Control of noise-induced motion

Oscillations are everywhere, ranging from perfectly ordered periodic and quasiperiodic to completely disordered, irregular ones, often described by probabilistic laws. Turbulence, climate changes, neuron spiking, heart electrical activity / all of these are examples of processes where irregular oscillations are possible and play a prominent role.The irregularity of oscillations can have two different origins: deterministic and stochastic.In the former case, although the dynamics of the oscillating system is defined by deterministic laws, the oscillations themselves are very sensitive to the initial state of the system: even a very small change in this initial state can lead to a substantial difference in the behaviour. This kind of oscillations is usually called deterministic chaos. Another type of irregular oscillations occurs when the dynamics of the system is defined by random fluctuations existing within, or applied to, the system. Notably, in spite of their randomness, the noise-induced oscillations in some systems can look quite regular and resemble very much the deterministic ones. One of the brightest examples is a sensory neuron which demonstrates no oscillations unless the signal on its input exceeds a certain level, after which the neuron generates one electrical pulse whose shape and duration are defined deterministically and almost do not depend on the input. The usual signal coming from an environment is in fact a random signal, that being applied to the neuron can generate a sequence of pulses looking quite coherent. Amazingly, irregular oscillations of this kind are widely spread in nature and technology. Besides the neurons and neuron networks, such oscillations can arise in semiconductor nanostructures, chemical reactions, some engineering mechanisms like drill string and many others. It is obvious that our ability to control irregular oscillations for example by making them more predictable is hugely beneficial to industry, technology, medicine, etc. Control means that by imposing some, preferrably small, forcing or feedback on the system, one is able to change the amplitude, timescale or regularity of oscillations, or even to cease them altogether. In the last decade a good progress was made in the control of deterministic chaos. The advanced nonlinear control methods exploit the fact that deterministic trajectories with the desired timescales already exist in the system, but are unstable and thus invisible in experiment, and the control tools just stabilize them. However, the systems where oscillations occur only due to noise have no such trajectories and no deterministic timescales. All timescales or orbits can be introduced only in the statistical sense. The control of oscillations that are purely random has never (or rarely, depending on what one means by control) been addressed previously.The main aim of the current research is to develop a general effective method for the control of oscillations induced merely by external random fluctuations that would be feasible as applied to real-life problems. As a control tool delayed feedback will be considered, which looks the most promising from the viewpoint of simplicity and efficiency. The main objectives are: (i) To develop qualitative theories for the delayed feedback as applied to minimal models that describe a large class of nonlinear systems in which noise can induce oscillations. (ii) To establish if the method is applicable to more realistic models like neuron-like networks. (iii) To verify if the delayed feedback can work in a real experiment on control of heart rate and of its variability in experiments with healthy human volunteers. The work will be carried out in Loughborough University in collaboration with Technical University of Berlin and the Department of Cardiovascular Sciences of Leicester University.