Mathematical Analysis of Complex Cellular Activity

Bertram, Richard; Tabak, Joël; Teka, Wondimu; Vo, T. T. N.; Wechselberger, Martin; Kirk, Vivien; Sneyd, James

doi:10.1007/978-3-319-18114-1

Cited by 7 publications

(12 citation statements)

References 169 publications

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“…This new class of bursting, which we will henceforth denote as folded-node bursting , should not be confused with the recent work on so-called “pseudo-plateau bursting”. Initially thought of as a bursting scenario [ 37 ], the “pseudo-plateau bursting” case was subsequently shown, in various biophysical models of pituitary cells as well as idealized systems, to be better understood as a MMO mechanism [ 38 , 39 ]. Indeed, the small-amplitude oscillations generated by its folded node were first thought to correspond to a bursting phase.…”

Section: Introductionmentioning

confidence: 99%

Classification of bursting patterns: A tale of two ducks

2022

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Bursting is one of the fundamental rhythms that excitable cells can generate either in response to incoming stimuli or intrinsically. It has been a topic of intense research in computational biology for several decades. The classification of bursting oscillations in excitable systems has been the subject of active research since the early 1980s and is still ongoing. As a by-product, it establishes analytical and numerical foundations for studying complex temporal behaviors in multiple timescale models of cellular activity. In this review, we first present the seminal works of Rinzel and Izhikevich in classifying bursting patterns of excitable systems. We recall a complementary mathematical classification approach by Bertram and colleagues, and then by Golubitsky and colleagues, which, together with the Rinzel-Izhikevich proposals, provide the state-of-the-art foundations to these classifications. Beyond classical approaches, we review a recent bursting example that falls outside the previous classification systems. Generalizing this example leads us to propose an extended classification, which requires the analysis of both fast and slow subsystems of an underlying slow-fast model and allows the dissection of a larger class of bursters. Namely, we provide a general framework for bursting systems with both subthreshold and superthreshold oscillations. A new class of bursters with at least 2 slow variables is then added, which we denote folded-node bursters, to convey the idea that the bursts are initiated or annihilated via a folded-node singularity. Key to this mechanism are so-called canard or duck orbits, organizing the underpinning excitability structure. We describe the 2 main families of folded-node bursters, depending upon the phase (active/spiking or silent/nonspiking) of the bursting cycle during which folded-node dynamics occurs. We classify both families and give examples of minimal systems displaying these novel bursting patterns. Finally, we provide a biophysical example by reinterpreting a generic conductance-based episodic burster as a folded-node burster, showing that the associated framework can explain its subthreshold oscillations over a larger parameter region than the fast subsystem approach.

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Section: Introductionmentioning

confidence: 99%

Classification of bursting patterns: A tale of two ducks

2022

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“…MMOs due to a folded node have been previously discussed, e.g., in the context of electrical bursting in pituitary cells [25], [26], [27] or climate change models [28], [29]. However, to the best of our knowledge, this study is the first that connects this particular MMO mechanism with EADs in cardiac action potentials and hence makes a novel contribution to EADs analysis.…”

Section: Introductionmentioning

confidence: 84%

“…EADs in cardiac action potential models with different time scales may be considered as mixed mode oscillations of the form 1 s . MMOs arise in a variety of scientific areas [25], [26], [27], [28], [29], and MMO theory [21], [42] discusses at least four different local mechanisms that give rise to such a behaviour: i) the tourbillon mechanism of a dynamic Hopf bifurcation, ii) a saddle focus equilibrium that goes through a singular Hopf bifurcation, iii) the passage through a folded node, and iv) three-time-scale problems with a singular Hopf bifurcation. Previous results of EADs analysis can either be linked with the tourbillon mechanism, see [17], [18], [20] for the case of a dynamic supercritical Hopf bifurcation and [20] for the case of a dynamic subcritical Hopf bifurcation, or with the saddle focus mechanism, see [20] and also the discussion below.…”

Section: Discussionmentioning

confidence: 99%

Early afterdepolarizations in cardiac action potentials as mixed mode oscillations due to a folded node singularity

2018

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Early afterdepolarizations (EADs) are pathological voltage oscillations during the repolarization phase of cardiac action potentials. They are considered as potential precursors to cardiac arrhythmias and have recently gained much attention in the context of preclinical drug safety testing under the Comprehensive in vitro Proarrhythmia Assay (CiPA) paradigm. From the viewpoint of multiple time scales theory, the onset of EADs has previously been studied by means of mathematical action potential models with one slow ion channel gating variable. In this article, we for the first time associate EADs with mixed mode oscillations in dynamical systems with two slow gating variables and present a folded node singularity of the slow flow as a novel mechanism for EADs genesis. We derive regions of the pharmacology parameter space in which EADs occur using both the folded node analysis and a full system bifurcation analysis, and we suggest the normal distance to the boundary of the EADs region as a mechanism-based risk metric to computationally estimate a drug’s proarrhythmic liability.

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“…Furthermore, a feedforward network is one of fundamental motifs of gene and protein regulatory networks [ 64 , 65 ]. Interestingly, a feedforward network is also observed in the networks of neuronal cells where stochastic multiscale dynamics of ion channels exist [ 75 – 78 ]. Extension of our work to such multiscale stochastic neuronal networks would be an interesting direction for future work.…”

Section: Discussionmentioning

confidence: 99%

Reduction of multiscale stochastic biochemical reaction networks using exact moment derivation

Kim

Sontag

2017

PLoS Comput Biol

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Biochemical reaction networks (BRNs) in a cell frequently consist of reactions with disparate timescales. The stochastic simulations of such multiscale BRNs are prohibitively slow due to high computational cost for the simulations of fast reactions. One way to resolve this problem uses the fact that fast species regulated by fast reactions quickly equilibrate to their stationary distribution while slow species are unlikely to be changed. Thus, on a slow timescale, fast species can be replaced by their quasi-steady state (QSS): their stationary conditional expectation values for given slow species. As the QSS are determined solely by the state of slow species, such replacement leads to a reduced model, where fast species are eliminated. However, it is challenging to derive the QSS in the presence of nonlinear reactions. While various approximation schemes for the QSS have been developed, they often lead to considerable errors. Here, we propose two classes of multiscale BRNs which can be reduced by deriving an exact QSS rather than approximations. Specifically, if fast species constitute either a feedforward network or a complex balanced network, the reduced model based on the exact QSS can be derived. Such BRNs are frequently observed in a cell as the feedforward network is one of fundamental motifs of gene or protein regulatory networks. Furthermore, complex balanced networks also include various types of fast reversible bindings such as bindings between transcriptional factors and gene regulatory sites. The reduced models based on exact QSS, which can be calculated by the computational packages provided in this work, accurately approximate the slow scale dynamics of the original full model with much lower computational cost.

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Mathematical Analysis of Complex Cellular Activity

Cited by 7 publications

References 169 publications

Classification of bursting patterns: A tale of two ducks

Classification of bursting patterns: A tale of two ducks

Early afterdepolarizations in cardiac action potentials as mixed mode oscillations due to a folded node singularity

Reduction of multiscale stochastic biochemical reaction networks using exact moment derivation

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