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Modeling and simulating the adaptive electrical properties of stochastic polymeric 3D

networks

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2013 Modelling Simul. Mater. Sci. Eng. 21 075007

(http://iopscience.iop.org/0965-0393/21/7/075007)

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Modelling Simul. Mater. Sci. Eng. 21 (2013) 075007 (17pp) doi:10.1088/0965-0393/21/7/075007

Modeling and simulating the adaptive electricalproperties of stochastic polymeric 3D networks

R Sigala1,2, A Smerieri3, A Schuz2, P Camorani3 and V Erokhin4

1 Department of Neurology, Charite Universitaetsmedizin, Berlin, Germany Chariteplatz 1,10117 Berlin, Germany2 Department of Physiology of Cognitive Processes, Max Planck Institute for BiologicalCybernetics, Spemannstr. 38 Tuebingen 72076, Germany3 Department of Physics, University of Parma Viale Usberti 7A, 43100 Parma, Italy4 CNR-IMEM and Department of Physics, University of Parma Viale Usberti 7A,43100 Parma, Italy

E-mail: rodrigo.sigala@charite.de, anteo.smerieri@gmail.com and victor.erokhin@fis.unipr.it

Received 1 July 2013, in final form 18 June 2013Published 18 September 2013Online at stacks.iop.org/MSMSE/21/075007

AbstractMemristors are passive two-terminal circuit elements that combine resistanceand memory. Although in theory memristors are a very promising approachto fabricate hardware with adaptive properties, there are only very fewimplementations able to show their basic properties. We recently developedstochastic polymeric matrices with a functionality that evidences the formationof self-assembled three-dimensional (3D) networks of memristors. Wedemonstrated that those networks show the typical hysteretic behavior observedin the ‘one input-one output’ memristive configuration. Interestingly, usingdifferent protocols to electrically stimulate the networks, we also observedthat their adaptive properties are similar to those present in the nervoussystem. Here, we model and simulate the electrical properties of these self-assembled polymeric networks of memristors, the topology of which is definedstochastically. First, we show that the model recreates the hysteretic behaviorobserved in the real experiments. Second, we demonstrate that the networksmodeled indeed have a 3D instead of a planar functionality. Finally, we showthat the adaptive properties of the networks depend on their connectivity pattern.Our model was able to replicate fundamental qualitative behavior of the realorganic 3D memristor networks; yet, through the simulations, we also exploredother interesting properties, such as the relation between connectivity patternsand adaptive properties. Our model and simulations represent an interestingtool to understand the very complex behavior of self-assembled memristornetworks, which can finally help to predict and formulate hypotheses for futureexperiments.

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0965-0393/13/075007+17$33.00 © 2013 IOP Publishing Ltd Printed in the UK & the USA 1

Modelling Simul. Mater. Sci. Eng. 21 (2013) 075007 R Sigala et al

1. Introduction

A ‘memristor’ (portmanteau from ‘memory resistor’) is a passive two-terminal circuit element,the electric resistance of which depends on the history of charge that has passed throughit [1]. Memristors combine memory and resistance, therefore they can store and processinformation at the same time. This unique property of the memristor has recently attractedthe interest of a wide range of scientists interested in information processing. Although theexistence of such a fundamental electronic device was pointed out theoretically in 1971,only very recently have practical implementations of the basic memristor’s functions beenachieved [2, 3]. One of the first implementations that succeeded in replicating the typicalbehavior of the memristor is based on functional molecular assemblies able to store andtransmit information [2]. Figure 1(a) shows a scheme of the organic memristor device. Themain element of such devices is a polymer-based material that connects two terminals througha thin polyaniline (PANI) film in its emeraldine salt (acid-doped) form. Additionally a solidelectrolyte, lithium doped solid polyethylene oxide (PEO), is attached to the central part ofthe PANI layer and connected to one of the terminals through a silver wire. One interestingproperty of this configuration is that the thin PANI film connecting the two terminals canbecome conductive or isolating if the appropriate electric potential is applied to the terminals.Since lithium ions can be exchanged between PEO and the PANI layer, the conductivity of thePANI film can vary within a range of up to four orders of magnitude [4].

Generally there are two approaches to build adaptive networks of memristors. Thefirst approach is based on hybrid semiconductor/nanowire/molecular (‘CMOL’) integratedcircuits [3, 5]. CMOL circuits can be used in artificial neural networks to build memristivesynapsis, implemented as crossbar arrays on top of complimentary metal-oxide-semiconductor(CMOS) circuits. These types of neuromorphic ‘active’ networks (‘Distributed CrosspointNetworks’ or ‘CrossNets’ for short) are suitable for information processing, pattern recognitionand classification [6, 7]. The second approach, which our research group follows, consists ofpassive networks of interconnected memristors without neurons or active elements. Thesetypes of networks have been implemented using functional molecular links able to performlogic computations [2, 8–10].

While in conventional computers the concepts of ‘memory’ and ‘processing’ are physicallyseparated, in the nervous system single processing units carry out both the storing and theprocessing of information together. Such a unit is the neuron, which is able to store andprocess information in the form of electric and chemical signals by way of its synapses,dendrites, cell body and axon [11]. In correspondence to this, devices based on memristorsseem to be suitable for the implementation of machines with highly adaptive properties likethe nervous system. Actually, our group has recently shown that networks of memristors withfew elements and a deterministic connectivity can implement some features observed in theneuronal circuits of a simple nervous system, like the pond snail, Lymnea stagnalis [12].

The nervous system has the ability to adapt, learn and perform computations with ahigh degree of efficiency and complexity. A key element that facilitates these functions isits self-assembled network structure [13]. Interestingly, polymeric-based memristors canalso build self-assembled random memristor networks. Using PANI and PEO alone or incombination with other inorganic materials that work as network nodes, it is possible to createfibrillar [14, 15] or self-assembled networks of memristors [16, 17] at a nanoscale level. Thisresults in self-assembled networks that can be potentially used as innovative complex elementsfor the processing of information.

The main objective of this study is to model and simulate the dynamics of polymericself-assembled memristor networks [18]; the qualitative explanation of their function is

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Figure 1. Schematic view (a) and electric scheme (b) of the single organic memristor. (a) In thisconfiguration the active channel is formed by PANI on top of a support and two electrodes. In thecenter, PEO constitutes the material of the electrolyte matrix. The area of contact between PANIand PEO is called the ‘active zone’, since almost all redox reactions, and therefore conductivitytransformations, occur there. D and S indicate ‘drain’ and ‘source’ terminals respectively.(b) Electrical model of the device, with one capacitor Cg and two variable resistors Rs and Rdlinked together with a ‘star connection’.

presented in [19]. Given the self-assembled nature of such networks, there is no control oftheir exact structure and connectivity pattern, and each device produced in the lab is a uniqueelement with a structure that depends on the actual conditions of each experiment. In our modelwe considered these constraints and defined networks of memristors built stochastically, i.e.applying some general parameters (e.g. number of nodes, number of connections, connectivitypattern, etc) but letting the final configuration (e.g. position of the nodes in space, initialmemristor values, connections, etc.) be established in a probabilistic way. Here we focusedon investigating how connectivity affects the adaptive properties (unsupervised learning) ofthe networks, reproducing experimental results and testing the three-dimensional (3D or non-planar) function of such networks. By providing a computational model that captures importantexperimental constraints, we expect to gain a better understanding of the complex adaptivebehavior of self-assembled memristor networks.

2. Materials and methods

The model of the memristor networks has a modular organization with different granularitylevels, going from single memristors up to large-scale networks of interconnected memristors.The model was divided into four main parts: (1) single memristor, (2) network structure,(3) network dynamics and (4) visualization and evaluation. The first module defines thebehavior of every single memristor. The second module generates a 3D arrangement of nodesand defines the connections between them. The third module calculates the conductivity valuesof all memristors at each time step according to the information provided by the first threemodules as well as to the input applied. Finally, the evaluation module quantifies and displaysthe activity of the network. Each of the four modules uses a set of independent parameters.Figure 2 illustrates the modules of the model and their interactions. In the following sectionswe describe in detail each of the first three modules. Additionally, in sections 2.4 and 2.5, weexplain the procedure used to simulate and quantify the behavior of the networks during thesimulations. All simulations were done in MATLAB (R2009b).

2.1. Single memristor

We simulated the behavior of a single memristor using a model that takes into account theelectrical properties of the constituent materials and the redox reactions involved. Such a model

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Figure 2. Illustration of the simulation environment divided into three basic modules. The‘Network Structure’ module defines the connectivity properties of the network, the ‘NetworkDynamics’ module simulates the evolution of the network states, while the ‘Network Evaluation’module calculates some global statistics after running the simulations. This module also displaysan interactive graph of the network. A sub-module that defines the behavior of a single memristorserves as input to the ‘Network Dynamics’.

has been used to reproduce the experimental behavior of the single macroscopic memristivedevice [20] illustrated in figure 1(a). This device may be considered as a star connection of twovariable resistances (PANI conductive channel) and a capacitor formed at the junction betweenPANI and PEO which represent the active zone of the device. The electrical description of thedevice consists, therefore, of a capacitor Cg and two variable resistors Rs and Rd linked througha star connection, as illustrated in figure 1(b). The two resistors Rs and Rd are respectivelylinked to the source (S in figure 1(a)) and drain terminals (D in figure 1(a)), while capacitorCg is connected to the gate electrode. In this paper we only recall the main features of themodel, and refer the reader to [14, 20] for a detailed description.

In this model, the voltage V drops across the two terminals of the memristor, called theInput and Output terminals, with a third electrode at the reference potential connected to theoutput terminal. The device electrical resistance is a function of the polyaniline redox stateq, defined as q = qox/qmax or ratio between oxidized sites qox and the total number of sitesavailable for redox reaction qmax. The junction between PANI and PEO is the area where theoxidation or reduction events occur, determining the value R(q) of resistance of the device.Such reactions are induced by the charge in the capacitor and the actual value of electricalresistance.

Since the relative amount q of oxidized PANI in the active zone of the device determinesboth the charge on the capacitor and the device resistance R(q), the memristor state can bedescribed at any time by this single parameter.

The differential equation used to calculate the time evolution of q in our simulation (inthe case of oxidation) was defined as follows:

q = (1 − q) · P(Vj − Vox) · Vj

R(q), (1)

where the time derivative of the oxidation state q (ionic current) is given by the products ofthree terms:

(1) (1-q) represents the available sites for oxidation.

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Figure 3. Three-dimensional visualization of memristor networks with different kinds ofconnectivity. (a) Example of a fully-randomly connected network, the connections (thin lines)of which were established independently of the distance between nodes (dots). (b) Exampleof a distance-dependent connected network. In this kind of network the probability that twonodes were connected was higher if the distance between them was short. This probabilitydecayed logarithmically as the distance increased. Such networks were dominated by short-rangeconnections thus favoring local transmission of information. In both types of networks, we used afixed number of nodes (250) and a fixed number of connections (1500).

(2) P is the probability of an oxidation reaction in the active zone, which is a function of thedifference of junction voltage Vj (q) and the oxidation potential Vox. We used a sigmoidfunction centered at Vox.

(3) Finally the last term is the Ohm law determining the ionic current in the junction.

In turn the junction voltage Vj is calculated from the applied voltage V to the device andthe combination of input and output resistances.

In our simulation we use for the reduction process the symmetrical equation:

q = q · P(Vj − Vred) · Vj

R(q)(2)

and we set Vox = 0.4 V and Vred = 0.2 V according to our experimental measurements on thematerial.

2.2. Network structure

To generate the memristor networks, 250 nodes and 1500 connections were distributed withina certain area depending on the conditions simulated (see sections 2.4 and 2.5). To definethe connectivity of the networks we used a so-called ‘distance-dependent’ connectivity, whichrelates logarithmically the distance between the nodes with the likelihood that they connect.Using this rule, networks ended up having a higher number of short connections and a lowernumber of long ones. For the set of simulations in which we investigated how the adaptiveproperties of the networks depended on their connectivity pattern, we also used a so-called‘fully-random’ connectivity rule (see section 2.5). Figure 3 shows an example of each classof networks, plotted in a 3D space.

Based on experimental observations, it is very likely that within the self-assembledmemristor networks created in the lab, some conducting PANI filaments do not contact enoughPEO to noticeably change their conductivity, resulting in connections with fixed resistance.To account for this phenomenon in our model, the probability of a connection being resistive

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instead of memristive was 0.2. At the last step of the procedure that builds the network,two nodes were randomly chosen as inputs and two others as outputs. The output nodeswere set to the ground potential, while the input nodes received constantly varying inputpotentials. For both memristors and resistors the initial resistance value was drawn from therange of possible resistance values of our PANI-PEO device, i.e. 50 k�–1 M�. The evolutionof each memristor under an external voltage occurred then according to the model of the singlememristor presented before.

2.3. Network dynamics

To define the dynamics of the networks, the potential distribution at each time stepwas calculated by solving the circuits composed by memristors and resistors using themodified nodal analysis (MNA) method. The MNA technique can solve any resistor circuit,independently of the topology, even in the presence of floating current or voltage sources[21, 22]. MNA solves the circuit by calculating the voltage at each circuit node usingKirchhoff’s first rule, which states that the algebraic sum of the currents into or out of anode must be equal to the total current injected by current generators, if they are present, inthat node. For a detailed explanation of this technique, we refer the reader to the appendix.

2.4. Simulation to reproduced the I–V characteristics of the real 3D memristor matrix

Recently, we fabricated a stochastic hybrid 3D network of memristors, the electrical propertiesof which showed learning and adaptive behavior [18]. It was also shown experimentally in thesame paper that some properties can be observed only if the network has a 3D organization.The experimental setup used to test the properties of that network is shown in figure 4(a). Twostrips of PANI-PEO copolymer were deposited so that they contacted four electrodes at the fourcorners of a square glass support. The whole structure was then covered with Li-doped PEO,contained with a plastic ring, in which several silver wires are immersed. We chose this solutionto give the same reference potential to all the memristors that are present in the network, inorder to keep the experiment simple and understandable. The I–V characteristics presentedin figure 4(c) were measured in the real 3D memristive matrix using the protocol describedin [18]. These characteristics represent the typical hysteretic behavior observed in the ‘oneinput–one output’ configuration of standard organic memristive devices [2] (for more detailson the experimental work, please refer to [18]). The presence of hysteresis indicates memoryin the system. The first goal of our simulations was to reproduce the I–V characteristics ofthe 3D network created in the lab.

To reproduce the behavior of the memristor matrix fabricated in the lab [18], we definedthe memristic networks trying to resemble as much as possible the experimental conditions,shown in figure 4(a). To this effect we imposed some important constraints on the definition ofthe networks. The first experimental constraint involved the region that the networks occupied.Networks were defined within a cuboid, two parallel sides of which were one-tenth of the otherfour. This region is represented with a square in the center of the illustration in figure 4(b),which also presents a 3D view of the networks within this space. This square region aims atrecreating the central space formed by the two crossing polymer stripes, as in the real setupdepicted in figure 4(a).

The second important constraint regarded the gate (reference) electrode of the memristors.Based on the model of a single memristor, described in section 2.1, if each memristor has itsgate electrode connected to one of its two terminals, the voltage used in equations (1) and (2) iscalculated by taking the difference between the voltage at the two terminals of the memristor,

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Figure 4. Simulating the crossed-configuration tested in the laboratory. (a) Illustration of theexperimental setup. (b) Schematic view of the experimental configuration and a 3D plot generatedduring the simulation of the same protocol. The red and blue circles represent inputs and outputs(gnd) respectively. (c) I–V characteristics for two opposite electrodes in the real experimentalsetup. (d) I–V characteristics of the simulated network of memristors. The left part shows the(drain or electronic) current flowing from one electrode to the opposite one, as a function of thevoltage applied to the input nodes. The right part shows the equivalent resistance as a function ofthe voltage. The hysteresis observed in the system indicates the effect of memory.

considering the right polarity. In the experimental set up, however, the gate electrode iscommon for all the memristors and the silver wires were kept at ground potential. Therefore,the voltage used in equations (1) and (2) to calculate the resistance variations of each memristorwas the average between the voltages at its two nodes.

Finally, the last constraint in this simulation regarded the input and output of the networks.As shown in figure 4(a), in the real configuration voltage was applied to the networks via twoopponent pairs of terminals, located at the tip of the two polymer stripes. To mimic thiscondition, the two input nodes were chosen to be the closest node to the central part of twoadjacent long sides of the cuboid, indicated by the red circles in figure 4(b). Analogously,the two output nodes were the two closest nodes to the central part of the two opposite sides,indicated by the blue circles in the figure.

2.5. Simulation to demonstrate adaptive behavior (unsupervised learning)

In contrast to the procedure in section 2.4, the goal of these simulations was to generallyinvestigate the adaptive properties of our 3D networks of memristors, instead of replicatingspecific experimental conditions. We therefore disregarded some of the strict experimentalconstraints imposed in section 2.4. For the space of the networks, the gate reference ofthe memristors and the position of input and output nodes, we used the following genericconditions: first, the space in which networks were defined was an equilateral cubic volume.Second, gate (reference) electrodes of each memristor were connected to one of its two

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Figure 5. Illustration of the stimulation procedure used to test the adaptive properties of thenetworks. After having initialized a network (white circle on the left side), we stimulated it during100 s (epoch duration). As input to the networks we used two approaches (named A and B),illustrated on the top and bottom part of the figure. ‘Input A’ were variations of only positivevoltages. ‘Input B’ were variations of positive and negative voltages. Depending on its initial stateand input history, a network was driven to different states, represented by the gray (input A) andblack (input B) dots. The procedure, referred to here as an epoch, was repeated 50 times. Networkstates were vectors with all resistance values and were labeled as EXi(tj ), x being the type of input,i the epoch number (1 : 50) and tj the time-step sampled.

terminals (section 2.1). Finally, the position of the input and output nodes was picked randomlyfrom the population of network nodes.

Twenty networks of each of the two connectivity types were created for these simulations.Input signals to the networks were electric potentials independently applied to the two pairsof input and output nodes during 100 s (epochs). To introduce some variability in the inputpatterns, the absolute value of the potentials was randomly chosen every 20 s between a givenrange, calculated for each single network. The limits of this voltage range were defined,such that any voltage within this range could, alone and disregarding the polarities of thememristor, raise the conductivities in 10–20% of the network connections at the beginning ofeach simulation. Input signals were of two classes: (A) positive voltages with respect to theground level, applied to both pairs. (B) Positive voltages were applied to one pair and negativevoltages to the other. Afterwards, the initial value of all memristors was initialized (randomlychosen a value) and the whole procedure was repeated 50 times (epochs). Figure 5 illustratesthis procedure.

To measure the degree of similarity of two networks at a given time we used the Pearsoncorrelation coefficient. For example, the similarity between two given epochs i and j at the

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time t of the training of a network, stimulated with independent random patterns of the type A,was defined by the function corr(EAi(t), EAj (t)). The vectors EAk(t) contain the resistancevalues of all the memristors and resistors in the network at time t of epoch k, as these valuesunivocally identify the state of the network. The overall similarity (indicated here as S) asa function of time, using the input pattern A, was given by the average correlation across allpossible paired comparisons between the simulated epochs or experiments:

SA(t) = 1

N

∑ij

corr(EAi, EAj ), (3)

where N is the number of possible i, j choices. In an analogous manner we can comparenetworks stimulated with inputs of type A to the networks stimulated with inputs of type Busing the same approach:

SAB(t) = 1

M

∑ij

corr(EAi, EBj ). (4)

Finally, both parameters SA and SAB only refer to a single network; to have an idea of theunsupervised learning capabilities of a generic memristor network, it is necessary to averageSA and SAB over different networks, randomly generated.

3. Results

3.1. Reproducing the I–V characteristics of the real 3D memristor network

As figure 4(d) demonstrates, the I–V characteristics of the simulated memristor network showthe same hysteretic behavior as the experimental ones in figure 4(c), confirming that the modelused for the simulations captures the essential functional aspects of the structure used in theexperiments, however we should keep in mind a couple of important differences. First, thevoltage was ramped up to 4 V and down to −4 V, instead of the range of 1.2 and −1.2 V usedin the real experiments. Second, as the model discarded gate currents, as described in themethods in section 2.1, we could not observe the drain and gate currents. However, we haddirect access to the differential current (i.e. the current that flows through the polymer filmwithout entering the PEO layer) and the equivalent resistance. It should also be noted that thesepreliminary results are an attempt to capture the qualitative behavior of the network withoutattempting to actually fit the data of the simulations.

In this simulation we replicated the hysteretic behavior of the real network of memristors.However, we focused on the qualitative and not on the quantitative aspect of this behavior, fortwo reasons: first, although the qualitative behavior is constant in the various experiments, thenumerical data change widely between one experiment and another; second, many parametersthat describe the network structure in our model are still educated guesses that have not yetbeen confirmed by data coming from a structural analysis of the networks actually built.

3.2. Testing the three-dimensionality of the simulated matrix

Finally, an interesting issue still open for the self-assembled block copolymer memristornetwork, which owns a 3D physical structure, is to verify whether the resulting circuit hasindeed a 3D functionality, or if it is instead merely a planar circuit. While having a 3Dstructure constitutes an innovative achievement with a wide range of possible applications,having a planar circuit would represent an important limitation of our implementation. Toverify this aspect of the simulated networks, we proposed the following scenario. If we apply

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Modelling Simul. Mater. Sci. Eng. 21 (2013) 075007 R Sigala et al

Figure 6. Response of a simulated four terminal random memristor circuit to a series of voltagesapplied to the terminals. (a) View of the simulation during its whole duration. The box in thetop and left part is a simplified layout for an equivalent circuit of the experimental setup shownin figures 5(a) and (b). (b) Zoomed view of the middle part of the simulation, indicated by thedashed box in A. In the upper graph, the blue and the green line represent I13 and I24 respectively,the red and the cyan line represent I14 and I23; in the bottom graphs, the red line represents thepotential applied to terminals 1 and 3, the cyan line the potential applied to terminals 2 and 4. Thered arrows indicate when I13 and I24 are higher than the other two currents.

training voltages in two opponent pairs of a four terminal ‘cross’ circuit, like the one shownin figures 4(a) and (b) and represented in a simplified form in the gray box on the left cornerof figure 6, a sure sign of non-planarity would be if the conductivity between both couplesof opposite terminals were greater than the conductivity between adjacent terminals. In otherwords, if we number the terminals 1,2,3,4 going clockwise for any terminal in the circuit, andwe denote with IAB the current that flows in the circuit between the terminal A and the terminalB, a circuit that satisfies the conditions

I13 > I23,

I13 > I14,

I24 > I23,

I24 > I14

(5)

is surely non-planar. The proof of this statement is that if the network was planar, the pathlinking terminal 1 with terminal 3 would need to contact the path between terminals 2 and 4.

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Modelling Simul. Mater. Sci. Eng. 21 (2013) 075007 R Sigala et al

Now if the total resistance was the same for the two ‘opponent’ paths, this would lead to thecreation of equally conductive ‘adjacent’ paths, i.e. between 2 and 3 or between 1 and 4. If oneof the opponent paths (from 1 to 3 or from 2 to 4) was instead more conductive than the other,creating a contact between both would drive the conductivity of the two newly created adjacentpaths (from 2 to 3 or from 1 to 4) at a conductivity level somewhere between the opponentpaths. In other words, the conditions in equation (5) can only be satisfied if the opponent paths(from 1 to 3 and from 2 to 4) do not make contact, which is not possible if the circuit is planar.Any further direct path between, for example, terminals 1 and 4, would only be used if theiroverall resistance was lower than opponent paths, leading to even higher values of I14, Thisshows that, while the conditions dictated in equation (5) are not necessary to have a circuitwith 3D paths that do not cross themselves, they surely are sufficient.

In figure 6 we can see the effect in a simulated four terminal circuit of a series of trainingvoltages that are applied at two terminals at once: the gate electrode in this experiment isalways at ground level, while terminals 1 and 3 are always at the same potential compared toterminals 2 and 4. Potentials can be positive or negative. The currents I13, I24, I14 and I23 aremeasured at every instant during the training, by applying a sample potential to the first of thetwo terminals, keeping the other terminal at ground level and measuring the resulting current,without modifying any resistance in the circuit.

From the figure it is possible to observe that, although at the beginning of the simulationI13 has the least magnitude, as the training proceeds, there are times, indicated in figure 6(b)by the red arrows, where both the I13 and the I24 get higher than the other two currents, thussatisfying the conditions dictated by equation (5) and proving that this particular simulatednetwork has indeed a non-planar functionality. Although we have not found a protocol thatcan show this kind of behavior in every circuit that we simulated or built, the correspondencebetween the I–V characteristics of the simulated circuit and the ones measured experimentallysuggest to us that the non-planar characteristics could be found also in some of the circuitscreated in the lab. Nevertheless, we stress the importance of a careful interpretation of thesimulation results, keeping in mind that assuming a non-planar characteristic in every realdevice could be misleading (see last part of the Conclusions).

The non-planar functionality of the memristor networks has recently been tested inthe experimental setup [18]. This constitutes a very promising characteristic of thepolymeric memristor networks when considering signal propagation and finally, processing ofinformation. In a 3D configuration the number of short-range connections can be significantlyincreased compared to a planar configuration. As we have already demonstrated, short-range connections can positively influence adaptability, efficiency and robustness of memristornetworks.

3.3. Adaptive behavior (via unsupervised learning) of the networks depends on theirconnectivity and type of input applied

We have recently shown that polymeric self-assembled memristor networks fabricated in thelaboratory show adaptive properties such as unsupervised learning [18], but how these adaptiveproperties depend on the topology of this type of network is unknown. In this set of simulationswe asked how the adaptive properties of this type of network are influenced by the structureof their connectivity. In one case, the connectivity of the networks was defined in a ‘fully-random’ fashion, while in the other case we imposed some structure in the topology by lettingthe connectivity depend upon a ‘distance-dependent’ rule. With this rule, the number ofconnections was a (log) function of the connection length, resulting in networks with a highernumber of short connections and a lower number of long ones.

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Figure 7. Adaptive behavior in stochastic memristor networks. Evolution of the average similarityas a function of the stimulation time of networks with a fully-random (a) (c) and a distance-dependent connectivity (b) (d). To calculate the evolution of similarity in a network, all possiblepair correlations between networks states of different epochs at a given time point were averaged.The decreasing correlation values show how network states deviate more the longer we stimulate.We compared epochs in which different (a) (b) as well as the same type of inputs (c) (d) wereapplied. (c) (d) Black and dashed curves represent the results according to the type of input used.In contrast to the fully-random case, networks with a distance-dependent connectivity seem todeviate faster from the initial state depending on the input used.

In a typical unsupervised learning protocol, a large batch of data belonging to two or more‘classes’ is used as input of a system without any corresponding target values. The goal of asystem in such unsupervised problem is to find a given structure in the input space, such thatfurther examples can be labeled as belonging to any of the given input classes. We appliedto the networks two different classes of inputs; in one case we used competing excitatory andinhibitory signals (A: positive potential in one node and negative in the other) while in theother we applied complementary reinforcing signals (B: positive potentials in both nodes).This procedure (epoch) was repeated 50 times after reinitializing all resistance values. Theconceptual scheme of this procedure, explained in detail in section 2.5, is shown in figure 5.

For each network we looked at the evolution of all resistance values (network states), bycomparing how similar these values were across the 50 epochs and along the stimulation time.The more adaptive a network is, the more dissimilar its states become as the simulation goes on.To check the similarity between network states, we used a measurement based on correlations(see section 2.5), where high correlation values indicating similar network states and lowvalues indicating dissimilar states. For each of the two classes of networks we compared thesimilarity across states produced by either the two different types of inputs, or by the sameclass of inputs. The results of these simulations are presented in figure 7. Correlations on

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Modelling Simul. Mater. Sci. Eng. 21 (2013) 075007 R Sigala et al

networks with fully-random and distance-dependent connectivity are shown in the left andright panels respectively. Each plot shows the average correlation values (S) of 20 differentnetworks as a function of time. Plots on the upper part show correlation values across networkstates corresponding to different types of input (A versus B). Plots on the bottom part showmean correlation values when the same type of input was used (input A, dashed curve; input B,non-dashed curve). In general the monotonic and decreasing correlation values as a functionof time indicate that the network configurations tend to diverge the longer we stimulate. Thisbehavior is stable regardless of the type of input and network connectivity, which confirm theadaptive properties of the memristor networks. We observed, however, a difference betweenfully random and distance-dependent connected networks on how fast their epochs decorrelate,depending on the input used. Networks with a distance-dependent connectivity decorrelatedfaster when the input was conflicting, i.e. positive in one node and negative in the other (inputA), than when both inputs were positive (input B). As observed in figure 7(c), this is not the casefor the fully randomly connected networks. This indicated that the efficacy of the stimulationdepended on both the input pattern and the connectivity of the networks.

We compared two different kinds of networks that resulted from varying the rules used todefine their connectivity: (a) fully randomly connected and (b) distance-dependent connectednetworks. Networks with a distance-dependent connectivity (dominated by short-rangeconnections) adapted faster to the inputs compared to fully randomly connected networks.It is known that many real networks found in biological, cognitive and social systems have adistance-dependent connectivity structure. A structure of that kind can be generally describedby relating logarithmically the probability of finding a connection between two given nodesand the distance between them, such as by noting that increasing the distance between twonodes reduces the probability that they connect [23]. The fact that many complex self-organizedsystems share similar connectivity structures might indicate an evolutionary advantage reflectedin their efficiency and robustness to spread information [24]. In mammalian brains, probablythe best example of efficiency and robustness regarding information processing, networks withdistance-dependent connectivity also exist: this has been shown for the short-range connectivitywithin the gray matter of the mammalian cerebral cortex [25, 26], and to a certain degreealso for its long-range connectivity via the white matter [27, 28]. The brain and many of itsimportant functions can only be explained by understanding brain connectivity [13]. Althoughhere we present some results that indicate that connectivity influences the adaptive propertiesof memristor networks, the effect of connectivity in information transmission and learningin these self-organized networks requires further investigation. In particular, it is necessaryto study connectivity in relation to other parameters like network size and connection/noderatio.

3.4. Localization of information processing elements in memristor networks

As indicated by the correlation values shown in figure 7(d), networks with a distance-dependentconnectivity evolved faster when inputs were competitive (input B), than when using tworeinforcing inputs (input A). Since the correlation evaluates the resistances of each and everyconnection in the networks (see equations (3) and (4)), an interesting question is to lookat the proportion of connections that contribute to the differences in the evolution. Onepossibility is that the differences in the resistances were homogeneously distributed acrossall the connections. Another possibility is to have some parts of the network more sensitive tofeatures in the input space than others, even with some parts that did not evolve at all. To lookat which parts of the networks were sensitive to the inputs applied, we looked at the averagechange of resistance during stimulation.

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Modelling Simul. Mater. Sci. Eng. 21 (2013) 075007 R Sigala et al

Figure 8. Evolution of the resistance of every connection of an exemplar network (distance-dependent connectivity) stimulated with input type A. The color code represents mean resistance(a) and standard deviation (b) calculated over 50 epochs. Connections were ranked in the verticalaxis according to the difference between the final and the initial resistance. Connections on topbecame insulating while connections on the bottom became conductive. This figure clearly showsthat while the majority of memristors became insulating, only a few of them became conductive.

Figure 8 shows the average behavior in the evolution of each network connection withinone epoch, calculated over 50 epochs in which the network was presented with inputs of typeA (both voltages positive). The left panel shows the mean resistance values of each link in thenetwork. Each line represents one element, and the lines are ordered according to the averagechange of resistance. On top of the graph, connections are shown that on average had muchhigher resistance at the end of the epochs than at the beginning. At the bottom, connectionsthat on average became more conductive during one epoch are presented. The right panelshows the variance for the same evolution, using the same ordering as the left panel.

The black band in the right panel of figure 8 corresponds to the 20% of resistors in thenetwork for which the resistance never changes, therefore presenting zero variance during thestimulation. A big majority of the memristors, located above the black band with zero variance,became on average more insulating. On the other hand, only a relatively small proportion ofmemristors, located below the black band, became more conducting.

When localizing elements of information processing in memristor networks, wedistinguished two general properties, as illustrated in figure 8. The first one is that thememristors that become increasingly more insulating are much greater in number than theones that become more and more conducting. The second one is that the variance in theevolution is much higher in the memristors that become more conducting than in the ones thatbecome more insulating. In other words, while each memristor becomes more insulating moreor less in the same manner, every memristor that on average becomes more conducting doesso in very different ways from one epoch to the other. Based on this fact, we can also concludethat information about the input is ‘stored’ in the memristors that become more conductive,which represent the most adaptive part of the networks.

4. Conclusions

We implemented a model to simulate and evaluate adaptive properties of stochastic memristornetworks. In this model, network dynamics could be parametrically simulated with the control

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Modelling Simul. Mater. Sci. Eng. 21 (2013) 075007 R Sigala et al

at different granularity levels, going from single memristors to large-scale configurations.The modular design of this simulation environment allowed us to easily exchange, extendor substitute any of its elements to perform a wide range of simulations. Here, we firstinvestigated the behavior of the networks recreating real experimental conditions. We showedthat the simulated networks resemble the same hysteretic behavior as the experimental ones.Additionally, we demonstrated that the simulated networks have indeed a 3D functionality,in contrast to a planar circuit. Finally, in the last group of simulations, we studied theadaptive properties of 3D (three-dimensional) stochastic memristor networks, asking howthe connectivity properties of the networks, together with the input, influence their adaptiveperformance. We showed that memristor networks stimulated with random noise follow astable behavior that diverges from their initial state according to the history of stimulation.Interestingly, when varying the connectivity of the networks (from fully-random to distance-dependent), we observed differences in the dynamics of the network that depended on the typeof input used.

Our model was able to capture fundamental qualitative aspects of real self-organizednetworks of memristors. We used a statistical approach to produce a big variability in thestructure of the networks, focusing on the general behavior of the matrix. Our model and thesimulation environment can contribute to understanding the behavior of complex memristornetworks; at the same time, they can help to formulate new hypotheses for future experiments.However, it is important to keep in mind that the model still does not capture importantelements of the real experiments, such as the enormous variability among devices, given theparticular conditions present in every experiment. Every device is different and every step ofits implementation can be the source of variability. More simulations in close relation withthe experiments are necessary to further limit and define the adaptive properties of these typesof memristor networks.

Appendix

This equation can be written for the ith node in the circuit as

∑j

Iij =∑

j

Vi − Vj

Rij

=∑

j

1

Rij

Vi −∑

j

Vj

Rij

= IGi , (6)

where the sum is over all the nodes j connected to the node i with a resistance Rij and IGi is

the sum of the currents injected into the node by all the current generators connected to thatnode (IG

i = 0 if no current generator is connected to the node). There is one such equationfor every node, so we can put them together in a system

AV = I, V = IA−1 (7)

where the known term column vector I contains all the currents generated by current generatorsin the circuit, the unknown vector V contains the voltage at all nodes, and the matrix elementAij is defined as

Aij ={∑

k 1Rik

if i = j,

− 1Rij

if i < > j.(8)

In other words, in the main diagonal of A, at row i, there is the sum of the conductances ofall the elements connected to the node i, while any off-diagonal element of A in position ij issimply the conductance of the element connecting i with j , with a negative sign.

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Modelling Simul. Mater. Sci. Eng. 21 (2013) 075007 R Sigala et al

MNA can also take care of floating voltage generators. Those bring a new equation in thesystem, of the type Vi − Vj = Vgen, but also new unknowns Ik, the currents flowing throughthem. The new equations are handled by rewriting the system into a larger one,

GE = K, E = KG−1. (9)

E is the new unknown vectors, which contain all the voltages in each node followed by all thecurrents in each voltage generator. The new known term vector K contains all the currentsinjected in each node, followed by all the potential differences created by each generator. Foursubmatrices compose the new G matrix,

G =(

A BT

B C

), (10)

where A is the matrix defined in equation (8). B has as many columns as A and as many rowsas the number of voltage generators; the entries of B are all zeros, except for the entries Bkp,where the positive terminals of the generators are connected, that are equal to 1, and the entriesBkn, where the negative terminals are connected, that are equal to −1. BT is the transposeof B, and C is a matrix with all entries equal to zero. The system described in equation (9)recovers with the last lines of matrix G all the equations added by the voltage generators andincludes in equation (6) the currents through the generators, using the matrix BT. It should benoted that solving a circuit using MNA requires inverting a matrix of size n + k −g, where n isthe number of nodes in the circuit, k is the number of voltage generators, and g is the numberof nodes connected to the ground potential, which do not generate equations to be included inthe matrix A, as we already know their voltage.

Solving the equations for a given time step t simulated the evolution in time of the circuit.This means that at each simulation step we considered our circuit as a resistor only circuit,which means that our time step must be short compared to the characteristic times of thememristor evolution. As those were of the order of hundreds of seconds, we found that a timestep of one second is usually short enough, and that further shortening does not change thesimulation results.

As the potential distribution was calculated at each time step, we were not limited toconstant voltage and current generators, but we could also simulate generators whose outputis an arbitrary function of time. Finally, it should be noted that, instead of adding the gatecurrents to the source currents, we discarded them during the simulations. In the real devicethese currents are two orders of magnitude smaller than the source-drain currents, so that theeffect of the gate-source currents on the current distribution in the circuit can be considerednegligible.

Acknowledgments

The authors thank Prof. Marco Fontana for his fundamental contribution to the project andProf. Valentino Braitenberg for being a daily source of inspiration. We acknowledge thefinancial support of the Future and Emerging Technologies (FET) program within the SeventhFramework Program for Research of the European Commission, under the FET-OPEN grantagreement BION, number 213219.

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