  Simplified neuronal models capture the essence of the electrical activity of
a generic neuron, besides being more interesting from the computational point
of view when compared to higher dimensional models such as the Hodgkin-Huxley
one. In this work, we propose a generalized resonate-and-fire model described
by a generalized Langevin equation that takes into account memory effects and
colored noise. We perform a comprehensive numerical analysis to study the
dynamics and the point process statistics of the proposed model, highlighting
interesting new features like: i) non-monotonic behavior (emergence of peak
structures, enhanced by the choice of colored noise characteristic time-scale)
of the coefficient of variation (CV) as a function of memory characteristic
time-scale, ii) colored noise-induced shift in the CV, and iii) emergence and
suppression of multimodality in the interspike interval (ISI) distribution due
to memory-induced subthreshold oscillations. Moreover, in the noise-induced
spike regime, we study how memory and colored noise affects the coherence
resonance (CR) phenomenon. We found that for sufficiently long memory, CR is
not only suppressed, but also the minimum of the CV $\times$ noise intensity
curve that characterizes the presence of CR may be replaced by a maximum. The
aforementioned features allow to interpret the interplay between memory and
colored noise as an effective control mechanism to neuronal variability. Since
both variability and non-trivial temporal patterns in the ISI distribution are
ubiquitous in biological cells, we hope the present model can be useful in
modeling real aspects of neurons.
