Baseline Synaptic Configuration and Post-Stimulation Retention Under Homeostatic Scaling: A Reduced Model Motivated by Repetitive Transcranial Magnetic Stimulation

Patients respond to a course of repetitive transcranial magnetic stimulation (rTMS) for depression in very different ways, and much of this variability is unexplained. Homeostatic synaptic scaling, which in cultured and rodent neurons adjusts excitatory synaptic strengths over hours to days to return activity toward a set point, could affect how long a stimulation-induced change lasts. Computational work has already linked rTMS to homeostatic changes in synapse number. This paper asks a narrower question about existing synapses: when scaling restores a neuron's activity after a stimulation-induced change, does what remains of that change depend on the starting synaptic configuration, even when starting activity is the same? I address it in a reduced theoretical model built from published findings; no new data were collected. Uniform scaling multiplies every excitatory synaptic strength by one factor, so it preserves their ratios and cannot undo a change that altered them. If activity is also linear in synaptic strength and exactly restored, two configurations with equal baseline activity that receive the same change are rescaled by the same factor, and their difference survives in proportion. In a worked example, one configuration keeps an illustrative target property after restoration and the other loses it, although the one that loses it keeps the larger residual change. A repeated-update control shows the limits of the result. With identical changes and full restoration in every round, the configurations converge, so this version of the model predicts different times of reaching the target rather than lasting nonresponse. I outline observations that could separate this mechanism from weak initial induction and from changes in synapse number. Whether intermittent clinical rTMS recruits synaptic scaling in human cortex is unknown; the model generates hypotheses and does not account for treatment response.