Geodesic
Excitability of neuron-like generator under pulse stimulation
Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 26 (2018) no. 5, pp. 6-19.

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Subject of the study. Excitable dynamic systems are the systems having a stable equilibrium and capable of generating a large amplitude response to a weak stimulation. Excitable dynamic systems research is one of the most interesting and actual problems of modern nonlinear science. In the present paper dynamics of phase-locked loop with bandpass filter is studied under external pulse stimulation. Novelty. Excitability of the phase-controlled generator is studied under external pulse stimulation. The parameters of stimulation to excite a large amplitude response (super-threshold response) are found. Methods. Qualitative theory of dynamic systems, numerical simulations based on nonlinear oscillations theory. Results. The model of phase-controlled generator based on phase-locked loop is studied in excitable state. The analysis of equilibrium states of the autonomic model shows presence of equilibrium only with $\gamma = 0$. The number of the equilibrium is continuum and all of them are non-robust. The structure of hyperbolic manifold depends on phase variable 3. The diapason of 3 is found where the manifold is stable and equilibrium states define a stable stationary state of the generator. Excitability of the phase-controlled generator is studied under external pulse stimulation. The super-threshold responses of the generator are qualitatively similar to spikes and bursts of neuron's membrane potential. The amplitude of stimulus required for the appearance of the super-threshold response is determined. The dependence of this amplitude on initial conditions is shown. Both the amplitude and duration of the stimulus have the effect on response appearance and the key factor is the square of the stimulus that could be a sum of several consecutive pulses. Discussion. The phase-controlled oscillator is an excitable dynamic system capable to response on external pulse stimulation. These responses are qualitatively similar to spikes and bursts of neuron's membrane potential. The phase-locked loop with bandpass filter could be considered as a neuron-like generator.
Keywords: phase-controlled generator, oscillation excitation, pulse stimulation, excitable system. 
@article{IVP_2018_26_5_a1,
     author = {M. A. Mishenko and N. S. Zhukova and V. V. Matrosov},
     title = {Excitability of neuron-like generator under pulse stimulation},
     journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},
     pages = {6--19},
     publisher = {mathdoc},
     volume = {26},
     number = {5},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVP_2018_26_5_a1/}
}
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M. A. Mishenko; N. S. Zhukova; V. V. Matrosov. Excitability of neuron-like generator under pulse stimulation. Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 26 (2018) no. 5, pp. 6-19. http://geodesic.mathdoc.fr/item/IVP_2018_26_5_a1/