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Mishchenko M. A., Kovaleva N. S., Matrosov V. V. Excitability of neuron-like generator under pulse stimulation. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2018, vol. 26, iss. 5, pp. 5-19. DOI: 10.18500/0869-6632-2018-26-5-5-19

# Excitability of neuron-like generator under pulse stimulation

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 γ = 0. The number of the equilibrium is continuum and all of them are non-robust. The structure of hyperbolic manifold depends on phase variable φ. The diapason of φ 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.

- Izhikevich E.M. Neural excitability, spiking and bursting. Int. J. Bifurc. Chaos, 2000, vol. 10, iss. 6, pp. 1171–1266.
- Izhikevich E.M. Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. Cambridge, The MIT Press, 2007.
- Rabinovich M.I., Varona P., Selverston A.I. Dynamical principles in neuroscience. Rev. Mod. Phys., 2006, vol. 78, iss. 4, pp. 1213–1265.
- Takahashi N., Hanyu Y., Musha T., Kubo R., Matsumoto G. Global bifurcation structure in periodically stimulated giant axons of squid. Phys. D: Nonlinear Phenom., 1990, vol. 43, iss. 2–3, pp. 318–334.
- Kaplan D.T., Clay J.R., Manning T., Glass L., Guevara M.R., Shrier A. Subthreshold dynamics in periodically stimulated squid giant axons. Phys. Rev. Lett., 1996, vol. 76, iss. 21, pp. 4074–4077.
- Sato S., Doi S. Response characteristics of the BVP neuron model to periodic pulse inputs. Math. Biosci., 1992, vol. 112, pp. 243–259.
- Doi S., Sato S. The global bifurcation structure of the BVP neuronal model driven by periodic pulse trains. Math. Biosci., 1995, vol. 125, iss. 2, pp. 229–250.
- Yoshino K., Nomura T., Pakdaman K., Sato S. Synthetic analysis of periodically stimulated excitable and oscillatory membrane models. Phys. Rev. E, 1999, vol. 59, iss. 1, pp. 956–969.
- Croisier H. Continuation and bifurcation analyses of a periodically forced slow-fast system. Diss. Phd thesis, Academie Wallonie-Europe, University de Liege, 2009.
- Farokhniaee A.A., Large E.W. Mode-locking behavior of Izhikevich neurons under periodic external forcing. Phys. Rev. E, 2017, vol. 95, iss. 6, pp. 1–9.
- Kazantsev V.B., Tchakoutio A.S., Jacquir S., Binczak S., Bilbault J.M. Active spike transmission in the neuron model with a winding threshold manifold. Neurocomputing, 2012, vol. 83, pp. 205–211.
- Tchakoutio A.S., Binczak S., Kazantsev V.B., Jacquir S., Bilbault J.M. Experimental active spike responses of analog electrical neuron: Beyond «integrate-and-fire» transmission. Nonlinear Dyn., 2015, vol. 82, iss. 3, pp. 1595–1604.
- Mischenko M.A., Shalfeev V.D., Matrosov V.V. Neuron-like dynamics in phaselocked loop. Izvestiya VUZ. Applied Nonlinear Dynamics, 2012, vol. 20, iss. 4, pp. 122 (in Russian).
- Matrosov V.V., Mishchenko M.A., Shalfeev V.D. Neuron-like dynamics of a phaselocked loop. Eur. Phys. J. Spec. Top., 2013, vol. 222, iss. 10, pp. 2399–2405.
- Ermentrout B. Ermentrout–Kopell canonical model. Scholarpedia, 2008, 3(3):1398, revision 122128
- Hoppensteadt F. Voltage-controlled oscillations in neurons. Scholarpedia, 2006, 1(11):1599, revision 129939
- Mishchenko M.A., Bolshakov D.I., Matrosov V.V. Instrumental implementation of a neuronlike generator with spiking and bursting dynamics based on a phase-locked loop. Tech. Phys. Lett. Pleiades Publishing, 2017, vol. 43, iss. 7, pp. 596–599.

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