Hodge A, Edwards R, Paul D H, van den Driessche P
Centre for Nonlinear Dynamics in Physiology and Medicine, McGill University, Montréal, QC, Canada.
Biol Cybern. 2006 Jul;95(1):55-68. doi: 10.1007/s00422-006-0065-9. Epub 2006 May 4.
Ordinary differential equations are used to model a peculiar motor behaviour in the anomuran decapod crustacean Emerita analoga. Little is known about the neural circuitry that permits E. analoga to control the phase relationships between movements of the fourth legs and pair of uropods as it digs into sand, so mathematical models might aid in identifying features of the neural structures involved. The geometric arrangement of segmental ganglia controlling the movements of each limb provides an intuitive framework for modelling. Specifically, due to the rhythmic nature of movement, the network controlling the fourth legs and uropods is viewed as three coupled identical oscillators, one dedicated to the control of each fourth leg and one for the pair of uropods, which always move in bilateral synchrony. Systems of Morris-Lecar equations describe the voltage and ion channel dynamics of neurons. Each central pattern generator for a limb is first modelled as a single neuron and then, more realistically as a multi-neuron oscillator. This process results in high-dimensional systems of equations that are difficult to analyse. In either case, reduction to phase equations by averaging yields a two-dimensional system of equations where variables describe only each oscillator's phase along its limit cycle. The behaviour observed in the reduced equations approximates that of the original system. Results suggest that the phase response function in the two dimensional system, together with minimal input from asymmetric bilateral coupling parameters, is sufficient to account for the observed behaviour.
常微分方程被用于模拟异尾类十足目甲壳动物艾氏活额寄居蟹（Emerita analoga）一种特殊的运动行为。对于艾氏活额寄居蟹在挖掘沙子时控制第四对步足和尾肢运动之间相位关系的神经回路，人们了解甚少，因此数学模型可能有助于识别所涉及神经结构的特征。控制每个肢体运动的节段神经节的几何排列为建模提供了一个直观的框架。具体而言，由于运动具有节律性，控制第四对步足和尾肢的网络被视为三个耦合的相同振荡器，一个专门控制每条第四对步足，一个控制一对尾肢，尾肢总是双侧同步运动。莫里斯 - 勒卡尔方程系统描述了神经元的电压和离子通道动力学。每个肢体的中枢模式发生器首先被建模为单个神经元，然后更实际地建模为多神经元振荡器。这一过程导致了难以分析的高维方程组。在任何一种情况下，通过平均简化为相位方程会得到一个二维方程组系统，其中变量仅描述每个振荡器沿其极限环的相位。在简化方程中观察到的行为近似于原始系统的行为。结果表明，二维系统中的相位响应函数，连同来自不对称双侧耦合参数的最小输入，足以解释所观察到的行为。
