  Recent experiments show that quasi-one-dimensional lattices of self-propelled
droplets exhibit collective instabilities in the form of out-of-phase
oscillations and solitary-like waves. This hydrodynamic lattice is driven by
the external forcing of a vertically vibrating fluid bath, which invokes a
field of subcritical Faraday waves on the bath surface, mediating the
spatio-temporal droplet coupling.By modelling the droplet lattice as a
memory-endowed system with spatially nonlocal coupling, we herein rationalise
the form and onset of instability in this new class of dynamical oscillator. We
identify the memory-driven instability of the lattice as a function of the
number of droplets, and determine equispaced lattice configurations precluded
by geometrical constraints. Each memory-driven instability is then classified
as either a super- or sub-critical Hopf bifurcation \emph{via} a systematic
weakly nonlinear analysis, rationalising experimental observations. We further
discover a previously unreported symmetry-breaking instability, manifest as an
oscillatory-rotary motion of the lattice. Numerical simulations support our
findings and prompt further investigations of this nonlinear dynamical system.
