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Modeling electrical spiking, bursting and calcium dynamics in gonadotropin releasing hormone (GnRH) secreting… Fletcher, Patrick Allen 2008

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Modeling Electrical Spiking, Bursting, and Calcium Dynamics in Gonadotropin-Releasing Hormone (GnRH) Secreting Neurons by Patrick Allen Fletcher B.Sc., The University of British Columbia, 2005 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in The Faculty of Graduate Studies (Mathematics)  The University Of British Columbia (Vancouver) October, 2008 c Patrick Allen Fletcher 2008  Abstract The plasma membrane electrical activities of neurons that secrete gonadotropin releasing hormone (GnRH), referred to as GnRH neurons hereafter, have been studied extensively. A couple of mathematical models have been developed previously to explain different aspects of these activities including spontaneous spiking and responses to stimuli such as current injections, GnRH, thapsigargin (Tg) and apamin. The goal of this paper is to develop one single, minimal model that accounts for the experimental results reproduced by previously existing models and results that were not accounted for by these models. The latter includes two types of membrane potential bursting mechanisms and the associated calcium oscillations in the cytosol. One of them has not been reported in experimental literatures on GnRH neurons and is thus regarded as a model prediction. Other improvements achieved in this model include the incorporation of a more detailed description of calcium dynamics in a three dimensional cell body with the ion channels evenly distributed on the cell surface. Although the model is mainly based on data collected in cultured GnRH cell lines, we show that it is capable of explaining some properties of GnRH neurons observed in several of other preparations including mature GnRH neurons in hypothalamic slices. One potential explanation is suggested. A phenomenological reduction of this model into a simplified form is presented. The simplified model will facilitate the study of the roles of plasma membrane electrical activities on the pulsatile release of GnRH by these neurons when it is coupled with a model of pulsatile GnRH release based on the autoregulation mechanism.  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  v  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vii  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  viii  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ix  Statement of Co-Authorship . . . . . . . . . . . . . . . . . . . . . . . . . . .  x  1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  Abstract  List of Tables  List of Abbreviations  Dedication  1.1  GnRH Pulsatility is Required for Reproduction . . . . . . . . . . . . . .  1  1.2  GnRH Neuron Anatomy  2  1.2.1 1.3  . . . . . . . . . . . . . . . . . . . . . . . . . .  Experimental Models  . . . . . . . . . . . . . . . . . . . . . . . .  3  GnRH Neuron Physiology . . . . . . . . . . . . . . . . . . . . . . . . . .  5  1.3.1  Electrical Activities . . . . . . . . . . . . . . . . . . . . . . . . .  5  1.3.2  Calcium Oscillations and Electrical Bursting . . . . . . . . . . .  5  1.3.3  GnRH Secretion . . . . . . . . . . . . . . . . . . . . . . . . . . .  7  1.3.4  Autocrine Feedback and GnRH Receptor Signaling  . . . . . . .  7  . . . . . . . . . .  11  1.4  Existing Models of Electrical and Calcium Dynamics  1.5  Objectives  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  13  1.6  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  16  2 Modeling Electrical Spiking and Bursting of GnRH Neurons . . . .  22  2.1  Introduction  2.2  Objectives of the Paper and the Model 2.2.1  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Experimental Observations the Model Aims to Reproduce  . . .  22 24 24  iii  2.2.2 2.3  The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  26  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  28  Results 2.3.1  Responses to Current Injections  . . . . . . . . . . . . . . . . . .  28  2.3.2  Responses to GnRH . . . . . . . . . . . . . . . . . . . . . . . . .  29  2.3.3  Responses to Thapsigargin . . . . . . . . . . . . . . . . . . . . .  30  2.3.4  Responses to Forskolin  . . . . . . . . . . . . . . . . . . . . . . .  31  2.3.5  Bursting Mechanisms . . . . . . . . . . . . . . . . . . . . . . . .  31  2.4  The Simplified Model  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  32  2.5  Discussion  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  33  2.6  Figures  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  37  2.7  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  46  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  50  3.1  Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  50  3.2  Unexplored Features of GnRH Neurons . . . . . . . . . . . . . . . . . .  51  3.2.1  Electrophysiological Details . . . . . . . . . . . . . . . . . . . . .  51  3.2.2  Second Messenger Signaling  52  3 Conclusion  3.3  3.4  . . . . . . . . . . . . . . . . . . . .  Future Directions for Modeling of GnRH Neurons  . . . . . . . . . . . .  3.3.1  Characterization of Calcium Oscillations and Bursting  3.3.2  Stimulus-Secretion Coupling  3.3.3  A Single Cell Model for GnRH Pulsatility  3.3.4  A Network Model of GnRH Pulsatility  54  . . . . .  54  . . . . . . . . . . . . . . . . . . . .  55  . . . . . . . . . . . .  55  . . . . . . . . . . . . . .  55  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  57  Appendices . . . . . . . . . . . . . . . .  62  A.1 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  66  A Model Description and Parameter Values  iv  List of Tables A.1 Table of Some Standard Model Parameter Values. . . . . . . . . . . . .  65  v  List of Figures 2.1  Responses of GT1 Neurons to GnRH, Tg, and Apamin . . . . . . . . . .  37  2.2  Diagram of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .  38  2.3  Model Response to Current Injection . . . . . . . . . . . . . . . . . . . .  39  2.4  Model Response to GnRH and Apamin . . . . . . . . . . . . . . . . . .  40  2.5  Model Response to Tg and Apamin . . . . . . . . . . . . . . . . . . . . .  41  2.6  Model Response to Forskolin . . . . . . . . . . . . . . . . . . . . . . . .  42  [Ca2+ ]i  2.7  IP3 Induced  Oscillations . . . . . . . . . . . . . . . . . . . . . . .  43  2.8  Store-Operated Bursting . . . . . . . . . . . . . . . . . . . . . . . . . . .  44  2.9  Simplified Model Responses to GnRH and Tg . . . . . . . . . . . . . . .  45  vi  List of Abbreviations [Ca2+ ]ER . . . . . . . . Endoplasmic Reticulum Calcium Concentration [Ca2+ ]i . . . . . . . . . . Cytosolic Calcium Concentration Gβγ . . . . . . . . . . . . . G Protein αβγ Subunit Gi/o . . . . . . . . . . . . . G Protein αi/o Subunit Gq/11 . . . . . . . . . . . . G Protein αq/11 Subunit Gs . . . . . . . . . . . . . . . G Protein αs Subunit IP3 . . . . . . . . . . . . . . Inositol Triphosphate AC . . . . . . . . . . . . . . Adenylyl Cyclase ADP . . . . . . . . . . . . After-Depolarizing Potential AHP . . . . . . . . . . . . After-Hyperpolarization AP . . . . . . . . . . . . . . Action Potential cAMP . . . . . . . . . . . Cyclic Adenosine Monophosphate CNG Channel . . . Cyclic Nucleotide Gated channel DAG . . . . . . . . . . . . Diacylglycerol ER . . . . . . . . . . . . . . Endoplasmic Reticulum Fsk . . . . . . . . . . . . . . Forskolin GFP . . . . . . . . . . . . . Green Fluorescent Protein GIRK Channel . . G Protein Gated Inwardly Rectifying Potassium Channel GPCR . . . . . . . . . . . G Protein Coupled Receptor HCN Channel . . . Hyperpolarization Activated, Cyclic Nucleotide Gated channel NCX . . . . . . . . . . . . Sodium Calcium Exchanger PLC . . . . . . . . . . . . . Phospholipase C PMCA . . . . . . . . . . Plasma Membrane Calcium ATPase SERCA . . . . . . . . . . Sarcoplasmic-Endoplasmic Reticulum ATPase SK Channel . . . . . Small Conductance Calcium Activated Potassium Channel Tg . . . . . . . . . . . . . . . Thapsigargin TTX . . . . . . . . . . . . Tetrodotoxin  vii  Acknowledgements I would like to thank Yue-Xian Li for introducing me to Neuroendocrinology. I am very grateful to Yue-Xian Li and Eric Cytrynbaum for the many helpful discussions we’ve had, and for their direction and support during the course of my degree program. I would also like to thank Anmar Khadra for helpful discussions, and the graduate students of the IAM for helping to make the past two years a lot of fun.  viii  To my family, to Kristi, and to Grandad  ix  Statement of Co-Authorship Chapter 2 of this thesis was co-written with Yue-Xian Li. Yue-Xian Li identified and designed the research program, and wrote the Abstract, Introduction, and Discussion of the manuscript. Patrick Fletcher performed the research with the guidance of YueXian Li, including numerical simulations, generation of the figures, and writing of the Objectives, Model Description, Results, and Appendix of the manuscript. Editing of manuscript was performed by both Yue-Xian Li and Patrick Fletcher.  x  Chapter 1  Introduction In all vertebrates, normal reproductive function depends on the interaction of hormonal signals generated in at least three major endocrine tissues: the hypothalamus, the pituitary, and the gonads. In the hypothalamus, a small, sparse population of neurons secrete GnRH in a rhythmic pattern. This periodic GnRH signal, which travels by blood from the hypothalamus to the nearby anterior pituitary, stimulates pituitary gonadotropes to secrete the gonadotropins, follicle stimulating hormone (FSH) and luteinizing hormone (LH). These hormones regulate gamete production and secretion of the sex hormones in the gonads. The sex hormones, the most well known of which are testosterone in males and estrogen and progesterone in females, act on many tissues in the body, including feedback to the hypothalamus and pituitary to regulate GnRH and gonadotropin secretion (1). This introductory chapter is intended to highlight the importance of GnRH neurons in animal reproduction, and to familiarize the reader with GnRH neuron anatomy and physiology. We will introduce one plausible mechanisms for generating the GnRH pulsatile rhythm, referred to here as the autocrine mechanism, by which GnRH can regulate GnRH neuron secretion via GnRH receptors. Keeping this mechanism in mind, we review relevant aspects of GnRH electrophysiology, calcium dynamics, and GnRH receptor mediated signaling. These considerations motivate the work presented in Chapter 2, namely the simplification and unification of the previous models of GnRH neuron electrical activities and calcium dynamics. This work represents a important step in working towards an integrated model of GnRH neurons capable of explaining experimentally observed Ca2+ oscillations, electrical activities, and pulsatility.  1.1  GnRH Pulsatility is Required for Reproduction  A defining feature of the population of GnRH neurons is that they secrete GnRH in a so-called pulsatile manner. This pulsatility is characterized by brief (∼5 minute) periods of coordinated secretion, which yields a bolus of GnRH concentrations in the portal capillary blood that will travel from the hypothalamus to the pituitary. These pulses are separated by long intervals of low basal hormone levels. The frequency and profile of such pulsatility is relatively consistent across species, with most exhibiting a 1  circhoral rhythm: one 5 minute pulse per 30-60 minutes (1, 2). In 1980, Knobil et al. (3) clearly demonstrated that the pulsatile pattern of GnRH release is necessary for normal reproductive function. In monkeys with hypothalamic lesions that abolished normal GnRH secretion, extrinsic replacement of the normal one GnRH pulse per hour restored ovulatory menstrual cycles. Replacement regimens with higher or lower frequency, or with constant levels of GnRH, were not successful in restoring normal cycles. Since an unchanging pattern of GnRH pulsatility was sufficient to restore normal menstrual cycles in rhesus monkeys, Knobil et al.’s results (3) also showed that the pulsatile GnRH rhythm need not change to explain the variations in pituitary and gonadal hormones across the cycle. This suggests that the changes in the pituitary’s secretion throughout the cycle occur due to hormonal interactions between the pituitary and gonads. This does not preclude the notion that regulation of GnRH pulsatility could provide a means to control an organism’s fertility. Wildt et al. (4) showed, in the same experimental preparation, that increases in pulse frequency or decreases in pulse amplitude led to a decline in secretion of both gonadotropins, while decreased frequency resulted in a predominance of FSH secretion from the pituitary. Indeed, GnRH neurons are clearly poised to integrate a variety of extrinsic and homeostatic signals in order to ensure appropriate reproductive status of an animal. Another line of evidence critically implicating GnRH in the control of the reproductive system comes from a human disease called Kallmann’s syndrome, or from so called hypogonadal mice. In both cases there is a complete lack of GnRH released from the hypothalamus, in the first case due to lack of GnRH neuron migration into the hypothalamus during development, and in the second case due to a mutation in the GnRH gene that renders it inactive. These individuals fail to develop functional reproductive systems, but may be rescued by appropriate GnRH replacement therapy (1). Given that GnRH neurons play such a critical role in maintaining reproductive function in vertebrates, it is of great interest to uncover the mechanism by which GnRH pulsatility is generated, and how it can be regulated by relevant reproductive stimuli. Before we go on to consider the details of its function, let us first familiarize ourselves with the protagonist of this story, the GnRH neuron.  1.2  GnRH Neuron Anatomy  GnRH neurons are unique in that they originate outside the central nervous system in the olfactory placode1 , which is also the origin of the olfactory sensory neurons. They first differentiate into the GnRH phenotype in the olfactory placode, then migrate into 1  A placode is a thickening of embryonic epithelium, from which some organ or structure later develops; in this case, the olfactory sensory epithelium  2  their final locations in the central nervous system during prenatal development (5). After migration, GnRH cell bodies end up sparsely distributed across hypothalamic and preoptic nuclei, not themselves organized into distinct nuclei as are other hypothalamic neurons, such as oxytocin2 and vasopressin3 neurons. Given their importance in the development and maintenance of the vertebrate reproductive system, it is perhaps surprising that GnRH neurons are very few in numbers: estimates range from ∼800 to 2000 neurons per animal (1). These neurons have oval, round, or fusiform4 cell bodies, most (∼60%) exhibiting a bipolar morphology. The dendrites of GnRH neurons can be quite long; recent reports indicate that dendrites may extend to at least 1.5 mm in coronal slice preparations of adult mice (6). The dendrites of GnRH neurons have been shown to be excitable and capable of being the site of generation of action potentials (APs) (7, 8). In mammals, GnRH neurons project axons to the median eminence of the hypothalamus, where GnRH is secreted into the capillaries of a portal venous system5 connecting the hypothalamus to the pituitary (1).  1.2.1  Experimental Models  It is worth mentioning at this point some of the main experimental preparations used to study GnRH neurons. Immortalized GnRH Neurons, or GT1 neurons In 1990, transgenic mice were created in which a tumor-inducing gene was specifically expressed in GnRH neurons under the control of the GnRH promoter. A hypothalamic tumor was harvested from one such embryonic mouse and cultured, yielding the GT1 line of immortalized mouse GnRH neurons (9). With these neurons, cultured in vitro, it became much more tractable to study a variety of features previously inaccessible in vivo. Belief in data obtained by studying such a model system relies on the assumption that these neurons have a similar morphology and phenotype to native embryonic neurons. Indeed, these cells have been shown to exhibit neuronal phenotype, including the extension of processes, expression of many ion channels and receptors, and responses to various relevant stimuli (which will be described in more detail below) (1). Furthermore, they have been shown to possess the GnRH receptor (10), and cultures of such neurons exhibit pulsatile GnRH release (11–13). It must be stressed that GT1 neurons are not the same GnRH neurons as those in the intact hypothalamus. The key difference here 2  Oxytocin is best known for its roles in birth and breastfeeding Arginine Vasopressin, also known as anti-diuretic hormone, promotes water retention by the kidneys 4 Spindle shaped. 5 A portal venous system is a pair of capillary beds connected by veins, whereby one capillary bed drains into the other. 3  3  is that cultures of GT1 neurons do not contain any other cell type, and so lack many of the cell-cell interactions that GnRH neurons expect in vivo. For this reason, it is likely that there will be many differences in gene expression in GT1 neurons, compared with native GnRH neurons. That being said, many of the results obtained using GT1 neurons may still be valid in vivo, and when possible we shall document reported results that are common across multiple preparations. Hereafter, we will refer specifically to immortalized GnRH neurons as GT1 neurons. Olfactory Placode Explant Cultures. Before GnRH neurons migrate, the olfactory placode can be removed from embryos and cultured. The GnRH phenotype is still attained by some neurons in the preparation, and after 2-4 weeks in culture, there is pulsatile release of GnRH in such preparations; this has been shown in olfactory placode explant cultures from rhesus monkey, sheep and rat (14–16). This preparation has the advantage that it can be removed without disturbing the normal cell-cell associations that occur in the olfactory placode. It must be borne in mind, however, that these neurons are fetal, and the culture preparation lacks the appropriate brain structures for normal migration and maturation. Thus, they likely develop differently and thus do not share all the same properties as native GnRH neurons (1). Transgenic Models. Adult GnRH neurons have been difficult to study experimentally due to their sparse distribution and low numbers in vivo. With the help of recently developed transgenic techniques, animals have been created in which reporter molecules, such as green fluorescent protein (GFP) and even a calcium sensitive form of GFP called Pericam used for monitoring intracellular calcium concentration ([Ca2+ ]i ), have been specifically targeted to GnRH neurons (For example: 17–19). By allowing easy identification in situ, these advances have significantly facilitated direct electrophysiological measurements in adult GnRH neurons in slice preparations, and dissociated explants or short term tissue cultures. This technology allows the closest experimental models to the in vivo case so far achieved. The caveat here is that slice preparations (100-300µm thick) invariably damage cells by cutting their dendrites and axons, and dissociation of tissue with proteases invariably will change some cell signaling, due to interruption of cell-cell contacts. We refer to GnRH neurons from such animals with the reporter molecule as prefix to GnRH: for example, GFP-GnRH neurons are GFP expressing GnRH neurons.  4  1.3  GnRH Neuron Physiology  GnRH neurons come equipped with a toolkit of ion channels and pumps, receptors, and signaling pathways. Many of these components have been well studied in these neurons or others, but exactly how GnRH neurons collectively use their toolkits to build their hormonal rhythm remains elusive. Here we describe some typical behavior of GnRH neurons, as well as the autocrine feedback mechanism.  1.3.1  Electrical Activities  Perhaps the one thing that as been consistent about reports of GnRH neuron electrical activity is the marked heterogeneity of firing patterns observed within any given experimental preparation (2). The firing patterns can be broadly classified as silent, tonic spiking, and phasic spiking or bursting. The most commonly reported behavior is low frequency irregular tonic spiking (for example, 80% of wild type adult GnRH neurons in the mouse slice study of Sim et al. (20)). A series of detailed electrophysiological experiments were performed in GT1 neurons that have elucidated a basic feature of the spiking mechanism in these neurons (21, 22). When depolarized by various means, these neurons have the ability to shift from firing high amplitude tetrodotoxin (TTX) sensitive action potentials of short duration, to low amplitude action potentials of long duration. The second spike profile was shown to yield higher calcium influx than the first. This behavior was reproduced by Van Goor et al. (22), as described below. Electrical bursts which have periods of a few to tens of seconds, and are insensitive to the block of synaptic inputs have also been reported (23–27). There are also reports of organization of such bursting into longer periods of activity with intervening periods of silence, which had periodicity of minutes (”clusters”), and even tens of minutes (”episodes”) (28, 29). Since action potential firing drives Ca2+ influx, phasic spiking results in significant changes in [Ca2+ ]i . This is likely the electrical correlate to another phenomenon, observed when measuring [Ca2+ ]i by fluorescent indicators in these cells: Ca2+ oscillations that are sensitive to electrical activity. The next section is devoted to summarizing results pertaining to Ca2+ oscillations and phasic activity.  1.3.2  Calcium Oscillations and Electrical Bursting  Spontaneous Ca2+ oscillations in the cytosol have been observed in GT1 neurons, olfactory placode derived embryonic GnRH neurons, and adult GnRH neurons. In GT1 neurons, these oscillations are characterized by 5-30 second duration, 3-120 second period,  5  and 100-350 nM amplitude (30). In embryonic neurons, Terasawa et al. (14) reported oscillations with a duration of 90 seconds, period of about 8 minutes, and amplitude of 300 nM . Both these reports indicated a sensitivity to TTX and L-type Ca2+ channel blocker nifedipine, thus indicating their dependence on voltage gated Ca2+ entry. Recently, Jasoni et al. (19) reported oscillations related to IP3 receptor channel activity, and independent of membrane electrical activity in adult mouse GnRH neurons. These oscillations seem to have properties of both the previous reports, including 10-15 second duration and 8.5 minute period, although periods as low as one minute were reported. Although reported as not statistically significant, the oscillations reported by Jasoni et al. (19) were reduced in frequency by TTX and nifedipine. It is not known what mechanisms contribute to this variety of Ca2+ oscillatory behavior, and it remains to be shown conclusively what is their role in GnRH pulsatility. Unfortunately, most reports of Ca2+ oscillations lack simultaneous membrane potential recordings. There are a few, however, and these further support the electrical nature of short period Ca2+ oscillations. Charles and Hales (30) showed that bursts of action currents in their preparation had a similar period (∼3-5 seconds) as the [Ca2+ ]i oscillations they observed. This was also the case in olfactory placode cultures studied by Constantin and Wray (31), where they demonstrated bursts of action currents to be simultaneous with [Ca2+ ]i oscillations. In GT1 neurons, Van Goor et al. (21) reported larger changes in Ca2+ due to phasic firing, which had a periodicity of about 10 seconds, patterns than individual spikes. Furthermore, there is a tantalizing correlation between the ”clusters” of electrical activity described above and the Ca2+ oscillations with a period of 8 minutes reported in both olfactory placode preparations and adult slice experiments. An interesting feature of the 8 minute oscillations observed in the olfactory placode cultures from rhesus monkey and from mice is that the population of cells in the culture synchronize their calcium transients roughly every 50 minutes (14) or 20 minutes (32), which correlates well with the periodicity of GnRH pulsatility in those preparations. The oscillations in between such synchrony are completely uncorrelated. This is the basis for a conjecture proposed by Yue-Xian Li that it is the GnRH pulses that cause this synchrony, as shall be discussed further in §2.5. Based on these reports, there could be several qualitatively different types of Ca2+ oscillations have been observed in GnRH neurons. Some factors that could discriminate them include IP3 dependence, duration and period, and sensitivity to various pharmacological stimuli. Indeed, Li and Rinzel (33) showed that IP3 receptor dynamics could allow IP3 sensitive Ca2+ oscillations of periods of seconds to minutes, with and without sensitivity to voltage gated Ca2+ entry. Further studies are required to come to a mechanistic description of the relationship between electrical activities and Ca2+ dynamics in 6  GnRH neurons. Because of the rarity of simultaneous [Ca2+ ]i and membrane potential recordings in the literature, a model would be useful testing the feasibility of possible mechanisms. Preliminary results towards this end will be presented in Chapter 2.  1.3.3  GnRH Secretion  GnRH is packaged into large dense core vesicles typical of peptidergic neurons (34). GnRH secretion could occur in at least two spatially distinct locations in GnRH neurons (35). First, the axon terminals in the median eminence, which are outside the blood brain barrier, are responsible for secreting the GnRH that makes up pulses and act on the pituitary. This secretion is driven by the electrical signals traveling down the axons of each GnRH neuron. It is likely modulated by the activity of other neurons that also terminate nearby, as well as by non-neuronal cells in the region (2). The second site of GnRH secretion is in the large dendrites of these neurons. This form of release is poorly studied thus far in GnRH neurons, but has been well studied in the oxytocin and vasopressin neurons of the magnocellular system, which are also peptidergic neurons in the hypothalamus. Such a site of secretion, as in the magnocellular system, could be involved in the communication between GnRH neurons (35). It is not known how such dendritic release might affect the electrical activites and Ca2+ dynamics in GnRH neurons, including the electrical signal that is ultimately sent to the median eminence. For the purposes of the present study, we shall not consider the details of GnRH secretion, deferring further discussion of this topic to §3.3.2. If it turns out to be true that GnRH neurons secrete GnRH significantly from their dendrites, there are direct implications for experimental preparations. In many experiments on GnRH neurons, in particular adult preparations with fully developed dendrites, no effort has been reported to control the levels of free GnRH. As we will see below, GnRH can have potent direct actions on GnRH neurons, and thus it could be that un-controlled GnRH concentrations in some case confounds reported results.  1.3.4  Autocrine Feedback and GnRH Receptor Signaling  How GnRH neurons coordinate their secretory activities to give rise to their crucial hormonal rhythm remains to be conclusively shown. Several possibilities exist, but for the present work, we shall focus on one experimentally well supported mechanism: the so-called autocrine feedback mechanism. This will serve as the context for the present work, and will motivate the selection of physiological processes we include in the model.  7  The GnRH receptor The GnRH receptor is a G protein coupled receptor (GPCR). Such receptors, when their ligand is bound, are able to catalyze the binding of guanosine-5’-triphosphate (GTP) to so-called G proteins. This activates the G protein, which then breaks apart into two subunits. The α subunit, which comes in at least three flavors (s, q, and i) is soluble, and goes on to influence other enzymes. The βγ subunit is membrane bound and can affect other transmembrane proteins. We shall refer to the G protein subunits with a capital G and a subscript. For α subunits, we omit the α for brevity. An Autocrine Mechanism of GnRH Pulsatility Autocrine regulation can occur when a cell secretes a molecule for which it expresses the receptor. A crucial experiment that motivates the hypothesis of autocrine regulation by GnRH was performed by Martinez de la Escalera et al. (11). In cultures with two subpopulations of GT1 neurons allowed to communicate only through the bathing medium, coherent GnRH pulsatility was observed. The lack of cell to cell contact between the two populations of neurons indicated the presence of a diffusible synchronizing agent (11). One possible candidate is GnRH itself, acting via GnRH receptors. Indeed, GnRH receptor expression has been demonstrated in GT1 neurons (10), olfactory placode cultures in rats (36), cultured fetal rat hypothalamic GnRH neurons (37, 38), and in subpopulations of mouse neurons in vivo (39, 40). Further evidence for GnRH neurons being the pulse generators comes from olfactory placode cultures. Theses cultures do not contain the hypothalamic network, and yet are capable of pulsatile release which is comparable to that of intact animals (14–16). The autocrine mechanism is consistent with another experimental preparation used by Woller et al. (41). They enzymatically dissociated explanted hypothalami, presumably interrupting most direct cell-cell interactions. In these preparations, pulsatility is retained suggesting that a diffusible (autocrine or paracrine) factor is involved in coordinating GnRH neuron secretion (41). The autocrine mechanism for GnRH pulsatility involves the direct feedback of GnRH onto GnRH neurons via GnRH receptors. Based on the results showing that GnRH receptor coupling undergoes dose dependent switching between the PLC and AC signaling pathways, Krsmanovic et al. (42) proposed the following mechanism. Pulses are generated via sequential activation of GnRH receptor coupled secretion stimulating and inhibiting signaling pathways. Activation occurs in two stages. First, when GnRH concentration is low, cAMP production is increased by Gs coupled stimulation of AC. This increases secretion, and thus extracellular GnRH concentration. Once GnRH increases sufficiently, coupling of the GnRH receptor switches to Gq/11 which leads to increased 8  IP3 and Ca2+ mobilization from internal stores. This again boosts secretion, further raising GnRH levels. Finally, at high GnRH concentration, GnRH receptor coupling switches to Gi/o which ultimately shuts down the GnRH pulse (42). A mathematical model of the sequential activation autocrine mechanism was developed by Khadra and Li (43). As in the GT1 neuron culture experiment of Martinez de la Escalera et al. (11) described above, the solution is well stirred so that each GnRH neuron secretes GnRH into the medium, and in turn respond to the common pool of GnRH. Mathematically, this mechanism was demonstrated not only to be feasible, but incredibly robust (43, 44). Since an in vivo correlate for the common pool of GnRH present in cultures and the model has yet to be found, it is not known whether such a mechanism could be acting in vivo. It is well known that GnRH neuronal electrical activities and calcium dynamics are inextricably linked to GnRH secretion. In experiments that measure the summation of the activity of many neurons (multiple unit activity) with multiple electrode arrays, a jump in activity has been shown to precedes LH pulses (1). It has also been shown that GnRH release from incubated medio-basal hypothalami can be stimulated by electrical stimulation of frequencies of at least ∼10 Hz (45). A similar result was found in oxytocin and vasopressin neurons; 13 Hz stimulation yielded significant release of those hormones in vitro(46). Thus it seems logical to postulate that GnRH pulses are the result of coordinated high frequency (∼10 to 20 Hz) action potential firing of GnRH neurons for roughly 5 minutes per 30 to 60 minute cycle. Accordingly, GnRH neurons are capable of fire spontaneous bursts of action potentials of ∼10 to 20 Hz (23, 24, 27, 47), and can be stimulated, using current injections or pharmacological stimuli (see below) to increase their basal firing rate. Khadra and Li (43) postulated that the membrane electrical activity is not a direct participant in pulsatility. For simplicity, the model treats the neurons as though they are voltage clamped such that there is a constant calcium influx. Therefore, the Khadra and Li model cannot predict the details of electrical behavior of GnRH neurons during pulsatility (43). Aspects of both the PLC and AC pathways active in GnRH receptor signaling were included in the model by Khadra and Li (43), and both these pathways have been shown to alter electrical activity and calcium dynamics in GnRH neurons. We are thus motivated to develop a simple mathematical model of electrical activities and calcium dynamics of GnRH neurons that is compatible with the autocrine mechanism of GnRH pulsatility. With this in mind, a brief overview of the influences of GnRH receptor coupled signaling on electrophysiology and calcium dynamics in GnRH is now presented. Other G protein coupled receptors are suggested to act via the same pathways, which although not considered in the present model, will be discussed further in §3.2.2.  9  Phospholipase C Coupled Signaling Phospholipase C (PLC) is a membrane bound enzyme whose main function is to cleave the membrane lipid phosphatidylinositol bisphosphate (PIP2 ) into the soluble ionsitol triphosphate (IP3 ) and the lipid diacylglycerol (DAG). IP3 diffuses rapidly (48) and binds to its receptor, which is a Ca2+ channel on the endoplasmic reticulum, and leads to intracellular Ca2+ release. GnRH receptor activation was shown by Krsmanovic et al. (42) to elicit a dose dependent monotonic increase in IP3 production via Gq/11 coupling. The half maximal activation of this pathway occurred at a GnRH concentration of ∼30 nM , with a dose of 100 nM GnRH nearly saturating this response. Indeed, a large biphasic release with peak [Ca2+ ]i of low µM levels is observed when GT1 neurons are challenged with 100 nM GnRH (See Figure 2.1A) (10, 21, 49, 50). DAG can activate some ion channels (see §3.2.1), and protein kinase C (PKC) which is also activated by [Ca2+ ]i . Activated PKC can phosphorylate various enzymes and proteins in the cell which changes their function. For simplicity, we ignore these effects in the present work, and focus on the more prominent role of IP3 . Adenylyl Cyclase Coupled Signaling Adenylyl cyclase (AC) is another membrane bound enzyme that converts adenosine triphosphate (ATP) into cyclic adenosine monophosphate (cAMP). The soluble signaling molecule cAMP may then directly activate some ion channels, as well as protein kinase A (PKA), which in turn may phosphorylate many other proteins to change their function. AC is dually regulated by G proteins; Gs activates AC, increasing cAMP production, while Gi/o inhibits AC, leading to decreased intracellular cAMP levels. An important activator of AC is a drug called forskolin (Fsk). In GT1 neurons and dissociated fetal rat GnRH neurons, low doses of GnRH (∼1 nM ) stimulated increases cAMP (42). Such increases in cAMP result in PKA independent increases in excitability, Ca2+ oscillation frequency, Ca2+ influx, and GnRH secretion are reported (31, 50–54), suggesting a direct effect of cAMP to increase excitability. GnRH receptor activation of Gi/o has been demonstrated to occur above ∼56 nM GnRH (42), leading to decreases in cAMP and GnRH secretion. Interestingly, the βγ subunits released from Gi/o activation are known to activate inwardly rectifying potassium channels (GIRK), a result that has been demonstrated in GT1 cells and acutely dissociated fetal GnRH neurons (54). Such an effect has not been considered in the previous or present models. Two well known ion channels directly activated by cAMP are the voltage insensitive cyclic nucleotide gated channels and hyperpolarization activated cyclic nucleotide gated channels, CNG and HCN respectively. The expression of CNG channels has been shown in GT1 neurons (51), and in adult rat GnRH neurons (55), and a variety of experimental 10  results support their functional presence in GT1 neurons. These include single channel recordings (51, 52), the inhibitory effect of CNG channel blocker L-cis-diltiazem (LCD) (51–53) and the transient expression of antisense CNG channel pore forming subunit CNGA2 (56). Due to their origin in the olfactory placode, it is perhaps not surprising that GnRH neurons express the same olfactory isomer of the CNG channel as do olfactory sensory neurons (51). As such, these channels are purely Ca2+ conducting at physiological levels of extracellular Ca2+ (57). The HCN mediated current, often referred to as Ih , is an inward sodium current, activated by both hyperpolarization and cAMP. HCN channel transcripts have been reported in both GT1 neurons and adult female rat GnRH neurons (58). A hyperpolarization activated inward current characteristic of HCN channels, with sensitivity to the HCN channel blocker ZD-7288, has been reported in adult and juvenile mice (20, 59). There is, however, some controversy over the effector of cAMP dependent increased activity, since mice lacking CNG (31) or HCN channels (60) show no change in Ca2+ oscillations. As mentioned above in §1.3.2, the mechanism for generation of such Ca2+ oscillations, and the role such channels would play, has not been described in detail. This will be the topic of some discussion in Chapters 2 and 3. In the present work, we have not made an effort to distinguish the contributions of such channels, opting instead as have LeBeau et al. (50) for a simplistic description (see §2.2.2 and Appendix A). Explaining such seemingly paradoxical results as those outlined in this section is a strong motivation for the use of a mathematical modeling approach, since non-linear dynamical interactions can often be counter-intuitive.  1.4  Existing Models of Electrical and Calcium Dynamics  A set of two physiological models of immortalized GnRH neuron electrical activities and Ca2+ dynamics have been previously published (22, 50). The first was a conductance based model of the electrical activities of a GT1 neuron. The second model extended the first by included calcium dynamics and calcium sensitive currents. Both models are successful in achieving the goals of those studies, and in the present work we largely follow the mechanisms they described. This section is devoted to explaining the inner workings of the previous models relevant to the present study. The model of Van Goor et al. (22) was a purely electrophysiological description of GnRH neurons. One of their main goals was explaining how GnRH neuron spiking behavior shifts under depolarization. They include a fast TTX-sensitive Na+ current, L-type and T-type Ca2+ currents, delayed rectifier, M-type, and inward rectifying K+ (Kir ) currents, and a linear depolarizing non-specific cation current Id . They used great care to match the amplitudes and waveforms of the various currents they recorded 11  experimentally. Using a four-state Na+ channel model, the authors confirm their experimental results that the shift from sharp to broad action potentials is due to reduction of Na+ current. This occurs because of loss of inactivation of the sodium current, which we describe further in §2.3. LeBeau et al. (50) then extended their previous model, including Ca2+ dynamics and the Ca2+ dependent currents to model the responses to GnRH, Tg, apamin, intracellular buffering, and forskolin (See Fig. 2.1 for the experimental results of GnRH and Tg application). To the electrophysiology, they added calcium dependent currents: ISK (small conductance, Ca2+ activated K+ current) and ISOC (store operated calcium current), based on previous experiments (49). The motivation for SK channels is due to apamin sensitivity of hyperpolarization observed during response to GnRH (49). Motivation for the SOC current, which is inhibited by high levels of ER Ca2+ ([Ca2+ ]ER ) is indirect, and summarized as follows. GnRH- or thapsigargin-induced inward current occurs at the reversal potential for K+ , suggesting activation of inward current, not inhibition of outward current is responsible for increased frequency and baseline. This current is abolished in zero Ca2+ medium. Furthermore, Van Goor et al. (21) reported that sodium and calcium currents measured before and after GnRH application were identical, suggesting that they are not modulated by GnRH receptor signaling. Likewise, Van Goor et al. (49) reported no change in the GnRH response when Kir was blocked with an appropriate dose of Cs+ . Thus, no modulation of such channels was considered to be the observed net inward current. Such modulation was therefore not considered by LeBeau et al. (50), nor will it be in our model. A final piece of evidence comes from the fact that cells dialyzed with Ca2+ buffers EGTA and BAPTA prior to GnRH stimulation also showed an increase in firing rate, suggesting that [Ca2+ ]i plays no role in causing the increase. The calcium dynamics involved two diffusively coupled subcompartments, shell and bulk, for each of cytosol and endoplasmic reticulum. The calcium sensitive currents above are coupled to the appropriate shell compartment’s Ca2+ concentration. With their small volume shell compartment, they are able to achieve realistic Ca2+ spike levels due to membrane driven Ca2+ influx. They also added plasma membrane ATPase and sodium calcium exchangers (PMCA and NCX), sarcoplasmic-endoplasmic reticulum ATPase (SERCA), and mitochondrial uptake and release for Ca2+ extrusion from the cell and sequestration into the endoplasmic reticulum (ER) and mitochondria. Using their model, they report that interplay between ISK , ISOC , and Id can explain responses to the aforementioned stimuli. GnRH Application. Refer to Figure 2.1A for an experimental record of the voltage and [Ca2+ ]i of a GT1 neuron’s response to 100 nM GnRH. In the model of LeBeau et al. (50), GnRH application was modeled simply by increased ER leak. This 12  yielded a biphasic release. During the spike phase, [Ca2+ ]i is high, so ISK activates and hyperpolarizes the membrane. Once [Ca2+ ]i drops due to store depletion, ISK drops as well. Meanwhile, ISOC increases due to the depletion of stores. The increase depolarization due to ISOC causes return to firing with increased firing frequency once ISK is reduced. When the same experiment is performed with pretreatment of apamin, no hyperpolarization is seen, but firing still pauses briefly before resuming at increased frequency. The authors thus postulated the existence of Ca2+ dependent inhibition of their non-specific cation channel to explain this pause in firing. Thapsigargin Application. Thapsigargin inhibits SERCAs, thus causing a slow leak of Ca2+ from the stores. Refer to Figure 2.1B for an experimental record of the voltage and [Ca2+ ]i of a GT1 neuron’s response to 5 µM . In the model of LeBeau et al. (50), Tg is modeled by setting the SERCA pump rate to zero. Due to their choice of leak rate from the ER, they obtain a modest rise in [Ca2+ ]i , which weakly activates ISK during spikes and associated increase in the depth of after-hyperpolarizations (AHPs). The slow activation of ISOC again results in a gradual increase in firing rate. Intracellular Buffers. The model yields the experimentally observed increased firing frequency following GnRH application to BAPTA dialyzed cells by the activation of ISOC . A discrepancy between the model result and the experimental traces (49, 50) is that in the model response there is a brief transient phase of very rapid firing, due to the fast depletion of the ER shell compartment and resulting activation of ISOC . See Figure 1C of Van Goor et al. (49) or Figure 4A of LeBeau et al. (50) for experimental records of such responses. Forskolin Application. Fsk is simulated by increased conductance of non-specific cation channel Id . This increases firing rate, but fails to yield increases in Ca2+ in the cytosol as has been reported experimentally (see Figure 2 of Kaneishi et al. (53)). While doing a good job at explaining many features of the described stimuli, these models are not ideal for exploring either IP3 dependent Ca2+ oscillations or the autocrine mechanism of GnRH pulsatility because they lack an accurate IP3 receptor description. These models also contain some detailed descriptions for channel dynamics that we suspect are superfluous for our purposes. Finally, the arbitrary splitting of the Ca2+ compartments prevents an accurate account of the total cell calcium, which we feel will be important for our future use of such a model. The modifications and differences between our approach and the approach outlined heres shall be discussed in Chapter 2.  1.5  Objectives  Understanding a phenomenon such as GnRH pulsatility poses some unique challenges. Since GnRH pulses have a long period, physiological processes acting on a wide range of 13  timescales, from milliseconds to hours, must be considered as possible components of the pulse generation mechanism. Furthermore, GnRH neurons have a complex anatomical structure that includes axons that converge on the median eminence, somata that are sparsely distributed in the hypothalamus, and long excitable dendrites. A crucial step in understanding GnRH pulsatility is the development of an accurate, yet simple model of each of the relevant components. Enough detail must be retained so that meaningful predications can be made, and yet the model should be simple enough to ensure the tractability and understandability of an integrated model of circhoral GnRH pulsatility. The present work is focussed on electrophysiology and calcium dynamics in GnRH neurons. With the autocrine mechanism for GnRH pulsatility in mind, we present a model in Chapter 2 in which electrophysiology and calcium dynamics are parameterized by the two important second messengers, IP3 and cAMP. This will allow future integration with a mechanism for pulsatility such as that of Khadra and Li (43). We use a spatio-temporal description of Ca2+ dynamics in order to achieve an accurate representation of the [Ca2+ ]i in the cell. For the sake of simplicity, we neglect the spatial structure of the dendrites, focussing instead on an idealized spherical cell body. We follow, for the most part, the mechanisms outlined by LeBeau et al. (50) to explain several informative experimental results from GT1 neurons, including depolarizing current injections, acute application of GnRH, store depletion by Thapsigargin (Tg), the SK channel blocker apamin, and increases in excitability caused by activation of AC. We make an effort to reduce the complexity of the model wherever possible, while maintaining the desired behavior. This includes a phenomenological simplification from the spatio-temporal description to a lumped calcium model with an approximation of [Ca2+ ]i at the membrane. Motivated by both the autocrine mechanism of pulsatility and the recent reports of IP3 sensitive [Ca2+ ]i transients in adult GnRH neurons (19), we include IP3 receptor dynamics. This allows for use of the present model in the study of IP3 dependent [Ca2+ ]i oscillations and electrical bursting. The model also predicts another mechanism for [Ca2+ ]i oscillations that is voltage gated calcium entry dependent. Thus, we propose two mechanisms for short period bursting and [Ca2+ ]i oscillations. These mechanisms are also capable of longer period bursting as well; the treatment of such issues here, however, is a preliminary one. The model exhibits, with few changes in relevant physiological parameter values, a variety of different behaviors that mimic those seen in experiments, such as quiescence, tonic spiking, and phasic spiking. Because of the parametrization by important second messengers, the model is not specific to GnRH receptor signaling. One could easily adapt the model to explore the effects of other PLC or AC coupled signaling events 14  on the presented electrical and calcium dynamics. 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Localization of olfactory cyclic nucleotidegated channels in rat gonadotropin-releasing hormone neurons. Endocrinology 143:2441–4. 56. Blackman, B. E., H. Yoshida, S. Paruthiyil, and R. I. Weiner, 2007. Frequency of intrinsic pulsatile gonadotropin-releasing hormone secretion is regulated by the expression of cyclic nucleotide-gated channels in GT1 cells. Endocrinology 148:3299– 306. 57. Frings, S., R. Seifert, M. Godde, and U. B. Kaupp, 1995. Profoundly different calcium permeation and blockage determine the specific function of distinct cyclic nucleotide-gated channels. Neuron 15:169–79. 58. Arroyo, A., B. Kim, R. L. Rasmusson, G. Bett, and J. Yeh, 2006. Hyperpolarizationactivated cation channels are expressed in rat hypothalamic gonadotropin-releasing hormone (GnRH) neurons and immortalized GnRH neurons. J Soc Gynecol Investig 13:442–50. 59. Zhang, C., M. A. Bosch, J. E. Levine, O. K. Ronnekleiv, and M. J. Kelly, 2007. Gonadotropin-releasing hormone neurons express K(ATP) channels that are regulated by estrogen and responsive to glucose and metabolic inhibition. J Neurosci 27:10153–64. 20  60. Constantin, S., and S. Wray, 2008. Gonadotropin-Releasing Hormone-1 Neuronal Activity Is Independent of Hyperpolarization-Activated Cyclic NucleotideModulated Channels but Is Sensitive to Protein Kinase A-Dependent Phosphorylation. Endocrinology 149:3500–3511.  21  Chapter 2  A Minimal Model of GnRH Neuron Electrical Spiking and Bursting 1 2.1  Introduction  The regulation of reproductive function and fertility is ultimately controlled by a pulsatile signal of gonadotropin-releasing hormone (GnRH). GnRH is synthesized and secreted by a group of a few hundred diffusely distributed GnRH neurons that work in synchrony to produce the pulsatile rhythm with a period ranging from 30 minutes to about one hour. GnRH stimulates the release of gonadotropins from the pituitary and thus regulates downstream gonadal activities indirectly. In rhesus monkey, the pulsatile GnRH signal with the appropriate period (i.e. one pulse per hour or circhoral) is both necessary and sufficient for sexual maturation and fertility, while constant GnRH signal or pulsatile signals with inappropriate periods fail to sustain the function of the reproductive system (61). Extensive efforts have been devoted to the understanding of the origin of this rhythm. The mysteries surrounding the rhythmogenesis of GnRH neurons remain defiant to any conventional knowledge of rhythmogenesis in neuronal networks. Unlike most other networks of neurons that are supposed to work in synchrony, GnRH neurons are distributed in a scattered manner with relatively few synaptic coupling between each other (62–64). There is evidence in cultured GnRH neurons showing that their plasma membrane electrical activities are not synchronized most of the time except for a brief period of a few minutes per hour (65). The other clear difference is the long period of the GnRH rhythmicity (about one hour) which suggests that cellular events with time scales ranging from milliseconds up to one hour can be involved in this rhythmogenesis. This pushes the frontier of conventional neuroscience that traditionally only considered ionotropic channel events with time scales of milliseconds to the inclusion of metabotropic receptors, second messengers, vesicle transportation 1 A version of this chapter will be submitted for publication. Fletcher, P.A. and Li, Y.X. A Minimal Model of GnRH Neuron Electrical Spiking and Bursting  22  and docking, autocrine/parcrine regulations, as well as genetic events. As a result, an integrated understanding of a process as complicated as the GnRH pulse generation strongly depends on detailed understanding of each composing part of this complicated system. This study is aimed at developing a mathematical model of reduced complexity that describes the electrical activities and calcium handling of a GnRH neuron while maintaining the ability of reproducing all the important properties of the neuron that are experimentally observed. This will facilitate the integration of this model to models of other parts of the GnRH neuron that we believe are directly responsible for the generation of the circhoral secretory rhythm. These include a recently proposed autocrine regulation mechanism that was revealed in cultured hypothalamic tissue (66), placode derived GnRH neurons (67), and GT1 neurons (67, 68) and mathematically modeled by treating the neurons as being clamped at a constant voltage (69, 70). Special care was taken to correctly model the points of interactions between the autocrine regulation mechanism and membrane electrical activities. The experimental data based on which the model is developed are mainly obtained from cultured GnRH cell lines called GT1 cells. Some data from cultured embryonic GnR neurons and hypothalamic slices are also taken into account. One of the goals is to show that it is rather easy to use the model based on data collected in GT1 cells to reproduce phenomena observed in mature slice experiments by slightly changing some parameter values. The model presented here is based predominantly on a couple of models previously developed by Van Goor et al. (71) and LeBeau et al. (72) . These were the pioneering physiological models of the electrical activities of GnRH neurons. The first model was aimed at modeling the ionic basis of action potential generation and spontaneous spiking activities while the second was focused mainly on the responses of an isolated GnRH neuron to a variety of electrophysiological and pharmacological stimuli. The two models are not identical and exhibit qualitatively different bifurcation structures. As a result, the second model does not reproduce all the results of the first one for the published set of parameter values. The Na+ and the nonspecific cationic currents were modeled in a more complicated way than appears necessary for reproducing the observed phenomena. A few currents that are known to exist in these neurons were included in the previous model, but turned out to be nonessential for the observed behaviors. With the ultimate goal of constructing an integrated model of the GnRH neuronal network that incorporates all important aspects of the pulse generator, we seek minimized realistic models for each of the component subsystems. This approach reduces the complexity, making the integrated model more tractable and understandable to a broader audience. With this goal in mind, we here present an integrated and minimized version of the two existing models previously developed (71, 72). The new model is capable of consistently reproducing all the experimen23  tal results previously reproduced by the two previous models. It can also reproduce some important experimental results the previous models did not reproduce. The previously developed two-compartment model for both the cytosol and the ER is replaced by a realistic description of the spatio-temporal distribution of intracellular calcium concentration ([Ca2+ ]i ) which turns out to be crucial in explaining some important experimental observations. The addition of the ER Ca2+ release through a realistic model of IP3 -receptor channels also allows to explain the occurrence of IP3 -dependent Ca2+ oscillations recently observed in mature GnRH neurons from slice preparations (73) and predicts the form of the associated bursting in the membrane potential. Another type of membrane potential bursting involving the store-operated Ca2+ current (ISOC ) that has not been reported in experimental literature is predicted by this model. Since the ultimate goal is to incorporate this plasma membrane model into the model of autocrine regulations and GnRH pulse generation, we propose a phenomenological simplification of this spatiotemporal model that retains the features of the full model. The remaining part of the paper is organized as follows. Section 2 outlines the list of the major experimental observations that the model is targeted to reproduce. Based on these objectives, the model is presented with basic model assumptions transparently stated. The main results of the model are presented in Section 3. In Section 4, a simplified version of the model is introduced. The physiological relevance of this model will be discussed in Section 5.  2.2 2.2.1  Objectives of the Paper and the Model Experimental Observations the Model Aims to Reproduce  A mathematical model is most useful if it enables one to pinpoint the minimal elements involved in the generation of one specific phenomenon. This is similar to what a typical electrophysiologist does by using specific channel blockers. The challenge a computational modeler faces is not to make the model reproduce one specific observation but to make it reproduce all known observations that are important to the function of a cell. By demanding a model to reproduce all these important experiments in a physiologically coherent way using parameter values within physiological ranges, one increases the reliability of any new predictions the model makes. This is based on the belief that a simplified dynamical system such as the model behaves reasonably closely to the realistic system if the model based on known facts mimics all the known aspects of the real system. Depending on the objectives, the models of a real cell can look very different. For example, the two previous models of the GnRH electrical activities by Van Goor et al. (71) and LeBeau et al. (72) were different because the goals were  24  different. Therefore, one should clearly state the objectives of a modeling work before presenting the model. A list of experimental observations that the model is aimed at reproducing is presented below. Keeping in mind that the ultimate goal is to integrate this model with the autocrine regulation model, we put special emphasis on the mechanisms through which the plasma membrane electrical activities are coupled to the G-protein activated second messengers in GnRH neurons as recently revealed in Krsmanovic et al. (68). E1. An unstimulated GnRH neuron often exhibits spontaneous firing of narrow, Na+ current-dependent APs at frequencies between 0.5 to 1 Hz. A depolarizing current injection triggers higher frequency spikes at decreased amplitudes with broader width and higher Ca2+ entry. These spikes are characterized by declining peak values and elevated baselines (71, 74–76). E2. A large dose of GnRH, as shown in Figure 2.1A, induces a biphasic release of calcium from intracellular stores causing a transient hyperpolarization with a depth of ∼14 mV and a duration of ∼15 seconds. The ability of apamin and calcium buffers to abolish this hyperpolarization suggests that the SK-type K+ channels are responsible (72, 74, 77). The GnRH-induced hyperpolarization is followed by spiking with an increased frequency and decreased amplitude that resembles the response to a depolarizing current injection. The enhanced inward current that causes this response is most likely the store-operated Ca2+ current (ISOC ) (72, 77). E3. Thapsigargin causes a slow increase in cytosolic calcium, an increase in the firing rate, and a deeper after-hyperpolarization (AHP) following each spike (Fig. 2.1). The increase in frequency increase is probably caused by inward currents that must overcome the hyperpolarizing effect of SK channels. This inward current is best explained by the ISOC . The deepened AHP is due to SK since apamin abolishes it resulting in higher spiking frequency (77). E4. Activation of adenylyl cyclase by forskolin (Fsk) increases AP firing rate, Ca2+ influx, Ca2+ oscillation frequency, and GnRH secretion. These increases are insensitive to protein kinase A (PKA) inhibitors, suggesting a direct mechanism of cAMP dependent increase in excitability (72, 76, 78–81). The identity of such a direct effector of the adenylyl cyclase pathway on membrane excitability is unknown, although a few candidates exist in GnRH neurons, including CNG channels and HCN channels (78, 82–84). E5. There are several reports of membrane potential burst firing patterns, typically described as irregular with variable burst duration (∼1-20 seconds) (64, 85–88). 25  There are also reports of transitions from non-firing (quiescence) to tonic or phasic modes of firing, in ”clusters” of a few minutes or ”episodes” of tens of minutes (64, 85, 86, 89, 90). Spontaneous Ca2+ oscillations in the cytosol have also been observed (65, 73, 91, 92). In GT1 neurons, these oscillations are characterized by 5-30 second duration, 3-120 second period, and 100-350 nM amplitude (91). Recently, Jasoni et al. (73) have also reported oscillations related to IP3 receptor channel activity, and independent of membrane electrical activity in adult mouse GnRH neurons. The mechanisms of such electrical bursting and calcium transients remain unknown. The model that we present below is capable of generating all the observed experiments listed above. E1-E4 are reproduced with mechanisms derived from and supported by experiments using similar mechanisms to those proposed by LeBeau et al. (72). E5 occurs as direct consequences of the mechanism that are already present in the model based on experiments supporting the explanation of E1-E4. Therefore, these results should be regarded as model predictions before they are confirmed by experiments beyond doubts.  2.2.2  The Model  We shall consider the case of calcium dynamics in the soma of GnRH neurons. For simplicity, we adopt a spherical geometry in which two calcium-containing compartments, cytosol and GnRH-sensitive endoplasmic reticulum (ER), are continuously distributed. The two compartments are separated by the ER membrane, which contains IP3 receptors, SERCAs (sarco- and endoplasmic reticulum calcium ATPases), and a non-specific calcium leak. The plasma membrane contains ion channels, PMCAs (plasma membrane calcium ATPases), and NCXs (sodium-calcium exchangers). Plasma membrane electrical activity drives Ca2+ ions into the cytosol, where they can diffuse through a heavily buffered cytosolic medium, be pumped into the ER or be extruded from the cell. There is no direct path for Ca2+ to enter the ER from the extracellular space. The calcium dynamics, illustrated in Figure 2.2, are governed by the following equations and boundary conditions: fcyt ∂C = (jrel − jfil ) + D∇2 C ∂t Vcyt ∂Ce fER = (jfil − jrel ) + DER ∇2 Ce ∂t VER ∂C ∂Ce Do = jin − jout , =0 ∂r r=R ∂r r=R  in cytosol  (2.1)  in ER  (2.2)  at cell surface  (2.3)  26  where C and Ce represent the [Ca2+ ]i and [Ca2+ ]ER respectively, and are functions of the spatial location and time. By assuming spherical symmetry, the partial differential equations are reduced to a quasi-one-dimensional model. In polar coordinates, the spatial variable is the radius r. This treatment is essential for explaining the observations reported in E3. This also eliminates the arbitrariness of introducing two diffusive coupling constants between the core and shell cytosolic and ER compartments (72). This approach has been adopted previously in a model of plasma membrane electrical activities in pituitary gonadotrophs (93, 94). We later demonstrate that results obtained by the PDEs 2.1-2.3 can be reproduced by a simplified system of ordinary differential equations with one phenomenological equation for the Ca2+ concentration near the plasma membrane. The local flux densities of ER Ca2+ release, jrel , and Ca2+ uptake, jfil , are mediated by passive leak and flux through the IP3 receptor, and by the SERCA pumps, respectively. The details of these fluxes are modeled based on the Li-Rinzel model derived from the De Young-Keizer model (95)(See Appendix A for details). The terms jin and jout (in µM ·µm ·ms−1 ) are the flux densities of Ca2+ entry and extrusion at the plasma membrane, respectively. The cytosolic volume Vcyt is assumed to take up 85% of the total cell volume, with the ER volume VER taking up the rest. The fluxes are scaled by fcyt or fER (both 0.01) to account for buffering in the cytosol and ER respectively. D and DER are the Ca2+ diffusion coefficients in the cytosol and ER respectively, and Do is the Ca2+ diffusion coefficient in a buffer-free medium. The values of these parameters can be found in Table A.1 in the Appendix. The spatial averages of C calculated in the model correspond roughly to the [Ca2+ ]i levels that are observable experimentally. The membrane electrical activity is governed by the current balance equation: Cm  dV = Iapp − (INa + ICaL + IK + Iir + INSC + ISK + ISOC ) dt  (2.4)  Here, INa is a TTX-sensitive transient Na+ current, ICaL is an L-type Ca2+ current, IK is a delayed rectifier K+ current, and Iir is an inward rectifier K+ current. INSC represents a cAMP activated non-specific cation current, ISK represents a small-conductance calcium activated K+ current, and ISOC represents a store operated calcium current. Iapp is zero unless otherwise stated. These currents represent a minimized set of the currents previously modeled in Van Goor et al. (71) and LeBeau et al. (72). They are qualitatively identical to the corresponding currents in the previous models, although some are expressed in simpler forms. Detailed mathematical expressions of the currents and parameter values are given in Appendix A. Numerical Methods. By assuming spherical symmetry, we are able to use a transformation of variables to rewrite the PDE system with one spatial dimension, the radius 27  from the center of the cell. This system is then discretized in space with a second order central difference approximation and integrated using MatLab R (2007b, The Mathworks, Natick, MA). Code for both MatLab and XPP available upon request.  2.3  Results  We wish to show here that our model reproduces the important experiments explained by the previous models (71, 72). We found that these results did not depend on a sophisticated description of the fast sodium or non-specific cation current, or the inclusion of the T-type Ca2+ current or the M-type K+ current. We also explore some behaviors not touched upon by the previous modeling efforts, such as bursting and Ca2+ oscillations. We first use the spatio-temporal description to obtain an accurate description of the calcium dynamics near the plasma membrane. Then, in order to reduce the complexity of the model, we use a phenomenological simplification that mimics the calcium profile at the membrane, as described in section 2.4.  2.3.1  Responses to Current Injections  Figure 2.3 shows the model’s ability to reproduce responses to current injection, described in E1 above. Our idealized GnRH neuron exhibits spontaneous low frequency firing (∼0.7Hz) of spikes whose amplitude is ∼75 mV and duration is ∼9 ms. The experimentally reported changes in amplitude, frequency, and spike width (71, 74, 77) during depolarizing current injection were also reproduced. With 5 and 15 pA applied current, amplitude decreased from 75 mV to 62 and 46 mV , frequency rose to 15 and 22 Hz, and duration rose to 12 and 15 ms, respectively (Fig. 2.3A,B). Furthermore, integrated calcium per spike also increases, measure as ICa integrated over a time interval twice the spike duration, centered at the spike peak time (data not shown). Care was taken to ensure that current amplitudes and waveforms were similar to reported experimental results (71, 74). As was demonstrated in the model of Van Goor et al. (71), the shift from high amplitude Na+ current dependent spikes to broad Ca2+ current based spikes is explained by the loss of deinactivation of the sodium channels. For our purposes, we found it sufficient to use a simple Hodgkin and Huxley-like description (96) to achieve this result. The inactivation variable h is plotted in Figure 2.3C as a function of membrane potential, so that during the course of an AP, h travels clockwise along the trajectories shown. During spontaneous activity, h is high during the rising phase of the AP. Under depolarizing drive, less time is spent in a hyperpolarized state, leading to a lower value of h during spikes. As a result, there is a 2-fold and 5 fold reduction in Na+ current  28  when 5 and 15 pA of current is injected, respectively (Fig. 2.3D).  2.3.2  Responses to GnRH  The voltage and calcium response of GnRH neurons challenged with GnRH (Fig. 2.1A) is reproduced by the model. An important detail offered by the spatial treatment of [Ca2+ ]i is that voltage gated calcium entry during each AP results in brief rises in Ca2+ near the membrane to ∼0.5 µM . The Ca2+ profile at the membrane closely follows the AP profile because the bulk of calcium influx is contributed by the L-type Ca2+ channels. Due to relatively slow cytosolic diffusion, and since Ca2+ is sequestered into the ER everywhere inside the cell, these Ca2+ transients are strongly localized near the membrane (Fig. 2.4C). It should also be noted that the [Ca2+ ]i at the membrane is lower than the average value due to the actions of PMCAs and NCXs. Application of GnRH is modeled as a global rise in IP3 concentration, since the concentration of IP3 should rapidly equilibrate in our 10 µm cell (97). Thus, in contrast to voltage driven Ca2+ influx, GnRH-induced release from stores occurs everywhere in the cell (Fig. 2.4C). We use an exponential timecourse to transition from the basal level of IP3 (0.01 µM ) to the new level (1 µM ) in order to time the peak at 3 seconds, as is reported experimentally. The exponential onset is not essential to the qualitative dynamics, however, and is used only to match the experimental result more quantitatively. For simplicity, at this point we do not consider effects that may be due to GnRH receptor coupling to adenylyl cyclase and cAMP during the application of 100 nM GnRH. To explain the response to GnRH, we assume calcium is released through IP3 receptor channels. We found that calcium-dependent inactivation of INSC postulated by LeBeau et al. (72) was not necessary for replication of the GnRH or Tg responses, and so for simplicity we did not include this feature. The cessation of firing during GnRH application in cells pretreated with apamin reported by Van Goor et al. (77) and LeBeau et al. (72) can be reproduced in our model by reducing the SK channel conductance, gsk , to 5% of original value instead of zero (not shown). By assuming that the inactivation of the IP3 receptor is fast enough, we obtain a biphasic calcium response to a rise in IP3 . The rising phase is due to the fast activation by IP3 and regenerative activation by rising [Ca2+ ]i . During the rising phase of the [Ca2+ ]i response, SK channels are activated and initiate the transient ∼15 mV hyperpolarization. ISK , however, stalls despite the continued rise in [Ca2+ ]i because the membrane potential quickly drops near potassium’s reversal potential, EK (Note the step-like shape of ISK in Figure 2.4B). Concurrently, dropping store Ca2+ level gradually activates ISOC (Fig. 2.4B,D). Shortly after ISK stalls, the net current switches  29  from outward to inward allowing the membrane potential to rise. The increasing depolarizing drive from ISOC is counteracted by ISK , which, under continued activation by [Ca2+ ]i , is allowed to increase as membrane potential rises away from EK . This balance is responsible for the slow rise of membrane potential, and persists until the spike phase of the Ca2+ response subsides and [Ca2+ ]i drops again. Delayed inactivation of the IP3 receptor, and with the help of SERCAs and NCXs, terminates the [Ca2+ ]i spike. The action of Ca2+ ATPases and the NCXs gradually extrudes Ca2+ from the cell, depleting both the ER and cytosol. As SK channel activation wanes as a consequence of falling [Ca2+ ]i , the increased ISOC accelerates the rise to threshold, and AP firing is resumed at a higher frequency. The plateau phase of the calcium response is also a balance; ISOC , ICaL during spikes, and continued release from the stores drive the cytosolic calcium up, while SERCAs, NCXs, and PMCAs remove calcium from the cytosol. Application of apamin during this phase yields a further increase in firing rate as in the experimental record in Figure 2.1A. The faster fluctuations of membrane potential drive global calcium levels to rise above the GnRH-induced plateau (Fig. 2.4C).  2.3.3  Responses to Thapsigargin  When SERCA pumps are inhibited, as in the experimental record above (Fig. 2.1B), we observe an immediate rise, followed by a transient drop, and ultimately a gradual increase in firing frequency (Fig. 2.5). Deep AHPs are observed after each spike despite the relatively smooth rise in average [Ca2+ ]i , which is a clear indication of the rapid, localized effect of voltage gated calcium influx near the membrane. The transient behavior of the membrane potential can be explained by the dynamic activation of ISOC and ISK by [Ca2+ ]i and [Ca2+ ]ER . Immediately as Tg is applied, the store level begins to drop and activate ISOC . Due to the slow leak, however, cytosolic Ca2+ concentration takes ∼5 seconds to reach levels that can activate SK significantly in between spikes. Thus, the initial transient behavior is an increase in frequency. SK channel activation soon catches up, however, and significantly slows depolarization. The average cytosolic Ca2+ concentration rises to a plateau of nearly 1µM in about ten seconds as Ca2+ leaks from the store, at which time SK channel activation also plateaus. Due to continued depletion of the stores, ISOC continues to rise yielding the log term increase in frequency. Again, we use an exponential timecourse as a transition of SERCA pump rate from normal to zero. A qualitatively similar result can be obtained by transition to a small non-zero pump rate.  30  2.3.4  Responses to Forskolin  Application of Fsk is modeled by increasing the level of cAMP, which activates INSC . As in the model of LeBeau et al. (72), firing frequency rises due to the increased inward current (Fig. 2.6A). A notable difference is that here, we observe a modest calcium accumulation comparable to that seen in GT1 cells (80). Concurrently, [Ca2+ ]ER rises due to the action of the SERCAs. The presence of ISOC in the model means that such a loading of the stores reduces membrane excitability. This becomes obvious when Fsk is removed, and there is hyperpolarization of ∼5 mV . This hyperpolarization persists until store Ca2+ leaks back to pre-stimulus levels, at which time spiking resumes (not shown). The same behavior occurs when a current injection is halted, yielding an apamin-insensitive mechanism for after-hyperpolarizations. Such a mechanism is a prediction of the model, and as such requires experimental validation.  2.3.5  Bursting Mechanisms  The components used in the model to obtain the previous results, after minor changes in parameter values, offer the possibility of at least two qualitatively distinct types of membrane potential bursting and associated Ca2+ oscillations. The first is an IP3 dependent mechanism, as has previously been well studied in many systems, such as the pituitary gonadotroph (94) (Fig. 2.7). The second is an electrical bursting mechanism involving ISOC and the ER Ca2+ load, hereafter referred to as store operated bursting (Fig. 2.8). The inclusion of IP3 receptor dynamics is essential for the first type of bursting. When inactivation of the IP3 receptor by cytosolic calcium is sufficiently slow, release from the stores becomes oscillatory. The amplitude, duration, and frequency of such oscillations can be controlled by tuning the parameters of release, and a wide range of burst durations and periods should be attainable. For example, Figure 2.7 shows an example of oscillations with ∼8 second bursts, and 3 second interburst intervals. The frequency of APs during bursts starts at ∼10 Hz, dropping to 8 Hz. During each IP3 induced Ca2+ oscillation, which reaches µM levels (Fig. 2.7C), SK channels are activated and hyperpolarize the membrane (Fig. 2.7A,B). Note that the ER does not deplete in this case, but instead oscillates about an average value that is lower than pre-stimulus levels: first releasing via IP3 receptors, then refilling due to the increased spiking activity (Fig. 2.7D). A second mode of burst firing was observed, in which IP3 dependent release from internal stores was not important. An example with 22 second burst period, 4 second duration, 5 Hz peak firing rate and spike frequency adaptation during the burst can be seen in Figure 2.8A. In this case, [Ca2+ ]ER plays the role of a slow variable, slowly 31  activating and inactivating ISOC . When stores load is high, ISOC activation is low, and the membrane is at rest. Ca2+ leaks from the ER, slowly activating ISOC , until it is sufficiently activated to drive Vm to the threshold for spiking. The high frequency AP firing rapidly refills the stores, inactivating ISOC once again. Note that during the electrical bursting, Ca2+ is high enough to activate some SK channels (Fig. 2.8B,C), which contribute to spike frequency adaptation and the short duration of the burst. Furthermore, the change in average cytosolic Ca2+ is modest, ∼100 nM and highly localized to the plasma membrane. Interestingly, as reported by Liu and Herbison (88), apamin was found to increase the number of spikes per burst and duration of the burst (data not shown). In the absence of SOC, this type of bursting would not be possible in this model. Because ISOC was also included in the model by LeBeau et al. (72), it could be able to reproduce such oscillations, but such issues were not explored in that study.  2.4  The Simplified Model  Motivated by the long term goal of coupling a model of electrical activities and calcium dynamics to mechanisms for generating GnRH pulsatility, we wish to simplify the spatial model described above. We obtain the simplified model by assuming that the calcium in the cytosol and ER are well mixed. The model is then reduced to a system of ordinary differential equations that accurately model the whole cell average value of Ca2+ in each compartment. The calcium dynamics in the simplified model thus are governed by: fcyt dC = fcyt β(jin − jout ) + (Jrel − Jref ) dt Vcyt dCe fER = (Jref − Jrel ) dt VER where the J now represent whole cell fluxes. The calcium fluxes across the plasma membrane must be rescaled by the factor β = Acell /Vcyt (0.35 µm−1 ) to convert from the per unit area fluxes required by the spatial model’s boundary condition (Eqn. 2.3), to the whole cell fluxes desired here. Instead of dividing the Ca2+ compartments into shell and bulk as did LeBeau et al. (72), we keep the whole cell Ca2+ equations intact and instead use a phenomenological equation to model [Ca2+ ]i at the membrane. We invoke a domain-like approximation, similar to the approach used by Van Goor et al. (98). It must be emphasized that the approach used here is phenomenological and serves to reproduce the timecourse of C(R), the calcium concentration at the membrane in the spatio-temporal model. We use Cm to represents the [Ca2+ ]i at the membrane, near the voltage-gated calcium channels. 32  This quantity evolves as τm  dCm = Cm,∞ − Cm dt  where Cm,∞ = pa2 − Km CR Here, a is the L-Type Ca2+ channel gating variable, τm is the timeconstant (17 ms) with which Cm relaxes to Cm,∞ , p and Km are free parameters (1.46 and 0.123 respectively). The electrical activities are modeled exactly as before, except the calcium sensitive currents; SK channels now respond to the superposition of the bulk and domain calcium: CR = Cm + C, while ISOC is activated by the whole cell [Ca2+ ]ER . We found that the qualitative results obtained here do not depend on whether ISOC is coupled to the average or the membrane calcium (data not shown), so no effort has been made to model [Ca2+ ]ER near the membrane for the simplified model. All other parameters and function definitions are as in the full model, and are given in the Appendix. The simplified model is able to reproduce the same results as the full model, demonstrating its utility for future goals of mathematical modeling in GnRH neurons. For brevity, we show in Figure 2.9 only the responses to GnRH, Tg and apamin. Qualitatively, the mechanism is the identical to that described above.  2.5  Discussion  The mechanism underling the origin of the GnRH pulse generation remains a mystery although tremendous effort has been devoted toward revealing it. Most researchers in the field believe that the rhythm occurs through synchronized secretory activities of hundreds of GnRH neurons. Yet it appears that the synchronization is not accomplished through close contacts or synapses between these neurons. GnRH neurons are distributed in a scattered manner in the hypothalamus, with few synaptic connections between them were detected (62, 63). Unlike the spike-by-spike synchrony in which all synchronized neurons spike in-phase and reach peak membrane potential at the same time, the electrical activities of GnRH neurons do not seem to synchronize in this manner. As a matter of fact, measurement of time variations of intracellular Ca2+ concentrations of cultured GnRH neurons taken from the olfactory placode region of monkey embryos showed that it is most likely that they are not spike-to-spike synchronized (65). This is because the observed cytosolic Ca2+ oscillations with a period of a few minutes which were known to be associated with plasma membrane electrical 33  activities are not synchronized cycle after cycle except for one peak once every hour. If synaptic coupling plays an important role between GnRH neurons, these Ca2+ oscillations which are indirect measures of the plasma membrane electrical activities are expected to synchronize cycle after cycle. The observations reported in (65) seem to suggest that the membrane electrical activities are synchronized once every hour probably by a mechanism independent from synaptic interactions. Similar observations were reported in mice (92). Such Ca2+ oscillations and hourly synchrony were also observed in the non-neuronal cells in the same culture (99). However, the synchronization is absent in the absence of GnRH neurons in such cultures. Although the authors suggest that these data support a role of non-neuronal cells in producing the synchronization of Ca2+ oscillations once every hour, the same data could be explained even better by the existence of a Ca2+ oscillation-independent mechanism for the hourly rhythm that synchronizes the Ca2+ oscillations in both GnRH neurons and non-neuronal cells. This explains why synchrony does not occur in the absence of GnRH neurons because only the GnRH neurons participate in the GnRH pulse generation. It is the hourly pulses of GnRH observed by the same group in the same system (100) that synchronizes these Ca2+ oscillations but not the opposite as suggested by these authors. One mechanism for GnRH pulse generation that is independent of the minute time scale Ca2+ oscillations is reported in cultured GnRH neurons and GT1 cells (68) and enzymatically dispersed hypothalamic tissue cultures containing GnRH neurons (66, 101). These works revealed that GnRH neurons express GnRH receptors thus allowing autocrine regulations of GnRH secretion by its own release. Mathematical modeling studies demonstrated that such a mechanism is viable and extremely robust (69, 70). In this autocrine mechanism, electrical activities of the plasma membrane is not a direct player for the pulse generation although its role for maintaining the intracellular Ca2+ stores and for sustaining the readily releasable pool of GnRH containing vesicles could not be dismissed. GnRH binding is known to activate G-proteins that trigger Ca2+ release from the IP3 -sensitive stores and increased cAMP production. Both second messengers can reset the plasma membrane potential through a number of known Ca2+ -dependent and cAMP-dependent currents. This new explanation of existing data remains a conjecture until firmly backed by further modeling and experimental supports. We would like to refer to this conjecture Li’s Conjecture since it was first formulated by Li immediately after reading the paper by Terasawa et al. (65). This requires a direct coupling between a good model of the plasma membrane electrical activities and the autocrine regulation model. In an attempt to couple the existing models of plasma membrane electrical activities in GnRH neurons (72) to the autocrine regulation model of GnRH pulse generation, we realized that the two models are not identical and there is space for further improvements. This paper presents a combined version of existing models 34  that coherently reproduces all the important experimental data that were previously previously modeled in two different models as well as data that were not accounted for by them. It reduced the number of ion channels to a number that is believed minimal for reproducing the experimental data outlined in the objectives of the modeling study. As a result, the number of equations are also reduced in the simplified version. Two new types of membrane bursting modes are predicted by this model that have not yet been reported in experimental literature on GnRH neurons. However, a rich variety of membrane potential bursting has been reported (64, 85–90) although no specific underlying mechanism has been rigorously derived. Models of other endocrine cells such the pancreatic beta cells (102) and pituitary gonadotrophs (94) have explored the role of store operated Ca2+ entry in similar kinds of bursting and the associated [Ca2+ ]i oscillations. There often exists a debate concerning the validity of applying conclusions based on data collected in cultured GnRH neurons from embryonic tissues and immortalized cell lines such as GT1 cells to GnRH neurons in mature slices and even in vivo. So far as the modeling is concerned, the model presented here as well as the previously published ones were based almost uniquely on data collected from GT1 cells. We soon discover that this model can easily be applied to observations obtained in mature slices. Often a slight change in the values of some parameter is enough to switch from the behavior of one to the other. The lesson we learned here is that many observed behaviors in these cells are generated by mechanisms that are shared by all these cells but not by the specific features of them. The fact that all the currents that we introduced in this model and that ligand binding to membrane receptors activates a similar set of G-proteins that triggers a similar set of second messenger cascades in all these preparations seems sufficient to make their dynamical behaviors similar. In this sense, mathematical models provide valuable insights into the potential similarities in the behaviors in diversely different preparations of GnRH neurons and the reasons for such similarities. Sometimes, such conclusions could even go beyond GnRH neurons. One finds striking similarity in electrical behaviors between different endocrine cells such as pancreatic beta cells, pituitary gonadotrophs, lactotrophs, somatotrophs, oxytocin and vasopressin secreting neurons, and GnRH neurons. The immediate follow-up to this work is to couple the membrane electrical model to the autocrine regulation model previously proposed. The goal is to show that Li’s Conjecture is viable at least in the modeling frame work. As far as we know now, there no reason why this would not work. A far more challenging goal is to modify the Khadra-Li model (69) to reflect a more realistic situation under in vivo conditions. Recent data seem to suggest that GnRH neurons possess huge dendrites that could extend in excess of 1 mm from the cell body (103). They contain large dense core 35  vesicles (LDCV) each containing gigantic amount of neuropeptides. The dendrites are predominant bipolar and some contain 3 to 5 large branches. Dendrites from different GnRH neurons show occasional apposition/overlapping with each other, sometimes over significant distances. There is even report on direct bridging between the dendrites of GnRH neurons (63) allowing GnRH neurons to form a network through interconnecting and sharing their dendritic spaces and contents. If the Khadra-Li model survives such a realistic network structure with the realistic model of membrane potentials presented here, Li’s Conjecture will be finally tested. It does not matter whether the conjecture is proved right or wrong, more in-depth understanding of GnRH pulse generator will be achieved. In particular, the exact roles played by the membrane electrical activities in the GnRH pulse generation will be better understood.  36  2.6  Figures  Figure 2.1: Simultaneous Vm and [Ca2+ ]i responses of GT1 neurons following (A), the application of 100 nM GnRH followed by 100 nM apamin; and (B), the application of 5 µM Tg followed by 100 nM apamin. Modified from Van Goor et al. (77) (Kindly provided by Stanko Stojilkovic. Copyright 1999, National Academy of Sciences, U.S.A.)  37  Figure 2.2: Diagram of the model, showing the influence of GnRH receptor signaling on electrical activities and Ca2+ dynamics. IP3 activates Ca2+ release from stores via IP3 receptors (IP3 R), cAMP activates NSC channels, cytosolic Ca2+ activates SK channels, and ER Ca2+ inhibits SOC channels. The Ca2+ fluxes jin , jout , jref , and jrel are indicated with arrows showing the direction of the flux via its mediator.  38  20  ↓ 5 pA  a  ↓ 15 pA  A  b  c  Vm (mV)  0 −20 −40 −60  20  2  a  8  h 0.4  20  40 t (ms)  60  0  D b  −200 −300  0.2  −60  12  c  −100  0.6  −40  10  0  C  0.8  c  −20  0  6 t (s) 1  B  b  0 Vm (mV)  4  INa (pA)  −80 0  −60 −40 −20 V (mV)  0  20  −400 0  a 20  40 t (ms)  60  Figure 2.3: Model response to current injection. (A) Membrane potential showing the reduction in spike amplitude and the increase in spike frequency after current injections of 5 and 15 pA at 6 and 9 seconds, respectively. (B-D) Membrane potential waveforms, inactivation gating variable h, TTX-sensitive Na+ current waveforms for spikes labeled a-c in (A). Note that in (C), h is plotted as a function of membrane potential, so that during the spike the trajectory plotted travels clockwise, reaching the spike peak on the right.  39  Figure 2.4: Model response to GnRH and apamin. GnRH is modeled as an increase in IP3 to 1 µM beginning at t=10 s, and apamin is modeled by setting gSK to zero at t=30 s. The formatting of Figures 2.4 - 2.8 is the same. (A) Membrane potential. (B) Ca2+ sensitive currents ISK (upper trace) and ISOC (lower trace). (C,D) Spatio-temporal profile of [Ca2+ ]i and [Ca2+ ]ER , respectively. The spatial average of Ca2+ concentration and the Ca2+ concentration at the membrane in each compartment are overlayed as a thick red and thin black line, respectively, at r=10 µm. Time axis (t) is identical for all panels. Radius axis (r) is the same in C and D.  40  Figure 2.5: Model response to Tg and apamin. Tg is modeled by reducing the value of νe to zero beginning at t=10 s, while apamin is modeled as in the GnRH response (Fig. 2.4).  41  Figure 2.6: Model response to Forskolin. Forskolin is modeled by increasing in cAMP to 1 µM . (C) Note the localized [Ca2+ ]i response. (D) ER loading due to the high frequency AP firing.  42  Figure 2.7: Model response to IP3 when IP3 receptor inactivation is slow. The following parameter values differ from those listed in the Appendix: τhi =2 s and P =0.5 pL·ms−1 .  43  Figure 2.8: Spontaneous store operated bursting. The following parameter values differ from those listed in the Appendix: ka =15 mV , gCaL =1.7 nS, L=0.004 pL · ms−1 , νe =2 µM · pL · ms−1 , and cAMP=0.5 µM .  44  ↓ GnRH  20  ↓ Apa  A  V (mV)  0 −20 −40 −60 −80  B  [Ca2+]  i,av  (µM)  2 1 0 ↓ Tg  20  ↓ Apa  C  V (mV)  0 −20 −40 −60 −80  [Ca2+]  i,av  (µM)  D 1  0 0  5  10  15  20  25  30  35  t (s)  Figure 2.9: Response of the simplified model to GnRH, Tg, and apamin. (A) Membrane potential and (B) cytosolic Ca2+ response to GnRH and apamin, modeled as in the spatial model (Fig. 2.4). (C) Membrane potential and (D) cytosolic Ca2+ response to Tg and apamin, modeled as in the spatial model (Fig. 2.5). In (B) and (D), the lower red trace is the [Ca2+ ]i at the membrane, CR , while the black trace is the average [Ca2+ ]i .  45  2.7  Bibliography  61. Knobil, E., T. M. Plant, L. Wildt, P. E. Belchetz, and G. Marshall, 1980. Control of the rhesus monkey menstrual cycle: permissive role of hypothalamic gonadotropinreleasing hormone. 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Dissecting autocrine effects on pulsatile release of gonadotropin-releasing hormone in cultured rat hypothalamic tissue. Exp Biol Med (Maywood) 229:56–64. 102. Chay, T. R., 1996. Modeling slowly bursting neurons via calcium store and voltageindependent calcium current. Neural Comput 8:951–78. 103. Campbell, R. E., S. K. Han, and A. E. Herbison, 2005. Biocytin filling of adult gonadotropin-releasing hormone neurons in situ reveals extensive, spiny, dendritic processes. Endocrinology 146:1163–9.  49  Chapter 3  Conclusion 3.1  Summary of Results  Based on the previous work of LeBeau et al. (104), we’ve stripped down the mechanisms required to reproduce some key experimental results that are intimately related to GnRH pulse generation. Our spatio-temporal approach yields an accurate treatment of Ca2+ handling. It highlights the difference between Ca2+ profiles localized near the membrane, generally associated with electrical activity, and Ca2+ responses that are global in the cell, such as release from stores. With the phenomenological simplification of our spatial model (see §2.4), we are left with a much simpler model that preserves the accuracy of total cell calcium as well as the [Ca2+ ]i profile near the membrane. We used a much simpler description of the fast sodium current and the non-specific cation current. Furthermore, we did not find a need to include the T-type Ca2+ current or the M-type K+ current. We thus reduced the electrophysiology of the GnRH neuron model to what we believe is a minimal state. The accurate handling of Ca2+ has allowed a realistic reproduction of the behavior of Ca2+ sensitive currents, such as the effect of the SK K+ channels (see 2.1B). With our simplified electrophysiology, however, we were unable to account for some details of the electrical response. First, GnRH-induced depolarization (see 2.1A) was not achieved. It is not known exactly which features are lacking in the electrophysiology that would allow such a response to be achieved, although it seems that a GnRH receptor coupled effect is likely. Second, features that may be important in terms of bursting, such as slow afterdepolarizing potentials (sADPs) have not been accounted for. Some potential candidates are listed below in §3.2.1. The utility of this model hinges on the fact that it is simpler to use than previous models, while the Ca2+ dynamics are more accurate. Furthermore, it has coupling points to two major second messengers, IP3 and cAMP, allowing easy coupling to various GPCR mediated signaling events or to models that include such messengers as dynamic variables, such as the model of GnRH pulsatility by Khadra and Li (105). These coupling points also allow a preliminary exploration into mechanisms for bursting, and the model predicts two possible mechanisms here. For more discussion of these points, see §3.3. 50  3.2  Unexplored Features of GnRH Neurons  There are several theoretical endeavors, however, that deserve a more in depth treatment. A more thorough investigation of a detailed physiological model containing all the reported ionic channels of GnRH neurons in vivo should be undertaken. Such a model could help to shed light on the role each channel type plays in shaping the electrical activity. Next, in §3.2.2, we describe experimental reports of other signaling mechanism not considered in the present work. A thorough evaluation of the interaction of various second messenger systems with electrical and Ca2+ handling behavior would shed light on the potential importance to GnRH pulsatility of each of such effects.  3.2.1  Electrophysiological Details  GnRH neurons, as do many neurons, are complex, with many different types of ion channels and signaling systems having been reported experimentally. In the present work, we have simplified the details of the electrophysiology of GnRH neurons. For our purposes, and in the context of the future work we hope to achieve with this model, such details are not of crucial importance. Instead, we require a basic spiking mechanism that does a reasonable job at describing Ca2+ influx due to electrical activities. That being said, some of the experimentally reported ion channels may turn out to play a larger role than previously thought. Of particular interest in the context of GnRH pulsatility are those reports of ion channels whose activity can be regulated by various second messengers, such as [Ca2+ ]i , [Ca2+ ]ER , cAMP, or levels of protein kinase activity. We considered in our model a simple delayed rectifier IK , inward rectifier Kir , and SK current. A few of the other K+ currents with potential relevance to GnRH receptor do exist. M-type potassium currents have been demonstrated in GnRH neurons, and were considered by Van Goor et al. (116) and LeBeau et al. (104), as described in §1.4. We found the M-current to be unnecessary for qualitative reproduction of the results in Chapter 2. Interestingly, however, a recent report showed that these channels are activated by GnRH in adult transgenic mouse GnRH neurons (111). The fast transient potassium channel has been reported by many authors. Its role in many systems is to increase latency to firing, as well as to aid in repolarization of action potentials, which has been demonstrated in adult GnRH neurons by DeFazio et al. (114). They noted that IA is much more prominent in adult neurons than it appears to be in embryonic or immortalized GnRH neurons. This current, due to its fast inactivation, could work with the sodium current to shift from sharp neuron-like to broad endocrine-like APs when GnRH neurons are depolarized. This would occur for a similar reason to what was demonstrated in §2.3.1; the loss of inactivation due to 51  depolarized baseline would lead to reduced IA , and a reduced action potential fall rate, thus increasing the duration of a spike. BK channels, calcium and voltage activated K+ channels, have been reported in GT1 neurons (117, 118) and adult rats (115). These channels can have a large impact due to their large conductance, and may prove to play a role in bursting. Indeed, such a channel, under the control of localized domains of high [Ca2+ ]i generated by opening of nearby Ca2+ channels, was shown to be important in conferring somatotropes in the pituitary with a plateau bursting phenotype (119). GIRK channels, which are activated by the βγ subunits liberated by activation Gi/o , have been shown to be expressed in GT1 neurons by Hu et al. (112). They showed that an inward rectifying current is substantially activated by LH, for which the receptor is Gi/o coupled. This mechanism would yield a similar effect to the reduction of cAMP and cAMP activated current that is also expected from Gi/o coupling. KATP channels are a candidate for the Kir channel observed in many GnRH neuron preparations. It’s expression has been documented by Zhang et al. (113) in adult transgenic mice. This channel confers metabolic control of excitability, since the KATP channel is activated when the ADP/ATP ratio is high. Such a channel is very important for normal insulin secretion in pancreatic β-cells (120), and has been implicated in their bursting behavior (see for example (143). A variety of Ca2+ currents have been demonstrated in GT1 neurons (121, 122), embryonic, and adult neurons. Both High Voltage (L, N, R, and P/Q-type) and Low Voltage transient (T-type) calcium currents are reported. Different subtypes may have different sub-cellular localizations, and may even bind to macromolecular complexes to allow signaling via very highly regulated Ca2+ signals, as is often the case with Ca2+ activated K+ channels like BK (123). Lewis and Ikeda (124) report that GnRH receptor activation inhibits N-type Ca2+ channels in dissociated adult rat superior cervical ganglion neurons. Furthermore, Haneda and Oka (125) have reported, in a teleost fish model, that N- and R-type calcium channels are inhibited in GnRH neurons, but not L-type or T-type currents. The recent study by Hiraizumi et al. (115) in adult short term cultured GnRH neurons indeed showed that about half of BK current recorded was sensitive to both N- and R-type currents.  3.2.2  Second Messenger Signaling  The effects of the second messenger dependent changes in electrical and calcium dynamics remain to be explored in depth theoretically. As described in the previous section (§3.2.1), there are several ion channels that are coupled directly to GPCR signaling via [Ca2+ ]i , cAMP, or βγ subunits of G proteins. Indeed, Han et al. (130) found that firing rate was much slower in whole cell versus perforated patch mode of patch clamp  52  recording, suggesting that soluble intracellular signaling molecules are important for maintaining normal firing rates. Given the evidence for cAMP stimulating firing rate, this could be consistent with normal levels of cAMP being higher in the cell than in the patch pipette. A recent study using microarray technology reported expression of a total of 50 subunit mRNAs for a variety of ionotropic and G protein coupled receptors, several of which were previously not known to exist in GnRH neurons. Further studies in GnRH physiology modeling could consider exploring such features by aiming to reproduce the responses to several GCPR mediated signaling events that have been reported experimentally, some of which are presented here. Adenylyl Cyclase. Besides the GnRH receptor, several other receptors have been shown to be expressed and functionally coupled to AC. Gi/o coupled receptors, which reduce cAMP, excitability, and GnRH secretion, include (but are not limited to): • Serotonin 5HT1A receptor (131) • LH receptor (132) • M2 muscarinic receptor (133) • Estrogen receptors (134) Gs coupled receptors, which increase cAMP, excitability, and GnRH secretion, include (but are not limited to): • Serotonin 5HT4 receptor (131) • LH receptor (132) • β 1 -adrenergic receptor (135) Calcium regulates some isoforms of adenylyl cyclase. Three studies have reported on the expression of AC subtypes in GT1 cells (136–138). All reports thus far agree on the presence of calcium inhibited isoforms AC III, V, VI, and IX. Krsmanovic et al. (137) report a predominance of expression of AC I, a calcium activated isoform, and an associated positive relationship between cAMP and [Ca2+ ]i . More recently, Martin et al. (138) used quantitative reverse-transcriptase PCR to quantify the relative abundance of each isoform, reporting 67% calcium insensitive isoforms and 32% calcium inhibited isoforms, while finding very little expression of ACI. Despite the conflicting reports, these results suggest that crosstalk between [Ca2+ ]i and cAMP levels via AC may be worth considering in a detailed model. 53  Another possible way to get a slow depolarizing current such as the one due to ISOC previously described could involve the calcium activated isoform of adenylyl cyclase. Calcium release from stores, either receptor mediated or thapsigargin induced, could lead to activation of AC and production of cAMP. This would slowly increase excitability via a cAMP activated inward current. Phospholipase C There are also reports of PLC coupled receptors besides GnRH receptors. These include kisspeptin receptor GPR54 (139), serotonin 5HT2C receptor (131) and M1 muscarinic receptors (133). Todman et al. (140) reported a a potent hyperpolarization of adult GnRH neuron membrane potential due to somatostatin, the mechanism for which is unknown at this time.  3.3 3.3.1  Future Directions for Modeling of GnRH Neurons Characterization of Calcium Oscillations and Bursting  An in depth characterization of possible mechanisms for Ca2+ transients and bursting in GnRH neurons remains to be undertaken. Such a study appears to be necessary to make sense of the wide variety of bursting modes, as well as the long term modulation, such as clusters and episodes of bursting interspersed with periods of quiescence. Indeed, it may be that the mechanism of GnRH pulsatility plays a large role in modulating such electrical activity and Ca2+ oscillations (see §3.3.3 and §3.3.4). A model such as the present one is a starting point, from which exploration of various mechanisms for bursting could be undertaken. Key goals for such a research program would be to test the feasibility of various burst generation mechanisms for Ca2+ oscillations and electrical bursting observed experimentally, and thereby predict the outcomes of experiments that would help to discriminate between mechanisms. One specific example goal would be to determine the role CNG or HCN channels in bursting, and to test whether sufficient machinery is left to generate bursting in their absence, as seen experimentally Constantin and Wray (141, 142). In the exploration of bursting in GnRH neurons, it would be informative to study the literature in pancreatic β-cells. Many similar components of electrophysiology and Ca2+ handling are common in the two systems, and β cells have been studied in much more depth (see for example, Bertram and Sherman (143)). A second source of valuable analogies could lie in the magnocellular system, where the bursting of vasopressin and oxytocin neurons is also better studied (144).  54  3.3.2  Stimulus-Secretion Coupling  Another important physiological process that must be characterized if an accurate model of GnRH pulsatility is to be achieved is the process of secretion in these neurons. Like other peptidergic neurons, GnRH neurons secrete their namesake hormone in large dense core vesicles. Drawing upon literature from oxytocin neurons (144), and gonadotropes (145), the roles of electrical activity and calcium dynamics in regulating large dense core vesicle cycling, priming, and secretion should be explored. Furthermore, the role of second messengers, such as cAMP and IP3 will need to be elucidated. Of particular importance to the in vivo situation, as discussed below in §3.3.4, is the concept of dendritic secretion. As reviewed in Ludwig and Leng (144), many peptidergic neurons, such as the oxytocin and vasopressin neurons of the magnocellular system in the hypothalamus, exhibit dendritic release of their respective peptide hormones. Furthermore, they have receptors for molecules that they release, and are thus able to respond physiologically to such secretory events. Another important feature of such systems is that they exhibit a process called priming. Priming occurs when a stimulus is able to activate mechanisms that ready dense core vesicles for release by a subsequent stimulus (144). Gonadotropin vesicles in pituitary gonadotropes are primed by GnRH, and the effect of such priming has been explored theoretically by Scullion et al. (145). Such secretory activity may be important for synchronizing the GnRH neuronal network, as has been demonstrated in oxytocinergic neurons in the hypothalamus (146).  3.3.3  A Single Cell Model for GnRH Pulsatility  An immediate application of the model presented in this thesis is to couple it to the pulsatility model of Khadra and Li (105). The simplest case involves the hypothetical situation of a single GnRH neuron in a droplet (105). Assuming that the volume of the droplet is small enough that the concentration of GnRH in the extracellular medium achieves physiological levels, the autocrine mechanism will generate pulsatility in a single cell. Such a model will be crucial for exploring the dynamical interactions of autocrine signaling of GnRH neurons with their electrical activities and calcium dynamics. Again, of particular interest will be to explore the resulting electrical, Ca2+ and cAMP signals and the constraints placed upon them by stimulus-secretion coupling.  3.3.4  A Network Model of GnRH Pulsatility  Despite the lack of evidence for a common pool of GnRH, the evidence reported does not exclude the possibility that GnRH autocrine effects could be important in GnRH pulsatility in vivo. Several experimental findings support the hypothesis that GnRH  55  neurons use GnRH to communicate directly with each other and coordinate their behavior. First, GnRH receptor expression has been shown in vivo (140, 147). Second, direct electrophysiological responses to GnRH have also been reported in adult transgenic mice expressing GFP in GnRH neurons (111, 140, 147). Although somewhat rare, there are reports of candidates for intercellular communication between GnRH neurons in vivo. GT1 neurons, given time in culture together, develop gap junction coupling as demonstrated functionally and by the expression of connexin-26 (148). Electron microscopic studies have reported up to 10% of synapses impinging on GnRH neuron cell bodies are GnRH immunoreactive (149–151). Close associations between dendrites of GnRH neurons have been observed, and even in some instances direct bridges between GnRH neurons has been reported (152). Thus, the feasibility of the autocrine mechanism in vivo relies on GnRH neurons being able to secrete GnRH at the level of their extensive dendrites, as in other hypothalamic peptidergic neurons (for a review, see Ludwig and Leng (144)). The topology of the network remains to be explored. Some constraints include the fact that most GnRH neurons are bipolar, with one dendrite and one axon. Perhaps the biggest challenge will be to determine how GnRH neurons, through dendritic GnRH release, will communicate amongst themselves. Dendrites are long, and far from spherical, the geometry will be very different from the sphere used in the present model. However, since the electrical activities are fast compared to processes such as vesicle priming, secretion, and GPCR signaling, we can use the simplified point neuron model for electrical activities. An analogy for this type of network synchronization by autocrine signaling has been far better studied in similar neurons in the hypothalamus: oxytocin neuron synchronized bursting. Rossoni et al. (146) have modeled the autocrine regulation that occurs in bundles of oxytocin neurons, and have shown that such a mechanism can lead to the synchronization of their behavior. It would be very interesting to adapt such an approach to the GnRH neuronal network, using the GnRH receptor based autocrine mechanism studied by Khadra and Li (105), to evaluate this as a potential mechanism for pulsatility. Furthermore, the use of the model presented here could allow predictions to be made about the associated electrical activities of GnRH neurons that should be observed during pulsatility.  56  3.4  Bibliography  104. LeBeau, A. P., F. Van Goor, S. S. Stojilkovic, and A. Sherman, 2000. Modeling of membrane excitability in gonadotropin-releasing hormone-secreting hypothalamic neurons regulated by Ca2+-mobilizing and adenylyl cyclase-coupled receptors. J Neurosci 20:9290–7. 105. Khadra, A., and Y. X. Li, 2006. A model for the pulsatile secretion of gonadotropinreleasing hormone from synchronized hypothalamic neurons. Biophys J 91:74–83. 106. Chu, Z., and S. M. Moenter, 2006. Physiologic regulation of a tetrodotoxinsensitive sodium influx that mediates a slow afterdepolarization potential in gonadotropin-releasing hormone neurons: possible implications for the central regulation of fertility. J Neurosci 26:11961–73. 107. Izhikevich, E., 2007. 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Catt, 2006. Essential role of G protein-gated inwardly rectifying potassium channels in gonadotropininduced regulation of GnRH neuronal firing and pulsatile neurosecretion. J Biol Chem 281:25231–40. 113. Zhang, C., M. A. Bosch, J. E. Levine, O. K. Ronnekleiv, and M. J. Kelly, 2007. Gonadotropin-releasing hormone neurons express K(ATP) channels that are regulated by estrogen and responsive to glucose and metabolic inhibition. J Neurosci 27:10153–64. 114. DeFazio, R. A., S. Heger, S. R. Ojeda, and S. M. Moenter, 2002. Activation of A-type gamma-aminobutyric acid receptors excites gonadotropin-releasing hormone neurons. Mol Endocrinol 16:2872–91.  57  115. Hiraizumi, Y., I. Nishimura, H. Ishii, N. Tanaka, T. Takeshita, Y. Sakuma, and M. Kato, 2008. Rat GnRH Neurons Exhibit Large Conductance Voltage- and Ca(2+)-Activated K(+) (BK) Currents and Express BK Channel mRNAs. J Physiol Sci 58:21–9. 116. Van Goor, F., A. P. LeBeau, L. Z. Krsmanovic, A. Sherman, K. J. Catt, and S. S. Stojilkovic, 2000. Amplitude-dependent spike-broadening and enhanced Ca(2+) signaling in GnRH-secreting neurons. Biophys J 79:1310–23. 117. Spergel, D. J., K. J. Catt, and E. Rojas, 1996. Immortalized GnRH neurons express large-conductance calcium-activated potassium channels. Neuroendocrinology 63:101–11. 118. Nishimura, I., K. Ui-Tei, K. Saigo, H. Ishii, Y. Sakuma, and M. Kato, 2008. 17betaestradiol at physiological concentrations augments Ca(2+) -activated K+ currents via estrogen receptor beta in the gonadotropin-releasing hormone neuronal cell line GT1-7. Endocrinology 149:774–82. 119. Van Goor, F., Y.-X. Li, and S. S. Stojilkovic, 2001. Paradoxical Role of Large-Conductance Calcium-Activated K+ (BK) Channels in Controlling Action Potential-Driven Ca2+ Entry in Anterior Pituitary Cells. J. Neurosci. 21:5902– 5915. 120. Miki, T., K. Nagashima, F. Tashiro, K. Kotake, H. Yoshitomi, A. Tamamoto, T. Gonoi, T. Iwanaga, J. Miyazaki, and S. Seino, 1998. 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Selective modulation of voltage-gated calcium channels in the terminal nerve gonadotropin-releasing hormone neurons of a teleost, the dwarf gourami (Colisa lalia). Endocrinology 145:4489–99. 126. Zhang, C., T. A. Roepke, M. J. Kelly, and O. K. Ronnekleiv, 2008. Kisspeptin depolarizes gonadotropin-releasing hormone neurons through activation of TRPClike cationic channels. J Neurosci 28:4423–34. 58  127. Lewis, R. S., 2007. The molecular choreography of a store-operated calcium channel. Nature 446:284–7. 128. Yuan, J. P., W. Zeng, G. N. Huang, P. F. Worley, and S. Muallem, 2007. STIM1 heteromultimerizes TRPC channels to determine their function as store-operated channels. Nat Cell Biol 9:636–45. 129. Van Goor, F., L. Z. Krsmanovic, K. J. Catt, and S. S. Stojilkovic, 1999. Coordinate regulation of gonadotropin-releasing hormone neuronal firing patterns by cytosolic calcium and store depletion. Proc Natl Acad Sci U S A 96:4101–6. 130. Han, S. K., I. M. Abraham, and A. E. Herbison, 2002. Effect of GABA on GnRH neurons switches from depolarization to hyperpolarization at puberty in the female mouse. Endocrinology 143:1459–66. 131. Wada, K., L. Hu, N. Mores, C. E. Navarro, H. Fuda, L. Z. Krsmanovic, and K. J. Catt, 2006. Serotonin (5-HT) receptor subtypes mediate specific modes of 5-HT-induced signaling and regulation of neurosecretion in gonadotropin-releasing hormone neurons. Mol Endocrinol 20:125–35. 132. Mores, N., L. Z. Krsmanovic, and K. J. Catt, 1996. Activation of LH receptors expressed in GnRH neurons stimulates cyclic AMP production and inhibits pulsatile neuropeptide release. Endocrinology 137:5731–4. 133. Krsmanovic, L. Z., N. Mores, C. E. Navarro, S. A. Saeed, K. K. Arora, and K. J. Catt, 1998. Muscarinic regulation of intracellular signaling and neurosecretion in gonadotropin-releasing hormone neurons. Endocrinology 139:4037–43. 134. Navarro, C. E., S. Abdul Saeed, C. Murdock, A. J. Martinez-Fuentes, K. K. Arora, L. Z. Krsmanovic, and K. J. Catt, 2003. Regulation of cyclic adenosine 3’,5’-monophosphate signaling and pulsatile neurosecretion by Gi-coupled plasma membrane estrogen receptors in immortalized gonadotropin-releasing hormone neurons. Mol Endocrinol 17:1792–804. 135. Martinez de la Escalera, G., A. L. Choi, and R. I. Weiner, 1992. Generation and synchronization of gonadotropin-releasing hormone (GnRH) pulses: intrinsic properties of the GT1-1 GnRH neuronal cell line. Proc Natl Acad Sci U S A 89:1852– 5. 136. Vitalis, E. A., J. L. Costantin, P. S. Tsai, H. Sakakibara, S. Paruthiyil, T. Iiri, J. F. Martini, M. Taga, A. L. Choi, A. C. Charles, and R. I. Weiner, 2000. Role of the cAMP signaling pathway in the regulation of gonadotropin-releasing hormone secretion in GT1 cells. Proc Natl Acad Sci U S A 97:1861–6. 137. Krsmanovic, L. Z., N. Mores, C. E. Navarro, M. Tomic, and K. J. Catt, 2001. Regulation of Ca2+-sensitive adenylyl cyclase in gonadotropin-releasing hormone neurons. Mol Endocrinol 15:429–40. 138. Martin, C., J. S. Jacobi, G. Nava, M. C. Jeziorski, C. Clapp, and G. Martinez de la Escalera, 2007. GABA inhibition of cyclic AMP production in immortalized 59  GnRH neurons is mediated by calcineurin-dependent dephosphorylation of adenylyl cyclase 9. Neuroendocrinology 85:257–66. 139. Liu, X., K. Lee, and A. E. Herbison, 2008. Kisspeptin excites gonadotropinreleasing hormone neurons through a phospholipase C/calcium-dependent pathway regulating multiple ion channels. Endocrinology 149:4605–14. 140. Todman, M. G., S. K. Han, and A. E. Herbison, 2005. Profiling neurotransmitter receptor expression in mouse gonadotropin-releasing hormone neurons using green fluorescent protein-promoter transgenics and microarrays. Neuroscience 132:703– 12. 141. Constantin, S., and S. Wray, 2008. Gonadotropin-releasing hormone-1 neuronal activity is independent of cyclic nucleotide-gated channels. Endocrinology 149:279– 90. 142. Constantin, S., and S. Wray, 2008. Gonadotropin-Releasing Hormone-1 Neuronal Activity Is Independent of Hyperpolarization-Activated Cyclic NucleotideModulated Channels but Is Sensitive to Protein Kinase A-Dependent Phosphorylation. Endocrinology 149:3500–3511. 143. Bertram, R., and A. Sherman, 2005. Negative Calcium Feedback: The Road from Chay-Keizer. In S. Coombes, and P. Bressloff, editors, Bursting: The Genesis of Rhythm in the Nervous System, World Scientific, London, chapter 2, 12–48. 144. Ludwig, M., and G. Leng, 2006. Dendritic peptide release and peptide-dependent behaviours. Nat Rev Neurosci 7:126–36. 145. Scullion, S., D. Brown, and G. Leng, 2004. Modelling the pituitary response to luteinizing hormone-releasing hormone. J Neuroendocrinol 16:265–71. 146. Rossoni, E., J. Feng, B. Tirozzi, D. Brown, G. Leng, and F. Moos, 2008. Emergent synchronous bursting of oxytocin neuronal network. PLoS Comput Biol 4. 147. Xu, C., X. Z. Xu, C. S. Nunemaker, and S. M. Moenter, 2004. Dose-dependent switch in response of gonadotropin-releasing hormone (GnRH) neurons to GnRH mediated through the type I GnRH receptor. Endocrinology 145:728–35. 148. Hu, L., A. J. Olson, R. I. Weiner, and P. C. Goldsmith, 1999. Connexin 26 expression and extensive gap junctional coupling in cultures of GT1-7 cells secreting gonadotropin-releasing hormone. Neuroendocrinology 70:221–7. 149. Chen, W. P., J. W. Witkin, and A. J. Silverman, 1989. beta-Endorphin and gonadotropin-releasing hormone synaptic input to gonadotropin-releasing hormone neurosecretory cells in the male rat. J Comp Neurol 286:85–95. 150. Leranth, C., L. M. Segura, M. Palkovits, N. J. MacLusky, M. Shanabrough, and F. Naftolin, 1985. The LH-RH-containing neuronal network in the preoptic area of the rat: demonstration of LH-RH-containing nerve terminals in synaptic contact with LH-RH neurons. Brain Res 345:332–6. 60  151. Pelletier, G., 1987. Demonstration of contacts between neurons staining for LHRH in the preoptic area of the rat brain. Neuroendocrinology 46:457–9. 152. Witkin, J. W., 1999. Synchronized neuronal networks: the GnRH system. Microsc Res Tech 44:11–8.  61  Appendix A  Model Description and Parameter Values Here we define functions and parameters not described in the text above. The following definitions are the same in both the spatial and simplified models, except where indicated. Table A.1 gives basic parameters including cell geometry, and conductances. The calcium flux densities are given by: jin = −αICa = −α(ICaL + ISOC + γINSC ) jout = jfil =  νp C 2 νn C 4 + C 2 + Kp2 C 4 + Kn4 νe C 2 + Ke2  C2  jrel = (L + P OI )(Ce − C). Calcium entry at the plasma membrane is obtained by scaling the whole-cell calcium current, ICa by α = (2F Acell )−1 (4.12e−3 µM · µm · ms−1 · pA−1 ) to obtain the per unit area flux required by the boundary condition at the membrane (See Eq. 2.3). The parameter γ (0.3) accounts for the fractional calcium conductance of the INSC channel as in LeBeau et al. (153). The νx and Kx , where x=p, n, and e are the maximal pump rates, and the cytosolic Ca2+ concentrations at which half-maximal activation is reached, respectively, for the plasma membrane calcium ATPase, sodium-calcium exchanger and sarcoplasmic-endoplasmic reticulum calcium ATPase (νe =1.3 µM · pL · ms−1 ; (in µm · µM · ms−1 ): νn =0.13, νx =0.04; (in µM ): Ke =0.2, Kn =1.3, Kx =0.1). The calcium release from stores is the sum of a leak and flux through the IP3 receptor, with flux rates L and P (0.0021 and 15 pL · ms−1 , respectively). Following Li and Rinzel (154), the open probability of the IP3 receptor (OI ) depends on cytosolic Ca2+ and the inactivation variable hi and is parameterized by IP3 concentration (I) as follows: OI =  I I + Ki  3  C C + Kca  3  h3i .  where Ki =0.1 µM and Ki =0.4 µM . The rate of change of the variable hi at a given 62  radius depends on the local cytosolic Ca2+ concentration at that radius as follows: τ hi  dhi = (Kd − (C + Kd )hi ), dt  where Kd =0.4 µM and τ hi , the time constant of IP3 receptor inactivation, is fast (2 ms) for the biphasic response. We keep the dynamics so that, by increasing τ hi , oscillatory release from the stores can be achieved. The voltage gated ionic currents are given by: INa = gNa m3∞ h(V − ENa ) ICaL = gCaL a2 (V − ECa ) IK = gK n4 (V − EK ) Iir = gir b∞ (V − EK ) where the activation variable m for INa is assumed to be fast and takes its equilibrium value, m∞ . The whole cell conductances are (in nS): gNa =11, gCaL =1.2, gK =25, and gir =1. The gating variables q ≡ h, a, and n are governed by τq  dq = q∞ − q. dt  The q∞ (V ) have the form q∞ = qmax /(1 + exp((Vm − Vq )/kq )) + qmin , with s = −1 for m∞ , a∞ and n∞ , s = 1 for h∞ and b∞ , and qmax and qmin are 1 and 0 respectively for all gating variables except b, which takes the values 0.8 and 0.2 respectively. Except for as indicated for Figure 2.8, the parameter values for the q∞ (V ) are identical for generation of all results presented (in mV ): Vm =-43, Vh =-55, Va =-29, Vn =-27, Vb =-80, km =6, kh =6, ka =15, kn =15, and kb =12. The τq (V ) have the form τq = τq /(exp((Vm − Vq )/kτq ) + zexp(−z(Vm − Vq )/kτq )), with z = 2 for τh , and z = 1 for τa , and τn . The parameter values for the τq (V ) are identical for generation of all results presented (in ms): τh =150, τa =10, and τn =40; (in mV ): Vτh =-65, Vτa =-29, Vτn =-33, kτh =15, kτa =25, and kτn =23. We use a Hill function to parameterize the activation of the NSC current by cAMP: INSC = gNSC  A2 (Vm − ENSC ) 2 KNSC + A2  where gNSC =0.3 nS is the whole cell conductance, A is the cytosolic cAMP concentration in µM , KNSC =2 µM is the cAMP concentration at which INSC is half maximally activated, and ENSC =72 mV is the Nernst potential based on the fractional conductance of Ca2+ and Na+ , given as ENSC = ENa + γ(ECa − ENa ). At this point, the hill 63  coefficient of 2 is not critical for any of the model behavior present, but is inspired from the cAMP dependence of CNG channels (155). Calcium dependent currents include an SK-type calcium-activated potassium current and a store-operated calcium current (SOC). The calcium-activated potassium current is given by ISK = gSK  CR8 8 (V − EK ), CR8 + KSK  where CR is the [Ca2+ ]i at the cell membrane, given by either C(R) in the spatial model or its approximation CR in the simplified model. gSK =1.5 nS is the whole cell conductance, and KSK =1 µM is the cytosolic Ca2+ concentration at which ISK is half maximally activated. The high hill coefficient is not critical to the described results. The store operated current is inhibited by high ER calcium, and is given by ISOC = gSOC  4 KSOC 4 4 (Vm − ECa ) KSOC + CeR  where CeR is the [Ca2+ ]ER near the cell membrane in the spatial model, or just the average value of [Ca2+ ]ER in the simplified model. gSOC =0.03 nS is the whole cell conductance, and KSOC =100 µM is the [Ca2+ ]ER at which ISOC is half maximally activated, and the Hill coefficient comes from Luik et al. (156).  64  Parameter  Symbol  basal [cAMP] basal [IP3 ] Cell Radius Cell Area Cell Volume Cytoplasmic Volume ER Volume Ca2+ Diffusion Coefficients: in Buffer Free Medium, Cytosol, ER Membrane Capacitance K+ , Na+ , Ca2+ Reversal Potentials  Value  A I R Acell (4πR2 ) Vcell (4/3πR3 · 1e−3 ) Vcyt (0.85·Vcell ) VER (0.15·Vcell ) Do , D, DER Cm EK , ENa , ECa  0.7 µM 0.01 µM 10 µm 1257 µm2 4.19 pL 3.56 pL 0.63 pL 300, 15, 1 µm2 s−1 14 pF -80, 60, 100 mV  Table A.1: Table of Some Standard Model Parameter Values.  65  A.1  Bibliography  153. LeBeau, A. P., F. Van Goor, S. S. Stojilkovic, and A. Sherman, 2000. Modeling of membrane excitability in gonadotropin-releasing hormone-secreting hypothalamic neurons regulated by Ca2+-mobilizing and adenylyl cyclase-coupled receptors. J Neurosci 20:9290–7. 154. Li, Y. X., and J. Rinzel, 1994. Equations for InsP3 receptor-mediated [Ca2+]i oscillations derived from a detailed kinetic model: a Hodgkin-Huxley like formalism. J Theor Biol 166:461–73. 155. Nakamura, T., and G. H. Gold, 1987. A cyclic nucleotide-gated conductance in olfactory receptor cilia. Nature 325:442–4. 156. Luik, R. M., B. Wang, M. Prakriya, M. M. Wu, and R. S. Lewis, 2008. Oligomerization of STIM1 couples ER calcium depletion to CRAC channel activation. Nature 454:538–542.  66  

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