Squaring the Circle in Neural Engineering: The Rectangular Waveform as a Historical Artefact and the Cost of Convenience

The rectangular (square) waveform universally used in neurostimulation and electrotherapy was adopted for reasons of technological convenience, not biological validation. We demonstrate that it constitutes a fundamental error of physical, mathematical, and neurophysiological nature, and propose a physiologically optimal alternative. Fourier spectral analysis shows that a 600 µs rectangular pulse at 50 Hz generates harmonics extending to 81,650 Hz with a calculated peak power of 7.75 × 108 W— a factor of 1,273 above the sinusoidal equivalent. The Hodgkin-Huxley model (Nobel Prize, 1963) establishes that voltage-gated ion channels respond to dV/dt, not to absolute voltage: the rectangular wavefront is neurophysiologically incoherent. We propose an optimal biomimetic signal described by a parametric Bézier curve calibrated on the conformational time constants of sodium and potassium channels, scalable within the physiologically bounded range 200 µs ≲ τ ≲ 1,000 µs. Combined with a capacitive electrode (CNT/aPDMS or TiN), this waveform incurs only +61% in initial energy relative to the faradaic-rectangular reference — an overhead that remains stable over time, unlike the progressive energy escalation imposed by rectangular-induced peri-electrode fibrosis. Independent experimental evidence (Lembcke et al., 2026) corroborates the proposed mechanistic framework. The hypothesis has direct clinical implications for millions of patients carrying deep brain stimulators, cardiac pacemakers, and cochlear implants, in whom peri-electrode fibrosis and impedance drift are downstream consequences of waveform-electrode mismatch. The corrected system is firmware-implementable without hardware redesign.