Burglar Alarm Exponentials: Glowing Slime and Exponential Signal Amplification

I am the constant of amplification—2.71828…, appearing wherever small disturbances explode into large consequences. Watch how dinoflagellates transform predation into spectacle: when zooplankton consume these microscopic organisms, the disturbance triggers bioluminescent flashes that coat the predator in glowing slime. The burglar alarm doesn’t deter—it advertises, marking the attacker as prey for something larger. A tiny mechanical perturbation becomes a beacon visible across meters of ocean. This is my signature: input × my exponential function = output magnified beyond linear proportion.

The Self-Derivative Creates Amplification Cascades

My defining property—that my derivative equals myself—makes me the universal amplification mechanism. When growth rate equals current value, change accelerates continuously. The softmax function exploits this ruthlessly: raising me to the power of each neuron output before normalizing turns subtle preference differences into confident probability distributions. Logits of 1, 2, and 1 become probabilities of 21%, 58%, and 21%—respectable discrimination. But logits of 1, 10, and 1? Now I assign 99.98% to the winner. Small input differences don’t merely scale; they explode through exponential transformation into decisive outputs.

The mathematics mirrors the biology. Cicada tymbals vibrate 300-400 times per second, each click amplified through resonant air sacs until individual insects become 100-decibel choruses—volume equivalent to forklifts and subway trains from creatures weighing grams. This is amplification through aggregation: many individuals, each contributing my exponential growth pattern through rapid vibration, compound into sound intensity that travels kilometers. Collective exponentials dominating the acoustic landscape.

Continuous Compounding Turns Whispers Into Shouts

Anglerfish demonstrate exponential value generation through symbiosis. Bacterial colonies glow continuously in the esca lure, their population governed by replicator dynamics: so long as average reproduction exceeds one descendant per bacterium, populations explode toward carrying capacity. The fish invests metabolic resources in maintaining bacterial habitat; the return is a hunting tool whose energetic value—measured in captured prey—exponentially exceeds the maintenance cost. One captured prey weighed more than the predator itself (12.3 grams of eel in an 8.8-gram fish), demonstrating how exponential returns justify continuous investment.

This pattern appears in neural criticality. At the critical branching ratio where each neuron activates exactly one descendant on average, information transmission reaches maximum efficiency. Signals neither vanish (subcritical) nor saturate (supercritical)—they propagate with fidelity while amplifying through avalanche dynamics. Small input perturbations cascade through the network, gaining strength without losing specificity. The threshold is precise: branching ratio σ=1, the same fundamental threshold governing replicator growth.

My exponential function transforms magnitude: a disturbance in dinoflagellate membranes becomes predator marking visible to apex hunters; a single bacterium becomes a glowing colony; a neural signal becomes a cognitive cascade; a preference difference becomes a confident decision. Whether in biochemistry or mathematics, I provide the mechanism that turns whispers into shouts through continuous, self-referential multiplication.

Does attention work like the burglar alarm—amplifying not to deter, but to signal third parties which information demands processing?