The ability of the brain to encode and store information depends on the plastic nature of the individual synapses. The increase and decrease in synaptic strength, mediated through the structural plasticity of the spine, are important for learning, memory, and cognitive function. Dendritic spines are small structures that contain the synapse. They come in a variety of shapes (stubby, thin, or mushroom-shaped) and a wide range of sizes that protrude from the dendrite. These spines are the regions where the postsynaptic biochemical machinery responds to the neurotransmitters. Spines are dynamic structures, changing in size, shape, and number during development and aging. While spines and synapses have inspired neuromorphic engineering, the biophysical events underlying synaptic and structural plasticity of single spines remain poorly understood. Our current focus is on understanding the biophysical events underlying structural plasticity. I will discuss recent efforts from my group — first, a systems biology approach to construct a mathematical model of biochemical signaling and actin-mediated transient spine expansion in response to calcium influx caused by NMDA receptor activation and a series of spatial models to study the role of spine geometry and organelle location within the spine for calcium and cyclic AMP signaling. Second, I will discuss how mechanics of membrane-cytoskeleton interactions can give insight into spine shape region. And I will conclude with some new efforts in using reconstructions from electron microscopy to inform computational domains. I will conclude with how geometry and mechanics plays an important role in our understanding of fundamental biological phenomena and some general ideas on bio-inspired engineering.