Models of electrical activity in excitable cells involve nonlinear interactions between many ionic currents. Changing parameters in these models can produce a variety of activity patterns with sometimes unexpected effects. Further more, introducing new currents will have different effects depending on the initial parameter set. In this study we combined global sampling of parameter space and local analysis of representative parameter sets in a pituitary cell model to understand the effects of adding K (+) conductances, which mediate some effects of hormone action on these cells. Global sampling ensured that the effects of introducing K (+) conductances were captured across a wide variety of contexts of model parameters. For each type of K (+) conductance we determined the types of behavioral transition that it evoked. Some transitions were counterintuitive, and may have been missed without the use of global sampling. In general, the wide range of transitions that occurred when the same current was applied to the model cell at different locations in parameter space highlight the challenge of making accurate model predictions in light of cell-to-cell heterogeneity. Finally, we used bifurcation analysis and fast/slow analysis to investigate why specific transitions occur in representative individual models. This approach relies on the use of a graphics processing unit (GPU) to quickly map parameter space to model behavior and identify parameter sets for further analysis. Acceleration with modern low-cost GPUs is particularly well suited to exploring the moderate-sized (5-20) parameter spaces of excitable cell and signaling models.

Corticotroph cells from the anterior pituitary are an integral component of the hypothalamic-pituitary-adrenal (HPA) axis, which governs the neuroendocrine response to stress. Corticotrophs are electrically excitable and fire spontaneous single-spike action potentials and also display secretagogue-induced bursting behavior. The HPA axis function is dependent on effective negative feedback in which elevated plasma glucocorticoids result in inhibition at the level of both the pituitary and the hypothalamus. In this study, we have used an electrophysiological approach coupled with mathematical modeling to investigate the regulation of spontaneous and CRH/arginine vasopressin-induced activity of corticotrophs by glucocorticoids. We reveal that pretreatment of corticotrophs with 100 nM corticosterone (CORT; 90 and 150 min) reduces spontaneous activity and prevents a transition from spiking to bursting after CRH/arginine vasopressin stimulation. In addition, previous studies have identified a role for large-conductance calcium- and voltage-activated potassium (BK) channels in the generation of secretagogue-induced bursting in corticotrophs. Using the dynamic clamp technique, we demonstrated that CRH-induced bursting can be switched to spiking by subtracting a fast BK current, whereas the addition of a fast BK current can induce bursting in CORT-treated cells. In addition, recordings from BK knockout mice (BK(-/-)) revealed that CORT can also inhibit excitability through BK-independent mechanisms to control spike frequency. Thus, we have established that glucocorticoids can modulate multiple properties of corticotroph electrical excitability through both BK-dependent and BK-independent mechanisms.

The electrical activity pattern of endocrine pituitary cells regulates their basal secretion level. Rat somatotrophs and lactotrophs exhibit spontaneous bursting and have high basal levels of hormone secretion, while gonadotrophs exhibit spontaneous spiking and have low basal hormone secretion. It has been proposed that the difference in electrical activity between bursting somatotrophs and spiking gonadotrophs is due to the presence of large conductance potassium (BK) channels on somatotrophs but not on gonadotrophs. This is one example where the role of an ion channel type may be clearly established. We demonstrate here that BK channels indeed promote bursting activity in pituitary cells. Blocking BK channels in bursting lacto-somatotroph GH4C1 cells changes their firing activity to spiking, while further adding an artificial BK conductance via dynamic clamp restores bursting. Importantly, this burst-promoting effect requires a relatively fast BK activation/deactivation, as predicted by computational models. We also show that adding a fast-activating BK conductance to spiking gonadotrophs converts the activity of these cells to bursting. Together, our results suggest that differences in BK channel expression may underlie the differences in electrical activity and basal hormone secretion levels among pituitary cell types and that the rapid rate of BK channel activation is key to its role in burst promotion.

Biological systems are characterized by a high number of interacting components. Determining the role of each component is difficult, addressed here in the context of biological oscillations. Rhythmic behavior can result from the interplay of positive feedback that promotes bistability between high and low activity, and slow negative feedback that switches the system between the high and low activity states. Many biological oscillators include two types of negative feedback processes: divisive (decreases the gain of the positive feedback loop) and subtractive (increases the input threshold) that both contribute to slowly move the system between the high- and low-activity states. Can we determine the relative contribution of each type of negative feedback process to the rhythmic activity? Does one dominate? Do they control the active and silent phase equally? to answer these questions we use a neural network model with excitatory coupling, regulated by synaptic depression (divisive) and cellular adaptation (subtractive feedback). We first attempt to apply standard experimental methodologies: either passive observation to correlate the variations of a variable of interest to system behavior, or deletion of a component to establish whether a component is critical for the system. We find that these two strategies can lead to contradictory conclusions, and at best their interpretive power is limited. We instead develop a computational measure of the contribution of a process, by evaluating the sensitivity of the active (high activity) and silent (low activity) phase durations to the time constant of the process. The measure shows that both processes control the active phase, in proportion to their speed and relative weight. However, only the subtractive process plays a major role in setting the duration of the silent phase. This computational method can be used to analyze the role of negative feedback processes in a wide range of biological rhythms.

Cell responses are commonly heterogeneous, even within a subpopulation. In the present study, we investigate the source of heterogeneity in the Ca(2+) response of anterior pituitary lactotrophs to a Ca(2+) mobilisation agonist, thyrotrophin-releasing hormone. This response is characterised by a sharp increase of cytosolic Ca(2+) concentration as a result of mobilisation of Ca(2+) from intracellular stores, followed by a decrease to an elevated plateau level that results from Ca(2+) influx. We focus on heterogeneity of the evoked Ca(2+) spike under extracellular Ca(2+) free conditions. We introduce a method that uses the information provided by a mathematical model to characterise the source of heterogeneity. This method compares scatter plots of features of the Ca(2+) response obtained experimentally with those made from the mathematical model. The model scatter plots reflect random variation of parameters over different ranges, and matching the experimental and model scatter plots allows us to predict which parameters are most variable. We find that a large degree of variation in Ca(2+) efflux is a likely key contributor to the heterogeneity of Ca(2+) responses to thyrotrophin-releasing hormone in lactotrophs. This technique is applicable to any situation in which the heterogeneous biological response is described by a mathematical model.

In Neuroscience, mathematical modelling involving multiple spatial and temporal scales can unveil complex oscillatory activity such as excitable responses to an input current, subthreshold oscillations, spiking or bursting. While the number of slow and fast variables and the geometry of the system determine the type of the complex oscillations, canard structures define boundaries between them. In this study, we use geometric singular perturbation theory to identify and characterise boundaries between different dynamical regimes in multiple-timescale firing rate models of the developing spinal cord. These rate models are either three or four dimensional with state variables chosen within an overall group of two slow and two fast variables. The fast subsystem corresponds to a recurrent excitatory network with fast activity-dependent synaptic depression, and the slow variables represent the cell firing threshold and slow activity-dependent synaptic depression, respectively. We start by demonstrating canard-induced bursting and mixed-mode oscillations in two different three-dimensional rate models. Then, in the full four-dimensional model we show that a canard-mediated slow passage creates dynamics that combine these complex oscillations and give rise to mixed-mode bursting oscillations (MMBOs). We unveil complicated isolas along which MMBOs exist in parameter space. The profile of solutions along each isola undergoes canard-mediated transitions between the sub-threshold regime and the bursting regime; these explosive transitions change the number of oscillations in each regime. Finally, we relate the MMBO dynamics to experimental recordings and discuss their effects on the silent phases of bursting patterns as well as their potential role in creating subthreshold fluctuations that are often interpreted as noise. The mathematical framework used in this paper is relevant for modelling multiple timescale dynamics in excitable systems.

Endocrine cells in the pituitary gland typically display either spiking or bursting electrical activity, which is related to the level of hormone secretion. Recent work, which combines mathematical modelling with dynamic clamp experiments, suggests the difference is due to the presence or absence of a few large-conductance potassium channels. Since endocrine cells only contain a handful of these channels, it is likely that stochastic effects play an important role in the pattern of electrical activity. Here, for the first time, we explicitly determine the effect of such noise by studying a mathematical model that includes the realistic noisy opening and closing of ion channels. This allows us to investigate how noise affects the electrical activity, examine the origin of spiking and bursting, and determine which channel types are responsible for the greatest noise. Further, for the first time, we address the role of cell size in endocrine cell electrical activity, finding that larger cells typically display more bursting, while the smallest cells almost always only exhibit spiking behaviour.

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.

Models of electrical activity in excitable cells involve nonlinear interactions between many ionic currents. Changing parameters in these models can produce a variety of activity patterns with sometimes unexpected effects. Further more, introducing new currents will have different effects depending on the initial parameter set. In this study we combined global sampling of parameter space and local analysis of representative parameter sets in a pituitary cell model to understand the effects of adding K (+) conductances, which mediate some effects of hormone action on these cells. Global sampling ensured that the effects of introducing K (+) conductances were captured across a wide variety of contexts of model parameters. For each type of K (+) conductance we determined the types of behavioral transition that it evoked. Some transitions were counterintuitive, and may have been missed without the use of global sampling. In general, the wide range of transitions that occurred when the same current was applied to the model cell at different locations in parameter space highlight the challenge of making accurate model predictions in light of cell-to-cell heterogeneity. Finally, we used bifurcation analysis and fast/slow analysis to investigate why specific transitions occur in representative individual models. This approach relies on the use of a graphics processing unit (GPU) to quickly map parameter space to model behavior and identify parameter sets for further analysis. Acceleration with modern low-cost GPUs is particularly well suited to exploring the moderate-sized (5-20) parameter spaces of excitable cell and signaling models.

Corticotroph cells from the anterior pituitary are an integral component of the hypothalamic-pituitary-adrenal (HPA) axis, which governs the neuroendocrine response to stress. Corticotrophs are electrically excitable and fire spontaneous single-spike action potentials and also display secretagogue-induced bursting behavior. The HPA axis function is dependent on effective negative feedback in which elevated plasma glucocorticoids result in inhibition at the level of both the pituitary and the hypothalamus. In this study, we have used an electrophysiological approach coupled with mathematical modeling to investigate the regulation of spontaneous and CRH/arginine vasopressin-induced activity of corticotrophs by glucocorticoids. We reveal that pretreatment of corticotrophs with 100 nM corticosterone (CORT; 90 and 150 min) reduces spontaneous activity and prevents a transition from spiking to bursting after CRH/arginine vasopressin stimulation. In addition, previous studies have identified a role for large-conductance calcium- and voltage-activated potassium (BK) channels in the generation of secretagogue-induced bursting in corticotrophs. Using the dynamic clamp technique, we demonstrated that CRH-induced bursting can be switched to spiking by subtracting a fast BK current, whereas the addition of a fast BK current can induce bursting in CORT-treated cells. In addition, recordings from BK knockout mice (BK(-/-)) revealed that CORT can also inhibit excitability through BK-independent mechanisms to control spike frequency. Thus, we have established that glucocorticoids can modulate multiple properties of corticotroph electrical excitability through both BK-dependent and BK-independent mechanisms.

Endocrine cells of the pituitary gland secrete a number of hormones, and the amount of hormone released by a cell is controlled in large part by the cell's electrical activity and subsequent Ca2+ influx. Typical electrical behaviors of pituitary cells include continuous spiking and so-called pseudo-plateau bursting. It has been shown that the amplitude of Ca2+ fluctuations is greater in bursting cells, leading to the hypothesis that bursting cells release more hormone than spiking cells. In this work, we apply computer simulations to test this hypothesis. We use experimental recordings of electrical activity as input to mathematical models of Ca2+ channel activity, buffered Ca2+ diffusion, and Ca2+-driven exocytosis. To compare the efficacy of spiking and bursting on the same cell, we pharmacologically block the large-conductance potassium (BK) current from a bursting cell or add a BK current to a spiking cell via dynamic clamp. We find that bursting is generally at least as effective as spiking at evoking hormone release and is often considerably more effective, even when normalizing to Ca2+ influx. Our hybrid experimental/modeling approach confirms that adding a BK-type K+ current, which is typically associated with decreased cell activity and reduced secretion, can actually produce an increase in hormone secretion, as suggested earlier.

The Hodgkin-Huxley (HH) model is the basis for numerous neural models. There are two negative feedback processes in the HH model that regulate rhythmic spiking. The first is an outward current with an activation variable n that has an opposite influence to the excitatory inward current and therefore provides subtractive negative feedback. The other is the inactivation of an inward current with an inactivation variable h that reduces the amount of positive feedback and therefore provides divisive feedback. Rhythmic spiking can be obtained with either negative feedback process, so we ask what is gained by having two feedback processes. We also ask how the different negative feedback processes contribute to spiking. We show that having two negative feedback processes makes the HH model more robust to changes in applied currents and conductance densities than models that possess only one negative feedback variable. We also show that the contributions made by the subtractive and divisive feedback variables are not static, but depend on time scales and conductance values. In particular, they contribute differently to the dynamics in Type I versus Type II neurons.

Many hormones are released in pulsatile patterns. This pattern can be modified, for instance by changing pulse frequency, to encode relevant physiological information. Often other properties of the pulse pattern will also change with frequency. How do signaling pathways of cells targeted by these hormones respond to different input patterns? in this study, we examine how a given dose of hormone can induce different outputs from the target system, depending on how this dose is distributed in time. We use simple mathematical models of feedforward signaling motifs to understand how the properties of the target system give rise to preferences in input pulse pattern. We frame these problems in terms of frequency responses to pulsatile inputs, where the amplitude or duration of the pulses is varied along with frequency to conserve input dose. We find that the form of the nonlinearity in the steady state input-output function of the system predicts the optimal input pattern. It does so by selecting an optimal input signal amplitude. Our results predict the behavior of common signaling motifs such as receptor binding with dimerization, and protein phosphorylation. The findings have implications for experiments aimed at studying the frequency response to pulsatile inputs, as well as for understanding how pulsatile patterns drive biological responses via feedforward signaling pathways.

Mathematical models are increasingly important in biology, and testability is becoming a critical issue. One limitation is that one model simulation tests a parameter set representing one instance of the biological counterpart, whereas biological systems are heterogeneous in their properties and behavior, and a model often is fitted to represent an ideal average. This is also true for models of a cell's electrical activity; even within a narrowly defined population there can be considerable variation in electrophysiological phenotype. Here, we describe a computational experimental approach for parameterizing a model of the electrical activity of a cell in real time. We combine the inexpensive parallel computational power of a programmable graphics processing unit with the flexibility of the dynamic clamp method. The approach involves 1), recording a cell's electrical activity, 2), parameterizing a model to the recording, 3), generating predictions, and 4), testing the predictions on the same cell used for the calibration. We demonstrate the experimental feasibility of our approach using a cell line (GH4C1). These cells are electrically active, and they display tonic spiking or bursting. We use our approach to predict parameter changes that can convert one pattern to the other.

Bursting oscillations in excitable systems reflect multi-timescale dynamics. These oscillations have often been studied in mathematical models by splitting the equations into fast and slow subsystems. Typically, one treats the slow variables as parameters of the fast subsystem and studies the bifurcation structure of this subsystem. This has key features such as a z-curve (stationary branch) and a Hopf bifurcation that gives rise to a branch of periodic spiking solutions. In models of bursting in pituitary cells, we have recently used a different approach that focuses on the dynamics of the slow subsystem. Characteristic features of this approach are folded node singularities and a critical manifold. In this article, we investigate the relationships between the key structures of the two analysis techniques. We find that the z-curve and Hopf bifurcation of the two-fast/one-slow decomposition are closely related to the voltage nullcline and folded node singularity of the one-fast/two-slow decomposition, respectively. They become identical in the double singular limit in which voltage is infinitely fast and calcium is infinitely slow.

The electrical activity pattern of endocrine pituitary cells regulates their basal secretion level. Rat somatotrophs and lactotrophs exhibit spontaneous bursting and have high basal levels of hormone secretion, while gonadotrophs exhibit spontaneous spiking and have low basal hormone secretion. It has been proposed that the difference in electrical activity between bursting somatotrophs and spiking gonadotrophs is due to the presence of large conductance potassium (BK) channels on somatotrophs but not on gonadotrophs. This is one example where the role of an ion channel type may be clearly established. We demonstrate here that BK channels indeed promote bursting activity in pituitary cells. Blocking BK channels in bursting lacto-somatotroph GH4C1 cells changes their firing activity to spiking, while further adding an artificial BK conductance via dynamic clamp restores bursting. Importantly, this burst-promoting effect requires a relatively fast BK activation/deactivation, as predicted by computational models. We also show that adding a fast-activating BK conductance to spiking gonadotrophs converts the activity of these cells to bursting. Together, our results suggest that differences in BK channel expression may underlie the differences in electrical activity and basal hormone secretion levels among pituitary cell types and that the rapid rate of BK channel activation is key to its role in burst promotion.

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

Pancreatic islets exhibit bursting oscillations that give rise to oscillatory Ca (2+) entry and insulin secretion from β-cells. These oscillations are driven by a slowly activating K (+) current, Kslow, which is composed of two components: an ATP-sensitive K (+) current and a Ca (2+) -activated K (+) current through SK4 channels. Using a mathematical model of pancreatic β-cells, we analyze how the factors that comprise Kslow can contribute to bursting. We employ the dominance factor technique developed recently to do this and demonstrate that the contributions the slow processes make to bursting are non-obvious and often counterintuitive, and that their contributions vary with parameter values and are thus adjustable.

Biological systems are characterized by a high number of interacting components. Determining the role of each component is difficult, addressed here in the context of biological oscillations. Rhythmic behavior can result from the interplay of positive feedback that promotes bistability between high and low activity, and slow negative feedback that switches the system between the high and low activity states. Many biological oscillators include two types of negative feedback processes: divisive (decreases the gain of the positive feedback loop) and subtractive (increases the input threshold) that both contribute to slowly move the system between the high- and low-activity states. Can we determine the relative contribution of each type of negative feedback process to the rhythmic activity? Does one dominate? Do they control the active and silent phase equally? to answer these questions we use a neural network model with excitatory coupling, regulated by synaptic depression (divisive) and cellular adaptation (subtractive feedback). We first attempt to apply standard experimental methodologies: either passive observation to correlate the variations of a variable of interest to system behavior, or deletion of a component to establish whether a component is critical for the system. We find that these two strategies can lead to contradictory conclusions, and at best their interpretive power is limited. We instead develop a computational measure of the contribution of a process, by evaluating the sensitivity of the active (high activity) and silent (low activity) phase durations to the time constant of the process. The measure shows that both processes control the active phase, in proportion to their speed and relative weight. However, only the subtractive process plays a major role in setting the duration of the silent phase. This computational method can be used to analyze the role of negative feedback processes in a wide range of biological rhythms.

Bursting oscillations are common in neurons and endocrine cells. One type of bursting model with two slow variables has been called 'phantom bursting' since the burst period is a blend of the time constants of the slow variables. A phantom bursting model can produce bursting with a wide range of periods: fast (short period), medium, and slow (long period). We describe a measure, which we call the 'dominance factor', of the relative contributions of the two slow variables to the bursting produced by a simple phantom bursting model. Using this tool, we demonstrate how the control of different phases of the burst can be shifted from one slow variable to another by changing a model parameter. We then show that the dominance curves obtained as a parameter is varied can be useful in making predictions about the resetting properties of the model cells. Finally, we demonstrate two mechanisms by which phase-independent resetting of a burst can be achieved, as has been shown to occur in the electrical activity of pancreatic islets.

We have shown previously that an intravenous injection of oxytocin (OT) in ovariectomized (OVX) rats initiates a circadian rhythm of prolactin (PRL) secretion similar to that observed after cervical stimulation (CS). In this study, we investigated the pathway through which OT triggers the PRL rhythm. We first tested whether an intracerebroventricular injection of OT could trigger the PRL secretory rhythm. As it did not, we injected OT intravenously while an OT receptor antagonist was infused intravenously. This antagonist completely abolished the PRL surges, suggesting that a peripheral target of OT is necessary for triggering the PRL rhythm. We hypothesized that OT may induce PRL release, which would be transported into the brain and trigger the rhythm. In agreement with this, OT injection increased circulating PRL by 5 min. To test whether this acute increase in PRL release would induce the PRL rhythm, we compared the effect of intravenously administered thyrotropin-releasing hormone (TRH) and OT. Although TRH injection also increased PRL to a comparable level after 5 min, only OT-injected animals expressed the PRL secretory rhythm. Motivated by prior findings that bilateral resection of the pelvic nerve blocks CS-induced pseudopregnancy and OT-induced facilitation of lordosis, we then hypothesized that the OT signal may be transmitted through the pelvic nerve. In fact, OT injection failed to induce a PRL secretory rhythm in pelvic-neurectomized animals, suggesting that the integrity of the pelvic nerve is necessary for the systemic OT induction of the PRL secretory rhythm in OVX rats.

Pituitary cells of the anterior pituitary gland secrete hormones in response to patterns of electrical activity. Several types of pituitary cells produce short bursts of electrical activity which are more effective than single spikes in evoking hormone release. These bursts, called pseudo-plateau bursts, are unlike bursts studied mathematically in neurons (plateau bursting) and the standard fast-slow analysis used for plateau bursting is of limited use. Using an alternative fast-slow analysis, with one fast and two slow variables, we show that pseudo-plateau bursting is a canard-induced mixed mode oscillation. Using this technique, it is possible to determine the region of parameter space where bursting occurs as well as salient properties of the burst such as the number of spikes in the burst. The information gained from this one-fast/two-slow decomposition complements the information obtained from a two-fast/one-slow decomposition.

Hormone secretion often occurs in a pulsatile manner. In this review, we discuss two rhythms of in vivo prolactin release in female rats and the ongoing research that we and others have performed aiming to understand the mechanisms underlying them. The peptide hormone oxytocin appears to play an important role in both rhythms. One rhythm occurs during the first half of pregnancy, but can also be induced in ovariectomised rats. This is characterised by a circadian pattern with two prolactin surges per day. Two methods for triggering this rhythm are discussed, each utilising a unique physiological pathway that includes oxytocin action, presumably on pituitary lactotrophs. The second rhythm occurs during the oestrous cycle and is characterised by a surge of prolactin on the afternoon of pro-oestrus. We discuss recent findings that oxytocin is more effective at stimulating prolactin release from lactotrophs taken from animals on the afternoon of pro-oestrus than from those of animals on the morning of dioestrus 1, raising the possibility that this hormone plays a physiological role in the regulation of prolactin secretion during the oestrous cycle.

Cell responses are commonly heterogeneous, even within a subpopulation. In the present study, we investigate the source of heterogeneity in the Ca(2+) response of anterior pituitary lactotrophs to a Ca(2+) mobilisation agonist, thyrotrophin-releasing hormone. This response is characterised by a sharp increase of cytosolic Ca(2+) concentration as a result of mobilisation of Ca(2+) from intracellular stores, followed by a decrease to an elevated plateau level that results from Ca(2+) influx. We focus on heterogeneity of the evoked Ca(2+) spike under extracellular Ca(2+) free conditions. We introduce a method that uses the information provided by a mathematical model to characterise the source of heterogeneity. This method compares scatter plots of features of the Ca(2+) response obtained experimentally with those made from the mathematical model. The model scatter plots reflect random variation of parameters over different ranges, and matching the experimental and model scatter plots allows us to predict which parameters are most variable. We find that a large degree of variation in Ca(2+) efflux is a likely key contributor to the heterogeneity of Ca(2+) responses to thyrotrophin-releasing hormone in lactotrophs. This technique is applicable to any situation in which the heterogeneous biological response is described by a mathematical model.

Spontaneous episodic activity is a fundamental mode of operation of developing networks. Surprisingly, the duration of an episode of activity correlates with the length of the silent interval that precedes it, but not with the interval that follows. Here we use a modeling approach to explain this characteristic, but thus far unexplained, feature of developing networks. Because the correlation pattern is observed in networks with different structures and components, a satisfactory model needs to generate the right pattern of activity regardless of the details of network architecture or individual cell properties. We thus developed simple models incorporating excitatory coupling between heterogeneous neurons and activity-dependent synaptic depression. These models robustly generated episodic activity with the correct correlation pattern. The correlation pattern resulted from episodes being triggered at random levels of recovery from depression while they terminated around the same level of depression. To explain this fundamental difference between episode onset and termination, we used a mean field model, where only average activity and average level of recovery from synaptic depression are considered. In this model, episode onset is highly sensitive to inputs. Thus noise resulting from random coincidences in the spike times of individual neurons led to the high variability at episode onset and to the observed correlation pattern. This work further shows that networks with widely different architectures, different cell types, and different functions all operate according to the same general mechanism early in their development.

Secretion is a fundamental cellular process involving the regulated release of intracellular products from cells. Physiological functions such as neurotransmission, or the release of hormones and digestive enzymes, are all governed by cell secretion. Anomalies in the processes involved in secretion contribute to the development and progression of diseases such as diabetes and other hormonal disorders. To unravel the mechanisms that govern such diseases, it is essential to understand how hormones, growth factors and neurotransmitters are synthesized and processed, and how their signals are recognized, amplified and transmitted by intracellular signaling pathways in the target cells. Here, we discuss diverse aspects of the detailed mechanisms involved in secretion based on mathematical models. The models range from stochastic ones describing the trafficking of secretory vesicles to deterministic ones investigating the regulation of cellular processes that underlie hormonal secretion. In all cases, the models are closely related to experimental results and suggest theoretical predictions for the secretion mechanisms.

Although removal of dopamine inhibition is established as a major factor in prolactin (PRL) release, a large body of evidence suggests that hypothalamic oxytocin (OT) may serve as a PRL-releasing hormone in the rat. PRL release is modulated by estradiol (E2), which rises between diestrus and proestrus of the estrous cycle, causing a PRL surge in the afternoon of proestrus. Given that E2 strongly modulates OT actions in both central and peripheral tissues, OT action on lactotrophs might also be modulated by the stage of the estrous cycle. To test this hypothesis, we have monitored PRL release and intracellular calcium levels ([Ca(2+)](i)) induced by OT in pituitary lactotrophs obtained from female rats in either diestrus 1 or proestrus. We found that both secretory and [Ca(2+)](i) responses to OT are significantly increased in lactotrophs obtained on proestrus. Moreover, we show that these differences are due to an increase in both the number of OT-responding lactotrophs and the magnitude of their individual [Ca(2+)](i) responses. Both secretory and [Ca(2+)](i) responses were abolished by a specific OT antagonist. Finally, dose-dependent studies show that the increased PRL-releasing effect of OT on proestrus is significant over a wide range of concentrations, particularly those observed in hypophyseal portal plasma. These results suggest that the rising E2 titers that culminate on proestrus facilitate the stimulatory action of OT on lactotrophs and support the notion that OT is a PRL-releasing hormone with an important role in the production of the proestrous surge of PRL.

We describe a new technique for comparing mathematical models to the biological systems that are described. This technique is appropriate for systems that produce relaxation oscillations or bursting oscillations, and takes advantage of noise that is inherent to all biological systems. Both types of oscillations are composed of active phases of activity followed by silent phases, repeating periodically. The presence of noise adds variability to the durations of the different phases. The central idea of the technique is that the active phase duration may be correlated with either/both the previous or next silent phase duration, and the resulting correlation pattern provides information about the dynamic structure of the system. Correlation patterns can easily be determined by making scatter plots and applying correlation analysis to the cluster of data points. This could be done both with experimental data and with model simulation data. If the model correlation pattern is in general agreement with the experimental data, then this adds support for the validity of the model. Otherwise, the model must be corrected. While this tool is only one test of many required to validate a mathematical model, it is easy to implement and is noninvasive.

Models of bursting in single cells typically include two subsystems with different timescales. Variations in one or more slow variables switch the system between a silent and a spiking state. We have developed a model for bursting in the pituitary lactotroph that does not include any slow variable. The model incorporates fast, noninactivating calcium and potassium currents (the spike-generating mechanism), as well as the fast, inactivating A-type potassium current (I(A)). I(A) is active only briefly at the beginning of a burst, but this brief impulse of I(A) acts as a burst trigger, injecting the spike trajectory close to an unstable steady state. The spiraling of the trajectory away from the steady state produces a period of low-amplitude spiking typical of lactotrophs. Increasing the conductance of A-type potassium current brings the trajectory closer to the unstable steady state, increasing burst duration. However, this also increases interburst interval, and for larger conductance values, all activity stops. To our knowledge, this is the first example of a physiologically based, single-compartmental model of bursting with no slow subsystem.

Dopamine (DA) released from the hypothalamus tonically inhibits pituitary lactotrophs. DA (at micromolar concentration) opens potassium channels, hyperpolarizing the lactotrophs and thus preventing the calcium influx that triggers prolactin hormone release. Surprisingly, at concentrations approximately 1000 lower, DA can stimulate prolactin secretion. Here, we investigated whether an increase in a K+ current could mediate this stimulatory effect. We considered the fast K+ currents flowing through large-conductance BK channels and through A-type channels. We developed a minimal lactotroph model to investigate the effects of these two currents. Both IBK and IA could transform the electrical pattern of activity from spiking to bursting, but through distinct mechanisms. IBK always increased the intracellular Ca2+ concentration, while IA could either increase or decrease it. Thus, the stimulatory effects of DA could be mediated by a fast K+ conductance which converts tonically spiking cells to bursters. In addition, the study illustrates that

The endothelins are a family of hormones that have a biphasic action on pituitary lactotrophs. The initial effect is stimulatory, followed later by inhibition that persists long after the agonist has been removed. Recent research has uncovered several G protein pathways that mediate these effects.

Simple neural network models of the Xenopus embryo swimming CPG, based on the one originally developed by Roberts and Tunstall (1990), were used to investigate the role of the voltage-dependent N-methyl-D-aspartate (NMDA) receptor channels, in conjunction with faster non-NMDA components of synaptic excitation, in rhythm generation. The voltage-dependent NMDA current "follows" the membrane potential, leading to a postinhibitory rebound that is more efficient than one without voltage dependency and allows neurons to fire more than one action potential per cycle. Furthermore, the model demonstrated limited rhythmic activity in the absence of synaptic inhibition, supporting the hypothesis that the NMDA channels provide a basic mechanism for rhythmicity. However, the rhythmic properties induced by the NMDA current were observed only when there was moderate activation of the non-NMDA synaptic channels, suggesting a modulatory role for this component. The simulations also show that the voltage dependency of the NMDA conductance, as well as the fast non-NMDA current, stabilizes the alternation pattern versus synchrony. To verify that these effects and their implications on the mechanism of swimming and transition to other types of activity take place in the real preparation, constraints on parameter values have to be specified. A method to estimate synaptic parameters was tested with generated data. It is shown that a global analysis, based on multiple iterations of the optimization process (Foster et al. 1993), gives a better understanding of the parameter subspace describing network activity than a standard fit with a sensitivity analysis for an individual solution.