Differential Equations of Ideal Memristors

Ideal memristor is a resistor with a memory, which adds dynamics to its behavior. The most usual characteristics describing this dynamics are the constitutive relation (i.e. the relation between flux and charge), or Parameter-vs-state- map (PSM), mostly represented by the memristance-to-charge dependence. One of the so far unheeded tools for memristor description is its differential equation (DEM), composed exclusively of instantaneous values of voltage, current, and their derivatives. The article derives a general form of DEM that holds for any ideal memristor and shows that it is always a nonlinear equation of the first order; the PSM forms are found for memristors which are governed by DEMs of the Bernoulli and the Riccati types; a classification of memristors according to the type of their dynamics with respect to voltage and current is carried out

A mathematical framework for the analysis and modelling of memristor nanodevices

This work presents a set of mathematical tools for the analysis and modelling of memristor devices. The mathematical framework takes advantage of the compliance of the memristor's output dynamics with the family of Bernoulli differential equations which can always be linearised under an appropriate transformation. Based on this property, a set of conditionally solvable general solutions are defined for obtaining analytically the output for all possible types of ideal memristors. To demonstrate its usefulness, the framework is applied on HP's memristor model for obtaining analytical expressions describing its output for a set of different input signals. It is shown that the output expressions can lead to the identification of a parameter which represents the collective effect of all the model's parameters on the nonlinearity of the memristor's response. The corresponding conclusions are presented for series and parallel networks of memristors as well. The analytic output expressions enable also the study of several device properties of memristors. In particular, the hysteresis of the current-voltage response and the harmonic distortion introduced by the device are investigated and both interlinked with the nonlinearity of the system. Moreover, the reciprocity principle, a property form classical circuit theory, is shown to hold for ideal memristors under specific conditions. Based on the insights gained through the analysis of the ideal element, this work takes a step further into the modelling of memristive devices in an effort to improve some of the macroscopic models currently used. In particular, a method is proposed for extracting the window function directly from experimentally acquired input-output measurements. The method is based on a simple mathematical transformation which relates window to sigmoidal functions and a set of assumptions which allow the mapping of the sigmoidal to current-voltage measurements. The equivalence between the two representations is demonstrated through a new generalised window function and several existing sigmoidals and windows. The proposed method is applied on three sets of experimental measurements which demonstrate the usefulness of the window modelling approach and the newly proposed window function. Based on this method the extracted windows are tailored to the device under investigation. The analysis also reveals a set of non-idealities which lead to the introduction of a new model for memristive devices whose response cannot be captured by the window-based approach.Open Acces

Topological electronics: from infinity to six

Topology captures the essence of what remains unchanged under a transformation. This study was motivated by a newly found topological invariant called super conformality that leads to local activity of a higher-integral-order electric element. As a result, the traditional periodic table of the electric elements can be dramatically reduced to have only 6 passive ones (resistor, inductor, capacitor, memristor, meminductor, and memcapacitor), in contrast to the unbounded table predicted 40 years ago. Our claim was experimentally verified by the fact that the two higher-integral-order memristors in the famous Hodgkin-Huxley circuit are locally active with an internal battery

Beyond Memristors: Neuromorphic Computing Using Meminductors

Resistors with memory (memristors), inductors with memory (meminductors) and capacitors with memory (memcapacitors) play different roles in novel computing architectures. We found that a coil with a magnetic core is an inductor with memory (meminductor) in terms of its inductance L(q) being a function of charge q. The history of the current passing through the coil is remembered by the magnetization inside the magnetic core. Such a meminductor can play a unique role (that cannot be played by a memristor) in neuromorphic computing, deep learning and brain-inspired computers since the time constant of a neuromorphic RLC circuit is jointly determined by the inductance L and capacitance C, rather than the resistance R. As an experimental verification, this newly invented meminductor was used to reproduce the observed biological behavior of amoebae (the memorizing, timing and anticipating mechanisms). In conclusion, a beyond-memristor computing paradigm is theoretically sensible and experimentally practical

The Effects of Radiation on Memristor-Based Electronic Spiking Neural Networks

In this dissertation, memristor-based spiking neural networks (SNNs) are used to analyze the effect of radiation on the spatio-temporal pattern recognition (STPR) capability of the networks. Two-terminal resistive memory devices (memristors) are used as synapses to manipulate conductivity paths in the network. Spike-timing-dependent plasticity (STDP) learning behavior results in pattern learning and is achieved using biphasic shaped pre- and post-synaptic spikes. A TiO2 based non-linear drift memristor model designed in Verilog-A implements synaptic behavior and is modified to include experimentally observed effects of state-altering, ionizing, and off-state degradation radiation on the device. The impact of neuron “death” (disabled neuron circuits) due to radiation is also examined. In general, radiation interaction events distort the STDP learning curve undesirably, favoring synaptic potentiation. At lower short-term flux, the network is able to recover and relearn the pattern with consistent training, although some pixels may be affected due to stability issues. As the radiation flux and duration increases, it can overwhelm the leaky integrate-and-fire (LIF) post-synaptic neuron circuit, and the network does not learn the pattern. On the other hand, in the absence of the pattern, the radiation effects cumulate, and the system never regains stability. Neuron-death simulation results emphasize the importance of non-participating neurons during the learning process, concluding that non-participating afferents contribute to improving the learning ability of the neural network. Instantaneous neuron death proves to be more detrimental for the network compared to when the afferents die over time thus, retaining the network’s pattern learning capability