Research

Bursting, Chaos

M. Marhl, M. Perc,

Determining the flexibility of regular and chaotic attractors

Chaos, Slotitons & Fractals
28 (2006) 822-833

Abstract: We present an overview of measures that are appropriate for determining the flexibility of regular and chaotic attractors. In particular, we focus on those system properties that constitute its responses to external perturbations. We deploy a systematic approach, first introducing the simplest measure given by the local divergence of the system along the attractor, and then develop more rigorous mathematical tools for estimating the flexibility of the system’s dynamics. The presented measures are tested on the regular Brusselator and chaotic Hindmarsh–Rose model of an excitable neuron with equal success, thus indicating the overall effectiveness and wide applicability range of the proposed theory. Since responses of dynamical systems to external signals are crucial in several scientific disciplines, and especially in natural sciences, we discuss several important aspects and biological implications of obtained results.

M. Perc, M. Marhl

Chaos in temporarily destabilized regular systems with the slow passage effect

Chaos, Slotitons & Fractals
27 (2006) 395-403

Abstract: We provide evidences for chaotic behaviour in temporarily destabilized regular systems. In particular, we focus on time-continuous systems with the slow passage effect. The extreme sensitivity of the slow passage phase enables the existence of long chaotic transients induced by random pulsatile perturbations, thereby evoking chaotic behaviour in an initially regular system. We confirm the chaotic behaviour of the temporarily destabilized system by calculating the largest Lyapunov exponent. Moreover, we show that the newly obtained unstable periodic orbits can be easily controlled with conventional chaos control techniques, thereby guaranteeing a rich diversity of accessible dynamical states that is usually expected only in intrinsically chaotic systems. Additionally, we discuss the biological importance of presented results.

S. Schuster, B. Knoke, M. Marhl

Differential regulation of proteins by bursting calcium oscillations - a theoretical study

Biosystems
81 (2005) 49-63

M. Perc and M. Marhl

Amplification of information transfer in excitable systems that reside in a steady state near a bifurcation point to complex oscilatory behavior

Physical Review E
71 (2005) art. no. 026229

Abstract: We study the amplification of information transfer in excitable systems. We show that excitable systems residing in a steady state near a bifurcation point to complex oscillatory behavior incorporate several frequencies that can be exploited for a resonant amplification of information transfer. In particular, for excitable neurons that reside in a steady state near a bifurcation point to elliptic bursting oscillations, we show that in addition to the resonant frequency of damped oscillations around the stable focus, another frequency exists that resonantly enhances large amplitude bursts and thus amplifies the information transfer in the system. This additional frequency cannot be found by the local stability analysis and has never been used for amplifying the information transfer in a system. The results obtained for elliptic bursting oscillations can be generalized also to other complex oscillators, such as parabolic or square-wave bursters. Additionally, the biological importance of presented results in the field of neuroscience is outlined.

M. Perc and M. Marhl

Resonance effects determine the frequency of bursting Ca2+ oscillations

Chemical Physics Letters
376 (2003) 432-437

Abstract: A mathematical model for bursting Ca2+ oscillations is analysed from a physical point of view as a system of internally coupled fast and slow oscillators. We show that the fast subsystem determines the interburst frequency, whereas altering the kinetics of the slow processes changes the duration of the bursting phase in a resonant manner. The resonance effect appears between two oscillatory Ca2+-buffering mechanisms. This may be biologically important for a highly selective Ca2+ signal transduction from cell receptors to target proteins.

M. Perc and M. Marhl

Different types of bursting calcium oscillations in non-excitable cells

Chaos, Solitons & Fractals
18 (2003) 759-773

Abstract: In the paper different types of bursting Ca2+ Oscillations are presented. We analyse bursting behaviour in four recent mathematical models for Ca2+ oscillations in non-excitable cells. Separately, regular, quasi-periodic, and chaotic bursting Ca2+ oscillations are classified into several subtypes. The classification is based on the dynamics of separated fast and slow subsystems, the so-called fast-slow burster analysis. For regular bursting Ca2+ oscillations two types of bursting are specified: Point-Point and Point-Cycle bursting. In particular, the slow passage effect, important for the Hopf-Hopf and SubHopf-SubHopf bursting subtypes, is explained by local divergence calculated for the fast sub-system. Quasi-periodic bursting Ca2+ oscillations can be found in only one of the four studied mathematical models and appear via a homoclinic bifurcation with a homoclinic torus structure. For chaotic bursting Ca2+ oscillations, we found that bursting patterns resulting from the period doubling root to chaos considerably differ from those appearing via intermittency and have to be treated separately. The analysis and classification of different types of bursting Ca2+ oscillations provides better insight into mechanisms of complex intra- and intercellular Ca2+ signalling. This improves our understanding of several important biological phenomena in cellular signalling like complex frequency-amplitude signal encoding and synchronisation of intercellular signal transduction between coupled cells in tissue.

T. Haberichter, M. Marhl, R. Heinrich

Birhythmicity, trirhythmicity and chaos in bursting calcium oscillations

Biophysical Chemistry
90 (2001) 17-30

M. Marhl, T. Haberichter, M. Brumen, R. Heinrich

Complex calcium oscillations and the role of mitochondria and cytosolic proteins

Biosystems
57 (2000) 75-86

Elsevier	*M. Marhl, M. Perc,* Determining the flexibility of regular and chaotic attractors Chaos, Slotitons & Fractals 28 (2006) 822-833	Abstract: We present an overview of measures that are appropriate for determining the flexibility of regular and chaotic attractors. In particular, we focus on those system properties that constitute its responses to external perturbations. We deploy a systematic approach, first introducing the simplest measure given by the local divergence of the system along the attractor, and then develop more rigorous mathematical tools for estimating the flexibility of the system’s dynamics. The presented measures are tested on the regular Brusselator and chaotic Hindmarsh–Rose model of an excitable neuron with equal success, thus indicating the overall effectiveness and wide applicability range of the proposed theory. Since responses of dynamical systems to external signals are crucial in several scientific disciplines, and especially in natural sciences, we discuss several important aspects and biological implications of obtained results.

Elsevier	*M. Perc, M. Marhl* Chaos in temporarily destabilized regular systems with the slow passage effect Chaos, Slotitons & Fractals 27 (2006) 395-403	Abstract: We provide evidences for chaotic behaviour in temporarily destabilized regular systems. In particular, we focus on time-continuous systems with the slow passage effect. The extreme sensitivity of the slow passage phase enables the existence of long chaotic transients induced by random pulsatile perturbations, thereby evoking chaotic behaviour in an initially regular system. We confirm the chaotic behaviour of the temporarily destabilized system by calculating the largest Lyapunov exponent. Moreover, we show that the newly obtained unstable periodic orbits can be easily controlled with conventional chaos control techniques, thereby guaranteeing a rich diversity of accessible dynamical states that is usually expected only in intrinsically chaotic systems. Additionally, we discuss the biological importance of presented results.

Elsevier	*S. Schuster, B. Knoke, M. Marhl* Differential regulation of proteins by bursting calcium oscillations - a theoretical study Biosystems 81 (2005) 49-63	Abstract: Calcium in ionic form is a second messenger connecting several input signals to several target processes in the cell. The question arises how one second messenger can transmit more than one signal simultaneously (bow-tie structure of signalling). Experimental data on calcium dynamics often show patterns of successive low-peak and high-peak oscillatory phases, known as bursting. Here, we propose that bursting calcium oscillations can perform the function of simultaneous transmission of two signals at physiological calcium concentrations, for example, by selective activation of two calcium-binding proteins. This differential regulation by periodic bursting is investigated in a theoretical model. The two proteins are assumed to be activated by calcium, and one of them is assumed to be subject to biphasic regulation due to additional inhibitory binding sites. To explore which characteristics of the complex signal could be responsible for independent regulation of low-peak activated and spike activated targets, different bursting patterns of simplified square pulses are applied. Depending on the change in the bursting pattern, one protein can be gradually activated at a constant level of the other protein's activity, or the two proteins can be activated simultaneously, or one protein can be activated while the other one is deactivated simultaneously. Thus, the two proteins can be regulated virtually independently.

APS	M. Perc and M. Marhl Amplification of information transfer in excitable systems that reside in a steady state near a bifurcation point to complex oscilatory behavior Physical Review E 71 (2005) art. no. 026229	Abstract: We study the amplification of information transfer in excitable systems. We show that excitable systems residing in a steady state near a bifurcation point to complex oscillatory behavior incorporate several frequencies that can be exploited for a resonant amplification of information transfer. In particular, for excitable neurons that reside in a steady state near a bifurcation point to elliptic bursting oscillations, we show that in addition to the resonant frequency of damped oscillations around the stable focus, another frequency exists that resonantly enhances large amplitude bursts and thus amplifies the information transfer in the system. This additional frequency cannot be found by the local stability analysis and has never been used for amplifying the information transfer in a system. The results obtained for elliptic bursting oscillations can be generalized also to other complex oscillators, such as parabolic or square-wave bursters. Additionally, the biological importance of presented results in the field of neuroscience is outlined.

Elsevier	M. Perc and M. Marhl Resonance effects determine the frequency of bursting Ca2+ oscillations Chemical Physics Letters 376 (2003) 432-437	Abstract: A mathematical model for bursting Ca2+ oscillations is analysed from a physical point of view as a system of internally coupled fast and slow oscillators. We show that the fast subsystem determines the interburst frequency, whereas altering the kinetics of the slow processes changes the duration of the bursting phase in a resonant manner. The resonance effect appears between two oscillatory Ca2+-buffering mechanisms. This may be biologically important for a highly selective Ca2+ signal transduction from cell receptors to target proteins.

Elsevier	*M. Perc and M. Marhl* Different types of bursting calcium oscillations in non-excitable cells Chaos, Solitons & Fractals 18 (2003) 759-773	Abstract: In the paper different types of bursting Ca2+ Oscillations are presented. We analyse bursting behaviour in four recent mathematical models for Ca2+ oscillations in non-excitable cells. Separately, regular, quasi-periodic, and chaotic bursting Ca2+ oscillations are classified into several subtypes. The classification is based on the dynamics of separated fast and slow subsystems, the so-called fast-slow burster analysis. For regular bursting Ca2+ oscillations two types of bursting are specified: Point-Point and Point-Cycle bursting. In particular, the slow passage effect, important for the Hopf-Hopf and SubHopf-SubHopf bursting subtypes, is explained by local divergence calculated for the fast sub-system. Quasi-periodic bursting Ca2+ oscillations can be found in only one of the four studied mathematical models and appear via a homoclinic bifurcation with a homoclinic torus structure. For chaotic bursting Ca2+ oscillations, we found that bursting patterns resulting from the period doubling root to chaos considerably differ from those appearing via intermittency and have to be treated separately. The analysis and classification of different types of bursting Ca2+ oscillations provides better insight into mechanisms of complex intra- and intercellular Ca2+ signalling. This improves our understanding of several important biological phenomena in cellular signalling like complex frequency-amplitude signal encoding and synchronisation of intercellular signal transduction between coupled cells in tissue.

Elsevier	*T. Haberichter, M. Marhl, R. Heinrich* Birhythmicity, trirhythmicity and chaos in bursting calcium oscillations Biophysical Chemistry 90 (2001) 17-30	Abstract: We have analyzed various types of complex calcium oscillations. The oscillations are explained with a model based on calcium-induced calcium release (CICR). In addition to the endoplasmic reticulum as the main intracellular Ca2+ store, mitochondrial and cytosolic Ca2+ binding proteins are also taken into account. This model was previously proposed for the study of the physiological role of mitochondria and the cytosolic proteins in gene rating complex Ca2+ oscillations [1]. Here, we investigated the occurrence of different types of Ca2+ oscillations obtained by the model, i.e. simple oscillations, bursting, and chaos. In a bifurcation diagram, we have shown that all these various modes of oscillatory behavior are obtained by a change of only one model parameter, which corresponds to the physiological variability of an agonist. Bursting oscillations were studied in more detail because they express birhythmicity, trirhythmicity and chaotic behavior. Two different routes to chaos are observed in the model: in addition to the usual period doubling cascade, we also show intermittency. For the characterization of the chaotic behavior, we made use of return maps and Lyapunov exponents. The potential biological role of chaos in intracellular signaling is discussed.

Elsevier	*M. Marhl, T. Haberichter, M. Brumen, R. Heinrich* Complex calcium oscillations and the role of mitochondria and cytosolic proteins Biosystems 57 (2000) 75-86	Abstract: Intracellular calcium oscillations, which are oscillatory changes of cytosolic calcium concentration in response to agonist stimulation, are experimentally well observed in various living cells. Simple calcium oscillations represent the most common pattern and many mathematical models have been published to describe this type of oscillation. On the other hand, relatively few theoretical studies have been proposed to give an explanation of complex intracellular calcium oscillations, such as bursting and chaos. In this paper, we develop a new possible mechanism for complex calcium oscillations based on the interplay between three calcium stores in the cell: the endoplasmic reticulum (ER), mitochondria and cytosolic proteins. The majority (approximate to 80%) of calcium released from the ER is first very quickly sequestered by mitochondria. Afterwards, a much slower release of calcium from the mitochondria serves as the calcium supply for the intermediate calcium exchanges between the ER and the cytosolic proteins causing bursting calcium oscillations. Depending on the permeability of the ER channels and on the kinetic properties of calcium binding to the cytosolic proteins, different patterns of complex calcium oscillations appear. With our model, we are able to explain simple calcium oscillations, bursting and chaos. Chaos is also observed for calcium oscillations in the bursting mode.