Reliability of unstable periodic orbit based control strategies in biological systems

Abstract

Presence of recurrent and statistically significant unstable periodic orbits (UPOs) in time series obtained from biological systems are now routinely used as evidence for low dimensional chaos . Extracting accurate dynamical information from the detected UPO trajectories are vital for successful control strategies that either aim to stabilize the system near the fixed point or steer the system away from the periodic orbits. A hybrid UPO detection method from return maps that combines topological recurrence criterion, matrix fit algorithm and stringent criterion for fixed point location gives accurate and statistically significant UPOs even in the presence of significant noise. Geometry of the return map, frequency of UPOs visiting the same trajectory, length of the data set, strength of the noise and degree of nonstationarity affect the efficacy of the proposed method. Results suggest that establishing determinism from unambiguous UPO detection is often possible in short data sets with significant noise, but derived dynamical properties are rarely accurate and adequate for controlling the dynamics around these UPOs. A repeat chaos control experiment on epileptic hippocampal slices through more stringent control strategy and adaptive UPO tracking is reinterpreted in this context through simulation of similar control experiments on an analogous but stochastic computer model of epileptic brain slices. Reproduction of equivalent results suggests that far more stringent criteria are needed for linking apparent success of control in such experiments with possible determinism in the underlying dynamics. Unstable periodic orbits (UPOs) analysis has now become an established tool for detecting determinism in biological systems. If the system is truly deterministic, accurate UPO characteristics can be further used for application of control or anti control strategies with potential medical application. However, accurate biological time series analysis for UPOs critically depend on the length of data set, strength of noise, type of geometry, nonstationarity etc. In this work we show that it is possible to establish determinism through UPO analysis in presence of all these factors. However, the inaccuracy in the measured UPO parameters and non-stationarity nature of biological systems mean only adaptive control strategies are useful for successful chaos control in such systems. At the same time, comparable success of adaptive control strategies on a stochastic neural network model of epileptic slices demand more stringent criteria for linking adaptive control success with determinism.

Capturing the continuous complexity of behaviour in Caenorhabditis elegans

“…3e). A number of biological systems have been investigated using periodic orbits, including neuronal activities 47,48 , human electroencephalograms 47 , crayfish photoreceptors 49 and cardiac arrhythmias 50 .…”

Capturing the Continuous Complexity of Behavior inC. elegans

“…The presence of such UPOs suggests an intriguing view of the worm's foraging dynamics as following a complex landscape composed of unstable orbits, a picture that is rigorously correct for chaotic systems [65,66]. Periodic orbits have also been investigated in a number of biological systems, including neuronal activities [67], human electroencephalograms [67], crayfish photoreceptors [68,69], as well as cardiac arrhythmias and seizures [70][71][72].…”

Predicting the unstable period-1 orbit (UPO-1) of DC–DC converters by extracting harmonic content

“…Conceptually, one can reconstruct the chaotic trajectories by UPOs (Ma et al ., 2013; Teramoto et al ., 2015; Ma et al ., 2015). Up to now, many approaches for locating UPOs are put forward, the classical recurrence plot method (Lathrop and Kostelich, 1989), the dynamic transformation method (So et al ., 1996), the commonly used Schmelcher and Diakonos’s method (the SD method) (Schmelcher and Diakonos, 1997) and some methods proposed recently (Saiki and Ishiyama, 2011; Saiki et al ., 2015; Rajpathak et al ., 2015; Mishra et al ., 2015; Ding et al ., 2016), to name a few. Generally, these methods are usually applied in discrete maps or data sequences, and most of these methods for extracting UPOs have something in common with target-oriented chaos control algorithms.…”