We consider a quantum system periodically driven with a strength which varies slowly on the scale of the driving period. The analysis is based on a general formulation of the Floquet theory relying on the extended Hilbert space. It is shown that the dynamics of the system can be described in terms of a slowly varying effective Floquet Hamiltonian that captures the long-term evolution, as well as rapidly oscillating micromotion operators. We obtain a systematic high-frequency expansion of all these operators. Generalizing the previous studies, the expanded effective Hamiltonian is now time-dependent and contains extra terms appearing due to changes in the periodic driving. The same applies to the micromotion operators which exhibit a slow temporal dependence in addition to the rapid oscillations. As an illustration, we consider a quantum-mechanical spin in an oscillating magnetic field with a slowly changing direction. The effective evolution of the spin is then associated with non-Abelian geometric phases reflecting the geometry of the extended Floquet space. The developed formalism is general and also applies to other periodically driven systems, such as shaken optical lattices with a time-dependent shaking strength, a situation relevant to the cold atom experiments.Comment: already published pape

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Using Hodgkin–Huxley and isolated subthalamic nucleus (STN) model neurons as examples, we show that electrical high-frequency stimulation (HFS) suppresses sustained neuronal spiking. The mechanism of suppression is explained on the basis of averaged equations derived from the original neuron equations in the limit of high frequencies. We show that for frequencies considerably greater than the reciprocal of the neuron’s characteristic time scale, the result of action of HFS is defined by the ratio between the amplitude and the frequency of the stimulating signal. The effect of suppression emerges due to a stabilization of the neuron’s resting state or due to a stabilization of a low-amplitude subthreshold oscillation of its membrane potential. Intriguingly, although we neglect synaptic dynamics, neural circuity as well as contribution of glial cells, the results obtained with the isolated high-frequency stimulated STN model neuron resemble the clinically observed relations between stimulation amplitude and stimulation frequency required to suppress Parkinsonian tremor.

The phase reduction method for a limit cycle oscillator subjected to a strong amplitude-modulated high-frequency force is developed. An equation for the phase dynamics is derived by introducing a new, effective phase response curve. We show that if the effective phase response curve is everywhere positive (negative), then an entrainment of the oscillator to an envelope frequency is possible only when this frequency is higher (lower) than the natural frequency of the oscillator. Also, by using the Pontryagin maximum principle, we have derived an optimal waveform of the perturbation that ensures an entrainment of the oscillator with minimal power. The theoretical results are demonstrated with the Stuart-Landau oscillator and model neurons.

The delayed feedback control (DFC) methods are noninvasive, which means that the control signal vanishes if the delay time is adjusted to be equal to the period of a target unstable periodic orbit (UPO). If the delay time differs slightly from the UPO period, a nonvanishing periodic control signal is observed. We derive an analytical expression for this period for a general class of multiple-input multiple-output systems controlled by an extended DFC algorithm. Our approach is based on the phase-reduction theory adapted to systems with time delay. The analytical results are supported by numerical simulations of the controlled Rössler system.