In this paper we describe a combined combinatorial/numerical approach to studying equilibria and bifurcations in network models arising in Systems Biology. ODE models of the dynamics suffer from high dimensional parameters which presents a significant obstruction to studying the global dynamics via numerical methods. The main point of this paper is to demonstrate that adapting and combining classical techniques with recently developed combinatorial methods provides a richer picture of the global dynamics despite the high parameter dimension. Given a network topology describing state variables which regulate one another via monotone and bounded functions, we first use the {\em Dynamic Signatures Generated by Regulatory Networks} (DSGRN) software to obtain a combinatorial summary of the dynamics. This summary is coarse but global and we use this information as a first pass to identify ``interesting'' subsets of parameters in which to focus. We construct an associated ODE model with high parameter dimension using our {\em Network Dynamics Modeling and Analysis} (NDMA) Python library. We introduce algorithms for efficiently investigating the dynamics in these ODE models restricted to these parameter subsets. Finally, we perform a statistical validation of the method and several interesting dynamical applications including finding saddle-node bifurcations in a $54$ parameter model.

翻译：本文提出了一种组合与数值相结合的方法，用于研究系统生物学中网络模型的平衡点与分岔问题。描述动力学的常微分方程模型具有高维参数，这给通过数值方法研究全局动力学带来了显著障碍。本文的核心在于证明，尽管参数维度较高，但将经典技术与近期发展的组合方法相结合并加以调整，能够为全局动力学提供更丰富的图景。给定一个描述状态变量通过单调有界函数相互调控的网络拓扑，我们首先使用{\em 调控网络生成的动态特征}（DSGRN）软件获取动力学的组合摘要。该摘要虽粗略但具有全局性，我们利用此信息作为初步筛选，以确定值得关注的“有趣”参数子集。随后，我们使用{\em 网络动力学建模与分析}（NDMA）Python库构建了一个具有高维参数的关联常微分方程模型。我们提出了高效研究这些常微分方程模型在限定参数子集内动力学的算法。最后，我们对该方法进行了统计验证，并展示了若干有趣的动力学应用，包括在一个包含$54$个参数的模型中寻找鞍结分岔。