The basic idea was to create a simple LED fade in/out effect circuit. I have used here a single dual op-amp NJM4556 which first stage is configured in comparator mode and second stage is configured as integrator. Triangular wave from integrator output is used to drive four LEDs. NJM4556 op-amp is able to deliver around 70 mA. It is enough to drive four LEDs with current 10 mA for each. Frequency of circuit is set to around 0.5 Hz. Below there is schematic of this circuit:
Here you could find details about this circuit and amplitude and frequency calculations:
I have made small modification for this circuit. I have removed four LEDs present on the PCB and added one NPN transistor which allow me to drive Cotton Ball Lights. I have used existing battery supply from this LED lights (3 * 1.5 V batteries). Below there is short video with final effect.