Hafnium oxide non-volatile memories have shown promise as an artificial synapse in neuromorphic computing architectures. However, there is still a need to fundamentally understand how to reliably control the analog resistance change induced by oxygen ions that partially rupture or re-form the conductive filament. In this work, the impact of measurement conditions (pulse amplitude and pulse width) and titanium dopants on the analog resistance change of atomic layer deposited hafnium oxide memristor synapses are studied. A lower pulse amplitude improves the linearity of resistance change as a function of the number of pulses but results in a smaller memory window. The addition of titanium dopants does not substantively change the analog resistance modulation of hafnium oxide. Density functional theory calculations show that titanium strongly impacts oxygen ion motion in the HfxTiyOz matrix but does not impact significantly in the HfTi metallic filament. This study demonstrates that the analog characteristic of HfxTiyOz artificial synapses is largely independent of the titanium doped bulk oxide since the resistance change is primarily controlled by the HfTi metallic conducting filament.