As spiking-based deep learning inference applications are increasing in embedded systems, these systems tend to integrate neuromorphic accelerators such as $μ$Brain to improve energy efficiency. We propose a $μ$Brain-based scalable many-core neuromorphic hardware design to accelerate the computations of spiking deep convolutional neural networks (SDCNNs). To increase energy efficiency, cores are designed to be heterogeneous in terms of their neuron and synapse capacity (big cores have higher capacity than the little ones), and they are interconnected using a parallel segmented bus interconnect, which leads to lower latency and energy compared to a traditional mesh-based Network-on-Chip (NoC). We propose a system software framework called SentryOS to map SDCNN inference applications to the proposed design. SentryOS consists of a compiler and a run-time manager. The compiler compiles an SDCNN application into subnetworks by exploiting the internal architecture of big and little $μ$Brain cores. The run-time manager schedules these sub-networks onto cores and pipeline their execution to improve throughput. We evaluate the proposed big little many-core neuromorphic design and the system software framework with five commonlyused SDCNN inference applications and show that the proposed solution reduces energy (between 37% and 98%), reduces latency (between 9% and 25%), and increases application throughput (between 20% and 36%). We also show that SentryOS can be easily extended for other spiking neuromorphic accelerators.