MBE
Communication and Synchronization in Disconnected Networks with Dynamic Topology: Moving Neighborhood Networks
Joseph D. Skufca Erik M. Bollt
Mathematical Biosciences & Engineering 2004, 1(2): 347-359 doi: 10.3934/mbe.2004.1.347
We consider systems that are well modelled as networks that evolve in time, which we call Moving Neighborhood Networks. These models are relevant in studying cooperative behavior of swarms and other phenomena where emergent interactions arise from ad hoc networks. In a natural way, the time-averaged degree distribution gives rise to a scale-free network. Simulations show that although the network may have many noncommunicating components, the recent weighted time-averaged communication is sufficient to yield robust synchronization of chaotic oscillators. In particular, we contend that such time-varying networks are important to model in the situation where each agent carries a pathogen (such as a disease) in which the pathogen's life-cycle has a natural time-scale which competes with the time-scale of movement of the agents, and thus with the networks communication channels.
keywords: communication in complex networks. mathematical model Disease spread in communities nonlinear dynamics self-organization epidemiology
MBE
A statistical approach to the use of control entropy identifies differences in constraints of gait in highly trained versus untrained runners
Rana D. Parshad Stephen J. McGregor Michael A. Busa Joseph D. Skufca Erik Bollt
Mathematical Biosciences & Engineering 2012, 9(1): 123-145 doi: 10.3934/mbe.2012.9.123
Control entropy (CE) is a complexity analysis suitable for dynamic, non-stationary conditions which allows the inference of the control effort of a dynamical system generating the signal [4]. These characteristics make CE a highly relevant time varying quantity relevant to the dynamic physiological responses associated with running. Using High Resolution Accelerometry (HRA) signals we evaluate here constraints of running gait, from two different groups of runners, highly trained collegiate and untrained runners. To this end, we further develop the control entropy (CE) statistic to allow for group analysis to examine the non-linear characteristics of movement patterns in highly trained runners with those of untrained runners, to gain insight regarding gaits that are optimal for running. Specifically, CE develops response time series of individuals descriptive of the control effort; a group analysis of these shapes developed here uses Karhunen Loeve Analysis (KL) modes of these time series which are compared between groups by application of a Hotelling $T^{2}$ test to these group response shapes. We find that differences in the shape of the CE response exist within groups, between axes for untrained runners (vertical vs anterior-posterior and mediolateral vs anterior-posterior) and trained runners (mediolateral vs anterior-posterior). Also shape differences exist between groups by axes (vertical vs mediolateral). Further, the CE, as a whole, was higher in each axis in trained vs untrained runners. These results indicate that the approach can provide unique insight regarding the differing constraints on running gait in highly trained and untrained runners when running under dynamic conditions. Further, the final point indicates trained runners are less constrained than untrained runners across all running speeds.
keywords: statistical hypothesis testing Control entropy gait analysis.
MBE
Control entropy: A complexity measure for nonstationary signals
Erik M. Bollt Joseph D. Skufca Stephen J . McGregor
Mathematical Biosciences & Engineering 2009, 6(1): 1-25 doi: 10.3934/mbe.2009.6.1
We propose an entropy statistic designed to assess the behavior of slowly varying parameters of real systems. Based on correlation entropy, the method uses symbol dynamics and analysis of increments to achieve sufficient recurrence in a short time series to enable entropy measurements on small data sets. We analyze entropy along a moving window of a time series, the entropy statistic tracking the behavior of slow variables of the data series. We employ the technique against several physiological time series to illustrate its utility in characterizing the constraints on a physiological time series. We propose that changes in the entropy of measured physiological signal (e.g. power output) during dynamic exercise will indicate changes in underlying constraint of the system of interest. This is compelling because CE may serve as a non-invasive, objective means of determining physiological stress under non-steady state conditions such as competition or acute clinical pathologies. If so, CE could serve as a valuable tool for dynamically monitoring health status in a wide range of non-stationary systems.
keywords: Entropy signal analysis Physiology
MBE
Heart rate variability as determinism with jump stochastic parameters
Jiongxuan Zheng Joseph D. Skufca Erik M. Bollt
Mathematical Biosciences & Engineering 2013, 10(4): 1253-1264 doi: 10.3934/mbe.2013.10.1253
We use measured heart rate information (RR intervals) to develop a one-dimensional nonlinear map that describes short term deterministic behavior in the data. Our study suggests that there is a stochastic parameter with persistence which causes the heart rate and rhythm system to wander about a bifurcation point. We propose a modified circle map with a jump process noise term as a model which can qualitatively capture such this behavior of low dimensional transient determinism with occasional (stochastically defined) jumps from one deterministic system to another within a one parameter family of deterministic systems.
keywords: circle map next angle map electrocardiography jump process. Heart rate variability

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