A simple integrative electrophysiological model of bursting GnRH neurons

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

A basic biophysical model for bursting neurons

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

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.     

Electrical connection between two bursting neurons inHelix pomatia

In some preparations of the CNS ofHelix pomatia, two neurons with bursting activity may be present in the right parietal ganglion, where usually there is only one bursting neuron RPal. If electrical activity of these neurons is recorded simultaneously, fluctuations of membrane potential are almost completely synchronized. Artificial depolarization and hyperpolarization of the membrane of one neuron caused depolarization or hyperpolarization of the other neuron. During long-term recording of the activity of both neurons synchronous modulation of their bursting activity was observed. Modulating factor (a peptide fraction obtained from the water-soluble part of snail brain homogenate) led to potentiation of the bursting activity of both neurons. It is concluded from the results of these experiments that two bursting RPal neurons, connected electrically with one another, may exist in the snail nervous system. In cases when the parameters of pacemaker activity of these two neurons are closely similar, electrical connection guarantees synchronization of their bursting activity and ensures a common frequency of changes in their membrane potential.     

An integrate-and-fire model for synchronized bursting in a network of cultured cortical neurons

It has been suggested that spontaneous synchronous neuronal activity is an essential step in the formation of functional networks in the central nervous system. The key features of this type of activity consist of bursts of action potentials with associated spikes of elevated cytoplasmic calcium. These features are also observed in networks of rat cortical neurons that have been formed in culture. Experimental studies of these cultured networks have led to several hypotheses for the mechanisms underlying the observed synchronized oscillations. In this paper, bursting integrate-and-fire type mathematical models for regular spiking (RS) and intrinsic bursting (IB) neurons are introduced and incorporated through a small-world connection scheme into a two-dimensional excitatory network similar to those in the cultured network. This computer model exhibits spontaneous synchronous activity through mechanisms similar to those hypothesized for the cultured experimental networks. Traces of the membrane potential and cytoplasmic calcium from the model closely match those obtained from experiments. We also consider the impact on network behavior of the IB neurons, the geometry and the small world connection scheme. Action Editor: David Golomb     

A calcium-based phantom bursting model for pancreatic islets

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

A mathematical model of adult GnRH neurons in mouse brain and its bifurcation analysis

GnRH neurons are hypothalamic neurons that secrete gonadotropin-releasing hormone (GnRH) which stimulates the release of gonadotropins, one of the crucial hormones for sexual development, fertility and maturation. A mathematical model was built to help elucidate the mechanisms underlying electrical bursting and synchronous [Ca²⁺] transients in GnRH neurons (Lee et al., 2010). The model predicted that bursting in GnRH neurons (at least of the short-bursting type) requires the existence of a [Ca²⁺]-dependent slow after-hyperpolarisation current (sI_AHP-UCL), and this predicted current was found experimentally. GnRH behaviour under a wide range of conditions (inhibition of Na⁺ channels, IP₃ receptors, [Ca²⁺]-dependent K⁺ channels, or Ca²⁺ pumps, or in the presence of zero extracellular [Ca²⁺]) is successfully reproduced by the model. In this paper, a simplified version of the previous model, with the same qualitative behaviour, is constructed and studied using timescale separation techniques and bifurcation analysis.     

A unified model for budburst of trees

Accurate plant phenology (seasonal plant activity driven by environmental factors) models are vital tools for ecosystem simulation models and for predicting the response of ecosystems to climate change. Since the early 1970s, efforts have concentrated on predicting phenology of the temperate and boreal forests because they represent one-third of the carbon captured in plant ecosystems and they are the principal ecosystems with seasonal patterns of growth on Earth (one-fifth of the plant ecosystems area). Numerous phenological models have been developed to predict the growth timing of temperate or boreal trees. They are in general empirical, nonlinear and non-nested. For these reasons they are particularly difficult to fit, to test and to compare with each other. The methodological difficulties as well as the diversity of models used have greatly slowed down their improvement. The aim of this study was to show that the most widely used models simulating vegetative or reproductive phenology of trees are particular cases of a more general model. This unified model has three main advantages. First, it allows for a direct estimation of (i) the response of bud growth to either chilling or forcing temperatures and (ii) the periods when these temperatures affect the bud growth. Second, it can be simplified according to standard statistical tests for any particular species. Third, it provides a standardized framework for phenological models, which is essential for comparative studies as well as for robust model identification.     

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.     

Ionic requirements for bursting activity in lobster stomatogastric neurons

Summary The ionic requirements for bursting activity have been investigated in the electrically coupled PD-AB cells group of the Stomatogastric ganglion in the lobster.The passive electrical properties and coupling parameters have been determined in either current or voltage clamp conditions. In voltage clamped cells, the current displayed slow inward transients with superimposed fast transients corresponding respectively to the slow waves and spikes of the coupled undamped cells. The amplitude and frequency of the slow transients were reduced upon hyperpolarization.Cyclic conductance changes were observed with short current pulses, the coupling ratio also changes cyclically being lower during the bursts and slowly increasing during the interburst interval.The slow wave amplitude increased in low K-saline. The post-burst hyperpolarization but not the top level of the wave behaved like a potassium electrode for [K]_o higher than 10 mM/l.TEA at low concentration (1 to 5 mM/l) increased the slow wave amplitude by lifting its top level by 10 to 20 mV. The post-burst hyperpolarization remained almost unchanged and its K-dependence was not altered by TEA.Low Na-saline reduced the slow wave amplitude (6 to 11 mV per decade). The Na-dependence increased in the presence of TEA. Slow waves devoid of spikes persisted in 10% Na saline containing TEA. 10^–9 M/1 TTX blocked the spikes. 10^–7 M/1 TTX blocked the slow waves.Mg-free saline had no effect on the slow wave. In Ca-free saline the cells depolarized and the bursting activity tended to vanish. Repolarization with current led to long lasting slow waves deprived of post-burst hyperpolarization. The bursting ceased when EDTA was added to the Ca-free saline.Cobalt (up to 10 mM/l) was similar to Ca-free saline in its effects; lengthening of the wave and blockage of the repolarization. Replacing Ba for Ca produced large (up to 70 mV) slow waves which were reduced by Co and Ca.Bistable states were observed in various experimental conditions. It is concluded that the slow waves are produced by activation of sodium and calcium currents. The amplitude of the slow wave is modulated by the simultaneous activation of a TEA-sensitive K current. The repolarization is caused by increased K current activated by the inward calcium current. The slow pacemaker potential in the interburst interval corresponds to the slow disappearance of the K current.This work was supported by N.I.H. grant no. 09322, NSF grant no. 00250, and a Guggenheim Foundation Fellowship to A.D.S. and by the CNRS and a DGRST grant no. 16501891 to M. Gola. We are grateful to Stuart Thompson and Felix Strumwasser for helpful comments and to Barbara McLean for technical assistance.     

A unified model for apical caspase activation

Apoptosis is orchestrated by the concerted action of caspases, activated in a minimal two-step proteolytic cascade. Existing data suggests that apical caspases are activated by adaptor-mediated clustering of inactive zymogens. However, the mechanism by which apical caspases achieve catalytic competence in their recruitment/activation complexes remains unresolved. We explain that proximity-induced activation of apical caspases is attributable to dimerization. Internal proteolysis does not activate these apical caspases but is a secondary event resulting in partial stabilization of activated dimers. Activation of caspases-8 and -9 occurs by dimerization that is fully recapitulated in vitro by kosmotropes, salts with the ability to stabilize the structure of proteins. Further, single amino acid substitutions at the dimer interface abrogate the activity of caspases-8 and -9 introduced into recipient mammalian cells. We propose a unified caspase activation hypothesis whereby apical caspases are activated by dimerization of monomeric zymogens.     

Predictive features of persistent activity emergence in regular spiking and intrinsic bursting model neurons

Proper functioning of working memory involves the expression of stimulus-selective persistent activity in pyramidal neurons of the prefrontal cortex (PFC), which refers to neural activity that persists for seconds beyond the end of the stimulus. The mechanisms which PFC pyramidal neurons use to discriminate between preferred vs. neutral inputs at the cellular level are largely unknown. Moreover, the presence of pyramidal cell subtypes with different firing patterns, such as regular spiking and intrinsic bursting, raises the question as to what their distinct role might be in persistent firing in the PFC. Here, we use a compartmental modeling approach to search for discriminatory features in the properties of incoming stimuli to a PFC pyramidal neuron and/or its response that signal which of these stimuli will result in persistent activity emergence. Furthermore, we use our modeling approach to study cell-type specific differences in persistent activity properties, via implementing a regular spiking (RS) and an intrinsic bursting (IB) model neuron. We identify synaptic location within the basal dendrites as a feature of stimulus selectivity. Specifically, persistent activity-inducing stimuli consist of activated synapses that are located more distally from the soma compared to non-inducing stimuli, in both model cells. In addition, the action potential (AP) latency and the first few inter-spike-intervals of the neuronal response can be used to reliably detect inducing vs. non-inducing inputs, suggesting a potential mechanism by which downstream neurons can rapidly decode the upcoming emergence of persistent activity. While the two model neurons did not differ in the coding features of persistent activity emergence, the properties of persistent activity, such as the firing pattern and the duration of temporally-restricted persistent activity were distinct. Collectively, our results pinpoint to specific features of the neuronal response to a given stimulus that code for its ability to induce persistent activity and predict differential roles of RS and IB neurons in persistent activity expression.     

A unified model for complex segregation analysis.

Various methods have been proposed for statistical inference of major genes by segregation analysis of human familial data. An attempt is made to resolve some divergences that have occurred in this context by the consideration of a unified model, with some practical applications.     

Possible mechanism of bursting suppression in nociceptive neurons

The use of the mathematical model of rat nociceptive neuron membrane allowed us to predict a new mechanism of suppression of ectopic bursting discharges, which arise in neurons of dorsal root ganglia and are one of the causes of neuropathic pain. The treatment with comenic acid leads to switching off the ectopic bursting discharges due to a decrease in the effective charge transferring via the activation gating structure of the slow sodium channels (Na _V1.8a). Comenic acid is a drug substance of a new non-opioid analgesic [1] Thus, this analgesic not only reduces the frequency of rhythmic discharges of nociceptive neuron membrane [2] but also it suppresses its ectopic bursting discharges.     

Interval analysis of bursting discharges in aplysia neurons

Time intervals of 12 records of bursting discharges in Aplysia neurons were analysed by digital computer to determine the interrelations between the burst period, the interburst interval and the burst duration. The effects of membrane potential changes on the parameters of bursting discharges were examined also. A low correlation was found between burst duration and burst period in the majority of cases, and this was interpreted as an indication of probable independence between the mechanisms governing these parameters. Also, a specific temporal organization of interspike intervals seems to be present in each type of neuron. The results suggest that the mechanism governing the burst period is characterized by a slow membrane potential oscillation resembling that observed in bursting neurons when actions potentials are blocked by tetrodotoxin. The burst duration would be determined by the response of the neuron to suprathreshold depolarization.     

Calcium current subtypes in GnRH neurons

Calcium plays roles in excitability, rhythm generation, and neurosecretion. Identifying channel subtypes that regulate calcium influx is thus important to understanding rhythmic GnRH secretion, which is a prerequisite for reproduction. Whole-cell voltage-clamp recordings were made from short-term dissociated GnRH adult ovariectomized (OVX) mice (n = 21) to identify channel subtypes that carry calcium current using selective channel blockers and voltage characteristics. Low-voltage activated (LVA) currents were not observed in 42 GnRH neurons tested, although most non-GnRH neurons (4/6) displayed LVA current. The L-type component of the high-voltage activated (HVA) calcium current was 25% +/- 2%. The remaining HVA calcium current passed through N-type (27% +/- 3%), P-type (15% +/- 1%), Q-type (18% +/- 3%), and R-type (15% +/- 1%) channels. Because these data differ substantially from reports on cultured GnRH neurons, which may represent reproductively immature models, we also examined GnRH neurons from gonadal-intact young (Postnatal Days 4-10, n = 8 mice) mice. LVA currents were still rare (2/28) in young mice. Although the same HVA components were observed, the proportions were shifted toward significantly more L-type and less N-type current, suggesting a possible developmental shift in calcium currents in GnRH neurons. These data suggest that calcium channel subtypes in GnRH neurons prepared in the short term from brain slices differ substantially from those in long-term cultured GnRH models. These findings provide a vital foundation to examine the role of calcium channels in the secretory and rhythmic machinery of GnRH neurons.     

Ionic mechanisms of endogenous bursting in CA3 hippocampal pyramidal neurons: a model study

A critical property of some neurons is burst firing, which in the hippocampus plays a primary role in reliable transmission of electrical signals. However, bursting may also contribute to synchronization of electrical activity in networks of neurons, a hallmark of epilepsy. Understanding the ionic mechanisms of bursting in a single neuron, and how mutations associated with epilepsy modify these mechanisms, is an important building block for understanding the emergent network behaviors. We present a single-compartment model of a CA3 hippocampal pyramidal neuron based on recent experimental data. We then use the model to determine the roles of primary depolarizing currents in burst generation. The single compartment model incorporates accurate representations of sodium (Na(+)) channels (Na(V)1.1) and T-type calcium (Ca(2+)) channel subtypes (Ca(V)3.1, Ca(V)3.2, and Ca(V)3.3). Our simulations predict the importance of Na(+) and T-type Ca(2+) channels in hippocampal pyramidal cell bursting and reveal the distinct contribution of each subtype to burst morphology. We also performed fast-slow analysis in a reduced comparable model, which shows that our model burst is generated as a result of the interaction of two slow variables, the T-type Ca(2+) channel activation gate and the Ca(2+)-dependent potassium (K(+)) channel activation gate. The model reproduces a range of experimentally observed phenomena including afterdepolarizing potentials, spike widening at the end of the burst, and rebound. Finally, we use the model to simulate the effects of two epilepsy-linked mutations: R1648H in Na(V)1.1 and C456S in Ca(V)3.2, both of which result in increased cellular excitability.