interneurons. Thus, we explored the sensitivity of period to changes in maximal conductances of _Leak, _K2, and _P, and we found that for our realistic bursters the effect of these parameters on period could not be assessed because when varied individually bursting activity was not maintained.     

Multiple timescale mixed bursting dynamics in a respiratory neuron model

Experimental results in rodent medullary slices containing the pre-Bötzinger complex (pre-BötC) have identified multiple bursting mechanisms based on persistent sodium current (I _NaP) and intracellular Ca²⁺. The classic two-timescale approach to the analysis of pre-BötC bursting treats the inactivation of I _NaP, the calcium concentration, as well as the Ca²⁺-dependent inactivation of IP ₃ as slow variables and considers other evolving quantities as fast variables. Based on its time course, however, it appears that a novel mixed bursting (MB) solution, observed both in recordings and in model pre-BötC neurons, involves at least three timescales. In this work, we consider a single-compartment model of a pre-BötC inspiratory neuron that can exhibit both I _NaP and Ca²⁺ oscillations and has the ability to produce MB solutions. We use methods of dynamical systems theory, such as phase plane analysis, fast-slow decomposition, and bifurcation analysis, to better understand the mechanisms underlying the MB solution pattern. Rather surprisingly, we discover that a third timescale is not actually required to generate mixed bursting solutions. Through our analysis of timescales, we also elucidate how the pre-BötC neuron model can be tuned to improve the robustness of the MB solution.     

Glucose response to bursting-spiking pancreatic β-cells by a barrier kinetic model

A mathematical model of the pancreatic -cell electrical activity was developed using a barrier kinetic model. Our model incorpolates the glucose sensitive channel which is known to conduct K⁺ in the absence of glucose. The model also incorporates Ca _i sensitive K⁺ channels which are inhibited by intracellular H⁺ ions. It is described by three non-linear simultaneous differential equations. Numerical integration of these equations allowed us to examine the effect of glucose and of external Ca²⁺ ions on the electrical and cellular activity of the -cell. Our results show that the contribution of glucose-sensitive channel activity, if not completely inhibited, plays a very important role in determining the bursting periodicity. Our results also shows that even a small decrease in pH _i is sufficient to change a bursting -cell to a spiking one. The voltage dependence of calcium sensitive K⁺ channels, however, affects little to the bursting mode of the electrical activity. Our simulation supports an incomplete selectivity of the voltage dependent calcium channel for calcium ions with low external [Ca²⁺]. It also supports the role of [Ca] _i as an inhibitor of this channel when [Ca] _i becomes unusually high.This work was supported by NSF PCM 82 15583     

Modelling intrinsic electrophysiological properties of ON and OFF retinal ganglion cells

ON and OFF retinal ganglion cells (RGCs) display differences in their intrinsic electrophysiology: OFF cells maintain spontaneous activity in the absence of any input, exhibit subthreshold membrane potential oscillations, rebound excitation and burst firing; ON cells require excitatory input to drive their activity and display none of the aforementioned phenomena. The goal of this study was to identify and characterize ionic currents that explain these intrinsic electrophysiological differences between ON and OFF RGCs. A mathematical model of the electrophysiological properties of ON and OFF RGCs was constructed and validated using published patch-clamp data from isolated intact mouse retina. The model incorporates three ionic currents hypothesized to play a role in generating behaviors that are different between ON and OFF RGCs. These currents are persistent Na⁺, I _NaP, hyperpolarization-activated, I _h, and low voltage activated Ca^2 +, I _T, currents. Using computer simulations of Hodgkin-Huxley type neuron with a single compartment model we found two distinct sets of I _NaP, I _h, I _T conductances that correspond to ON and OFF RGCs populations. Simulations indicated that special properties of I _T explain the differences in intrinsic electrophysiology between ON and OFF RGCs examined here. The modelling shows that the maximum conductance of I _T is higher in OFF than in ON cells, in agreement with recent experimental data.     

Thermal Sensitivity of Lateral Inhibition in Limulus Eye

The effectiveness of lateral inhibition, measured as spike response decrement in a test ommatidium, produced by activity in a group of neighboring ommatidia, decreases as temperature decreases (Q₁₀ of 2.6). The corresponding sensory transducer-spike encoding processes have a weaker temperature dependence (Q₁₀ of 1.6). Relative synaptic delay, the time difference between the latency of inhibition onset and the latency of test receptor excitation, has a strong temperature dependence (Q₁₀ of 5), while receptor potential onset latency (Q₁₀ of 1.4) and optic nerve spike conduction velocity (Q₁₀ of 1.7), two factors inherent in relative synaptic delay, are less temperature sensitive. Oscillations of optic nerve spike response ("bursting") may be produced by thermal adjustment of temperature-sensitive parameters of the lateral inhibitory network in the retina. Burst interval has a strong temperature dependence (Q₁₀ of 2.4) and broad interspike interval distribution compared to the thermal sensitivity (Q₁₀ of 1.4) and narrow spike interval spectrum of the response of a single unit within the bursting group.     

The role of extracellular potassium dynamics in the different stages of ictal bursting and spreading depression: A computational study

Experimental evidences point out the participation of nonsynaptic mechanisms (e.g., fluctuations in extracellular ions) in epileptiform bursting and spreading depression (SD). During these abnormal oscillatory patterns, it is observed an increase of extracellular potassium concentration [K⁺]_o and a decrease of extracellular calcium concentration [Ca²⁺]_o which raises the neuronal excitability. However, whether the high [K⁺]_o triggers and propagates these abnormal neuronal activities or plays a secondary role into this process is unclear. To better understand the influence of extracellular potassium dynamics in these oscillatory patterns, the experimental conditions of high [K⁺]_o and zero [Ca²⁺]_o were replicated in an extended Golomb model where we added important regulatory mechanisms of ion concentration as Na⁺-K⁺ pump, ion diffusion and glial buffering. Within these conditions, simulations of the cell model exhibit seizure-like discharges (ictal bursting). The SD was elicited by the interruption of the Na⁺−K⁺ pump activity, mimicking the effect of cellular hypoxia (an experimental protocol to elicit SD, the hypoxia-induced SD). We used the bifurcation theory and the fast-slow method to analyze the interference of K⁺ dynamics in the cellular excitability. This analysis indicates that the system loses its stability at a high [K⁺]_o, transiting to an elevated state of neuronal excitability. Effects of high [K⁺]_o are observed in different stages of ictal bursting and SD. In the initial stage, the increase of [K⁺]_o creates favorable conditions to trigger both oscillatory patterns. During the neuronal activity, a continuous growth of [K⁺]_o by outward K⁺ flow depresses K⁺ currents in a positive feedback way. At the last stage, due to the depression of K⁺ currents, the Na⁺-K⁺ pump is the main mechanism in the end of neuronal activity. Thus, this work suggests that [K⁺]_o dynamics may play a fundamental role in these abnormal oscillatory patterns.     

Cellular properties and modulation of the stomatogastric ganglion neurons of a stomatopod,Squilla oratoria

Cellular properties and modulation of the identified neurons of the posterior cardiac plate-pyloric system in the stomatogastric ganglion of a stomatopod, Squilla oratoria, were studied electrophysiologically. Each class of neurons involved in the cyclic bursting activity was able to trigger an endogenous, slow depolarizing potential (termed a driver potential) which sustained bursting. Endogenous oscillatory properties were demonstrated by the phase reset behavior in response to brief stimuli during ongoing rhythm. The driver potential was produced by membrane voltage-dependent activation and terminated by an active repolarization. Striking enhancement of bursting properties of all the cell types was induced by synaptic activation via extrinsic nerves, seen as increases in amplitude or duration of driver potentials, spiking rate during a burst, and bursting rate. The motor pattern produced under the influence of extrinsic modulatory inputs continued for a long time, relative to that in the absence of activation of modulatory inputs. Voltage-dependent conductance mechanisms underlying postinhibitory rebound and driver potential responses were modified by inputs. It is concluded that endogenous cellular properties, as well as synaptic circuitry and extrinsic inputs, contribute to generation of the rhythmic motor pattern, and that a motor system and its component neurons have been highly conserved during evolution between stomatopods and decapods.Abbreviations AB anterior burster neuron - CoG commissural ganglion - CPG central pattern generator - lvn lateral ventricular nerve - OG oesophageal ganglion - pcp posterior cardiac plate - PCP pcp constrictor neuron - PD pyloric dilator neuron - PY pyloric constrictor neuron - son superior oesophageal nerve - STG stomatogastric ganglion - stn stomatogastric nerve     

Inhibition modifies the effects of slow calcium-activated potassium channels on epileptiform activity in a neuronal network model

Generation of epileptiform activity typically results from a change in the balance between network excitation and inhibition. Experimental evidence indicates that alterations of either synaptic activity or intrinsic membrane properties can produce increased network excitation. The slow Ca²⁺-activated K⁺ currents (sI_AHP) are important modulators of neuronal firing rate and excitability and have important established and potential roles in epileptogenesis. While the effects of changes in sI_AHP on individual neuronal excitability are readily studied and well established, the effects of such changes on network behavior are less well known. The experiments here utilize a defined small network model of multicompartment pyramidal cells and an inhibitory interneuron to study the effects of changes in sI_AHP on network behavior. The benefits of this model system include the ability to observe activity in all cells in a network and the effects of interactions of multiple simultaneous influences. In the model with no inhibitory interneuron, increasing sI_AHP results in progressively decreasing burst activity. Adding an inhibitory interneuron changes the observed effects; at modest inhibitory strengths, increasing sI_AHP in all network neurons actually results in increased network bursting (except at very high values). The duration of the burst activity is influenced by the length of delay in a feedback loop, with longer loops resulting in more prolonged bursting. These observations illustrate that the study of potential antiepileptogenic membrane effects must be extended to realistic networks. Network inhibition can dramatically alter the observations seen in pure excitatory networks.     

Interactions of persistent sodium and calcium-activated nonspecific cationic currents yield dynamically distinct bursting regimes in a model of respiratory neurons

The preBötzinger complex (preBötC) is a heterogeneous neuronal network within the mammalian brainstem that has been experimentally found to generate robust, synchronous bursts that drive the inspiratory phase of the respiratory rhythm. The persistent sodium (NaP) current is observed in every preBötC neuron, and significant modeling effort has characterized its contribution to square-wave bursting in the preBötC. Recent experimental work demonstrated that neurons within the preBötC are endowed with a calcium-activated nonspecific cationic (CAN) current that is activated by a signaling cascade initiated by glutamate. In a preBötC model, the CAN current was shown to promote robust bursts that experience depolarization block (DB bursts). We consider a self-coupled model neuron, which we represent as a single compartment based on our experimental finding of electrotonic compactness, under variation of g _NaP, the conductance of the NaP current, and g _CAN, the conductance of the CAN current. Varying these two conductances yields a spectrum of activity patterns, including quiescence, tonic activity, square-wave bursting, DB bursting, and a novel mixture of square-wave and DB bursts, which match well with activity that we observe in experimental preparations. We elucidate the mechanisms underlying these dynamics, as well as the transitions between these regimes and the occurrence of bistability, by applying the mathematical tools of bifurcation analysis and slow-fast decomposition. Based on the prevalence of NaP and CAN currents, we expect that the generalizable framework for modeling their interactions that we present may be relevant to the rhythmicity of other brain areas beyond the preBötC as well.