In the brain, most synapses are formed on minute protrusions known as dendritic spines. Unlike their artificial intelligence counterparts, spines are not merely tuneable memory elements: they also embody algorithms that implement the brain's ability to learn from experience and cope with new challenges. Importantly, they exhibit structural dynamics that depend on activity, excitatory input and inhibitory input (synaptic plasticity or 'extrinsic' dynamics) and dynamics independent of activity ('intrinsic' dynamics), both of which are subject to neuromodulatory influences and reinforcers such as dopamine. Here we succinctly review extrinsic and intrinsic dynamics, compare these with parallels in machine learning where they exist, describe the importance of intrinsic dynamics for memory management and adaptation, and speculate on how disruption of extrinsic and intrinsic dynamics may give rise to mental disorders. Throughout, we also highlight algorithmic features of spine dynamics that may be relevant to future artificial intelligence developments.