Neuromorphic computing is an efficient solution for large-scale associative learning problems such as pattern recognition, but its hardware implementation is stymied by the need for low-power, scalable faux neurons, typically built using relaxation oscillators. This work proposes relaxation oscillators using VO${}_{2}$-based heterostructures with optimized thermal time constants for low-power operation at microwave frequencies. Synchronization behavior between two coupled oscillators is also investigated. This study offers a theoretical foundation for the use of such oscillators in neuromorphic computing.