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

1. Opening: the Quadrature of the Circle

1.1. A Mathematical Problem Two Thousand Years Old

The quadrature of the circle is one of the founding problems of ancient geometry: to construct, using only a ruler and compass, a square of equal area to a given circle. For two millennia, mathematicians and philosophers attempted to solve it. In 1882, Ferdinand von Lindemann demonstrated its impossibility once and for all: $\pi$ is a transcendental number, the root of no polynomial with rational coefficients [17]. No finite algebraic construction can bridge the circle from the square.

1.2. Vitruvian Man: The Square and the Circle as Geometries of Life

Vitruvius, in his De architectura (1st century BCE), describes man as the measure of all things, inscribable both in a square and in a circle [22]. Leonardo da Vinci illustrated this duality around 1490 in what remains the most celebrated image in the history of human anatomy. Two superimposed silhouettes: one inscribed in the circle and the other in the square. The circle is not an approximation of an improved square, it is its primary form, the form of movement, rotation, and life.

1.3. The Missed Quadrature of Electrotherapy

Electrotherapy has for 150 years pursued its own quadrature of the circle: substituting a rectangular signal for the curved, continuous, periodic form of the neuronal action potential.

Lindemann’s result applies here by analogy: substitution is impossible without loss. One cannot replace the circle with the square without leaving something behind. That “something” is what we shall now quantify.

2. What the Founders Already Knew

2.1. Du Bois-Reymond and the Wave of Negativity (1843)

Emil Heinrich Du Bois-Reymond (1818–1896), working in Berlin under Johannes Müller, demonstrated in 1843 that a stimulus applied to the surface of a nerve produces a local decrease in electrical potential — which he named the wave of relative negativity (negative Schwankung) — propagating along the fiber [7]. This discovery founded the field of scientific electrophysiology.

What is decisive for our argument: Du Bois-Reymond stimulated using mechanical interruption circuits and induction coils. The signals produced had mechanically constrained rise times, rounded, damped, quasi-sinusoidal. Involuntarily, instruments of the era produced physiological signals close to the target. And stimulation worked, at modest voltages.

2.2. Helmholtz and the Measurement of Conduction Velocity (1850)

Hermann von Helmholtz (1821–1894) measured in 1850 the propagation velocity of the nerve impulse in frog nerve-muscle preparations, obtaining values between 25 and 43 m/s [13]. His apparatus used induction coils (Ruhmkorff spirals) for stimulation. Again, the stimulation signal was a quasi-sinusoidal impulse envelope — the natural product of an inductance discharge.

Helmholtz did not choose the sinusoid through physiological reasoning. He was constrained to it by the available technology. That constraint was a stroke of luck.

2.3. The Historical Rupture

The discoveries of Du Bois-Reymond and his contemporaries could have provided a rigorous physiological foundation for electrotherapeutic practice from the mid-nineteenth century onwards. They were “coldly received by electrotherapists, who criticised them severely for several decades” [16]. The dissociation between fundamental electrophysiology and clinical practice is ancient. It is not the product of a lack of data. It is a refusal.

3. The Technological Stratigraphy of the Error

The fundamental error of electrotherapy was not born from a single bad decision. It was built up in successive layers, each technological advance perfecting the wrong target.

3.1. Level 1: The Manual Switch and the Induction Coil (19th Century)

The first stimulators produced signals with mechanically constrained rise times, rounded, and damped. These instruments had stimulated effectively at modest energies. The error did not yet exist: technology was involuntarily correct.

3.2. Level 2: Vacuum Tube Electronics (1920–1960)

The first electronic generators allowed for the first time the choice of waveform. The square wave became possible , and immediately desirable, because it appeared to be precise, reproducible, and parameterizable. Its duration could be read directly from an oscilloscope screen.

This is the bifurcation point. The rectangular signal was not adopted following a physiological demonstration. It was adopted because it was convenient for the human operator. The quality criterion became visual legibility, not biological adequacy.

3.3. Level 3: The Operational Amplifier (1960–1980)

The advent of $\mu$A741 and its successors provided slew rates of several V / $\mu$ s. Rise times could now be quasi-vertical. The plateaus were perfectly flat. Technology reached the perfection of the ideal square wave.

The ideal square wave is, as we shall demonstrate, the worst possible neurophysiological signal.

3.4. Level 4: The Fast MOSFET and the Microprocessor (1980–2000)

The generators of the era of foundational publications (Dumoulin & de Bisschop, 1987; Crépon, 1994) produced wavefronts in a few hundred nanoseconds. The engineering was remarkable. The target was still the square wave.

3.5. Level 5: High-Voltage Digital Stimulation (2000 – Present)

Deep brain stimulation (DBS), cochlear implants, and spinal neurostimulators operate with pulses whose fronts reach a few tens of nanoseconds, at voltages that can exceed 100 V. Jamali et al. [15] proposed in 2019 a stimulation method at 1 MHz with sub-microsecond pulses, reporting a reduction in charge per pulse from 60 nC to 0.3 nC. At these frequencies, energy is predominantly dissipated as heat. The device resembles a low-power electrosurgical unit more than a neurostimulator.

The law of this stratigraphy: at each technological generation, the quality criterion was fidelity to the rectangle — steeper rise, flatter plateau, sharper descent. Nobody asked whether the rectangle was the right target. This is the error of perfection: to optimise brilliantly in the wrong direction.

4. Codification of the Error in the Foundational Literature

4.1. Two Assertions, One Generation of Error

Assertion 1 — Dumoulin & de Bisschop (1987).

In Électrothérapie [9], p. 33: direct current is presented as the ideal waveform because “it is easy to establish and break a circuit.”

This claim confuses the electrical properties of a resistive load with the physiology of an excitable membrane. In terms of pure resistance, the establishment of a direct current does indeed produce sustained depolarisation. On a neurological membrane, the same manoeuvre produces a single initial depolarisation, followed by accommodation and then electrical silence – as Hodgkin and Huxley had demonstrated 35 years earlier [14]. The neuron responds to the variation of the potential ($\mathrm{d}V/\mathrm{d}t$), not to its absolute value.

Assertion 2 — Crépon (1994).

In Électrophysiothérapie [5], p. 23: the advantage of the rectangular signal is that “it is possible to rise to the plateau value in zero time.”

A zero-time rise is a Heaviside discontinuity. Its Fourier series expansion is an infinite sum of sinusoids whose amplitudes never tend to zero. In real devices, a 100 ns front corresponds to a cutoff frequency of approximately 3.5 MHz — several orders of magnitude above the physiologically relevant bandwidth of the neuron.

4.2. A Matter of Training, Not Negligence

Guy de Bisschop held positions in clinical neurophysiology at the hospitals of Marseille and Martigues, and lectured at Paris V. Francis Crépon authored major pedagogical references, still in print in 2014. These authors were not lacking in rigour — they lacked specific conceptual tools: Fourier spectral analysis applied to non-sinusoidal signals, the concept of biological bandwidth, and the molecular biophysics of ion channels. These tools are taught in undergraduate physics and biomedical engineering. They were not and are still not systematically included in medical and paramedical curricula.

4.3. The Responsibility of Manufacturers

Neurostimulation device manufacturers employ engineers in electronics and signal processing. These engineers know the spectral content of their signals. They measure load impedance. They calculate peak power delivered.

The technical manuals of neurostimulation devices do not to this day specify the spectral content of the delivered signal, the cutoff frequency $-3$ dB, the calculated maximum power, nor the physiological limits beyond which injected energy ceases to be neurophysiologically relevant.

This information was available. We will leave it to the reader to appreciate why it did not find its way into user manuals.

5. Physical and Mathematical Demonstration

5.1. Spectral Analysis of the Rectangular Signal

A rectangular signal of period T, amplitude A, and duty cycle $1/2$ decomposes into a Fourier series:

$$f\left(t\right)={\displaystyle \frac{4A}{\pi }}\sum _{n=1,3,5,\dots }^{\infty }{\displaystyle \frac{1}{n}}sin\left({\displaystyle \frac{2\pi nt}{T}}\right)$$

(1)

For a pulse of duration $\tau =600\mu \mathrm{s}$ at frequency $f_0=50$ Hz:

• Fundamental harmonic: $f_1=50$ Hz
• Significant harmonics extending to $f_n\approx \mathrm{81,650}$ Hz (for $n\approx 1633$)
• Wavefront duration $\Delta t\approx 100$ ns (modern device): $f_{max}\approx 1/(2\Delta t)\approx 5$ MHz

The physiologically relevant bandwidth of a neuron is determined by the duration of the action potential: 1–2 ms, corresponding to a cutoff frequency of approximately 500–1000 Hz. The vast majority of the spectral energy of the rectangular signal is deposited at frequencies to which the membrane is blind — it cannot respond to them through selective ionic mechanisms. This energy is dissipated as heat.

5.2. Comparative Peak Power Calculation

Two key observations:

(1) The peak power of the rectangular signal is approximately 1273 times greater than that of the sinusoidal signal for comparable clinical parameters. The commercial claim that the rectangular signal “uses less energy” is doubly incorrect: the mean rectangular amplitude is 13% higher than the sinusoidal amplitude, and the peak power exceeds it by more than three orders of magnitude.

(2) The impedance varies by a factor of 64: the actual charge delivered to the membrane varies proportionally between frequencies. The rectangular signal is intrinsically unstable against a biological load, unlike the sinusoidal signal, whose impedance is constant.

Table 1. Comparative energy parameters — rectangular vs. sinusoidal signal.

Parameter	Rectangular	Sinusoidal
Minimum impedance	$31\Omega$	$187\Omega$
Maximum impedance	$2,000\Omega$	$187\Omega$
Impedance variation	$\times 64$	constant
Peak power ($I_{max}^2\times Z_{min}$)	$7.75\times {10}^8$ W	$6.1\times {10}^5$ W
Mean amplitude	3.92 mA	3.46 mA
Ratio $P_{c,\mathrm{rect}}/P_{c,\sin }$	$\approx 1273$

Table 1. Comparative energy parameters — rectangular vs. sinusoidal signal.

Parameter	Rectangular	Sinusoidal
Minimum impedance	$31\Omega$	$187\Omega$
Maximum impedance	$2,000\Omega$	$187\Omega$
Impedance variation	$\times 64$	constant
Peak power ($I_{max}^2\times Z_{min}$)	$7.75\times {10}^8$ W	$6.1\times {10}^5$ W
Mean amplitude	3.92 mA	3.46 mA
Ratio $P_{c,\mathrm{rect}}/P_{c,\sin }$	$\approx 1273$

5.3. The Sinusoid: Omnipresent and Unrecognized

There is a profound irony in the situation described. The sinusoidal signal is:

• The form of every oscillatory motion (pendulum, spring, tide)
• The projection of uniform circular motion — Vitruvian Man’s circle
• The form of sound waves, light, and electromagnetic radiation.
• The form of the alternating current that powers the stimulation devices themselves
• The approximate form of the action potential

Yet, inside the neurostimulation device itself, the mains sinusoid is rectified, chopped, and deformed — to produce a rectangular signal before it reaches the patient. The physiological signal is deliberately destroyed at the input of the device.

One of the authors verified experimentally, during a 50 Hz mains electrification incident (1985), that the pure mains sinusoid produces effective and reproducible muscular tetanisation at modest voltage, exactly what Du Bois-Reymond observed with his induction coils.

We optimised for the human observer. We forgot to optimise for the neuron.

6. Membrane Neurophysiology: What the Neuron Requires

6.1. The Hodgkin-Huxley Model (1952)

The Hodgkin-Huxley model [14], Nobel Prize 1963, describes the dynamics of the action potential:

$$C_m{\displaystyle \frac{\mathrm{d}V}{\mathrm{d}t}}=-g_{\mathrm{Na}}(V-E_{\mathrm{Na}})-g_{\mathrm{K}}(V-E_{\mathrm{K}})-g_L(V-E_L)+I_{\mathrm{ext}}$$

(2)

The conductances $g_{\mathrm{Na}}$ and $g_{\mathrm{K}}$ are governed by activation and inactivation variables (m, h, n) obeying first-order differential equations with voltage-dependent time constants ${\tau }_m\left(V\right)$, ${\tau }_h\left(V\right)$, ${\tau }_n\left(V\right)$ — the conformational time constants of the channel proteins.

6.2. What the Square Wave Imposes on the Membrane

Rise front ($\Delta t\to 0$): theoretically infinite $\mathrm{d}V/\mathrm{d}t$. Voltage-gated sodium channels are transmembrane proteins — mechanical objects with finite conformational time constants. Their opening cannot be instantaneous. The imposed electrical constraint has no relation to any physiological stimulation.

Plateau ($\mathrm{d}V/\mathrm{d}t=0$): the potential is kept constant. However, it is precisely $\mathrm{d}V/\mathrm{d}t$ that is the relevant signal for sodium channels. A constant potential produces accommodation: channels progressively inactivate. Stimulation becomes ineffective.

Falling front: a second discontinuity of opposite sign. A second mechanical constraint on the membrane proteins.

In chronic stimulation (DBS, prolonged TENS), these repeated mechanical constraints on channel proteins may induce conformational fatigue, local neuroinflammation, and peri-electrode fibrotic reaction.

6.3. Membrane Accommodation: A Neglected Phenomenon

Membrane accommodation, the progressive decrease of excitability under constant current, has been described in the electrophysiological literature since the 19th century. Pflüger’s law (1859) [19] states that it is the variation of current, not its intensity, that determines the contractile response. This law underpins the concepts of chronaxie and rheobase — both of which, it should be noted, are measured using rectangular signals, introducing a systematic measurement bias that the literature has never corrected.

7. Documented Clinical Consequences

7.1. Peri-Electrode Fibrosis in DBS

Deep brain stimulation uses chronically implanted electrodes that deliver rectangular high-frequency pulses. The literature documents a progressive impedance drift after implantation [2,12,20]:

• Impedance doubling within 12 days [2]
• Factor of 10 within 22 days under continuous stimulation
• Correlation between impedance increase ($>4000\Omega$) and loss of clinical efficacy [12]
• Post-mortem studies: peri-electrode fibrous sheath, reactive astrocytosis, neuronal loss, neuroinflammation

The cause of this fibrosis is described as unknown in recent meta-analyses [11]. Our hypothesis: chronic membrane aggression by an inadequate signal → local neuroinflammation → reactive fibrosis → impedance increase → compensatory voltage increase → aggravation of aggression. A vicious cycle not identified as such in the literature.

7.2. Cardiac Pacemakers and Implantable Defibrillators: The Weightiest Argument

Chronic cardiac pacing by implanted pacemakers may constitute the most widespread and clinically critical application of the rectangular waveform in medicine. More than five million new devices are implanted annually worldwide [18]. Tens of millions of patients carry, permanently implanted in direct contact with myocardial or nodal tissue, an electrode that delivers continuous rectangular impulses — 24 hours a day, 365 days a year, for 8 to 15 years.

Myocardial tissue is excitable tissue in the full sense: it possesses voltage-gated ion channels, conformational time constants, a refractory period, and susceptibility to accommodation. The arguments developed in this article therefore apply to it in their entirety.

Peri-electrode impedance drift in chronic cardiac stimulation.

The formation of a fibrous capsule around the tip of the pacemaker electrode after implantation is a well-documented clinical phenomenon, universally known to cardiologists [10]. It manifests itself as an increase in stimulation impedance in the weeks following implantation, followed by stabilisation at a level above the initial baseline. The practical consequences are:

• Increased stimulation threshold that requires reprogramming of the energy delivered
• Increased battery consumption, reducing device longevity
• In extreme cases, loss of capture requiring surgical re-intervention

The official cause of this fibrosis: foreign body inflammatory reaction. This explanation, while partially accurate, does not explain why inflammation is systematically localised to the active electrode-tissue interface, nor why it is more pronounced under active stimulation than in its absence [1].

Our hypothesis applied to cardiac pacing.

The mechanism proposed in this article – chronic membrane aggression by an inadequate signal → inflammation → fibrosis → impedance increase → compensatory energy increase → aggravation — applies to myocardial tissue with an additional dimension: resting heart rate (60–80 bpm) implies chronic stimulation at relatively low pulse frequency, but the rectangular rise fronts remain identical. Each impulse imposes on myocardial tissue the same spectral discontinuities as on the neuronal membrane.

The energetic argument reversed.

Pacemaker manufacturers justify the use of ultra-short rectangular pulses (typically 400–500 $\mu$s) by battery longevity optimisation: a short pulse consumes less energy than a long one. This argument is valid in terms of the charge per pulse.

However, if a significant fraction of each pulse’s energy is dissipated as heat in biologically out-of-band harmonics, the useful energetic efficiency — energy actually converted into myocardial depolarisation — is lower than the charge calculation suggests. Furthermore, the peri-electrode fibrosis induced by spectral mismatch forces progressive energy increases over time, gradually negating the benefit of the short pulse.

The energetic logic of the rectangular signal is circular: it creates the very conditions that require its intensification.

Cochlear implants.

The same reasoning applies to cochlear implants, whose electrodes deliver rectangular biphasic pulses to the auditory nerve fibres. The progressive loss of efficacy and peri-electrode histological changes observed after cochlear implantation [21] deserve to be re-examined through the lens of spectral mismatch.

In terms of public health, the question of the optimal waveform in chronic cardiac pacing exceeds by several orders of magnitude that of TENS or even DBS. This is the weightiest argument in this article and, to our knowledge, one that has never been formulated in the literature.

7.3. Consumer TENS Devices

Tens of millions of TENS (Transcutaneous Electrical Nerve Stimulation) devices are sold annually worldwide, all using rectangular signals, all direct heirs of the chain of errors described in this article. The clinical evaluation of their efficacy in chronic pain remains controversial [4]. To date, no rigorous comparative study has compared rectangular and biomimetic signals with identical physiological parameters.

8. The Optimal Biomimetic Signal: Complete Description

8.1. Design Principles

An optimal neurostimulation signal must satisfy the following constraints:

1.

Zero net charge — to prevent tissue electrolysis and net ionic migration

2.

Spectrum concentrated within the biological bandwidth — useful energy between ∼1 Hz and ∼2 kHz

3.

Rise time with respect to conformational time constants of Na⁺ channels (${\tau }_m\approx 0.1$ – $0.5$ ms)

4.

Repolarization phase with respect to the kinetics of K⁺ (${\tau }_n\approx 1$–5 ms)

5.

Post-potential hyperpolarization reproducing the relative refractory period

8.2. The Bézier Curve as a Modeling Tool

A cubic Bézier curve is defined by four control points $P_0,P_1,P_2,P_3$:

$$\mathbf{B}\left(t\right)={(1-t)}^3{P}_0+3{(1-t)}^{2}t{P}_1+3(1-t)t^2{P}_2+t^3{P}_3,t\in [0,1]$$

(3)

The power of this representation for our application is twofold:

(a) Each control point $P_i$ corresponds to an identifiable ionic event: Na⁺ channel opening ($P_1$), depolarisation peak ($P_2$), Na⁺ inactivation, and K⁺ activation ($P_3$). The coordinates are therefore biologically interpretable and measurable by electrophysiology.

(b) The curve is $C^{\infty }$ — infinitely differentiable. There are no discontinuities, no fronts, no parasitic high-frequency harmonics. The spectrum is naturally bounded.

8.3. Parametric Description of the Optimal Signal

The complete signal is composed of three concatenated Bézier segments, ensuring continuity in value and first derivative at junction points.

Segment 1 — Depolarization (duration $T_1\approx 0.5$–1 ms).

$$\begin{array}{cccc}\hfill P_0& =(0,0)\hfill & \hfill & \mathrm{resting}\mathrm{potential}\hfill \\ \hfill P_1& =(0.1T_1,0.05)\hfill & \hfill & \mathrm{onset}\mathrm{of}{\mathrm{Na}}^{+}\mathrm{channel}\mathrm{opening}\hfill \\ \hfill P_2& =(0.4T_1,0.85)\hfill & \hfill & \mathrm{rapid}{\mathrm{Na}}^{+}\mathrm{activation}\hfill \\ \hfill P_3& =(T_1,1.0)\hfill & \hfill & \mathrm{depolarization}\mathrm{peak}\hfill \end{array}$$

This segment reproduces the apparent exponential activation of sodium channels – rapid but not discontinuous.

Segment 2 — Rapid repolarization (duration $T_2\approx 1$–2 ms).

$$\begin{array}{cc}\hfill P_0& =(T_1,1.0)\hfill \\ \hfill P_1& =(T_1+0.2T_2,0.9)\hfill & \hfill & {\mathrm{Na}}^{+}\mathrm{inactivation}\hfill \\ \hfill P_2& =(T_1+0.6T_2,-0.1)\hfill & \hfill & {\mathrm{K}}^{+}\mathrm{activation}\hfill \\ \hfill P_3& =(T_1+T_2,-A_h)\hfill & \hfill & \mathrm{hyperpolarization}\mathrm{trough}\hfill \end{array}$$

where $A_h\approx 0.15$–$0.25$ (post-potential hyperpolarization, 15–25% of depolarization amplitude).

Segment 3 — Return to rest (duration $T_3\approx 3$–8 ms).

$$\begin{array}{cc}\hfill P_0& =(T_1+T_2,-A_h)\hfill \\ \hfill P_1& =(T_1+T_2+0.3T_3,-0.7A_h)\hfill & \hfill & {\mathrm{K}}^{+}\mathrm{inactivation}\hfill \\ \hfill P_2& =(T_1+T_2+0.7T_3,-0.1A_h)\hfill & \hfill & \mathrm{Na}/\mathrm{K}\mathrm{pump}\hfill \\ \hfill P_3& =(T_1+T_2+T_3,0)\hfill & \hfill & \mathrm{return}\mathrm{to}\mathrm{rest}\hfill \end{array}$$

Zero net charge condition.

$${\int }_0^{T_1+T_2+T_3}\mathbf{B}\left(t\right)\mathrm{d}t=0$$

(4)

satisfied by numerically adjusting $A_h$ and $T_3$.

8.4. Comparative Visualization

Figure 1. Comparison of the proposed biomimetic signal (green) and the standard rectangular signal (red, dashed). The biomimetic signal reproduces the phases of the action potential: progressive depolarization respecting Na⁺ kinetics, peak, K⁺ repolarization, post-potential hyperpolarisation, and return to rest. Zero net charge is guaranteed. The rectangular signal presents non-physiological discontinuities at the rising and falling fronts.

Figure 1. Comparison of the proposed biomimetic signal (green) and the standard rectangular signal (red, dashed). The biomimetic signal reproduces the phases of the action potential: progressive depolarization respecting Na⁺ kinetics, peak, K⁺ repolarization, post-potential hyperpolarisation, and return to rest. Zero net charge is guaranteed. The rectangular signal presents non-physiological discontinuities at the rising and falling fronts.

8.5. Advantages of the Biomimetic Signal

1.

Concentrated spectrum: absence of discontinuities ⇒ energy naturally limited to the biological bandwidth

2.

Respect for ionic kinetics: the signal invites channels to open according to their own natural dynamics, rather than forcing them through a discontinuity

3.

Stable impedance: absence of high-frequency components ⇒ predictable and constant load

4.

Reduced thermal energy: minimisation of energy deposited outside the biological band

5.

Biologically interpretable parameters: each control point corresponds to a measurable and adjustable membrane time constant, tunable to the properties of the target tissue

8.6. Waveform Parameterization and Clinical Adaptability

The Bézier-optimal waveform described in this article should not be understood as a fixed pulse of predetermined duration, but rather as a family of physiologically constrained waveforms sharing a common geometric architecture.

The essential properties of the waveform are carried by its shape – the progressive rise calibrated on the conformational time constants of voltage-gated ion channels, the smooth peak, and the return to zero ensuring net charge balance – not by any particular absolute duration. The control points of the Bézier curve define ratios, not absolute time values. Therefore, the total duration of the impulse $\tau$ remains a free parameter, adjustable without compromising the neurophysiological rationale of the waveform.

Formally, if the reference waveform is defined over $[0,T_0]$, any scaled version over $[0,\lambda T_0]$ with $\lambda >0$ preserves:

• the shape of the $\mathrm{d}V/\mathrm{d}t$ profile relative to the channel time constants,
• the zero net charge condition $\left({\int }_0^{\lambda T_0}V\left(t\right)\mathrm{d}t=0\right)$,
• the harmonic content structure, scaled proportionally in frequency.

This parameterisation restores to clinicians and manufacturers a degree of freedom they already possess with rectangular waveforms — pulse duration adjustment — while embedding it within a physiologically rational framework. Duration $\tau$ can be adapted to the depth of the target tissue, the diameter of the fibre, the local impedance, or the therapeutic indication, exactly as current practice allows, without reverting to the rectangular approximation.

Physiological bounds on $\tau$.

However, the free parameter $\tau$ is not unbounded. The physiological rationale of the waveform imposes a hard lower limit: the total impulse duration must remain compatible with the conformational dynamics of the target ion channels. For sodium channels controlled by voltage (Na_v), the activation time constant ${\tau }_m$ is in the range of 50–100 $\mu$s at physiological temperature [14]. A waveform compressed below $\tau \approx 200\mu$s would present a $\mathrm{d}V/\mathrm{d}t$ profile too brief for channel activation to follow, effectively decoupling the stimulus from the gating mechanism it is designed to engage – entering a regime of electrochemical shock rather than physiological recruitment.

The physiologically valid range for $\tau$ is therefore bounded:

$$200\mu \mathrm{s}\lesssim \tau \lesssim 1,000\mu \mathrm{s}$$

(5)

where the upper bound reflects the onset of membrane accommodation – the progressive inactivation of Na_v channels under sustained depolarisation, quantified by the time constant ${\tau }_h$ [14].

Within this window, $\tau$ remains a genuine clinical variable. Shorter durations favour the recruitment of large myelinated fibres (A$\beta$, A$\alpha$); longer durations extend the recruitment to smaller fibres (A$\delta$, C), following the strength-duration relationship described by Lapicque [20] and confirmed by contemporary chronaxie measurements. The Bézier waveform family thus subsumes the classical strength-duration framework within a physiologically coherent signal architecture.

For implantable device manufacturers, this architecture offers a direct implementation pathway: the Bézier waveform family can be encoded as a lookup table parameterized by $\tau$, requiring no continuous computation and adding negligible complexity to existing pulse generation hardware. The transition from rectangular to physiologically shaped stimulation is therefore not a hardware redesign — it is, in most cases, a firmware decision.

9. The Electrode-Tissue Interface: Faradaic vs. Capacitive Regimes

9.1. Two Charge Injection Mechanisms

Every stimulation electrode operates at the boundary between the electronic domain (electron flow in metal) and the ionic domain (ion flow in tissue). At this interface, charge transfer occurs through two fundamentally distinct mechanisms:

Capacitive injection — redistribution of charge at the electric double layer, without transfer of matter. No electrochemical reaction occurs. The process is perfectly reversible.

Faradaic injection — electron transfer across the interface, driving reduction-oxidation reactions: water electrolysis, metal oxidation, generation of reactive oxygen species (H₂O₂, OH^•), local pH variations.

The equivalent circuit of the electrode-tissue interface (Randles model) places these two mechanisms in parallel: a double-layer capacitance $C_{dl}$ and a charge transfer resistance $R_{ct}$ in series with a Warburg diffusion impedance $Z_W$.

9.2. Why the Rectangular Waveform Drives Faradaic Reactions

The steep rising front of the rectangular waveform instantaneously imposes the maximum current density at the electrode surface. This transient peak drives the interfacial potential beyond the Flade potential – the threshold above which faradaic reactions become thermodynamically favourable – at every single pulse.

The smooth rising front of the Bézier waveform, by contrast, allows the double-layer capacitance to charge progressively. The interfacial potential increases slowly and remains within the capacitive window for a larger fraction of the pulse duration.

9.3. Clinical Consequences: The Faradaic Spiral

The products of faradaic reactions, oxidative radicals, pH changes, and dissolved metal ions, are direct pro-inflammatory agents. In chronically implanted devices, this produces a self-reinforcing cascade: faradaic oxidative stress triggers local neuroinflammation, progressing to peri-electrode fibrosis, raising interfacial impedance Z, forcing compensatory increases in stimulation voltage $V_{\mathrm{stim}}$, which in turn intensifies the faradaic reaction at the next pulse.

This vicious cycle – already qualitatively described in the literature as the peri-electrode impedance drift of unknown cause [11] – finds here a mechanistic explanation rooted in the waveform-interface mismatch.

9.4. Comparative Energetic Analysis: Three Configurations

To quantify the cost and benefit of electrode-waveform combinations, we performed a simplified analytical calculation using a lumped RC membrane model (node of Ranvier, ${\tau }_m=1$ ms, depolarisation threshold $\Delta V_m=20$ mV, pulse duration $\tau =600\mu$s, tissue resistivity $\rho =300$ $\Omega ·$cm).

Three configurations were compared, defined by their electrode-waveform coupling coefficient $\alpha$ (fraction of electrode voltage reaching the membrane):

Table 2. Comparative energetic analysis — three electrode/waveform configurations.

Configuration	${\mathit{V}}_0$ (V)	E ($\mathit{\mu }$J)	$\mathit{V}/{\mathit{V}}_{\mathbf{ref}}$	$\mathit{E}/{\mathit{E}}_{\mathbf{ref}}$
Faradaic Pt + Square (ref.)	2.2	12.35	1.00	1.00
Capacitive Pt + Square	7.4	137.2	3.33	11.1
Capacitive TiN + Bézier	4.9	19.85	2.20	1.61

Table 2. Comparative energetic analysis — three electrode/waveform configurations.

Configuration	${\mathit{V}}_0$ (V)	E ($\mathit{\mu }$J)	$\mathit{V}/{\mathit{V}}_{\mathbf{ref}}$	$\mathit{E}/{\mathit{E}}_{\mathbf{ref}}$
Faradaic Pt + Square (ref.)	2.2	12.35	1.00	1.00
Capacitive Pt + Square	7.4	137.2	3.33	11.1
Capacitive TiN + Bézier	4.9	19.85	2.20	1.61

Two conclusions follow directly:

(1) A capacitive electrode combined with a rectangular waveform is energetically catastrophic (×11 in energy): the mismatch between a waveform optimised for DC charge transfer and an interface designed for AC coupling is total.

(2) A capacitive high-performance electrode (TiN, CNT/aPDMS) combined with the Bézier waveform incurs only $+61\%$ in initial energy relative to the faradaic reference, an overhead that, crucially, does not compound over time. As the peri-electrode impedance remains stable (no fibrosis), this modest initial cost is never escalated by the compensatory voltage increases that progressively burden rectangular-faradaic systems in chronic use.

9.5. CNT/aPDMS Electrodes: The Natural Substrate for the Biomimetic Waveform

Among emerging capacitive electrode materials, carbon nanotube / adhesive polydimethylsiloxane (CNT/aPDMS) composites represent a particularly compelling match for the Bézier waveform. Their relevant properties include

• Purely capacitive by construction: the insulating PDMS matrix prevents faradaic reactions structurally, not merely kinetically
• Exceptional charge injection capacity: functionalised CNT electrodes reach 1.0–1.6 mC / cm² without faradaic reactions [21], compared to ∼0.05 mC / cm ² for bare platinum
• Mechanical compliance: Young’s modulus of CNT/PDMS composites approaches that of neural tissue, reducing the mechanical component of inflammatory response
• Progressive tissue integration: CNT assemblies have been shown to undergo gradual structural integration with neural tissue over days, with stimulation thresholds decreasing over time [22] — the inverse of the fibrotic spiral
• Validated durability: Mechanical stability has been demonstrated over $>4\times {10}^6$ biphasic stimulation pulses and 10,000 stretch cycles at 20% strain [23]

The CNT/aPDMS electrode is not just a material improvement. It is the physical substrate that the smooth waveform demands. Applied to such a purely capacitive interface, the rectangular waveform would be doubly aberrant: electrochemically incoherent and neurophysiologically wasteful. The Bézier waveform finds here its natural receiver.

10. Discussion

10.1. The Irony of the Unrecognized Sinusoid

The sinusoidal signal is the most universal mathematical form in nature. It is present in the main current that powers the stimulation device itself. And at the final step of the chain, it is deliberately destroyed to produce a rectangle.

The sinusoid is paradoxically foreign to us — not because it is rare, but because its “useful” duration cannot be read directly from an oscilloscope screen. We preferred the signal whose duration can be measured with a ruler on the display. The square wave has a visual advantage: the pulse duration is immediately apparent. The sinusoid does not offer this legibility.

We optimised for the human observer. We forgot to optimise for the neuron.

10.2. Toward Spectral Normalization of Neurostimulation Signals

This article argues for the introduction, in medical device safety standards for neurostimulation, of a minimum spectral specification:

• Mandatory declaration of the spectral content of the delivered signal ($-3$ dB cutoff frequency; energy fraction in the band $>10$ kHz)
• Declaration of the calculated peak power
• Justification of spectral adequacy with respect to the biological bandwidth of the target tissue

These requirements are technically trivial for any manufacturer equipped with a spectrum analyser. Their absence from current standards is the regulatory gap that has allowed the error to persist.

10.3. Limitations and Future Directions

The biomimetic signal proposed in this article is described on theoretical grounds and from available literature on ion channel time constants. Experimental validation requires:

1.

In vitro electrophysiological studies comparing the efficiency of action potential triggering across waveforms (rectangular vs. Bézier biomimetic)

2.

In vivo animal studies measuring stimulation threshold, fiber selectivity, and peri-electrode inflammatory response

3.

Clinical studies comparing the analgesic efficacy and tolerability of biomimetic versus rectangular TENS

4.

Systematic characterisation of DBS peri-electrode fibrosis as a function of waveform

These studies are feasible with current technology. Arbitrary waveform generators can reproduce the described signal exactly. Digital signal processing enables real-time Bézier implementation at frequencies well above those required.

11. Perspectives

The convergence of evidence presented in this article points toward a coherent and actionable therapeutic horizon. The argument is no longer purely critical: the physical, neurophysiological, electrochemical, and material foundations of an optimised neurostimulation system are now simultaneously available.

11.1. A complete System, Not an Isolated Correction

The rectangular waveform error was never an isolated error. It emerged from a technological stratigraphy in which each generation of engineers optimised the wrong target with increasing precision. Correcting it therefore requires a systemic response: a physiologically shaped waveform acting upon a capacitive electrode interface designed to receive it.

The Bézier-optimal waveform described in this article and the CNT/aPDMS electrode family represent precisely this complementary pair. Applied to a purely capacitive interface, the rectangular waveform would be doubly aberrant: electrochemically inappropriate and neurophysiologically wasteful. Conversely, the Bézier waveform finds in the CNT/aPDMS electrode its natural receiver: a progressive rise remaining within the capacitive regime, zero faradaic reaction, zero chronic oxidative stress, and a stable impedance that does not drift upward under the pressure of peri-electrode fibrosis.

The energetic calculation presented above quantifies what this stability means in practice. The initial apparent cost of the capacitive-Bézier configuration — approximately $+61\%$ in energy relative to the faradaic platinum reference — is not a penalty. It is a fixed overhead that, unlike the rectangular-faradaic combination, does not compound over time. In chronically implanted devices, where peri-electrode fibrosis forces progressive energy compensation that can reach multiples of the original threshold over years, the smooth-capacitive system offers something the current standard cannot: a stable operating point for the lifetime of the device.

11.2. For Patients

The clinical consequences of this stability are direct and measurable. Millions of patients currently live with implanted neurostimulation devices – deep brain stimulators, cochlear implants, cardiac pacemakers, implantable defibrillators, retinal prostheses – whose therapeutic efficacy degrades silently as peri-electrode impedance rises. The clinical response to this degradation is invariably the same: increase stimulation energy, accelerate battery depletion, and eventually schedule surgical revision.

Each of these consequences is, at least in part, a downstream effect of the waveform-electrode mismatch described in this article. An optimised system would not eliminate all causes of implant degradation, but it would remove the electrochemical cause, the one that operates at every single pulse, millions of times per year, for the entire duration of implantation.

For the patient, this translates into a prospect that is modest in its technical requirements and significant in its human consequences: longer battery life, more stable therapeutic effect, and fewer surgical revisions.

11.3. For Clinicians

The practitioners who have applied rectangular waveforms throughout their careers have done so in good faith on the basis of training that reflected the available consensus. This article does not challenge their competence; it challenges the foundation on which that consensus was based.

The perspective that opens for clinicians is one of recovered agency. Understanding that the waveform shape is a neurophysiological variable, not a fixed technical parameter, restores the possibility of rational, biology-driven parameter selection. The transition from empirical adjustment of pulse width and amplitude to principled waveform design, within the physiologically bounded range $200\mu \mathrm{s}\lesssim \tau \lesssim 1,000\mu \mathrm{s}$, represents a qualitative shift in the clinical relationship with electrophysiological tools.

For rehabilitation practitioners in particular, where electrostimulation is applied to intact or recovering peripheral nervous systems, the margin for improvement is immediate and requires no implantable device: existing external stimulators could, in principle, implement Bézier-shaped waveforms through firmware update alone.

11.4. For Manufacturers

The industrial argument requires careful reframing. The rectangular waveform has long been presented as an energy-efficient choice, and the capacitive electrode as an unnecessary complication. The analysis presented here inverts both claims.

A stimulation system combining a high-capacitance electrode — CNT/aPDMS, TiN, or PEDOT-based — with a physiologically optimised waveform offers a genuinely differentiated product: lower chronic energy consumption, extended battery life, reduced peri-electrode tissue reaction, and a more stable long-term therapeutic profile. Each of these properties directly translates into competitive advantage, reduced surgical revision rates, and a stronger medico-legal position.

The materials exist. The fabrication processes are documented. Electrochemical characterisation has been performed. The waveform is freely published. What remains is the decision to align the engineering target with the biology it was always meant to serve.

11.5. An Open Invitation

The waveform described in this article is not a proprietary solution. It is a physical consequence of the conformational dynamics of voltage-gated ion channels, first characterised by Hodgkin and Huxley in 1952 and honoured by the Nobel Committee in 1963. Mathematics has been available for decades. The technology to implement them in real time is available today.

The quadrature of the circle was proven impossible in 1882. The stimulation of living neural tissue with a rectangular pulse was never proven optimal — it was simply never seriously questioned. This article is an invitation to begin that question and to follow it wherever the physics leads.

12. Open Publication and Priority

All parameters of the biomimetic signal described in this article are published under the Creative Commons Attribution 4.0 International Licence (CC-BY 4.0). This publication establishes permanent and opposable prior art. No patent application claiming this signal, its parameters, its calculation method, or its electronic implementation may be validly filed after the deposit date of this preprint.

Any manufacturer is free to implement this signal without royalties, provided the source is cited. Universal dissemination of the optimal signal is preferable to any industrial protection.

13. Conclusion

Neurostimulation has attempted for 150 years to achieve the quadrature of the circle: substituting a rectangular signal for the curved form of the neuronal action potential. Lindemann’s 1882 proof of the mathematical impossibility of the quadrature finds its biophysical analogue here: one cannot replace the curved signal of the neuron with a rectangular signal without energetic loss, membrane aggression, and clinical consequences.

This error is the product of a convergence: the visual convenience of the rectangular signal on an oscilloscope; technological drift toward ever-steeper wavefronts; insufficient signal theory training; the absence of spectral requirements in medical device standards; and the systematic choice of faradaic metal electrodes that amplify rather than attenuate the mismatch.

The proposed biomimetic signal, a Bézier curve whose control points correspond to the time constants of voltage-gated ion channels, scalable within a physiologically bounded duration range, is now implementable in real time with available technology. The electrode materials to receive it correctly already exist. Both are published freely.

The circle is not an approximation of an improved square. It is the form the neuron has been waiting for all along.