Competing memristors for brain-inspired computing

Nanoscale Titanium Oxide Memristive Structures for Neuromorphic Applications: Atomic Force Anodization Techniques, Modeling, Chemical Composition, and Resistive Switching Properties

This paper presents the results of a study on the formation of nanostructures of electrochemical titanium oxide for neuromorphic applications. Three anodization synthesis techniques were considered to allow the formation of structures with different sizes and productivity: nanodot, lateral, and imprint. The mathematical model allowed us to calculate the processes of oxygen ion transfer to the reaction zone; the growth of the nanostructure due to the oxidation of the titanium film; and the formation of TiO, Ti2O3, and TiO2 oxides in the volume of the growing nanostructure and the redistribution of oxygen vacancies and conduction channel. Modeling of the nanodot structure synthesis process showed that at the initial stages of growth, a conductivity channel was formed, connecting the top and bottom of the nanostructure, which became thinner over time; at approximately 640 ms, this channel broke into upper and lower nuclei, after which the upper part disappeared. Modeling of the lateral nanostructure synthesis process showed that at the initial stages of growth, a conductivity channel was also formed, which quickly disappeared and left a nucleus that moved after the moving AFM tip. The simulation of the imprint nanostructure synthesis process showed the formation of two conductivity channels at a distance corresponding to the dimensions of the template tip. After about 460 ms, both channels broke, leaving behind embryos. The nanodot, lateral, and imprint nanostructure XPS spectra confirmed the theoretical calculations presented earlier: in the near-surface layers, the TiO2 oxide was observed, with the subsequent titanium oxide nanostructure surface etching proportion of TiO2 decreasing, and proportions of Ti2O3 and TiO oxides increasing. All nanodot, lateral, and imprint nanostructures showed reproducible resistive switching over 1000 switching cycles and holding their state for 10,000 s at read operation.

Memristive Artificial Synapses Based on Brownmillerite for Endurable Weight Modulation

Exploring a computing paradigm that blends memory and computation functions is essential for artificial synapses. While memristors for artificial synapses are widely studied due to their energy-efficient structures, random filament conduction in general memristors makes them less preferred for endurability in long-term synaptic modulation. Herein, the topotactic phase transition (TPT) in brownmillerite-phased (110)-SrCoO2.5 (SCO2.5) is harnessed to enhance the reversibility of oxygen ion migration through 1-D oxygen vacancy channels. By employing a heteroepitaxial structured 2-terminal configuration of Au/SCO2.5/SrRuO3/SrTiO3, the brownmillerite SCO2.5-based synapse artificial synapses are exploited. Demonstration of the TPT behavior is corroborated by comparing oxygen migration energy by density-functional theory calculations and experimental results, and by monitoring the voltage pulse-induced peak shift in the Raman spectra of SCO2.5. With the voltage pulse-driven TPT behaviors, it is reliably characterized by linear, symmetric, and endurable long-term potentiation and depression performances. Notably, the durability of the TPT-based weight control mechanism is demonstrated by achieving consistent and noise-free weight updates over 32 000 iterations across 640 cycles. Furthermore, learning performances based on deep neural networks and convolutional neural networks on various image datasets yielded very high recognition accuracy. The work offers valuable insights into designing memristive synapses that enable reliable weight updates in neural networks.

Memristive Ion Dynamics to Enable Biorealistic Computing

Conventional artificial intelligence (AI) systems are facing bottlenecks due to the fundamental mismatches between AI models, which rely on parallel, in-memory, and dynamic computation, and traditional transistors, which have been designed and optimized for sequential logic operations. This calls for the development of novel computing units beyond transistors. Inspired by the high efficiency and adaptability of biological neural networks, computing systems mimicking the capabilities of biological structures are gaining more attention. Ion-based memristive devices (IMDs), owing to the intrinsic functional similarities to their biological counterparts, hold significant promise for implementing emerging neuromorphic learning and computing algorithms. In this article, we review the fundamental mechanisms of IMDs based on ion drift and diffusion to elucidate the origins of their diverse dynamics. We then examine how these mechanisms operate within different materials to enable IMDs with various types of switching behaviors, leading to a wide range of applications, from emulating biological components to realizing specialized computing requirements. Furthermore, we explore the potential for IMDs to be modified and tuned to achieve customized dynamics, which positions them as one of the most promising hardware candidates for executing bioinspired algorithms with unique specifications. Finally, we identify the challenges currently facing IMDs that hinder their widespread usage and highlight emerging research directions that could significantly benefit from incorporating IMDs.

Bioactive Ion-Confined Ultracapacitive Memristors with Neuromorphic Functions

The field of bioinspired iontronics, bridging electronic devices and ionic systems, has multiple biological applications. Carbon-based ultracapacitive devices hold promise for controlling bioactive ions via electric double layers due to their high-surface-area and biocompatible porous carbon electrodes. However, the interplay between complex bioactive ions and porous carbons remains unclear due to the variety of structures of bioactive ions present in biological systems. Herein, we investigate the adsorption behavior of a series of bioactive ammonium-based cations with varying alkyl chain lengths in nanoporous carbons. We find that strong physisorption results from the synergistic hydrophobic interaction and electrostatic attraction between porous carbons (with a negative zeta potential) and bioactive cations. Bioactive cations with varying alkyl chain lengths can be irreversibly physically adsorbed and confined within nanoporous carbons resulting in anion enrichment and depletion during electric polarization. This situation, in turn, results in a characteristic memristive behavior in all-carbon capacitive ionic memristor devices. Our findings highlight the relationship between the resistance state of the memristor and ion adsorption mechanisms in all-carbon capacitive devices, which hold potential for future transmitter delivery, biointerfacing, and neuromorphic devices.

Linearly programmable two-dimensional halide perovskite memristor arrays for neuromorphic computing

The exotic properties of three-dimensional halide perovskites, such as mixed ionic-electronic conductivity and feasible ion migration, have enabled them to challenge traditional memristive materials. However, the poor moisture stability and difficulty in controlling ion transport due to their polycrystalline nature have hindered their use as a neuromorphic hardware. Recently, two-dimensional (2D) halide perovskites have emerged as promising artificial synapses owing to their phase versatility, microstructural anisotropy in electrical and optoelectronic properties, and excellent moisture resistance. However, their asymmetrical and nonlinear conductance changes still limit the efficiency of training and accuracy of inference. Here we achieve highly linear and symmetrical conductance changes in Dion-Jacobson 2D perovskites. We further build a 7 × 7 crossbar array based on analogue perovskite synapses, achieving a high device yield, low variation with synaptic weight storing capability, multi-level analogue states with long retention, and moisture stability over 7 months. We explore the potential of such devices in large-scale image inference via simulations and show an accuracy within 0.08% of the theoretical limit. The excellent device performance is attributed to the elimination of gaps between inorganic layers, allowing the halide vacancies to migrate homogeneously regardless of grain boundaries. This was confirmed by first-principles calculations and experimental analysis.

Alzheimer's disease and other memory disorders in the age of AI: reflection and perspectives on the 120th anniversary of the birth of Dr. John von Neumann

Two themes are coming to the forefront in this decade: Cognitive impairment of an aging population and the quantum leap in developing artificial intelligence (AI). Both can be described as growing exponentially and presenting serious challenges. Although many questions have been addressed about the dangers of AI, we want to go beyond the fearful aspects of this topic and focus on the possible contribution of AI to solve the problem of chronic disorders of the elderly leading to cognitive impairment, like Alzheimer's disease, Parkinson's disease, and Lewy body dementia. Our second goal is to look at the ways in which modern neuroscience can influence the future design of computers and the development of AI. We wish to honor the memory of Dr. John von Neumann, who came up with many breakthrough details of the first electronic computer. Remarkably, Dr. von Neumann dedicated his last book to the comparison of the human brain and the computer as it stood in those years of the mid-1950s. We will point out how his ideas are more relevant than ever in the age of supercomputers, AI and brain implants.

Reliable and Robust Two-Dimensional Perovskite Memristors for Flexible-Resistive Random-Access Memory Array

Two-dimensional (2D) halide perovskites have become a promising class of memristive materials due to their low power consumption, compositional versatility, and microstructural anisotropy in electronics. However, implementing high-performance resistive random-access memory requires a higher reliability and moisture resistance. To address these issues, component studies and attempts to improve the phase stability have been reported but have not been able to achieve sufficient reliability. Here, highly textured thin films grown perpendicular to the substrate in Ruddlesden-Popper 2D perovskites exhibited highly stable and reliable binary memory performance. We further built a flexible crossbar array to verify data storage capability, achieving a high device yield, robust endurance, long retention, reliability to operate under bending conditions, and moisture stability over a year. These device performances are attributed to preformed vertically oriented nanocrystals that allow the conductive filaments to operate reliably. Our finding provides the material design strategy that can be extended to the development of semiconductor materials for next-generation memory devices.

Self-Induced Light Emission in Solid-State Memristors Replicates Neuronal Biophotons

Key neuronal functions have been successfully replicated in various hardware systems. Noticeable examples are neuronal networks constructed from memristors, which emulate complex electrochemical biological dynamics such as the efficacy and plasticity of a neuron. Neurons are highly active cells, communicating with chemical and electrical stimuli, but also emit light. These so-called biophotons are suspected to be a complementary vehicle to transport information across the brain. Here, we show that a memristor also releases photons during its operation akin to the production of neuronal light. Critical attributes of biophotons, such as self-generation, stochasticity, spectral coverage, sparsity, and correlation with the neuron's electrical activity, are replicated by our solid-state approach. Importantly, our time-resolved analysis of the correlated current transport and photon activity shows that emission takes place within a nanometer-sized active area and relies on electrically induced single-to-few active electroluminescent centers excited with moderate voltage (<3 V). Our findings further extend the emulating capability of a memristor to encompass neuronal optical activity and allow to construct memristive atomic-scale devices capable of handling simultaneously electrons and photons as information carriers.

Advances on MXene-Based Memristors for Neuromorphic Computing: A Review on Synthesis, Mechanisms, and Future Directions

Neuromorphic computing seeks to replicate the capabilities of parallel processing, progressive learning, and inference while retaining low power consumption by drawing inspiration from the human brain. By further overcoming the constraints imposed by the traditional von Neumann architecture, this innovative approach has the potential to revolutionize modern computing systems. Memristors have emerged as a solution to implement neuromorphic computing in hardware, with research based on developing functional materials for resistive switching performance enhancement. Recently, two-dimensional MXenes, a family of transition metal carbides, nitrides, and carbonitrides, have begun to be integrated into these devices to achieve synaptic emulation. MXene-based memristors have already demonstrated diverse neuromorphic characteristics while enhancing the stability and reducing power consumption. The possibility of changing the physicochemical properties through modifications of the surface terminations, bandgap, interlayer spacing, and oxidation for each existing MXene makes them very promising. Here, recent advancements in MXene synthesis, device fabrication, and characterization of MXene-based neuromorphic artificial synapses are discussed. Then, we focus on understanding the resistive switching mechanisms and how they connect with theoretical and experimental data, along with the innovations made during the fabrication process. Additionally, we provide an in-depth review of the neuromorphic performance, making a connection with the resistive switching mechanism, along with a compendium of each relevant performance factor for nonvolatile and volatile applications. Finally, we state the remaining challenges in MXene-based devices for artificial synapses and the next steps that could be taken for future development.

Investigation of Resistance Switching and Synaptic Properties of VO x for Neuromorphic Applications

The taking run on artificial intelligence in the last decades is based on the von Neumann architecture where memory and computation units are separately located from each other. This configuration causes a large amount of energy and time to be dissipated during data transfer between these two units, in contrast to synapses in biological neurons. A new paradigm has been proposed inspired by biological neurons in human brains, known as neuromorphic computing. Due to the unusual current-voltage characteristic of memristor devices such as pinched hysteresis loops, memristors are considered a key element of neuromorphic architecture. In this study, we report the basic current-voltage characteristic of the memristor devices in the form of Si/SiO2/Pt(30 nm)/VO x (3, 13, 25 nm)/Pt (30 nm) sandwich structure. Synaptic functions such as spike-time-dependent plasticity (STDP), paired-pulse facilitation (PPF), long-term potentiation (LTP), and long-term depression (LTD) of memristor devices were examined in detail. The oxide layer VO x has been grown by using the VO2 target in a pulsed laser deposition (PLD) chamber. The composition and oxidation states of the oxide layer were examined using the X-ray photoelectron spectroscopy (XPS) technique. The status of oxygen vacancies, which play an active role in the operation of the devices, was examined with a photoluminescence (PL) technique. The experimental results showed that the thickness of the oxide layer can significantly influence the synaptic and resistive switching properties of the devices.

Recent Advance in Synaptic Plasticity Modulation Techniques for Neuromorphic Applications

Manipulating the expression of synaptic plasticity of neuromorphic devices provides fascinating opportunities to develop hardware platforms for artificial intelligence. However, great efforts have been devoted to exploring biomimetic mechanisms of plasticity simulation in the last few years. Recent progress in various plasticity modulation techniques has pushed the research of synaptic electronics from static plasticity simulation to dynamic plasticity modulation, improving the accuracy of neuromorphic computing and providing strategies for implementing neuromorphic sensing functions. Herein, several fascinating strategies for synaptic plasticity modulation through chemical techniques, device structure design, and physical signal sensing are reviewed. For chemical techniques, the underlying mechanisms for the modification of functional materials were clarified and its effect on the expression of synaptic plasticity was also highlighted. Based on device structure design, the reconfigurable operation of neuromorphic devices was well demonstrated to achieve programmable neuromorphic functions. Besides, integrating the sensory units with neuromorphic processing circuits paved a new way to achieve human-like intelligent perception under the modulation of physical signals such as light, strain, and temperature. Finally, considering that the relevant technology is still in the basic exploration stage, some prospects or development suggestions are put forward to promote the development of neuromorphic devices.

Control-Etched Ti3C2Tx MXene Nanosheets for a Low-Voltage-Operating Flexible Memristor for Efficient Neuromorphic Computation

Hardware neural networks with mechanical flexibility are promising next-generation computing systems for smart wearable electronics. Overcoming the challenge of developing a fully synaptic plastic network, we demonstrate a low-operating-voltage PET/ITO/p-MXene/Ag flexible memristor device by controlling the etching of aluminum metal ions in Ti3C2Tx MXene. The presence of a small fraction of Al ions in partially etched MXene (p-Ti3C2Tx) significantly suppresses the operating voltage to 1 V compared to 7 V from fully Al etched MXene (f-Ti3C2Tx)-based devices. Former devices exhibit excellent non-volatile data storage properties, with a robust ∼103 ON/OFF ratio, high endurance of ∼104 cycles, multilevel resistance states, and long data retention measured up to ∼106 s. High mechanical stability up to ∼73° bending angle and environmental robustness are confirmed with consistent switching characteristics under increasing temperature and humid conditions. Furthermore, a p-Ti3C2Tx MXene memristor is employed to mimic the biological synapse by measuring the learning-forgetting pattern for ∼104 cycles as potentiation and depression. Spike time-dependent plasticity (STDP) based on Hebb's Learning rules is also successfully demonstrated. Moreover, a remarkable accuracy of ∼95% in recognizing modified patterns from the National Institute of Standards and Technology (MNIST) data set with just 29 training epochs is achieved in simulation. Ultimately, our findings underscore the potential of MXene-based flexible memristor devices as versatile components for data storage and neuromorphic computing.

Halide Perovskites-Based Diffusive Memristors for Artificial Mechano-Nociceptive System

Numerous efforts for emulating organ systems comprised of multiple functional units have driven substantial advancements in bio-realistic electronics and systems. The resistance change behavior observed in diffusive memristors shares similarities with the potential change in biological neurons. Here, the diffusive threshold switching phenomenon in Ag-incorporated organometallic halide perovskites is utilized to demonstrate the functions of afferent neurons. Halide perovskites-based diffusive memristors show a low threshold voltage of ≈0.2 V with little variation, attributed to the facile migration of Ag ions uniformly dispersed within the halide matrix. Based on the reversible and reliable volatile threshold switching, the memristors successfully demonstrate fundamental nociceptive functions including threshold firing, relaxation, and sensitization. Furthermore, to replicate the biological mechano-nociceptive phenomenon at a system level, an artificial mechano-nociceptive system is built by integrating a diffusive memristor with a force-sensing resistor. The presented system is capable of detecting and discerning the detrimental impact caused by a heavy steel ball, effectively exhibiting the corresponding sensitization response. By further extending the single nociceptive system into a 5 × 5 array, successful stereoscopic nociception of uneven impulses is achieved in the artificial skin system through array-scale sensitization. These results represent significant progress in the field of bio-inspired electronics and systems.

Enhanced electrical and magnetic properties of (Co, Yb) co-doped ZnO memristor for neuromorphic computing

We investigate the morphological, electrical, magnetic, and resistive switching properties of (Co, Yb) co-ZnO for neuromorphic computing. By using hydrothermal synthesized nanoparticles and their corresponding sputtering target, we introduce Co and Yb into the ZnO structure, leading to increased oxygen vacancies and grain volume, indicating grain growth. This growth reduces grain boundaries, enhancing electrical conductivity and room-temperature ferromagnetism in Co and Yb-doped ZnO nanoparticles. We present a sputter-grown memristor with a (Co, Yb) co-ZnO layer between Au electrodes. Characterization confirms the ZnO layer's presence and 100 nm-thick Au electrodes. The memristor exhibits repeatable analog resistance switching, allowing manipulation of conductance between low and high resistance states. Statistical endurance tests show stable resistive switching with minimal dispersion over 100 pulse cycles at room temperature. Retention properties of the current states are maintained for up to 1000 seconds, demonstrating excellent thermal stability. A physical model explains the switching mechanism, involving Au ion migration during "set" and filament disruption during "reset." Current-voltage curves suggest space-charge limited current, emphasizing conductive filament formation. All these results shows good electronic devices and systems towards neuromorphic computing.

Reconfigurable Neuromorphic Computing: Materials, Devices, and Integration

Neuromorphic computing has been attracting ever-increasing attention due to superior energy efficiency, with great promise to promote the next wave of artificial general intelligence in the post-Moore era. Current approaches are, however, broadly designed for stationary and unitary assignments, thus encountering reluctant interconnections, power consumption, and data-intensive computing in that domain. Reconfigurable neuromorphic computing, an on-demand paradigm inspired by the inherent programmability of brain, can maximally reallocate finite resources to perform the proliferation of reproducibly brain-inspired functions, highlighting a disruptive framework for bridging the gap between different primitives. Although relevant research has flourished in diverse materials and devices with novel mechanisms and architectures, a precise overview remains blank and urgently desirable. Herein, the recent strides along this pursuit are systematically reviewed from material, device, and integration perspectives. At the material and device level, one comprehensively conclude the dominant mechanisms for reconfigurability, categorized into ion migration, carrier migration, phase transition, spintronics, and photonics. Integration-level developments for reconfigurable neuromorphic computing are also exhibited. Finally, a perspective on the future challenges for reconfigurable neuromorphic computing is discussed, definitely expanding its horizon for scientific communities.

Harnessing the Metal-Insulator Transition of VO2 in Neuromorphic Computing

Future-generation neuromorphic computing seeks to overcome the limitations of von Neumann architectures by colocating logic and memory functions, thereby emulating the function of neurons and synapses in the human brain. Despite remarkable demonstrations of high-fidelity neuronal emulation, the predictive design of neuromorphic circuits starting from knowledge of material transformations remains challenging. VO2 is an attractive candidate since it manifests a near-room-temperature, discontinuous, and hysteretic metal-insulator transition. The transition provides a nonlinear dynamical response to input signals, as needed to construct neuronal circuit elements. Strategies for tuning the transformation characteristics of VO2 based on modification of material properties, interfacial structure, and field couplings, are discussed. Dynamical modulation of transformation characteristics through in situ processing is discussed as a means of imbuing synaptic function. Mechanistic understanding of site-selective modification; external, epitaxial, and chemical strain; defect dynamics; and interfacial field coupling in modifying local atomistic structure, the implications therein for electronic structure, and ultimately, the tuning of transformation characteristics, is emphasized. Opportunities are highlighted for inverse design and for using design principles related to thermodynamics and kinetics of electronic transitions learned from VO2 to inform the design of new Mott materials, as well as to go beyond energy-efficient computation to manifest intelligence.

Sprayed FeWO4 thin film-based memristive device with negative differential resistance effect for non-volatile memory and synaptic learning applications

Resistive switching (RS) memories have attracted great attention as promising solutions to next-generation non-volatile memories and computing technologies because of their simple device configuration, high on/off ratio, low power consumption, fast switching, long retention, and significant cyclic stability. In this work, uniform and adherent iron tungstate (FeWO₄) thin films were synthesized by the spray pyrolysis method with various precursor solution volumes, and these were tested as a switching layer for the fabrication of Ag/FWO/FTO memristive devices. The detailed structural investigation was done through various analytical and physio-chemical characterizations viz. X-ray diffraction (XRD) and its Rietveld refinement, Raman spectroscopy, and X-ray photoelectron spectroscopy (XPS) techniques. The results reveal the pure and single-phase FeWO₄ compound thin film formation. Surface morphological study shows the spherical particle formation having a diameter in the range of 20 to 40 nm. The RS characteristics of the Ag/FWO/FTO memristive device demonstrate non-volatile memory characteristics with significant endurance and retention properties. Interestingly, the memory devices show stable and reproducible negative differential resistance (NDR) effects. The in-depth statistical analysis suggests the good operational uniformity of the device. Moreover, the switching voltages of the Ag/FWO/FTO memristive device were modeled using the time series analysis technique by utilizing Holt's Winter Exponential Smoothing (HWES) approach. Additionally, the device mimics bio-synaptic properties such as potentiation/depression, excitatory post-synaptic current (EPSC), and spike-timing-dependent plasticity (STDP) learning rules. For the present device, the space-charge-limited current (SCLC) and trap-controlled-SCLC effects dominated during positive and negative bias I-V characteristics, respectively. The RS mechanism dominated in the low resistance state (LRS), and the high resistance state (HRS) was explained based on the formation and rupture of conductive filament composed of Ag ions and oxygen vacancies. This work demonstrates the RS in the metal tungstate-based memristive devices and demonstrates a low-cost approach for fabricating memristive devices.

Second-order associative memory circuit hardware implemented by the evolution from battery-like capacitance to resistive switching memory

Memristor-based Pavlov associative memory circuit presented today only realizes the simple condition reflex process. The secondary condition reflex endows the simple condition reflex process with more bionic, but it is only demonstrated in design and involves the large number of redundant circuits. A FeOx-based memristor exhibits an evolution process from battery-like capacitance (BLC) state to resistive switching (RS) memory as the I-V sweeping increase. The BLC is triggered by the active metal ion and hydroxide ion originated from water molecule splitting at different interfaces, while the RS memory behavior is dominated by the diffusion and migration of ion in the FeOx switching function layer. The evolution processes share the nearly same biophysical mechanism with the second-order conditioning. It enables a hardware-implemented second-order associative memory circuit to be feasible and simple. This work provides a novel path to realize the associative memory circuit with the second-order conditioning at hardware level.

Reliable multilevel memristive neuromorphic devices based on amorphous matrix via quasi-1D filament confinement and buffer layer

Conductive-bridging random access memory (CBRAM) has garnered attention as a building block of non-von Neumann architectures because of scalability and parallel processing on the crossbar array. To integrate CBRAM into the back-end-of-line (BEOL) process, amorphous switching materials have been investigated for practical usage. However, both the inherent randomness of filaments and disorders of amorphous material lead to poor reliability. In this study, a highly reliable nanoporous-defective bottom layer (NP-DBL) structure based on amorphous TiO₂ is demonstrated (Ag/a-TiO₂/a-TiO_x/p-Si). The stoichiometries of DBL and the pore size can be manipulated to achieve the analog conductance updates and multilevel conductance by 300 states with 1.3% variation, and 10 levels, respectively. Compared with nonporous TiO₂ CBRAM, endurance, retention, and uniformity can be improved by 10⁶ pulses, 28 days at 85°C, and 6.7 times, respectively. These results suggest even amorphous-based systems, elaborately tuned structural variables, can help design more reliable CBRAMs.

Dynamic Analysis of the Switched-Inductor Buck-Boost Converter Based on the Memristor

The direct current (DC)–DC converter presents abundant nonlinear phenomena, such as periodic bifurcation and chaotic motion, under certain conditions. For a switched-inductor buck-boost (SIBB) converter with the memristive load, this paper constructs its state equation model under two operating statuses, investigates its chaotic dynamic characteristics, and draws and analyzes the bifurcation diagrams of the inductive current and phase portraits, under some parameter changing by the MATLAB simulation based on the state equation. Then, by applying certain minor perturbations to parameters, the chaotic phenomenon suppression method is explored by controlling peak current in continuous current mode (CCM) to keep the converter run normally. Finally, the power simulation (PSIM) verifies that the waveforms and the phase portraits controlling the corresponding parameters are consistent with those of the MATLAB simulation.