Page 1

Intrinsic Stability of Temporally Shifted Spike-Timing

Dependent Plasticity

Baktash Babadi1*, L. F. Abbott1,2

1Center for Theoretical Neuroscience, Department of Neuroscience, Columbia University, New York, New York, United States of America, 2Department of Physiology and

Cellular Biophysics, Columbia University College of Physicians and Surgeons, New York, New York, United States of America

Abstract

Spike-timing dependent plasticity (STDP), a widespread synaptic modification mechanism, is sensitive to correlations between

presynapticspiketrainsanditgeneratescompetitionamongsynapses.However,STDPhasaninherentinstabilitybecausestrong

synapses are more likely to be strengthened than weak ones, causing them to grow in strength until some biophysical limit is

reached. Through simulations and analytic calculations, we show that a small temporal shift in the STDP window that causes

synchronous, or nearly synchronous, pre- and postsynaptic action potentials to induce long-term depression can stabilize

synaptic strengths. Shifted STDP also stabilizes the postsynaptic firing rate and can implement both Hebbian and anti-Hebbian

forms of competitive synaptic plasticity. Interestingly, the overall level of inhibition determines whether plasticity is Hebbian or

anti-Hebbian. Even a random symmetric jitter of a few milliseconds in the STDP window can stabilize synaptic strengths while

retaining these features. The same results hold for a shifted version of the more recent ‘‘triplet’’ model of STDP. Our results

indicatethatthedetailedshapeoftheSTDPwindowfunctionnearthetransitionfromdepressiontopotentiationisoftheutmost

importance in determining the consequences of STDP, suggesting that this region warrants further experimental study.

Citation: Babadi B, Abbott LF (2010) Intrinsic Stability of Temporally Shifted Spike-Timing Dependent Plasticity. PLoS Comput Biol 6(11): e1000961. doi:10.1371/

journal.pcbi.1000961

Editor: Lyle J. Graham, Universite ´ Paris Descartes, Centre National de la Recherche Scientifique, France

Received April 23, 2010; Accepted September 17, 2010; Published November 4, 2010

Copyright: ? 2010 Babadi, Abbott. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits

unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: Research supported by the National Institute of Mental Health (MH-58754) and by an NIH Director’s Pioneer Award, part of the NIH Roadmap for

Medical Research, through grant number 5-DP1-OD114-02. The funders had no role in study design, data collection and analysis, decision to publish, or

preparation of the manuscript.

Competing Interests: The authors have declared that no competing interests exist.

* E-mail: bb2280@columbia.edu

Introduction

Hebbian synaptic plasticity can effectively organize neural

circuits in functionally useful ways, but only when implemented in

a manner that induces competition among synapses [1]. Spike-

timing dependent synaptic plasticity (STDP), which has been

observed in a wide variety of preparations (see [2] for a review),

appears to provide such an implementation by forcing synapses to

compete for control of the timing of postsynaptic action potentials

while being strengthened or weakened. In STDP, a synapse is

potentiated when a presynaptic action potential precedes a

postsynaptic spike, and depressed otherwise (see [3] for a review).

STDP has been shown to induce a competitive form of Hebbian

plasticity that is useful for a variety of neuro-computational

problems (see [4] for a review). However, this form of STDP has

an inherent instability in that strong synapses get stronger and weak

synapses get weaker. This instability can be tamed by biophysical

limitations on synaptic strengths, resulting in a U-shaped distribu-

tion of synaptic efficacies [5]. Nevertheless, it is interesting to

examinemodelsthatdonot requiresuchconstraintsforstabilization

and that generate unimodal distributions of synaptic strengths

resembling those measured in cultured and cortical networks [6–8].

Synaptic competition and synaptic stability (meaning that

synapses reach a stable equilibrium distribution independent of

bounds on their strengths) are desirable but conflicting features of

Hebbian synaptic plasticity. For example, the instability of STDP

mentioned in the previous paragraph can be eliminated by

introducing strength-dependent modification [9,10], but at the

expense of eliminating synaptic competition. By interpolating

between stable and unstable models of STDP, it is possible to

obtain both synaptic competition and stability, but over a limited

parameter range [11]. Here we propose an alternative solution

inspired by the slow kinetics of NMDA receptors. We show that

STDP can be stabilized if the boundary separating potentiation and

depression does not occur for simultaneous pre- and postsynaptic

spikes, but rather for spikes separated by a small time interval.

Through simulation as well as by solving the Fokker-Planck

equation governing the distribution of synaptic strengths, we show

that any positive shift of the STDP window can stabilize the

distribution of synaptic strengths while preserving synaptic

competition. These properties also hold for a multi-spike STDP

rule in which triplets of pre- and postsynaptic spikes are the key

events in determining the synaptic change [12], as opposed to pair-

based STDP in which pairs of pre- and postsynaptic spikes govern

the plasticity process. Moreover, our simulations show that even a

random symmetric jitter of a few milliseconds in the STDP window

can stabilize synaptic strengths while retaining these features.

Results

To study the effects of STDP on synaptic strengths, we

simulated a single spiking neuron that receives excitatory and

inhibitory presynaptic spike trains with Poisson statistics at rates

rexand rin, respectively (Methods). The strengths of the excitatory

synapses, denoted by w, change due to STDP, while the strengths

of the inhibitory synapses remain constant. We first consider the

PLoS Computational Biology | www.ploscompbiol.org1 November 2010 | Volume 6 | Issue 11 | e1000961

Page 2

pair-based model of STDP. A more complicated multi-spike

model will be studied afterward. In the pair-based model the

change in synaptic strength, Dw, induced by a pair of pre- and

postsynaptic action potentials with time difference Dt~tpost{tpre

is determined by

Dw~F(Dt)~

{A{e(Dt{d)=t{

Aze{(Dt{d)=tz

if Dtƒd

if Dtwd:

(

ð1Þ

The parameters Azand A{, both positive, determine the

maximum amount of synaptic potentiation and depression,

respectively. We define synaptic strengths in units of membrane

potential depolarization (mV), so Azand A{have mV units as

well (Methods). The time constants tzand t{determine the

temporal extent of the STDP window for potentiation and

depression. The parameter d, also positive, introduces a shift in the

STDP window such that even in cases where a presynaptic action

potential precedes the postsynaptic spike by a short interval

(0vDtvd), the corresponding synapse gets depressed. Note that

we recover conventional pair-based STDP by setting d~0.

Further details of the synaptic modification procedure appear in

the Methods, and the numerical values of the STDP parameters

are given in Table 1. An important feature of the pair-based model

we use is that STDP arises solely from pairs of pre- and

postsynaptic spikes that are nearest neighbors in time, in

agreement with experimental results [13]. Specifically, each

postsynaptic action potential can only potentiate a synapses on

the basis of the interval to the presynaptic spike immediately

preceding it, and each presynaptic action potential can only

depress a synapses on the basis of the timing interval to the

immediately preceding postsynaptic spike. This assumption is

important for the results we obtain using the pair-based STDP

model, as discussed below.

Stability of synaptic strengths

With conventional, unshifted STDP (d~0), synaptic strengths

grow or shrink indefinitely unless limits are imposed. These limits

produce a U-shaped distribution of synaptic strengths (figure 1A,

[5]). However, if we introduce a d~2ms shift into the STDP

window, the steady-state distribution of synaptic strengths is

unimodal and stable even when no limits are imposed (figure 1B).

Why does this occur?

The total effect of a sequence of pre- and postsynaptic action

potentials on the strength of a synapse can be computed by

multiplying the STDP window function by the probability of a

spike pair appearing with time difference Dt and then integrating

over all values of Dt. If we assume Poisson spike trains and ignore

the effects of the synapse, the probability distribution of nearest-

neighbor pre-post pairs is an exponentially decaying function of

the magnitude of the interval between them (figure 1C). The decay

rate of this exponential is equal to the sum of the pre- and

postsynaptic firing rates (Methods). The presence of a synapse

induces an additional contribution to this distribution for small

positive Dt arising from postsynaptic spikes induced by the

synaptic input (figure 1C). The size of this ‘‘causal bump’’ is

proportional to the probability of a presynaptic action potential

evoking a postsynaptic response, and hence to the strength of the

synapse. The stronger the synapse, the larger the bump. In

addition, because the postsynaptic spike latency is shorter for

stronger synapses, the bump moves closer to Dt~0 as the synaptic

strength increases (figure 1D). These features of the pre-post

interval distribution are crucial for our analyses.

When there is no shift in the STDP window, the causal bump

falls entirely within the potentiation domain (figure 1E), which is

why synaptic strengths grow until something else stops them

(figure 1F). When the STDP window is shifted, part of the causal

bump falls into the region where depression occurs (figure 1G).

Furthermore as the synapse gets stronger, a larger portion of the

causal bump falls into the depression domain, both because the

causal bump gets bigger and because it moves closer to Dt~0

(figure 1H). This prevents further growth of the synaptic strength

and explains why a shift stabilizes synaptic growth through STDP.

Stabilization of synaptic weights occurs for any positive value of

the delay (d), but larger delays result in lower mean values and

sharper distributions for the weights (figure 2).

For a more quantitative evaluation of shifted STDP, we

computed the steady-state solution of the Fokker-Planck equation

governing the distribution of synaptic strengths [14–16] (Methods).

Table 1. Neuronal, synaptic, and plasticity parameters.

Parameter Symbol Default value

Membrane time constant

tm

20ms

Spiking threshold

Vth

{40mV

Resting membrane potential

Vr

{60mV

Maximum potentiation amplitude

Az

0:006mV

Maximum depression amplitude

A{

0:005mV

Potentiation time constant

tz

20ms

Depression time constant

t{

20ms

Window shift

d

2ms

Synaptic time constant

ts

5ms

Number of excitatory synapses

Nex

1000

Number of inhibitory synapses

Nin

250

Inhibitory synaptic strength

win

4mV

Excitatory input rate

rex

10Hz

Inhibitory input rate

rin

10Hz

doi:10.1371/journal.pcbi.1000961.t001

Author Summary

Synaptic plasticity is believed to be a fundamental

mechanism of learning and memory. In spike-timing

dependent synaptic plasticity (STDP), the temporal order

of pre- and postsynaptic spiking across a synapse

determines whether it is strengthened or weakened. STDP

can induce competition between the different inputs

synapsing onto a neuron, which is crucial for the formation

of functional neuronal circuits. However, strong synaptic

competition is often incompatible with inherent synaptic

stability. Synaptic modification by STDP is controlled by a

so-called temporal window function that determines how

synaptic modification depends on spike timing. We show

that a small shift, or random jitter, in the conventional

temporal window function used for STDP that is compat-

ible with the underlying molecular kinetics of STDP, can

both stabilize synapses and maintain competition. The

outcome of the competition is determined by the level of

inhibitory input to the postsynaptic neuron. We conclude

that the detailed shape of the temporal window function is

critical in determining the functional consequences of

STDP and thus deserves further experimental study.

Intrinsic Stability of Shifted STDP

PLoS Computational Biology | www.ploscompbiol.org2 November 2010 | Volume 6 | Issue 11 | e1000961

Page 3

Figure 1. Comparison of unshifted and shifted STDP. A. The U-shaped steady-state distribution of synaptic strengths for conventional

unshifted STDP. B. The unimodal steady-state distribution of synaptic strengths for shifted STDP (d~2ms). C. The probability density of pairing

intervals for presynaptic and postsynaptic spike trains. The blue area is the symmetric acausal contribution, and the pink area is the additional causal

bump arising from postsynaptic spikes induced by the presynaptic input. D. Same as C, but for a stronger synapse. The causal bump is larger and

closer to Dt~0. E. The causal bump superimposed on the unshifted STDP window. The potentiation part of the STDP curve is red and the depression

part blue. The causal bump falls entirely within the potentiation domain (red shading). F. Same as E, but for a stronger synapse. The causal bump still

falls within the potentiation region. G. Same as E, but for shifted STDP. Part of the causal bump falls into the depression region (blue shading). H.

Same as G, but for a stronger synapse. More of the causal bump falls into the depression region.

doi:10.1371/journal.pcbi.1000961.g001

Intrinsic Stability of Shifted STDP

PLoS Computational Biology | www.ploscompbiol.org3 November 2010 | Volume 6 | Issue 11 | e1000961

Page 4

With a few reasonable approximations and ignoring any limits or

bounds, the steady-state distribution of synaptic strengths has the

form of a gamma distribution,

r(w)~N0(wzm)k{1exp {wzm

h

??

,

ð2Þ

where N0is a normalization constant and m, h and k are

computed parameters. If either k or h is negative, this distribution

cannot be normalized, implying unstable synaptic strengths. The

calculations indicate that h is positive for any positive shift (dw0,

Methods). Positivity of k requires that AztzwA{t{. Note that

this is opposite to the condition required of conventional, unshifted

STDP (see for example [5]). Because it is easier to do the analytic

calculations without imposing strict boundary conditions on the

synaptic strengths, the analytic formula sometimes includes a small

probability for negative strength synapses, which is not allowed in

the simulations. Other than this small discrepancy, the agreement

between the analytic distribution and the simulation results is

good (figures 2 & 3). In what follows, d~2ms, Az~0:006mV,

A{~0:005mV, and tz~t{~20ms, unless stated otherwise.

Steady-state firing rate

STDP has an interesting regulatory effect on the steady-state

firing rate of a neuron [5,15]. With unshifted STDP, this is a

buffering effect making the steady-state postsynaptic firing rate

relatively insensitive to the firing rates of excitatory and inhibitory

inputs. Shifted STDP also buffers the postsynaptic firing rate, but

the residual dependence on the presynaptic rates displays an

interesting effect. Although the steady-state firing rate decreases

when the inhibitory input rates are increased, it has a surprising

non-monotonic dependence on the rates of excitatory inputs

(figure 3).

The stabilization of synaptic strengths discussed in the previous

section arises from the change of size and shape of the causal

bump seen in figure 1C & D. Buffering of the steady-state

postsynaptic firing rate is affected primarily by the shape of the

symmetric, non-causal component of the spike-timing probability.

As mentioned previously, this component falls off exponentially,

for either positive or negative spike-timing differences, at a rate

given by the sum of the presynaptic and postsynaptic firing rates

(Methods). If this sum grows, the acausal part of the distribution

gets more peaked near zero, bringing more spike pairs into the

region of the STDP window where the shift leads to synaptic

depression. The resulting reduction in synaptic strength then

lowers the postsynaptic firing rate. This form of buffering would

not be present if all spike pairs, rather than only nearest-neighbor

pairs, were involved in STDP. If we allowed all spike pairs to

induce synaptic plasticity the relevant symmetric, non-causal

distribution would be flat, rather than exponentially decaying. In

this case, there is no analogous stabilization and, in fact,

postsynaptic rates slowly rise, making the plasticity unstable, even

with shifted STDP. This is why we require shifted STDP to be

based only on nearest-neighbor spike pairs.

Figure 2. Shifted STDP stabilizes the distribution of synaptic strengths. The horizontal axis is the value of the shift, the vertical axis is the

synaptic strength and the gray level is the probability density of strengths, obtained by simulation. Solid line is the analytically calculated mean and

dashed lines show the analytically calculated standard deviation around the mean. Insets show the distribution of synaptic strengths for different

values of the shift. Solid curves are analytically calculated distributions. The arrows at the bottom of the horizontal axis of the main plot show the shift

values corresponding to the insets.

doi:10.1371/journal.pcbi.1000961.g002

Intrinsic Stability of Shifted STDP

PLoS Computational Biology | www.ploscompbiol.org4 November 2010 | Volume 6 | Issue 11 | e1000961

Page 5

In general, we expect the firing rate of a neuron to increase

when its excitatory inputs fire more rapidly, and this is exactly

what occurs for excitatory input rates below about 10 Hz in

figure 3. However, for excitatory input rates higher than this, the

steady-state (after STDP has equilibrated) postsynaptic firing rate

decreases. This occurs for the reason outlined in the previous

paragraph. Increasing the presynaptic rate causes the acausal

distribution to sharpen and induces synaptic depression. This slows

the postsynaptic rate, broadening the acausal distribution until the

spike intervals in the delay region are sufficiently reduced in

number. This is what causes the steady-state postsynaptic firing

rate to drop when the excitatory presynaptic rates are raised to

high levels.

Shifted STDP also has a buffering property on changes in the

inhibitory input rate. In presence of strong inhibitory input, the

postsynaptic firing rate falls. This broadens the acausal part of the

spike-pair distribution, lowering the chance for pairs to fall into the

depression domain caused by the shift and, thus, resulting in more

potentiation. However, in this case, the effect is not strong enough

to overcome the expected tendency of the postsynaptic rate to be

suppressed by inhibition (figure 3).

Synaptic competition

Hebbian plasticity in general and STDP in particular allows

neurons to become selective to correlated subsets of their inputs,

but this requires synaptic competition [1]. We call synaptic

plasticity ‘‘competitive’’ if correlating a subset of synaptic inputs

causes both that set and the remaining synapses to change their

strengths in an opposing manner, so than either the correlated or

the uncorrelated set of synapses gains control of the postsynaptic

firing (see for example [11]). In particular, if STDP is competitive,

the strengths of either the correlated or uncorrelated subgroup of

synapses should cluster near zero. To determine whether the

necessary competition exists with shifted STDP, we imposed

pairwise correlations with a coefficient of 0.2 on one half of the

incoming excitatory spike trains while leaving the other half

uncorrelated (Methods). With unshifted STDP, this arrangement

induces a competition that correlated synapses always win [5]. In

Figure 3. The steady-state postsynaptic firing rate. The steady state firing rate is plotted as a function of the input rates for excitation and

inhibition. The inset shows the corresponding analytic result.

doi:10.1371/journal.pcbi.1000961.g003

Intrinsic Stability of Shifted STDP

PLoS Computational Biology | www.ploscompbiol.org5 November 2010 | Volume 6 | Issue 11 | e1000961