  A widely accepted view of computations in the brain relies on population
coding, where the neural ensemble firing rate is modulated in a stable manner
to transmit information and perform various cognitive tasks. At the same time,
oscillatory neural activity is specifically modulated in frequency, coherence
and power during cognitive performance. How the firing rate and oscillations
interact remains a salient question. In this paper, we develop a theory for the
interactions between oscillatory signals and the firing rate of neural
populations based on activity of non-linear voltage-dependent NMDA synapses.
Notably, we show under which conditions oscillatory inputs can control the mean
firing rate without loss of stability. Using mathematical analysis and
simulations of mean-field models, we demonstrate that presence of NMDA synapses
on both the excitatory and the inhibitory neurons is critical for sinusoidal
oscillations to significantly and stably increase the firing rate. We
characterize the oscillation-induced mean firing rate shift as a function of
the fast and slow synaptic weights and demonstrate the parameter region, in
which the effect under investigation is mostly pronounced. Results of our work
may help identify the properties of neural circuits that allow for constructive
control of the firing rate codes by large-scale neural oscillations.
