Window functions and sigmoidal behaviour of memristive systems

0.85 MB·

by  Panayiotis S. Georgiou

Window functions and sigmoidal behaviour of memristive systems
Panayiotis S. Georgiou,1,∗ Sophia N. Yaliraki,2 Emmanuel M. Drakakis,1 and Mauricio Barahona3,†
1Department of Bioengineering, Imperial College London, London SW7 2AZ, United Kingdom
2Department of Chemistry, Imperial College London, London SW7 2AZ, United Kingdom
6 3Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
1
0 A common approach to model memristive systems is to include empirical window functions to
2 describe edge effects and non-linearities in the change of the memristance. We demonstrate that
under quite general conditions, each window function can be associated with a sigmoidal curve
n
relating the normalised time-dependent memristance to the time integral of the input. Conversely,
a
this explicit relation allows us to derive window functions suitable for the mesoscopic modelling of
J
memristive systems from a variety of well-known sigmoidals. Such sigmoidal curves are defined in
4
terms of measured variables and can thus be extracted from input and output signals of a device
1
and then transformed to its corresponding window. We also introduce a new generalised window

function that allows the flexible modelling of asymmetric edge effects in a simple manner.
]
l
l
a Keywords: memristor; memristive system; window; sigmoidal; modelling
h
-
s I. INTRODUCTION ever, the underlying mechanisms for these induced
e
m changes are complex and differ significantly among
The memristor, which was originally defined by devices [5]. As a result, a myriad of models con-
.
t Leon Chua in 1971 based on symmetry arguments, structed from microscopic physical mechanisms or
a
m is the one-port element that relates charge and flux- based on fitting to macroscopic experimental mea-
linkage [1]. The memristor completes the tetrad surements (or combinations of both) have been pro-
-
d of ideal passive elements together with the resis- posed [9–16]. As an alternative to such detailed
n tor, capacitor and inductor. For decades, memris- models,macroscopicempiricalmodelshavebeende-
o tor research was mostly of theoretical interest until veloped to capture input-output responses, and in
c in 2008 researchers at Hewlett Packard (HP) fabri- particular, properties related to hysteresis, memory,
[
cated a nano-scale device whose behaviour was de- edge effects, and nonlinearities in the change of the

1 scribed with a memristor model [2]. The experi- memristance. Inthissetting,empiricalwindowfunc-
v mental and theoretical interest in memristors has tions are commonly used to encapsulate the nonlin-
1 since grown rapidly, motivated by their possible ear mechanisms that affect the change in the mem-
2
use to improve existing circuits, as well as enabling ristance and limit its range of values. Following the
0
2 novelapplications. Theirintrinsicnon-volatilemem- window function introduced by the HP group [2],
0 ory [3], in combination with their low-power con- several groups have since introduced window func-
. sumption [4], nano-scale size [5], high switching tionscapturingdifferentobservedproperties[17–22].
5
speed[6]andsynapse-likebehaviour[7],arealldesir- Forareviewofexistingwindowfunctionsandrelated
0
6 able properties for potential uses in computer mem- models the reader may refer to Refs. [23–25].
1 ories, as well as reconfigurable, neuromorphic and
Herewestudythemathematicalpropertiesofwin-
: learning/adaptive circuits [8].
v dow functions in terms of the associated saturat-
i A wide range of resistance-switching devices have ing behaviour of the memristor. In particular, we
X been classified as memristive. In most cases, the
show that, under some general assumptions, a win-
r device exhibits two limiting resistance states (‘high’
a dow function leads to a sigmoidal in the observable
and‘low’)withanelectrically-inducedchangeincon-
signal of the system. This sigmoidal function can
ductance(i.e.,byapplyingvoltageorcurrent). How-
thus be obtained from input-output measurements
and used to infer the underlying window function
from the data. We also introduce a generalised win-
∗ [email protected] dow function with adjustable parameters that allow
† [email protected] us to flexibly describe asymmetric boundary effects
2
1 which relates the charge q and flux-linkage ϕ. The
memristorissaidtobecharge-controlledifitispossi-
ble to express the implicit relation (1) as an explicit
function of the charge: ϕ = ϕˆ(q). It then follows
that
0 v =M(q)i(t), (2)
0 0.5 1
whichdescribestheoutputvofthecharge-controlled
memristor with memristance M(q) under the time-
varying current input
P t
P
t
dq(t)
i= . (3)
dt
NotethatthememristanceM(q)=dϕ/dq isafunc-
tion only of the time integral of the input and is
equal to the ratio between voltage and current (i.e.,
outputandinput)ateachtime,t. Thedescriptionof
a flux-controlled memristor, defined by the explicit
function q =qˆ(ϕ), follows analogous arguments.
An ideal memristor, as defined originally by
Figure 1. Memristive model and window function.
Chua [1], is characterised by a unique and time-
Whenaninput(currentorvoltage)isapplied,thebound-
invariant q −ϕ function with the following proper-
ary between the doped and undoped regions moves, in-
ducing a change in the memristance, i.e., a change in z. ties [26, 27]:
Theedgeeffectsarerepresentedbythefunctionf(z): as
A.1 nonlinear
z approaches either of its limiting values z=0 or z=1,
it becomes increasingly harder for the memristance to A.2 continuously differentiable
change. The device saturates at z = 0 or z = 1 cor- A.3 strictly monotonically increasing,
respondingtothelimitingresistancesR orR ,re-
OFF ON
spectively. ThedeviceschematicisadaptedfromStrukov and on the i−v plane its canonical form is obtained
et al [2]. from(2)and(3). Undertheseconditions,thecharge-
controlledqˆ(ϕ)andflux-controlledϕˆ(q)functionsare
invertible,andthesetworepresentationsofthemem-
ristor are equivalent [28–30].
in the device, and we use it to illustrate the math- To enable the modelling of a broader class of
ematical link between window and sigmoidal func- systems with memristor-like properties, Chua and
tions in simulated data. Finally, we exploit the ex- Kanglaterintroducedtheconceptofmemristivesys-
tensive results on sigmoidal functions to construct tems [31],inwhichthememristancemaydependnot
a list of possible window functions associated with only on internal state variables of the system, but
well-known sigmoidals, which can be used to model also can be a direct function of the input drive and
input-output memristive responses. of time.
In this work, we consider a particular form of
memristive systems that has been used widely to
II. MEMRISTORS AND MEMRISTIVE modelrecentexperiments[23]. (Henceforthwefocus
SYSTEMS on the current-driven system, but our work applies
equally to the voltage-driven case). The model as-
A memristor is defined by the constitutive rela- sumes that the memristance corresponds to a mixed
tion: [1] resistor formed by a low resistance RON and a high
resistance R (R > R ) in series, with the
OFF OFF ON
f (q,ϕ)=0, (1) total effective resistance equal to the weighted sum
M
3
of the two: III. FROM WINDOW FUNCTIONS TO
SIGMOIDALS
M(z)=zR +(1−z)R , (4)
ON OFF A. Properties of window functions and a new
generalised window
where z ∈ [0,1] is an internal (dimensionless) state
variable whose dynamics represents the nonlinear The window function f(z) in (5a) is introduced
change in the memristance. The equations for the as a means to describe a variety of effects, including
memristive system are then: edge effects and nonlinearities in the drift, and is ei-
ther obtained from microscopic considerations [11–
14] or, more empirically, from macroscopic argu-
z˙ =αf(z)i(t) (5a)
ments[2,17–22]. Initsverysimplestform,f(z)=1,
v =M(z)i(t), (5b) as in the first HP memristor model, which assumed
lineardriftandneglectedboundaryeffects[2]. How-
ever, already in Ref. [2], a second model with a
where α is a constant of appropriate dimensions
parabolic window was proposed to reflect boundary
(here Coulomb−1), and f(z) is the dimensionless
effectsanddriftnonlinearities. Subsequently, avari-
window function that encapsulates boundary effects
ety of window functions have been introduced based
andnon-linearitiesinthechangeofthememristance.
on different semi-empirical principles to reflect dif-
Such non-linearities may arise because of different
ferent underlying properties [17–22].
underlying mechanisms, for example, due to non-
Most of the proposed window functions [2, 17,
linear ionic drift. In the original HP memristor
19, 20] (although not all [18, 21, 22]) share a com-
model [2], the internal state variable z was associ-
monmathematicaldescriptiongivenbythefollowing
ated with the position of the front separating the
specifications:
doped and undoped regions in the TiO nanoscale
2
structure, and a window f(z) with parabolic shape
B.1 f(z):[0,1]→[0,1]
was adopted to represent edge effects (Figure 1).
B.2 f(0)=f(1)=0
To avoid notational confusion, we remark that in B.3 single maximum f(z )=1
max
his recent new classification of memristors [32, 33], B.4 continuous differentiability of f(z),
LeonChuadefinesthedevicescharacterisedby(5)as
i.e., the window is a strictly concave function in the
ideal generic memristors. Additionally, in those pa-
normalised internal state variable z ∈[0,1] with two
pers, Chua relaxes the requirements for ideal mem-
roots at z = 0 and z = 1 and a single maximum at
ristors allowing them to be not only passive devices
z =z .
(as originally defined in [1]) but also active devices. max
In this work, however, we follow strictly the defi- Within this general form, we combine and extend
nitions given by Chua in the original Refs. [1, 31] severalfeaturesofpreviouslyproposedwindowfunc-
and restrict our analysis to (passive) ideal memris- tions into the following generalised window:
tors, since (A.1)–(A.3) constitute the minimal set of
f(z;r,p)=(cid:2)ψ(r)z(1−zr−1)(cid:3)p, r (cid:54)=1 (6a)
properties giving rise to memristive behaviour (e.g.
hysteresis, memory, zero-crossing) [27]. Under the 1
ψ(r)= (6b)
new classification, the system in (5) falls into the z (1−zr−1)
max max
class of ideal generic memristors. It was recently
shown [32, 33], as demonstrated here, that under where the parameters p ∈ (0,1] and r ∈ R+ −{1}
certain conditions this class reduces to ideal mem- offertheflexibilitytoadjusttheasymmetryandflat-
ristors. For the sake of clarity, we will henceforth ness of the window. Here ψ(r) is a normalization
follow strictly the original terminology in [1, 31]. constant such that f(z ;r,p) = 1, and the maxi-
max
We remark that our aim here is not to introduce a mum is at z = r1/(1−r). Note that (6) satisfies
max
new class of memristors, but to explore the full im- the properties (B.1)-(B.4).
plications of considering fundamental properties of Thewindowfunction(6)providesageneralization
strictly passive ideal memristors (see also [34, 35]). of previous windows [2, 17, 19, 20], so that it can
4
f(z) f(z) f(z)
1 1 1
z z z
z z z
0 max 1 0 max 1 0 max 1
Figure 2. The effect of the parameters r and p on the window (6). The parameter r controls the position of the
maximum z and, as a result, the asymmetry of the window: (a) 0 < r < 2, left skewed; (b) r = 2, centred; (c)
max
r>2,rightskewed. Theparameterpcontrolstheflatnessoftheregionaroundz ,hencethedepthoftheboundary
max
effects. Thisisillustratedinallthreeplotswherethewindowisplottedfordifferentpvaluesforeachvalueofr. Asp
approaches0thewindowbecomesincreasinglyflatter. Thespecificvaluesusedarer=0.8in(a)andr=12.2in(c).
account for the asymmetry of the nonlinear profile Any non-unit sigmoidal S˜ can be converted to its
characterizing the change in memristance and the unit sigmoidal using S(x)=(S˜(x)−L)/(U −L).
varying depth of the boundary effects, as illustrated Consider now a window function f(z) complying
in Figure 2. In particular, the parameter p deter- with (B.1)-(B.4). Then its reciprocal
minestheflatnessoftheregionaroundz ,whereas
max 1
the parameter r controls the asymmetry of the win- f¯(z):= (7)
dow,i.e.,for0<r <2(excludingr =1)thewindow f(z)
is left skewed, while for r >2 it is right skewed.
has the following properties:
D.1 f¯(z):(0,1)→[1/f(z ),+∞)
max
D.2 lim f¯(z)=lim f¯(z)=+∞
B. Relating window functions to sigmoidals z→0+ z→1−
D.3 single minimum f¯(z )=1
max
D.4 continuous differentiability of f¯(z),
Wenowshowthattheclassofwindows(B.1)-(B.4)
can be related to sigmoidal functions. Sigmoidal i.e., f¯(z) is a strictly convex function in z ∈ (0,1)
functions are elongated S-shaped curves, which are with a minimum at z = z and asymptotes at
max
usedasagenericrepresentationofsaturation-related z =0,1.
phenomena in a variety of fields, from neuroscience We can use these properties to characterise the
andbiologytophysics. Formally,afunctionS :R→ solution of the memristive differential equation (5a)
R is a sigmoidal function if, and only if [36]: governingthedynamicsoftheinternalstatevariable.
Using the separability of (5a), it follows that:1
C.1 S is monotonic;
C.2 infRS =L, supRS =U. F(z):=(cid:90) zf¯(ξ)dξ =α(cid:90) ti(τ)dτ =α(q(t)−q ),
0
0 0
Hence as x increases, the sigmoidal increases mono- (8)
tonically from a lower asymptotic value (L) towards
its upper asymptote (U), and the slope of the curve
increases until it reaches a maximum at the point
of inflection (xσ) after which it decreases. In this 1 To remove the singularity of (8) at z = 0 and z = 1, in
work, all the sigmoidals S are (normalised) unit sig- practicetheintegralisevaluatedintherangez∈[δ,1−δ]
moidals, i.e., bounded between L = 0 and U = 1. where0<δ(cid:28)1,asdetailedin[37,38].
5
whereq(t)isthetotalchargethathaspassedthrough these assumptions, the system is also an ideal mem-
thedevicetakingintoaccountthepolarityofthein- ristor,asspecifiedby(A.1)-(A.3).2 Toseethis,inte-
put with q := q(0) the initial charge and z = z(t) grate(11)withrespecttottoobtaintheconstitutive
0
with z(0)=0. It follows from (D.1)-(D.4) that F(z) relation in the q−ϕ plane:
is a strictly increasing function since f¯(z) > 0 for
(cid:90) q
z ∈ (0,1) with a single point of inflection at z .
max ϕ(q)= M(S(ξ)) dξ. (12)
It is also bounded between the two asymptotes at
q0
z = 0 and z = 1. Furthermore, since F(z) is
strictly increasing, its unique inverse exists in the Because the memristance M(S(q)) is a positive
range of interest z ∈ (0,1) and is itself monotonic function bounded between R and R the sys-
ON OFF
andboundedbetweenz =0andz =1. Insummary, tem is passive [1]. It also follows that each window
the state variable satisfies the requirements of a sig- satisfying (B.1)-(B.4) corresponds to a unique ϕ(q),
moidal C.1–C.2. Consequently, F(z) is the inverse whichisastrictlyincreasingnonlinearfunctionofq,
of a scaled and shifted sigmoidal of the charge. Let thus satisfying all the requirements of an ideal pas-
z = S(q) indicate the sigmoidal curve, we can then sive memristor. We remark that in the context of
rewrite (8) as: Chua’s new classification of memristors [32, 33], the
procedure detailed above converts an ideal generic
F(z)=α(cid:0)S−1(z)−q0(cid:1), (9) memristor into its corresponding ideal memristor.
In conclusion, although the memristive sys-
andthesigmoidalofthechargeisobtainedfrom(9):
tem (5a) incorporates a window function to model
(cid:18) (cid:19) nonlinearities and boundary effects, the separability
F(z)
z =F−1(α(q−q0))=S α +q0 =S(q). (10) of the state equation together with the form of the
window implies that the system is equivalent to an
Substituting (10) in (4) and (5b) gives the output ideal memristor. Note that the window functions in
voltage in terms of the input and its integral: Strukov et al [2], Benderli et al [19], Prodromakis
et al [20] and Joglekar and Wolf [17], as well as the
v =M(S(q))i(t)=[S(q)R +(1−S(q))R ]i(t). newwindowproposedhereinEq.(6),fallunderthis
ON OFF
(11) class, and the corresponding memristive systems are
Thisexpressionshowsthatthenonlinearityofthere- ideal memristors. On the other hand, the window
sponse is encapsulated in the sigmoidal curve S(q). function in Ref. [18] does not fulfil (B.3) and, as a
Inparticular,thememristanceM(S(q))isalinearly result, the characteristic q − ϕ curve is a double-
transformedsigmoidalwithasymptoticvaluescorre- valued function [21] which cannot be classified as an
sponding to the limiting resistances of the device, idealmemristor. Thewindowfunctionsproposedby
R and R . The sigmoidal indicates that the Takahashi et al [22] and Corinto et al [21] are also
OFF ON
effect of the integrated input (i.e. charge) on the excludedfromouranalysissincetheydonotcomply
stateofthesystembecomessmallerclosertoitstwo with(B.3). ThewindowbyCorintoetal [21,24]may
saturation boundaries (z =0 or z =1). After a lim- also introduce step discontinuities, violating (B.4)
itedamountofchargeisinjected,thesystemreaches and leading to a non-ideal memristor. Nevertheless,
saturationandbehavesasalinearresistorwithresis- if such windows are carefully defined to avoid the
tance R for z = 0, or, R for z = 1. Between introduction of discontinuities, the procedure pro-
OFF ON
the two saturation limits, the sigmoidal curve deter- posed above is still applicable and can lead to an
mines the amount of integrated input necessary to ideal memristor.
induce a change in the state of the system. There- Example 1: We use the new generalised win-
fore, the window in the state equation enforces the dow(6)toillustratetheconnectionbetweenwindow
boundary conditions (by limiting z) as well as mod-
eling the nonlinear change of z.
From (11) it is clear that the memristive sys-
tem (4)–(5) with a window (B.1)-(B.4) is a charge- 2 Note that the properties (B.1)-(B.4) of f(z) assumed here
controlled, current-driven memristor as defined are sufficient conditions for ideal generic memristors, yet
in (2). In addition, it can be shown that, under constituteastrictersubsetofthoseproposedin[33].
6
Sigmoidal
1
0
Window
1
Input Output
0
0 0
Memristance
0
Input
Output
0
Figure 3. Edge effects in the response of the memristive system (4)-(5) with window function (6) to a sinusoidal
inputcurrent,i(t)=i sin(2πt/T ). Symmetricwindows(r=2)withvaryingdepthsofedgeeffects(p=0.05,0.5,0.9)
0 0
are presented together with their corresponding sigmoidal curves. The diagram illustrates the relation between the
sigmoidalandthewindowfunction,aswellastheinput/outputsignalsandthememristance. Asseeninthesigmoidals,
higher values of p (which correspond to deeper edge effects) result in a higher charge threshold for the memristance.
The red dots indicate the point at which the rate of change is maximal; for higher p, this point is delayed towards
the end of the period T and, as a result, the memristance changes noticeably later. Model and input parameters:
0
i =0.3mA, T =8s, α=5000C−1, R =50Ω, R =2.5kΩ and z =10−4.
0 0 ON OFF 0
7
Sigmoidal
0
Window
Input Output
0
0 0
Memristance
0
Input
Output
0
Figure4. Asymmetryeffectsintheresponseofthememristivesystem(4)-(5)withwindowfunction(6)toasinusoidal
input current, i(t) = i sin(2πt/T ). In this case, the parameter p controlling the depth of the edge effects is kept
0 0
fixedatp=0.5andtheeffectoftheasymmetryisexplored,withvaluesr=0.5(leftskewed),r=2(symmetric),and
r = 8.4 (right skewed). As in Figure 3, the diagram illustrates the relation between the sigmoidal and the window
function, as well as the input/output signals and the memristance. The effect of asymmetry in the sigmoidals is to
modifythepositionoftheinflectionpoint,whichalsomodifiesthecharacteristicshapeofthei−v curve. Modeland
input parameters as in Figure 3.
8
functions and sigmoidals. Following (8), we first in- which is then differentiated to obtain
tegrate the reciprocal of (6) to obtain:
d[F(z)] 1
(cid:16) (cid:17) = , (18)
z1−pr F p,1−pr,1+ 1−pr,z1−r dz 4z(1−z)
2 1 1−r 1−r
F(z)= , (13)
(1−pr) [−ψ(r)]p yielding the window
where F (·) is the Gaussian hypergeometric func-
2 1 f(z)=4z(1−z). (19)
tion [39]. The corresponding sigmoidal can then be
obtained by numerically inverting this function. In Hence the corresponding window for (16) is a scaled
Figure 3 we use these formulas to illustrate the re- versionofthewindowinStrukovetal [2]andclosely
sponse of the system (4)-(5) with window (6) to a related to the logistic curve [40, 41]. Note that by
sinusoidal input. The window is chosen to be sym- setting p=1 and r =2 in (6) it can be verified that
metric (r =2) and we vary the value of the parame- this window is a subclass of the generalised window
terpdeterminingtheflatnessofthewindow,andwe proposed here.
also show the corresponding sigmoidals. Note how
higher values of p, corresponding to deeper edge ef-
fects,leadtolargerchargethresholdforthememris- C. Summary of transformations and examples
tance to rise. Similarly, Figure 4 demonstrates how
the response of the same system changes when the
Thetransformationsbetweenthewindowfunction
window (6) is made asymmetric by keeping the pa-
and the sigmoidal can be summarised as follows.
rameter p fixed while varying r.
Given a memristive system (4)–(5) with window
Ifwefixp=1,wecanobtainthesigmoidalforour
function f(z) satisfying (B.1)–(B.4), then the inter-
window (6) analytically. In this case, (13) simplifies
nal state variable z follows a sigmoidal curve
to
ln(cid:12)(cid:12)1−z1−r(cid:12)(cid:12) z =S(q)=F−1(α(q−q0)), (20)
F(z)= =αq(t), (14)
(1−r)ψ(r) (cid:82)
where F(z)= 1/f(z)dz and z ∈(0,1).
where we assume q = 0 for simplicity, and the in- Conversely, if the sigmoidal z = S(q) is known,
0
verse of (14) then gives the sigmoidal: then we compute the corresponding window f(z) by
applying:
(cid:104) (cid:105) 1
S(q)= 1±eαq(1−r)ψ(r) 1−r . (15)
1 =α d (cid:2)S−1(z)(cid:3), (21)
f(z) dz
Example 2: Conversely, we can convert sig-
moidal curves to their corresponding window func-
which follows from inverting (10) and consider-
tions. This reverse transformation can be used to
ing (8). An illustration of these transformations is
suggest possible window functions based on the ex-
shown in Figure 5.
tensive range of sigmoidals studied in the literature.
Thesetransformationsallowustoobtainwindows
To illustrate this point, consider one of the most
from the wide range of sigmoidal functions exten-
commonly used sigmoidals, i.e., the hyperbolic tan-
sively used in the modelling of various physical and
gent standardised to the unit interval
biological processes [40, 42, 43]. The windows ob-
1 tainedcanthenbeusedwhenfittingdata,aswellas
S(q)= [tanh(2αq+α )+1], (16)
2 1 giving insight to the underlying processes responsi-
ble for the observed dynamics. Without attempting
where α ∈R is a dimensionless constant determin-
1 to be exhaustive, we have computed windows from
ing the horizontal shift and α ∈ R+ controls the
some of the most commonly used sigmoidal curves
growth rate of S(q). Assuming again q = 0, it fol-
0 with different characteristics. The calculations lead-
lows from (9) that
ing to these windows follow the same pattern as
1 above (hence not shown), and we list the results in
F(z)=αS−1(z)= 2(artanh(2z−1)−α1), (17) Table I.
9
1
1
1
0 0 0
0 1 0 1 0 1
Figure5. Illustrationofthetransformationsrequiredtoconvertawindowfunctionf(z)toasigmoidalz=S(q)(from
left to right, red arrows) and vice-versa (from right to left, green arrows).
IV. WINDOW FUNCTIONS FROM resistance of the device. This leads us to consider a
EXPERIMENTAL SIGMOIDALS normalisedinstantaneoustime-dependentresistance:
[v(t)/i(t)]−R
The foundations laid above suggest a series of z(t)= OFF, (22)
R −R
steps through which suitable window functions can ON OFF
befittedtoexperimentaldataundertheassumptions which scales the resistance with respect to the mini-
thatthemeasureddevicewillconformtotheclassof mum(R )andmaximum(R )resistancestates
ON OFF
windowed memristors studied above. In particular, ofthedevice. Theselimitingstatesneedtobedeter-
the experimental input-output data can be used to mined by driving the system to both saturation lim-
extractthesigmoidalofthetime-integraloftheinput its in a sustained manner with two input signals of
z = S(q), which can then be related to the window opposite polarity. This process calibrates the range
function by using the transformation (21). This re- z ∈ [0,1] by mapping the range of memristances to
constructed window function empirically models the M∈[R ,R ].
OFF ON
nonlinearity of z˙, and can be tailored to the specific Once R and R have been obtained, we ini-
ON OFF
device under investigation. In particular, different tialise the device to R or R and apply an in-
ON OFF
sigmoidals could be fitted and compared to select put i(t) measuring the instantaneous output voltage
the most appropriate description and window func- v(t) at sampling times t , i=1,...,N. Any type of
i
tion. Conversely,ifaphysicalmodelexiststoinform inputcanbeusedprovideditforcesthedevicetoex-
thechoiceofwindow,thetransformation(20)canbe plore the entire range of the memristance. For each
used to obtain the corresponding sigmoidal, which sampling time, we evaluate the accumulated charge
can then be fitted to the data. q(t) = (cid:82)tii(τ)dτ and the state variable z(t ) using
0 i
The relation between the sigmoidal, its cor- (22). We then fit the N sampled pairs (q(ti),z(ti))
responding window function, and the underlying to a suitable sigmoidal function, such as those in
input-outputbehaviourofthedeviceisillustratedin Table I. The data-fitted sigmoidal can then be con-
Figures 3 and 4. The internal state z is an abstract verted to its corresponding window function by ap-
variable which in general is not directly related to plying the transformation (21).
any measurable quantities unless certain aspects of
the system are pre-specified. However, if we assume
that the memristance is modelled by (4) then z can V. DISCUSSION
be mapped to measurable quantities. As described
above, the cumulative effect of the input becomes Window functions are commonly used in the
smallerthecloserthesystemistoanyofthebound- macroscopicmodellingofmemristivedevicesinorder
aries (z = 0 or z = 1) at which the system reaches to incorporate edge effects and other nonlinearities
saturation and behaves as a linear resistor with re- in a semi-empirical manner. Here we have shown
sistanceR orR ,respectively. Hence,whilein that, under quite general assumptions for the win-
OFF ON
saturation, the integrated input has no effect on the dow function and the memristive system, such de-
10
TableI.Commonlyusedsigmoidals[42,43]andtheircorrespondingwindowfunctions. Allthesigmoidalsareinunit
form (see Sec. IIIB) and the window functions comply with (B.1)–(B.4). The parameter r affects various features
of the sigmoidals (e.g., inflection point, growth rate, horizontal shift). The normalization ψ(r) is defined such that
f(z ) = 1. The Weibull and Morgan-Mercer-Flodin (MMF) functions can only be viewed as sigmoidals within a
max
restricted range of values [42, 43]. Note that the charge q is always scaled by an overall normalization constant α of
appropriate dimension as in Eq. (6).
Sigmoidal, S(q) Window function, f(z) Scaling, ψ(r) Parameters
(6) Inverse of Eq. (13) (cid:2)ψ(r)z(1−zr−1)(cid:3)p rr−r1 r∈R+−{1},
q. r−1 p∈(0,1]
E
(6) (cid:104)1±e(1−r)ψ(r)αq(cid:105)1−1r ψ(r)z(1−zr−1) rr−r1 r∈R+−{1},
q. r−1 p=1
E
z
t
r
pe e−re(−eαq) −ezln(z) — r>0
m
o
G
n
a 1 1
t + arctan(παq) sin2(πz) — —
rc 2 π
A
bull 1−e[−ψ(r)(αq)r] rψ(r) (cid:20)−ln(1−z)(cid:21)1−r1 er−1 −1≤r<0,
ei (1−z)−1 ψ(r) r(r−1)r−1 r>1
W
F 1 (cid:20) z1−r (cid:21)−1/r (r+1)r+1
M r ψ(r) r>1
M 1+ψ(r)(αq)−r (1−z)r+1 (r−1)1−r(4r)r
vices are characterised by an internal state variable sigmoidal functions for their potential use in meso-
that exhibits a sigmoidal response to the time inte- scopic models. Conversely, based on the physical
graloftheinput. Thecharacteristicsigmoidalofthe properties of the device, a window function maybe
device is linked to the underlying window function favoured. In that case, the corresponding sigmoidal
via a bidirectional transformation that can be used can be computed and fitted to the data.
to obtain the corresponding sigmoidal for a given
In addition, we have proposed a generalised win-
window and vice versa.
dowwhichextendssomeofthefeaturesofpreviously
proposedwindows[17,19,20]toallowforadditional
Thesigmoidalprovidesadescriptionwhichismore
flexibility when fitting data. The proposed window
easily linked to measured observables and which
isrelatedtoaknownfamilyoffunctions,thegeneric
takesintoaccounttheeffectoftheinputsignalonthe
growth functions, from which most of the commonly
stateofthesystem. Specifically, thesigmoidalcurve
usedsigmoidalscanbederived,includingthelogistic
can be extracted from input/output measurements
and Richards curves [40, 41].
and reflects the accumulated effect of the input and
the saturation of the response. Such sigmoidal func- Finally, we have shown that, under the general
tionsfittedtoexperimentaldatacanbeconvertedto assumptions considered here, the incorporation of
awindowfunctiontoobtainamemristivemodeltai- a separable window function for modelling nonlin-
loredtotheexperimentaldeviceunderinvestigation. earities and boundary effects results in a system
Withinourframework,avarietyofplausiblewindow which is still an ideal memristor. In contrast with
functions can be obtained using the wide range of Ref[44], whereidealityisassessedbasedonwhether