Waveform Driven Plasticity in BiFeO3 Memristive
Devices: Model and Implementation
Christian Mayr, Paul Staerke, Johannes Partzsch, Rene Schueffny
Institute of Circuits and Systems
TU Dresden, Dresden, Germany
{christian.mayr,johannes.partzsch,rene.schueffny}@tu-dresden.de
Love Cederstroem
Zentrum Mikroelektronik Dresden AG
Dresden, Germany
love.cederstroem@zmdi.com
Yao Shuai
Inst. of Ion Beam Physics and Materials Res.
Helmholtz-Zentrum Dresden-Rossendorf e.V.
Dresden, Germany
y.shuai@hzdr.de
Nan Du, Heidemarie Schmidt
Professur Materialsysteme der Nanoelektronik
TU Chemnitz, Chemnitz, Germany
nan.du@s2012.tu-chemnitz.de,Heidemarie.Schmidt@etit.tu-chemnitz.de
Abstract
Memristive devices have recently been proposed as efficient implementations of
plastic synapses in neuromorphic systems. The plasticity in these memristive devices, i.e. their resistance change, is defined by the applied waveforms. This behavior resembles biological synapses, whose plasticity is also triggered by mechanisms that are determined by local waveforms. However, learning in memristive
devices has so far been approached mostly on a pragmatic technological level. The
focus seems to be on finding any waveform that achieves spike-timing-dependent
plasticity (STDP), without regard to the biological veracity of said waveforms or
to further important forms of plasticity. Bridging this gap, we make use of a plasticity model driven by neuron waveforms that explains a large number of experimental observations and adapt it to the characteristics of the recently introduced
BiFeO3 memristive material. Based on this approach, we show STDP for the
first time for this material, with learning window replication superior to previous
memristor-based STDP implementations. We also demonstrate in measurements
that it is possible to overlay short and long term plasticity at a memristive device
in the form of the well-known triplet plasticity. To the best of our knowledge, this
is the first implementations of triplet plasticity on any physical memristive device.
1 Introduction
Neuromorphic systems try to replicate cognitive processing functions in integrated circuits. Their
complexity/size is largely determined by the synapse implementation, as synapses are significantly
more numerous than neurons [1]. With the recent push towards larger neuromorphic systems and
higher integration density of these systems, this has resulted in novel approaches especially for the
synapse realization. Proposed solutions on the one hand employ nanoscale devices in conjuction
with conventional circuits [1] and on the other hand try to integrate as much synaptic functionality
(short- and long term plasticity, pulse shaping, etc) in as small a number of devices as possible. In
1
this context, memristive devices 1 as introduced by L. Chua [2] have recently been proposed as efficient implementations of plastic synapses in neuromorphic systems. Memristive devices offer the
possibility of having the actual learning mechanism, synaptic weight storage and synaptic weight
effect (i.e. amplification of the presynaptic current) all in one device, compared to the distributed
mechanisms in conventional circuit implementations [3]. Moreover, a high-density passive array
on top of a conventional semiconductor chip is possible [1]. The plasticity in these memristors,
i.e. their resistance change, is defined by the applied waveforms [4], which are fed into the rows
and columns of the memristive array by CMOS pre- and postsynaptic neurons [1]. This resembles biological synapses, whose plasticity is also triggered by mechanisms that are determined by
local waveforms [5, 6]. However, learning in memristors has so far been approached mostly on
a pragmatic technological level. The goal seems to be to find any waveform that achieves spiketiming-dependent plasticity (STDP) [4], without regard to the biological veracity of said waveforms
or to further important forms of plasticity [7].
Bridging this gap, we make use of a plasticity rule introduced by Mayr and Partzsch [6] which is
driven in a biologically realistic way by neuron waveforms and which explains a large number of
experimental observations. We adapt it to a model of the recently introduced BiFeO3 memristive
material [8]. Measurement results of the modified plasticity rule implemented on a sample device
are given, exhbiting configurable STDP behaviour and pulse triplet [7] reproduction.
2 Materials and Methods
2.1 Local Correlation Plasticity (LCP)
The LCP rule as introduced by Mayr and Partzsch [6] combines two local waveforms, the synaptic
conductance g(t) and the membrane potential u(t). Presynaptic activity is encoded in g(t), which
determines the conductance change due to presynaptic spiking. Postsynaptic activity in turn is signaled to the synapse by u(t). The LCP rule combines both in a formulation for the change of the
synaptic weight w that is similar to the well-known Bienenstock-Cooper-Munroe rule [9]:
dw
= B · g(t) · (u(t) − Θu )
(1)
dt
In this equation, Θu denotes the voltage threshold between weight potentiation and depression,
which is normally set to the resting potential. Please note that coincident pre- and postsynaptic
activities are detected in this rule by multiplication: A weight change only occurs if both presynaptic
conductance is elevated and postsynaptic membrane potential is away from rest.
The waveforms for g(t) and u(t) are determined by the employed neuron model. Mayr et al. [6] use
a spike response model [10], with waveforms triggered at times of pre- and postsynaptic spikes:
g(t) = Ĝ · e
−
t−tpre
n
τpre
u(t) = Up,n · δ(t − tpost
n ) + Urefr · e
tpre
n
t−tpost
− τ n
post
pre
for tpre
n ≤ t < tn+1 ,
(2)
for tpost
≤ t < tpost
n
n+1 ,
(3)
tpost
n
where
and
denote the n-th pre- and postsynaptic spike, respectively. The presynaptic conductance waveform is an exponential with height Ĝ and decay time constant τpre . The postsynaptic
potential at a spike is defined by a Dirac pulse with integral Up,n , followed by an exponential decay
with height Urefr (< 0) and membrane time constant τpost .
Following [6], postsynaptic adaptation is realised in the value of Up,n . For this, Up,n is decreased
from a nominal value Up if the postsynaptic pulse occurs shortly after another postsynaptic pulse:
−
Up,n = Up · (1 − e
post
tpost
−tn−1
n
τpost
)
(4)
The time constant for the exponential decay in this equation is the same as the membrane time
constant.
1
In 1971 Leon Chua postulated the existence of a device where the current or voltage is directly controlled
by voltage flux or charge respectively, this was called a memristor. Using a general state space description
Chua and Kang later extended the theory to cover the very broad class of memristive devices [2]. Even though
the two terms are used interchangeably in other studies, since the devices used in this study do not fit the strict
definition of memristor, we will refer to them as memristive devices in the following.
2
g in nS
u in mV
∆w in %
1.0
0.8
0.6
0.4
0.2
0.0
15
10
5
0
−5
−10
2.0
1.5
1.0
0.5
0.0
0
20
40
60
t in ms
80
100
120
Figure 1: Progression of the conductance g, the membrane potential u and the synapse weight w for
a sample spike pattern.
Figure 1 shows the pre- and postsynaptic waveforms, as well as the synaptic weight for a sample
spike train. For the simple waveforms, two principal weight change mechanisms are present: If
the presynaptic side is active at a postsynaptic spike, the weight is instantaneously increased by
the large elevation of the membrane potential. In contrast, all presynaptic activity falling into the
refractoriness period of the neuron (exponential decay after spike) integrates as a weight decrease.
As shown in [6], this simple model can replicate a multitude of experimental evidence, on par
with the most advanced (and complex) phenomenological plasticity models currently available. In
addition, the LCP rule directly links synaptic plasticity to other pre- and postsynaptic adaptation
processes by their influence on the local waveforms. This can be used to explain further experimental
results [6]. In Sec. 3.1, we will adapt the above rule equations to the characteristics of our memristive
device, which is introduced in the next section.
2.2
Memristive Device
Non-volatile passive analog memory has often been discussed for applications in neuromorphic
systems because of the space limitations of analog circuitry. However, until recently only a few
groups had access to sufficient materials and devices. Developments in the field of nano material
science, especially in the last decade, opened new possibilities for creating compact circuit elements
with unique properties.
Most notably after HP released information about their so-called Memristor [11] much effort has
been put in the analysis of thin film semiconductor-metal-metaloxide compounds. One of the commonly used materials in this class is BiFeO3 (BFO). The complete conducting mechanisms in BFO
are not fully understood yet, with partly contradictory results reported in literature, but it has been
confirmed that different physical effects are overlayed and dominate in different states. Particularly
the resistive switching effect seems promising for neuromorphic devices and will be discussed in
more detail. It has been shown in [12, 8] that the effect can appear uni- or bipolar and is highly
dependent on the processing regarding the substrate, growth method, doping, etc. [13].
We use BFO grown by pulsed laser deposition on Pt/Ti/SiO2 /Si substrate with an Au top contact,
see in Fig. 2. Memristors were fabricated with circular top plates, which were contacted with needle
probes, whereas the continuous bottom plate was contacted at one edge of the die. The BFO films
have a thickness of some 100nm. The created devices show a unipolar resistive switching with a
rectifying behavior. For a positive bias the device goes into a low resistive state (LRS) and stays
there until a negative bias is applied which resets it back to a high resistive state (HRS). The state
can be measured without influencing it by applying a low voltage of under 2V.
Figure 3 shows a voltage-current-diagram which indicates some of the characteristics of the device.
The measurement consists of three parts: 1) A rising negative voltage is applied which resets the
device from an intermediate level to HRS. 2) A rising voltage lowers the resistance exponentially.
3
Figure 2: Photograph of the fabricated memristive material that was used for the measurements.
3) A falling positive voltage does not affect the resistance anymore and the relation is nearly ohmic.
Because of the rectifying characteristic the current in LRS and HRS for negative voltages does not
exhibit as large a dynamic range as for positive voltages.
800
10−3
700
10−4
600
10−5
abs(I) in A
Im in uA
500
400
300
200
10−6
10−7
100
10−8
0
−100
−6
−4
−2
0
Vm in V
2
4
6
10−9−6
−4
−2
0
Vm in V
2
4
6
Figure 3: Voltage-current diagram of the device as linear and log-scale plot
2.3
Phenomenological Device Model
To apply the LCP model to the BFO device and enable circuit design, a simplified device model
is required. We have based our model on the framework of Chua and Kang [2]; that is, using an
output function (i.e., for current Im ) dependent on time, state and input (i.e., voltage Vm ). Recently,
this has been widely used for the modeling of memristive devices [11, 14, 15]. In contrast to many
memristive device models which are based on a sinh function for the output relationship (following
Yang et al. [14]), we model the BFO device as two semiconductor junctions. The junctions can
abstractly be described by a diode equation: Id = I0 (exp(qV /kT ) − 1) [16]. In an attempt to catch
the basic characteristics, our device could be modeled employing two diode equations letting a state
variable, x, influence the output and roughly represent the conductance:
(
)
Im = h(x, Vm , t) = I01 · (ed1 ·Vm (t) − 1) − I02 · (e−d2 ·Vm (t) − 1) · x(t)
(5)
where Vm is the voltage over the device2 and the diode like equations guarantee a zero crossing
hysteresis. The use of parameters I0i and di now allows individual control of current characteristics
for negative and positive voltages, and as shown in the previous section these are rather asymmetric
for our BFO devices. For the purpose of modeling plasticity, our focus has been on the dynamic
behavior of the conductance change; this was investigated in some detail by Querlioz et al. [15] and
has served as the basis for our model of the state variable:
dx
= f (x, Vm , t) = Γ(x) · Ψ(Vm )
dt
2
(6a)
With sinh(z) = 1/2 · (ez − e−z ), our approach is not fundamentally different from using a sinh function.
4
In the above the functions Γ(x) and Ψ(Vm ) relate to how the current state affects the state development and the effect of the applied voltage, respectively. Γ(x) is described by an exponential function.
x−Gmin
e−β1 Gmax −Gmin , Vm (t) > 0,
Gmax −x
−β
Γ(x) =
(6b)
e 2 Gmax −Gmin , Vm (t) ≤ 0, x > Gmin ,
0,
else
In Ψ(Vm ) we again favor using separate exponential over sinh functions for increased controllability
of the different voltage domains (positive and negative). Here the parameters ϕ1 and ϕ2 govern the
voltage dependence of the state modification, with α1 and α2 scaling the result. With β1 and β2 , the
speed of state saturation is set:
{
(
)
α1 · (eϕ1 Vm − 1 ,) Vm (t) ≥ 0,
(6c)
Ψ(Vm ) =
α2 · 1 − e−ϕ2 Vm , Vm (t) < 0,
For implementation, we have used one of the most prominent commercially available simulators
R
R
for custom analog and mixed-signal integrated circuit design, the Cadence⃝
Spectre⃝
. Using behavioral current sources, the equations for h(x, Vm , t) and f (x, Vm , t) can be implemented and
simulated with feasibility for circuit design. Depicted in Fig. 4 are the conductance change over
time, at different voltages, for model (Fig. 4a) and measurements (Fig. 4b). It can be seen how the
exponential dependency on device voltage gives rise to different levels of operation (Equations (5)
and (6c)). Also the saturation of conductance change for a given voltage is visible (Equation (6b)).
The sharp changes of current seen in the model are a result of our simplistic approach, whereas the
real devices show slower transitions. In addition, it can be noted that above 5 V the real device
appears to experience a significantly steeper rise in current. However, the target is to have reasonable characteristics in the region of operation below 5 V which is relevant in our plasticity rule
experiments.
1.6
Im (t)
Vm (t)
6
5
1.0
4
0.8
3
0.6
0.4
20
30
40
50
60
5
1.0
4
0.8
3
0.6
2
0.2
1
10
6
0.4
2
0.2
Im (t)
Vm (t)
1.2
Im in mA
Im in mA
1.2
0.0
0
1.4
Vm in Volt
1.4
0.0
0
70
t in s
Vm in Volt
1.6
1
10
20
30
40
50
60
70
t in s
(a)
(b)
Figure 4: Device current for different applied voltages for model (a) and measurement (b).
3 Results
3.1
Modified LCP
A nonlinearity or learning threshold is required in order to carry out the correlation operation
between pre- and postsynaptic waveforms that characterizes various forms of long term learning
[9, 17]. In the original LCP rule, this is done by the multiplication of pre- and postsynaptic waveforms, i.e. only coincident activity results in learning. Memristive devices are usually operated in
an additive manner, i.e. the pre- and postsynaptic waveforms are applied to both terminals of the
device, thus adding/subtracting their voltage curves. In order for the state of the memristive device
to only be affected by an overlap of both waveforms, a positive and negative modification threshold
is required [4]. As can be seen from equation 6c, the internal voltage driven state change Ψ(Vm )
is affected by two different parameters ϕ1 and ϕ2 which govern the thresholds for negative and
positive voltages. For our devices, these work out to effective modification thresholds of -2V and
5
Vpre in V
Vpost in V
−2
−1
0
1
2
Vpre-Vpost in V
4
2
0
−2
∆Im in %
2
1
0
−1
−2
50
40
30
20
10
0
0
20
40
60
t in ms
80
100
120
Figure 5: Modification of the original LCP rule for the BFO memristive device, from top to bottom:
pre- and postsynaptic voltages/waveforms, exponential decay with τpre resp. τpost (postsynaptic
waveform plotted as inverse to illustrate waveform function); resultant voltage difference across
memristive device and corresponding memristance modification thresholds (horizontal grey lines);
and memristance change as computed from the model of sec. 2.3
+2.3V. Thus, we need waveforms where coincident activity causes a voltage rise above the positive
threshold resp. a voltage drop below the negative threshold. In addition, we need a dependence
between voltage level and weight change, as the simplest method to differentiate between weights is
the voltage saturation characteristic in Fig. 3. That is, a single stimulus (e.g. pulse pairing in STDP)
should result in a distinctive memristive programming voltage, driving the memristive device into
the corresponding voltage saturation level via the (for typical experiments) 60 stimulus repetitions.
Apart from quantitative adjustments to the original LCP rule, this requires one qualitative adjustment. The presynaptic conductance waveform is now taken as a voltage trace and a short rectangular
pulse is added immediately before the exponential downward trace, arriving at a waveform similar
to the spike response model for the postsynaptic trace, see uppermost curve in Fig. 5. We call this
the modified LCP rule. For overlapping pre- and postsynaptic waveforms, the rectangular pulses of
both waveforms ’ride up’ on the exponential slopes of their counterparts when looking at the voltage
difference Vm = Vpre − Vpost across the memristive device for pre- and postsynaptic waveforms
applied to both terminals of the device (see third curve from top in Fig. 5). Since the rectangular
pulses are short compared to the exponential waveforms, they represent a constant voltage whose
amplitude depends on the time difference between both waveforms (as expressed by the exponential
slopes) as required above. Thus, as in the original LCP rule, the exponential slopes of pre- and postsynaptic neuron govern the STDP time windows. Repeated application of such a pre-post pairing
drives the memristive device in its corresponding voltage-dependent saturation level.
Similar to the original LCP rule, short term plasticity of the postsynaptic action potentials can now
be added to make the model more biologically realistic (e.g. with respect to the triplet learning
protocol [6]). We employ the same attenuation function as in equation 4, adjusting the duration of
the postsynaptic action potential, see second curve from top in Fig. 5.
Please note: One further important advantage of using this modified LCP rule is that both preand postsynaptic waveform are causal, i.e. they start only at the pre- respectively postsynaptic
pulse. This is in contrast to most currently proposed waveforms for memristive learning, i.e. these
waveforms have to start well in advance of the actual pulse [4], which requires preknowledge of a
pulse occurrence. Especially in an unsupervised learning context with self-driven neuron spiking,
this preknowledge is simply not existent.
6
120
120
τpre=15ms, τpost=35ms
τpre=30ms, τpost=50ms
80
60
60
40
40
20
0
20
0
−20
−20
−40
−40
−60
−60
−80
−200
−150
−100
−50
τpre=15ms, τpost=35ms
τpre=30ms, τpost=50ms
100
80
∆Im in %
∆Im in %
100
0
∆t in ms
50
100
150
−80
−200
200
−150
−100
−50
(a)
0
∆t in ms
50
100
150
200
(b)
Figure 6: Results for STDP protocol: (a) model simulation, (b) measurement with BFO memristive
device.
3.2
Measurement results
The waveforms developed in the previous section can be tested in actual protocols for synaptic plasticity. As a first step, we investigate the behaviour of the BFO memristive device in a standard
pair-based STDP experiment. For this, we apply 60 spike pairings of different relative timings at
a low repetition frequency (4Hz), comparable to biological measurement protocols [17]. Measurements were performed with a BFO memristive device as shown in Fig. 2. As shown in the model
simulations in Fig. 6a, the developed waveforms are transformed by the memristive device into
approx. exponentially decaying conductance changes. This is in good agreement with biological
measurements [17] and common STDP models [7]. The model results are confirmed in measurements for the BFO memristive device, as shown in Fig. 6b. Notably, the measurements result in
smooth, continuous curves. This is an expression of the continuous resistance change in the BFO
material, which results in a large number of stable resistance levels. This is in contrast e.g. to
memristive materials that rely on ferroelectric switching, which exhibit a limited number of discrete
resistance levels [18, 1]. Moreover, the nonlinear behaviour of the BFO memristive device has only
limited effect on the resulting STDP learning window. The resistance change is directly linked to
the applied waveforms. For example, as shown in Fig. 6, an increase in time constants results in correspondingly longer STDP time windows. Following our modeling approach, these time constants
are directly linked to the time constants of the underlying neuron and synapse model.
60
30
45
20
∆t1 in ms
15
0
0
−10
−20
−15
−30
−30
−30
(a)
−20
0
−10
10
∆t2 t in ms
20
∆ Im in %
30
10
30
(b)
Figure 7: Measurement results for the triplet protocol of Froemke and Dan [7]. (a) biological measurement data, adapted from [7], (b) measurement with BFO memristive device.
7
Experiments have shown that weight changes of single spike pairings, as expressed by STDP, are
nonlinearly integrated when occuring shortly after one another. Commonly, triplets of spikes are
used to investigate this effect, as carried out by [7]. The main deviation of these experimental results
compared to a pure STDP rule occur for the post-pre-post triplet [6], which can be attributed to
postsynaptic adaptation [7]. With this adaptation included in our waveforms (equation 4, as seen in
the action potential duration in the second curve from the top of Fig. 5), the BFO memristive device
measurements well resemble the post-pre-post results of [7]. The measurement results in Fig. 7b
show more depression than the biological data for the pre-post-pre triplet (upper left quadrant).
This is because changes in resistance need some time to build up after a stimulating pulse. In the
pre-post-pre case, the weight increase has not fully developed when it is overwritten by the second
presynaptic pulse, which results in weight decrease. This effect is dependent on the measured device
and the parameters of the stimulation waveforms (cf. Supplementary Material).
For keeping the stimulation waveforms as simple as possible, only postsynaptic adaptation has been
included. However, it has been shown that presynaptic short-term plasticity also has a strong influence on long-term learning [19, 6]. With our modeling approach, a model of short-term plasticity
can be easily connected to the stimulation waveforms by modulating the length of the presynaptic
pulse. Along the same lines, the postsynaptic waveform can be shifted by a slowly changing voltage
analogous to the original LCP rule (cf. Eq. 1) to introduce a metaplastic regulation of weight potentiation and depression [6]. Together, these extensions open up an avenue for the seamless integration
of different forms of plasticity in learning memristive devices.
3.3
Conclusion
Starting from a waveform-based general plasticity rule and a model of the memristive device, we
have shown a direct way to go from these premises to biologically realistic learning in a BiFeO3
memristive device. Employing the LCP rule for memristive learning has several advantages. As
a memristor is a two-terminal device, the separation of the learning in two waveforms in the LCP
rule lends itself naturally to employing it in a passive array of memristors [1, 4]. In addition, this
waveform-defined plasticity behaviour enables easy control of the STDP time windows, which is
further aided by the excellent multi-level memristive programming capability of the BiFeO3 memristive devices. There is only a very small number of memristors where plasticity has been shown at
actual devices at all [18, 1]. Among those, our highly-configurable, finely grained learning curves
are unique, other implementations exhibit statistical variations [1], can only assume a few discrete
levels [18] or the learning windows are device-inherent, i.e. cannot be adjusted [20]. This comes at
the price that in contrast to e.g. phase-change materials, BiFeO3 is not easily integrated on top of
CMOS [8].
The waveform-defined plasticity of the LCP rule enables the explicit inclusion of short term plasticity in long term memristive learning, as shown for the triplet protocol. As the pre- and postsynaptic
waveforms are generated in the CMOS neuron circuits below the memristive array [1], short term
plasticity can thus be added at little extra overall circuit cost and without modification of the memristive array itself. In contrast to our easily controlled short term plasticity, the only previous work
targeting memristive short term plasticity employed intrinsic (i.e. non-controllable) device properties [20]. To the best of our knowledge, this is the first time triplets or other higher-order forms of
plasticity have been shown for a physical memristive device.
In a wider neuroscience context, waveform defined plasticity as shown here could be seen as a
general computational principle, i.e. synapses are not likely to measure time differences as in naive
forms of STDP rules, they are more likely to react to local static [21] and dynamic [5] state variables.
Some interesting predictions could be derived from that, e.g. STDP time constants that are linked to
synaptic conductance changes or to the membrane time constant [22, 6]. These predictions could be
easily verified experimentally.
Acknowledgments
The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007- 2013) under grant agreement no. 269459 (Coronet).
8
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