将力场分解为随机轨迹重构图上的流

Ramón Nartallo-Kaluarachchi, Paul Expert, David Beers, Alexander Strang, Morten L. Kringelbach, Renaud Lambiotte, Alain Goriely
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引用次数: 0

摘要

将不可逆力和可逆力从随机波动中分离出来,是分析从真实世界动态系统测量的随机轨迹时面临的一个挑战性问题。我们提出了一种方法,通过随机轨迹重构,将静止朗文过程的动力学近似为离散状态马尔可夫过程在相空间的图表示上的演化。接下来,我们利用亥姆霍兹-霍德格德(Helmholtz-Hodged)分解可收缩单纯复合物上的边流与随机过程分解为不可逆和可逆部分的类比。这样,我们就能对重建的流进行分解,并区分随机轨迹下的不可逆流和可逆梯度流。我们在一系列可解和非线性系统上验证了我们的方法,并将其用于深入了解闪烁的红血细胞以及健康和心律失常的心跳动态。特别是,我们捕捉到了健康细胞与被动细胞、健康心跳与心律失常心跳之间不可逆循环电流的差异。我们的方法在数据驱动的随机动力学和图信号处理方法的界面上开辟了新天地,有望进一步应用于生物实验和生理记录的分析。最后,它促使我们在未来对离散和连续空间中的赫姆霍兹-霍奇分解的收敛性进行分析。
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Decomposing force fields as flows on graphs reconstructed from stochastic trajectories
Disentangling irreversible and reversible forces from random fluctuations is a challenging problem in the analysis of stochastic trajectories measured from real-world dynamical systems. We present an approach to approximate the dynamics of a stationary Langevin process as a discrete-state Markov process evolving over a graph-representation of phase-space, reconstructed from stochastic trajectories. Next, we utilise the analogy of the Helmholtz-Hodge decomposition of an edge-flow on a contractible simplicial complex with the associated decomposition of a stochastic process into its irreversible and reversible parts. This allows us to decompose our reconstructed flow and to differentiate between the irreversible currents and reversible gradient flows underlying the stochastic trajectories. We validate our approach on a range of solvable and nonlinear systems and apply it to derive insight into the dynamics of flickering red-blood cells and healthy and arrhythmic heartbeats. In particular, we capture the difference in irreversible circulating currents between healthy and passive cells and healthy and arrhythmic heartbeats. Our method breaks new ground at the interface of data-driven approaches to stochastic dynamics and graph signal processing, with the potential for further applications in the analysis of biological experiments and physiological recordings. Finally, it prompts future analysis of the convergence of the Helmholtz-Hodge decomposition in discrete and continuous spaces.
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