A database of computational models of a half-center oscillator for analyzing how neuronal parameters influence network activity

A half-center oscillator (HCO) is a common circuit building block of central pattern generator networks that produce rhythmic motor patterns in animals. Here we constructed an efficient relational database table with the resulting characteristics of the Hill et al.’s (J Comput Neurosci 10:281–302, 2001) HCO simple conductance-based model. The model consists of two reciprocally inhibitory neurons and replicates the electrical activity of the oscillator interneurons of the leech heartbeat central pattern generator under a variety of experimental conditions. Our long-range goal is to understand how this basic circuit building block produces functional activity under a variety of parameter regimes and how different parameter regimes influence stability and modulatability. By using the latest developments in computer technology, we simulated and stored large amounts of data (on the order of terabytes). We systematically explored the parameter space of the HCO and corresponding isolated neuron models using a brute-force approach. We varied a set of selected parameters (maximal conductance of intrinsic and synaptic currents) in all combinations, resulting in about 10 million simulations. We classified these HCO and isolated neuron model simulations by their activity characteristics into identifiable groups and quantified their prevalence. By querying the database, we compared the activity characteristics of the identified groups of our simulated HCO models with those of our simulated isolated neuron models and found that regularly bursting neurons compose only a small minority of functional HCO models; the vast majority was composed of spiking neurons.     

Asymmetric and transient properties of reciprocal activity of antagonists during the paw-shake response in the cat

Mutually inhibitory populations of neurons, half-center oscillators (HCOs), are commonly involved in the dynamics of the central pattern generators (CPGs) driving various rhythmic movements. Previously, we developed a multifunctional, multistable symmetric HCO model which produced slow locomotor-like and fast paw-shake-like activity patterns. Here, we describe asymmetric features of paw-shake responses in a symmetric HCO model and test these predictions experimentally. We considered bursting properties of the two model half-centers during transient paw-shake-like responses to short perturbations during locomotor-like activity. We found that when a current pulse was applied during the spiking phase of one half-center, let’s call it #1, the consecutive burst durations (BDs) of that half-center increased throughout the paw-shake response, while BDs of the other half-center, let’s call it #2, only changed slightly. In contrast, the consecutive interburst intervals (IBIs) of half-center #1 changed little, while IBIs of half-center #2 increased. We demonstrated that this asymmetry between the half-centers depends on the phase of the locomotor-like rhythm at which the perturbation was applied. We suggest that the fast transient response reflects functional asymmetries of slow processes that underly the locomotor-like pattern; e.g., asymmetric levels of inactivation across the two half-centers for a slowly inactivating inward current. We compared model results with those of in-vivo paw-shake responses evoked in locomoting cats and found similar asymmetries. Electromyographic (EMG) BDs of anterior hindlimb muscles with flexor-related activity increased in consecutive paw-shake cycles, while BD of posterior muscles with extensor-related activity did not change, and vice versa for IBIs of anterior flexors and posterior extensors. We conclude that EMG activity patterns during paw-shaking are consistent with the proposed mechanism producing transient paw-shake-like bursting patterns found in our multistable HCO model. We suggest that the described asymmetry of paw-shaking responses could implicate a multifunctional CPG controlling both locomotion and paw-shaking.     

Correlated Conductance Parameters in Leech Heart Motor Neurons Contribute to Motor Pattern Formation

Neurons can have widely differing intrinsic membrane properties, in particular the density of specific conductances, but how these contribute to characteristic neuronal activity or pattern formation is not well understood. To explore the relationship between conductances, and in particular how they influence the activity of motor neurons in the well characterized leech heartbeat system, we developed a new multi-compartmental Hodgkin-Huxley style leech heart motor neuron model. To do so, we evolved a population of model instances, which differed in the density of specific conductances, capable of achieving specific output activity targets given an associated input pattern. We then examined the sensitivity of measures of output activity to conductances and how the model instances responded to hyperpolarizing current injections. We found that the strengths of many conductances, including those with differing dynamics, had strong partial correlations and that these relationships appeared to be linked by their influence on heart motor neuron activity. Conductances that had positive correlations opposed one another and had the opposite effects on activity metrics when perturbed whereas conductances that had negative correlations could compensate for one another and had similar effects on activity metrics.     

Modeling the leech heartbeat elemental oscillator I. Interactions of intrinsic and synaptic currents

We have developed a biophysical model of a pair of reciprocally inhibitory interneurons comprising an elemental heartbeat oscillator of the leech. We incorporate various intrinsic and synaptic ionic currents based on voltage-clamp data. Synaptic transmission between the interneurons consists of both a graded and a spike-mediated component. By using maximal conductances as parameters, we have constructed a canonical model whose activity appears close to the real neurons. Oscillations in the model arise from interactions between synaptic and intrinsic currents. The inhibitory synaptic currents hyperpolarize the cell, resulting in activation of a hyperpolarization-activated inward currentI _h and the removal of inactivation from regenerative inward currents. These inward currents depolarize the cell to produce spiking and inhibit the opposite cell. Spike-mediated IPSPs in the inhibited neuron cause inactivation of low-threshold Ca⁺⁺ currents that are responsible for generating the graded synaptic inhibition in the opposite cell. Thus, although the model cells can potentially generate large graded IPSPs, synaptic inhibition during canonical oscillations is dominated by the spike-mediated component.     

High Prevalence of Multistability of Rest States and Bursting in a Database of a Model Neuron

Flexibility in neuronal circuits has its roots in the dynamical richness of their neurons. Depending on their membrane properties single neurons can produce a plethora of activity regimes including silence, spiking and bursting. What is less appreciated is that these regimes can coexist with each other so that a transient stimulus can cause persistent change in the activity of a given neuron. Such multistability of the neuronal dynamics has been shown in a variety of neurons under different modulatory conditions. It can play either a functional role or present a substrate for dynamical diseases. We considered a database of an isolated leech heart interneuron model that can display silent, tonic spiking and bursting regimes. We analyzed only the cases of endogenous bursters producing functional half-center oscillators (HCOs). Using a one parameter (the leak conductance ()) bifurcation analysis, we extended the database to include silent regimes (stationary states) and systematically classified cases for the coexistence of silent and bursting regimes. We showed that different cases could exhibit two stable depolarized stationary states and two hyperpolarized stationary states in addition to various spiking and bursting regimes. We analyzed all cases of endogenous bursters and found that 18% of the cases were multistable, exhibiting coexistences of stationary states and bursting. Moreover, 91% of the cases exhibited multistability in some range of . We also explored HCOs built of multistable neuron cases with coexisting stationary states and a bursting regime. In 96% of cases analyzed, the HCOs resumed normal alternating bursting after one of the neurons was reset to a stationary state, proving themselves robust against this perturbation.     

Diversity and modulation of ionic conductances in leech neurons

A complete understanding of animal behavior at the cellular level requires detailed information on the intrinsic biophysical properties of neurons, muscles, and the synaptic connections they make. In the past 10 to 15 years, electrophysiological studies of leech neurons have revealed a diverse array of voltage-gated ionic conductances distinguished by their pharmacological sensitivity to classic ion channel blockers. Voltage-clamp studies have provided new information about the kinetics and voltage-dependence of Na⁺ conductances, several K⁺ currents, including I_A, I_K and I_K(Ca.)' and high- and low-voltage-gated Ca²⁺ conductances. These studies showed that the action potentials of most leech neurons result from the usual sequence of permeability changes to Na⁺, K⁺, and Ca²⁺ ions. They also added insight as to the role played by particular combinations of conductances in providing individual neurons with electrical properties appropriate for the particular information they encode. Evidence is accumulating on the modulatory actions of endogenous neurotransmitters such as FMRFamide, serotonin, and octopamine on motor behaviors in the animal. Parallel studies suggest that changes in behavior can be explained, at least in part, by the alteration of firing patterns of selected neurons and muscles resulting form modulation of multiple ion conductances. This makes the leech exceptionally attractive for neuroethological studies because it is one of the simplest organisms in which the methods of psychology and neurobiology can be combined. Information gathered from this animal will therefore increase our understanding regarding general principles underlying the cellular basis of behavior. © 1995 John Wiley & Sons, Inc.     

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.     

Modeling the leech heartbeat elemental oscillator II. Exploring the parameter space

In the previous paper, we described a model of the elemental heartbeat oscillator in the leech. Here, the parameters of our model are explored around the baseline canonical model. The maximal conductances of the currents and the reversal potential of the leak current are varied to reveal the effects of individual currents and the interaction between synaptic and intrinsic currents in the model. The model produces two distinct modes of oscillation as the parameters are varied, S-mode and G-mode. These two modes are defined, their origin is identified, and the parameter space is mapped into S-mode and G-mode oscillation and no oscillation. Finally, we will make predictions for how the period can be modulated in heart interneurons.     

Alterations produced in the quantitative characteristics of efferent activity parameters of the locomotor generator by decerebellation

Changes in the parameters of activity in hindlimb locomotor generators following decerebellation were quantified during experiments on decerebrate immobilized cats. Eliminating modulating cerebellar influences on nuclei of descending systems was found to lead to a slight increase in the length of activity in the flexor generator half-center and less intensive activity, as well as shortening of the period and more intensive activity in the extensor half-center, together with increased instability in generator operation, reduced statistical dependence between alterations in parameters of activity at the hindlimb half-center generators, and finally intensified effects of afferent inputs on generator activity. A comparison is drawn between the functional role of the spino-cerebellar loop in the operation of locomotor and scratch generators.A. A. Bogomolets Institute of Physiology, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Neirofiziologiya, Vol. 23, No. 2, pp. 142–150, March–April, 1991.     

The phantom burster model for pancreatic beta-cells

Pancreatic beta-cells exhibit bursting oscillations with a wide range of periods. Whereas periods in isolated cells are generally either a few seconds or a few minutes, in intact islets of Langerhans they are intermediate (10-60 s). We develop a mathematical model for beta-cell electrical activity capable of generating this wide range of bursting oscillations. Unlike previous models, bursting is driven by the interaction of two slow processes, one with a relatively small time constant (1-5 s) and the other with a much larger time constant (1-2 min). Bursting on the intermediate time scale is generated without need for a slow process having an intermediate time constant, hence phantom bursting. The model suggests that isolated cells exhibiting a fast pattern may nonetheless possess slower processes that can be brought out by injecting suitable exogenous currents. Guided by this, we devise an experimental protocol using the dynamic clamp technique that reliably elicits islet-like, medium period oscillations from isolated cells. Finally, we show that strong electrical coupling between a fast burster and a slow burster can produce synchronized medium bursting, suggesting that islets may be composed of cells that are intrinsically either fast or slow, with few or none that are intrinsically medium.     

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.     

Membrane electrical parameters in turtle bladder measured using impedance-analysis techniques

Summary Equivalent-circuit impedance analysis experiments were performed on the urinary bladders of freshwater turtles in order to quantify membrane ionic conductances and areas, and to investigate how changes in these parameters are associated with changes in the rate of proton secretion in this tissue. In all experiments, sodium reabsorption was inhibited thereby unmasking the electrogenic proton secretion process. We report the following: (1) transepithelial impedance is represented exceptionally well by a simple equivalent-circuit model, which results in estimates of the apical and basolateral membrane ionic conductances and capacitances; (2) when sodium transport is inhibited with mucosal amiloride and serosal ouabain, the apical and basolateral membrane conductances and capacitances exhibit a continual decline with time; (3) this decline in the membrane parameters is most likely caused by subtle time-dependent changes in cell volume, resulting in changes in the areas of the apical and basolateral membranes; (4) stable membrane parameters are obtained if the tissue is not treated with ouabain, and if the oncotic pressure of the serosal solution is increased by the addition of 2% albumin; (5) inhibition of proton secretion using acetazolamide in CO₂ and HCO ₃ ^– -free bathing solutions results in a decrease in the area of the apical membrane, with no significant change in its specific conductance; (6) stimulation of proton transport with CO₂ and HCO ₃ ^– -containing serosal solution results in an increase in the apical membrane area and specific conductance. These results show that our methods can be used to measure changes in the membrane electrophysiological parameters that are related to changes in the rate of proton transport. Notably, they can be used to quantify in the live tissue, changes in membrane area resulting from changes in the net rates of endocytosis and exocytosis which are postulated to be intimately involved in the regulation of proton transport.     

Elevated temperature alters the ionic dependence of amine-induced pacemaker activity in a conditional burster neuron

Summary The anterior burster neuron of the lobster (Panulirus interruptus) stomatogastric ganglion is a conditional burster that functions as the primary pacemaker for the pyloric motor network. When modulatory inputs to this cell are blocked, it loses its bursting properties and becomes quiescent. Applications of the monoamines, dopamine, octopamine or serotonin restore rhythmic bursting in this cell (Flamm and Harris-Warrick 1986). At 15 °C, serotonin- and octopamine-induced oscillations depend critically upon sodium entry (blocked by low sodium saline or tetrodotoxin); dopamine-induced oscillations depend upon calcium entry (blocked by reduced extracellular calcium; Harris-Warrick and Flamm 1987). We show here that the ionic dependence of amine-induced oscillations in the anterior burster cell differs at 15 and 21 °C. At 21 °C, all amines have the potential to induce rhythmic oscillations in saline containing tetrodotoxin. At the elevated temperature and in tetrodotoxin, both calcium and sodium currents are essential for the maintenance of dopamine-induced oscillaions; serotonin-induced oscillations do not depend upon either calcium or sodium alone; octopamine-induced oscillations do not depend upon calcium and show a variable dependence upon sodium. Thus, multiple ionic mechanisms, which vary with both the modulator and the ambient temperature, can be recruited to support rhythmic activity in a conditional burster neuron.Abbreviations AB anterior burster - PD pyloric dilator - PY pyloric constrictor - DA dopamine - 5HT serotonin - Oct octopamine - STG stomatogastric ganglion - TTX tetrodotoxin - GSP graded synaptic potential     

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.

     

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.     

Inferring and quantifying the role of an intrinsic current in a mechanism for a half-center bursting oscillation

This paper illustrates an informatic technique for inferring and quantifying the dynamic role of a single intrinsic current in a mechanism of neural bursting activity. We analyze the patterns of the most dominant currents in a model of half-center oscillation in the leech heartbeat central pattern generator. We find that the patterns of dominance change substantially over a cycle, allowing different local reductions to be applied to the model. The result is a hybrid dynamical systems model, which is a piecewise representation of the mechanism combining multiple vector fields and discrete state changes. The simulation of such a model tests explicit hypotheses about the mechanism and is a novel way to retain both mathematical clarity and scientific detail in answering mechanistic questions about a complex model. Several insights into the central mechanism of “escape-release” in the model are elucidated by this analysis and compared with previous studies. The broader application and extension of this technique is also discussed.     

Dual oscillator model of the respiratory neuronal network generating quantal slowing of respiratory rhythm

We developed a dual oscillator model to facilitate the understanding of dynamic interactions between the parafacial respiratory group (pFRG) and the preBötzinger complex (preBötC) neurons in the respiratory rhythm generation. Both neuronal groups were modeled as groups of 81 interconnected pacemaker neurons; the bursting cell model described by Butera and others [model 1 in Butera et al. (J Neurophysiol 81:382–397, 1999a)] were used to model the pacemaker neurons. We assumed (1) both pFRG and preBötC networks are rhythm generators, (2) preBötC receives excitatory inputs from pFRG, and pFRG receives inhibitory inputs from preBötC, and (3) persistent Na⁺ current conductance and synaptic current conductances are randomly distributed within each population. Our model could reproduce 1:1 coupling of bursting rhythms between pFRG and preBötC with the characteristic biphasic firing pattern of pFRG neurons, i.e., firings during pre-inspiratory and post-inspiratory phases. Compatible with experimental results, the model predicted the changes in firing pattern of pFRG neurons from biphasic expiratory to monophasic inspiratory, synchronous with preBötC neurons. Quantal slowing, a phenomena of prolonged respiratory period that jumps non-deterministically to integer multiples of the control period, was observed when the excitability of preBötC network decreased while strengths of synaptic connections between the two groups remained unchanged, suggesting that, in contrast to the earlier suggestions (Mellen et al., Neuron 37:821–826, 2003; Wittmeier et al., Proc Natl Acad Sci USA 105(46):18000–18005, 2008), quantal slowing could occur without suppressed or stochastic excitatory synaptic transmission. With a reduced excitability of preBötC network, the breakdown of synchronous bursting of preBötC neurons was predicted by simulation. We suggest that quantal slowing could result from a breakdown of synchronized bursting within the preBötC.     

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.     

Oscillations in K(ATP) conductance drive slow calcium oscillations in pancreatic β-cells

ATP-sensitive K⁺ (K(ATP)) channels were first reported in the β-cells of pancreatic islets in 1984, and it was soon established that they are the primary means by which the blood glucose level is transduced to cellular electrical activity and consequently insulin secretion. However, the role that the K(ATP) channels play in driving the bursting electrical activity of islet β-cells, which drives pulsatile insulin secretion, remains unclear. One difficulty is that bursting is abolished when several different ion channel types are blocked pharmacologically or genetically, making it challenging to distinguish causation from correlation. Here, we demonstrate a means for determining whether activity-dependent oscillations in K(ATP) conductance play the primary role in driving electrical bursting in β-cells. We use mathematical models to predict that if K(ATP) is the driver, then contrary to intuition, the mean, peak, and nadir levels of ATP/ADP should be invariant to changes in glucose within the concentration range that supports bursting. We test this in islets using Perceval-HR to image oscillations in ATP/ADP. We find that mean, peak, and nadir levels are indeed approximately invariant, supporting the hypothesis that oscillations in K(ATP) conductance are the main drivers of the slow bursting oscillations typically seen at stimulatory glucose levels in mouse islets. In conclusion, we provide, for the first time to our knowledge, causal evidence for the role of K(ATP) channels not only as the primary target for glucose regulation but also for their role in driving bursting electrical activity and pulsatile insulin secretion.