Invariant manifold reductions for Markovian ion channel dynamics

Keener, James P.

doi:10.1007/s00285-008-0199-6

Cited by 36 publications

(56 citation statements)

References 9 publications

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“…A comparison of the deterministic limits obtained for these models is provided in Appendix A and establishes an equivalence between these deterministic versions as soon as their initial conditions satisfy a combinatorial relationship. This question is further studied in [19], where the reduction of the law of jump Markov processes to invariant manifolds is investigated. We recall that every single ion channel process is independent within a subpopulation and between subpopulations.…”

Section: Application Of the Law Of Large Numbers To The Hodgkin-huxlementioning

confidence: 99%

Fluid limit theorems for stochastic hybrid systems with application to neuron models

Pakdaman¹,

2010

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In this paper we establish limit theorems for a class of stochastic hybrid systems (continuous deterministic dynamics coupled with jump Markov processes) in the fluid limit (small jumps at high frequency), thus extending known results for jump Markov processes. We prove a functional law of large numbers with exponential convergence speed, derive a diffusion approximation, and establish a functional central limit theorem. We apply these results to neuron models with stochastic ion channels, as the number of channels goes to infinity, estimating the convergence to the deterministic model. In terms of neural coding, we apply our central limit theorems to numerically estimate the impact of channel noise both on frequency and spike timing coding.

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Section: Application Of the Law Of Large Numbers To The Hodgkin-huxlementioning

confidence: 99%

Fluid limit theorems for stochastic hybrid systems with application to neuron models

Pakdaman¹,

2010

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“…(4)- (7) can be simplified using the fact that it possesses two invariant manifolds 26 , so its dynamics can be reduced to a two dimensional dynamical system for two variables p and q, defined through P C = pq,…”

Section: Model and Approachmentioning

confidence: 99%

Minimal model for calcium alternans due to SR release refractoriness

Cantalapiedra

Alvarez-Lacalle

Peñaranda

et al. 2017

Chaos: An Interdisciplinary Journal of Nonlinear Science

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In the heart, rapid pacing rates may induce alternations in the strength of cardiac contraction, termed pulsus alternans. Often, this is due to an instability in the dynamics of intracellular calcium concentration, whose transients become larger and smaller at consecutive beats. This alternation has been linked experimentally and theoretically to two different mechanisms: an instability due to 1) a strong dependence of calcium release with sarcoplasmic reticulum (SR) load, together with a slow calcium reuptake into the SR or 2) to SR release refractoriness, due to a slow recovery of the ryanodine receptors (RyR2) from inactivation. The relationship between calcium alternans and refractoriness of the RyR2 has been more elusive than the corresponding SR Ca load mechanism. To study the former, we reduce a general calcium model, which mimics the deterministic evolution of a calcium release unit, to its most basic elements. We show that calcium alternans can be understood using a simple nonlinear equation for calcium concentration at the dyadic space, coupled to a relaxation equation for the number of recovered RyR2s. Depending on the number of RyR2s that are recovered at the beginning of a stimulation, the increase in calcium concentration may pass, or not, over an excitability threshold that limits the occurrence of a large calcium transient. When the recovery of the RyR2 is slow, this produces naturally a period doubling bifurcation, resulting in calcium alternans. We then study the effects of inactivation, calcium diffusion and release conductance for the onset of alternans.We find that the onset of alternans requires a well-defined value of diffusion while it is less sensitive to the values of inactivation or release conductance.PACS numbers: 47.55. 47.20.Ma, 68.03.Cd ionic models; ventricular arrhythmias; calcium alternans 1 Cardiac alternans is a dysfunction that has been identified as a risk factor for cardiac arrhythmias. During alternans, beat-to-beat alternations in action potential duration (APD) appear alongside alternations in cytosolic calcium concentration. Despite been tightly coupled, voltage clamp experiments have shown that, often, the originating mechanism for alternans stems from an instability in calcium handling. Two alternative scenarios may explain this instability, either a slow reuptake of calcium after a release, or a long refractory period in release. In this paper we study the conditions for alternans due to refractoriness in calcium release, by reducing the dynamics of calcium to a simple two-variable model. The model includes a nonlinear dependence of release with calcium concentration and a slow recovery from refractoriness. This simple model presents excitability with a threshold that depends on the state of the ryanodine receptors (RyRs), i.e., calcium sensitive channels through which calcium is released to the cytosol. Calcium alternans appears as a bifurcation in which only one out of every two pulses is able to cross this excitability limit.

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“…In this sense, both stochastic HH models can be considered as compatible with the original HH model as long as one is interested in the dynamics for theses special initial conditions. In Keener (2009), another result is obtained in a slightly different setting: assuming the transition coefficients to be only function of time, and forgetting V, it is shown that the corresponding invariant manifold is globally attracting under eigenvalues conditions.…”

Section: Comparison Between Two-state and Multistate Gating Schemesmentioning

confidence: 95%

Reduction of stochastic conductance-based neuron models with time-scales separation

2011

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We introduce a method for systematically reducing the dimension of biophysically realistic neuron models with stochastic ion channels exploiting time-scales separation. Based on a combination of singular perturbation methods for kinetic Markov schemes with some recent mathematical developments of the averaging method, the techniques are general and applicable to a large class of models. As an example, we derive and analyze reductions of different stochastic versions of the Hodgkin Huxley (HH) model, leading to distinct reduced models. The bifurcation analysis of one of the reduced models with the number of channels as a parameter provides new insights into some features of noisy discharge patterns, such as the bimodality of interspike intervals distribution. Our analysis of the stochastic HH model shows that, besides being a method to reduce the number of variables of neuronal models, our reduction scheme is a powerful method for gaining understanding on the impact of fluctuations due to finite size effects on the dynamics of slow fast systems. Our analysis of the reduced model reveals that decreasing the number of sodium channels in the HH model leads to a transition in the dynamics reminiscent of the Hopf bifurcation and that this transition accounts for changes in characteristics of the spike train generated by the model. Finally, we also examine the impact of these results on neuronal coding, notably, reliability of discharge times and spike latency, showing that reducing the number of channels can enhance discharge time reliability in response to weak inputs and that this phenomenon can be accounted for through the analysis of the reduced model.

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Invariant manifold reductions for Markovian ion channel dynamics

Cited by 36 publications

References 9 publications

Fluid limit theorems for stochastic hybrid systems with application to neuron models

Fluid limit theorems for stochastic hybrid systems with application to neuron models

Minimal model for calcium alternans due to SR release refractoriness

Reduction of stochastic conductance-based neuron models with time-scales separation

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