Timofeeva Y, Coombes S
Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU, UK.
J Math Biol. 2003 Sep;47(3):249-69. doi: 10.1007/s00285-003-0205-y. Epub 2003 May 15.
The De Young Keizer model for intracellular calcium oscillations is based around a detailed description of the dynamics for inositol trisphosphate (IP(3)) receptors. Systematic reductions of the kinetic schemes for IP(3) dynamics have proved especially fruitful in understanding the transition from excitable to oscillatory behaviour. With the inclusion of diffusive transport of calcium ions the model also supports wave propagation. The analysis of waves, even in reduced models, is typically only possible with the use of numerical bifurcation techniques. In this paper we review the travelling wave properties of the biophysical De Young Keizer model and show that much of its behaviour can be reproduced by a much simpler Fire-Diffuse-Fire (FDF) type model. The FDF model includes both a refractory process and an IP(3) dependent threshold. Parameters of the FDF model are constrained using a comprehensive numerical bifurcation analysis of solitary pulses and periodic waves in the De Young Keizer model. The linear stability of numerically constructed solution branches is calculated using pseudospectral techniques. The combination of numerical bifurcation and stability analysis also allows us to highlight the mechanisms that give rise to propagation failure. Moreover, a kinematic theory of wave propagation, based around numerically computed dispersion curves is used to predict waves which connect periodic orbits. Direct numerical simulations of the De Young Keizer model confirm this prediction. Corresponding travelling wave solutions of the FDF model are obtained analytically and are shown to be in good qualitative agreement with those of the De Young Keizer model. Moreover, the FDF model may be naturally extended to include the discrete nature of calcium stores within a cell, without the loss of analytical tractability. By considering calcium stores as idealised point sources we are able to explicitly construct solutions of the FDF model that correspond to saltatory periodic travelling waves.
德扬 - 凯泽尔细胞内钙振荡模型基于对肌醇三磷酸（IP(3)）受体动力学的详细描述。事实证明，对IP(3)动力学的动力学方案进行系统简化在理解从可兴奋行为到振荡行为的转变方面特别有成效。通过纳入钙离子的扩散传输，该模型还支持波传播。即使在简化模型中，对波的分析通常也只有使用数值分岔技术才有可能。在本文中，我们回顾了生物物理德扬 - 凯泽尔模型的行波特性，并表明其许多行为可以由一个简单得多的“激发 - 扩散 - 激发”（FDF）型模型重现。FDF模型包括一个不应期过程和一个依赖IP(3)的阈值。FDF模型的参数通过对德扬 - 凯泽尔模型中的孤立脉冲和周期波进行全面的数值分岔分析来约束。使用伪谱技术计算数值构建的解分支的线性稳定性。数值分岔和稳定性分析的结合还使我们能够突出导致传播失败的机制。此外，基于数值计算的色散曲线的波传播运动学理论用于预测连接周期轨道的波。德扬 - 凯泽尔模型的直接数值模拟证实了这一预测。FDF模型的相应行波解通过解析得到，并显示与德扬 - 凯泽尔模型的解在定性上有很好的一致性。此外，FDF模型可以自然地扩展以包括细胞内钙库的离散性质，而不会损失解析可处理性。通过将钙库视为理想化的点源，我们能够明确构建与跳跃式周期行波相对应的FDF模型的解。
