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### Time-delayed feedback control of noise-induced oscillations

A main focus of our work during the last proposal period has been the control of noise-induced oscillations through time-delayed feedback. In collaboration with the Visiting Scientists Dr. N. Janson and Dr. A. Balanov we were able to show, for the first time, that in nonlinear stochastic systems which in the absence of noise exhibit no autonomous oscillations whereas in the presence of noise oscillate in an irregular manner, the regularity (coherence) as well as the timescales of these oscillations could be tuned by time-delayed feedback.

To begin with, we studied simple generic models, namely the Van der Pol oscillator below a subcritical Hopf bifurcation, as a model for a system in the vicinity of a bifurcation, and the FitzHugh-Nagumo model as prototype example of an excitable system. Suitable as quantitative measure for the coherence of the oscillations, depending on the system, is either the correlation time, with the normalized autocorrelation function, the normalized variance of the interspike intervals, or the coherence factor (the ratio between the height and the half-width of the dominant spectral peak). For both classes of models, we studied, in detail, the dependence of the correlation time and the spectral properties on the noise intensity and the control parameters.

In the Van der Pol oscillator, we could explain the behaviour of the system quantitatively through an analysis in the vicinity of the fixed point, since the system is slightly below a local bifurcation. In the absence of noise and control, a supercritical Hopf bifurcation takes place and the fixed point is a stable focus below the bifurcation. The complex eigenvalues of the fixed point of the delay equation determine which modes can be excited by noise. The eigenvalue with the smallest absolute value of the real part, i. e. the least stable one, determines the correlation time as well as the dominant period of the noise-induced oscilations. When the real parts of two modes cross, the dominant peak jumps to the new eigenvalue branch. The noisy spectrum, as well as the correlation time, can be analytically calculated in a linear approximation.

Moreover, we have developed a self-consistent mean field approximation, which is in good agreement with the numerical results, upto relatively large noise intensity. Here, the nonlinear term is substituted by a rescaled bifurcation parameter and determined from the variance of the multivariate Ornstein-Uhlenbeck process. For this effective linear process, the correlation time for optimal time delay and the noisy spectrum can be calculated explicitly analytically. The corrlelation time is substantially increased for optimal time delay, while for badly chosen time delay it becomes practically zero. In contrary to the local dynamics of the Van der Pol oscillator, the FitzHugh-Nagumo model describes an excitable system with global dynamics and is often used to explains the spiking behavior of neurons: for certain choice of parameters, the only attractor in the deterministic system is a stable node. The time-delayed feedback control here cannot be treated through a local analysis, however in collaboration with subdivision A4 we have developed a discrete state model which allows for a semi-analytical treatment,

An important extension is the time-delayed feedback control of coupled neuron systems,

which we studied by means of two symmetrically-diffusively coupled FitzHugh Namumo equations, as minimal model for two interacting neurons. The influence upon the coupled dynamics through local control is of practical relevance. Therefore we studied to what extent can the feedback in one of the two sub-systems control the coherence- and synchronization-behaviour of the whole system. We found that the correlation time and the mean interspike interval of both sub-systems can indeed be controlled. A further interesting result is that the stochastic synchronization - measured by the frequency ratio (frequency synchronization) as well as by the synchronization index (phase synchronization) - can be changed in both directions. That means, the synchronization can be amplified or suppressed, depending on the choice of the time delay and the control amplitude. Here, there is again a resonance-like dependence on the time delay. This result is especially interesting in view of possible neural applications, where a high synchronization is often related to a sick state (Parkinson, epilepsy), which can obviously be suppressed through local feedback control. Similar effects have been found in Subdivision C3 for systems of many globally coupled neurons with global feedback.