  We consider a system consisting of two delay differential equations with a
large parameter, modeling the association of a pair of neurooscillators. The
unknown functions describe the changes in the normalized membrane potentials of
neurons over time, with the large parameter characterizing the speed of
electrical processes. The first equation is separated from the system and
represents a generalized Hutchinson equation. This equation, as known,
possesses periodic solutions with high peaks over the period. The second
equation is also based on the generalized Hutchinson equation, but with an
additional term, linking it to an oscillator satisfying the first equation. For
the second equation, it is possible to asymptotically construct the so-called
fading neuron mode, which is as follows: for any natural number $n$, one can
adjust the parameters of the problem in such a way that the solution is
asymptotically close to a periodic function with high peaks over $n$ periods,
and then, after a transient process represented by decreasing peaks, becomes
asymptotically small.
