ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)


For citation:

Levanova T., Kazakov A. O., Korotkov A. G., Osipov G. V. The impact of electrical couplings on the dynamics of the ensemble of inhibitory coupled neuron-like elements. Izvestiya VUZ. Applied Nonlinear Dynamics, 2018, vol. 26, iss. 5, pp. 101-112. DOI: 10.18500/0869-6632-2018-26-5-101-112

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
(downloads: 514)
Full text PDF(En):
(downloads: 92)
Language: 
Russian
Article type: 
Article
UDC: 
517.925 + 517.93

The impact of electrical couplings on the dynamics of the ensemble of inhibitory coupled neuron-like elements

Autors: 
Levanova Tatiana, Lobachevsky State University of Nizhny Novgorod
Kazakov Aleksej Olegovich, National Research University "Higher School of Economics"
Korotkov Aleksandr Gennadevich, Lobachevsky State University of Nizhny Novgorod
Osipov Grigorij Vladimirovich, Lobachevsky State University of Nizhny Novgorod
Abstract: 

Topic. The phenomenological model of ensemble of three neurons coupled by chemical (synaptic) and electrical couplings is studied. Single neuron is modeled by van der Pol oscillator. Aim of work is to study of influence of coupling strength and frequency detuning between elements in the case of regime of sequential activity that is observed in ensemble of neuronlike elements with chemical inhibitory couplings. Method. The research is made with usage of analytical methods of nonlinear dynamics and computer modeling. Results. It was shown that adding of arbitrarily small electrical coupling to ensemble of van der Pol oscillators with chemical synaptic inhibitory couplings leads to the destruction of a stable heteroclinic contour between saddle limit cycles. It was also shown that nonidentity of elements (while electrical couplings are absent) do not lead to destruction of heteroclinic contour. This situation, in general, is not typical for such systems. Discussion. We suggest to consider the ensemble of elements as phenomenological model of neuronal network. Such approach has the following advantage: it is possible to study low-dimensional neuronal models and reproduce the main effects that are observed in more complex models, for example, in biologically realistic model of Hodgkin–Huxley and also in real experiments.

Reference: 
  1. Birmingham K., Gradinaru V., Anikeeva P., Grill W.M., Pikov V., McLaughlin B., Pasricha P., Weber D., Ludwig K., Famm K. Bioelectronic medicines: A research roadmap. Nature Reviews Drug Discover, 2014, vol. 13, p. 399.
  2. Seo D., Neely R.M., Shen K., Singhal U., Alon E., Rabaey J.M., Carmena J.M., Maharbiz M. Wireless recording in the peripheral nervous system with ultrasonic neural dust. Neuron, 2016, vol. 91(3), p. 529.
  3. Sacramento J.F., Chew D.J., Melo B.F., Doneg M., Dopson W., Guarino M.P., 1. Birmingham K., Gradinaru V., Anikeeva P., Grill W.M., Pikov V., McLaughlin B., Pasricha P., Weber D., Ludwig K., Famm K. Bioelectronic medicines: A research roadmap. Nature Reviews Drug Discover, 2014, vol. 13, p. 399. 2. Seo D., Neely R.M., Shen K., Singhal U., Alon E., Rabaey J.M., Carmena J.M., Maharbiz M. Wireless recording in the peripheral nervous system with ultrasonic neural dust. Neuron, 2016, vol. 91(3), p. 529. 3. Sacramento J.F., Chew D.J., Melo B.F., Doneg M., Dopson W., Guarino M.P.,
  4. Afraimovich V.S., Zhigulin V.P., Rabinovich M.I. On the origin of reproducible sequential activity in neural circuits. Chaos, 2004, vol. 14(4), p. 1123.
  5. Levanova T.A., Komarov M.A., Osipov G.V. Sequential activity and multistability in an ensemble of coupled Van der Pol oscillators. Eur. Phys. J. Special Topics, 2013, vol. 222, p. 2417.
  6. Mikhaylov A. O., Komarov M.A., Levanova T.A., Osipov G.V. Sequential switching activity in ensembles of inhibitory coupled oscillators. Europhys. Lett., 2013, vol. 101(2), p. 20009.
  7. Levanova T.A., Kazakov A.O., Osipov G.V., Kurths J. Dynamics of ensemble of inhibitory coupled Rulkov maps. Eur. Phys. J. Special Topics, 2016, vol. 225, p. 147.
  8. Mikhaylov A.O., Komarov M.A., Osipov G.V. Sequential switching activity in the ensemble of nonidentical Poincare systems. Izvestiya VUZ, Applied Nonlinear Dynamics, 2013, vol. 21(5), p. 79 (in Russian).
  9. Nicholls J.G., Martin A.R., Brown D.A., Diamond M.E., Weisblat D.A., Fuchs P.A. From Neuron to brain. 5th ed. Sinauer Associates, 2011. 621 p.
  10. Andronov A.A., Vitt A.A., Khaikin S.E. Theory of Oscillations. New York: Pergamon Press, 1966.
  11. Benettin G., Galgani L., Giorgilli A., Strelcyn J.-M. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems. A method for computing all of them. Part 1: Theory. Meccanica, 1980, vol. 15(1), p. 9.
  12. Benettin G., Galgani L., Giorgilli A., Strelcyn J.-M. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems. A method for computing all of them. Part 2: Numerical application. Meccanica, 1980, vol. 15(1), p. 21.
  13. Kuznetsov S.P. Dynamical chaos. Izvestiya VUZ, Applied Nonlinear Dynamics, 2002, vol. 10(1–2), p. 189 (in Russian).
  14. Zwillinger D. Handbook of Differential Equations, 3rd ed. Boston: Academic Press, 1997.
  15. Komarov M.A., Osipov G.V., Suykens J.A.K. Sequentially activated groups in neural networks. Europhys. Lett., 2009. vol. 86. P. 60006. 
Received: 
29.03.2018
Accepted: 
23.05.2018
Published: 
31.10.2018
Short text (in English):
(downloads: 86)