Neuromorphic is a relatively new interdisciplinary research topic, which employs various fields of science and technology, such as electronic, computer, and biology. Neuromorphic systems consist of software/hardware systems, which are utilized to implement the neural networks based on human brain functionalities. The goal of neuromorphic systems is to mimic the biologically inspired concepts of the nervous systems, envisioned to provide advantages, such as lower power consumption, fault tolerance, and massive parallelism for the next generation of computers. This brief presents a neural computing hardware unit and a neuromorphic system architecture based on a modified leaky integrate and fire neuron model in a spiking neural network for a pattern recognition task in register-transfer level. The neuron model and the spiking network are explored, considering digital implementation, targeting low-cost high-speed large-scale systems. Results of the hardware synthesis and implementation on field-programmable gate array are presented as a proof of concept. Accordingly, the maximum frequency of the implemented neuron model and spiking network are 412.371 MHz and 189.071 MHz, respectively.