Spiking Neural Networks (SNN)-based neuromorphic computing system is proposed as an alternate approach for future computation to overcome the memory bottleneck issue in modern computer design. Computing technology today is constrained by constraints similar with those faced by biological systems, such as the number of transistors that can be accommodated per unit area, the resources and power required for communication between those transistors, and the increasing impact of noise as device sizes shrink. The goal of this thesis is to look into essential aspects of biological neurological systems and come up with approaches to apply these concepts to artificial systems. The work discussed here is organized around the development of a neuromorphic system that is designed to emulate the behavior of biological neurons and synapses in the human brain. The synapse and neuron mimicking circuit are developed using CMOS transistor and capacitor.