PHYSICAL REVIEW E 85, 041918 (2012)

Biomimetic model of the outer plexiform layer by incorporating memristive devices

A. Gelencser,1,2,* T. Prodromakis,2,† C. Toumazou,2 and T. Roska1

1Interdisciplinary Technical Sciences Doctoral School, Pazmany Peter Catholic University, 1088 Budapest, Hungary2Centre for Bio-inspired Technology, Department of Electrical and Electronic Engineering, Imperial College London,

London SW7 2AZ, United Kingdom(Received 8 December 2011; revised manuscript received 13 March 2012; published 26 April 2012)

In this paper we present a biorealistic model for the first part of the early vision of processing by incorporatingmemristive nanodevices. The architecture of the proposed network is based on the organization and functioningof the outer plexiform layer (OPL) in the vertebrate retina. We demonstrate that memristive devices are indeed avaluable building block for neuromorphic architectures, as their highly nonlinear and adaptive response could beexploited for establishing ultradense networks with dynamics similar to that of their biological counterparts. Weparticularly show that hexagonal memristive grids can be employed for faithfully emulating the smoothing effectoccurring in the OPL to enhance the dynamic range of the system. In addition, we employ a memristor-basedthresholding scheme for detecting the edges of grayscale images, while the proposed system is also evaluated forits adaptation and fault tolerance capacity against different light or noise conditions as well as its distinct deviceyields.

DOI: 10.1103/PhysRevE.85.041918 PACS number(s): 87.18.Sn, 02.70.−c

I. INTRODUCTION

Over the past years, the performance and efficiency of bio-logical systems have inspired many researchers and engineers,giving birth to the emerging fields of biomimetics [1] andbioinspiration [2]. The human retina is anything but a simplepassive relay station, as it pre-processes and compresses allsensed information through an immensely complex neuronalnetwork that on average contains 4.6 million cones, 92 millionrods [3], and 1 million ganglion cells [4]. If we take intoaccount the remaining bipolar, horizontal, and amacrine cellsand the fact that they are highly interconnected, we get a par-allel network of grand complexity. This complexity is furtherelucidated in recent studies where evidence is provided that atleast 10 parallel signals arise from a single visual point [5] andwe get a much more accurate view of the functioning of thevertebrate retina. In fact, nowadays it is believed that althoughbiological systems are based on relatively primitive elements,it is this naturally occurring interconnection complexity thatfacilitates higher order functioning.

In this paper we demonstrate the potential of emergingnanoscale elements as synapse emulators for mimickingcomplex biological functions. We specifically focus on theconnection of the sensory and consecutive system of the retina,which is the first and common step in the visual informationflow. These connections take place on the outer plexiform layer(OPL) in the vertebrate retina; they form a highly dynamicsystem that enables the smoothing of optical inputs and thuscatalyzes the enhancement of the retina’s dynamic range, whilethe different parallel channels emerge after this point.

Our approach is distinct from the already existing modelsdescribed in Refs. [6–22]. The use of time-variant resistorsto model a biological network raises principal and practicalconcerns. Our approach alleviates these issues by employingthe latest biological knowledge [5] and an emerging nanoscale

*gelencser.andras@itk.ppke.hu†t.prodromakis@imperial.ac.uk

device that is used as a more adequate synapse emulator [23],the memristor. This device exhibits a highly nonlinear dynamicbehavior, which, along its infinitesimal dimensions, serves asan excellent building block for facilitating practical realiza-tions of the highly complex synaptic networks constitutingthe OPL. We further expand this approach by utilizing amemristive-based thresholding scheme for performing edgedetection. Finally, we demonstrate that this platform exhibitsattributes similar to those of naturally occurring systems suchas noise resilience, self-adaptation, and fault tolerance.

II. METHODS

The decomposition of the employed images into equivalentbiasing voltages, as well as the hexagonal and rectangularimplementations of the memristive grids, was performedin MATLAB. The constructed systems were simulated inPSPICE and relatively small networks were utilized to min-imize the computation requirements. In the case of largememristive networks, Biolek’s memristor spice model [41]was utilized, whereas for single-node simulations we usedProdromakis’s model [42], as it also considers the nonlineardopant kinetics of the employed elements. For ultradenseimplementations the model described in Ref. [53] could alsobe adapted to relax the excessive computation requirements.

III. BIOLOGICAL NETWORK

Neurobiologists have identified five major classes of neu-rons, which, in the mammalian retina, are divided into about60 anatomically different types of cells [24], distributed acrossseven distinct layers. We particularly focus on the OPL and,eminently, on the synaptic connections and correspondingsignal propagation through them.

A. The mammalian retina

Figure 1 shows a schematic cross section of the mammalianretina. Five major neuron types are denoted: photoreceptors,bipolar, ganglion, horizontal, and amacrine cells. Manifold

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GELENCSER, PRODROMAKIS, TOUMAZOU, AND ROSKA PHYSICAL REVIEW E 85, 041918 (2012)

FIG. 1. (Color online) Conceptual representation of the mammalian retina. To avoid confusion, only five major cell types are depicted, withthe main interconnection between them organized across the seven layers. The visual information flows through the retina via bipolar cells andtravels to the thalamus via the axons of ganglion cells. Lateral interconnections also exist between the bipolar and the horizontal cells in theOPL and the bipolar and the amacrine cells in the IPL.

approaches exist for studying the retina, nevertheless, in thispaper we tend to follow the information flow. In the verticaldirection, photoreceptors detect the incidental light throughthe night vision rods and the cones that are responsiblefor daylight and color vision. These are in direct contactwith bipolar cells, which relay the visual information inthe form of membrane depolarization or hyperpolarization tothe ganglion cells. The axons of the latter cells propagate thepreprocessed visual information toward the brain, essentiallyforming the optic nerve, with on cells depolarizing and offcells hyperpolarizing in accordance with the correspondingimpulses.

On the other hand, the information flow in the lateraldirection is mainly mediated by the horizontal and amacrinecells, as shown in Fig. 1. The horizontal cells build synapticconnections with the photoreceptors and the input of bipolarcells in the OPL. At the same time, the interconnections amongthe output of bipolar cells, amacrine cells, and retinal ganglioncells take place in the inner plexiform layer (IPL).

Cells in the retina use two modes to transmit theirsignals. Photoreceptors, bipolar and horizontal cells, and someamacrine cells, when exposed to a stimulus, produce gradedchanges in their membrane potential [25,26], permitting a fastand continuous signal flow. On the other hand, all ganglioncells and some amacrine cells produce action potentials [27].And although the attainable propagation velocity is muchslower, it is rather robust, allowing the information to travelover long distances.

B. Outer plexiform layer

Horizontal cells play an important role in synaptic in-teractions within the OPL. These cells are electrically in-terconnected through gap junctions, as shown in Fig. 2(a),forming a complex lateral network. These cells do not justreceive information from the photoreceptors; a feedbackmechanism [28] exists that pools the information from the

horizontal cell network over a wide spatial area of the OPLthat adjusts the photoreceptors’ gain. In the case where all thephotoreceptor cells are evenly illuminated, then the horizontalcells will be accordingly stimulated evenly and all nodes inthe OPL will essentially be equipotential. Therefore thereis no current flowing between them. If there is, however, asmall spotlight stimulus, then the membrane potential of theunaffected horizontal cells will differ, allowing the conductionof electrical currents between these cells and essentiallyshunting the responses of the illuminated ones. This hastwo main ramifications: first, a photoreceptor’s response isproportional to the ratio between its photo input and the localaverage of the surrounding region of the retina; and second,this local circuit provides bipolar cells with a center-surroundorganization.

The current flow between the horizontal cells diminishesthe gradient between the adjacent photo inputs and essentiallyfacilitates local Gaussian filtering to smooth the input image.The smoothed visual signal then splits into two separatechannels of information [29], detecting the lighter and thedarker objects of the background via on and off pathways. Thecombination of these pathways creates simultaneous contrast(light and dark boundaries) of visual objects (center-surroundreceptive field) in deeper layers [30]. Amacrine cells in theIPL play a critical role in detecting the speed and orientationof motion [31,32] as well as determining spatial color visionor nocturnal vision [33].

We scrutinize the architecture of synaptic interconnectionsin the OPL to create our model. These interconnectionsoccur in the lower part of the cone (the cone pedicle) andare often called triads. The central element of a triad is aninvaginating bipolar dendrite, originating from an on bipolarcell, while the lateral elements are invaginating horizontalcell dendrites or axons. The triad also comprises bipolardendrites, originating from off bipolar cells, that establish basaljunctions at the bottom of the cone pedicle [34], as depictedin Fig. 2(b).

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FIG. 2. (Color online) (a) Illustration of the OPL architecture,with on and off bipolar cells contracted for simplicity. In theOPL, photoreceptor, horizontal, and bipolar cells are interconnectedthrough so-called triads. Horizontal cells are interlinked through gapjunctions, forming an extended lateral network. (b) A triad from theOPL is formed by the cone pedicle, on and off bipolar cell dendrites,and horizontal cell dendrites or axons. The incoming visual signalis modulated at these nodes through the interconnecting neurons,essentially establishing the first layer of visual processing.

IV. RETINOMORPHIC MEMRISTIVE GRIDS

Photoreceptor cells of more developed species are typicallytessellated in an hexagonal manner, as can be seen in Fig. 2of Ref. [35], with the layers underneath following this samehexagonal arrangement to ensure an effective spatial usage[36]. Modeling the triads occurring in the OPL necessitatesthe employment of elements with infinitesimal dimensions thatare capable of dynamically adapting their weight in accordancewith the current flowing through these. Here we describe howthe attributes of the recently discovered memristor can beutilized in this particular bio-inspired system.

A. Memristive components

The recent nanoscale implementation of the memristor byHewlett-Packard (HP) [37], particularly its application as asynapse emulating device (demonstrated in Ref. [23] and alsoRefs. [38] and [39]), has created a lot of excitement withinthe neuromorphic community. The device was theoreticallypredicted in 1971 by L. Chua in his seminal paper [40] andwas postulated as a variable resistor with a state-dependentdynamic response,

v(t) = M(q)i(t), (1)

where M(q) is the charge-dependent resistance, or, moreappropriately, the memristance; v(t) is the applied bias voltage;

FIG. 3. Schematic illustration of the memristive fuse. Two iden-tical memristors of opposing polarities are connected serially.

and i(t) is the current flowing through the memristor. At thelowest approximation where a linear ionic dopand drift isconsidered, the memristance will take any value between twolimiting resistive states, a high-resistive state (HRS) Roff anda low-resistive state (LRS) Ron, described as

M(q) = Ronw(t)

D+ Roff

(1 − w(t)

D

), (2)

where w(t) is the width of the doped region and D is thetotal thickness of the bilayer or active region of the device.Ron is essentially the net resistance of the device when theactive region is completely doped (w = D), and Roff when theopposite occurs (w → 0). More information on the intrinsicproperties of the models we employed can be found inRefs. [41] and [42] (also see Sec. II).

The strength (conductance) modulation of a single mem-ristive element depends not only on the charge that has flownthrough the device but also on the direction in which it hasflown. This polarity dependence is, however, an undesirablefeature for the proposed OPL model, and a typical way aroundthis is the utilization of two devices in series with opposingpolarities, as shown in Fig. 3, also denoted a memristive fuse[43]. The beauty of this combination is that it overall maintainsthe nonlinear relationship between the time integrals of currentand voltage, without demonstrating any polarity dependence.This is a very useful characteristic for preserving the edgeson the input image; more details are given in Sec. V A.

B. Biomimetic OPL triads

Although we briefly describe the function of horizontal,bipolar, and photoreceptor cells, in this work we do notexplicitly model these neurons. Instead, this paper focuses onthe synaptic interconnections of such neurons and how theseadapt to different stimuli for inherently performing smoothing,as well as detecting edges. Any visual input is captured via thephotoreceptor cells of the retina. As this is out of the scope ofthis paper, we opted not to emulate the exact functioning ofthese receptors but, rather, the effect in translating any lightstimuli into an appropriate current bias. We thus represent thecells’ signaling with equivalent voltage sources to bias theunderlying memristive network. Clearly, this approach is onlyapplicable for colorless image inputs.

As illustrated in Fig. 4, every voltage source connects via aresistor to a discrete node of an hexagonal memristive network.The value of this resistance is comparable to that of theinitial memristive state (1 k�), since the lower this resistanceis, the larger the current in the underlying node will be. Ittherefore serves as a control parameter of the time evolution

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FIG. 4. (Color online) Closeup view of a single node of theproposed grid. Voltage sources and serial resistors are representationsof the output signals of photoreceptors, when subjected to anoptical stimulus. Every node establishes a triad through hexagonallyinterconnected memristive fuses for imitating the dendrites or axonsof horizontal cells, one memristor that represents the dendrite ofbipolar cells, and bias sources and resistors at the input stage thatdenote the cone pedicle.

of our system. These nodes are essentially representationsof the triads comprising the OPL [44], with the memristivefuses employed to emulate the synaptic interconnects of thehorizontal cell network. In a similar manner to Fig. 2(b), singlememristors were used to emulate the bipolar cells’ dendrites,while the biasing provided by the voltage sources and theirseries resistors resembles the activity occurring at the conepedicles.

Here we have ascertained an hexagonal topology forcomplying with the biological counterpart system. Every nodewithin the grid is linked with six neighboring nodes throughmemristive fuses. This approach is essentially similar to theone employed by C. Mead in Ref. [6], with our system beingdistinct in that we employed nonlinear memristive devicesinstead of linear resistors. This option facilitates the proposedmodel with an inherent local Gaussian filtering functioning toany input image.

C. Localized smoothing

In order to demonstrate the network’s adaptation capacitywe first examine the response of a relatively small networkwhile subjected to an optical stimuli comprising of a largecontrast. This is achieved by employing a central biasingscheme where only a center node is excited. This stimulusis illustrated in the inset in Fig. 5(a), with the center nodebeing biased with a dc voltage of 30 mV (black pixel), whilethe neighboring pixels were all grounded. This essentiallyestablishes a large potential difference across the devicesthat are affiliated with this particular node [denoted as greencircles in inset A in Fig. 5(b)]. Clearly, all devices that linkthe nodes adjacent to the center node are equipotential andthus no memristance modulation is expected to occur at anytime. Nevertheless, small static potentials are established onthe remaining nodes due to the time evolution of the system.All memristive fuses that are subjected to similar biasingconditions are represented by the same color and mark, asillustrated in Fig. 5(b). The larger the current flow through theadjacent nodes, the higher the state modulation rate.

FIG. 5. (Color online) Effect of single-node biasing with a largecontrast. (a) Static voltage distribution across center-surroundingnodes of the OPL network after t = 30 s. Inset: Biasing scheme.(b) Memristance transient of OPL memristive fuses. Time evolutionof all memristive fuse states affiliated with this node. Inset: Distinctcolors and marks correspond to the neighboring devices to the centernode.

This simple network essentially adapts to the applied biasthrough the establishment of distinct time-dependent currentsthat modulate the corresponding states of the devices, as illus-trated in Fig. 5(b). In turn, the corresponding static potentialsof the nodes evolve after t = 30 s in the distribution shown inFig. 5(a), with the central, neighboring, and peripheral nodes’potentials being 21.9, 0.6, and 0.4 mV, respectively. The timeevolution of the nodes’ potential is further illustrated in S.1(see Supplemental Material [50]).

Figure 5(b) shows that the yellow (upward-pointingtriangle) and purple (triangle) fuses will saturate like thegreen (circle) fuses with time, but to achieve this state theydefinitely need much more time. This occurs because thereis still sufficient current flow through the green (circle) fuses,after they saturate, and a part of this current reaches the yellow(upward-pointing triangle) and purple (triangle) fuses, whichin turn modulates their memristance until they are saturated.The distinct resistive states are indeed important for detectingthe edges on the image, however, we are mainly concernedwith how quickly the saturation occurs. If we apply theappropriate threshold(s), we can obtain the eligible edges ofthe image at every given time. Of course, it will be increasinglymore complicated to distinguish the edge pixels, as the fusesbecome saturated.

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The response of the center node can be analyzed byobserving the current flows through the adjacent memristivefuses. Node N is surrounded by six adjacent nodes and links tothese via memristive fuses M1, M2, M3, M4, M5, and M6. Theinput voltage bias Vp(i,j ) is conveyed to a stimulating currentthrough a series resistor Ri,j . Let us consider the voltage atnode N to be Vout, while the corresponding voltages at the sixneighboring nodes are V1, V2, V3, V4, V5, and V6, respectively.Kirchhoff’s current law reads

Iin =6∑

i=1

Ii, (3)

and according to Ohm’s law,

Vp(i,j ) − Vout

Ri,j

=6∑

i=1

Vout − Vi

Mi

. (4)

Rearranging Eq. (4) yields

Vout =Vp(i,j ) + Ri,j

∑6i=1

Vi

Mi

1 + Ri,j

∑6i=1

1Mi

. (5)

Therefore, the corresponding output voltage Vout at node Nat any point in time is related to the weighted average of thepotentials established in the surrounding nodes.

V. BIOMIMETIC OUTER PLEXIFORM LAYER

The typical resting potential of neuron cells is −65 mV,with the photoreceptors’ resting membrane potential beingabout −40 mV. When some light stimulus is present, thesecells use the absorbed photons’ energy to hyperpolarize theirinherent membrane potential; i.e., the light intensity of anoptical stimulus will cause photoreceptor cells to hyperpolarizein relation to the light intensity. On the other hand, a darkintensity will have an opposing effect, causing the cells todepolarize, as illustrated in Fig. 6(a).

In this approach the stimulating potential resulting fromphotoreceptor cells being either hyperpolarized or depolarizedwas modulated arbitrarily by the employed voltage sourcesVp(i,j ). Every pixel of an input image was represented bya corresponding voltage level, as depicted in the inset inFig. 6(a). The pixel intensity of all stimuli was set on a 16-level grayscale, where 0 represents the highest possible lightintensity and 15 corresponds to a dark current. Since mostlyall images exhibit a rectangular format, we defined an effectivemapping to accommodate an hexagonal topology. Consideringevery even row of images to be shifted to the right by ahalf-pixel, one can get a pseudohexagonal image [Fig. 6(b)]where every pixel has six neighbors. Following this, all opticalinputs were transcribed into a two-dimensional vector ofbiasing voltages that, through a serial resistor, establish theOPL’s stimulating currents. In a similar manner, the recordedpotentials of the memristive OPL nodes are monitored asthe network adjusts the corresponding memristive weightsand are transcribed back to the matching pixel intensities, toobtain some meaningful data. We should note here that as the

FIG. 6. (Color online) Mapping mechanisms between the inputimage and the OPL model. (a) Simulated photoreceptor depolar-ization due to distinct grayscale levels. Every pixel of an inputimage is represented by a corresponding voltage level. (b) Mappingbetween rectangular image and the hexagonal grid. All images werehandled as pseudohexagonals to assign every pixel to a node of thegrid.

employed depolarization potentials are in the millivolt range,they will provoke a relatively slow state modulation of thememristive network. As a consequence, the system evolvementis better observed in a time period of a few seconds rather thanmilliseconds, which, however, can be adjusted by modifyingthe stimulus amplitudes. At this early stage of processing thereis no need to train any neurons, as the system does not yet knowwhat it is looking at. Clearly, this is not a classical neuralnetwork but, rather, a synaptic network whose dynamics ismodeled via memristive grids [45].

For all cases presented in this paper, all memristivedevices were set to the following parameters: RON = 100 �,ROFF = 16 k�, and Rinit = 200 �, where Rinit is the initialmemristance, and we employ a nonlinear boundary conditiondescribed in Ref. [42]. Finally, throughout all evaluations weutilized a scheme that comprises clear edges along with areasof finer detail and a simplified picture of a Rubik’s Cube thathas clear patterns. In these simulations we biased every nodein the system, the whole network, with a specific input voltageaccording to the corresponding image pixel.

A. Smoothing and local Gaussian filtering

Most of the existing edge-detection schemes demonstratevarious issues when the optical input is distorted [46]. Toaddress these, Gaussian filtering was proposed to suppressnoise through the smoothing (blurring) of the image. A

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one-dimensional Gaussian function can be described as

g(x) = 1√2πσ

e− x2

2σ2 , (6)

where x and y are the distance from the origin and σ is thestandard deviation of the Gaussian distribution, which can alsobe expanded into two dimensions through

g(x,y) = 1

2πσ 2e− x2+y2

2σ2 . (7)

Nevertheless, a uniform Gaussian blur across the wholeimage can cause the displacement of edges, the vanishing ofless intense edges, and the creation of edge artifacts [47,48].The occurrence of this effect, however, is diminishable, in thecase of local Gaussian filtering [49]. In our approach, mem-ristive dynamics are employed to achieve this performanceintrinsically. The filtering variance is dynamically adapted tothe local variance of the image and the smoothing alleviatesany nonuniformities.

If we use an input image to bias the whole network, then thefollowing happens: at the beginning of the simulation everymemristor fuse is set to an LRS. All node currents flow not onlytoward the output nodes, but also to the neighboring nodes,according to the established voltage gradients. Therefore theoutput image, as transcribed by the OPL node potentials, getsvery blurred. The amplitude of the current flowing throughany memristive fuse in the OPL network depends on theweighted sum of the current flow through the neighboringcells. If the potential difference between adjacent nodes ishigh, then the current flow gets higher; if the difference is low,however, then the current flow is significantly less. In otherwords, a clear edge on the input image causes a big intensitygradient and, consequently, a faster memristance modulation.Such memristive fuses drift toward an HRS in a faster mannerthan the neighboring cells and the lateral current flow from thehigh pixel gradient essentially diminishes. In consequence,the edge is preserved, while the smoothing effect is vigorouslydecreased. On the other hand, if the contrast between adjacentpixels is low, the corresponding memristance change will besignificantly slower. In this case, this particular memristivefuse will allow a larger lateral current flow and thus theresulting image will be more homogeneous over this area.Hence, small intensity variations in an image tend to smoothout, meaning that added noise is also annihilated, proving the

feasibility of the proposed memristive-based local Gaussianfiltering.

This is particularly demonstrated through the examplesdepicted in Figs. 7(a) and 7(c), with Fig. 7(a) being the originalinput image and Fig. 7(c) a distorted version of the originalimage with additive white noise with a Gaussian distribution(μ = 0 and σ = 0.3). In both cases the smoothing causedby this network after 30 s is illustrated in Figs. 7(b) and7(d), respectively. These figures depict the static voltagesmeasured at the OPL’s nodes after being transcribed back tothe corresponding grayscale intensities in accordance with thescale shown in the inset in Fig. 6(a). By observation, thetwo figures do not show any considerable difference, andthis was quantified in terms of grade intensity mismatchto be approximately 3%, as shown in Fig. 7(e). In bothcases, the smoothed versions preserved the main edges, whilewherever there was an insignificant intensity gradient or smallintensity variation caused by the addition of noise, the systemsmoothed this out. The smoothing transient of both cases is alsoillustrated dynamically in S.2a and S.2b (see SupplementalMaterial [50]).

B. Edge detection

The intensity contrast between adjacent pixels imposes thebiasing of the underlying memristive fuses with correspondingpotential differences, and as such, an edge can easily bedetected by monitoring the outgoing current flow at the OPLnodes. In the counterpart biological system, this informationis conveyed to the IPL through the dendrites of on/off bipolarcells. It is important to mention that this edge-detectionphenomenon is a part of the early vision processing, whilethe main output of the edge information occurs later in theretina. Here, we utilize two approaches: (1) we employ singlememristors in the output stage of the OPL to facilitate aresistive thresholding scheme, and (2) we monitor the statevariance of the OPL memristive fuses.

At any given time, the relative memristance change, in boththe memristive fuses and the single devices, is a measure of thecurrent flowing through these devices. As the system evolves,the devices associated with nodes that are exposed to largepotentials, i.e., in neighboring pixels at an edge, will drifttoward lower conductive states at a rate set by the overlyingintensity contrast. By monitoring the transient memristance

FIG. 7. Demonstration of local Gaussian filtering with our proposed OPL model. (a) The original input image. (b) The extrapolatedsmoothed version of (a) after t = 30 s. (c) A distorted version of (a); (d) the corresponding smoothed output after t = 30 s. (e) The accentuatedintensity mismatch of the smoothed outputs with versus without distortion.

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FIG. 8. Detection of the edges of the utilized input image shown in (a). Edge detection was achieved via (b) a bipolar threshold schemeat the output of the OPL nodes, with 600 � � MT � 2 k�, and (c) a thresholding scheme at the memristive fuses with MT = 1.6 k�. Theseresults are compared with conventional edge-detection algorithms: (d) Prewitt, (e) Sobel, and (f) Canny.

change of the devices in the OPL output nodes, we associateappropriate thresholding values for defining clear edges. Alldevices falling between these thresholds will thus indicate theexistence of an edge. In the case where Fig. 8(a) is the sourceimage and the threshold is bounded within 600 � � MT �2 k�, the detected edges will correspond to what is shown inFig. 8(b). Clearly, these thresholds could be manually adjustedto attain more or less edge detail.

In the second approach, a more elaborate thresholdingscheme involves the monitoring of the states of all memristivefuses associated with a node. If at least three devices areexceeding a preset threshold, this particular node is thendenoted an edge pixel. Figure 8(c) illustrates the edges as de-tected through this approach for MT = 1.6 k�. SupplementalVideos S.3a and S.3b [50] demonstrate the transient responseof our system; Figs. 7(a) and 7(c) are used to bias thememristive network, respectively. Additionally, as a figure ofmerit, Figs. 8(d)–8(f) illustrate the edges detected by employ-ing conventional algorithms, specifically Prewitt, Sobel, andCanny, respectively.

Similarly, this method was also exploited with a Rubik’sCube image, as shown in Fig. 9(a), with the memristivethreshold being set at MT = 3 k�. In this example, thesmoothing process has caused some distortion of the inhibitedpattern on the front sides of the cube, as this is erroneouslyconsidered to be noise due to the small contrast differenceexisting between these single-pixel lines and their background.As a result, this pattern tends to be smoothed out, asillustrated in Fig. 9(b) as well as in Supplemental VideoS.4 [50]. Nevertheless, our model manages to distinguishthe main edges of the cube, as well as the finer edges thatare inhibited on the top side of the cube. This is clearly

illustrated in Fig. 9(c), where it appears to attain cleareredges compared to the conventional edge detectors Prewitt,Sobel, and Canny, whose results are correspondingly shownin Figs. 9(d)–9(f), respectively. Supplemental Videos S.5a andS.5b [50] demonstrate the transient responses of the system asit detects the edges of the Rubik’s Cube shown in Fig. 9(a) andof a noise version [the input image was distorted as done inFig. 7(c)] of the same, respectively.

C. Adaptation to light conditions

The vertebrate retina is capable of self-adapting to maintainthe retinal response to visual objects approximately the samewhen the level of illumination changes. Here we demonstratethat our model behaves in a similar manner to its biologi-cal counterpart when subjected to distinct light conditions.Figures 10(b) and 10(c) demonstrate that the proposed mem-ristive network is capable of detecting the edges inhibitedin the original image despite the 2× light variance in theoriginal figure. In these examples we have manually adjustedthe memristive threshold to achieve edge detection similarto that of the original system. When the image is brighter,the difference between a contour’s pixels will be relativelyhigher, meaning that there will be more current flowing throughthe memristive devices that correspond to this edge, thustheir state will be altered in a faster manner. In this case,matching the edge-detection performance of our system, asshown in Fig. 9(c), necessitates the use of a higher memristancethreshold, MT L = 6.35 k�.

On the other hand, when the image has a darker tone,the contrast between the pixels defining an edge will be lesssignificant. In consequence, smaller potentials are established

FIG. 9. Detection of the edges of the utilized input image shown in (a). (b) The corresponding smoothed output after t = 30 s. (c) Edgedetection was achieved by applying a thresholding scheme with MT = 3 k�. This result is also compared against conventional edge-detectionalgorithms: (d) Prewitt, (e) Sobel, and (f) Canny.

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FIG. 10. Evaluation of the memristive platform against distinctlight conditions. (a, d) Representations of the employed input image[Fig. 9(a)] in brighter and lighter tones. The detected edges for bothconditions when corresponding memristive thresholds are utilized:(b) MT L = 6.35 k� and (e) MT D = 1.2 k�. Corresponding resultswhen MT L = 3 k� is the same for both cases, after the system hasevolved for (c) tL = 21.8 s and (f) tD = 44.7 s.

across the corresponding memristive fuses and their state willchange in a slower manner. Likewise, the lower memristancethreshold of MT D = 1.2 k� is required to achieve a perfor-mance similar to that shown in Fig. 9(c). Clearly, a lowerthreshold implies that more memristive fuses will exceed thisthreshold at any given time, justifying the detection of thickeredges as illustrated in Fig. 10(e). Supplemental videos S.6a andS.6b illustrate the transients shown in Figs. 10(b) and 10(e),respectively.

Since the relative contrast in the pixels of a “light” and“dark” tone image will be rather similar to that in the originalimage, if one maintains the same threshold (MT = 3 k�) forboth light conditions, the proposed system can, in principle,detect the same edges as shown in Fig. 9(c) (t = 30 s).However, in the case of a lighter environment the system willreach a similar solution at t = 21.8 s, as shown in Fig. 10(c),while in the case of a darker environment the system willrequire double the time (t = 44.7 s) to reach a similar solution[Fig. 10(f)].

D. Fault tolerance

Biological systems depend on rather primitive elementswhose properties often vary randomly. Yet nature is capable ofperforming highly complex functions in a very reliable mannerby employing redundancy. Similarly, solid-state devices, par-ticularly memristive devices of deep submicron dimensions,demonstrate a very poor yield. Given the fact that memristorsare a disruptive technology, the reliability and robustness of thedevices become a significant burden. In this view, we extendour investigation of the effect that defective devices couldpotentially have on our model.

We consider that for a 100% yield, all memristive fusesare reliably set to RON = 100 �, ROFF = 16 k�, and

FIG. 11. (Color online) Test conditions for evaluating the faulttolerance capacity of this model. The distribution of memristiveelements whose initial state (Minit) was altered by (a) 25% and (b)50% of the total memristors in the network is shown.

Rinit = 200 �. In order to test the robustness of our system wemodel different yields, by assigning erratic initial states to anumber of randomly selected memristive elements. This meansthat the conductance of these memristive elements differs fromthe normal one. When a device is considered to be faulty,its RON can vary from 50% to 400% compared to the idealscenario. Similarly, ROFF may vary from 62.5% to 125% andRinit can take any value from 50% to 4000% compared withthe ideal values. Figure 11 shows two circuit maps, where 25%[Fig. 11(a)] and 50% [Fig. 11(b)] of the total memristors inthe network were randomly affected. The employed bluescale(grayscale) mapping corresponds to the randomly distributedinitial states Minit with midblue (midgray) hexagons markingthe unaffected devices, and lighter and darker hexagons

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FIG. 12. Smoothing and edge-detection performance for varyingmemristor yields: (a) ideal scenario, (b) 75% yield, and (c) 50%yield. (d, e) The accentuated difference between the ideal and therandomly affected networks. The corresponding detected edges areillustrated for yields of (f) 100%, (g) 75%, and (h) 50%.

the affected ones. (Striped hexagons symbolize the networknodes.)

When the same optical conditions as in Fig. 9(a) are applied,the OPL will produce a smoothed equivalent as shown inFig. 12(a). When, however, the network’s yield is set to 75%and 50%, the memristive grid acquires an uneven initial weightdistribution that produces the smoothed versions shown inFigs. 12(b) and 12(c), respectively. The relative differencebetween the flawless and the affected memristive fuses isshown in Figs. 12(d) and 12(e), respectively. We can observethat the smoothing of the image will decrease, because of thehigh number of defective memristors. Yet our edge-detectionmethod still holds and is capable of detecting most of thecorrect edges. Supplemental videos S.7a and S.7b demonstratethe time evolution of the network that causes the smoothing,while S.7c and S.7d illustrate dynamically the edges asdetected for a yield of 75% and 50%, respectively [50].

Regardless of the low yield values we tested for, theproposed hexagonal memristive network appears to be pro-ficient in detecting the inhibited edges effectively. However,

when the same conditions are employed in a rectangularmemristive architecture, the results are not as encouraging asin the hexagonal topology. Besides the geometrical advantagethat allows more unit cells to be tessellated per unit area,the hexagonal topology bares two extra interconnections pernode, enhancing the system’s redundancy. Therefore, the localaveraging occurs with two more spatial partners, accountingfor the introduced faults in the device’s characteristics.

Although we have demonstrated the potential of usingmemristive networks to mimic the retina dynamics, physicalimplementations can be rather challenging, requiring theemployment of complex interconnection schemes [51,52] thatcould impose challenging processing (e.g., the use of three-dimensional CMOS). Such demanding interfacing schemesmay hinder the device’s reliability substantially, which is alsothe main reason we investigated the effect of low yields on theproposed application.

VI. CONCLUSIONS

We have presented a biorealistic model of the OPL ofthe vertebrate retina, based on an hexagonal memristive gridimplementation, by exploiting the highly nonlinear dynamicproperties that memristors possess intrinsically. This imple-mentation assists in minimizing the overall complexity ofother previously reported systems, while at the same time itachieves local Gaussian filtering that facilitates an adaptivesmoothing of both distorted and undistorted optical stimuli.Moreover, it was demonstrated that edge detection can beachieved by means of a simple memristor thresholding schemeimplemented at the OPL’s nodes outputs as well as through thecollective evaluation of states of the memristive elements pernode. Both smoothing and edge detection were assessed versusdistinct light conditions and it was shown that the proposedplatform behaves in a manner similar to that of its biologicalcounterpart.

Finally, with yield being considered a very important aspectin deep submicron technologies and, particularly, practicalmemristive implementations, we have demonstrated that theproposed bio-inspired OPL model can effectively manage arelatively large variation in the device’s properties. Here wehave specifically focused on emulating the functioning of theOPL of the retina. Nonetheless, more interesting processestake place beyond this layer, especially in the IPL, and suchgrids could be potentially applied to attain more complexfunctionalities such as the tracking of moving objects.

ACKNOWLEDGMENTS

The authors wish to acknowledge the ERASMUS schemeas well as the financial contribution of Dr. Wilf Corrigan andthe CHIST-ERA net project “Plasticity in NEUral MemrstiveArchitectures.”

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