  Natural and technological networks exhibit dynamics that can lead to complex
cooperative behaviors, such as synchronization in coupled oscillators and
rhythmic activity in neuronal networks. Understanding these collective dynamics
is crucial for deciphering a range of phenomena from brain activity to power
grid stability. Recent interest in co-evolutionary networks has highlighted the
intricate interplay between dynamics on and of the network with mixed time
scales. Here, we explore the collective behavior of excitable oscillators in a
simple networks of two Theta neurons with adaptive coupling without
self-interaction. Through a combination of bifurcation analysis and numerical
simulations, we seek to understand how the level of adaptivity in the coupling
strength, $a$, influences the dynamics. We first investigate the dynamics
possible in the non-adaptive limit; our bifurcation analysis reveals stability
regions of quiescence and spiking behaviors, where the spiking frequencies
mode-lock in a variety of configurations. Second, as we increase the adaptivity
$a$, we observe a widening of the associated Arnol'd tongues, which may overlap
and give room for multi-stable configurations. For larger adaptivity, the
mode-locked regions may further undergo a period-doubling cascade into chaos.
Our findings contribute to the mathematical theory of adaptive networks and
offer insights into the potential mechanisms underlying neuronal communication
and synchronization.
