We study the synchronization of two model neurons coupled through a synapse
having an activity-dependent strength. Our synapse follows the rules of
Spike-Timing Dependent Plasticity (STDP). We show that this plasticity of the
coupling between neurons produces enlarged frequency locking zones and results
in synchronization that is more rapid and much more robust against noise than
classical synchronization arising from connections with constant strength. We
also present a simple discrete map model that demonstrates the generality of
the phenomenon.Comment: 4 pages, accepted for publication in PR

C. Bell

G. Laurent

G.-Q. Bi

H. Markram

J. Fell

L. Glass

M. Rabinovich

R. Elson

S. Coombes

W. Miltner

Y. Perez

English

arXiv

Crossref

Robustness and enhancement of neural synchronization by activity-dependent coupling

We study the synchronization of two model neurons coupled through a synapse having an activity-dependent strength. Our synapse follows the rules of spike-timing dependent plasticity. We show that this plasticity of the coupling between neurons produces enlarged frequency-locking zones and results in synchronization that is more rapid and much more robust against noise than classical synchronization arising from connections with constant strength. We also present a simple discrete map model that demonstrates the generality of the phenomenon

Zhigluin, V. P.

Rabinovich, M. I.

Huerta, R.

Abarbanel, H. D. I.

Caltech Authors - Main

ego,
PHYSICAL REVIEW E 67, 021901 ~2003!Robustness and enhancement of neural synchronization by activity-dependent coupling
V. P. Zhigulin,1,2,* M. I. Rabinovich,2 R. Huerta,2,3 and H. D. I. Abarbanel2,4
1Department of Physics, MC 103-33, California Institute of Technology, Pasadena, California 91125
2Institute for Nonlinear Science, University of California, San Diego, La Jolla, California 92093-0402
3GNB, ETS de Ingenierı´a Informática, Universidad Auto´noma de Madrid, 28049 Madrid, Spain
4Department of Physics and Marine Physical Laboratory (Scripps Institution of Oceanography), University of California, San Di
La Jolla, California 93093-0402
~Received 7 February 2002; revised manuscript received 3 October 2002; published 6 February 2003!
We study the synchronization of two model neurons coupled through a synapse having an activity-dependent
strength. Our synapse follows the rules of spike-timing dependent plasticity. We show that this plasticity of the
coupling between neurons produces enlarged frequency-locking zones and results in synchronization that is
more rapid and much more robust against noise than classical synchronization arising from connections with
constant strength. We also present a simple discrete map model that demonstrates the generality of the phe-
nomenon.
DOI: 10.1103/PhysRevE.67.021901 PACS number~s!: 87.18.Sn, 87.18.Bb, 05.45.Xtei
io
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yn-
entSynchronous activity among individual neurons or th
ensembles is a robust phenomenon observed in many reg
of the brain, in sensory systems and in other neural netwo
With constant synaptic connections, the regions of neu
synchronization are quite narrow in parameter space and
origin of the observed robustness of synchronization is
clear. It is known that many neurons in the cortex, in t
cerebellum, and in other neural systems are coupled thro
excitatory synaptic connections whose strength can be
tered through activity-dependent plasticity. Indeed, this p
ticity is widely thought to underlie learning processes, and
itself constitutes a broadly interesting phenomenon. Here
discuss its role in the synchronization of neurons in a n
work.
There have been recent experimental advances in the
derstanding of such plasticity, and, in particular, of the cr
cal dependence on timing in presynaptic and postsyna
signaling. Two manifestations of this kind of synaptic pla
ticity are the spike-timing dependent plasticity~STDP! @1,2#,
seen in excitatory connections between neurons, and its
verse, observed, for example, in the connections betw
excitatory and inhibitory neurons in the electrosensory lo
of fish @3#. The connections between excitatory neuro
through inhibitory interneurons are typical in sensory s
tems @4,5# and the cerebral cortex@6#. These also expres
synaptic plasticity@7# and play an important role in the con
trol and synchronization of neural ensembles in the hipp
ampus.
We report here on the synchronization of two model n
rons coupled through STDP or inverse STDP synapses.
demonstrate that such coupling leads to neural synchron
tion that is more rapid, more flexible, and much more rob
against noise than synchronization mediated by cons
strength connections.~For reviews, see@8–10#.! We also
build a simple discrete map that illustrates the enhancem
of synchronization by activity-dependent coupling. The m
*Electronic address: zhigulin@caltech.edu1063-651X/2003/67~2!/021901~4!/$20.00 67 0219r
ns
s.
al
he
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gh
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n-
-
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-
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e
s
-
-
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e
a-
t
nt
nt
p
allows us to speculate about the general applicability
learning-enhanced synchronization.
We consider here the simplest neural network: two n
rons with unidirectional, activity-dependent excitatory sy
aptic coupling. Each neuron is described by the Hodgk
Huxley equations with standard Na, K, and ‘‘leak’’ curren
@11#,
C
dVi~ t !
dt
52gNami~ t !
3hi~ t !@Vi~ t !2ENa#2gKni~ t !
4
3@Vi~ t !2EK#2gL@Vi~ t !2EL#2I syn~ t !1I stim,
~1!
wherei 51,2.
Each of the activation and inactivation variablesyi(t)
5$ni(t),mi(t),hi(t)% satisfies first-order kinetics,
dyi~ t !
dt
5ay@Vi~ t !#@12yi~ t !#2by@Vi~ t !#yi~ t !. ~2!
The parameters in these equations are given in@12#.
Each neuron receives a constant inputI stim forcing it to
spike with a constantI stim-dependent frequency. The secon
neuron is synaptically driven by the first via an excitato
current dependent on the postsynapticV2(t) and presynaptic
V1(t) membrane voltages,
I syn~ t !5g~ t !S~ t !V2~ t !. ~3!
S(t) is the fraction of open synaptic channels. It satisfi
first-order kinetics,
dS~ t !
dt
5a@12S~ t !#H„V1~ t !…2bS~ t !, ~4!
with H„V1(t)…5$11tanh@10V1(t)#%/4.
The time-dependent synaptic coupling strength(t) is
conditioned by the dynamics of the presynaptic and posts
aptic neurons. We consider two types of activity-depend
couplings:~i! an excitatory synapse with STDP, and~ii ! an©2003 The American Physical Society01-1
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ZHIGULIN et al. PHYSICAL REVIEW E 67, 021901 ~2003!excitatory synapse with inverse STDP. Through STDPg(t)
changes byDg(t), which is a function of the time differenc
Dt5tpost2tpre between the times of postsynaptic and pres
aptic spikes. We use the additive update rule
Dg~ t !5G~Dt !5Asgn~Dt !exp~2guDtu! ~5!
for STDP andDg(t)52G(Dt) for inverse STDP. We used
A50.004mS andg50.15 ms21.
We studied the synchronization properties of this coup
system by setting the autonomous period of the postsyna
neuron to 15 ms, then evaluating the actual period of
oscillationT2 as a function of the imposed autonomous o
cillation periodT1 of the presynaptic neuron. In Fig. 1, w
showT1 /T2 as a function ofT1 in two cases:~a! a synaptic
coupling with constant strength 0.008mS and~b! a synaptic
coupling with inverse STDP. In the latter case, the stea
state coupling strength depends on the ratio of neuronal
quencies~c!. Its average over allT1 values is 0.002mS,
which is much lower than the strength in the case of cons
coupling.
In Fig. 1~a!, we see the familiar ‘‘devil’s staircase’’ asso
ciated with frequency-locking domains of a driven nonline
oscillator. Only frequency locking with ratios 1:1, 2:1, 3:
and 4:1 leads to synchronization plateaus with signific
width. In Fig. 1~b!, we see that the synchronization domai
are substantially broadened due to activity-dependent c
pling, especially forT1 /T251. Some synchronization pla
teaus exhibit multistability, which we confirmed by obser
ing the associated hysteresis. These results show that e
FIG. 1. Devil’s staircase for~a! constant synaptic strength an
~b! synaptic strength varying according to inverse STDP coupli
T1 and T2 are the observed periods of the presynaptic~driving!
neuron and postsynaptic~driven! neuron, respectively. In~c! the
final value of synaptic strength is displayed.02190-
d
tic
s
-
-
e-
nt
r
t
u-
n a
weak but adaptive connection with strength that is de
mined dynamically is able to greatly enhance and enrich s
chronization.
We also studied the robustness of this enhanced sync
nization in the presence of noise by adding zero me
Gaussian, white noise to the membrane currents of each
ron. We examined the behavior of the system with RM
noise amplitudes 50.01, 0.05, 0.1, and 0.5 nA.
For s50.01 nA, no phase-locking plateaus were d
stroyed. Ats50.05 nA, the 4:1 plateau became distorte
Larger s sequentially eliminated synchronization platea
until only the 1:1 plateau remained. The 1:1 plateau was s
for all s. In Fig. 2, we illustrate the effect of the noise o
synchronization whens50.1 nA with ~a! constant and~b!
inverse STDP coupling. While in~a! most of the plateaus
have disappeared, in~b! the 1:1, the 2:1, and even the 3
frequency-locking regimes remained. In sharp distinction
classical synchronization, frequency locking throu
activity-dependent coupling is significantly more robust
the presence of noise.
To understand the mechanisms behind such a remark
robustness, we studied the diffusion of oscillation pha
caused by noise. Fors50.5 nA, in Fig. 3~a! we show that in
the case of 1:1 synchronization and coupling with const
strength 0.008mS, noise-induced phase diffusion results
2p phase slips that destroy synchronized state. Quite to
contrary, Fig. 3~b! shows that in the case of activity
dependent coupling, phase slips are absent and the p
difference does not increase. In this particular case,
strength of coupling varied around the mean of 0.0064mS
with a standard deviation of 0.0026mS.
In Fig. 4, we plot the average rate of phase slips for d
ferent amplitudes of the noise. In line with the above obs
.
FIG. 2. Same as Fig. 1, but with zero mean, Gaussian, w
noise withs50.1 nA added to the membrane currents.1-2
lin
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ROBUSTNESS AND ENHANCEMENT OF NEURAL . . . PHYSICAL REVIEW E 67, 021901 ~2003!vation, we see that in the case of activity-dependent coup
~dashed line!, phase slips are suppressed in a wide range
noise amplitudes. We argue here that this suppressio
phase slips is the primary mechanism responsible for rob
ness of synchronization mediated by activity-dependent c
pling. After the introduction of a discrete map model, we w
discuss this mechanism in more detail.
We also considered synchronization through an activ
dependent synapse in the interesting case in which
presynaptic neuron produces bursts of spikes and
postsynaptic neuron spikes irregularly. We found that s
chronization through an STDP synapse is very fast; eve
few spikes are enough for the frequency locking to estab
itself. Neurons in the same setup with constant coupling s
chronize much more slowly andonly if the strength of the
connection is appropriate for the given ratio of their freque
cies. Hence, activity-dependent synapses allow adapta
‘‘on the run,’’ synching a postsynaptic neuron to the firin
properties of its presynaptic partner.
To understand the above results in a general way, we h
constructed a discrete time map model of periodic genera
with STDP-like coupling. This map accounts for the depe
dence of the coupling strength on the activity of generato
FIG. 3. The difference of oscillation phases of two neurons a
function of time in the cases of~a! constant and~b! activity-
dependent coupling.
FIG. 4. Average rate of phase slips as a function of rms no
amplitude for the case of 1:1 synchronization and constant~solid
line! or activity-dependent~dashed line! coupling.02190g
of
of
t-
u-
-
e
e
-
a
h
n-
-
on
ve
rs
-
s.
Take T1
0 and T2
0 as the autonomous periods of the first a
second generators. As a result of unidirectional coupling,
period of the second generator will change by some amo
DT each time it receives a spike from the first genera
Assuming initial phases to be 0, the times of the (n11)st
spike of the first generator and the (m11)st spike of the
second generator are taken to satisfy
tn11
(1) 5tn
(1)1T1
0 , ~6a!
tm11
(2) 5tm
(2)1T2
02DTm,n , ~6b!
where n and m are such that m
(2)<tn
(1)<tm11
(2) . In general,
DTm,n would be a function ofT1
0 , T2
0 , tn
(1) , tm
(2) , and the
coupling strengthgm,n . We argue that the two main variable
here aretn
(1)2tm
(2) andgm,n . In the simplest case,DTm,n can
be approximated by
DTm,n5gm,nF~ tn
(1)2tm
(2)!, ~7!
where the functionF(x) is the analog of a phase respon
curve @13# for our model. To obtain results quantitative
comparable with our neuronal model, we fit it by a no
negative quadratic function that describes the phase resp
of our model neurons:F(x)5835163x29x2 for 0<x<T2
0
and 0 otherwise.gm,n obeys the inverse STDP update rule
gm11,n5gm,n2G~ tm11
(2) 2tn
(1)!, ~8a!
gm,n5gm,n212G~ tm
(2)2tn
(1)!. ~8b!
In Fig. 5, we show the Arnol’d tongues calculated for t
map ~6a!–~8b! in the cases of~a! constant and~b! inverse
STDP coupling. As with the model neurons, we see t
activity-dependent coupling greatly enlarges the zones
synchronization.
This discrete map can be further analyzed to find its fix
points corresponding ton:m synchronization and to examin
a
e
FIG. 5. Arnol’d tongues calculated for the discrete map mo
with ~a! constant and~b! activity-dependent coupling.T2
0513 ms.1-3
ro
ty
la
i
r
r
ty
a
ss
ty
le
ta
s
u
nes
of
tion
dea
he
ing.
that
In
ced
of
of
3-
A-
1-
-1-
ks
ZHIGULIN et al. PHYSICAL REVIEW E 67, 021901 ~2003!their stability. We present here only the case of 1:1 synch
nization. Thenm5n, and the system of equations~6!–~8!
can be written in the following simple form:
tn115tn1T1
02T2
01gnF~tn!, ~9a!
gn115gn2G~T1
02tn11!2G~2tn11!, ~9b!
wheretn5tn
(1)2tn
(2) . The fixed points of Eqs.~9! are given
by gn
f 5(T2
02T1
0)/F(tn
f ) andtn
f 5T1
0/2. Stability calculations
show that for suchF(t) and G(t) these fixed points are
stable. The second fixed point illustrates that activi
dependent coupling introduces a new limitation on the re
tionship between the phases of two oscillators. It is this lim
tation that causes the suppression of phase slips unde
influence of noise. Detailed analysis shows that in the cou
of noise-affected synchronization, the strength of activi
dependent coupling adjusts dynamically to keep this ph
relationship close to satisfaction and, hence, suppre
phase slips.
In conclusion, we have analyzed the effects of activi
dependent coupling on synchronization properties of coup
neurons. We showed that such coupling results in a subs
tial extension of the temporal synchronization zones, lead
more rapid synchronization, and makes it much more robnc
r-
,
ci
h
-
02190-
-
-
-
the
se
-
se
es
-
d
n-
to
st
against noise. The enlargement of synchronization zo
means that with STDP-like learning rules, the number
synchronized neurons in a large heterogeneous popula
must increase. In fact, this is an aspect of the popular i
due to Hebb@14#. It is supported by the results in@15,16#,
which indicate that the coherence of fast EEG activity in t
gamma band increases in a process of associative learn
Based on our discrete map model results, we argue
the particular details of the signal-generating devices~e.g.,
neurons! and their connections~e.g., synapses! are not essen-
tial and the obtained results have general applicability.
fact, we observed similar phenomena of robust and enhan
synchronization in computer simulations of other types
periodic generators~such as Van-der-Pol andu oscillators!
with STDP-like activity-dependent coupling.
ACKNOWLEDGMENTS
This work was partially supported by U.S. Department
Energy Grant Nos. DE-FG03-90ER14138 and DE-FG0
96ER14592, NSF Grant Nos. PHY0097134 and EI
0130708, Army Research Office Grant No. DAAD19-01-
0026, Office of Naval Research Grant No. N00014-00
0181, and NIH Grant No. R01 NS40110-01A2. R.H. than
MCyT ~Spain! for Grant No. BFI2000-0157.,
C.@1# H. Markram, J. Lubke, M. Frotscher, and B. Sakmann, Scie
275, 213 ~1997!.
@2# G.-Q. Bi and M.-M. Poo, J. Neurosci.18, 10464~1998!.
@3# C. Bell, V. Han, Y. Sugavara, and K. Grant, J. Exp. Biol.202,
1339 ~1999!.
@4# G. Laurent, Science286, 723 ~1999!.
@5# M. Rabinovich, A. Volkovskii, P. Lecanda, R. Huerta, H. Aba
banel, and G. Laurent, Phys. Rev. Lett.87, 068102~2001!.
@6# M. Bear and D. Linden, inSynapses, edited by W.M. Cowan,
T.C. Sdhof, and C.F. Stevens~Johns Hopkins University Press
Baltimore, MD, 2001!, p. 455.
@7# Y. Perez, F. Morin, and J.-C. Lacaille, Proc. Natl. Acad. S
U.S.A. 98, 9401~2001!.
@8# L. Glass, Nature~London! 410, 277 ~2001!.
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and H. Abarbanel, Phys. Rev. Lett.81, 5692~1998!.
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@11# R. D. Traub and R. Miles,Neuronal Networks of the Hippoe
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,
campus~Cambridge University Press, New York, 1991!, Chap.
4.
@12# The following values were used:gL50.027mS, EL5
264 mV, ENa550 mV, gNa57.15mS, gK51.43mS, EK5
295 mV, a510 ms21, b50.2 ms21, C51.4331024
mF, an50.032(2502V) / $exp@(2502V)/5#21%; bn5
0.5 exp@(2552V) / 40#, am50.32(2522V) / $exp@(252
2V)/4#21%, bm50.28(251V)/$exp@(251V)/5#21%, ah
50.128 exp@(2482V)/18#, and bh54/$exp@(2252V)/5#
11%.
@13# A. Winfree, The Geometry of Biological Time~Springer-
Verlag, New York, 1980!.
@14# D. Hebb, The Organization of Behavior~Wiley, New York,
1949!.
@15# W. Miltner, C. Braun, M. Arnold, H. Witte, and E. Traub
Nature~London! 397, 434 ~1999!.
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Elgar, and G. Fernandez, Nat. Neurosci.4, 1259~2001!.1-4


PHYSICAL REVIEW E 67, 021901 ~2003!Robustness and enhancement of neural synchronization by activity-dependent coupling
V. P. Zhigulin,1,2,* M. I. Rabinovich,2 R. Huerta,2,3 and H. D. I. Abarbanel2,4
1Department of Physics, MC 103-33, California Institute of Technology, Pasadena, California 91125
2Institute for Nonlinear Science, University of California, San Diego, La Jolla, California 92093-0402
3GNB, ETS de Ingenierı´a Informa´tica, Universidad Auto´noma de Madrid, 28049 Madrid, Spain
4Department of Physics and Marine Physical Laboratory (Scripps Institution of Oceanography), University of California, San Diego,
La Jolla, California 93093-0402
~Received 7 February 2002; revised manuscript received 3 October 2002; published 6 February 2003!
We study the synchronization of two model neurons coupled through a synapse having an activity-dependent
strength. Our synapse follows the rules of spike-timing dependent plasticity. We show that this plasticity of the
coupling between neurons produces enlarged frequency-locking zones and results in synchronization that is
more rapid and much more robust against noise than classical synchronization arising from connections with
constant strength. We also present a simple discrete map model that demonstrates the generality of the phe-
nomenon.
DOI: 10.1103/PhysRevE.67.021901 PACS number~s!: 87.18.Sn, 87.18.Bb, 05.45.XtSynchronous activity among individual neurons or their
ensembles is a robust phenomenon observed in many regions
of the brain, in sensory systems and in other neural networks.
With constant synaptic connections, the regions of neural
synchronization are quite narrow in parameter space and the
origin of the observed robustness of synchronization is not
clear. It is known that many neurons in the cortex, in the
cerebellum, and in other neural systems are coupled through
excitatory synaptic connections whose strength can be al-
tered through activity-dependent plasticity. Indeed, this plas-
ticity is widely thought to underlie learning processes, and in
itself constitutes a broadly interesting phenomenon. Here we
discuss its role in the synchronization of neurons in a net-
work.
There have been recent experimental advances in the un-
derstanding of such plasticity, and, in particular, of the criti-
cal dependence on timing in presynaptic and postsynaptic
signaling. Two manifestations of this kind of synaptic plas-
ticity are the spike-timing dependent plasticity ~STDP! @1,2#,
seen in excitatory connections between neurons, and its in-
verse, observed, for example, in the connections between
excitatory and inhibitory neurons in the electrosensory lobe
of fish @3#. The connections between excitatory neurons
through inhibitory interneurons are typical in sensory sys-
tems @4,5# and the cerebral cortex @6#. These also express
synaptic plasticity @7# and play an important role in the con-
trol and synchronization of neural ensembles in the hippoc-
ampus.
We report here on the synchronization of two model neu-
rons coupled through STDP or inverse STDP synapses. We
demonstrate that such coupling leads to neural synchroniza-
tion that is more rapid, more flexible, and much more robust
against noise than synchronization mediated by constant
strength connections. ~For reviews, see @8–10#.! We also
build a simple discrete map that illustrates the enhancement
of synchronization by activity-dependent coupling. The map
*Electronic address: zhigulin@caltech.edu1063-651X/2003/67~2!/021901~4!/$20.00 67 0219allows us to speculate about the general applicability of
learning-enhanced synchronization.
We consider here the simplest neural network: two neu-
rons with unidirectional, activity-dependent excitatory syn-
aptic coupling. Each neuron is described by the Hodgkin-
Huxley equations with standard Na, K, and ‘‘leak’’ currents
@11#,
C
dVi~ t !
dt 52gNami~ t !
3hi~ t !@Vi~ t !2ENa#2gKni~ t !4
3@Vi~ t !2EK#2gL@Vi~ t !2EL#2Isyn~ t !1Istim ,
~1!
where i51,2.
Each of the activation and inactivation variables yi(t)
5$ni(t),mi(t),hi(t)% satisfies first-order kinetics,
dyi~ t !
dt 5ay@Vi~ t !#@12yi~ t !#2by@Vi~ t !#yi~ t !. ~2!
The parameters in these equations are given in @12#.
Each neuron receives a constant input Istim forcing it to
spike with a constant Istim-dependent frequency. The second
neuron is synaptically driven by the first via an excitatory
current dependent on the postsynaptic V2(t) and presynaptic
V1(t) membrane voltages,
Isyn~ t !5g~ t !S~ t !V2~ t !. ~3!
S(t) is the fraction of open synaptic channels. It satisfies
first-order kinetics,
dS~ t !
dt 5a@12S~ t !#HV1~ t !2bS~ t !, ~4!
with HV1(t)5$11tanh@10V1(t)#%/4.
The time-dependent synaptic coupling strength g(t) is
conditioned by the dynamics of the presynaptic and postsyn-
aptic neurons. We consider two types of activity-dependent
couplings: ~i! an excitatory synapse with STDP, and ~ii! an©2003 The American Physical Society01-1
ZHIGULIN et al. PHYSICAL REVIEW E 67, 021901 ~2003!excitatory synapse with inverse STDP. Through STDP g(t)
changes by Dg(t), which is a function of the time difference
Dt5tpost2tpre between the times of postsynaptic and presyn-
aptic spikes. We use the additive update rule
Dg~ t !5G~Dt !5Asgn~Dt !exp~2guDtu! ~5!
for STDP and Dg(t)52G(Dt) for inverse STDP. We used
A50.004 mS and g50.15 ms21.
We studied the synchronization properties of this coupled
system by setting the autonomous period of the postsynaptic
neuron to 15 ms, then evaluating the actual period of its
oscillation T2 as a function of the imposed autonomous os-
cillation period T1 of the presynaptic neuron. In Fig. 1, we
show T1 /T2 as a function of T1 in two cases: ~a! a synaptic
coupling with constant strength 0.008 mS and ~b! a synaptic
coupling with inverse STDP. In the latter case, the steady-
state coupling strength depends on the ratio of neuronal fre-
quencies ~c!. Its average over all T1 values is 0.002 mS,
which is much lower than the strength in the case of constant
coupling.
In Fig. 1~a!, we see the familiar ‘‘devil’s staircase’’ asso-
ciated with frequency-locking domains of a driven nonlinear
oscillator. Only frequency locking with ratios 1:1, 2:1, 3:1,
and 4:1 leads to synchronization plateaus with significant
width. In Fig. 1~b!, we see that the synchronization domains
are substantially broadened due to activity-dependent cou-
pling, especially for T1 /T251. Some synchronization pla-
teaus exhibit multistability, which we confirmed by observ-
ing the associated hysteresis. These results show that even a
FIG. 1. Devil’s staircase for ~a! constant synaptic strength and
~b! synaptic strength varying according to inverse STDP coupling.
T1 and T2 are the observed periods of the presynaptic ~driving!
neuron and postsynaptic ~driven! neuron, respectively. In ~c! the
final value of synaptic strength is displayed.02190weak but adaptive connection with strength that is deter-
mined dynamically is able to greatly enhance and enrich syn-
chronization.
We also studied the robustness of this enhanced synchro-
nization in the presence of noise by adding zero mean,
Gaussian, white noise to the membrane currents of each neu-
ron. We examined the behavior of the system with RMS
noise amplitudes s50.01, 0.05, 0.1, and 0.5 nA.
For s50.01 nA, no phase-locking plateaus were de-
stroyed. At s50.05 nA, the 4:1 plateau became distorted.
Larger s sequentially eliminated synchronization plateaus
until only the 1:1 plateau remained. The 1:1 plateau was seen
for all s . In Fig. 2, we illustrate the effect of the noise on
synchronization when s50.1 nA with ~a! constant and ~b!
inverse STDP coupling. While in ~a! most of the plateaus
have disappeared, in ~b! the 1:1, the 2:1, and even the 3:1
frequency-locking regimes remained. In sharp distinction to
classical synchronization, frequency locking through
activity-dependent coupling is significantly more robust in
the presence of noise.
To understand the mechanisms behind such a remarkable
robustness, we studied the diffusion of oscillation phase
caused by noise. For s50.5 nA, in Fig. 3~a! we show that in
the case of 1:1 synchronization and coupling with constant
strength 0.008 mS, noise-induced phase diffusion results in
2p phase slips that destroy synchronized state. Quite to the
contrary, Fig. 3~b! shows that in the case of activity-
dependent coupling, phase slips are absent and the phase
difference does not increase. In this particular case, the
strength of coupling varied around the mean of 0.0064 mS
with a standard deviation of 0.0026 mS.
In Fig. 4, we plot the average rate of phase slips for dif-
ferent amplitudes of the noise. In line with the above obser-
FIG. 2. Same as Fig. 1, but with zero mean, Gaussian, white
noise with s50.1 nA added to the membrane currents.1-2
ROBUSTNESS AND ENHANCEMENT OF NEURAL . . . PHYSICAL REVIEW E 67, 021901 ~2003!vation, we see that in the case of activity-dependent coupling
~dashed line!, phase slips are suppressed in a wide range of
noise amplitudes. We argue here that this suppression of
phase slips is the primary mechanism responsible for robust-
ness of synchronization mediated by activity-dependent cou-
pling. After the introduction of a discrete map model, we will
discuss this mechanism in more detail.
We also considered synchronization through an activity-
dependent synapse in the interesting case in which the
presynaptic neuron produces bursts of spikes and the
postsynaptic neuron spikes irregularly. We found that syn-
chronization through an STDP synapse is very fast; even a
few spikes are enough for the frequency locking to establish
itself. Neurons in the same setup with constant coupling syn-
chronize much more slowly and only if the strength of the
connection is appropriate for the given ratio of their frequen-
cies. Hence, activity-dependent synapses allow adaptation
‘‘on the run,’’ synching a postsynaptic neuron to the firing
properties of its presynaptic partner.
To understand the above results in a general way, we have
constructed a discrete time map model of periodic generators
with STDP-like coupling. This map accounts for the depen-
dence of the coupling strength on the activity of generators.
FIG. 3. The difference of oscillation phases of two neurons as a
function of time in the cases of ~a! constant and ~b! activity-
dependent coupling.
FIG. 4. Average rate of phase slips as a function of rms noise
amplitude for the case of 1:1 synchronization and constant ~solid
line! or activity-dependent ~dashed line! coupling.02190Take T1
0 and T2
0 as the autonomous periods of the first and
second generators. As a result of unidirectional coupling, the
period of the second generator will change by some amount
DT each time it receives a spike from the first generator.
Assuming initial phases to be 0, the times of the (n11)st
spike of the first generator and the (m11)st spike of the
second generator are taken to satisfy
tn11
(1) 5tn
(1)1T1
0
, ~6a!
tm11
(2) 5tm
(2)1T2
02DTm ,n , ~6b!
where n and m are such that tm
(2)<tn
(1)<tm11
(2)
. In general,
DTm ,n would be a function of T1
0
, T2
0
, tn
(1)
, tm
(2)
, and the
coupling strength gm ,n . We argue that the two main variables
here are tn
(1)2tm
(2) and gm ,n . In the simplest case, DTm ,n can
be approximated by
DTm ,n5gm ,nF~ tn
(1)2tm
(2)!, ~7!
where the function F(x) is the analog of a phase response
curve @13# for our model. To obtain results quantitatively
comparable with our neuronal model, we fit it by a non-
negative quadratic function that describes the phase response
of our model neurons: F(x)5835163x29x2 for 0<x<T20
and 0 otherwise. gm ,n obeys the inverse STDP update rules,
gm11,n5gm ,n2G~ tm11
(2) 2tn
(1)!, ~8a!
gm ,n5gm ,n212G~ tm
(2)2tn
(1)!. ~8b!
In Fig. 5, we show the Arnol’d tongues calculated for the
map ~6a!–~8b! in the cases of ~a! constant and ~b! inverse
STDP coupling. As with the model neurons, we see that
activity-dependent coupling greatly enlarges the zones of
synchronization.
This discrete map can be further analyzed to find its fixed
points corresponding to n:m synchronization and to examine
FIG. 5. Arnol’d tongues calculated for the discrete map model
with ~a! constant and ~b! activity-dependent coupling. T2
0513 ms.1-3
ZHIGULIN et al. PHYSICAL REVIEW E 67, 021901 ~2003!their stability. We present here only the case of 1:1 synchro-
nization. Then m5n , and the system of equations ~6!–~8!
can be written in the following simple form:
tn115tn1T1
02T2
01gnF~tn!, ~9a!
gn115gn2G~T1
02tn11!2G~2tn11!, ~9b!
where tn5tn
(1)2tn
(2)
. The fixed points of Eqs. ~9! are given
by gn
f 5(T202T10)/F(tnf ) and tnf 5T10/2. Stability calculations
show that for such F(t) and G(t) these fixed points are
stable. The second fixed point illustrates that activity-
dependent coupling introduces a new limitation on the rela-
tionship between the phases of two oscillators. It is this limi-
tation that causes the suppression of phase slips under the
influence of noise. Detailed analysis shows that in the course
of noise-affected synchronization, the strength of activity-
dependent coupling adjusts dynamically to keep this phase
relationship close to satisfaction and, hence, suppresses
phase slips.
In conclusion, we have analyzed the effects of activity-
dependent coupling on synchronization properties of coupled
neurons. We showed that such coupling results in a substan-
tial extension of the temporal synchronization zones, leads to
more rapid synchronization, and makes it much more robust02190against noise. The enlargement of synchronization zones
means that with STDP-like learning rules, the number of
synchronized neurons in a large heterogeneous population
must increase. In fact, this is an aspect of the popular idea
due to Hebb @14#. It is supported by the results in @15,16#,
which indicate that the coherence of fast EEG activity in the
gamma band increases in a process of associative learning.
Based on our discrete map model results, we argue that
the particular details of the signal-generating devices ~e.g.,
neurons! and their connections ~e.g., synapses! are not essen-
tial and the obtained results have general applicability. In
fact, we observed similar phenomena of robust and enhanced
synchronization in computer simulations of other types of
periodic generators ~such as Van-der-Pol and u oscillators!
with STDP-like activity-dependent coupling.
ACKNOWLEDGMENTS
This work was partially supported by U.S. Department of
Energy Grant Nos. DE-FG03-90ER14138 and DE-FG03-
96ER14592, NSF Grant Nos. PHY0097134 and EIA-
0130708, Army Research Office Grant No. DAAD19-01-1-
0026, Office of Naval Research Grant No. N00014-00-1-
0181, and NIH Grant No. R01 NS40110-01A2. R.H. thanks
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0.5 exp@(2552V) / 40# , am50.32(2522V) / $exp@(252
2V)/4#21%, bm50.28(251V)/$exp@(251V)/5#21%, ah
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Robustness and Enhancement of Neural Synchronization by
  Activity-Dependent Coupling

Robustness and Enhancement of Neural Synchronization by Activity-Dependent Coupling

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