Mechanism of appearing complex relaxation oscillations in a system of two synaptically coupled neurons (Q2199024)

The authors consider the system of differential-difference equations \[ \dot u_1=\left(\lambda f(u_1(t-1))+bg(u_2(t-h))\ln(u_*/u_1)\right)u_1, \] \[ \dot u_2=\left(\lambda f(u_2(t-1))+bg(u_1(t-h))\ln(u_*/u_2)\right)u_2 \] as a phenomenological model of the synaptic coupling of a pair of impulse neurons. Here \(u_1(t),\) \(u_2(t)>0\) represent their normalized membrane potential and a additional delay \(h>1\) is caused by delays in the conduction of impulses between nerve cells. The singular perturbation parameter \(\lambda\gg 1\) characterizes the speed of electrical processes and \(u_*=\mathrm{exp}(c\lambda)\), \(c<0\) is the threshold value controlling the coupling. Making the exponential change of variables in the model, \(u_j=\mathrm{exp}(\lambda x_j)\), \(j=1,2\), the coexistence of (orbitally asymptotically stable) relaxation cycles of the transformed system with a given number of the positivity intervals in the period is studied.