Neuromorphic computing and spiking neural networks (SNN) mimic the behavior of biological systems and have drawn interest for their potential to perform cognitive tasks with high energy efficiency. However, some factors such as temporal dynamics and spike timings prove critical for information processing but are often ignored by existing works, limiting the performance and applications of neuromorphic computing. On one hand, due to the lack of effective SNN training algorithms, it is difficult to utilize the temporal neural dynamics. Many existing algorithms still treat neuron activation statistically. On the other hand, utilizing temporal neural dynamics also poses challenges to hardware design. Synapses exhibit temporal dynamics, serving as memory units that hold historical information, but are often simplified as a connection with weight. Most current models integrate synaptic activations in some storage medium to represent membrane potential and institute a hard reset of membrane potential after the neuron emits a spike. This is done for its simplicity in hardware, requiring only a 'clear' signal to wipe the storage medium, but destroys temporal information stored in the neuron.In this work, we derive an efficient training algorithm for Leaky Integrate and Fire neurons, which is capable of training a SNN to learn complex spatial temporal patterns. We achieved competitive accuracy on two complex datasets. We also demonstrate the advantage of our model by a novel temporal pattern association task. Codesigned with this algorithm, we have developed a CMOS circuit implementation for a memristor-based network of neuron and synapses which retains critical neural dynamics with reduced complexity. This circuit implementation of the neuron model is simulated to demonstrate its ability to react to temporal spiking patterns with an adaptive threshold.