  Spike-frequency adaptation (SFA) is a fundamental neuronal mechanism taking
into account the fatigue due to spike emissions and the consequent reduction of
the firing activity. We have studied the effect of this adaptation mechanism on
the macroscopic dynamics of excitatory and inhibitory networks of quadratic
integrate-and-fire (QIF) neurons coupled via exponentially decaying
post-synaptic potentials. In particular, we have studied the population
activities by employing an exact mean field reduction, which gives rise to next
generation neural mass models. This low-dimensional reduction allows for the
derivation of bifurcation diagrams and the identification of the possible
macroscopic regimes emerging both in a single and in two identically coupled
neural masses. In single populations SFA favours the emergence of population
bursts in excitatory networks, while it hinders tonic population spiking for
inhibitory ones. The symmetric coupling of two neural masses, in absence of
adaptation, leads to the emergence of macroscopic solutions with broken
symmetry: namely, chimera-like solutions in the inhibitory case and anti-phase
population spikes in the excitatory one. The addition of SFA leads to new
collective dynamical regimes exhibiting cross-frequency coupling (CFC) among
the fast synaptic time scale and the slow adaptation one, ranging from
anti-phase slow-fast nested oscillations to symmetric and asymmetric bursting
phenomena. The analysis of these CFC rhythms in the $\theta$-$\gamma$ range has
revealed that a reduction of SFA leads to an increase of the $\theta$ frequency
joined to a decrease of the $\gamma$ one. This is analogous to what reported
experimentally for the hippocampus and the olfactory cortex of rodents under
cholinergic modulation, that is known to reduce SFA.
