Article
Robust Perron cluster analysis in conformation dynamics
KonradZuseZentrum fuer Informationstechnik, Berlin D14195, Germany
Linear Algebra and its Applications (Impact Factor: 0.94). 01/2003; 398(1):161184. DOI: 10.1016/j.laa.2004.10.026 ABSTRACT
The key to molecular conformation dynamics is the direct identification of metastable conformations, which are almost invariant sets of molecular dynamical systems. Once some reversible Markov operator has been discretized, a generalized symmetric stochastic matrix arises. This matrix can be treated by Perron cluster analysis, a rather recent method involving a Perron cluster eigenproblem. The paper presents an improved Perron cluster analysis algorithm, which is more robust than earlier suggestions. Numerical examples are included.
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 "For the identification of metastability in time series, their characteristic slow time scales must be 47 separated from the fast dynamics of phase space trajectories. The method known as Perron clustering 48 [Deuflhard and Weber, 2005], starts with an ad hoc partitioning of the system's phase space that leads to 49 an approximate Markov chain description [Deuflhard and Weber, 2005; Allefeld et al., 2009; Larralde 50 and Leyvraz, 2005; Froyland, 2005; Gaveau and Schulman, 2006]. Applying spectral clustering 51 methods to the resulting transition matrix yields the time scales of the process, while their corresponding 52 (left)eigenvectors allow the unification of cells into a partition of metastable states [Allefeld et al., 2009; 53 Gaveau and Schulman, 2006]. "
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DESCRIPTION: Transients, metastable states (MS) and their temporal recurrences encode fundamental information of biophysiological system dynamics. To understand the system’s temporal dynamics, it is important to detect such events. In neural experimental data, it is challenging to extract these features due to the large trialtotrial variability. A proposed detection methodology extracts recurrent MS in time series (TS). It comprises a timefrequency embedding of TS and a novel statistical inference analysis for recurrence plots (RP). To this end, we propose to transform TS into their timefrequency representations and compute RPs based on the instantaneous spectral power values in various frequency bands. Additionaly, we introduce a new statistical test that compares trial RPs with corresponding surrogate RPs and obtains statistically significant information within RPs. The combination of methods is validated by applying it on two artificial datasets, commonly used in lowlevel brain dynamics modelling. In a final study of visually evoked Local Field Potentials, the methodology is able to reveal recurrence structures of neural responses in recordings with a trialtotrial variability. Focusing on different frequency bands, the deltaband activity is much less recurrent than alphaband activity in partially anesthetized ferrets. Moreover, alphaactivity is susceptible to prestimuli, while deltaactivity is much less sensitive to prestimuli. 
 "For the identification of metastability in time series, their characteristic slow time scales must be 47 separated from the fast dynamics of phase space trajectories. The method known as Perron clustering 48 [Deuflhard and Weber, 2005], starts with an ad hoc partitioning of the system's phase space that leads to 49 an approximate Markov chain description [Deuflhard and Weber, 2005; Allefeld et al., 2009; Larralde 50 and Leyvraz, 2005; Froyland, 2005; Gaveau and Schulman, 2006]. Applying spectral clustering 51 methods to the resulting transition matrix yields the time scales of the process, while their corresponding 52 (left)eigenvectors allow the unification of cells into a partition of metastable states [Allefeld et al., 2009; 53 Gaveau and Schulman, 2006]. "
Conference Paper: Dynamics analysis of neural univariate time series by recurrence plots

 "There are several methods that can be used to minimize Eq. 4. Since the cost function is a quadratic form it is natural to assume that s should be chosen to be in some sense close to the dominating eigenvectors of the matrix C 2 b 21 E responding to the largest eigenvalue, (Deuflhard and Weber 2005, Newman 2006, Nilsson Jacobi and Görnerup 2009). Unfortunately, in our case the spectral approach performs poorly primarily due to the instability of the eigenvalue problem for large matrices. "
Dataset: Nilsson Jacbi et al 2012