We introduce a mathematical framework for the analysis of the input-output dynamics of externally driven memristors. We show that, under general assumptions, their dynamics comply with a Bernoulli differential equation and hence can be nonlinearly transformed into a formally solvable linear equation. The Bernoulli formalism, which applies to both charge-and flux-controlled memristors when either current or voltage driven, can, in some cases, lead to expressions of the output of the device as an explicit function of the input. We apply our framework to obtain analytical solutions of the i-v characteristics of the recently proposed model of the Hewlett-Packard memristor under three different drives without the need for numerical simulations. Our explicit solutions allow us to identify a dimensionless lumped parameter that combines device-specific parameters with properties of the input drive. This parameter governs the memristive behaviour of the device and, consequently, the amount of hysteresis in the i-v. We proceed further by defining formally a quantitative measure for the hysteresis of the device, for which we obtain explicit formulas in terms of the aforementioned parameter, and we discuss the applicability of the analysis for the design and analysis of memristor devices.

No abstract

A common approach to model memristive systems is to include empirical window functions to describe edge effects and non-linearities in the change of the memristance. We demonstrate that under quite general conditions, each window function can be associated with a sigmoidal curve relating the normalised time-dependent memristance to the time integral of the input. Conversely, this explicit relation allows us to derive window functions suitable for the mesoscopic modelling of memristive systems from a variety of well-known sigmoidals. Such sigmoidal curves are defined in terms of measured variables and can thus be extracted from input and output signals of a device and then transformed to its corresponding window. We also introduce a new generalised window function that allows the flexible modelling of asymmetric edge effects in a simple manner.

10.11.14 KB. OK to add article in press to spiral, OA paper under cc by licens

The copyright line for this article was changed on March 17, 2017, after original online publication.Purpose: High intensity focused ultrasound (HIFU) provides a non-invasive salvage treatment option for patients with recurrence after external beam radiation therapy (EBRT). As part of EBRT the prostate is frequently implanted with permanent fiducial markers. To date, the impact of these markers on subsequent HIFU treatment is unknown. The objective of this work was to systematically investigate, using computational simulations, how these fiducial markers affect the delivery of HIFU treatment. Methods: A series of simulations was performed modelling the propagation of ultrasound pressure waves in the prostate with a single spherical or cylindrical gold marker at different positions and orientations. For each marker configuration, a set of metrics (spatial-peak temporal-average intensity, focus shift, focal volume) was evaluated to quantify the distortion introduced at the focus. An analytical model was also developed describing the marker effect on the intensity at the focus. The model was used to examine the marker's impact in a clinical setting through case studies. Results: The simulations show that the presence of the marker in the pre-focal region causes reflections which induce a decrease in the focal intensity and focal volume, and a shift of the maximum pressure point away from the transducer's focus. These effects depend on the shape and orientation of the marker and become more pronounced as its distance from the transducer's focus decreases, with the distortion introduced by the marker greatly increasing when placed within 5 mm of the focus. The analytical model approximates the marker's effect and can be used as an alternative method to the computationally intensive and time consuming simulations for quickly estimating the intensity at the focus. A retrospective review of a small patient cohort selected for focal HIFU after failed EBRT indicates that the presence of the marker may affect HIFU treatment delivery. Conclusions: The distortion introduced by the marker to the HIFU beam when positioned close to the focus may result in an undertreated region beyond the marker due to less energy arriving at the focus, and an overtreated region due to reflections. Further work is necessary to investigate whether the results presented here justify the revision of the patient selection criteria or the markers' placement protocol.