  Nonlinear oscillators have been used to model neurons that fire periodically
in the absence of input. These oscillators, which are called neuronal
oscillator, share some common response structures with other biological
oscillations. In this study, we analyze the dependence of the global dynamics
of an impulse-driven stochastic neuronal oscillator on the relaxation rate to
the limit cycle, the strength of the intrinsic noise, and the impulsive input
parameters. To do this, we use a Markov operator that both reflects the density
evolution of the oscillator and is an extension of the phase transition curve,
which describes the phase shift due to a single isolated impulse. Previously,
we derived the Markov operator for the finite relaxation rate that describes
the dynamics of the entire phase plane. Here, we construct a Markov operator
for the infinite relaxation rate that describes the stochastic dynamics
restricted to the limit cycle. In both cases, the response of the stochastic
neuronal oscillator to time-varying impulses is described by a product of
Markov operators. Furthermore, we calculate the number of spikes between two
consecutive impulses to relate the dynamics of the oscillator to the number of
spikes per unit time and the interspike interval density. Specifically, we
analyze the dynamics of the number of spikes per unit time based on the
properties of the Markov operators. Each Markov operator can be decomposed into
stationary and transient components based on the properties of the eigenvalues
and eigenfunctions. This allows us to evaluate the difference in the number of
spikes per unit time between the stationary and transient responses of the
oscillator, which we show to be based on the dependence of the oscillator on
past activity. Our analysis shows how the duration of the past neuronal
activity depends on the relaxation rate, the noise strength and the input
parameters.
