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Updated: Mar 8, 2026

Optogenetic Entrainment of Hippocampal Theta Oscillations in Behaving Mice
Published on: June 29, 2018
Optimal waveform for the entrainment of oscillators perturbed by an amplitude-modulated high-frequency force
Viktor Novičenko1, Irmantas Ratas2
1Institute of Theoretical Physics and Astronomy, Vilnius University, Saulėtekio Avenue 3, LT-10222 Vilnius, Lithuania.
Abstract:
We analyze limit cycle oscillators under perturbation constructed as a product of two signals, namely, an envelope with a period close to natural period of an oscillator and a high-frequency carrier signal. A theory for obtaining an envelope waveform that achieves the maximal frequency interval of entrained oscillators is presented. The optimization problem for fixed power and maximal allowed amplitude is solved by employing the phase reduction method and the Pontryagin's maximum principle. We have shown that the optimal envelope waveform is a bang-bang-type solution. Also, we have found "inversion" symmetry that relates two signals with different powers but the same interval of entrained frequencies. The theoretical results are confirmed numerically on FitzHugh-Nagumo oscillators.
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