István Z. Kiss

Engineering Complex Dynamical Structures: Sequential Patterns and Desynchronization

We used phase models to describe and tune complex dynamic structures to desired states; weak, nondestructive signals are used to alter interactions among nonlinear rhythmic elements. Experiments on electrochemical reactions on electrode arrays were used to demonstrate the power of mild model-engineered feedback to achieve a desired response. Applications are made to the generation of sequentially visited dynamic cluster patterns similar to reproducible sequences seen in biological systems and to the design of a nonlinear antipacemaker for the destruction of pathological synchronization of a population of interacting oscillators.

Experiments on arrays of globally coupled chaotic electrochemical oscillators: Synchronization and clustering

Experiments on chaotically oscillating arrays of 64 nickel electrodes in sulfuric acid were carried out. External resistors in parallel and series are added to vary the extent of global coupling among the oscillators without changing the other properties of the system. The array is heterogeneous due to small variations in the properties of the electrodes and there is also a small amount of noise. The addition of global coupling transforms a system of independent elements to a state of complete synchronization. At intermediate coupling strengths stable clusters, or condensates of elements, form. All the elements in a cluster follow the same chaotic trajectory but each cluster has its own dynamics; the system is thus temporally chaotic but spatially ordered. Many cluster configurations occur under the same conditions and transitions among them can be produced. For values of the coupling parameter on either side of the stable cluster region a non-stationary behavior occurs in which clustered and synchronized states alternately form and break up. Some statistical properties of the cluster states are determined. (c) 2000 American Institute of Physics.

Spatially Organized Dynamical States in Chemical Oscillator Networks: Synchronization, Dynamical Differentiation, and Chimera Patterns

Dynamical processes in many engineered and living systems take place on complex networks of discrete dynamical units. We present laboratory experiments with a networked chemical system of nickel electrodissolution in which synchronization patterns are recorded in systems with smooth periodic, relaxation periodic, and chaotic oscillators organized in networks composed of up to twenty dynamical units and 140 connections. The reaction system formed domains of synchronization patterns that are strongly affected by the architecture of the network. Spatially organized partial synchronization could be observed either due to densely connected network nodes or through the ‘chimera’ symmetry breaking mechanism. Relaxation periodic and chaotic oscillators formed structures by dynamical differentiation. We have identified effects of network structure on pattern selection (through permutation symmetry and coupling directness) and on formation of hierarchical and ‘fuzzy’ clusters. With chaotic oscillators we provide experimental evidence that critical coupling strengths at which transition to identical synchronization occurs can be interpreted by experiments with a pair of oscillators and analysis of the eigenvalues of the Laplacian connectivity matrix. The experiments thus provide an insight into the extent of the impact of the architecture of a network on self-organized synchronization patterns.

Restoration of rhythmicity in diffusively coupled dynamical networks

Oscillatory behaviour is essential for proper functioning of various physical and biological processes. However, diffusive coupling is capable of suppressing intrinsic oscillations due to the manifestation of the phenomena of amplitude and oscillation deaths. Here we present a scheme to revoke these quenching states in diffusively coupled dynamical networks, and demonstrate the approach in experiments with an oscillatory chemical reaction. By introducing a simple feedback factor in the diffusive coupling, we show that the stable (in)homogeneous steady states can be effectively destabilized to restore dynamic behaviours of coupled systems. Even a feeble deviation from the normal diffusive coupling drastically shrinks the death regions in the parameter space. The generality of our method is corroborated in diverse non-linear systems of diffusively coupled paradigmatic models with various death scenarios. Our study provides a general framework to strengthen the robustness of dynamic activity in diffusively coupled dynamical networks.

Optimal Waveform for the Entrainment of a Weakly Forced Oscillator

A theory for obtaining a waveform for the effective entrainment of a weakly forced oscillator is presented. Phase model analysis is combined with calculus of variation to derive a waveform with which entrainment of an oscillator is achieved with a minimum power forcing signal. Optimal waveforms are calculated from the phase response curve and a solution to a balancing condition. The theory is tested in chemical entrainment experiments in which oscillations close to and farther away from a Hopf bifurcation exhibited sinusoidal and higher harmonic nontrivial optimal waveforms, respectively.

Investigation of c-myc and K-ras amplification in renal clear cell adenocarcinoma.

Tumour DNA samples isolated from 36 patients with renal clear cell carcinoma were investigated for c-myc and K-ras amplification, using a quantitative dot-blot hybridization. The characteristic clinical and histological parameters involved in the statistical analysis were age, sex, histological grade of the tumour, the TNM staging system, tumour size and weight, vascular invasion and the quality of life. The goal of the study was to estimate the prevalence as well as the prognostic value of the amplification of the oncogenes in question. Amplified c-myc (2.47-fold on the average) was found in three specimens (8.3%), showing slight correlation with intravasation (P > 0.05, n.s.). K-ras amplification (2.93-fold) detected in six tumours (16.6%) was shown to significantly correlate with both histological grade (2.2 vs. 1.8, P < 0.05) and tumour size (15 vs. 8 cm, P < 0.05). In cases with amplified K-ras also lymph node involvement was somewhat more frequent (P > 0.05, n.s.). No coamplification of these oncogenes was observed. The results of the study suggest that K-ras amplification may account for a more rapid progression of the disease.

Synchronization engineering: Theoretical framework and application to dynamical clustering.

A method for engineering the global behavior of populations of rhythmic elements is presented. The framework, which is based on phase models, allows a nonlinear time-delayed global feedback signal to be constructed which produces an interaction function corresponding to the desired behavior of the system. It is shown theoretically and confirmed in numerical simulations that a polynomial, delayed feedback is a versatile tool to tune synchronization patterns. Dynamical states consisting of one to four clusters were engineered to demonstrate the application of synchronization engineering in an experimental electrochemical system.

Complexity of globally coupled chaotic electrochemical oscillators

Interactions among small sets of two to eight nickel electrodes undergoing chaotic electrodissolution in sulfuric acid were studied. A single oscillator under these conditions exhibits low-dimensional chaotic behavior. Global coupling among the electrodes was added with the use of external resistors in a manner such that the strength of the coupling could be varied while the other parameters of the system remained constant. Such global coupling is of course equivalent to an appropriate local coupling for the two-electrode system and even for a three-electrode system if arranged in a ring. We investigate the changes in complexities of both the individual oscillators and of the total current as functions of coupling strength and of array size. The dynamics of the individual oscillators are almost identical to those of the single oscillator at added coupling strengths of zero (where the oscillators are almost independent) and at maximum coupling strength (where they are synchronized). There are two trends (with exceptions) with changing coupling strength. (1) The complexity (information dimension) of the individual currents has a maximum at intermediate values of the coupling strength, i.e., at conditions in which interactions occur but where the coupling is not strong enough to produce synchronization. (2) An increase in global coupling decreases the complexity of the total current. The general trends with coupling strength are interrupted by clustering that occurs with the four and eight electrode arrays. Cluster configurations for the larger array exhibiting both chaotic (3,5 cluster) and periodic (4,4 cluster) dynamics were observed.

Noise enhanced phase synchronization and coherence resonance in sets of chaotic oscillators with weak global coupling

The effect of noise on phase synchronization in small sets and larger populations of weakly coupled chaotic oscillators is explored. Both independent and correlated noise are found to enhance phase synchronization of two coupled chaotic oscillators below the synchronization threshold; this is in contrast to the behavior of two coupled periodic oscillators. This constructive effect of noise results from the interplay between noise and the locking features of unstable periodic orbits. We show that in a population of nonidentical chaotic oscillators, correlated noise enhances synchronization in the weak coupling region. The interplay between noise and weak coupling induces a collective motion in which the coherence is maximal at an optimal noise intensity. Both the noise-enhanced phase synchronization and the coherence resonance numerically observed in coupled chaotic Rössler oscillators are verified experimentally with an array of chaotic electrochemical oscillators.

Phase synchronization of nonidentical chaotic electrochemical oscillators

Experiments on two coupled chaotic electrochemical oscillators with different frequencies are presented. We consider two types of coupling. The first is imposed through a set of external resistors. The second arises from the potential drop in the electrolyte and generally depends on cell geometry and reaction rate. Phase synchronization is observed with the addition of weak coupling of both types. Along with the onset of phase synchronization an increase in the amplitude of the mean current oscillations occurs. Simulations using a coupled form of an electrochemical model support the experimental findings.