Biological systems, and particularly neuronal circuits, embody a very high level of complexity. Mathematical modeling is therefore essential for understanding how large sets of neurons with complex multiple interconnections work as a functional system. With the increase in computing power, it is now possible to numerically integrate a model with many variables to simulate behavior. However, such analysis can be time-consuming and may not reveal the mechanisms underlying the observed phenomena. An alternative, complementary approach is mathematical analysis, which can demonstrate direct and explicit relationships between a property of interest and system parameters. This paper introduces a mathematical tool for analyzing neuronal oscillator circuits based on multivariable harmonic balance (MHB). The tool is applied to a model of the central pattern generator (CPG) for leech swimming, which comprises a chain of weakly coupled segmental oscillators. The results demonstrate the effectiveness of the MHB method and provide analytical explanations for some CPG properties. In particular, the intersegmental phase lag is estimated to be the sum of a nominal value and a perturbation, where the former depends on the structure and span of the neuronal connections and the latter is roughly proportional to the period gradient, communication delay, and the reciprocal of the intersegmental coupling strength.

中文翻译：

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用于水蛭游泳的神经元振荡器的多变量谐波平衡分析。

生物系统，尤其是神经元回路，具有非常高的复杂性。因此，数学建模对于理解具有复杂多重互连的大型神经元集如何作为一个功能系统工作至关重要。随着计算能力的提高，现在可以对具有许多变量的模型进行数值积分来模拟行为。然而，这种分析可能很耗时，并且可能无法揭示所观察到的现象背后的机制。另一种互补的方法是数学分析，它可以证明感兴趣的属性和系统参数之间的直接和明确的关系。本文介绍了一种用于分析基于多变量谐波平衡 (MHB) 的神经元振荡器电路的数学工具。该工具应用于水蛭游泳的中央模式生成器 (CPG) 模型，该模型包括弱耦合分段振荡器链。结果证明了 MHB 方法的有效性，并为某些 CPG 特性提供了分析解释。特别地，段间相位滞后估计为标称值和扰动的总和，其中前者取决于神经元连接的结构和跨度，后者大致与周期梯度、通信延迟和扰动成正比。段间耦合强度的倒数。