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The architecture is operated in the proportional limit (Ti\u2009=\u2009109 and Td\u2009=\u20090), so the switching dynamics are governed by the proportional gain Kp while retaining a mapping to the AD-IC resistor ratio. Promoting the diversion-branch conductance parameter \u03bb5 from a fixed scalar to a dynamical state driven by the controller output yields a six-dimensional closed-loop system. In the quasi-static limit, where \u03bb5 evolves slowly relative to the FHN oscillation period, the time to drive the circuit from its high-energy firing mode to the quiescent low-energy state satisfies the semi-analytical relation \u03c4switch=I(H\u2217)\/Kp, where H\u2217 is the target energy and I(H\u2217) is an energy-landscape path integral evaluated numerically from quasi-static steady-state measurements. For a selected target, the inverse dependence on Kp\u2009=\u2009R1\/R2 gives a closed-form gain-scaling rule that separates component selection from the application energy target. A feasibility boundary Hmax\u2217\u22481.87 emerges from the minimum of the energy landscape along the approach path to the fold bifurcation; a two-branch bifurcation diagram confirms the wide bistable region in which both the firing mode and the quiescent mode coexist as stable attractors. Validation against 15 Runge\u2013Kutta simulations across a two-dimensional (Kp,H\u2217) design space gives a mean absolute error of 14.7%, with systematic overestimation attributed to forcing-induced advance of the transition relative to the quasi-static fold point at larger gains.<\/jats:p>","DOI":"10.1063\/5.0352758","type":"journal-article","created":{"date-parts":[[2026,9,22]],"date-time":"2026-09-22T13:19:49Z","timestamp":1790083189000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["Semi-analytical switching-time control in a memristive FitzHugh\u2013Nagumo circuit via an AD-IC PID architecture operated in the proportional limit"],"prefix":"10.1063","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-6263-4636","authenticated-orcid":false,"given":"Yi\u011fit","family":"Aydo\u011fan","sequence":"first","affiliation":[{"name":"Department of Computer Science, Aberystwyth University 1 , Aberystwyth SY23 3DB,","place":["United Kingdom"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6572-215X","authenticated-orcid":false,"given":"Elham","family":"Minayi","sequence":"additional","affiliation":[{"name":"Department of Electrical and Electronics Engineering, Dogus University 2 , Istanbul 34775,","place":["T\u00fcrkiye"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2026,9,22]]},"reference":[{"key":"2026092209194411800_c1","doi-asserted-by":"publisher","first-page":"445","DOI":"10.1016\/S0006-3495(61)86902-6","article-title":"Impulses and physiological states in theoretical models of nerve membrane","volume":"1","year":"1961","journal-title":"Biophys. 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