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

Neurostimulation and electrotherapy universally employ the rectangular waveform (square) as their standard stimulation signal. This article demonstrates that this choice constitutes a fundamental error of physical, mathematical, and neurophysiological nature, perpetuated since the mid-twentieth cen-tury through three converging factors: insufficient signal theory training in medical and paramedical curricula; technological drift toward ever-steeper wavefronts perceived as progress; and inadequate spectral disclosure by medical device manufacturers. We recall that the founders of electrical neu-rophysiology – Du Bois-Reymond (1843) and Helmholtz (1850) – stimulated with smooth-envelope signals, involuntarily close to membrane physiological requirements. We analyse the technological stratigraphy that progressively established the square wave as the unquestioned norm, and identify two erroneous assertions in the French foundational literature (Dumoulin & de Bisschop, 1987; Crépon, 1994) as crystallisation points of the error in clinical practice. We present spectral and energetic calcu-lations demonstrating the inadequacy of the rectangular signal relative to the biological bandwidth of the excitable membrane: for a 600 µs rectangular pulse at 50 Hz, Fourier harmonics extend to 81,650 Hz, wavefront components exceed 5 MHz, and the calculated peak power reaches 7.75 × 10⁸ W, versus 6.1 × 10⁵ W for the equivalent sinusoidal signal. We propose an optimal biomimetic signal described by a parametric Bézier curve whose inflexion points correspond to the conformational time constants of voltage-gated ion channels as described by the Hodgkin-Huxley model (1952). This zero-mean signal respects the natural opening and inactivation kinetics of sodium and potassium channels, concentrating its energy within the physiologically relevant bandwidth. The waveform is described as a scalable family parameterized by pulse duration τ, bounded within the physiologically valid range 200 µs ≲ τ ≲ 1000 µs, preserving full clinical adaptability. We analyse the electrode-tissue electrochemical interface, distinguishing faradaic from capacitive charge injection regimes, and demonstrate that the rectangular waveform systematically drives the interface into the faradaic regime, generating oxidative species and chronic peri-electrode inflammation. We present a comparative energetic analysis of three electrode-waveform configurations, showing that a high-capacitance elec-trode (CNT/aPDMS or TiN) combined with the Bézier 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 fibrosis. We discuss the documented clinical consequences of the fundamental error: peri-electrode fibrosis in deep brain stimulation (DBS), progressive impedance drift in cardiac pacemakers and cochlear implants, and the relative inefficacy of consumer TENS devices. We conclude with a prospective section that addresses the concrete benefits this corrected approach offers to patients, clinicians, and manufacturers. This work is published open access under Creative Commons CC-BY 4.0. All parameters of the optimal signal are fully described herein, establishing permanent publication priority and excluding subsequent patent filing on this concept.