In recent experimental studies, cardio-respiratory synchronisation was observed when the respiration rate was higher than the resting heart rate. Under this condition, there is a strong asymmetry in the synchronisation tongue towards frequencies higher than that of an unperturbed self-oscillator. These experimental results are compelling because they support a novel hypothesis about the interaction of cardiac and respiratory centres in the brain. This hypothesis, however, required further validation; we aimed to observe the key experimental features in related dynamical models. A number of simple self-oscillatory systems under an external driving force have been considered in this work, and a persistent sodium plus potassium neuron model was found to display the strong asymmetry, which is the key feature of the experimental observations. Bifurcations and other changes in the model’s state space were considered by varying the amplitude and frequency of the external force. It was shown that the strong asymmetry in the synchronisation tongue is linked to a non-uniformity of the surface of the non-synchronous invariant torus, and, within a certain range of parameters, to the additional saddle–node bifurcations of cycles. Then, it was demonstrated that the torus non-uniformity leads to changes in the shape of the phase difference potential as the amplitude increases. To illustrate the latter effect, a numerical approach for estimating the one dimensional phase difference profile and providing the corresponding explicit analytical expression is developed. Also, a discrepancy in synchronisation boundaries obtained by two different approaches, one based on bifurcational analysis and the other on a time series technique, was observed. The latter observation stresses the necessity of considering the standard deviation of instantaneous frequency alongside the mean value for the non-ambiguous identification of synchronisation in periodically driven systems.

在最近的实验研究中，当呼吸率高于静息心率时观察到心肺同步。在这种情况下，同步舌对比未受干扰的自振器的频率更高的频率存在强烈的不对称性。这些实验结果令人信服，因为它们支持了关于大脑中心脏和呼吸中枢相互作用的新假设。然而，这个假设需要进一步验证。我们旨在观察相关动力学模型中的关键实验特征。在这项工作中已经考虑了许多在外部驱动力下的简单自振荡系统，并且发现了持久性钠钾神经元模型显示出强烈的不对称性，这是实验观察的关键特征。通过改变外力的幅度和频率来考虑模型状态空间的分叉和其他变化。结果表明，同步舌中的强不对称性与非同步不变环表面的不均匀性有关，并且在一定的参数范围内，与循环的附加鞍点分叉有关。然后，证明了环面的不均匀性导致相位差电位的形状随着振幅的增加而变化。为了说明后一种效果，开发了一种用于估计一维相位差轮廓并提供相应的显式解析表达式的数值方法。此外，通过两种不同的方法获得的同步边界的差异，观察到一个基于分岔分析，另一个基于时间序列技术。后一种观察强调必须考虑瞬时频率的标准偏差以及周期性驱动系统中同步的非模糊识别的平均值。