Research of dynamic modes in the mathematical model of elementary thalamocortical cell

A. V. Kolosov ; I. V. Nuidel ; V. G. Yakhno

Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 24 (2016) no. 5, pp. 72-83.

Résumé

In the work the mathematical model of the thalamocortical network’s unit cell and it’s characteristic dynamical modes in system, describing the interaction between a thalamus, thalamus reticular nucleus and a cortex, is studied. During normal information processing, input signal gating occurs in time in the thalamocortical network. The violation of the normal functioning leads to an epilepsy, when the perception of information is disrupted. The consideration of this system will lead to an understanding of the human perception’s violation regularities, which correspond to the thalamocortical network’s self-oscillation. The focused mathematical model is described by a system of three differential equations. For this system a three-dimensional phase space is constructed, enabling us to track changes of the system’s «equilibrium states» when parameters change. The analysis of the system for the first time is performed using a three-dimensional phase space; the behavior of the representative points’ trajectories is considered, and the representation of equilibrium states of the system becomes obvious. Due to the large number of parameters of the system to build that space makes it easier to predict the development of the system in subsequent time for any parameters. The classification of dynamical modes in system (unexcited, excited and a self-oscillation), depending on the magnitude of the constant external signal incoming to the thalamus, is carried out. It is shown that the response of the system consists of a first pulse and following it selfoscillation pulses, whose period is different from the duration of the first pulse. The dependence of characteristic times of the first impulse response and the period of the self-oscillation with the external signal is studied. The numerical analysis of the model revealed the existence of $U$-shaped dependence of the first pulse duration and a decreasing of the period of the self-oscillation in the response to the increase of the external signal value. The results are important for further consideration of the plausibility of the hypothesis, according to which the thalamocortical networks control the activity of the areas of the cerebral cortex and focused (normally) on the integration of the obtained results for decision-making at higher levels of neural network processing in brain structures.

Détail

Keywords: Neuronlike network, self-oscillation, relaxation oscillations, epilepsy, inhibitory connections, excitatory connections, signal gating, sensor signal.