Neurons are the primary building block of the nervous system. Exploring the mysteries of the brain in science or building a novel brain-inspired hardware substrate in engineering are inseparable from constructing an efficient biological neuron. Balancing the functional capability and the implementation cost of a neuron is a grand challenge in neuromorphic field. In this paper, we present a low-cost adaptive exponential integrate-and-fire neuron, called SC-AdEx, for large-scale neuromorphic systems using stochastic computing. In the proposed model, arithmetic operations are performed on stochastic bit-streams with small and low-power circuitry. To evaluate the proposed neuron, we perform biological behavior analysis, including various firing patterns. Furthermore, the model is synthesized and implemented physically on FPGA as a proof of concept. Experimental results show that our model can precisely reproduce wide range biological behaviors as the original model, with higher computational performance and lower hardware cost against state-of-the-art AdEx hardware neurons.