There has been significant recent interest towards achieving highly efficient deep neural network architectures. A promising paradigm for achieving this is the concept of evolutionary deep intelligence, which attempts to mimic biological evolution processes to synthesize highly-efficient deep neural networks over successive generations. An important aspect of evolutionary deep intelligence is the genetic encoding scheme used to mimic heredity, which can have a significant impact on the quality of offspring deep neural networks. Motivated by the neurobiological phenomenon of synaptic clustering, we introduce a new genetic encoding scheme where synaptic probability is driven towards the formation of a highly sparse set of synaptic clusters. Experimental results for the task of image classification demonstrated that the synthesized offspring networks using this synaptic cluster-driven genetic encoding scheme can achieve state-of-the-art performance while having network architectures that are not only significantly more efficient (with a ~125-fold decrease in synapses for MNIST) compared to the original ancestor network, but also tailored for GPU-accelerated machine learning applications.