This research introduces the first-ever implementation of non-linear relaxation oscillators and coupled networks utilising energy-efficient Recessed-Source/Drain (Re-S/D) Silicon-On-Insulator (SOI) Feedback Field Effect Transistors (FBFETs). FBFETs provide a subthreshold slope (SS) that is close to zero and a larger ON Current ( I-ON ) compared to traditional MOSFETs. As a result, power consumption is reduced. The behaviour of relaxation oscillators is precisely regulated by manipulating hysteresis in the I-D -V-S characteristics through device design parameters such as work function (W-F), Re-S/D thickness ( TRe-S/D ), and gate oxide thickness ( T-ox ). By altering the thickness of the Re-S/D layer, ranging from 0 nm to 50 nm, we achieved 61.96% decrease in energy per spike, 53.03% increase in spike frequency, and 42.7% decrease in power consumption compared with 50 nm Re-S/D thickness. Further, by varying the bias current of the Re-S/D SOI FBFET from 0.01 nA to 2.98 nA, the oscillating frequency of the relaxation oscillator ranged from 55.55 Hz to 27.5 KHz, and the power consumption ranged from 0.6166 pW to 91.895 pW. Furthermore, the behaviour of the relaxation oscillator is observed by varying the circuit parameters such as gate voltage ( V-GF ) of Re-S/D SOI FBFET, Supply Voltage ( V-DD ), and NMOS currents ( I-M1 , and I-M2 ), while keeping the device parameters constant. In addition, a coupled network was constructed with two relaxation oscillators, which exhibited synchronization dynamics in response to changes in the relaxation oscillators. Due to the controlled oscillations of a coupled network, our proposed Re-S/D SOI FBFET oscillator presents a promising avenue for efficiently implementing neuromorphic computing algorithms.