Mathematical modelling in neuroendocrinology

Accurate prediction of birth implementing a statistical model through the determination of steroid hormones in saliva

Steroidal hormone interaction in pregnancy is crucial for adequate fetal evolution and preparation for childbirth and extrauterine life. Estrone sulphate, estriol, progesterone and cortisol play important roles in the initiation of labour mechanism at the start of contractions and cervical effacement. However, their interaction remains uncertain. Although several studies regarding the hormonal mechanism of labour have been reported, the prediction of date of birth remains a challenge. In this study, we present for the first time machine learning algorithms for the prediction of whether spontaneous labour will occur from week 37 onwards. Estrone sulphate, estriol, progesterone and cortisol were analysed in saliva samples collected from 106 pregnant women since week 34 by enzyme-immunoassay (EIA) techniques. We compared a random forest model with a traditional logistic regression over a dataset constructed with the values observed of these measures. We observed that the results, evaluated in terms of accuracy and area under the curve (AUC) metrics, are sensibly better in the random forest model. For this reason, we consider that machine learning methods contribute in an important way to the obstetric practice.

Mathematical Modelling of Endocrine Systems

Hormone rhythms are ubiquitous and essential to sustain normal physiological functions. Combined mathematical modelling and experimental approaches have shown that these rhythms result from regulatory processes occurring at multiple levels of organisation and require continuous dynamic equilibration, particularly in response to stimuli. We review how such an interdisciplinary approach has been successfully applied to unravel complex regulatory mechanisms in the metabolic, stress, and reproductive axes. We discuss how this strategy is likely to be instrumental for making progress in emerging areas such as chronobiology and network physiology. Ultimately, we envisage that the insight provided by mathematical models could lead to novel experimental tools able to continuously adapt parameters to gradual physiological changes and the design of clinical interventions to restore normal endocrine function.

The determinants of food choice

Health nudge interventions to steer people into healthier lifestyles are increasingly applied by governments worldwide, and it is natural to look to such approaches to improve health by altering what people choose to eat. However, to produce policy recommendations that are likely to be effective, we need to be able to make valid predictions about the consequences of proposed interventions, and for this, we need a better understanding of the determinants of food choice. These determinants include dietary components (e.g. highly palatable foods and alcohol), but also diverse cultural and social pressures, cognitive-affective factors (perceived stress, health attitude, anxiety and depression), and familial, genetic and epigenetic influences on personality characteristics. In addition, our choices are influenced by an array of physiological mechanisms, including signals to the brain from the gastrointestinal tract and adipose tissue, which affect not only our hunger and satiety but also our motivation to eat particular nutrients, and the reward we experience from eating. Thus, to develop the evidence base necessary for effective policies, we need to build bridges across different levels of knowledge and understanding. This requires experimental models that can fill in the gaps in our understanding that are needed to inform policy, translational models that connect mechanistic understanding from laboratory studies to the real life human condition, and formal models that encapsulate scientific knowledge from diverse disciplines, and which embed understanding in a way that enables policy-relevant predictions to be made. Here we review recent developments in these areas.

Oxytocin Neurones: Intrinsic Mechanisms Governing the Regularity of Spiking Activity

Oxytocin neurones of the rat supraoptic nucleus are osmoresponsive and, with all other things being equal, they fire at a mean rate that is proportional to the plasma sodium concentration. However, individual spike times are governed by highly stochastic events, namely the random occurrences of excitatory synaptic inputs, the probability of which is increased by increasing extracellular osmotic pressure. Accordingly, interspike intervals (ISIs) are very irregular. In the present study, we show, by statistical analyses of firing patterns in oxytocin neurones, that the mean firing rate as measured in bins of a few seconds is more regular than expected from the variability of ISIs. This is consistent with an intrinsic activity-dependent negative-feedback mechanism. To test this, we compared observed neuronal firing patterns with firing patterns generated by a leaky integrate-and-fire model neurone, modified to exhibit activity-dependent mechanisms known to be present in oxytocin neurones. The presence of a prolonged afterhyperpolarisation (AHP) was critical for the ability to mimic the observed regularisation of mean firing rate, although we also had to add a depolarising afterpotential (DAP; sometimes called an afterdepolarisation) to the model to match the observed ISI distributions. We tested this model by comparing its behaviour with the behaviour of oxytocin neurones exposed to apamin, a blocker of the medium AHP. Good fits indicate that the medium AHP actively contributes to the firing patterns of oxytocin neurones during non-bursting activity, and that oxytocin neurones generally express a DAP, even though this is usually masked by superposition of a larger AHP.

Modelling the in vivo spike activity of phasically-firing vasopressin cells

A minimalist model of magnocellular vasopressin neurones was developed to examine the hypothesis that their phasic behaviour is the product of intrinsic voltage- and activity-dependent intracellular mechanisms that create a bistable dynamical system. The model can closely match a range of phasic behaviours recorded in vasopressin cells in vivo, as well as reproduce the three archetypal behaviours of vasopressin cells (continuous firing, sparse sporadic firing and phasic firing) by varying one of the fourteen model parameters. In addition, the mean and standard deviation of burst and silence periods can be matched by varying a further two parameters. In the model, the long-term behaviour (phasic characteristics) of cells is largely independent of the short-term behaviour (interspike intervals).

Investigating and modelling pituitary endocrine network function

Endocrine cells in the mammalian pituitary are arranged into three-dimensional homotypic networks that wire the gland and act to optimise hormone output by allowing the transmission of information between cell ensembles in a temporally precise manner. Despite this, the structure-function relationships that allow cells belonging to these networks to display coordinated activity remain relatively uncharacterised. This review discusses the recent technological advances that have allowed endocrine cell network structure and function to be probed and the mathematical models that can be used to analyse and present the resulting data. In particular, we focus on the mechanisms that allow endocrine cells to dynamically function as a population to drive hormone release as well as the experimental and theoretical methods that are used to track and model information flow through the network.

Neuroendocrinology and its quantitative development: a bioengineering view

Biomedical engineering is clearly present in modern neuroendocrinology, and indeed has come to embrace it in many respects. First, we briefly review the origins of endocrinology until neuroendocrinology, after a long saga, was established in the 1950's decade with quantified results made possible by the radioimmunoassay technique (RIA), a development contributed by the physical sciences. However, instrumentation was only one face of the quantification process, for mathematical models aiding in the study of negative feedback loops, first rather shyly and now at a growing rate, became means building the edifice of mathematical neuroendocrinology while computer assisted techniques help unravel the associated genetic aspects or the nature itself of endocrine bursts by numerical deconvolution analysis. To end the note, attention is called to the pleiotropic characteristics of neuroendocrinology, which keeps branching off almost endlessly as bioengineering does too.

Prolactin secretion patterns: basic mechanisms and clinical implications for reproduction

Prolactin (PRL) is one of the most versatile hormones in the mammalian body affecting reproductive, sexual, metabolic, immune, and other functions. It is therefore not surprising that the neural control of PRL secretion is complex, involving the coordinated actions of several hypothalamic nuclei. A plethora of experimental data exists on the hypothalamic control of hormone secretion under various physiological stimuli. There have been even mathematical models and computer studies published, which help to understand the complex hypothalamic–pituitary network. Nevertheless, the putative role of PRL for human reproduction still has to be clarified. Here, we review data on the underlying mechanisms controlling PRL secretion using both experimental and mathematical approaches. These investigations primarily focus on rhythmic secretion in rats during early pregnancy or pseudopregnancy, and they point to the important role of oxytocin as a crucial PRL-releasing factor. Recent data on human studies and their theoretical and clinical implications are reviewed as well. In particular, studies demonstrating a sustained PRL surge after sexual climax in males and females are presented, indicating possible implications for both sexual satiation and reproductive functions. Taking these data together, there is evidence for the hypothesis that the PRL surge induced by sexual activity, together with the altered PRL rhythmic pattern, is important for successful initialization of pregnancy not only in rodents but also possibly in humans. However, further investigations are needed to clarify such a role in humans.

A new method of spike modelling and interval analysis

Here we develop a new model of spike firing, based on the leaky integrate and fire model, modified to simulate afterpotentials. 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.