Understanding anomalous delays in a model of intracellular calcium dynamics

Abstract

In many cell types, oscillations in the concentration of free intracellular calcium ions are used to control a variety of cellular functions. It has been suggested [16] that the mechanisms underlying the generation and control of such oscillations can be determined by means of a simple experiment, whereby a single exogenous pulse of inositol trisphosphate (IP 3 ) is applied to the cell.However, more detailed mathematical investigations [6] have shown that this is not necessarily always true, and that the experimental data are more difficult to interpret than first thought. Here, we use geometric singular perturbation techniques to study the dynamics of Class I, Class II and Hybrid models of calcium oscillations. In particular, we show how recently developed canard theory for singularly perturbed systems systems with three or more slow variables [19] applies to these calcium models and how the presence of a curve of folded singularities and corresponding canards can result in anomalous delays in the response of these models to a pulse of IP 3 .Oscillations in the concentration of cytoplasmic free calcium are a signalling mechanism in practically every cell type, and control a wide variety of cellular functions, from secretion and movement to cell differentiation and gene 1 that a simple experiment (a single exogenous pulse of inositol trisphosphate, or IP 3 ) is able to distinguish between whether or not calcium oscillations in a cell are a result principally of a Class I or a Class II mechanism. However, more detailed mathematical studies of these models [6] have shown that the situation is more complex than appeared at first, and there are situations in which a Class I model can look remarkably like a Class II model, making interpretation of the experimental data ambiguous and difficult. Here we perform a detailed mathematical study of Class I and Class II models, investigate how they respond to a pulse of IP 3 , and provide a detailed explanation of why Class I and Class II models cannot necessarily be easily distinguished by the IP 3 pulse experiment. It turns out that this is related to the phenomenon of canards in systems with three slow variables, the theory of which has been developed recently [19].

Mixed-Mode Oscillations with Multiple Time Scales

“…Wechselberger [238] extends the theory of canards to such systems and Harvey et al [99] use these results to study models of intracellular calcium dynamics.…”

Canard Theory and Excitability

“…Excitability of neurons [18,29,52] and networks are prime examples of transient dynamics, especially as responses to brief or non-stationary time-varying inputs. Recent developments in canard theory [26,63,64] have provided a new direction for understanding these transient dynamics that are modelled as nonautonomous multiple time-scale systems. It is well known that a nonautonomous system can be viewed as an extended autonomous system by increasing the phase space dimension by one.…”

“…Recently, a canard mechanism was identified that leads to transient dynamics in multiple time-scales systems [26,41,44,64]. Canards are exceptional solutions in singularly perturbed systems which occur on boundaries of regions corresponding to different dynamic behaviours.…”

Multiple Geometric Viewpoints of Mixed Mode Dynamics Associated with Pseudo-plateau Bursting

“…In a neighborhood of the fold surface L or a folded node, the asymptotic error estimates were O(ε 2/3 ) [30,50] and O(ε 1/2 ) [9,62], respectively. Our work also adds to the currently scarce supply of examples of canard-induced MMOs in systems with more than two slow variables [22,23]. In both the 3-fast/1-slow and 1-fast/3-slow analyses, the underlying geometry influences the bursting through a combination of local and global mechanisms.…”

“…Such timescale splittings have been used to effectively explain neuronal dynamics [17,18,43,44,64] and intracellular calcium dynamics [22,23]. This section also adds to the currently sparse literature of examples dealing with systems that have more than two slow variables [22,23,63]. The 1-fast/k-slow approach has been applied to 3D reductions of (2.2) in which k = 2.…”