The Leaky Integrator Bound for Synaptic Residue: A Conditional Calibration and Falsification Protocol

The leaky integrator bound A_max = ((1−λ)/(1+λ)) K is a rigorous result for discrete-time systems with exponential forgetting and bounded state. We apply this bound to synaptic memory, specifically the CaMKII (calcium/calmodulin-dependent protein kinase II) molecular switch. CaMKII exhibits the three properties required by the theorem: exponential decay (protein turnover with time constant τ ≈ 50 hours), bounded state (saturation concentration K), and discrete activation events (synaptic stimulation). The bound predicts that the maximum sustainable average firing rate a synapse can support without CaMKII saturation is ((1−λ)/(1+λ)) K, where λ = e^{−Δt/τ}. For physiologically realistic parameters (λ ≈ 0.98, Δt = 1 hour), the sustainable rate is approximately 0.01 K — one percent of saturation capacity per time step. This prediction is testable using existing in vitro calcium imaging and optogenetic stimulation techniques. The paper does not prove any conjecture; it applies a known mathematical result to a specific biological system and identifies the experiment that would test it.