A Duck with a Head (and Neurons in Resonance)

She was just a neuroscientist studying the nonlinear dynamics of neural networks. He, like hundreds of others, was a mathematician specializing in stochastic processes and bifurcation theory. They met at an interdisciplinary conference on mathematical modeling of human behavior, and ever since, their conversations had tried to stay within the realm of pure science, where they frolicked merrily—while making sure their dual-control system, for the time being, stayed on the braking side of the equation. But, as folk wisdom tells us, a periodically forced system can transition to new states, and any orbitofrontal cortex can loosen its grip on inhibition… And so, he was already writing: “If I had to model my thoughts about you right now, I’d say my excitation trajectory is dangerously approaching a canard cycle. And, as you understand… this duck definitely has a head.” She:“Oh? So you’re saying that your trajectory is moving along the slow manifold, but now it’s unlikely to return to its previous state and will instead pass through a critical point and explode?” He:“Exactly. But be careful—if you overcomplicate me, my system will collapse into amplitude death before the transition even happens.” She:“Mmm. So the system doesn’t have to reach peak excitation to transition into another state? A small stochastic fluctuation is enough for you to experience a jump into a… deeper potential attractor?” He:“Damn. If you keep this up, I’m about to cross the separatrix.” She:“Patience, my brave chaotic oscillator. I prefer my systems in chimera states first—partially synchronized, partially unstable. It builds tension before total phase locking… and you know what happens after that.” He:“Full-scale neural synchronization leading to an explosive release of accumulated energy?” She:“That’s one way to put it. Or, to be more precise: a stochastic escape across the separatrix into the uniquely stable attractor of absolute satisfaction.” He:“God, I love it when you talk about nonlinear dynamics to me.” She:“Good. Now be a good mathematician and keep integrating the input function until my response curve reaches saturation.” He:“This is the best application of the Yerkes-Dodson law I’ve ever experienced. Optimal arousal for maximum performance.” She:“And control. Don’t forget control, or you’ll peak too soon and the post-climax B-process will hit too hard. Remember, we aim for sustained, well-tuned oscillations—not premature bifurcation.” He:“I think I’m experiencing… a Wiener process.” She:“Oh? Was that a small ‘w’ you just shakily whispered, all soft and uncertain—just a little stochastic teasing? Or are you implying a Wiener Process with a strong directional bias, an impressive—perhaps even dangerously persistent—W, one with highly constrained initial conditions and a hard-to-reverse trajectory?” He:“Both. Definitely both.” She:“Mmm. Then let’s synchronize our oscillators. I think it’s time for a nonlinear resonance event.” “And there’s the singular leap into a new attractor,”he did not think, as his temporal lobe temporarily ceased technical support. “And here’s the intuitively predictable chaotic finale,”she had just enough time to think, as their oscillators did not fully synchronize—but she, at least, was not cursed with a mandatory refractory period and could allow the system to transition into a new activation cycle without those pesky pauses.