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

For citation:

Glyzin S. D., Kolesov A. Y. Periodic modes of group dominance in fully coupled neural networks. Izvestiya VUZ. Applied Nonlinear Dynamics, 2021, vol. 29, iss. 5, pp. 775-798. DOI: 10.18500/0869-6632-2021-29-5-775-798

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: 316)
Article type: 

Periodic modes of group dominance in fully coupled neural networks

Glyzin Sergey Dmitrievich, P. G. Demidov Yaroslavl State University
Kolesov A. Yu., P. G. Demidov Yaroslavl State University

Nonlinear systems of differential equations with delay, which are mathematical models of fully connected networks of impulse neurons, are considered. Purpose of this work is to study the dynamic properties of one special class of solutions to these systems. Large parameter methods are used to study the existence and stability in сonsidered models of special periodic motions – the so-called group dominance or k-dominance modes, where k ∈ N. Results. It is shown that each such regime is a relaxation cycle, exactly k components of which perform synchronous impulse oscillations, and all other components are asymptotically small. The maximum number of stable coexisting group dominance cycles in the system with an appropriate choice of parameters is 2m − 1, where m is the number of network elements. Conclusion. Considered model with maximum possible number of couplings allows us to describe the most complex and diverse behavior that may be observed in biological neural associations. A feature of the k-dominance modes we have considered is that some of the network neurons are in a non-working (refractory) state. Each periodic k-dominance mode can be associated with a binary vector (α1, α2, . . . , αm), where αj = 1 if the j-th neuron is active and αj = 0 otherwise. Taking this into account, we come to the conclusion that these modes can be used to build devices with associative memory based on artificial neural networks.

This work was supported by Russian Foundation for Basic Research, grant No. 18-29-10055
  1. Hodgkin AL, Huxley AF. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 1952;117(4):500–544. DOI: 10.1113/jphysiol.1952.sp004764.
  2. Izhikevich EM. Dynamical Systems in Neuroscience. The Geometry of Excitability and Bursting. Cambridge, Mass.: MIT Press; 2006. 464 p.
  3. Kolesov AY, Kolesov YS. Relaxational oscillations in mathematical models of ecology. Proc. Steklov Inst. Math. 1995;199:1–126.
  4. Maiorov VV, Myshkin IY. Mathematical modeling of a neuron net on the basis of the equation with delays. Math. Models Comput. Simul. 1990;2(11):64–76 (in Russian).
  5. Glyzin SD, Kolesov AY, Rozov NK. Self-excited relaxation oscillations in networks of impulse neurons. Russian Math. Surveys. 2015;70(3):383—452. DOI: 10.1070/RM2015v070n03ABEH004951.
  6. Kolesov AY, Mishchenko EF, Rozov NK. A modification of Hutchinson’s equation. Comput. Math. Math. Phys. 2010;50(12):1990–2002. DOI: 10.1134/S0965542510120031.
  7. Hutchinson GE. Circular causal systems in ecology. Ann. N. Y. Acad. Sci. 1948;50(4):221—246. DOI: 10.1111/j.1749-6632.1948.tb39854.x.
  8. Glyzin SD, Kolesov AY, Rozov NK. On a method for mathematical modeling of chemical synapses. Diff. Equat. 2013;49(10):1193–1210. DOI: 10.1134/S0012266113100017. 
  9. Somers D, Kopell N. Rapid synchronization through fast threshold modulation. Biol. Cybern. 1993;68(5):393–407. DOI: 10.1007/BF00198772.
  10. Kolesov AY, Mishchenko EF, Rozov NK. A relay with delay and its C1-approximation. Proc. Steklov Inst. Math. 1997;216:119–146.