Pressing a physical button, but digitally using an Arduino in an existing circuit
Even simpler is just to have the Arduino or any other chip to connect the AD pin via 2k2 resistor to ground momentarily, just like what the button does.
You only need to have an open-drain output, which can be implemented in Arduino software, or with a suitable logic gate or comparator that has an open-drain output, or even with a transistor.
The timing circuit can also be an Arduino, or a 555, or simple RC time constant with schmitt trigger inverter gate or comparator.