#### For citation:

Mishchenko M. A., Kovaleva N. S., Polovinkin A. V., Matrosov V. V. Excitation of phase-controlled oscillator by pulse sequence. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2021, vol. 29, iss. 2, pp. 240-253. DOI: 10.18500/0869-6632-2021-29-2-240-253

# Excitation of phase-controlled oscillator by pulse sequence

The purpose of this work is to study the dynamics of phase-controlled oscillator in excitable mode under rectangular pulse train forcing. The excitable system is a system having a stable equilibrium and a large amplitude periodic pseudo-orbit passing near the equilibrium. Methods. In this paper, the dynamics of the generator under periodic or Poisson rectangular pulse train stimulation is studied by numerical simulation. Different values are proposed to characterize statistics of the oscillator’s responses on different number of stimulating pulses. Results. The role of amplitude and period of external forcing on generator excitation is studied. Relative frequency of the oscillator responses depends on the amplitude of stimulating pulses. Phase-controlled oscillator’s responses are synchronized with different rational relations to number of stimulating pulses depending on the amplitude of the pulses. The inter-pulse intervals of the oscillator are not completely rational to inter-pulse intervals of stimulating train in contrast to relative frequency of the oscillator responses. The Poisson pulse train stimulation gives nearly the same statistics of the oscillator’s responses as periodic forcing. Conclusion. Detailed study of the dynamics of phase-controlled oscillator in excitable mode under rectangular pulse train forcing is conducted. Different ways of the oscillator’s responses characterization shows the strong dependence on the amplitude of stimulating pulses but much weaker dependence on inter-pulse intervals of stimulating train. The most informative characteristics is a ratio between inter-pulse intervals of oscillator responses and stimulating train.

- Izhikevich EM. Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. Cambridge: The MIT Press; 2007. 441 p.
- Rabinovich MI et al. Dynamical principles in neuroscience. Rev. Mod. Phys. 2006;78(4): 1213–1265. DOI: 10.1103/RevModPhys.78.1213.
- Croisier H. Continuation and bifurcation analyses of a periodically forced slow-fast system. Diss. Phd thesis, Academie Wallonie-Europe, Universite de Liege; 2009. 126 p. ´
- Yoshino K et al. Synthetic analysis of periodically stimulated excitable and oscillatory membrane models. Phys. Rev. E. 1999;59(1):956–969. DOI: 10.1103/PhysRevE.59.956.
- Sato S, Doi S. Response characteristics of the BVP neuron model to periodic pulse inputs. Math. Biosci. 1992;112(2):243–259. DOI: 10.1016/0025-5564(92)90026-S.
- Doi S, Sato S. The global bifurcation structure of the BVP neuronal model driven by periodic pulse trains. Math. Biosci. 1995;125(2):229–250. DOI: 10.1016/0025-5564(94)00035-x.
- Kazantsev VB et al. Active spike transmission in the neuron model with a winding threshold manifold. Neurocomputing. 2012;83:205–211. DOI: 10.1016/j.neucom.2011.12.014.
- Nguetcho AST et al. Experimental active spike responses of analog electrical neuron: beyond «integrate-and-fire» transmission. Nonlinear Dyn. 2015;82(3):1595–1604. DOI: 10.1007/s11071-015-2263-2.
- Takahashi N et al. Global bifurcation structure in periodically stimulated giant axons of squid. Physica D: Nonlinear Phenomena. 1990;43(2–3):318–334. DOI: 10.1016/0167-2789(90)90140-K.
- Kaplan DT et al. Subthreshold dynamics in periodically stimulated squid giant axons. Phys. Rev. Lett. 1996;76(21):4074–4077. DOI: 10.1103/PhysRevLett.76.4074.
- Farokhniaee AA, Large EW. Mode-locking behavior of Izhikevich neurons under periodic external forcing. Phys. Rev. E. 2017;95(6):062414. DOI: 10.1103/PhysRevE.95.062414.
- Mishchenko MA. Neuron-like model on the basis of the phase-locked loop. Vestnik of Lobachevsky State University of Nizhni Novgorod. 2011;5(3):279–282 (in Russian).
- Mishchenko MA, Bolshakov DI, Matrosov VV. Instrumental implementation of a neuronlike generator with spiking and bursting dynamics based on a phase-locked loop. Tech. Phys. Lett. 2017;43(7):596–599. DOI: 10.1134/S1063785017070100.
- Izhikevich EM. Neural excitability, spiking and bursting. Int. J. Bifurc. Chaos. 2000;10(6): 1171–1266. DOI: 10.1142/S0218127400000840.
- Mishchenko MA, Shalfeev VD, Matrosov VV. Neuron-like dynamics in phase-locked loop. Izvestiya VUZ. Applied Nonlinear Dynamics. 2012;20(4):122–130 (in Russian). DOI: 10.18500/0869-6632-2012-20-4-122-130.
- Matrosov VV, Mishchenko MA, Shalfeev VD. Neuron-like dynamics of a phase-locked loop. Eur. Phys. J. Spec. Top. 2013;222(10):2399–2405. DOI: 10.1140/epjst/e2013-02024-9.
- Mishchenko MA, Zhukova NS, Matrosov VV. Excitability of neuron-like generator under pulse stimulation. Izvestiya VUZ. Applied Nonlinear Dynamics. 2018;26(5):6–19 (in Russian). DOI: 10.18500/0869-6632-2018-26-5-5-19.
- Cox DR, Smith WL. The superposition of several strictly periodic sequences of events. Biometrika. 1953;40(1–2):1–11. DOI: 10.2307/2333090.
- Feller W. An Introduction to Probability Theory and Its Applications. Vol. 1, 3rd Edition. Wiley; 1968. 528 p.
- Goldfinger MD. Poisson process stimulation of an excitable membrane cable model. Biophys. J. 1986;50(1):27–40. DOI: 10.1016/S0006-3495(86)83436-1.

- 2584 reads