From global to local: exploring the relationship between parameters and behaviors in models of electrical excitability

Machine learning provides insight into models of heterogeneous electrical activity in human beta-cells

Understanding how heterogeneous cellular responses emerge from cell-to-cell variations in expression and function of subcellular components is of general interest. Here, we focus on human insulin-secreting beta-cells, which are believed to constitute a population in which heterogeneity is of physiological importance. We exploit recent single-cell electrophysiological data that allow biologically realistic population modeling of human beta-cells that accounts for cellular heterogeneity and correlation between ion channel parameters. To investigate how ion channels influence the dynamics of our updated mathematical model of human pancreatic beta-cells, we explore several machine learning techniques to determine which model parameters are important for determining the qualitative patterns of electrical activity of the model cells. As expected, K^＋ channels promote absence of activity, but once a cell is active, they increase the likelihood of having action potential firing. HERG channels were of great importance for determining cell behavior in most of the investigated scenarios. Fast bursting is influenced by the time scales of ion channel activation and, interestingly, by the type of Ca^2＋ channels coupled to BK channels in BK-CaV complexes. Slow, metabolically driven oscillations are promoted mostly by K(ATP) channels. In summary, combining population modeling with machine learning analysis provides insight into the model and generates new hypotheses to be investigated both experimentally, via simulations and through mathematical analysis.

Network Properties of Electrically Coupled Bursting Pituitary Cells

The endocrine cells of the anterior pituitary gland are electrically active when stimulated or, in some cases, when not inhibited. The activity pattern thought to be most effective in releasing hormones is bursting, which consists of depolarization with small spikes that are much longer than single spikes. Although a majority of the research on cellular activity patterns has been performed on dispersed cells, the environment in situ is characterized by networks of coupled cells of the same type, at least in the case of somatotrophs and lactotrophs. This produces some degree of synchronization of their activity, which can be greatly increased by hormones and changes in the physiological state. In this computational study, we examine how electrical coupling among model cells influences synchronization of bursting oscillations among the population. We focus primarily on weak electrical coupling, since strong coupling leads to complete synchronization that is not characteristic of pituitary cell networks. We first look at small networks to point out several unexpected behaviors of the coupled system, and then consider a larger random scale-free network to determine what features of the structural network formed through gap junctional coupling among cells produce a high degree of functional coupling, i.e., clusters of synchronized cells. We employ several network centrality measures, and find that cells that are closely related in terms of their closeness centrality are most likely to be synchronized. We also find that structural hubs (cells with extensive coupling to other cells) are typically not functional hubs (cells synchronized with many other cells). Overall, in the case of weak electrical coupling, it is hard to predict the functional network that arises from a structural network, or to use a functional network as a means for determining the structural network that gives rise to it.

Quantitative roles of ion channel dynamics on ventricular action potential

Mathematical models for the action potential (AP) generation of the electrically excitable cells including the heart are involved different mechanisms including the voltage-dependent currents with nonlinear time- and voltage-gating properties. From the shape of the AP waveforms to the duration of the refractory periods or heart rhythms are greatly affected by the functions describing the features or the quantities of these ion channels. In this work, a mathematical measure to analyze the regional contributions of voltage-gated channels is defined by dividing the AP into phases, epochs, and intervals of interest. The contribution of each time-dependent current for the newly defined cardiomyocyte model is successfully calculated and it is found that the contribution of dominant ion channels changes substantially not only for each phase but also for different regions of the cardiac AP. Besides, the defined method can also be applied in all Hodgkin-Huxley types of electrically excitable cell models to be able to understand the underlying dynamics better.

Ion channel noise shapes the electrical activity of endocrine cells

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.

Heterogeneous alpha-cell population modeling of glucose-induced inhibition of electrical activity

Glucagon release from the pancreatic alpha-cells is regulated by glucose, but the underlying mechanisms are far from understood. It is known that the alpha-cell population is very heterogeneous, but - compared to the insulin-secreting beta-cells - the consequences of this cell-to-cell variation are much less studied. Since the alpha-cells are not electrically coupled, large differences in the single cell responses are to be expected, and this variation may contribute to the confusion regarding the mechanisms of glucose-induced suppression of glucagon release. Using mathematical modeling of alpha-cells with realistic cell-to-cell parameter variation based on recent experimental results, we show that the simulated alpha-cells exhibit great diversity in their electrophysiological behavior. To robustly reproduce experimental recordings from alpha-cell exposed to a rise in glucose levels, we must assume that both intrinsic mechanisms and paracrine signals contribute to glucose-induced changes in electrical activity. Our simulations suggest that the sum of different electrophysiological responses due to alpha-cell heterogeneity is involved in glucose-suppressed glucagon secretion, and that more than one mechanism contribute to control the alpha-cell populations' behavior. Finally, we apply regression analysis to our synthetic alpha-cell population to infer which membrane currents influence electrical activity in alpha-cells at different glucose levels. The results from such statistical modeling suggest possible disturbances underlying defect regulation of alpha-cell electrical behavior in diabetics. Thus, although alpha-cells appear to be inherently complex and heterogeneous as reflected in published data, realistic modeling of the alpha-cells at the population level provides insight into the mechanisms of glucagon release.

A computational model for gonadotropin releasing cells in the teleost fish medaka

Pituitary endocrine cells fire action potentials (APs) to regulate their cytosolic Ca²⁺ concentration and hormone secretion rate. Depending on animal species, cell type, and biological conditions, pituitary APs are generated either by TTX-sensitive Na⁺ currents (I_Na), high-voltage activated Ca²⁺ currents (I_Ca), or by a combination of the two. Previous computational models of pituitary cells have mainly been based on data from rats, where I_Na is largely inactivated at the resting potential, and spontaneous APs are predominantly mediated by I_Ca. Unlike in rats, spontaneous I_Na-mediated APs are consistently seen in pituitary cells of several other animal species, including several species of fish. In the current work we develop a computational model of gonadotropin releasing cells in the teleost fish medaka (Oryzias latipes). The model stands out from previous modeling efforts by being (1) the first model of a pituitary cell in teleosts, (2) the first pituitary cell model that fires sponateous APs that are predominantly mediated by I_Na, and (3) the first pituitary cell model where the kinetics of the depolarizing currents, I_Na and I_Ca, are directly fitted to voltage-clamp data. We explore the firing properties of the model, and compare it to the properties of previous models that fire I_Ca-based APs. We put a particular focus on how the big conductance K⁺ current (I_BK) modulates the AP shape. Interestingly, we find that I_BK can prolong AP duration in models that fire I_Ca-based APs, while it consistently shortens the duration of the predominantly I_Na-mediated APs in the medaka gonadotroph model. Although the model is constrained to experimental data from gonadotroph cells in medaka, it may likely provide insights also into other pituitary cell types that fire I_Na-mediated APs.

Excitable cells elicit electrical pulses called action potentials (APs), which are generated and shaped by a combination of ion channels in the cell membrane. Since one type of ion channels is permeable to Ca²⁺ ions, there is typically an influx of Ca²⁺ during an AP. Pituitary cells therefore use AP firing to regulate their cytosolic Ca²⁺ concentration, which in turn controls their hormone secretion rate. The amount of Ca²⁺ that enters during an AP depends strongly on how long it lasts, and it is therefore important to understand the mechanisms that control this. Pituitary APs are generally mediated by a combination of Ca²⁺ channels and Na⁺ channels, and the relative contributions of from the two vary between cell types, animal species and biological conditions. Previous computer models have predominantly been adapted to data from pituitary cells which tend to fire Ca²⁺-based APs. Here we develop a new model, adapted to data from pituitary cells in the fish medaka, which APs that are predominantly Na⁺-based, and compare its dynamical properties to the previous models that fire Ca²⁺-based APs.

Where to look and how to look: Combining global sensitivity analysis with fast/slow analysis to study multi-timescale oscillations

Parameterized systems of nonlinear ordinary differential equations, the type of system that is often used in mathematical models for biological systems, can be of sufficient complexity that it can take years to appreciate the full range of behaviors that can be produced. Global sensitivity analysis is one tool that has been developed for determining which parameters have the largest impact on the behavior of the model. Thus, it provides the user with a tool to know where to look in parameter space for important changes in behavior. However, it says nothing about the underlying mechanism mediating a change in behavior. For this, other tools exist. If the system dynamics occur over multiple highly-separated time scales then one useful analysis tool is fast/slow geometric analysis, also known as geometric singular perturbation analysis. This is based on bifurcation analysis of a fast or slow subsystem, and can shed light on the influence that a parameter has on structures of either subsystem, and thus on the system dynamics. Hence, once one knows where to look in parameter space for interesting behavior, this technique describes how to look at the system to extract information about how parameter changes influence the behavior of the system. In this study, we combine the two techniques in the analysis of bursting behavior in a model of insulin-secreting pancreatic β-cells, with the goal of determining the key parameters setting the period of the bursting oscillations, and understanding why they are so influential. This can be viewed as a case study for combining mathematical techniques to build on the strengths of each and thereby achieve a better understanding of what most influences the range of model behaviors and how this influence is brought about.

Common and diverse elements of ion channels and receptors underlying electrical activity in endocrine pituitary cells

The pituitary gland contains six types of endocrine cells defined by hormones they secrete: corticotrophs, melanotrophs, gonadotrophs, thyrotrophs, somatotrophs, and lactotrophs. All these cell types are electrically excitable, and voltage-gated calcium influx is the major trigger for their hormone secretion. Along with hormone intracellular content, G-protein-coupled receptor and ion channel expression can also be considered as defining cell type identity. While many aspects of the developmental and activity dependent regulation of hormone and G-protein-coupled receptor expression have been elucidated, much less is known about the regulation of the ion channels needed for excitation-secretion coupling in these cells. We compare the spontaneous and receptor-controlled patterns of electrical signaling among endocrine pituitary cell types, including insights gained from mathematical modeling. We argue that a common set of ionic currents unites these cells, while differential expression of another subset of ionic currents could underlie cell type-specific patterns. We demonstrate these ideas using a generic mathematical model, showing that it reproduces many observed features of pituitary electrical signaling. Mapping these observations to the developmental lineage suggests possible modes of regulation that may give rise to mature pituitary cell types.

The Effects of GABAergic Polarity Changes on Episodic Neural Network Activity in Developing Neural Systems

Early in development, neural systems have primarily excitatory coupling, where even GABAergic synapses are excitatory. Many of these systems exhibit spontaneous episodes of activity that have been characterized through both experimental and computational studies. As development progress the neural system goes through many changes, including synaptic remodeling, intrinsic plasticity in the ion channel expression, and a transformation of GABAergic synapses from excitatory to inhibitory. What effect each of these, and other, changes have on the network behavior is hard to know from experimental studies since they all happen in parallel. One advantage of a computational approach is that one has the ability to study developmental changes in isolation. Here, we examine the effects of GABAergic synapse polarity change on the spontaneous activity of both a mean field and a neural network model that has both glutamatergic and GABAergic coupling, representative of a developing neural network. We find some intuitive behavioral changes as the GABAergic neurons go from excitatory to inhibitory, shared by both models, such as a decrease in the duration of episodes. We also find some paradoxical changes in the activity that are only present in the neural network model. In particular, we find that during early development the inter-episode durations become longer on average, while later in development they become shorter. In addressing this unexpected finding, we uncover a priming effect that is particularly important for a small subset of neurons, called the “intermediate neurons.” We characterize these neurons and demonstrate why they are crucial to episode initiation, and why the paradoxical behavioral change result from priming of these neurons. The study illustrates how even arguably the simplest of developmental changes that occurs in neural systems can present non-intuitive behaviors. It also makes predictions about neural network behavioral changes that occur during development that may be observable even in actual neural systems where these changes are convoluted with changes in synaptic connectivity and intrinsic neural plasticity.

Modeling the diversity of spontaneous and agonist-induced electrical activity in anterior pituitary corticotrophs

Pituitary corticotrophs fire action potentials spontaneously and in response to stimulation with corticotropin-releasing hormone (CRH) and arginine vasopressin (AVP), and such electrical activity is critical for calcium signaling and calcium-dependent adrenocorticotropic hormone secretion. These cells typically fire tall, sharp action potentials when spontaneously active, but a variety of other spontaneous patterns have also been reported, including various modes of bursting. There is variability in reports of the fraction of corticotrophs that are electrically active, as well as their patterns of activity, and the sources of this variation are not well understood. The ionic mechanisms responsible for CRH- and AVP-triggered electrical activity in corticotrophs are also poorly characterized. We use electrophysiological measurements and mathematical modeling to investigate possible sources of variability in patterns of spontaneous and agonist-induced corticotroph electrical activity. In the model, variation in as few as two parameters can give rise to many of the types of patterns observed in electrophysiological recordings of corticotrophs. We compare the known mechanisms for CRH, AVP, and glucocorticoid actions and find that different ionic mechanisms can contribute in different but complementary ways to generate the complex time courses of CRH and AVP responses. In summary, our modeling suggests that corticotrophs have several mechanisms at their disposal to achieve their primary function of pacemaking depolarization and increased electrical activity in response to CRH and AVP.NEW & NOTEWORTHY We and others recently demonstrated that the electrical activity and calcium dynamics of corticotrophs are strikingly diverse, both spontaneously and in response to the agonists CRH and AVP. Here we demonstrate this diversity with electrophysiological measurements and use mathematical modeling to investigate its possible sources. We compare the known mechanisms of agonist-induced activity in the model, showing how the context of ionic conductances dictates the effects of agonists even when their target is fixed.