A non-autonomous system leading to cyclic chaotic sets to model physiological rhythms

It is proposed a nonlinear system to model highly complex states of rhythms, whose patterns of activity seem irregular. A non-autonomous system which takes into account both exogenous and endogenous influences. The dynamic behaviors of its stroboscopic map are investigated, by using triangular systems. The model provides a theoretical framework for addressing cyclic transitions between chaotic sets. The analysis underlines the role of the parameters in the structure and shape of the attractors, so to be in agreement with experimental data.

[1]  Jeffrey M. Hausdorff,et al.  Physionet: Components of a New Research Resource for Complex Physiologic Signals". Circu-lation Vol , 2000 .

[2]  C. Pennartz,et al.  A Mathematical Model for the Intracellular Circadian Rhythm Generator , 1999, The Journal of Neuroscience.

[3]  C. Mira,et al.  Chaotic Dynamics: From the One-Dimensional Endomorphism to the Two-Dimensional Diffeomorphism , 1987 .

[4]  G.B. Moody,et al.  PhysioNet: a Web-based resource for the study of physiologic signals , 2001, IEEE Engineering in Medicine and Biology Magazine.

[5]  Christian Mira,et al.  Chaotic Dynamics in Two-Dimensional Noninvertible Maps , 1996 .

[6]  Martin P Paulus,et al.  Heart rate variability in bipolar mania and schizophrenia. , 2010, Journal of psychiatric research.

[7]  Paulo C. Rech,et al.  Arnold tongues and the Devil's Staircase in a discrete-time Hindmarsh–Rose neuron model , 2015 .

[8]  Eileen M. Joyce,et al.  Sleep and circadian rhythm disruption in schizophrenia† , 2012, The British journal of psychiatry : the journal of mental science.

[9]  O. Kinouchi,et al.  A brief history of excitable map-based neurons and neural networks , 2013, Journal of Neuroscience Methods.

[10]  Nabil H. Farhat,et al.  Self-Organization in a Parametrically Coupled Logistic Map Network: A Model for Information Processing in the Visual Cortex , 2009, IEEE Transactions on Neural Networks.

[11]  A. Goldbeter,et al.  A Model for Circadian Rhythms in Drosophila Incorporating the Formation of a Complex between the PER and TIM Proteins , 1998, Journal of biological rhythms.

[12]  R. Dieci,et al.  From bi-stability to chaotic oscillations in a macroeconomic model , 2001 .

[13]  Patrick Gaydecki,et al.  The use of the Hilbert transform in ECG signal analysis , 2001, Comput. Biol. Medicine.

[14]  L. Glass.  Synchronization and rhythmic processes in physiology , 2001, Nature.

[15]  C Obcemea,et al.  Chaotic Dynamics of Tumor Growth and Regeneration , 2006 .

[16]  Andrey Shilnikov,et al.  Methods of the Qualitative Theory for the Hindmarsh-rose Model: a Case Study - a Tutorial , 2008, Int. J. Bifurc. Chaos.

[17]  Vladimir I. Nekorkin,et al.  Map Based Models in Neurodynamics , 2010, Int. J. Bifurc. Chaos.

[18]  Aushra Abouzeid,et al.  Definition and Empirical Characterization of Creative Processes , 2003, Nonlinear dynamics, psychology, and life sciences.

[19]  Fatemeh Hadaeghi,et al.  Towards a complex system understanding of bipolar disorder: A map based model of a complex winnerless competition. , 2015, Journal of theoretical biology.

[20]  S. Kolyada,et al.  On dynamics of triangular maps of the square , 1992, Ergodic Theory and Dynamical Systems.

[21]  Mario Markus,et al.  Lyapunov exponents of the logistic map with periodic forcing , 1989 .

[22]  Premananda Indic,et al.  Scaling Behavior of Human Locomotor Activity Amplitude: Association with Bipolar Disorder , 2011, PloS one.

[23]  Hans Thunberg,et al.  Periodicity versus Chaos in One-Dimensional Dynamics , 2001, SIAM Rev..

[24]  A. Wirz-Justice,et al.  Sleep–wake cycles and cognitive functioning in schizophrenia , 2011, British Journal of Psychiatry.

[25]  M. Sanjuán,et al.  Map-based models in neuronal dynamics , 2011 .

[26]  Benjamin Lindner,et al.  Spontaneous voltage oscillations and response dynamics of a Hodgkin-Huxley type model of sensory hair cells , 2011, Journal of mathematical neuroscience.

[27]  Huaguang Gu,et al.  A four-dimensional neuronal model to describe the complex nonlinear dynamics observed in the firing patterns of a sciatic nerve chronic constriction injury model , 2015 .

[28]  Robert M. May,et al.  Simple mathematical models with very complicated dynamics , 1976, Nature.

[29]  Andrey Shilnikov,et al.  Complete dynamical analysis of a neuron model , 2012 .

[30]  V I Nekorkin,et al.  Chaotic oscillations in a map-based model of neural activity. , 2007, Chaos.

[31]  H. Haken,et al.  Rhythms in physiological systems : proceedings of the international symposium at Schloß Elmau, Bavaria, October 22-25, 1990 , 1991 .

[32]  M. Feigenbaum.  Quantitative universality for a class of nonlinear transformations , 1978 .

[33]  J. M. Gonzalez-Miranda.  On the effect of circadian oscillations on biochemical cell signaling by NF-κB. , 2013, Journal of theoretical biology.