Energy-Optimal Electrical-Stimulation Pulses Shaped by the Least-Action Principle

Abstract

Electrical stimulation (ES) devices interact with excitable neural tissue toward eliciting action potentials (AP’s) by specific current patterns. Low-energy ES prevents tissue damage and loss of specificity. Hence to identify optimal stimulation-current waveforms is a relevant problem, whose solution may have significant impact on the related medical (e.g. minimized side-effects) and engineering (e.g. maximized battery-life) efficiency. This has typically been addressed by simulation (of a given excitable-tissue model) and iterative numerical optimization with hard discontinuous constraints - e.g. AP’s are all-or-none phenomena. Such approach is computationally expensive, while the solution is uncertain - e.g. may converge to local-only energy-minima and be model-specific. We exploit the Least-Action Principle (LAP). First, we derive in closed form the general template of the membrane-potential’s temporal trajectory, which minimizes the ES energy integral over time and over any space-clamp ionic current model. From the given model we then obtain the specific energy-efficient current waveform, which is demonstrated to be globally optimal. The solution is model-independent by construction. We illustrate the approach by a broad set of example situations with some of the most popular ionic current models from the literature. The proposed approach may result in the significant improvement of solution efficiency: cumbersome and uncertain iteration is replaced by a single quadrature of a system of ordinary differential equations. The approach is further validated by enabling a general comparison to the conventional simulation and optimization results from the literature, including one of our own, based on finite-horizon optimal control. Applying the LAP also resulted in a number of general ES optimality principles. One such succinct observation is that ES with long pulse durations is much more sensitive to the pulse’s shape whereas a rectangular pulse is most frequently optimal for short pulse durations.

Model-based analysis and design of waveforms for efficient neural stimulation

“…Krouchev et al (2014) used the Least Action Principle and a space clamped patch of neuronal membrane to determine the optimal trajectory of transmembrane potential and subsequently, for a given model, determine the energy optimal stimulation current to drive the transmembrane potential along that trajectory. Again, the resulting optimal current waveforms were approximately Gaussian in shape, reinforcing the findings of earlier studies that fortuitously selected this shape a priori or arrived at this shape via optimization.…”

An Investigation of Neural Stimulation Efficiency With Gaussian Waveforms

“…Following the findings of recent computational studies on energy optimal stimulation waveforms [19]- [21], Gaussian shaped stimuli were compared against standard rectangular stimulation in a nerve-muscle preparation in anaesthetized rats in terms of energy, charge and power efficiency. In order to attain results that can be applied for practical applications such as cochlear implants or neuromuscular prosthetics, both commonly-used electrode configurations, monopolar and bipolar stimulation, were tested.…”

“…Promisingly, three recent studies independently found a Gaussian shaped stimulation waveform to be optimal in terms of minimized energy consumption using model-based approaches probed by a genetic algorithm [19], the calculus of variation [20] and the least action principle [21]. While these studies agree that a Gaussian stimulation waveform shows increased energy efficiency over rectangular stimuli, there are considerable differences in the claimed potential to save energy by replacing simple rectangular waveforms with modified pulses.…”

In Vivo Measurements of the Frequency-Dependent Impedance of the Spinal Cord

“…�is is illustrated in figure 10. Some examples include the genetic algorithm-based method of Wongsarnpigoon and Grill [39], the least-action principle pulses of Krouchev et al [40], and the patient-specific waveform synthesis of Yalçinkaya and Erbaş [41]. An important goal of such work is the minimization of pulse energy (thus obtaining extended battery life) while maximizing physiological efectiveness [42].…”