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Hypothalamic vasopressin neurons are neuroendocrine cells which form part of the homeostatic systems that maintain osmotic pressure. In response to synaptic inputs encoding osmotic pressure and changes in plasma volume, they generate spike triggered secretion of peptide hormone vasopressin from axonal terminals in the posterior pituitary. The thousands of neurons’ secretory signals generate a summed plasma vasopressin signal acting at the kidneys to regulate water loss. Vasopressin is synthesised in cell bodies, packaged into vesicles, and transported to large stores in the pituitary terminals. Supported by activity-dependent upregulation of synthesis and transport, these stores can maintain a secretion response for several days of elevated osmotic pressure, tested by dehydration or salt loading. However, despite upregulated synthesis, stores gradually decline during sustained challenge, followed by a slow recovery. With no evidence of a store encoding feedback signal, previous modelling explained these synthesis dynamics based on activity-dependent upregulation of transcription and mRNA content. Here this model is adapted and integrated into our existing spiking and secretion model to generate a neuronal population model, able to simulate the secretion, store depletion, and replenishment, response to sustained osmotic challenge, matching the dynamics observed experimentally and making functional predictions for the cell body mechanisms.

Magnocellular oxytocin and vasopressin neurons of the hypothalamus project to the posterior pituitary where they secrete their peptide hormone signals directly into the bloodstream. Their large anatomically distinct secretory mechanisms provide a uniquely accessible system in which to unite experimental and modelling approaches in the investigation of how input signals and electrophysiological properties of neurons relate to physiological function. We describe how the mechanisms have been translated and assembled into a mathematical model representation that can explain and simulate the complex and highly non-linear stimulus-secretion coupling of these neurons, and how this model has been applied to further understand these systems.

The plain title might have been an almost retro sounding grumpy retort, but it has inspired a journey of sorts, and something along the way I hope you won’t have come across before. An opinionated exploration of the distinctive phasic spiking patterns of magnocellular vasopressin neurons of the supraoptic and paraventricular nuclei of the hypothalamus. A mostly life essential population of neurons that signal the kidneys to regulate water loss in response to signals that encode plasma volume and osmotic pressure, as well as regulating blood pressure, and possibly metabolism and social behaviour. The viewpoint of a modeller shorn of any explicit maths.

The neuroendocrine systems of the hypothalamus are critical for survival and reproduction, and are highly conserved throughout vertebrate evolution. Their roles in controlling body metabolism, growth and body composition, stress, electrolyte balance and reproduction have been intensively studied, and have yielded a rich crop of original and challenging insights into neuronal function, insights that circumscribe a vision of the brain that is quite different from conventional views. Despite the diverse physiological roles of pituitary hormones, most are secreted in a pulsatile pattern, but arising through a variety of mechanisms. An important exception is vasopressin which uses bursting neural activity, but produces a graded secretion response to osmotic pressure, a sustained robust linear response constructed from noisy, nonlinear components. Neuroendocrine systems have many features such as multiple temporal scales and nonlinearity that make their underlying mechanisms hard to understand without mathematical modelling. The models presented here cover the wide range of temporal scales involved in these systems, including models of single cell electrical activity and calcium dynamics, receptor signalling, gene expression, coordinated activity of neuronal networks, whole-organism hormone dynamics and feedback loops, and the menstrual cycle. Many interesting theoretical approaches have been applied to these systems, but important problems remain, at the core the question of what is the true advantage of pulsatility.

Oxytocin neurons of the rat hypothalamus project to the posterior pituitary, where they secrete their products into the bloodstream. The pattern and quantity of that release depends on the afferent inputs to the neurons, on their intrinsic membrane properties, and on nonlinear interactions between spiking activity and exocytosis: A given number of spikes will trigger more secretion when they arrive close together. Here we present a quantitative computational model of oxytocin neurons that can replicate the results of a wide variety of published experiments. The spiking model mimics electrophysiological data of oxytocin cells responding to cholecystokinin (CCK), a peptide produced in the gut after food intake. The secretion model matches results from in vitro experiments on stimulus-secretion coupling in the posterior pituitary. We mimic the plasma clearance of oxytocin with a two-compartment model, replicating the dynamics observed experimentally after infusion and injection of oxytocin. Combining these models allows us to infer, from measurements of oxytocin in plasma, the spiking activity of the oxytocin neurons that produced that secretion. We have tested these inferences with experimental data on oxytocin secretion and spiking activity in response to intravenous injections of CCK. We show how intrinsic mechanisms of the oxytocin neurons determine this relationship: In particular, we show that the presence of an afterhyperpolarization (AHP) in oxytocin neurons dramatically reduces the variability of their spiking activity and even more markedly reduces the variability of oxytocin secretion. The AHP thus acts as a filter, protecting the final product of oxytocin cells from noisy fluctuations., We present a model of oxytocin neurons that relates secretion to spike activity. We fit this to experimental data and use it to explore the significance of intrinsic membrane properties.

Here we develop a new model of spike firing, based on the leaky integrate and fire model, modified to simulate after potentials. We also develop new analysis techniques, applying these to recorded and model generated data in order to make a comparative analysis and develop the model as a hypothesis for the functional components of the neuron. The model is based in this first instance on hypothalamic oxytocin neurons. We demonstrate how model parameters and cell properties relate to features observed in inter-spike intervals histograms, and the limits of these in being able to detect patterning features in spike recordings. A new technique, spike train analysis, is able to detect previously unobserved patterning, showing a dependence of spike intervals on previous firing activity. This effect is reproduced in the model by adding the small amplitude but long lasting after hyper-polarising potential (AHP). A fit measure based on log likelihood is used to compare model generated data to recorded spike intervals, taking account of interval dependence on previous activity. This measure is used with the simplex multiple parameter search algorithm to develop an automated method for fitting the model to recorded data.

Recent evidence based on studies in hypothalamo-pituitary disconnected Soay sheep suggests that the generation of circannual rhythms may be localizable to specific tissues or physiological systems. Now, the authors present a physiological model of a circannual rhythm generator centered in the pituitary gland based on the interaction between melatonin-responsive cells in the pars tuberalis that act to decode photoperiod, and lactotroph cells of the adjacent pars distalis that secrete prolactin. The model produces a self-sustained, circannual rhythm in endocrine output that the authors explore by mathematical modeling. The circannual oscillation requires a delayed negative feedback mechanism. The authors highlight specific features of the pituitary dynamics as a guide to future research on circannual rhythms.

In neuroendocrinology, mathematical modelling is about formalising our understanding of the behaviour of the complex biological systems with which we deal. Formulating our explanations mathematically ensures their logical consistency, and makes them open to structured analysis; it is a stringent test of their intellectual coherence. In addition, however, modellers are seeking to extend our understanding in new ways, by seeking novel, simple explanations for complex behaviour. Here we discuss some styles of modelling as they have been applied to neuroendocrine systems, and discuss some of their strengths and limitations.

Integrate-and-fire (IF) models can provide close matches to the discharge activity of neurons, but do they oversimplify the biophysical properties of the neurons? A single compartment Hodgkin-Huxley (HH) model of the oxytocin neuron has previously been developed, incorporating biophysical measurements of channel properties obtained in vitro. A simpler modified integrate-and-fire model has also been developed, which can match well the characteristic spike patterning of oxytocin neurons as observed in vivo. Here, we extended the HH model to incorporate synaptic input, to enable us to compare spike activity in the model with experimental data obtained in vivo. We refined the HH model parameters to closely match the data, and then matched the same experimental data with a modified IF model, using an evolutionary algorithm to optimise parameter matching. Finally we compared the properties of the modified HH model with those of the IF model to seek an explanation for differences between spike patterning in vitro and in vivo. We show that, with slight modifications, the original HH model, like the IF model, is able to closely match both the interspike interval (ISI) distributions of oxytocin neurons and the observed variability of spike firing rates in vivo and in vitro. This close match of both models to data depends on the presence of a slow activity-dependent hyperpolarisation (AHP); this is represented in both models and the parameters used in the HH model representation match well with optimal parameters of the IF model found by an evolutionary algorithm. The ability of both models to fit data closely also depends on a shorter hyperpolarising after potential (HAP); this is explicitly represented in the IF model, but in the HH model, it emerges from a combination of several components. The critical elements of this combination are identified.

Vasopressin neurons generate distinctive phasic patterned spike activity in response to elevated extracellular osmotic pressure. These spikes are generated in the cell body and are conducted down the axon to the axonal terminals where they trigger Ca2+ entry and subsequent exocytosis of hormone-containing vesicles and secretion of vasopressin. This mechanism is highly non-linear, subject to both frequency facilitation and fatigue, such that the rate of secretion depends on both the rate and patterning of the spike activity. Here we used computational modelling to investigate this relationship and how it shapes the overall response of the neuronal population. We generated a concise single compartment model of the secretion mechanism, fitted to experimentally observed profiles of facilitation and fatigue, and based on representations of the hypothesised underlying mechanisms. These mechanisms include spike broadening, Ca2+ channel inactivation, a Ca2+ sensitive K+ current, and releasable and reserve pools of vesicles. We coupled the secretion model to an existing integrate-and-fire based spiking model in order to study the secretion response to increasing synaptic input, and compared phasic and non-phasic spiking models to assess the functional value of the phasic spiking pattern. The secretory response of individual phasic cells is very non-linear, but the response of a heterogeneous population of phasic cells shows a much more linear response to increasing input, matching the linear response we observe experimentally, though in this respect, phasic cells have no apparent advantage over non-phasic cells. Another challenge for the cells is maintaining this linear response during chronic stimulation, and we show that the activity-dependent fatigue mechanism has a potentially useful function in helping to maintain secretion despite depletion of stores. Without this mechanism, secretion in response to a steady stimulus declines as the stored content declines., Author Summary Vasopressin is a hormone that is secreted from specialised brain cells into the bloodstream; it acts at the kidneys to control water excretion, and thereby help to maintain a stable ‘osmotic pressure’. Specialised cells in the brain sense osmotic pressure, and generate electrical signals which the thousands of vasopressin neurons process and respond to by producing and secreting vasopressin. In response to these signals, vasopressin neurons generate complex “phasic” patterns of electrical activity, and this activity leads to vasopressin secretion in a complex way that depends on both the rate and pattern of this activity. We have now built a computational model that describes both how the vasopressin neurons generate electrical activity and also how that activity leads to secretion. The model, which gives a very close fit to experimental data, allows us to explore the adaptive advantages of particular features of the vasopressin neurons. This analysis reveals the importance of heterogeneity in the properties of vasopressin neurons, and shows how the vasopressin system is optimally designed to maintain a consistent hormonal output in conditions where its stores of releasable hormone are severely depleted.

The task of the vasopressin system is homeostasis, a type of process which is fundamental to the brain's regulation of the body, exists in many different systems, and is vital to health and survival. Many illnesses are related to the dysfunction of homeostatic systems, including high blood pressure, obesity and diabetes. Beyond the vasopressin system's own importance, in regulating osmotic pressure, it presents an accessible model where we can learn how the features of homeostatic systems generally relate to their function, and potentially develop treatments. The vasopressin system is an important model system in neuroscience because it presents an accessible system in which to investigate the function and importance of, for example, dendritic release and burst firing, both of which are found in many systems of the brain. We have only recently begun to understand the contribution of dendritic release to neuronal function and information processing. Burst firing has most commonly been associated with rhythm generation; in this system it clearly plays a different role, still to be understood fully.

Non-technical summary Our ability to respond to stress is critically dependent upon the release of the stress hormone adrenocorticotrophic hormone (ACTH) from corticotroph cells of the anterior pituitary gland. ACTH release is controlled by the electrical properties of corticotrophs that are determined by the movement of ions through channel pores in the plasma membrane. We show that a calcium-activated potassium ion channel called SK4 is expressed in corticotrophs and regulates ACTH release. We provide evidence of how SK4 channels control corticotroph function, which is essential for understanding homeostasis and for treating stress-related disorders. Abstract The anterior pituitary corticotroph is a major control point for the regulation of the hypothalamic–pituitary–adrenal (HPA) axis and the neuroendocrine response to stress. Although corticotrophs are known to be electrically excitable, ion channels controlling the electrical properties of corticotrophs are poorly understood. Here, we exploited a lentiviral transduction system to allow the unequivocal identification of live murine corticotrophs in culture. We demonstrate that corticotrophs display highly heterogeneous spontaneous action-potential firing patterns and their resting membrane potential is modulated by a background sodium conductance. Physiological concentrations of corticotrophin-releasing hormone (CRH) and arginine vasopressin (AVP) cause a depolarization of corticotrophs, leading to a sustained increase in action potential firing. A major component of the outward potassium conductance was mediated via intermediate conductance calcium-activated (SK4) potassium channels. Inhibition of SK4 channels with TRAM-34 resulted in an increase in corticotroph excitability and exaggerated CRH/AVP-stimulated ACTH secretion in vitro. In accordance with a physiological role for SK4 channels in vivo, restraint stress-induced plasma ACTH and corticosterone concentrations were significantly enhanced in gene-targeted mice lacking SK4 channels (Kcnn4−/−). In addition, Kcnn4−/− mutant mice displayed enhanced hypothalamic c-fos and nur77 mRNA expression following restraint, suggesting increased neuronal activation. Thus, stress hyperresponsiveness observed in Kcnn4−/− mice results from enhanced secretagogue-induced ACTH output from anterior pituitary corticotrophs and may also involve increased hypothalamic drive, thereby suggesting an important role for SK4 channels in HPA axis function.

This chapter describes a model of the hypothalamic and pituitary components involved in controlling growth hormone release. The model has been developed by gathering and attempting to formalise the experimental data on the system but has been kept as simple as possible, focusing on the functional rather than mechanical properties of its components. In this way it has shown that a relatively simple model can be capable of producing complex behaviour and accurately reproducing the behaviour and output of a real brain system.

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