Below is an illustration. The part on the left is just to simulate the charging phase of the capacitor. The transistor chosen there is just because I could easily get ahold of its Spice model (it's a power transistor, not needed for your application unless you need to discharge the capacitor at a very high current). On the right there's a basic negative voltage generator that should work fine here. The input signal should be an oscillator. If it oscillates, a negative voltage is generated (enough to block the transistor, around -2V); when it stops oscillating, the voltage quickly rises back to zero. If you're using an MCU, it will just be a matter of generating a clock on the controlling IO. Otherwise you can add a small oscillator, for instance with an inverter (schmitt trigger input), a resistor and a capacitor. No need for anything accurate here.
All this requires a few components but with the benefit of drawing little current. You could have used a normally-closed relay instead, but even a small reed relay will draw a lot more current than this and most will require a 5V supply, a transistor and a freewheeling diode, so...