Percolation with plasticity for neuromorphic systems
PPercolation with plasticity for neuromorphic systems
V. G. Karpov, ∗ G. Serpen, † and Maria Patmiou ‡ Department of Physics and Astronomy, University of Toledo, Toledo, OH 43606, USA Department of Electrical Engineering and Computer Science, University of Toledo, Toledo, OH 43606, USA (Dated: May 5, 2020)We develop a theory of percolation with plasticity media (PWPs) rendering properties of interestfor neuromorphic computing. Unlike the standard percolation, they have multiple ( N (cid:29)
1) inter-faces and exponentially large number ( N !) of conductive pathways between them. These pathwaysconsist of non-ohmic random resistors that can undergo bias induced nonvolatile modifications (plas-ticity). The neuromorphic properties of PWPs include: multi-valued memory, high dimensionalityand nonlinearity capable of transforming input data into spatiotemporal patterns, tunably fadingmemory ensuring outputs that depend more on recent inputs, and no need for massive interconnects.A few conceptual examples of functionality here are random number generation, matrix-vector mul-tiplication, and associative memory. Understanding PWP topology, statistics, and operations opensa field of its own calling upon further theoretical and experimental insights. I. INTRODUCTION
Devices for neuromorphic computing remain amongthe most active areas of research with a variety of mod-els for neurons, synapses and their networks.
They aretypically built of nonvolatile memory cells and intercon-nects wired in a certain architecture.Here we introduce a concept of neuromorphic deviceswhere neither artificial memory cells nor interconnectsare required. They are based on disordered materi-als with percolation conduction such as amorphous,polycrystalline, and doped semiconductors, or granularcompounds. We recall that percolation transport takesplace in systems of microscopic random resistors, and isdominated by the infinite cluster of smallest resistors al-lowing connectivity between the electrodes.Among possible percolation conduction materials, weconsider those exhibiting plasticity, i. e. exhibitingnonvolatile changes in their resistances in response tostrong enough electric field. They include metal oxidesand chalcogenide compounds used with resistive randomaccess memory (RRAM) and phase change memory(PCM), granular metals, and nano-composites. A PWP example in Fig. 1 shows some conductivepathways for the case of a relatively small number of elec-trodes. Anticipating a particular application below, Fig.1 assumes certain voltages E i applied to all electrodesbut one used to measure the electric current I . Otherimplementations would assume different circuitries withvarious power sources and meters attached to their mul-tiple electrodes. As explained in what follows, each ofthe pathways can undergo multiple field induced changesthereby presenting a multivalued memory unit.We note the following PWP features relevant for neu-romorphic applications:(i) The exponentially large combinatorial number, M = N ! ≈ exp( N ln N ) (cid:29)
1, of interelectrode resistances R ij that scales exponentially with the number ( N (cid:29)
1) ofelectrodes. For example, M ∼ in a design of Fig. 1.Such extremely high dimensionality in combination with FIG. 1: Schematic 2D illustration of percolation systems withmultiple local interfaces (electrodes). G ij stand for pathwayconductances. Only a small number of 12! ≈ pathwayspossible in the sketch are shown. The peripheral numbers1,2,..,12 refer to the assumed 12 electrodes. resistors’ nonlinearity (non-ohmicity) makes PWP idealobjects for the reservoir computing. (ii) Multivalued memory in conductive pathways oper-ated by electric pulses that modify R ij due to materialplasticity; they play the role of multiple microscopicmemory cells.(iii) Direct connectivity between the bond-forming mi-croscopic resistors eliminates the need for artificial inter-connects. In fact, each microscopic resistor in PWP canact as a nanometer sized memristor without artificial in-terconnects.(iv) The multivalued memory in combination with mul-tiplicity ( M (cid:29)
1) offers a platform for the in-memorycomputing. Furthermore, mathematically, series of cellsin PWP present multidimensional random vectors form-ing a base for hyperdimensional computing. (v) The randomness of PWP topology offers a natural im- a r X i v : . [ c ond - m a t . d i s - nn ] M a y plementation of the randomly wired neural networks out-performing ( at least in some applications) their regularlywired counterparts. That randomness can as well be-come beneficial with reservoir computing applications. In what follows we consider the physical parameters ofPWPs and some examples of their neuromorphic func-tionality. The paper is organized as follows. It starts witha purely qualitative discussion in Sec. II that explainswithout any math the model and the logic of the paper.Section III describes the standard percolation concept.The PWP systems, including their limiting cases of largeand small structures, are described in Sec. IV. Examplesof PWP’s neuromorphic functionality are presented inSec. V. Following the nomenclature established for otherneuromorphic systems, Sec. VI briefly discusses certainmetrics of the proposed PWP devices. Similarities andarchitectural differences between PWPs and biologicalneural networks are discussed in Sec. VII. Numerical es-timates in Sec. VIII suggest that the proposed systemscan be experimentally implemented allowing verificationsof their expected properties. We briefly touch upon theissue of inherent randomness of PWP systems in Sec. IX.The conclusions in Sec. X list this approachs capabilitiesand limitations.
II. QUALITATIVE DESCRIPTION
This section provides a simplified low resolution guidefor subsequent consideration. It offers a brief summaryof our work aimed at the backgrounds of electrical en-gineering and computer science researchers most signifi-cantly contributing to the fields of artificial intelligenceand neuromorphic computing.1. Underlying our approach is the classical conceptof percolation conduction as the electric transportthrough a network of exponentially different ran-dom resistors. It is dominated by the percolationcluster formed by conductive bonds connecting theelectrodes as illustrated in Fig. 2. The bondsare composed of the elements with minimum resis-tances whose total concentration is just sufficientto form the electrode connecting pathways. Thecharacteristic mesh size of the percolation cluster L c is much greater than the linear size a of a mi-croscopic resistor (see Fig. 2). The percolationcluster is effectively uniform over distances signifi-cantly exceeding L c .2. The percolation conduction is extremely nonohmicstarting from voltages U ∼ kT /q ∼ . − . ∼ L c /a (cid:29)
1) exponentially different resistors,voltages on some of them must be exponentiallyhigh to maintain the same current. As a result, theexponentially higher voltage, of the order of that
FIG. 2: A sketch of the percolation cluster between two elec-trodes. Shown under magnification is a bond domain consist-ing of microscopic resistances in series. across the entire L c distance, is localized on themost resistive element.3. The local high fields in the percolation cluster canbe strong enough to cause local structural transfor-mations (nonvolatile switching) and correspondingdecrease in local resistivity. The possibility of suchtransformations is customarily referred to as plas-ticity in the neuromorphic domain; hence, percola-tion with plasticity (PWP). Each drop in resistancecaused by plasticity can serve as a memory recorddetectable relative to the preceding value. Underthe same conditions, the voltage originally localizedon the highest of the series of exponentially randomresistors will concentrate on the next highest onewhen the former is switched to the low resistancestate, etc. causing a sequence of resistance drops.4. In addition to being a non-ohmic resistance, eachelement of a percolation bond possesses capacitiveproperties. It will perform as a capacitor if the elec-tric field pulse is fast enough to make its displace-ment current greater than the real one. Such capac-itive elements can be interpreted as ‘slow’ resistors.These elements possess Maxwell’s relaxation timesgreater than the pulse time. They are intuitivelyreferred to as capacitors because the RC time of anelement with resistivity ρ , dielectric permittivity ε ,length l and area A equals ερ/ π thus representingthe Maxwell relaxation time. In spite of their largeresistances, such capacitive elements do not accom-modate significant voltages because their currentsare due to the voltage rates of change rather thanvoltages themselves. These displacement currentsare physically related to the local charging pro-cesses, the duration of which is reciprocal in thecorresponding element’s resistance.5. If pulses arriving at the same element within its re-laxation time are in integral strong enough to turnit from the capacitive to resistive mode, the resis-tance of bond, in which it belongs, will change. Thechange becomes nonvolatile if the resulting field issufficiently strong for switching. That property issimilar to that of the spike timing dependent plas-ticity (STDP) central to neuromorphics.6. Unlike the standard two-electrode percolation sys-tems, PWPs have multiple ( N (cid:29)
1) electrodes asillustrated in Fig. 1. That creates N ! (cid:29) ∼
10) of non-volatile changes in resistance.8. Expected PWP applications include multi-valuedmemory, random number generation, associativelearning, and reservoir computing. The parametersof proposed systems fall in the domain of practicallyimplementable material systems.
III. STANDARD PERCOLATIONCONDUCTION
We start with a recap of the pertinent percolationconcepts.
Percolation conduction is dominated by thesparse infinite cluster of the exponentially different ran-dom resistors between the two large electrodes. The clus-ter bonds consist of the minimally strong random resis-tors with total concentration sufficient to form an in-finitely connected network. It is effectively uniform overlarge distances L (cid:29) L c where L c is the correlation radiusdetermining its characteristic mesh size.Each bond of the cluster consists of a large number ofmicroscopic resistors R i . Their exponential randomnessis described as R i = R exp( ξ i ) where quantities ξ i areuniformly distributed in the interval (0 , ξ max ) and i =1 , , .. .The physical meaning of ξ depends on the type of asystem. For definiteness, we assume here ξ i = V i /kT corresponding to random barriers V i in noncrystallinematerials where k is the Boltzmann’s constant and T is the temperature. In reality, the nature of percolationconduction can be more complex including e. g. finitesize effects and thermally assisted tunneling between themicroscopic sites in nanocomposites. These compli-cations will not qualitatively change our considerationbelow.The cluster constituting microscopic resistors exhibitnon-ohmicity due to the field induced suppression of theirbarriers V i , according to J i = J exp( − V i /kT ) sinh( qU i / kT ) (1)where J i is the resistor current, J is a constant, U i = E i a is the voltage applied to the barrier, a and E i are,respectively, the barrier width and local electric field, and q is the electron charge.An important conceptual point is that the ap-plied voltage concentrates first on the strongest resistor FIG. 3: A fragment of conductive pathways in the infinitepercolation cluster representative of polycrystalline or gran-ular materials. Numbers 1-6 represent random resistors indescending order. of a percolation cluster bond (resistor 1 in Fig. 3) sup-pressing it to the level of the next strongest (resistor 2in Fig. 3), so the two equally dominate the entire bondvoltage drop. It then suppresses the next-next strongestresistors (3,4,5,.. in Fig. 3), etc. As a result, the perco-lation cluster changes its structure as L c = a (cid:112) V /qEa, and ∆ V = (cid:112) V m qaE (2)resulting in the macroscopic non-ohmic conductivity, σ = σ exp (cid:16)(cid:112) ξ max qaE/kT (cid:17) . (3)Here L c and ∆ V are, respectively, the field dependentcorrelation radius and maximum barrier decrease in thepercolation cluster, and V is the amplitude of barriervariations.Two assumptions behind the above discussednonohmicity are: (a) The volatility of bias inducedchanges where each microscopic resistor adiabaticallyadjusts its resistance to the instantaneous bias. (b) Thequasistatic nature of biasing implying time intervals ex-ceeding the relaxation times of all microscopic resistors,i. e. t (cid:29) τ m = τ exp( ξ m ) where τ ∼ It was arguedthat for a cluster bond a bias pulse of length t generatesthe current I = I τ t exp (cid:32)(cid:114) ξ max N qVkT (cid:33) . (4)This result applies when t is shorter than the maximumrelaxation time τ max = τ exp( ξ max ) and is formally dif-ferent from that of dc analysis by the substitution ξ max → ξ t ≡ ln( t/τ ); the two results coincide when ξ t = ξ max . For the entire percolation cluster, the modifi-cation ξ max → ξ t predicts the current, I = I τ t exp (cid:32)(cid:114) a E q kT ln tτ (cid:33) . (5)To avoid any misunderstanding, we note that the fieldinduced changes in resistances of percolation clusters de-scribed in this section is volatile (i. e. it disappears whenthe field is removed). It should not be confused with thenonvolatile plasticity introduced here. IV. PERCOLATION WITH PLASTICITYSYSTEMS
PWP phenomenon differs from the standard perco-lation conduction in both topology and non-ohmicity.The former is such that the proposed PWP has multi-ple ( N (cid:29)
1) electrodes; the latter is due to the non-volatile nature of bias induced changes. These propertiescan be somewhat different for PWP systems with large( L (cid:29) L c ) and small ( L (cid:28) L c ) geometrical dimensions L as described next. A. General
We start with noting some general properties of PWPsrelevant to neuromorphic applications.(1) The variations ∆ R ij (with respect to the aver-ages (cid:104) R ij (cid:105) ) in the interelectrode resistances R ij are ran-dom quantities that are uncorrelated with any desiredaccuracy for not-too-close electrodes. Consider ∆ R ij = R ij − (cid:104) R ij (cid:105) with R ij = R (cid:80) N ij exp( ξ i ) for a bond of N ij resistors and ξ i uniformly distributed in the interval( ξ min , ξ max ). It is then straightforward to obtain the cor-relation coefficient between the resistances of ( i, j ) and( k, l ) bonds, C ≡ (cid:104) ∆ R ij ∆ R kl (cid:105) (cid:112) (cid:104) (∆ R ij ) (cid:105)(cid:104) (∆ R kl ) (cid:105) = N s (cid:112) N ij N kl (6)where N ij and N kl represent the numbers of micro-scopic resistances in those bonds, and N s is the numberof resistances shared between them. We describe eachbond as a random walk. Then, if the two bonds arenot close geometrically, separated by distances L ijkl ex-ceeding a (cid:112) N ij + N kl , then their overlap is exponentiallysmall, and C ∼ exp {− L ijkl / [ a ( N ij + N kl )] } (cid:28) . (7)The averages implied by the definition for ∆ R ij can bereadily measured for an ensemble of geometrically similarpairs, such as (1,9), (2,8), (3,7), (4,12), etc. in Fig. 1.(2) Related to the above item (1), there is a character-istic distance l ij , above which the electrodes i and j are electrically independent (no crosstalk between them). Toestimate that length, we use it in place of L ijkl in Eq. (7)setting also N ij ∼ N kl ∼ L/a where L is the sample sizeand a is the microscopic resistor length. This yields, l ij ∼ √ La. (8)Assuming as a rough estimate a ∼
10 nm and L ∼ l ij ∼ µ m.(3) High frequency inductive coupling of conductivepathways can be estimated based on a model of a wireand a loop of diameter L . Using the standard electrody-namics, the ratio of the induced current over the primarycurrent then becomes (in the Gaussian system) I (cid:48) I ∼ ωLc R (9)Here ω is the frequency, c is the speed of light, and R is the resistance of the bond. Assuming as a reference,values L ∼ R ∼ ω ∼ Ohm with PCM and RRAM applications) make that ra-tio small and acceptable.(4) The characteristic RC times related to the writ-ing and reading processes can be estimated as ∼
10 psfor R ∼ ∼ C ∼
10 pF (corresponding toa 1 cm sample with the same macroscopic resistivity asthat of 1 MOhm RRAM resistor with 10 nm linear di-mension), rather competitive against the background ofmodern technology.(5) The property of plasticity takes place when the lo-cal electric field exceeds its material dependent thresholdvalue for resistance switching as discussed next.
B. Large PWPs
The concept of infinite percolation cluster survives ifthe electrode sizes l and interelectrode distances L ij aremuch larger than L c , in which case (we call it ‘largePWP’) resistances R ij are determined, in the ohmicregime, by electrode geometry, R ij = ( σl ) − f ij ( l/L ij ) (10)similar to the case of steady currents between finite sizeelectrodes in massive conductors, such as grounding elec-trodes in a soil. Here f ij is a dimensionless functionwhose shape depends on the electrode locations throughthe confinement of electric currents by the sample bound-aries. The macroscopic conductivity σ in the equationfor R ij is taken in the limit of infinitely large percolationsystems where it is uniquely defined.While f ij can be numerically modeled with for par-ticular electrode configurations, some general statementscan be made based on the available examples and a sim-plified model. The latter presents an electrode as ametal hemisphere immersed into a macroscopically uni-form medium formed by the percolation cluster over largescales as illustrated in Fig. 4. A rough estimate is givenby f ij ≈ O ( l/L ij ) + O ( l/L ) (11)to the accuracy of a numerical multiplier that depends ona particular electrode geometry. Here O ( x ) means ‘of theorder of x ’ and it is assumed that l/L (cid:28) l/L ij (cid:28) R ij do not depend strongly on the interelectrode distancesin the ohmic regime. Yet, the differences in the inter-electrode resistances will exist due to statistical fluctua-tions in their connecting bonds. The bond of length L ij in a large PWP will contain L ij /L c quasi-independentcells of the percolation cluster. Each cell has on aver-age resistance R c making the average bond resistance (cid:104) R ij (cid:105) = R c L ij /L c . Taking into account that resistancesof individual cells exhibit random variation of the orderof R c , one thus arrives at the estimate for the character-istic relative variations of resistances, δR/ (cid:104) R ij (cid:105) ∼ (cid:113) L c /L ij . (12)While relatively small, the fluctuations δR/ (cid:104) R ij (cid:105) arestill significant enough to experimentally discriminate be-tween different interelectrode resistances. For example, δR/ (cid:104) R ij (cid:105) ∼ . L c ∼
10 nm and L ij ∼ µ m.Unlike the standard percolation between two flat elec-trodes, in large PWP the electric field systematically de-cays with distance r > ∼ l from a small electrode due to thecurrent spreading (again, similar to the case of ground-ing electrodes) as illustrated in Fig. 4. That geometri-cal effect will significantly alter the nature of nonohmic-ity making it most significant in the proximity of r ∼ l around the electrode. One can show that due to thatfield suppression, the steady state non-ohmic conductionwill be limited to r ∼ l (cid:112) ξ max qaU/LkT (13) FIG. 4: A large PWP geometry with two hemisphere elec-trodes and a fragment of percolation cluster resolved in themagnifying glass. Shown in dash are the equipotential (arcs)and electric field (arrows) lines in the proximity of the rightelectrode illustrating the geometrical effect of electric fieldspreading with distance from the electrode. where U is the total voltage applied to a sample of length L . The pulse regime limitation will be described by asimilar formula with the substitution ξ max → ξ t . C. Small PWPs
The concept of infinite cluster fails when l and/or L ij are smaller than L c (‘small PWP’), in which case one hasto consider multiple conductive paths unrelated to theinfinite cluster. Both the cases of large and small PWPare possible across a broad variety of percolation systems.For example, the current L ∼
10 nm-node technologybelongs in small PWP with a > ∼ . V /T < ∼ L c ∼
30 nm. With the latter parameter values,increasing L to microns and beyond will result in largePWP networks.The resistances of conductive pathways in small PWPexhibit significant variability. Next, we estimate theirstatistics. Based on Eq. (4) a chain resistance is esti-mated as R = R max exp( − δV /kT ) where δV = V max − v max and v max is the maximum barrier in that chain, R max = R exp( V max /kT ), and V max is the maximumbarrier in the entire system, V max ≥ v max . Assuminguniformly distributed barriers, the average number ofresistors with the barriers above a given v max in a n -resistor chain is n v = n ( V max − v max ) / ( V max − V min ),where V min is the minimum barrier. The Poisson prob-ability of a chain having no barriers greater than v max is P V ( n ) = exp( − n v ), and the probability of finding achain with a maximum barrier in the interval kT around v max is P ( v max ) = P V ( n ) kT / ( V max − V min ) . (14)Multiplying P ( v max ) by the probability P n ( L ) ∼ ( L/an ) exp( − L /na ) of an n-resistor chain connectingpoints distance L from each other, we obtain the prob-ability density of n-chain with a given barrier v max (tothe accuracy of a numerical multiplier in the exponent).Integrating that product over n by steepest descent andexpressing v max through R yields the probabilistic distri-bution density, P ( R ) ∝ R exp (cid:32) − La (cid:114) kTV max − V min ln R max R (cid:33) . (15)It follows that resistance spectrum is a gradual functionwith a certain characteristic width ∆ R . As estimatedseparately for the cases of small and large L , width ∆ R can be approximated for the entire range of L by thefollowing equation:∆ R ≈ R max (cid:34) (cid:18) La (cid:19) kTV max − V min (cid:35) − . (16)For all practical values, it encompasses multiple ordersof magnitude.Because of the dispersion in the values v max betweendifferent pathways, the nonohmicity exponents will varyfrom one R ij to another. More specifically, instead of ξ max in Eq. (4) the value v max /kT should be used withthe probability distribution of Eq. (14). That addition-ally broadens the distribution of path resistances in thenonohmic regime; we omit here the obvious formal de-scription of that effect. D. Plasticity by switching
A unique non-ohmicity feature of PWP is its non-volatile nature rendered by the underlying material (say,of PCM or RRAM type). Each microscopic elementof a conductive path can exist in either high or low-resistive state whose respective resistances, R > and R < ,are markedly different. R > resistances are random, allexceeding R < . The applied bias concentrated on thestrongest of R > resistors (in the manner of Fig. 3) willchange them to R < by switching , i. e. by long-livedstructural transformation not adaptable to subsequentvoltage variations. In the steady state bias regime, thenext strongest resistor will be stressed with practically the same voltage as opposed to the above discussed caseof volatile non-ohmicity in the standard percolation clus-ters. In the first approximation, an originally resistivepercolation bond will transform into its conductive stateby n discrete steps where n is the number of its micro-scopic resistances, similar to a falling row of dominoesarranged in the order of descending ξ ’s.The latter behavior can be more complex in largePWPs due to the geometrical field distortion illustratedin Fig. 4. There, the field will eliminate large resistancesin the region of characteristic length given by Eq. (13)which then becomes a sort of low resistive protrusioninto the bulk material. Such a protrusion will concen-trate the electric field similar to the lightning rod effect.As a result, switching will take place in the next highfield domain growing that protrusion further, etc., untilit reaches the opposite electrode. While the kinetics ofsuch a process can be readily described, we will omit ithere.For switching to occur, the local field on a microscopicresistor must exceed a certain critical value E c , typ-ically on the order of 10 − V/cm, and E c decreaseslogarithmically with the electric pulse (spike) length. That temporal dependence opens a venue to the spiketiming dependent plasticity (STDP), which is anotherimportant property of neural networks. Because almostthe entire voltage drops across a microscopic resistanceof small linear size a , the microscopic field E is signifi-cantly stronger than the apparent macroscopic field andwas shown to approach ∼
20 MV/cm, well above thevalues of E c sufficient for threshold switching. E. Plasticity in the pulse regime
Pulse excitation regime brings in additional physics. The random elements of a percolation bond will behaveas resistors when their relaxation times τ are shorter thanthe pulse duration t ; however the elements with τ (cid:29) t will act as capacitors causing no significant voltage dropin response to short pulses. The boundary between thesefast and slow elements is defined by τ = t , i. e. ξ = ξ t ≡ ln( t/τ ) given that τ = τ exp( ξ ).The resistance R t = R exp( ξ t ) will be the highest ofall bond elements resistances and the voltage pulse willconcentrate on it for its duration. Should the correspond-ing local field exceed E c , the resistance R t will switch to R < with other resistors intact, which leads to the inter-electrode path resistance decreasing by roughly a factorof [ e ] (base of natural logarithms) as illustrated in Fig.5. It is straightforward to show that the number of suchstepwise changes in a bond resistance is estimated as (cid:112) qV /kT ∼ , (17)which property can, in principle, be used to create multi-level memory operated by trains of pulses.Corresponding to Eq. (17), the characteristic voltageper microscopic resistor is given by, U ≈ U = U L (cid:115) kTqU L , (18)where U L is the macroscopic voltage across the bondof length L . According to the numerical estimates inSec. VIII below, U can create the microscopic field ofstrength exceeding the switching value. FIG. 5: A series of stepwise drops in a PWP bond resistancein response to a train of pulses.
The concept of slow resistors acting in a manner ofcapacitors has been proposed and verified earlier.
Itmay be appropriate to additionally explain here that ca-pacitors do not accommodate significant voltages whenin series with resistors because they conduct the displace-ment currents, j D = ( ε/ π )( d E /dt ), where ε is the dielec-tric permittivity. The overlay between the capacitor andresistor regimes takes place when j D = j = σ E where j is the real (charge transport) current, E is the electricfield strength, and σ is the conductivity. The displace-ment current through a capacitor is due to the rate offield change, unrelated to voltage, rather than the fielditself proportional to voltage in a resistor.The ratio of displacement vs. real current can be pre-sented as j D /j = ( ε/ πσ )( d ln E /dt ). The expression inthe first parenthesis represents the Maxwell’s relaxationtime τ , while the reciprocal of the second parenthesisgives the pulse duration. That takes us again to thecriterion τ (cid:29) t for the element of a percolation clusteroperating in the capacitive mode.Relating this understanding with microscopic models,we note that the displacement currents are due to charg-ing/discharging processes in, say, capacitor electrodes,or in certain defect configurations responsible for electricpotential distributions in percolation clusters. F. Reverse plasticity
The above description does not address the importantquestion of reversibility of switched structures back tohigh resistive states. The feasibility of such reverse pro-cess follows from the known practices of RRAM and PCMoperations. More specifically, two conceivable answersinclude thermal annealing towards the original high re-sistive state by Joule heat generated by relatively lowcurrents over considerable times. Another possibility isbased on PWPs built of materials with a degree of fer-roelectricity allowing reversibility in response to electricpolarity changes. V. EXAMPLES OF FUNCTIONALITY
Multivalued memory.
When high enough voltage is ap-plied between a pair of PWP electrodes, their connectingpath will change its resistance depending on the pulseduration as explained in Sec. IV D. Given for example N = 3 electrodes on each of the faces of 1x1x1 cm cube,the number of perceptive pathways M = N ! ∼ cm − is theoretically higher than that of human cortex resultingin a higher memory capacity than the current 3D crossbararchitecture. It will be further enhanced with the func-tionality of multiple records per one electrode pair. Weshould admit however that various unaccounted factorscould interfere and the claim of that superior memorycapacity remains to be validated experimentally. Generation of random numbers.
As shown in Sec.IV A, not-too-spatially-close pairs of electrodes have ran-dom resistances uncorrelated with any desired accuracywhen their spacial separation increases. They form amultitude of uncorrelated random numbers.
Matrix-vector multiplication.
The measurement basedoperation of matrix-vector multiplication follows fromFig. 1. Suppose that A i is the desired product of the vector J j and the matrix F ij . We rescale J j with acertain multiplier ( z ) to a convenient interval of elec-trode voltages E j . Secondly, using a proper multiplier( z ), we rescale F ij so that all its elements fall in theinterval of PWP system conductances, δG = δR/ (cid:104) R ij (cid:105) with δR from Eq. (12). The desired product becomes A i = z z I i where I i = (cid:80) j G ij E j is the current throughthe i th electrode in Fig. 1. Because the conductancematrix G ij = R − ij contains exponentially large number( M (cid:29)
1) of elements covering interval δG , any desiredvalue of G ij can be located at least approximately amongthe measured conductances with fairly good accuracy without any additional actions . After that, applying volt-age E j to the electrode j produces a measurable current G ij E j through electrode i , which can be stored e. g. as apartial charge on a certain capacitor C i . Measuring thetotal of all such partial contributions supplied by elec-trode i in response to various E j will give the componentof sought vector I i . That procedure can be further im-proved by choosing different approximate G ij s and usinglinear regression for all the chosen values. Brain-like associative learning commonly illustratedwith Pavlov’s dog salivation experiments (see e. g. Ref.42) is readily implemented utilizing shared portions be-tween bonds of a PWP cluster, such as bonds (1,5) and(1,7) in Fig. 1. Identifying the ‘sight of food’ and ‘sound’stimuli with signals on the electrodes 5 and 7, predictsthat properly and simultaneously triggering both willswitch their corresponding pathways (1,5) and (1,7) toa low resistive state making both salivation triggering(through the output on electrode 1). In general, conduc-tive pathways connecting various pairs of the electrodesand sharing the same portion of a PWP cluster will bemutually affected by a single bias-induced change. Thatdemonstrates a single-trial learning model for storage andretrieval of information resembling that of the cortex ofthe mammalian brain.
Other functionalities based on PCM and RRAM struc-tures for neuromorphic computing appear all attainablewith PWP systems. We note that even without utilizingtheir plasticity, PWPs can serve as high capacity tunablenonlinear reservoirs for reservoir computers. For exam-ple, introducing nonohmic (yet volatile) changes in theresistances of pathways (1,5), (11,5), and (5,7) will pro-duce measurable changes in the resistances of pathways(11,3), (1,7), and (11,7), and the latter will depend ontemporal order in which the former changes were intro-duced. Utilizing the system plasticity will significantlyadd to that functionality. In fact, our proposed PWPsrepresent exponentially more powerful reservoir comput-ing systems compared to the one built using a limitednumber of memristors. VI. PWP METRICS AND IMPLEMENTATIONS
We briefly mention several metrics of the proposedPWP devices following the nomenclature used for otherneuromorphic systems. (1) Dimensions and architec-ture.
The above estimate of a superior information den-sity may be reduced to account for larger physical di-mension of a single microscopic resistor. However, evenassuming a microscopic resistor of PWP a in the rangeof tens of microns yields the density ∼ cm − . (2) Energy consumption . Assuming a PWP strucure madeof the same materials as the existing PCM and RRAM,we expect its energy efficiency to be superior becauseof the lack of interconnects requiring costly energy. (3)
Operating speed/programming time . Generally, PWP de-vices RC times are greater than those of nano RRAMand PCM. Like other brain-inspired systems, their com-putational efficiency will be achieved through the highdegree of parallelism. We recall in this connection thatthe combinatorial huge number of memory units N ! isexponentially higher that the number of electrodes N .(4) Multi-level states . Assuming a ∼
10 nm microscopicresistor and 1 cm device, each bond in a PWP clusterwill contain hundreds of micro-resistors; hence, hundredsof multi-level states per typical bond, at the level al-lowing robust analog operations. (6)
Retention and en-durance . PWP systems can be superior to the existingPCM/RRAM based devices because of the lack of mul-tiple interconnects triggering degradation.
VII. SIMILARITY WITH BIOLOGICALSYSTEMS
The PWP architecture and functionality have similar-ities with that of biological neural networks. We re-call that the latter consists of individual neurons inte-grating input signals and firing pulses upon exceeding acertain threshold of integration. The neurons communi-cate with network by means of synapses that inject ionsthrough their membrane ionic channels then electricallyaltering adjacent neurons. The neuro-synaptic entitiesare connected with each other through axons providingpathways for electric pulses. A significant degree of ran-domness and stochasticity is introduced by ion channelswhose concentrations and characteristics vary even be-tween nominally similar biological membranes.
The property of firing pulses upon accumulatingenough electric charge is found in PWP’s slow elements( τ > t ) that operate as capacitors. Such elements turninto resistors when, through their multiple connectionsin PWP, they acquire voltages sufficient to make realcurrents larger than the displacement ones. Such accu-mulation becomes possible when the signals arrive withina certain time interval thus resembling STDP in neuralnetworks. The inherent randomness of PWP correlateswith that in biological neural networks.Note that the neuron analogy described here per-tains to the system functionality, but not the structure.In PWP, slow elements can be associated with any ofthe system elements, depending on the pulse duration.Therefore, the PWP neuron-like elements are in a sense distributed throughout the system and the same elementcan play the role of a synapse, or axon, or neuron de-pending on the excitation conditions. The concept offunctional similarities between PWP and biological neu-ral networks as outlined here remains in its infancy call-ing upon further analyses.
VIII. NUMERICAL ESTIMATES OF PWPPARAMETERS
Assuming a ∼ V max /kT ∼
100 yields L c ∼ aV max /kT ∼
100 nm. Empirically, the field reliablyleading to switching is < ∼ U /a , with U from Eq.(18), which yields U L < ∼ . < ∼
10 kVacross 1 cm thick samples.We conclude that multivalued memory by switching inPWP corresponds to rather moderate voltages at leastfor large PWP systems. Because the earlier estimated ∼
10 records corresponds to the bond of correlation ra-dius L c ∼
100 nm, the number of records per 1 cm thicksample is estimated as ∼ . The question of theirtemporal overlaps remains to be addressed based on theknown statistical properties of percolation clusters.For small PWP, using the same parameters and samplesize L < ∼ L c one obtains ∼ L/L c possible transitions,which however will be all distinctly different. Note thatfor 10 nm samples that estimate predicts one switchingevent, which on average is what is observed in the currentRRAM and PCM devices. IX. THE ROLE OF RANDOMNESS
A characteristic feature of our proposed PWPs is thatthey are generically random and possess exponentiallybroad spectra of parameters. As such, they are sugges-tive of probabilistic algorithms that employ a degree ofrandomness as part of its logic. We are not aware ofany general concept substantiating the advantages of ran-domness for computations, although the challenge of itis quite appreciated by now and became a hot topic.
The uniqueness of our approach is that it proposes agenerically ‘random hardware’ with which even a logi-cally deterministic algorithm becomes randomized. Thegeneral power of such approach remains to be tested yet,although some examples do point at its potential: superi-ority of randomly-wired networks, and intentionallyrandomized reservoir computers. Biological neural net-works provide of course an ultimate example of neuro-morphic computing leveraging randomness (we are notbuilt of well controlled silicon nano-chips).From that perspective, our approach proposes an in-teresting and practically implementable machinery fortesting the potential of ‘randomized neuromorphic hard-ware’. That goal can be achieved both through com-puter modeling of PWP that benefits from the earlierdeveloped percolation modeling algorithms, and throughdirect material implementations, such as e.g chalcogenidefilms with multiple electrodes; both directions are beingattempted by our group.
X. CONCLUSIONS
We have introduced the percolation with plasticity(PWP) systems that, while being essentially random,exhibit various neuromorphic functionalities and simi-larities to the cortex of mammalian brain. These sys-tems demonstrate rich physics, understanding of whichremains in its infancy.The outstanding challenges in theory go far beyondthe standard percolation paradigm including statisticalaspects, microscopic phase transformations, heat trans-fer, AC propagation and signal cross-coupling in randomsystems with multiple co-existing interfaces. Both ana-lytical research and numerical modeling will help to bet-ter understand PWP operations and functions. Multiple material bases can be used to fabricate PWP devices,ranging from the known PCM and RRAM materials tothe nanocomposites which beg to be further exploredfor memory and other neuromorphic applications. Theabove noted functional similarities with biological neuralnetworks suggests the need for further investigations.To avoid any misunderstanding we should emphasizethat the present work is limited to the basic theory ofPWP systems. It only tangentially addresses the re-lated neuromorphic applications that can be tried at boththe levels of software implementations and real materialdevices. Obvious experimental projects potentially trig-gered by this work include pulse regime nonohmicity indisordered systems, statistical properties of percolationclusters with multiple electrodes, and reservoir comput-ing using with percolation systems.
Acknowledgements
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