A biophysically-based neuromorphic model of spike rate- and timing-dependent plasticity.

Abstract

Current advances in neuromorphic engineering have made it possible to emulate complex neuronal ion channel and intracellular ionic dynamics in real time using highly compact and power-efficient complementary metal-oxide-semiconductor (CMOS) analog very-large-scale-integrated circuit technology. Recently, there has been growing interest in the neuromorphic emulation of the spike-timing-dependent plasticity (STDP) Hebbian learning rule by phenomenological modeling using CMOS, memristor or other analog devices. Here, we propose a CMOS circuit implementation of a biophysically grounded neuromorphic (iono-neuromorphic) model of synaptic plasticity that is capable of capturing both the spike rate-dependent plasticity (SRDP, of the Bienenstock-Cooper-Munro or BCM type) and STDP rules. The iono-neuromorphic model reproduces bidirectional synaptic changes with NMDA receptor-dependent and intracellular calcium-mediated long-term potentiation or long-term depression assuming retrograde endocannabinoid signaling as a second coincidence detector. Changes in excitatory or inhibitory synaptic weights are registered and stored in a nonvolatile and compact digital format analogous to the discrete insertion and removal of AMPA or GABA receptor channels. The versatile Hebbian synapse device is applicable to a variety of neuroprosthesis, brain-machine interface, neurorobotics, neuromimetic computation, machine learning, and neural-inspired adaptive control problems.

Full text

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A biophysically-based neuromorphic model of spike
rate- and timing-dependent plasticity
Guy Rachmutha,b, Harel Z. Shouvalc, Mark F. Beard, and Chi-Sang Poona,1
aHarvard-MIT Division of Health Sciences and Technology, Massachusetts Institute of Technology, Cambridge, MA 02139;
Applied Sciences, Harvard University, Cambridge, MA 02138;
Houston, TX 77030; and
Technology, Cambridge, MA 02139
bDivision of Engineering and
cDepartment of Neurobiology and Anatomy, University of Texas Medical School at Houston,
dThe Picower Institute for Learning and Memory and Department of Brain and Cognitive Sciences, Massachusetts Institute of
Edited by* Leon N. Cooper, Brown University, Providence, RI, and approved October 3, 2011 (received for review May 24, 2011)
Current advances in neuromorphic engineering have made it
possible to emulate complex neuronal ion channel and intracellular
ionic dynamics in real time using highly compact and power-
efficient complementary metal-oxide-semiconductor (CMOS) ana-
log very-large-scale-integrated circuit technology. Recently, there
has been growing interest in the neuromorphic emulation of the
spike-timing-dependent plasticity (STDP) Hebbian learning rule by
phenomenological modeling using CMOS, memristor or other ana-
log devices. Here, we propose a CMOS circuit implementation of a
biophysically grounded neuromorphic (iono-neuromorphic) model
of synaptic plasticity that is capable of capturing both the spike
rate-dependent plasticity (SRDP, of the Bienenstock-Cooper-Munro
or BCM type) and STDP rules. The iono-neuromorphic model repro-
duces bidirectional synaptic changes with NMDA receptor-depen-
dent and intracellular calcium-mediated long-term potentiation
or long-term depression assuming retrograde endocannabinoid
signaling as a second coincidence detector. Changes in excitatory
or inhibitory synaptic weights are registered and stored in a non-
volatile and compact digital format analogous to the discrete
insertion and removal of AMPA or GABA receptor channels. The
versatile Hebbian synapse device is applicable to a variety of neu-
roprosthesis, brain-machine interface, neurorobotics, neuromi-
meticcomputation,machinelearning, andneural-inspiredadaptive
control problems.
iono-neuromorphic modeling ∣ rate-based synaptic plasticity ∣
silicon neuron ∣ subthreshold microelectronics ∣ VLSI circuit
L
to changing environments. A putative neuronal mechanism of
learning and memory is Hebbian synaptic plasticity (1)—the
adaptive modification of excitatory synaptic strength following
paired activation of the pre- and postsynaptic neurons. Two clas-
sic paradigms for the induction of Hebbian synaptic plasticity in
the mammalian hippocampus and neocortex are rate-based plas-
ticity (2–4) [herein referred to as spike-rate-dependent plasticity
(SRDP)] and spike-timing-dependent plasticity (STDP) (5–7).
The SRDP induction protocols control presynaptic firing rate
in order to vary the sign and magnitude of synaptic plasticity (8):
a high-frequency (20–100 Hz) train of presynaptic pulses results
in long-term potentiation (LTP) of the synaptic strength, whereas
a low-frequency (1–5 Hz) train results in long-term depression
(LTD). These protocols are consistent with the theoretical learn-
ing rule (BCM rule) proposed by Bienenstock, Cooper, and Mun-
ro (9), in which the sign and magnitude of synaptic plasticity are
controlled solely by postsynaptic activity as determined by presy-
naptic firing rate: low postsynaptic activity weakens synaptic
efficacy and high postsynaptic activity strengthens it. By contrast,
the STDPinduction protocol stipulates that precise timing of pre-
and postsynaptic activities determines the direction and strength
of synaptic plasticity: repeated pairings of a presynaptic stimulus
followed by a postsynaptic spike (prepost pairing, Δt > 0) results
in LTP, whereas reversing the order of pairing (postpre pairing,
earning and memory are emergent animal behaviors governed
by activity-dependent neuronal adaptation rules in response
Δt < 0) results in LTD (10). Mechanistically, both the SRDP and
STDP induction protocols elicit NMDA receptor (NMDAR)-
mediated intracellular calcium dynamics (4, 11, 12), which acti-
vate downstream processes that either up- or downregulate
synaptic strength through the insertion or removal of individual
excitatory AMPA receptor channels (13, 14). This common
mechanistic link suggests a possible underlying interrelationship
between these two seemingly distinct forms of Hebbian synaptic
plasticity (15).
In an attempt to reconcile the SRDP and STDP rules, several
computational models have been proposed (16–22). Currently,
there is general agreement that LTP can result from both SRDP
and STDP learning rules via postsynaptic NMDAR-mediated
coincidence detection of prepostynaptic activities. However, the
mechanism of timing-based LTD is less clear cut. Previous mod-
eling studies have shown that a STDP learning rule involving a
single postsynaptic coincidence detection mechanism as with LTP
may induce LTD not only within the expected LTD window with
postpre pairing (Δt < 0) but also beyond the LTP window with
prepost pairing (Δt > 0) (17, 23–25). In order to robustly repro-
duce the canonical STDP curve with a single postpre (Δt < 0)
LTD window, it has been proposed that a second coincident
detector may be required (23, 24). However, although many bio-
physical mechanisms and models of second coincidence detection
for STDP have been proposed (25, 26), a general computational
model that consistently unifies the SRDP and STDP rules based
on an experimentally demonstrated second coincident detector
is currently lacking.
Previous theoretical analyses of the relationships between the
SRDP and STDP rules were mostly based on numerical simula-
tions of model equations on digital computers. Another approach
to neural modeling is via direct emulation of neuronal dynamics
on electronic devices such as complementary metal-oxide-semi-
conductor (CMOS) (27–29) or nanowire circuits (30–32); i.e.,
analog “neuromorphic” computation instead of digital model
simulation. Recently, there has been growing interest in the neu-
romorphic modeling and implementation of the STDP learning
rule using CMOS (32–42) or metal-oxide-metal circuits (43) or
memristor-based nanodevices. Compared to conventional soft-
ware-based computer modeling and simulation approaches, these
neuromorphic electronic circuits have extremely small size (mi-
cro- to nanoscale) and low power requirements (μAto pAcurrent
per unit device with 0.5–5 V power supply) for large scale neural
modeling and high speed simulation purposes. These capabil-
Author contributions: G.R., H.Z.S., M.F.B., and C.-S.P. designed research; G.R. and C.-S.P.
performed research; G.R. and C.-S.P. analyzed data; and G.R., H.Z.S., and C.-S.P. wrote
the paper.
The authors declare no conflict of interest.
*This Direct Submission article had a prearranged editor.
1To whom correspondence should be addressed. E-mail: cpoon@mit.edu.
See Author Summary on page 19453.
This article contains supporting information online at www.pnas.org/lookup/suppl/
doi:10.1073/pnas.1106161108/-/DCSupplemental.
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ities are critical for many real-time, portable/implantable neural
computing applications such as neuroprosthesis, brain-machine
interface, neurorobotics, neuromimetic computation, machine
learning, or neural-inspired adaptive control (44). However, most
such neuromorphic models emulate the temporally asymmetric
STDP characteristic by direct phenomenological curve fitting
(45) instead of biophysical modeling (26). It has been suggested
that nonmechanistic phenomenological modeling of STDP may
lead to many predictive failures especially when applied to other
forms of synaptic plasticity (25). To our knowledge, none of the
phenomenological neuromorphic STDP devices developed so far
can reproduce the SRDP learning rule; hence, they are inflexible
in responding to rate-based stimuli.
Another limitation of previous neuromorphic synaptic plasti-
city models is the difficulty of long term analog storage of sy-
naptic weights using electrical capacitors (32, 46), which are
volatile and bulky to implement on CMOS. Although compact
nonvolatile long term memory of synaptic weights can potentially
be achieved by using digital random-access memories (41) or
advanced floating-gate (47, 48) or memristor technologies (49),
these devices are not readily amenable to the biophysical model-
ing of NMDAR-mediated plasticity in a Hebbian synapse.
Here, we propose an iono-neuromorphic (i.e., biophysically
grounded) CMOS circuit implementation of Hebbian synaptic
plasticity that is capable of capturing both the NMDAR-depen-
dent SRDP and STDP learning rules. Our iono-neuromorphic
model is based on the hypothesis that retrograde endocannabi-
noid signaling and presynaptic NMDAR may provide a second
coincidence detector of pre- and postsynaptic activity in addition
to postsynaptic NMDAR (50–52). To emulate the underlying
biophysical mechanisms, we employ a recently proposed wide-dy-
namic-range iono-neuromorphic CMOS circuit design approach
that allows robust modeling of all types of voltage-dependent
or ligand-gated ion channel and intracellular ionic dynamics
on analog very-large-scale-integrated (aVLSI) circuits (53). We
show that our iono-neuromorphic model readily reproduces LTP
and LTD based on either the SRDP or STDP learning rules
implemented on the same CMOS chip. The proposed iono-neu-
romorphic model of LTP and LTD lends itself readily to long
term storage of synaptic weights in a nonvolatile digital format
that is analogous to the discrete insertion and removal of AMPA
or GABA receptor channels in real neurons, thus circumventing
the limitations of analog memory.
Methods and Results
Iono-Neuromorphic Model of Postsynaptic NMDAR-Dependent LTP
and LTD. Iono-neuromorphic model of NMDA and AMPA channels.
A“learning synapse” circuit model of an excitatory postsynaptic
hippocampal dendritic spine compartment is designed as follows
(Fig. 1A). A set of CMOS building block circuits biased in the
subthreshold regime for robust iono-neuromorphic modeling
[with wide input dynamic range to overcome device mismatch in
subthreshold circuits (44)] are configured to emulate fast AMPA
and slower NMDA channels, as described previously (53). The
output currents are sent to a membrane node circuit that keeps
the membrane potential VMEMnear the resting potential VREST
in the absence of stimulation (Fig. 1B). In response to a single
presynaptic stimulation, excitatory IAMPAand INMDAimpinge
on the membrane capacitor (CMEM) causing VMEMto generate
an excitatory postsynaptic potential (EPSP) that relaxes towards
VRESTat a rate determined by the membrane time constant
τMEM¼CMEM∕gleak(Fig. 1C). Importantly, several discrete AMPA
channels carry excitatory postsynaptic current (EPSC) in parallel,
and each channel is gated by a binary control variable Cn(where
n ¼ 1;2;…;N) that determines whether a particular AMPA chan-
nel is active. Thus, the number of active AMPA channels encodes
the synaptic weight.
NMDA channels—gated by both presynaptic glutamate and
postsynaptic VMEMcontrol over extracellular magnesium block
—have slower dynamics, and encode coincident pre- and postsy-
naptic activities by INMDAamplitude. Calcium influx via INMDA
(generated as its own current ICa
voltage converter circuit to generate intracellular calcium
(½Caþ2?i) signal (Fig. 1C). The calcium signal in turn activates
downstream circuits that adjust the number of active AMPA
channels (Cnvector) according to a learning rule implemented
by ½Caþ2?i-dependent plasticity circuits.
The iono-neuromorphic synapse design is biologically intuitive
and allows application of experimental manipulations to observe
emergent behaviors. For example, the model is capable of mod-
ifying hippocampal silent synapses expressing only NMDA chan-
nels into expressing AMPA channels following an induction
protocol (54). Additionally, the circuits allow tremendous flexibil-
ity in emulating synapses from various brain structures by simply
tuning a small (1–4) set of parameters such as maximum conduc-
tance or activation dynamics of both excitatory and inhibitory
channels.
þ2) is integrated on a current-
Iono-neuromorphic intracellular calcium-mediated plasticity model.
Models of synaptic plasticity posit an important role of calcium
in medicating downstream processing that results in expression of
potentiation or depression of the synaptic weight. The learning
rule implementation underlying on-chip synaptic plasticity is
an adaptation of a biophysical model proposed by Shouval, et al.
(17, 25) that relies on intracellular calcium dynamics to deter-
mine synaptic plasticity. The model computes the change in
synaptic weight (dw) by evaluating:
dw ¼ ηð½Ca?Þ · ðΩð½Ca?Þ − λwÞ;
[1]
where w is the present synaptic weight, Ωð½Ca?Þ is the calcium-
Fig. 1.
(C) Circuit outputs in response to a presynaptic action potential (AP) input (APPRE). See also Fig. S1.
(A) Simple synapse consisting of AMPA and NMDA channels, and calcium. (B) Circuit models of individual elements of the synapse, color coded with (A).
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dependent update rule, ηð½Ca?Þ is a monotonically increasing
(supralinear) function of ½Caþ2?ithat controls the learning rate,
and λ is a “forgetting” constant that assures that the synaptic
weight reverses back from saturation if it is not maintained at
its potentiated or depressed levels.
The Ω function employs LTP and LTD thresholds values (θLTP
and θLTD, respectively) to set potentiation and depression levels
as a function of ½Caþ2?i. By simply changing the θLTPand θLTD
thresholds, several plasticity vs. ½Caþ2?irules can be realized.
For our purposes, we implemented the SRDP learning rule that
has been observed experimentally in visual cortex and in hippo-
campus (55, 56). The rule postulates that for θLTD< ½Caþ2?i<
θLTP, the synaptic strength will depress; when ½Caþ2?i> θLTP,
the synaptic strength potentiates. It is important to note that this
rule was developed to explain SRDP results which involve long
trains of stimulation that generate a constant ½Caþ2?ilevel for a
relatively long period of time. This assumption is not true for
STDP protocols.
The Ω and η functions are expressed mathematically using
exponential functions, making them simple to implement using
subthreshold-biased transistors. Thus, it is practical to use aVLSI
technology to incorporate the learning rule into a large number of
synapses. The Ω circuit computes an output signal IΩas a func-
tion of ½Caþ2?isignal from the calcium circuit (Fig. 2A). The Ω
circuit is split into LTP and LTD sections. A cascade of differen-
tial-pair circuits compare ½Caþ2?iand a threshold (either θLTPor
θLTD) and compute the output current for each section. These
output currents are subtracted from each other, and the resultant
current is added to the naïve synaptic weight represented by
IΩ-CONS. The Ω circuit can be tuned to generate various Hebbian
learning rules as seen in the hippocampus (Fig. 2B), or anti-Heb-
bian learning rules observed in the cerebellum (57). Importantly,
θLTPand θLTDmay be modified dynamically via an internal circuit
to implement meta-plasticity (58), allowing synaptic plasticity
over longer time scales in an unsupervised manner.
The η circuit is designed to mimic the calcium-dependent
learning rate irrespective of the direction of plasticity. This circuit
captures the fact that while both LTD and LTP can be induced
using a 900 pulse train, longer stimulation trains are needed to
generate LTD (∼15 min at 1 Hz stimulation) compared to
LTP (9 s at 100 Hz). Because we update the synaptic weight with
enabling discrete AMPA channel circuits to fully turn on or off,
we converted the η circuit into a calcium-dependent digital
Enable signal. This approach allows the neromorphic synapse
to modify its synaptic weight only during induction protocols that
generate calcium influx that accumulates above a putative enable
threshold, θη. The circuit employs a transconductance circuit
with a calcium-dependent bias current IBIAS-Ca, which results
in different charging rates based on a dynamic calcium level
(Fig. 2C) and superlinear behavior of the η function.
The circuit generates an output current Iηproportional to the
difference of ½Caþ2?iand a rest voltage ηRESTwhenever ½Caþ2?i>
VηREST. Iηis converted to a voltage signal Vηvia a capacitor assur-
ing monotonically increasing output as a function ofthe induction
protocol. A leak transistor is included to keep Vηnear VηRESTfor
quiescent activity. Vηis sent to a pulse-generator circuit that
'compares Vηto the enable threshold θη. The circuit resets itself
whenever it goes HI generating an Enable pulse, and resets for
a period of time Δt (a time period that is not predictable due
to the subthreshold biasing). The circuit therefore signals that
an induction protocol has generated enough calcium to enable
the synapse to update its synaptic weight (Fig. 2D). Because dur-
ing an induction protocol, ½Caþ2?i> VηRESTfor a period of time,
Vηcan generate several Enable pulses during the induction pro-
tocol, allowing the synapse to update dynamically and then finally
locking in a value when the ½Caþ2?ifalls below VηREST.
This property is inherent in the model, suggesting that up-
dating the synaptic weight is strongly dependent on the exact
dynamics of the synaptic expression mechanism. In our circuits,
the expression dynamics are a function of ½Caþ2?iaccording to:
θη· Ceta
Iη· ½Ca2þ?;
where Cetais the capacitor of the η circuit, and θηis a threshold
voltage of the comparator. This equation suggests that induction
protocols must endure at least τupdateseconds for the synaptic
weight to update in an unsupervised manner. To simulate SRDT,
we set θηto a level that requires calcium levels to remain elevated
for at least several seconds or minutes before the Enable signal is
generated in order to reproduce experimental results (8).
τupdate∼
[2]
Fig. 2.
to both set the WR-TCA maximum current and as an input. The integrated Vηis sent to a comparator that generates a digital pulse when Vη> θη. (D) η circuit
output in response to ½Caþ2?itransient.
(A) Ωcircuit.Input½Caþ2?isignal used to compute output currentIΩ. (B)IΩ—½Caþ2?icurve parameterized byθLTPandθLTD. (C) ηcircuit.Input ½Caþ2?iis used
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Digital storage of synaptic weights for LTP and LTD. In our model,
synaptic weights are encoded iono-neuromorphically by a dis-
crete set of digitally gated AMPA channels controlled by Cn,
the updated synaptic strength. If Cnis “HI” (LO), the nthAMPA
conductance is activated (deactivated) and the synaptic weight is
potentiated (depressed) compared to its previous state. To con-
vert from an analog η and Ω to a digital vector Cn, we set λ ¼ 1,
digitized IΩusing a picoampere A/D converter (59, 60) and
paired the Enable signal from the η circuit to update a W-circuit
(e.g., dw from Eq. 1) following a particular induction protocol.
The W-circuit updates the weight in two steps. First, an asyn-
chronous digital finite state-machine [(FSM), see Fig. S1,
Table S1 for details] uses the digitized Ω signal (DΩ) to compute
future weight vector Cn, which are mapped to the inputs of dedi-
cated D-flipflop (DFF) circuits that control individual AMPA
channels. This vector then waits for an Enable (square pulse)
command from the digitized Vη, which causes the Cnvector to
update the synaptic weight, and resets Vηback to zero.
The FSM provides a convenient abstraction of multiple non-
linear and interacting cellular and molecular mechanisms
thought to be activated by postsynaptic Caþ2signal to drive sy-
naptic plasticity; e.g., calcinurin driving LTD (61) or CaMKII
driving LTP (62). The FSM can incorporate an arbitrary update
rule as a function of DΩ. By modifying the ratios of mirrored
IAMPA(see Fig. 1B), the synaptic weight can be varied in a non-
linear way. Interestingly, the inevitable mismatch across transis-
tors due to CMOS process variability (63) provides a natural
method for heterogeneous AMPA conductances across synapses.
Thus the fabrication process may generate synapses with different
maximum synaptic weight that, when applied in a large network,
give rise to interesting neuronal computations.
CMOS prototype. As a proof of concept, the CMOS iono-neuro-
morphic Hebbian synaptic plasticity chip was prototyped using
the AMI 1.5 μm process (Fig. S2A); use of deep-submicron pro-
cesses will further reduce the chip size at a higher cost. Currently,
the chip’s size was 4.6 mm × 4.6 mm with 116 I/O pins, although
the circuits consumed less than 50% of the available area. The
chip consisted of several ion-channel circuits (e.g., AMPA,
NMDA, voltage-gated calcium, etc.), the η and Ω circuits as well
as digital storage circuits, totaling 400 transistors and 10 capaci-
tors for implementation of both the NMDAR-dependent postsy-
naptic plasticity model and the endocannabinoid-dependent
presynaptic plasticity model (see below). The transistor count
reflects the relative complexity of our biophysically grounded
iono-neuromorphic model compared to phenomenological mod-
els. Additional transistors were also needed in our wide-dynamic-
range subthreshold CMOS circuit designs, which effectively
mitigated the effects of transistor mismatch and significantly
improved the robustness of aVLSI implementation (44, 53). The
capacitors occupied the bulk of the chip area as some of them
were relatively large (30 pF) in order to achieve a 1∶1 electronic-
to-biological time scale at nA level currents. The sizes of the
capacitors are comparable to those required in phenomenologi-
cal neuromorphic models. The circuits were interfaced with each
other via external pins so that each circuit could be tested inde-
pendently on a circuit board (Fig. S2B) to ensure proper opera-
tion. The chip consumed on the order of 100 nW of power (on a
5 V power supply) when only the analog circuits were operational
(majority of the time). When undergoing a synaptic plasticity
induction protocol, the activation of the digital portions of the
circuit increased the power consumption only transiently. All
circuits carried out simulations in real biological time.
Emulation of Postsynaptic NMDAR-Dependent SRDP Learning Rule.
The learning synapse circuit was tested using several well known
induction protocols to draw direct comparisons with biological
preparations. We first employed a pairing protocol used to iden-
tify the role of NMDAR-mediated calcium influx on synaptic
plasticity. Similar to experimental protocols, VMEMwas voltage
clamped at VRESTand the synaptic reversal potential ESYN(−70
and 0 mV, respectively) while presynaptic stimulation was applied
at a rate of 1 Hz. The results reveal differences in INMDA, ICa
and Caþ2accumulation depending on the clamping voltage
(Fig. 3A). Because INMDAconsists of Ca and Na ions, its reversal
potential is 0 mV. Therefore, when VMEMis clamped at ESYN,
INMDAis near 0. However, ICA, which has its own reversal poten-
tial ECa¼ 140 mV, is substantially larger. When VMEM¼ ESYN,
repeated stimulations resulted in elevated ½Caþ2?ilevel above
θLTPfor a period longer than τupdateand eventually resulted in
potentiation (Fig. 3B).
We next tested SRDP induction protocols aimed at showing
NMDA channel activity as a Hebbian coincidence detector of
pre- and postsynaptic activity and its control of synaptic plasticity.
SRDP protocols assume that the mean AP firing rate is the main
information transfer mode in neural networks, and so employ
trains of stimulations. We used a standard protocol of 900 pulses
at several presynaptic firing rates ranging from low- to high-
frequency stimulations (4). The resultant ½Caþ2?iaccumulation
was proportional to the input stimulus frequency such that for
higher frequencies, ½Caþ2?ilevels were higher than for lower
frequencies (Fig. 3C). The calcium conversion circuit (Fig. 1B)
saturates at different voltages for different stimulation protocols,
which are monotonically increasing as a function of frequency.
The signals in Fig. 3C are raw signals directly from the calcium
circuit. The Ω circuit was tuned to a standard BCM curve, which
transitioned from LTD to LTP at a calcium level that was elicited
by a 10 Hz stimulus train. The outputs were sent to the W-circuit
and resulted in graded potentiation for presynaptic input rates
>10 Hz, and depression for input rates <10 Hz (Fig. 3D). These
results reproduce experimentally recorded observations.
The results show that classical induction protocols require only
NMDAR-mediated calcium dynamics to express synaptic plasti-
þ2,
Fig. 3.
calcium current is still present when VMEMis clamped at ESYNleading to larger
calcium influx. (B) Synaptic (top) and somatic VMEM(bottom) represented by
two separate circuits. Synaptic EPSPs prior to an induction protocol failed to
elicit a somatic AP (black signals) from an integrate-and-fire neuron circuit
connected downstream to the synaptic circuit. Following an induction pro-
tocol, the higher synaptic VMEMelicited an AP in the soma (blue signals). This
shows that changing synaptic weights can influence AP firing. (C) Calcium
levels in response to 900 pulses at various frequencies. The y-axis is simulated
½Caþ2?iamplitude measured in volts (scale not shown). (D) Synaptic plasticity
following standard SRDP protocol which reproduces experimental data.
(A) Synaptic responses to a classical pairing protocol; A significant
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city. Similar to biological experiments, blocking on-chip NMDAR
activation also blocked rate-based synaptic plasticity.
Emulation of Postsynaptic NMDAR-Dependent STDP Learning Rule.
Recent studies show that when somatic APs back-propagate into
the dendritic arbor and are repeatedly paired with presynaptic
stimulation, they can induce synaptic plasticity depending on
precise arrival sequence (12, 64). Biophysically, STDP appears to
depend on NMDAR-mediated calcium influx, suggesting similar
induction mechanisms as SRDP plasticity. If STDP models as-
sume that maximum calcium levels drive the synaptic plasticity,
the function of maximum ½Caþ2?idriving STDP would be in the
shape of Fig. 4A, with graded increase from −100 ms < Δt <
þ100 ms, and a fast transition in the critical window between
[−5 ms 5 ms] (Fig. 4A). However, experimental STDP results
suggest that there may be two separate systems mediating the
postpre and prepost portions of STDP (23, 24), which should be
modeled as a discontinuity rather than a fast transition between
the lowest depression and highest potentiation levels.
We subjected the learning synapse to an STDP stimulation
protocol as described by Bi and Poo (12). In this protocol, 60 pre-
post or postpre pairings were applied at the synapse at a rate of
1 Hz [or lower (65)]. A standard presynaptic pulse activated the
postsynaptic compartment. A back-propagating AP (bAP) gener-
ated by an integrate-and-fire neuron circuit (66) was sent to the
postsynaptic compartment with a delay Δt of 1–100 ms, with Δt >
0 for prepost stimulations, and Δt < 0 for postpre stimulations. In
response to an STDP pairing, postsynaptic NMDAR-medicated
½Caþ2?iis sent to the Ω and η circuits, and the ½Caþ2?idynamics are
reflected in both IΩand Vη. Both the presence of glutamate and
the voltage spike (by the backpropagating AP) is required for
NMDA channel activation (and consequent calcium influx). Be-
cause a prepost stimulation should result in potentiation, we
tuned the Ω parameters such that for any Δt > 0, ½Caþ2?ielevated
above both θLTPand θLTDfor a brief period of time, but pairings
with increasing Δt generate ½Caþ2?isignal with lower maximum
amplitude (Fig. 4B). In the absence of another pairing, ½Caþ2?i
quickly decreased at a physiological rate to below θLTP, remains
above the θLTDlevels for some time, and decreased back to base-
line. Thus, to generate potentiation, the model predicts that
either a downstream mechanism should “lock in” the fact that
½Caþ2?icrossed θLTP(a low-pass filtering effect with a very long
time constant), or that τupdate(Eq. 2) is at the millisecond level.
This fast-acting coincidence detector may be reflected in dendri-
tic ion-channel activities (65, 67), or by downstream CaMKII
autophosphorylation in response to transient calcium elevation
(68). We tuned η circuit’s τupdateto be very short by modifying
θη, and reproduced the LTP portion of STDP window.
For LTD, however, the situation is much more complicated.
For the artificial synapse circuit with only the NMDA channel
as calcium source, STDP protocols did not display abrupt transi-
tion in the calcium level around Δt ∼ 0 (Fig. 4B) after tuning
the activation dynamics of the NMDA channel or modifying the
width of bAP, as suggested previously (17). Moreover, in a model
where ½Ca2þ?iis only generated by NMDA channels, the maxi-
mum ½Ca2þ?ireaches similar values for prepost pairing at +40 ms
as it does for postpre pairings at −10 ms (Fig. 4B). Hence, based
purely on NMDAR-initiated calcium dynamics there should be a
prepost window for LTD, as previously hypothesized (17, 23–25).
If the prepost arrivaldifference is −10 ms, then thebAParrives
10 ms before the glutamate, and so by the time the glutamate
binds, VMEMhas decayed significantly leading to a smaller frac-
tion of NMDA channel activation. If the prepost arrival timing is
+40 ms, the glutamate has arrived 40 ms before the bAP. With a
binding time constant of 80 ms, someof the glutamate would have
disassociated from the NMDA channels by the time the voltage
spike arrives, leading to a similarly smaller activation of NMDA
channels. Both situations lead to smaller calcium influx. In a mod-
el where only the maximum calcium level determines LTD or
LTP, this result predicts that if LTD occurs at −10 ms timing, then
LTD must occur at the +40 ms timing (due to the approximately
the same maximum calcium level), as it is an element of the
model itself irrespective of exact parameters. This prediction of
an LTD window at +40 ms (or so) has been suggested by
Shouval, et al. (17, 25).
The underlying assumption of most plasticity models is that
prolonged, moderate levels of calcium activate various intracel-
lular phosphatases (such as calcineurin) and lead to LTD. The
dependence on NMDAR activity implies coincident activity, but
the postpre scenario requires another calcium source. Recent
data suggest that voltage-gated L-type calcium channels (CaV-L)
contribute to calcium influx in postsynaptic neurons (69). We
therefore added circuit models of CaV−Lchannels to the postsy-
naptic compartment, which together with presynaptic-activated
NMDA channels combined to generate different ½Caþ2?idy-
namics in response to postpre and prepost simulations. However,
calcium influx via CaV-L is small compared with NMDAR-
mediated ICa(69), and its addition was not enough to generate
a large difference in the maximum ½Caþ2?ilevels. We added
L-type calcium channels to the postsynaptic neuron to generate
some calcium influx in response to a backpropagating AP, which
could potentially lead to a larger calcium level when the gluta-
mate arrives for postpre pairings. In this scenerio, the calcium
level for a Δt ¼ −10 ms pairing could be larger than a Δt ¼
þ40 ms pairing, leading to the calcium generated by a −10 ms
pairing crossing the LTD threshold, and calcium generated by
a +40 ms pairing not crossing the LTD threshold. However,
L-type calcium current is much smaller than NMDAR-mediated
calcium influx, and so the additional calcium provided by the
L-type channel does not significantly affect the postsynap-
tic ½Caþ2?i.
By using the chip, a purely postsynaptic mechanism of coin-
cidence detection cannot robustly express both LTP and LTD
under both SRDP and STDP protocols with the same model
parameters. The two major discrepancies for the prepost proto-
cols are: (i) θLTPand θLTDmust be decreased significantly from
SRDP levels, possibly implying separate downstream sensors of
postsynaptic ½Caþ2?iaccumulation; and (ii) the learning rate
τupdatemust be modified from operating on the order of seconds
for SRDP to possibly milliseconds to account for STDP. Tuning
the η and Ω circuits allowed modeling of prepost LTP. However,
the model did not generate a single postpre LTD behavior with
the same set of parameters even with the addition of CaV-L,
as discussed previously (17, 23–25). We hypothesize that there
may be a separate bAP-induced suppression of NMDA recep-
tor-mediated EPSCs which sets the spike-timing window for
LTD (64).
Emulation of STDP Learning Rule with Retrograde Endocannabinoid
Signaling. The need for a second coincidence detector beyond
Fig. 4.
tocol to accommodate the calcium-control hypothesis. Note the discontinuity
between maximum depression and maximum potentiation in the [−5 ms,
5 ms] window. (B) Maximum ½Caþ2?ilevels reached during an STDP protocol
for the simple synapse circuit.
(A) Maximum ½Caþ2?ilevels required during an STDP induction pro-
E1270
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www.pnas.org/cgi/doi/10.1073/pnas.1106161108Rachmuth et al.