Regulation of Electrical Bursting in a Spatiotemporal Model of a GnRH Neuron

Gonadotropin-releasing hormone (GnRH) neurons are hypothalamic neurons that control the pulsatile release of GnRH that governs fertility and reproduction in mammals. The mechanisms underlying the pulsatile release of GnRH are not well understood. Some mathematical models have been developed previously to explain different aspects of these activities, such as the properties of burst action potential firing and their associated Ca²⁺ transients. These previous studies were based on experimental recordings taken from the soma of GnRH neurons. However, some research groups have shown that the dendrites of GnRH neurons play very important roles. In particular, it is now known that the site of action potential initiation in these neurons is often in the dendrite, over 100 μm from the soma. This raises an important question. Since some of the mechanisms for controlling the burst length and interburst interval are located in the soma, how can electrical bursting be controlled when initiated at a site located some distance from these controlling mechanisms? In order to answer this question, we construct a spatio-temporal mathematical model that includes both the soma and the dendrite. Our model shows that the diffusion coefficient for the spread of electrical potentials in the dendrite is large enough to coordinate burst firing of action potentials when the initiation site is located at some distance from the soma.     

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 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.     

Hodgkin–Huxley type modelling and parameter estimation of GnRH neurons

In this paper a simple one compartment Hodgkin–Huxley type electrophysiological model of GnRH neurons is presented, that is able to reasonably reproduce the most important qualitative features of the firing pattern, such as baseline potential, depolarization amplitudes, sub-baseline hyperpolarization phenomenon and average firing frequency in response to excitatory current. In addition, the same model provides an acceptable numerical fit of voltage clamp (VC) measurement results. The parameters of the model have been estimated using averaged VC traces, and characteristic values of measured current clamp traces originating from GnRH neurons in hypothalamic slices. The resulting parameter values show a good agreement with literature data in most of the cases. Applying parametric changes, which lead to the increase of baseline potential and enhance cell excitability, the model becomes capable of bursting. The effects of various parameters to burst length have been analyzed by simulation.     

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.     

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     

Understanding calcium homeostasis in postnatal gonadotropin-releasing hormone neurons using cell-specific Pericam transgenics

The gonadotropin-releasing hormone (GnRH) neurons are the key output cells of a complex neuronal network controlling fertility in mammals. To examine calcium homeostasis in postnatal GnRH neurons, we generated a transgenic mouse line in which the genetically encodable calcium indicator ratiometric Pericam (rPericam) was targeted to the GnRH neurons. This mouse model enabled real-time imaging of calcium concentrations in GnRH neurons in the acute brain slice preparation. Investigations in GnRH-rPericam mice revealed that GnRH neurons exhibited spontaneous, long-duration (~8s) calcium transients. Dual electrical-calcium recordings revealed that the calcium transients were correlated perfectly with burst firing in GnRH neurons and that calcium transients in GnRH neurons regulated two calcium-activated potassium channels that, in turn, determined burst firing dynamics in these cells. Curiously, the occurrence of calcium transients in GnRH neurons across puberty or through the estrous cycle did not correlate well with the assumption that GnRH neuron burst firing was contributory to changing patterns of pulsatile GnRH release at these times. The GnRH-rPericam mouse was also valuable in determining differential mechanisms of GABA and glutamate control of calcium levels in GnRH neurons as well as effects of G-protein-coupled receptors for GnRH and kisspeptin. The simultaneous measurement of calcium levels in multiple GnRH neurons was hampered by variable rPericam fluorescence in different GnRH neurons. Nevertheless, in the multiple recordings that were achieved no evidence was found for synchronous calcium transients. Together, these observations show the great utility of transgenic targeting strategies for investigating the roles of calcium with specified neuronal cell types.     

Courtship interactions stimulate rapid changes in GnRH synthesis in male ring doves

Many birds and mammals show changes in the hypothalamo-pituitary-gonadal (HPG) axis in response to social or sexual interactions between breeding partners. While alterations in GnRH neuronal activity play an important role in stimulating these changes, it remains unclear if acute behaviorally-induced alterations in GnRH release are accompanied by parallel changes in GnRH synthesis. To investigate this relationship, we examined changes in the activity of GnRH neurons in the brains of male ring doves following brief periods of courtship interactions with females. Such interactions have been previously shown to increase plasma LH in courting male doves at 24 h, but not at 1 h, after pairing with females. In the first study, males allowed to court females for 2 h had 60% more cells that showed immunocytochemical labeling for GnRH-I in the preoptic area (POA) of the hypothalamus than did control males that remained isolated from females. To determine whether an increase in GnRH gene expression preceded this increase in GnRH immunoreactivity in the POA, changes in the number of cells with detectable GnRH-I mRNA in the POA were measured by in situ hybridization following a 1 h period of courtship interactions with females. In this second study, courting males exhibited 40% more cells with GnRH-I in this region than did isolated control males. GnRH-immunoreactive neurons in two other diencephalic regions failed to show these courtship-induced changes. Plasma LH was not elevated after 1 or 2 h of courtship. These results demonstrate that the release of GnRH-I in the POA that is presumably responsible for courtship-induced pituitary and gonadal activation is accompanied by a rapid increase in GnRH synthesis that occurs before plasma LH levels increase. We suggest that this increase in GnRH synthesis is necessary to support the extended period of HPG axis activation that is seen in this species during the 5–10 day period of courtship and nest building activity.     

Dynamics of in-phase and anti-phase bursting in the coupled pre-Bötzinger complex cells

Activity of neurons in the pre-Bötzinger complex within the mammalian brain stem has an important role in the generation of respiratory rhythms. Previous experimental results have shown that the dynamics of sodium and calcium within each cell may be responsible for various bursting mechanisms. In this paper, we study the bursting dynamics of the two-coupled pre-Bötzinger complex neurons. Using a combination of fast-slow decomposition and two-parameter bifurcation analysis, we explore the possible forms of dynamics that the model network can produce as well the transitions of in-phase and anti-phase bursting respectively.     

Courtship interactions stimulate rapid changes in GnRH synthesis in male ring doves

Many birds and mammals show changes in the hypothalamo-pituitary-gonadal (HPG) axis in response to social or sexual interactions between breeding partners. While alterations in GnRH neuronal activity play an important role in stimulating these changes, it remains unclear if acute behaviorally-induced alterations in GnRH release are accompanied by parallel changes in GnRH synthesis. To investigate this relationship, we examined changes in the activity of GnRH neurons in the brains of male ring doves following brief periods of courtship interactions with females. Such interactions have been previously shown to increase plasma LH in courting male doves at 24 h, but not at 1 h, after pairing with females. In the first study, males allowed to court females for 2 h had 60% more cells that showed immunocytochemical labeling for GnRH-I in the preoptic area (POA) of the hypothalamus than did control males that remained isolated from females. To determine whether an increase in GnRH gene expression preceded this increase in GnRH immunoreactivity in the POA, changes in the number of cells with detectable GnRH-I mRNA in the POA were measured by in situ hybridization following a 1 h period of courtship interactions with females. In this second study, courting males exhibited 40% more cells with GnRH-I in this region than did isolated control males. GnRH-immunoreactive neurons in two other diencephalic regions failed to show these courtship-induced changes. Plasma LH was not elevated after 1 or 2 h of courtship. These results demonstrate that the release of GnRH-I in the POA that is presumably responsible for courtship-induced pituitary and gonadal activation is accompanied by a rapid increase in GnRH synthesis that occurs before plasma LH levels increase. We suggest that this increase in GnRH synthesis is necessary to support the extended period of HPG axis activation that is seen in this species during the 5–10 day period of courtship and nest building activity.     

Calcium Involvement in Regulation of Neuronal Bursting in Disinhibited Neuronal Networks: Insights from Calcium Studies in a Spherical Cell Model

Cytosolic calcium is involved in the regulation of many intracellular processes. Intracellular calcium may therefore potentially affect the behavior of both single neurons and synaptically connected neuronal assemblies. In computer model studies, we investigated calcium dynamics in spherical neurons during periods of recurrent neuronal bursting that were simulated in a disinhibited neuronal network. The model takes into account calcium influx via voltage-gated calcium channels, extrusion through the cell membrane, and binding to two different buffers representing fixed and mobile endogenous calcium buffers. Throughout the duration of the simulated recurrent neuronal bursting, the concentration of free fixed buffers shows a hyperbolic decrease in time at a rate that is not uniform inside a neuron. Recurrent calcium influxes associated with bursting lead to the formation of gradients in the concentration of the fixed buffer in the radial direction, and are accompanied by the redistribution of mobile buffers acting to compensate for these gradients. Simulated intracellular calcium transients have a slow component characterized by a gradual increase in the calcium baseline level that reaches a plateau 120-200 s after the onset of recurrent bursting. Using this model, we demonstrate what we believe is a novel mechanism of regulation of network excitability that occurs in conditions of prolonged and recurrent neuronal bursting in disinhibited networks. This mechanism is expressed via interaction of calcium clearance systems inside neurons with calcium-dependent potassium regulation of neuronal excitability in membranes. This is a network phenomenon because it arises largely by synaptic interactions. Therefore, it can serve as a network safety mechanism to prevent excessive and uncontrolled neuronal firing resulting from the lack of inhibition or after acute suppression of the inhibitory drive.     

Firing pattern of bursting neurons under sinusoidal drive in mean-field modeling

Bursting has been observed in many sensory neurons, and is thought to be important in neural signaling, sleep, and some disorders of the brain. Bursting neurons have been studied via various types of conductance-based models at the single-neuron level. Important features of bursting have been reproduced by this type of model, but it is not certain how well the behavior of populations of bursting neurons can be represented solely by that of individual neurons. To study bursting neurons at the population level, a conductance-based model is incorporated into a mean-field model to yield a mean-field bursting model. The responses of the model to sinusoidal inputs are studied, showing that neurons with various different initial states are capable of phase-locked or intermittent firing, depending on their baseline voltage. Furthermore, depending on this voltage, the bursting frequency either slaves to the original unperturbed bursting frequency or approaches a steady value when the external driving frequency increases. Finally, use of white noise perturbations shows that the bursting frequency of the neurons remains the same even under a more general external stimulus.     

Evidence for multiple calcium response mechanisms in mammalian olfactory receptor neurons

Olfactory receptor neurons employ a diversity of signaling mechanisms for transducing and encoding odorant information. The simultaneous activation of subsets of receptor neurons provides a complex pattern of activation in the olfactory bulb that allows for the rapid discrimination of odorant mixtures. While some transduction elements are conserved among many species, some species-specificity occurs in certain features that may relate to their particular physiology and ecological niche. However, studies of olfactory transduction have been limited to a relatively small number of vertebrate and invertebrate species. To better understand the diversity and evolution of olfactory transduction mechanisms, we studied stimulus-elicited calcium fluxes in olfactory neurons from a previously unstudied mammalian species, the domestic cat. Isolated cells from cat olfactory epithelium were stimulated with odorant mixtures and biochemical agents, and cell responses were measured with calcium imaging techniques. Odorants elicited either increases or decreases in intracellular calcium; odorant-induced calcium increases were mediated either by calcium fluxes through the cell membrane or by mobilization of intracellular stores. Individual cells could employ multiple signaling mechanisms to mediate responses to different odorants. The physiological features of these olfactory neurons suggest greater complexity than previously recognized in the role of peripheral neurons in encoding complex odor stimuli. The investigation of novel and unstudied species is important for understanding the mechanisms of odorant signaling that apply to the olfactory system in general and suggests both broadly conserved and species-specific evolutionary adaptations.     

To Burst or Not to Burst?

It is well known that some neurons tend to fire packets of action potentials followed by periods of quiescence (bursts) while others within the same stage of sensory processing fire in a tonic manner. However, the respective computational advantages of bursting and tonic neurons for encoding time varying signals largely remain a mystery. Weakly electric fish use cutaneous electroreceptors to convey information about sensory stimuli and it has been shown that some electroreceptors exhibit bursting dynamics while others do not. In this study, we compare the neural coding capabilities of tonically firing and bursting electroreceptor model neurons using information theoretic measures. We find that both bursting and tonically firing model neurons efficiently transmit information about the stimulus. However, the decoding mechanisms that must be used for each differ greatly: a non-linear decoder would be required to extract all the available information transmitted by the bursting model neuron whereas a linear one might suffice for the tonically firing model neuron. Further investigations using stimulus reconstruction techniques reveal that, unlike the tonically firing model neuron, the bursting model neuron does not encode the detailed time course of the stimulus. A novel measure of feature detection reveals that the bursting neuron signals certain stimulus features. Finally, we show that feature extraction and stimulus estimation are mutually exclusive computations occurring in bursting and tonically firing model neurons, respectively. Our results therefore suggest that stimulus estimation and feature extraction might be parallel computations in certain sensory systems rather than being sequential as has been previously proposed.     

Alzheimer's beta-amyloid, human islet amylin, and prion protein fragment evoke intracellular free calcium elevations by a common mechanism in a hypothalamic GnRH neuronal cell line

A growing number of reports suggest that elevated levels of extracellular Alzheimer's beta-amyloid protein alter the homeostasis of free [Ca(2+)](i) in different cell types of the mammalian brain. In line with these results, we have previously shown that AbetaP[1-40] forms cation-selective channels (Ca(2+) included) across artificial planar bilayers formed from acidic phospholipids and across excised membrane patches from immortalized hypothalamic GnRH neurons (GT1-7 cells), suggesting that the nonregulated Ca(2+)-influx through these spontaneously formed "amyloid channels" may provide a mechanism to explain its toxicity (1). We have now found and report here that the application of AbetaP[1-40] to GT1-7 neurons consistently elevates [Ca(2+)](i) levels. We also found that human islet amylin and the prion protein fragment (PrP106-126), peptides that acquire beta-pleated sheet conformation in water solutions and have been reported to form ion channels across planar bilayer membranes, also increase cytosolic free calcium in GT1-7 neurons. Searching for protective agents, we found that soluble cholesterol, known to decrease the fluidity of the cell membrane, inhibits AbetaP[1-40]-evoked [Ca(2+)](i) rise. These results suggest that unregulated Ca(2+) entry across amyloid channels may be a common mechanism causing cell death, not only in diseases of the third age, including Alzheimer's disease and type 2 diabetes mellitus, but also in prion-induced diseases.     

Two types of independent bursting mechanisms in inspiratory neurons: an integrative model

The network of coupled neurons in the pre-Bötzinger complex (pBC) of the medulla generates a bursting rhythm, which underlies the inspiratory phase of respiration. In some of these neurons, bursting persists even when synaptic coupling in the network is blocked and respiratory rhythmic discharge stops. Bursting in inspiratory neurons has been extensively studied, and two classes of bursting neurons have been identified, with bursting mechanism depends on either persistent sodium current or changes in intracellular Ca²⁺, respectively. Motivated by experimental evidence from these intrinsically bursting neurons, we present a two-compartment mathematical model of an isolated pBC neuron with two independent bursting mechanisms. Bursting in the somatic compartment is modeled via inactivation of a persistent sodium current, whereas bursting in the dendritic compartment relies on Ca²⁺ oscillations, which are determined by the neuromodulatory tone. The model explains a number of conflicting experimental results and is able to generate a robust bursting rhythm, over a large range of parameters, with a frequency adjusted by neuromodulators.     

A mathematical model for LH release in response to continuous and pulsatile exposure of gonadotrophs to GnRH

In a previous study, a model was developed to investigate the release of luteinizing hormone (LH) from pituitary cells in response to a short pulse of gonadotropin-releasing hormone (GnRH). The model included: binding of GnRH to its receptor (R), dimerization and internalization of the hormone receptor complex, interaction with a G protein, production of inositol 1,4,5-trisphosphate (IP₃), release of calcium from the endoplasmic reticulum (ER), entrance of calcium into the cytosol via voltage gated membrane channels, pumping of calcium out of the cytosol via membrane and ER pumps, and release of LH. The extended model, presented in this paper, also includes the following physiologically important phenomena: desensitization of calcium channels; internalization of the dimerized receptors and recycling of some of the internalized receptors; an increase in G _q concentration near the plasma membrane in response to receptor dimerization; and basal rates of synthesis and degradation of the receptors. With suitable choices of the parameters, good agreement with a variety of experimental data of the LH release pattern in response to pulses of various durations, repetition rates, and concentrations of GnRH were obtained. The mathematical model allows us to assess the effects of internalization and desensitization on the shapes and time courses of LH response curves.     

How is bursting mode maintained under the action of exo- and endogenous factors?

The investigation is concerned with slow oscillations of the intracellular calcium concentration, fast oscillations of the membrane potential, and their interactions during bursting generation. The question associated with functioning of the bursting mode is considered: how is the condition of simultaneous existence of the slow and fast membrane potential oscillations (necessary for the bursting mode) realized under exo- and endogenous actions that result in changes in the neuron membrane parameters? To answer the question the regions of existence of fast, slow and bursting oscillations have been constructed on different parameter planes. The study has revealed that in spite of the fact the regions of the existence of the slow and fast oscillations are sensitive to parameter variations, the condition of the bursting mode existence remains due to invariance of the spatial vicinity of these regions. Furthermore, it has been demonstrated that there exists the possibility to enhance the maximal conductances of the calcium-dependent currents simultaneously such that even strong variations of their values do not induce changes of the oscillations mode (i.e. it remains bursting).     

From Plateau to Pseudo-Plateau Bursting: Making the Transition

Bursting electrical activity is ubiquitous in excitable cells such as neurons and many endocrine cells. The technique of fast/slow analysis, which takes advantage of time scale differences, is typically used to analyze the dynamics of bursting in mathematical models. Two classes of bursting oscillations that have been identified with this technique, plateau and pseudo-plateau bursting, are often observed in neurons and endocrine cells, respectively. These two types of bursting have very different properties and likely serve different functions. This latter point is supported by the divergent expression of the bursting patterns into different cell types, and raises the question of whether it is even possible for a model for one type of cell to produce bursting of the type seen in the other type without large changes to the model. Using fast/slow analysis, we show here that this is possible, and we provide a procedure for achieving this transition. This suggests that the design principles for bursting in endocrine cells are just quantitative variations of those for bursting in neurons.     

Synchronous and asynchronous bursting states: role of intrinsic neural dynamics

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