Simulating a pyroelectric sensor
Try a voltage of about 1V and blip it up or down by say 500mV for a second or so.

You can do this with a 74HC123 or a 555 and a few resistors.

That may or may not reliably trigger the downstream circuit. PIR sensors are normally paired with a Fresnel lens that causes an AC signal from motion and you may have to do more to stimulate it, depending on how immune to curtains moving etc. they have tried to make it. 

The cheap circuits just use a 5 cent LM324 quad op-amp and a few passive parts, no micro. Possibly that's what you have. The datasheet shows a sensitivity quite typical of consumer Chinese PIR sensors used in motion detector lamps etc. (3300V/W) 

You might want to create a more general purpose circuit with an Arduino and PWM and play with real sensors and an oscilloscope to simulate them.