Filamentary resistive random-access memory (ReRAM) employs a single nanoscale event to trigger a macroscopic state change. While fundamentally it involves a gradual electrochemical evolution in a nanoscale filament that culminates in an abrupt change in filament's resistance, understanding over many length and time scales from the filament level to the device level is needed to inform the device behavior. Here, we demonstrate the nanoscale elements have corresponding elements in an empirical equivalent circuit. Specifically, the filament contains a variable resistor and capacitor that switch at a critical voltage. This simple model explains several observations widely reported on disparate filamentary ReRAMs. In particular, its collective system dynamics incorporating the power-law time-relaxation of the variable capacitance can accurately account for the responses of variously sized single-filament HfOx ReRAMs to DC/quasi-static and pulse electrical stimulation, exhibiting Avrami-like switching kinetics and a pulse-rate dependence in on/off voltages.