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Living neural networks in our brains autonomously self-organize into large, complex architectures during early development to result in an organized and functional organic computational device. A key mechanism that enables the formation of complex architecture in the developing brain is the emergence of traveling spatio-temporal waves of neuronal activity across the growing brain. Inspired by this strategy, we attempt to efficiently self-organize large neural networks with an arbitrary number of layers into a wide variety of architectures. To achieve this, we propose a modular tool-kit in the form of a dynamical system that can be seamlessly stacked to assemble multi-layer neural networks. The dynamical system encapsulates the dynamics of spiking units, their inter/intra layer interactions as well as the plasticity rules that control the flow of information between layers. The key features of our tool-kit are (1) autonomous spatio-temporal waves across multiple layers triggered by activity in the preceding layer and (2) Spike-timing dependent plasticity (STDP) learning rules that update the inter-layer connectivity based on wave activity in the connecting layers. Our framework leads to the self-organization of a wide variety of architectures, ranging from multi-layer perceptrons to autoencoders. We also demonstrate that emergent waves can self-organize spiking network architecture to perform unsupervised learning, and networks can be coupled with a linear classifier to perform classification on classic image datasets like MNIST. Broadly, our work shows that a dynamical systems framework for learning can be used to self-organize large computational devices.