Recruited convergence sets the sign of plasticity under a single calcium rule: converging subicular and infralimbic inputs to the bed nucleus of the stria terminalis

The number of synapses recruited by a stimulus can give opposite signs of plasticity at two pathways governed by the same calcium-dependent rule. This paper quantifies that established cooperativity mechanism for converging subicular and infralimbic inputs to the bed nucleus of the stria terminalis. For an exponentially filtered periodic input, geometric summation gives the calcium-threshold recruitment N(θ) = θ(1 − r)/(γa) in the proportional, long-train regime, where r = exp(−T/τ), T is the pulse period, τ the decay time, a the contact size and γ the calcium gain. Ratios of these calcium-threshold recruitments equal ratios of the thresholds. A comparison theorem gives the conditions for strict depression and a necessary calcium crossing for potentiation; the complete calcium trace determines the final weight. Independent release gives an explicit mean, variance and concentration bound, and causal arrival jitter gives a truncated Gaussian moment at a fixed observation time. In a conductance-based model with a Shouval-type calcium rule, four trains of 100 pulses at 100 Hz produce changes within ±2 % for 1–2 recruited contacts, depression for 3–12, and potentiation for 13–40; the interpolated zero crossing is 12.24 contacts. Halving release probability moves it to 25.30 contacts. The crossing also depends strongly on stimulus frequency and NMDA decay, while varying the membrane time constant from 15 to 33 ms moves it by less than 0.1 contact. Equal summed drive can yield opposite signs when the individual contact sizes differ. These results establish sufficiency for opposite signs under separate stimulation. In the published preparation, depression follows stimulation of the other pathway; the transmitter-gated rule instead leaves a silent pathway stationary. The published infralimbic null also requires the actual burst protocol and recruitment to be resolved: in the model's 400 Hz series every tested positive convergence potentiates. Intensity, frequency and matched-calcium measurements distinguish these mechanisms. GPT-6 (OpenAI) assisted with the mathematical analysis, verification and preparation of the current revision. The scientific content and the responsibility for it rest with the author.