  We investigate a one-dimensional model of active motion, which takes into
account the effects of persistent self-propulsion through a memory function in
a dissipative-like term of the generalized Langevin equation for particle
swimming velocity. The proposed model is a generalization of the active
Ornstein-Uhlenbeck model introduced by G. Szamel [Phys. Rev. E {\bf 90}, 012111
(2014)]. We focus on two different kinds of memory which arise in many natural
systems: an exponential decay and a power law, supplemented with additive
colored noise. We provide analytical expressions for the velocity
autocorrelation function and the mean-squared displacement, which are in
excellent agreement with numerical simulations. For both models, damped
oscillatory solutions emerge due to the competition between the memory of the
system and the persistence of velocity fluctuations. In particular, for a
power-law model with fractional Brownian noise, we show that long-time active
subdiffusion occurs with increasing long-term memory.
