This paper introduces a comprehensive mechanistic model of a neuron with plasticity that explains the creation of engrams, the biophysical correlates of memory. In the context of a single neuron, this means clarifying how information conveyed as time-varying input signals is processed, stored, and subsequently recalled. Moreover, the model addresses two additional, long-standing, specific biological problems: how Hebbian plasticity, which amplifies synaptic weight, integrates with homeostatic plasticity, which stabilizes it, and how to identify a concise learning rule for synapses. In this study, a biologically accurate Hodgkin-Huxley-style electric-circuit equivalent is derived through a one-to-one mapping from the known properties of ion channels. The dynamics of the synaptic cleft, which is often overlooked, is found to be essential in this process. Analysis of the model reveals a simple and concise learning rule, indicating that the neuron functions as an internal-feedback adaptive filter, which is commonly used in signal processing. Simulation results confirm the circuit's functionality, stability, and convergence, demonstrating that even a single neuron without external feedback can function as a potent signal processor. The article is interdisciplinary and spans a broad range of subjects within the realm of biophysics, including neurobiology, electronics, and signal processing.