Fast-slow analysis of a stochastic mechanism for electrical bursting

Electrical bursting oscillations in neurons and endocrine cells are activity patterns that facilitate the secretion of neurotransmitters and hormones and have been the focus of study for several decades. Mathematical modeling has been an extremely useful tool in this effort, and the use of fast-slow analysis has made it possible to understand bursting from a dynamic perspective and to make testable predictions about changes in system parameters or the cellular environment. It is typically the case that the electrical impulses that occur during the active phase of a burst are due to stable limit cycles in the fast subsystem of equations or, in the case of so-called "pseudo-plateau bursting," canards that are induced by a folded node singularity. In this article, we show an entirely different mechanism for bursting that relies on stochastic opening and closing of a key ion channel. We demonstrate, using fast-slow analysis, how the short-lived stochastic channel openings can yield a much longer response in which single action potentials are converted into bursts of action potentials. Without this stochastic element, the system is incapable of bursting. This mechanism can describe stochastic bursting in pituitary corticotrophs, which are small cells that exhibit a great deal of noise as well as other pituitary cells, such as lactotrophs and somatotrophs that exhibit noisy bursts of electrical activity.

Excitability and Threshold Mechanism for Enhanced Neuronal Response Induced by Inhibition Preceding Excitation

Postinhibitory facilitation (PIF) of neural firing presents a paradoxical phenomenon that the inhibitory effect induces enhancement instead of reduction of the firing activity, which plays important roles in sound location of the auditory nervous system, awaited theoretical explanations. In the present paper, excitability and threshold mechanism for the PIF phenomenon is presented in the Morris-Lecar model with type I, II, and III excitabilities. Firstly, compared with the purely excitatory stimulations applied to the steady state, the inhibitory preceding excitatory stimulation to form pairs induces the firing rate increased for type II and III excitabilities instead of type I excitability, when the interval between the inhibitory and excitatory stimulation within each pair is suitable. Secondly, the threshold mechanism for the PIF phenomenon is acquired. For type II and III excitabilities, the inhibitory stimulation induces subthreshold oscillations around the steady state. During the middle and ending phase of the ascending part and the beginning phase of the descending part within a period of the subthreshold oscillations, the threshold to evoke an action potential by an excitatory stimulation becomes weaker, which is the cause for the PIF phenomenon. Last, a theoretical estimation for the range of the interval between the inhibitory and excitatory stimulation for the PIF phenomenon is acquired, which approximates half of the intrinsic period of the subthreshold oscillations for the relatively strong stimulations and becomes narrower for the relatively weak stimulations. The interval for the PIF phenomenon is much shorter for type III excitability, which is closer to the experiment observation, due to the shorter period of the subthreshold oscillations. The results present the excitability and threshold mechanism for the PIF phenomenon, which provide comprehensive and deep explanations to the PIF phenomenon.

Bursting in cerebellar stellate cells induced by pharmacological agents: Non-sequential spike adding

Cerebellar stellate cells (CSCs) are spontaneously active, tonically firing (5-30 Hz), inhibitory interneurons that synapse onto Purkinje cells. We previously analyzed the excitability properties of CSCs, focusing on four key features: type I excitability, non-monotonic first-spike latency, switching in responsiveness and runup (i.e., temporal increase in excitability during whole-cell configuration). In this study, we extend this analysis by using whole-cell configuration to show that these neurons can also burst when treated with certain pharmacological agents separately or jointly. Indeed, treatment with 4-Aminopyridine (4-AP), a partial blocker of delayed rectifier and A-type K⁺ channels, at low doses induces a bursting profile in CSCs significantly different than that produced at high doses or when it is applied at low doses but with cadmium (Cd²⁺), a blocker of high voltage-activated (HVA) Ca²⁺ channels. By expanding a previously revised Hodgkin–Huxley type model, through the inclusion of Ca²⁺-activated K⁺ (K(Ca)) and HVA currents, we explain how these bursts are generated and what their underlying dynamics are. Specifically, we demonstrate that the expanded model preserves the four excitability features of CSCs, as well as captures their bursting patterns induced by 4-AP and Cd²⁺. Model investigation reveals that 4-AP is potentiating HVA, inducing square-wave bursting at low doses and pseudo-plateau bursting at high doses, whereas Cd²⁺ is potentiating K(Ca), inducing pseudo-plateau bursting when applied in combination with low doses of 4-AP. Using bifurcation analysis, we show that spike adding in square-wave bursts is non-sequential when gradually changing HVA and K(Ca) maximum conductances, delayed Hopf is responsible for generating the plateau segment within the active phase of pseudo-plateau bursts, and bursting can become “chaotic” when HVA and K(Ca) maximum conductances are made low and high, respectively. These results highlight the secondary effects of the drugs applied and suggest that CSCs have all the ingredients needed for bursting.

Excitable cells, including neurons, fire action potentials (APs) in their membrane voltage that allow them to communicate with each other and to serve certain physiological purposes. They do so either tonically by firing APs periodically, or episodically by repeatedly firing clusters of APs (called bursts) separated by quiescent periods. Each one of those firing patterns can be neuron-specific and dependent on synaptic inputs and/or their physiological environment. Cerebellar stellate cells (CSCs) that synapse onto Purkinje cells, the sole output of the cerebellum responsible for motor control, are spontaneously active inhibitory interneurons that fire APs tonically. We previously studied the excitability properties of these neurons and showed that they possess several important key features, including type I excitability, runup, non-monotonic first spike latency and switching in responsiveness. In this study, we show that CSCs can also exhibit two modes of burst firing, called square-wave and pseudo-plateau, when treated with certain pharmacological agents. Using bifurcation theory, we demonstrate that spike adding in the square-wave burst is non-sequential, changing by several spikes when certain conductances are altered gradually. This study thus sheds lights onto the overall effects of the pharmacological agents and highlights the ability of CSCs to burst in certain biological conditions.

Deciphering the dynamics of lamellipodium in a fish keratocytes model

Motile cells depend on an intricate network of feedback loops that are essential in driving cell movement. Integrin-based focal adhesions (FAs) along with actin are the two key factors that mediate such motile behaviour. Together, they generate excitable dynamics that are essential for forming protrusions at the leading edge of the cell and, in certain cases, traveling waves along the membrane. A partial differential equation (PDE) model of a self-organizing lamellipodium in crawling keratocytes has been previously developed to understand how the three spatiotemporal patterns of activity observed in such cells, namely, stalling, waving and smooth motility, are produced. The model consisted of three key variables: the density of barbed actin filaments, newly formed FAs called nascent adhesions (NAs) and VASP, an anti-capping protein that gets sequestered by NAs during maturation. Using parameter sweeping techniques, the distinct regimes of behaviour associated with the three activity patterns were identified. In this study, we convert the PDE model into an ordinary differential equation (ODE) model to examine its excitability properties and determine all the patterns of activity exhibited by this system. Our results reveal that there are two additional regimes not previously identified, including bistability and oscillatory-like type IV excitability (generated by three steady states and their manifolds, rather than limit cycles). These regimes are also present in the PDE model. Applying slow-fast analysis on the ODE model shows that it exhibits a canard explosion through a folded-saddle and that rough motility seen in keratocytes is likely due to noise-dependent motility governed by dynamics near the interface of bistability and type IV excitability. The two parameter bifurcation suggests that the increase in the proportion of rough motion is due to a shift in activity towards the bistable and type IV excitable regimes induced by a decrease in NA maturation rate. Our results thus provide important insight into how microscopic mechanical effects are integrated to produce the observed modes of motility.