Self-Sustained Relaxation Oscillations in Time-Delay Neural Systems

7th International Workshop on MUlti-Rate Processes & HYSteresis; 2nd International Workshop on Hysteresis and Slow-Fast Systems

Glyzin, S.D.^1 ; Kolesov, A.Yu.^1 ; Rozov, N.Kh.^2

Faculty of Mathematics, Yarostavl State University, Sovetskaya str., 14, Yaroslavl

150000, Russia^1

Faculty of Mechanics and Mathematics, Moscow State University, Main Building, Leninskiye Gory, Moscow

119991, Russia^2

关键词: Coupled neurons;    Differential-difference equations;    Number of components;    One-dimensional chains;    Periodic motion;    Periodic solution;    Relaxation oscillation;    Singularly perturbed;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/727/1/012004/pdf
DOI  :  10.1088/1742-6596/727/1/012004

来源: IOP

【 摘 要 】

A new method to model the phenomena 'bursting' and 'buffering' in neural systems is represented. Namely, a singularly perturbed nonlinear scalar differential difference equation with two delays is introduced, which is a mathematical model of a single neuron. It is shown that for suitably chosen parameters this equation has a stable periodic solution with an arbitrary prescribed number of asymptotically high impulses (spikes) on a period interval. It is also shown that the buffering phenomenon occurs in a one-dimensional chain of diffusively coupled neurons of this type: as the number of components in the chain grows in a way compatible with a decrease of the diffusion coefficient, the number of co-existing stable periodic motions increases indefinitely.