Models of hypothalamus

• Richard Bertram, Department of Mathematics, Florida State University, Tallahassee, Florida

• Dr. Joel Tabak, Department of Biological Science, Florida State University

• Dr. Natalia Toporikova, Florida State University

The Hypothalamus and the Pituitary Gland

The hypothalamus is the region of the brain that controls hormone release, and is thus part of the neuroendocrine system. The pituitary gland is closely allied with the hypothalamus, both functionally and physically. Some hypothalamic neurons project directly to the neural lobe of the pituitary. Others project to the median eminence, where they connect with blood vessels that supply the pituitary. In either case, the hypothalamic neurons influence cells of the pituitary, either upregulating or downregulating their activity. Hormones secreted from the pituitary in turn influence neurons throughout the brain (including the hypothalamus), and endocrine glands located in other parts of the body such as the adrenal, pineal, thyroid, parathyroid, thymus, heart, stomach, duodenum, pancreas, testes, ovaries, and placenta.

Figure 1: Connections from the hypothalamus to pituitary, and some targets of pituitary hormones.

The pituitary gland has two well defined regions: the posterior pituitary (or neurohypophysis), and the anterior pituitary (or adenohypophysis). The posterior pituitary is innervated by the terminals of oxytocin and vasopressin neurons which originate in the hypothalamus and which synapse directly on capillaries that fenestrate the pituitary. However the anterior pituitary does not receive direct neural input from the brain, but instead is connected to the hypothalamus through a vascular network called the hypophyseal portal system. Specific releasing factors are secreted into this vascular system by the hypothalamus and subsequently act upon their target cells in the anterior pituitary.

The anterior pituitary gland contains several cell types, each secreting a specific hormone:

• Corticotrophs secrete adrenocorticotropic hormone (ACTH), which stimulates the synthesis and release of glucocorticoids from the adrenal cortex.
• Somatotrophs secrete growth hormone, which promotes the growth of most cells in the body.
• Lactotrophs secrete prolactin, which initiates milk synthesis in the mammary glands and has many other roles related to reproduction and parental behavior.
• Gonadotrophs secrete two hormones, luteinizing hormone (LH) and follicle stimulating hormone (FSH). In the female, LH stimulates ovulation and the formation of corpora lutea in the ovary. In the male, LH stimulates the Leydig cells to secrete androgens such as testosterone. FSH promotes the development of gametes and the secretion of gonadal hormones in males and females.
• Thyrotrophs secrete thyroid stimulating hormone (TSH), which stimulates the synthesis and release of thyroxine and triiodothyronine from the thyroid gland.

All of these cells are electrically excitable, and possess many of the same membrane currents as do neurons, however they do not have dendritic or axonal projections. In common with the neural release of neurotransmitter, hormonal secretion from pituitary cells often arises as a consequence of electrical activity. That is, increased electrical activity raises the intracellular levels of calcium and cAMP, and this stimulates secretion of the hormone. Hormone release can also occur as a consequence of calcium release from intracellular stores such as the endoplasmic reticulum. Furthermore most pituitary cells, and many neuroendocrine cells, discharge bursts of action potentials and one of the main contributions that modelling has made is to help understand the mechanisms by which these cells burst.

Secretion from both the hypothalamus and the pituitary is pulsatile (Knobil 1980; Brabant et al. 1986). This means that during homeostasis, release is not constant, but rather occurs in intermittent boluses that repeat periodically. The cycle time of this pulsatility can range from minutes (e.g. the release of oxytocin during lactation) to an hour (e.g. GnRH secretion) to a day (diurnal vasopressin release). It is often the case that the hormone rhythm is composed of several frequencies.

For some endocrine cells (e.g. oxytocin and GnRH neurons) each bolus is generated by the entire population firing bursts of action potentials in close synchrony. However, for others, typically those with slower cycle times (e.g. vasopressin neurons), pulsatility is mediated by a general increase in firing rate across the population, but there is little or no synchronization of individual firing times. Therefore it would appear that (i) all endocrine and neuroendocrine cells have intrinsic rhythmic properties; (ii) there is some form of communication between cells even though populations are spread bilaterally in the hypothalamus; and (iii) the rhythm is entrained to that of the suprachiasmatic nucleus.

What is the purpose of pulsatile hormone secretion? In a mathematical study, Li and Goldbeter (1989) demonstrated that if the target cell has a desensitizing receptor, as is typically the case, then hormone released in a constant manner is not optimally effective at stimulating the target cell. A periodic release pattern may not, however, be any better than a constant release pattern. What is important is the duty cycle of the periodic release pattern. That is, the ratio of hormone release duration to inter-release duration during each periodic cycle. Li and Goldbeter demonstrated that there is an optimal duty cycle for which the effect of the pulsatile hormone release has maximum impact on the target cell. The optimal duty cycle depends on the rate at which the hormone receptor desensitizes and resensitizes.

Models of Hypothalamic Neuron Interactions

The Milk-Letdown Reflex

When mammalian young suckle, cells of the mammary gland periodically release milk into a duct that is then available for extraction. This milk-letdown reflex is mediated by bursts of oxytocin (OT) secretion from magnocellular OT neurons. The bursts of secretion are due to episodes or bursts of electrical activity that are coordinated among the population of OT neurons. The mechanism behind this coordination of activity, producing a brief burst of spikes (duration of about 3 seconds) every 5 minutes or so, has been the focus of much attention over several decades.

• Emergent Bursting Through Dendritic Bundling

The modeling paper of Rossoni et al. (2008) suggests that the coordinated activity of OT neurons is due to the bundling of dendrites. This hypothesis is based on data showing that dendrites of OT neurons form bundles, and that OT can be secreted from the bundled dendrites and bind to receptors on dendrites of other neurons in the bundle. However, OT secretion from dendrites is most likely to occur when the secretion machinery has been primed by suckling. Rossoni et al. build these features into a network model in which neural activity is described by a modified leaky integrate-and-fire model. The spike threshold varies over time, reflecting several factors. Most importantly, OT lowers the threshold, so that the release of OT from one neuron in a bundle makes it more likely that other neurons in the bundle will fire and release more OT. This positive feedback, in the presence of synaptic noise to start the cascade, results in a burst of activity of the population of OT cells. The burst is terminated by depletion of the readily-releasable pool of OT-filled vesicles in the dendrites. This model is novel in that the coupling occurs exclusively through dendrites.

Models with Hypothalamus-Pituitary Interactions

The LH Surge

During the human menstrual cycle in women and the estrous cycle in many other female mammals pulses of LH release from gonadotrophs occur with a period that changes throughout the cycle, but is roughly once per hour. At the midpoint of the cycle, or during the afternoon of proestrus, a much larger release of LH occurs. This LH surge is a trigger for ovulation, and is thus critical for fertility. Both the LH pulses and surges are controlled, at least in part, by gonadotropin-releasing hormone (GnRH) which is secreted by GnRH neurons of the hypothalamus into the blood at the median eminence (ME). These neurons, which are widely distributed throughout the preoptic area of the hypothalamus, produce periodic bursts of action potentials that are synchronized across the entire population of GnRH cells and that repeat every hour or so. During each burst, a bolus of GnRH is secreted into the blood and is then conveyed to the rest of the body. When the blood reaches the pituitary the GnRH stimulates the release of LH from gonadotrophs. Several mathematical models have been developed for the LH pulsing and LH surge.

• Pulsatile Secretion of GnRH

What makes the GnRH neurons burst approximately once every hour? How are these bursts synchronized? Both questions are addressed in Khadra and Li (2006). In this model, the mechanism for GnRH bursting is autofeedback of GnRH onto the neurons themselves. These neurons are thought to secrete GnRH not only from their axonal terminals, but also somato-dendritically by dense core exocytosis (Ludwig and Leng, 2006). When GnRH binds to an autoreceptor it activates three types of G-proteins: G_s (stimulatory, activates adenylyl cyclase), G_q (stimulatory, produces IP₃), and G_i (inhibitory, inhibits adenylyl cyclase). The affinities of these G-proteins are such that G_s is activated at the lowest GnRH concentration, followed by G_q, followed by G_i. Thus, there is positive feedback at lower GnRH concentrations, followed by negative feedback at higher concentrations. By this mechanism, oscillations in GnRH secretion are achieved. Synchronization is achieved by assuming that all GnRH neurons contribute to a common GnRH pool. With this assumption, each GnRH neuron is affected by every other GnRH neuron (through the G-protein-coupled autoreceptors), so when one fires it prompts the others to fire.

• Sensitization and Desensitization

Although there is an increase in the amount of GnRH released during the LH surge, this extra GnRH is thought to be insufficient to account for the large increase in LH during the surge. The model of Scullian et al (2004) is based on the assumption that gonadotrophs become more sensitive to GnRH following prior exposure. That is, there is a self-priming effect. This amplifies the increase in GnRH, leading to the LH surge.

In the model of Tien et al. (2005) the LH surge involves desensitization of IP₃ receptors. Calcium oscillations occur when GnRH receptors are not desensitized, and it is during these oscillations that LH is released. The IP₃ receptors slowly desensitize, however, ending the surge. The time required for the receptors to resensitize is the duration of the intersurge interval.

Pulsatile Growth Hormone Secretion

In male rats, growth hormone (GH) is secreted from somatotrophs in large pulses with period of approximately 3 hr. These pulses result from episodic secretion of GH-releasing factor (GRF) by neurons of the arcuate nucleus, a region of the hypothalamus. GH secretion is inhibited by somatostatin, which is secreted from neurons of the periventricular nucleus, another region of the hypothalamus. An earlier model was developed by Martin (1979), proposing two potential oscillatory mechanisms. Later, Brown et al. (2004) developed a model incorporating positive effects of GRF and the negative effects of somatostatin. This model also includes desensitization of GRF receptors, and somatostatin-mediated acceleration of resensitization. Thus, somatostatin has two effects: it directly inhibits GH secretion, and it indirectly stimulates GH secretion by accelerating resensitization of GRF receptors. The authors show that the model is able to account for the time course of GH release during pulsatile GRF stimulation. A later extension of the model added dynamics for the populations of GRF and somatostatin secreting neurons (MacGregor and Leng, 2005).

Figure 2: PRL and DA oscillations due to delayed negative feedback.

Rhythmic Prolactin Secretion

During the first half of pregnancy in rats there is a circadian rhythm in prolactin (PRL) secretion from lactotrophs. There are two PRL surges each day, once in the morning and then again in the afternoon. Rhythmic PRL secretion is essential for a viable pregnancy. Bertram et al. (2006) developed a model in which the rhythm is caused by interaction between lactotrophs and dopaminergic (DA) neurons of the arcuate nucleus. Dopamine is an inhibitor of lactotroph activity. Prolactin is an activator of DA neurons. However, the PRL effect is slow, since it works primarily through modulation of gene expression in DA neurons. Thus, the PRL feedback onto DA neurons is delayed. This simple model is described by two ordinary differential equations, one for the activity level of lactotrophs (PRL) and the other for the activity level of DA neurons (DA): \[ \begin{matrix} \dot{PRL} & = & \frac{T_p}{1+k_d DA^2} - q PRL \\ \dot{DA} & = & T_d (1+k_p PRL_\tau^2) - q DA \end{matrix} \] Here \(T_p\ ,\) \(T_d\ ,\) \(k_d\ ,\) \(k_p\ ,\) and \(q\) are parameters, and \(PRL_\tau\) is the PRL variable delayed by \(\tau\) hrs. The Figure illustrates that this system produces oscillations in the PRL and DA variables. The PRL peaks precede the DA peaks, as has been demonstrated in in vivo measurements.

Detailed Single Cell Models

One class of mathematical models focuses on detailed cellular interactions, and includes variables such as the membrane potential, ion channel activation and inactivation variables, and calcium concentration. Some examples of such models are described below.

Hypothalamic Neurons

• Arginine Vasopressin Neurons

The Arginine Vasopressin neurons (AVP) neurons are located in the supraoptic nucleus. They project to the neural lobe of the pituitary, where they release the hormone AVP into the blood. In the absence of stimulation, most AVP neurons fire at a low rate and release AVP at a low level. During dehydration or a drop in blood pressure the firing pattern changes to bursting, with total burst period of approximately 40 sec. This is accompanied by an increase in the release of AVP. A model of the AVP neuron was developed by Roper and colleagues (Roper et al., 2003; Roper et al., 2004). This model incorporates standard ionic currents for the production of action potentials. In addition, it contains currents for the production of a depolarizing after potential (DAP) and an after-hyperpolarization (AHP). The DAP is a small (3mV), transient (2-4 seconds or so), exponentially decaying depolarization of the membrane and it follows each spike, while the AHP is a much larger hyperpolarization of the membrane but is only activated when several spikes are fired close together. Whether a DAP or an AHP is produced depends upon the firing frequency and the number of spikes.

In this model, the DAP is due to the Ca²⁺-dependent inactivation of a K⁺ leak current while the AHP is due primarily to a Ca²⁺-activated K⁺ current (SK current). If the cell is stimulated to fire a few high-frequency spikes, then a DAP is produced. This DAP induces more spikes, which brings in more calcium and so increase the size of the DAP until it eventually forms a plateau potential that sustains further spiking. Thus, a burst of spikes is generated via positive feedback involving the DAP. The AHP activates early in the burst, and slows down the spiking. However, it is not strong enough to terminate the active phase of the burst. Instead, dynorphin, an opioid that is co-released with AVP and is also released from the dendrites of the neuron, is responsible for terminating the active phase in this model. This opioid acts in an autocrine fashion, binding to k-opioid receptors in the neuron and reducing the sensitivity of the DAP-producing leak current for Ca²⁺. That is, the effect of dynorphin feedback is to disinhibit the K⁺ leak current. As a result, the size of the DAP declines during the active phase of a burst, and eventually is not large enough to sustain the burst. The cell then enters a silent phase, whose duration is determined by the time required for the effects of dynorphin to wear off. This model is an interesting example where autocrine feedback is essential for the generation of a bursting oscillation, a mechanism shared with that in the Khadra and Li (2006) model for GnRH neurons.

Pituitary Cells

• The Gonadotroph

The most extensively modeled pituitary cell is the gonadotroph. In the absence of the stimulatory factor GnRH, gonadotrophs often exhibit spontaneous activity consisting of a continuous train of action potentials. This spontaneous activity was modeled by Li et al. (1995). Here, spatial compartmentalization of Ca²⁺ within the cytoplasm of the cell is very important. One Ca²⁺ compartment (the shell calcium) is adjacent to the plasma membrane, and is primarily affected by the opening and closing of Ca²⁺ channels in the plasma membrane. Because of its small volume, the Ca²⁺ concentration in this compartment reaches a high value during an action potential and dissipates quickly as Ca²⁺ moves into the other compartment or out of the cell. Another channel type in the plasma membrane gives rise to the small conductance Ca²⁺-activated K⁺ current, I_SK. This current is activated by shell Ca²⁺ during an action potential, and is responsible for the hyperpolarizing after potential (HAP) following each spike. The HAPs slow down spiking, yielding a frequency consistent with the data. The second Ca²⁺ compartment (the bulk calcium) represents the Ca²⁺ concentration deeper within the cell. The bulk Ca²⁺ interacts with the endoplasmic reticulum (ER), an organelle that stores Ca²⁺. In the unstimulated case, the ER acts as a passive Ca²⁺ filter, taking up Ca²⁺ during the upstroke of an action potential and releasing it back to the bulk Ca²⁺ during the downstroke (Bertram and Sherman, 2004).

The interaction between bulk Ca²⁺ and the ER becomes very important when the gonadotroph is stimulated by GnRH. When GnRH binds to its receptor on the plasma membrane it results in the production of IP₃. This internal second messenger binds to and opens receptors/channels on the ER membrane, releasing Ca²⁺ into the bulk Ca²⁺ compartment. The IP₃ receptors are also activated rapidly and inactivated slowly by the bulk Ca²⁺. This combination of fast activation and slow inactivation can give rise to Ca²⁺ oscillations that are separate from Ca²⁺ oscillations driven by the plasma membrane electrical activity, provided that the GnRH concentration is sufficiently high. A mathematical model for these ER-driven oscillations is given in Li et al. (1994).

Figure 3: Li et al. (1997) model showing gonadotroph activity before and during GnRH stimulation.

Finally, in Li et al. (1997) a model was developed that combines the electrical activity of the plasma membrane with the active ER oscillator. In the unstimulated state, the model exhibits spontaneous activity (continuous spiking). With sufficient GnRH stimulation, the ER oscillator is activated and interacts with the membrane oscillator. During the upstroke of an ER oscillation a great deal of Ca²⁺ is released into the cytosol. This activates the SK current which hyperpolarizes the cell, turning off the spiking. The cytosolic Ca²⁺ level returns to a lower level (Ca²⁺ is transported out of the cell by pumps in the plasma membrane and is transported into the ER by SERCA pumps) at the end of an ER oscillation, deactivating I_SK and allowing the cell to spike once again. Thus, the model produces a bursting pattern of electrical activity in the stimulated state. During bursting the cytosolic Ca²⁺ concentration is high during the silent phase of the burst and low during the active phase, the opposite of what is seen in most biophysical models of bursting cells. This antiphase behavior has been verified in gonadotrophs in simultaneous recordings of membrane potential and Ca²⁺ concentration.

• Somatotrophs and Lactotrophs

Unlike the gonadotroph, which often spikes continuously in the unstimulated state, the somatotroph typically fires bursts of action potentials during basal activity. The result is that the basal Ca²⁺ level, and basal secretion, is higher in somatotrophs than in gonadotrophs. To investigate why somatotroph spontaneous activity is so different from that of the gonadotroph, Van Goor et al (2001) took a model of the gonadotroph and added a second type of K(Ca) current. The channels for this type, the BK (big conductance) current I_BK, are located very close to Ca²⁺ channels and the channel conductance depends on both Ca²⁺ and voltage. This BK current is known to be present in somatotrophs, but not in gonadotrophs, and if it is pharmacologically blocked the cell's activity switches from bursting to spiking. During the upstroke of an action potential I_BK activates quickly, which prevents significant activation of the delayed rectifier K⁺ current (I_Kdr) and so limits the amplitude of the spike. The result of the reduction in I_Kdr is that the action potentials are wider, and single spikes can be converted into bursts of spikes. Thus, there is a paradoxical effect that increasing a hyperpolarizing K⁺ current leads to an increase in the mean cytosolic Ca²⁺ concentration in the cell and an increase in hormone secretion.

Figure 4: Spiking converted to bursting with addition of a BK or an A-type K⁺ current

There is a similar paradoxical finding in lactotrophs. The primary hypothalamic inhibitor of lactotrophs is dopamine (DA). At micromolar concentrations DA stops all electrical activity and lowers the mean Ca²⁺ concentration in lactotrophs. However, at smaller nanomolar concentrations DA increases the Ca²⁺ concentration and increases prolactin secretion. To investigate this biphasic effect, Tabak et al (2006) developed a simple model of the lactotroph that contains differential equations for membrane potential, channel activation and inactivation variables, and the cytosolic Ca²⁺ concentration. It was shown that DA can act in a stimulatory manner if it either activates a BK current or if it activates an A-type K⁺ current (figure). There is experimental evidence supporting both effects of DA, although only at larger (micromolar) DA concentrations. Because the lactotroph model used by Tabak et al. is significantly simpler than the gonadotroph/somatotroph model used by Van Goor et al. (2001), it was possible to perform a fast/slow bifurcation analysis and thus understand the effect of simulated application of DA from a mathematical viewpoint. A related paper, Toporikova et al. (2006), contains a mathematical analysis of one form of bursting that arises in the lactotroph model. A novel feature of this bursting mechanism is that it does not involve a slow variable. This is illustrated in the figure, where in one case (addition of BK current) the bursting is terminated when [Ca], the slow variable, is clamped. In the other case (addition of A current) bursting persists when [Ca] is clamped.

Models of the Suprachiasmatic Nucleus

The suprachiasmatic nucleus (SCN) is the region of the hypothalamus that is primarily responsible for the generation of circadian rhythms in behavior. There are two nuclei, one in each hemisphere of the brain. The circadian rhythms generated by the SCN persist in constant darkness, although they are entrained by light. The light stimulus reaches the hypothalamus through the retino-hypothalamic tract. Two general classes of models have been developed for the SCN. One class aims to understand the rhythm generation within a single SCN neuron. The other class aims to understand how the rhythms are synchronized throughout the SCN.

Circadian Rhythm Generation

It is now known that circadian rhythms in SCN neurons are generated through negative transcriptional feedback. A similar mechanism is utilized in Drosophila and Neurospora, although the details differ in each case. The essential elements of the rhythm are the clock genes, which undergo circadian expression. In the SCN, these include period (Per), cryptochrome (Cry), Bmal1, and Rev-erb\(\alpha\). Another gene that is important in the rhythm generation, but that does not exhibit rhythmic expression itself, is Clock. The Figure illustrates the fundamental components of the rhythm generation mechanism. A CLOCK/BMAL1 dimer acts as a positive transcription factor for Cry and Per. These are then translated into CRY and PER protein within the cytosol, where they dimerize to form the complex CRY/PER. The dimer is transported into the nucleus, where it binds to the CLOCK/BMAL1 complex and inhibits its action as a transcription factor. There is a significant time delay between the transcription of Cry and Per and the inhibitory action of the CRY/PER complex. Thus, the system exhibits delayed negative feedback, and this is responsible for the circadian rhythm. Light can entrain this rhythm through its positive action on the Per transcription rate.

Figure 5: Components of the circadian rhythm.

• The Leloup-Goldbeter Model

The model of Leloup and Goldbeter (2003) consists of 19 differential equations, and describes the negative feedback loop illustrated in the figure, but in more detail. In particular, it includes variables for mRNA concentrations of clock genes, and phosphorylated and unphosphorylated protein concentrations, including PER, CRY, CRY/PER complex, BMAL1, and the CRY/PER-CLOCK/BMAL1 complex in the nucleus. There are also variables for REV-ERB\(\alpha\) (mRNA and protein concentrations). This model produces a circadian rhythm with the correct phase relation among the variables. It also shows that even in the absence of the PER negative feedback loop, another gene transcription loop, the negative feedback of BMAL1 on Bmal1 transcription, can produce a sustained rhythm in the clock genes, but with period less than 24 hrs.

• The Forger-Peskin Model

The model of Forger and Peskin (2003) consists of 73 differential equations, and is a very detailed model of the negative feedback loop that gives rise to circadian rhythms in SCN neurons. This model has variables for the same physical quantities as the Leloup-Goldbeter model, plus many others. The additional detail makes for a closer coupling with the biological system. A stochastic version of the model yields results consistent with the deterministic model.

Synchronization within the SCN

Each nucleus of the SCN consists of about 10,000 neurons. Many of these neurons produce endogenous circadian rhythms in clock gene expression, in ion channel density, and in electrical activity. How do these neurons synchronize their circadian rhythms? One possibility is through chemical synaptic connections. Many of the SCN neurons are GABAergic, and many express arginin-vasopressin (AVP) or vasointestinal polypeptide (VIP). These could all be synchronizing factors. Another possibility is that synchronization is through direct electrical coupling. There is experimental evidence for extensive gap junctional coupling within the SCN. Some models of SCN synchronization assume local coupling, while others assume global coupling. All models of SCN synchronization have used simple mathematical representations of the single-cell circadian oscillator, such as a van der Pol oscillator.

• Local Coupling

This would reflect the situation in which neurons are coupled through gap junctions or chemical synapses from nearby cells. In Kunz and Achermann (2003) a Kronauer model is used to represent the single-cell circadian oscillator. This is similar to a van der Pol oscillator, but with a term included for the effects of light. The coupling is to 4, 8, or 20 neighbors, and the coupling function uses the difference of the state of the oscillator to the weighted average of its neighbors' states. The values of the weights are decreasing functions of distance from the cell. The authors show that this local coupling can yield synchronization of the full population of cells (a 100x100 array), where individual oscillator periods are chosen randomly from a normal distribution. They also show that periodic light pulses can entrain the system.

• Global Coupling

This would reflect long-range coupling through chemical synapses. In Gonze et al. (2005) a Goodwin model is used to represent the single-cell circadian oscillator. This model has three variables: one for clock gene mRNA concentration, one for clock protein concentration, and one for active transcriptional inhibitor concentration. The effects of light come in through the mRNA equation. The model cell secretes neurotransmitter based on the value of the mRNA variable. The authors assume mean field coupling, where each oscillator feels the population average of the secreted transmitter concentration. It is shown that this mean field coupling is sufficient to synchronize a 10,000-oscillator population, where single-oscillator periods are chosen randomly from a normal distribution. It is also shown that the population can be entrained by light pulses.

References

Bertram R., Sherman A. (2004). Filtering of calcium transients by the endoplasmic reticulum. Biophys. J. 87:3775-3785.

Bertram R., Egli M., Toporikova N., Freeman M. E. (2006). A mathematical model for the mating-induced prolactin rhythm of female rats. Am. J. Physiol. 290:E573-E582, 2006.

Brabant G., Ranft U., Oran K., Hesch R. D., von zur Muhlen A. (1986). Thyrotropin-an episodically secreted hormone. Acta Endocrinol. 112:315-322.

Brown D., Stephens E.A., Smith R.G., Li. G, Leng G. (2004). Estimation of parameters for a mathematical model of growth hormone secretion. J. Neuroendocrinol. 16:936-946.

Forger D.B., Peskin C.S. (2003). A detailed predictive model of the mammalian circadian clock. Proc. Natl. Acad. Sci. USA. 100:14806-14811.

Gonze D., Bernard S., Waltermann C., Kramer A., Herzel H. (2005). Spontaneous synchronization of coupled circadian oscillators. Biophys. J. 89:120-129.

Khadra A., Li Y.-X. (2006). A model for the pulsatile secretion of gonadotropin-releasing hormone from synchronized hypothalamic neurons. Biophys. J. 91:74-83.

Knobil E. (1980). The neuroendocrine control of the menstrual cycle. Recent Prog. Horm. Res. 36:63-88.

Kunz H., Achermann P. (2003). Simulation of circadian rhythm generation in the suprachiasmatic nucleus with locally coupled self-sustained oscillators. J. Theor. Biol. 224:63-78.

Leloup J.-C., Goldbeter A. (2003). Toward a detailed computational model for the mammalian circadian clock. Proc. Natl. Acad. Sci. USA. 100:7051-7056.

Ludwig M., Leng G. (2006). Dendritic peptide release and peptide-dependent behaviours. Nature Reviews Neuroscience 7:126-136.

Li Y.-X., Goldbeter A. (1989). Frequency specificity in intercellular communication: Influence of patterns of periodic signaling on target cell responsiveness. Biophys. J. 55:125-145.

Li Y.-X., Rinzel J., Keizer J., Stojilkovic S.S. (1994). Calcium oscillations in pituitary gonadotrophs-Comparison of experiment and theory. Proc. Natl. Acad. Sci. USA. 91:58-62.

Li Y.-X., Rinzel J., Vergara L., Stojilkovic S.S. (1995). Spontaneous electrical and calcium oscillations in unstimulated pituitary gonadotrophs. Biophys. J. 69:785-795.

Li Y.-X., Stojilkovic S.S., Keizer J., Rinzel J. (1997). Sensing and refilling calcium stores in an excitable cell model. Biophys. J. 72:1080-1091.

MacGregor D.J., Leng G. (2005). Modelling the hypothalamic control of growth hormone secretion. J. Neuroendocrinol. 17:788-803.

Martin J. B. (1979). Brain mechanisms for integration of growth hormone secretion. Physiologist 22:23-29.

Roper P., Callaway J., Shevchenko T., Teruyama R., Armstrong W. (2003). AHP's, HAP's and DAP's: How potassium currents regulate the excitability of rat supraoptic neurones. J. Comput. Neurosci. 15:367-389.

Roper P., Callaway J., Armstrong W. (2004). Burst initiation and termination in phasic vasopressin cells of the rat supraoptic nucleus: A combined mathematical, electrical, and calcium fluorescence study. J. Neurosci. 24:4818-4831.

Rossoni E., Feng J., Tirozzi B., Brown D., Leng G., Moos F. (2008). Emergent synchronous bursting of oxytocin neuronal network. PLoS Comput. Biol. 4(7):e10000123.

Scullion S., Brown D., Leng G. (2004). Modelling the pituitary response to luteinizing hormone-releasing hormone. J. Neurophysiol. 16:265-271.

Tabak J., Toporikova N., Freeman M.E., Bertram R. (2006). Low dose of dopamine may stimulate prolactin secretion by increasing fast potassium currents. J. Comput. Neurosci. (in press).

Tien J.H., Lyles D., Zeeman M.L. (2005). A potential role of modulating inositol 1,4,5-trisphosphate receptor desensitization and recovery rates in regulating ovulation. J. Theor. Biol. 232:105-117.

Toporikova N., Tabak J., Freeman M.E., Bertram R. (2006). A-type K+ current can act as a trigger for bursting in the absence of a slow variable. Neural Comp. (submitted).

Van Goor F., Li Y.-X., Stojilkovic S. S. (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.

Internal references

• Eugene M. Izhikevich (2006) Bursting. Scholarpedia, 1(3):1300.
• Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.

External Links

Richard Bertram's web site

Supraoptic.com

See also

Bursting, Calcium Oscillations, Electrophysiology, Endocrine System, Hormones,Hypothalamus, Models of Calcium Oscillations, Neuroendocrine System, Neuron, Oscillations, pituitary gland, Synchronization

Sponsored by: Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia

Reviewed by: Dr. Yue-Xian Li, Mathematics and Zoology, UBC, Vancouver, Canada

Reviewed by: Dr. Peter Roper, Dept Mathematics, U. Utah

Accepted on: 2006-11-28 15:29:35 GMT