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Modern massively-parallel Graphics Processing Units (GPUs) and Machine Learning (ML) frameworks enable neural network implementations of unprecedented performance and sophistication. However, state-of-the-art GPU hardware platforms are extremely power-hungry, while microprocessors cannot achieve the performance requirements. Biologically-inspired Spiking Neural Networks (SNN) have inherent characteristics that lead to lower power consumption. We thus present a bit-serial SNN-like hardware architecture. By using counters, comparators, and an indexing scheme, the design effectively implements the sum-of-products inherent in neurons. In addition, we experimented with various strength-reduction methods to lower neural network resource usage. The proposed Spiking Hybrid Network (SHiNe), validated on an FPGA, has been found to achieve reasonable performance with a low resource utilization, with some trade-off with respect to hardware throughput and signal representation.