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Impact of spatially correlated noise on neuronal firing
Sentao Wang, Feng Liu,*and Wei Wang
National Laboratory of Solid State Microstructure and Department of Physics, Nanjing University, Nanjing 210093, China
Yuguo Yu
Center for the Neural Basis of Cognition, Carnegie Mellon University, Pittsburgh Pennsylvania 15213, USA
共Received 6 May 2003; revised manuscript received 30 September 2003; published 29 January 2004兲
We explore the impact of spatially correlated noise on neuronal firing when uncoupled Hodgkin-Huxley
model neurons are subjected to a common subthreshold signal. Noise can play a positive role in optimizing
neuronal behavior. Although the output signal-to-noise ratio decreases with enhanced noise correlation, both
the degree of synchronization among neurons and the spike timing precision are improved. This suggests that
there can exist precisely synchronized firings in the presence of correlated noise and that the nervous system
can exploit temporal patterns of neural activity to convey more information than just using rate codes. The
mechanisms underlying these noise-induced effects are also discussed in detail.
DOI: 10.1103/PhysRevE.69.011909 PACS number共s兲: 87.16.Ac, 05.40.Ca
I. INTRODUCTION
Cortical spike trains with high interspike interval 共ISI兲
variability have been observed in a wide range of stimulus-
evoked activity of pyramidal neurons 关1兴. Moreover, the neu-
ral responses to repeated presentations of the same stimulus
often vary largely from trial to trial 关2兴. An issue thus arises
concerning how the nervous system can precisely process
information. One often assumes that the response variability
represents ‘‘neural noise’’ and that the averaged response
over some populations of neurons may suppress the inherent
noise and enhance the extracted information about the stimu-
lus. But such a hypothesis is based on the condition that the
response variation in each neuron is more or less indepen-
dent of that in its neighbors 关2,3兴. However, it has been dem-
onstrated that neural populations can exhibit significant co-
variance both in their spontaneous and stimulus-evoked
activity 关4,5兴.
It has been suggested that, in the presence of correlation
in response variation, one possible function for neurons to
carry redundant message may be to improve the temporal
resolution in coding a rapidly changing variable 关6兴. That is,
temporal correlations are crucial for signal processing as part
of the information is encoded in the temporal structure of
activity patterns. Furthermore, the synchronized activities of
neurons with high temporal precision can be transmitted
more efficiently than the asynchronous ones 关7兴. It has also
been argued 关8兴that cortical neurons might act more as co-
incidence detectors preferentially relaying synchronized ac-
tivity, than as temporal integrators effectively summating in-
coming synaptic inputs. However, how the synchronization
and coincidence-detection mechanisms work in as noisy an
environment as a cortical circuit remains elusive.
It is well known that cortical neurons are subjected to
large numbers of random synaptic inputs and other endog-
enous noise. As a first approximation, we can model these
together as Gaussian noise. But as mentioned above, cortical
neurons display coherence in their firing activity, and thus
the correlation in input noise must be taken into account. It
has recently been shown that noise can play a constructive
role in weak signal detection, such as improving the output
signal-to-noise ratio 共SNR兲, in the context of stochastic reso-
nance 共SR兲关9兴. While we have previously discussed the im-
pact of spatially correlated noise on the output SNR 关10兴,
here we mainly explore its influence on neuronal synchroni-
zation and spike timing precision, as well as their biological
relevance.
Motivated by the aforementioned considerations, we con-
struct a network composed of uncoupled neurons which are
subjected to a common subthreshold 共local field potential兲
signal s(t) plus spatially correlated noise
(t). The neural
behavior depends remarkably on both noise intensity Dand
the measure Rof noise correlation. On the one hand, the
output SNR, the population coherence measure, and the
spike timing precision all go through a maximum as a func-
tion of D. On the other hand, the SNR monotonically de-
creases with increasing R, whereas both the degree of syn-
chronization among the neurons and the spike timing
precision are improved. This makes it possible for neural
networks to exploit precise spatiotemporal firing patterns to
encode the stimulus. The mechanisms underlying these
noise-induced effects are also discussed in detail. This paper
is organized as follows. The model is described in Sec. II,
while the results and discussion are presented in Sec. III.
Finally, a conclusion is given in Sec. IV.
II. MODEL
We consider a summing network composed of Hodgkin-
Huxley 共HH兲model neurons which are connected in parallel
and converge to a summing center ⌺, as shown in Fig. 1.
The dynamic equations for the network are presented as fol-
lows 关11兴:
Cm
dVi
dt ⫽⫺gNami
3hi共Vi⫺ENa兲⫺gKni
4共Vi⫺EK兲
⫺gl共Vi⫺El兲⫹I0⫹s共t兲⫹
i共t兲,共1兲
dmi
dt ⫽
␣
m共Vi兲共1⫺mi兲⫺

m共Vi兲mi,共2兲
*Corresponding author. Email address: fliu@nju.edu.cn
PHYSICAL REVIEW E 69, 011909 共2004兲
1063-651X/2004/69共1兲/011909共7兲/$22.50 ©2004 The American Physical Society69 011909-1
dhi
dt ⫽
␣
h共Vi兲共1⫺hi兲⫺

h共Vi兲hi,共3兲
dni
dt ⫽
␣
n共Vi兲共1⫺ni兲⫺

n共Vi兲ni,i⫽1,...,N.共4兲
Here Cm⫽1
F/cm2,ENa⫽50 mV, EK⫽⫺77 mV, El
⫽⫺54.4 mV, gNa⫽120 mS/cm2,gK⫽36 mS/cm2,gl
⫽0.3 mS/cm2, and
␣
m(V)⫽0.1(V⫹40)/(1⫺e⫺(V⫹40)/10),

m(V)⫽4e⫺(V⫹65)/18,
␣
h(V)⫽0.07e⫺(V⫹65)/20,

h(V)⫽1/
(1⫹e⫺(V⫹35)/10),
␣
n(V)⫽0.01(V⫹55)/(1⫺e⫺(V⫹55)/10),
and

n(V)⫽0.125e⫺(V⫹65)/80. All the currents are in units of
A/cm2.
I0is a constant bias and taken as 1
A/cm2.s(t)isa
subthreshold signal, Acos(2
fst), corresponding to the input
generated by the local field potential. The signal frequency is
set to fs⫽50 Hz unless specified otherwise, and the signal
amplitude is A⫽1
A/cm2. Similar to that in Refs. 关10,12兴,
the noise term is assumed to be
i(t)⫽
冑
1⫺Ri(t)
⫹
冑
R
(t) with
具
i共t兲
典
⫽0,
具
i共t1兲j共t2兲
典
⫽2D
␦
ij
␦
共t1⫺t2兲,共5兲
and
具
共t兲
典
⫽0,
具
共t1兲
共t2兲
典
⫽De⫺
兩
t1⫺t2
兩
.共6兲
Here
具典
represents the ensemble average and Dis referred to
as noise intensity. i(t) is the independent Gaussian white
noise, while
(t) is the Gaussian colored noise with the cor-
relation time ⫺1being 2 ms. Since
具
i共t1兲
j共t2兲
典
⫽2D共1⫺R兲
␦
ij
␦
共t1⫺t2兲⫹RDe⫺
兩
t1⫺t2
兩
,
共7兲
the control parameter R(0⭐R⭐1) measures the noise cor-
relation between a pair of neurons. In the above assumption,
we can consider that i(t) represents the internal noise,
while
(t) reflects the random synaptic input from neurons
beyond the system under study. This is plausible when we
model cortical neurons in the same column. It has been
stressed in Ref. 关5兴that common input, common stimulus
selectivity, and common noise are tightly linked in function-
ing cortical circuits.
The output of the network is defined as
Iout共t兲⫽1
N兺
i⫽1
N
„Vi共t兲⫺V*….共8兲
V*is the firing threshold taken as ⫺20 mV, and
(x)⫽1if
x⭓0 and
(x)⫽0ifx⬍0. Thus the summer (⌺) operates
by averaging the output spike train of each unit to obtain a
resultant output for the entire system. The number of neurons
in the network is set to N⫽500 unless specified otherwise.
The output SNR is defined as 10 log10(S/B) with Sand B
representing the signal peak and the mean amplitude of the
noise at the input signal frequency in the power spectrum for
Iout(t), respectively. Numerical integration is performed by a
second-order stochastic algorithm 关13兴, and the time step is
500/32 768 ms. An average over 50 different realizations is
taken to obtain reported results.
III. RESULTS AND DISCUSSION
First, we investigate the influence of spatially correlated
noise on neuronal firing. For R⫽1, since all the neurons are
subjected to an identical input, they discharge at the same
time but the firings exhibit skipping as seen in Fig. 2共a兲. That
is, Iout(t) is composed of irregular sequences of ones and
zeros. In contrast, for the case of independent noise (R
⫽0), while one neuron is responding poorly to the signal
without spiking, others may be responding well. As a result,
Iout(t) varies nearly periodically at the same frequency as
the signal though its peak values are small. For 0⬍R⬍1,
however, Iout(t) exhibits apparent fluctuations between dif-
ferent driving cycles; that is, Iout(t) is nearly zero in some
driving cycles, whereas it takes a relatively high value in
others. Such fluctuations become more remarkable with in-
creasing R. This means that more neurons tend to fire simul-
taneously as the correlation in input noise is enhanced. In
other words, the neurons exhibit synchronous firing but
meanwhile the ISIs are more variable. It is noted that
冑
R
(t) has a dominant impact on neuronal firing compared
to
冑
1⫺Ri(t) when R⬎0.5.
Figure 2共b兲depicts the output SNR against noise intensity
Dfor different values of R. Each curve presents a typical
characteristic of the SR, namely, the SNR goes through a
maximum with increasing D. The optimal noise intensity
slightly shifts rightward as Rincreases. Note that at each
noise level the SNR is rigorously a decreasing function of R.
This is clearly shown in Fig. 2共c兲. In fact, in the case of
independent input noise, population averaging can effec-
tively suppress the inherent noise, and thus the output signal
contains more information about the stimulus. In contrast, in
FIG. 1. A schematic diagram of the network composed of HH
neurons. The total noise is divided into two items: the common
noise
冑
R
(t) and the independent noise
冑
1⫺Ri(t).
WANG et al. PHYSICAL REVIEW E 69, 011909 共2004兲
011909-2
the case of spatially correlated noise, the averaged activity is
nearly as noisy and variable as that of individual neurons.
Thus the SNR decreases evidently compared to that for R
⫽0. These imply that, in terms of the SNR, improving the
correlation in input noise instead diminishes the beneficial
effect of population averaging as reported in Ref. 关10兴.Ithas
also been demonstrated that positive noise correlation de-
creases the estimation capacity of the network in the light of
a Fisher information measure 关14兴.
Moreover, pooling more neurons has a minor influence on
improving the SNR in the presence of noise correlation 关see
Fig. 2共d兲兴. For R⫽0, the SNR first increases apparently with
the ensemble size Nand is saturated at large N(⬎1000). But
provided there is little correlation in the noise, the SNR rises
slightly or nearly remains constant with increasing N. Thus it
seems unlikely to enhance the performance of the averaged
activity by pooling more neurons in the light of the SNR.
Nevertheless, this also indicates that the neurons exhibit
strong synchronization when subjected to correlated noise
input.
The most common way to characterize the ISI variability
is via the coefficient of variation (Cv) of interspike intervals,
which is defined as the ratio of the standard deviation to the
mean of ISI. Figure 3共a兲depicts Cvagainst noise intensity
for different values of R. For each R,Cvis a monotonically
decreasing function of D共for D⭓0.5). This is in agreement
with the results shown in Ref. 关15兴共cf. Fig. 4 therein兲.Itis
noted that Cvis an increasing function of Ras seen in Fig.
3共b兲. This means that the spike sequences become more vari-
able with increasing the correlation in input noise.
It is noted that the above conclusions also hold for various
signal frequencies. Figure 4共a兲depicts the SNR against Rfor
different values of fs. The SNR always declines monotoni-
cally with R. But the neurons display different firing coher-
ence with the signal. As a result, the SNR takes a relatively
large value for 30⭐fs⭐100 Hz 关see Fig. 4共b兲兴. That is, the
neurons are more sensitive to the signals with frequencies
ranging from 30 to 100 Hz. Such frequency sensitivity has
also been reported in Ref. 关16兴and results from the reso-
nance effect between the subthreshold oscillation of mem-
brane potential and the periodic signal. Resonance improves
the ability of neurons to respond selectively to inputs at pre-
ferred frequencies. As a matter of fact, the resonance and
frequency preference may be one of the basic principles un-
derlying cognitive and behavioral processes.
To quantify the synchronization between neurons, we use
a coherence measure based on the normalized cross correla-
tions of their spike trains at zero time lag 关17兴. To be specific,
supposing that a long time interval Tis divided into small
bins of
and that two spike trains are given by Xi(l)⫽0or
FIG. 2. 共a兲Iout(t) vs time 共with D⫽1) and 共b兲the output SNR
vs noise intensity for R⫽0.0, 0.3, 0.7, and 1.0, respectively. 共c兲The
output SNR vs the measure Rof noise correlation for D⫽0.1, 1,
and 10, respectively. 共d兲The output SNR vs the network size for
R⫽0.0, 0.01, 0.05, and 0.1, respectively, with D⫽1.
FIG. 3. 共a兲Cvvs noise intensity Dfor R⫽0.0, 0.3, 0.5, and 1.0,
respectively. 共b兲Cvvs Rfor D⫽0.5, 1, and 10, respectively.
IMPACT OF SPATIALLY CORRELATED NOISE ON . . . PHYSICAL REVIEW E 69, 011909 共2004兲
011909-3
1 and Xj(l)⫽0 or 1, with l⫽1,2,...,m共here T/m⫽
). The
coherence measure for the pair is then defined as
Kij共
兲⫽
兺
l⫽1
m
Xi共l兲Xj共l兲
冑
兺
l⫽1
m
Xi共l兲兺
l⫽1
m
Xj共l兲
.共9兲
The population coherence measure Kis obtained by averag-
ing Kij over all pairs of the neurons in the network. Here
is
taken as 2 ms.
Figure 5共a兲plots Kversus noise intensity D.Kgoes
through a maximum as a function of D, namely, there also
exists an optimal noise level for the neuronal synchroniza-
tion. Furthermore, this optimal noise intensity is nearly iden-
tical to that for the SNR shown in Fig. 2共b兲. Note that Kis
higher in the case of R⫽0.7. In fact, Kis an increasing
function of Ras seen in Fig. 5共b兲. Therefore, the level of
synchronization among the neurons is indeed improved by
increasing the noise correlation.
We have discussed the neural firing under different con-
ditions of input noise and found that the spatially correlated
noise enhances the degree of synchronized firing. Now we
examine in detail the impact of noise correlation on spike
timing. Figure 5共c兲plots poststimulus time histograms
共PSTHs兲, which characterize the number of spikes collected
at the summer per millisecond 关11兴. It is in essence equiva-
lent to Iout(t) showing how the neurons fire synchronously
over time. Obviously, there are many peaks located around
the maxima of the signal, indicating a phase locking to the
stimulus. In the case of independent noise, the peaks in each
cycle are nearly of the same short height, meaning that the
neurons show weak synchronization. When the noise corre-
lation is enhanced, the degree of synchronization among the
neurons is evidently improved as the PSTH takes a large
value in some driving cycles but a much smaller one in oth-
ers. As Rfurther increases, those high peaks prominently rise
while the fluctuations become more remarkable. These re-
sults are in agreement with those shown in Fig. 2共a兲.
As mentioned above, in temporal coding the precise tim-
ing of spikes is used to encode a stimulus. Thus the reliabil-
ity and precision of firing patterns is a dominant factor de-
termining the quality of a temporal code. Based on the shape
of a smoothed data set taken from a five-point moving aver-
age of the PSTH, the spike timing precision is defined as
关18兴
Pi⫽Hi/wi,共10兲
where Hiis the height of the ith peak in the smoothed PSTH,
and wiis the width at Hi/e. The mean precision Pis ob-
tained by an average over 200 driving cycles. Clearly, P
quantitatively characterizes the average number of spikes
and their coincidence in any firing event in the PSTH.
FIG. 4. D⫽1. 共a兲The output SNR vs Rfor the signals with
fs⫽20, 70, and 120 Hz, respectively. 共b兲The output SNR vs the
signal frequency for R⫽0.0, 0.3, 0.7, and 1.0, respectively.
FIG. 5. 共a兲The population coherence Kvs noise intensity for
R⫽0.0, 0.3, and 0.7, respectively. 共b兲Kvs Rwith D⫽1. 共c兲The
input signal s(t) and PSTHs for R⫽0.0, 0.3, and 0.7, respectively,
with D⫽1.
WANG et al. PHYSICAL REVIEW E 69, 011909 共2004兲
011909-4
Figure 6共a兲shows Pversus noise intensity Dfor various
values of R. Each curve displays a SR-like behavior. That is,
there exists an optimal noise level which maximizes the tim-
ing precision via the SR mechanism. This indicates that the
spike timing precision in response to subthreshold periodic
stimuli can be enhanced by input noise, as reported in Ref.
关18兴. The optimal noise intensity also slightly shifts right-
ward with increasing R. Moreover, for each Rthe maximum
SNR, K, and Poccur at the same noise intensity. These
verify that noise can play a positive role in weak signal pro-
cessing. The timing precision also monotonically increases
with R关see Fig. 6共b兲兴, which means that the correlated noise
does make the spike timing more precise. It is noted that P
increases more steeply when R⬎0.5 since
冑
R
(t), the
dominant part of the noise, makes the neurons prone to fire
synchronously. The inset of Fig. 6共b兲depicts Pagainst the
signal frequency for D⫽1. Clearly, Ptakes a relatively large
value for 50⭐fs⭐90 Hz. This indicates that the neurons
transmit these signals with a high precision and respond pref-
erentially to them.
It is worth noting that the timing precision linearly in-
creases with the number of neurons and that the slope rises
with increasing R, which is clearly seen in Fig. 6共c兲. This is
largely different from the dependence of the output SNR on
Nshown in Fig. 2共d兲. Here we see that pooling more neurons
is of functional significance in effectively firing postsynaptic
neurons, and this effect is more prominent in the case of
correlated noise.
It is of interest to investigate the mechanism underlying
these noise-induced effects. We first discuss the bifurcation
in the HH neuron to a constant bias I0in the absence of noise
(D⫽0) and input signal (A⫽0). As seen in Fig. 7共a兲, for
I0⬍Ic⫽6.2 there is only a globally stable fixed point. The
birth of stable and unstable limit cycles occurs at Icdue to
the saddle-node bifurcation. For Ic⬍I0⬍Ih⫽9.8, there exist
a stable fixed point, a stable limit cycle, and an unstable limit
cycle. The unstable limit cycle constitutes the boundary
separating the attractive basins corresponding to the fixed
FIG. 6. 共a兲The spike timing precision Pvs noise intensity for
R⫽0.0, 0.3, and 0.7, respectively. 共b兲Pvs Rfor D⫽0.5 and 1.0,
respectively. The inset is Pvs the signal frequency fsfor R⫽0.0
and 0.3, respectively, with D⫽1. 共c兲Pvs the network size for R
⫽0.0, 0.3, and 0.7, respectively, with D⫽1.
FIG. 7. 共a兲Deterministic bifurcation diagram of a HH neuron
under dc current input. Here I0is the bifurcation parameter and Vis
the membrane potential. The thick and dashed lines represent stable
and unstable fixed points, respectively. The filled and open circles
represent both maxima and minima of stable and unstable limit
cycles, respectively. Noise-induced bifurcation diagram for R⫽0
共b兲and R⫽1共c兲. The curves represent the upper (䊉) and lower
(⫻) bounds of the stationary distribution of membrane potential for
each noise intensity D.
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point and limit cycle. When the initial condition V0falls
inside this boundary, i.e., in the contraction region, the dy-
namics of system will be attracted to the point attractor. If V0
is outside the boundary, i.e., in the expansion region, the
system will be attracted to the stable limit cycle. As we shall
see, this feature has a large impact on neuronal synchroniza-
tion. At I0⫽Ihthe Hopf bifurcation occurs and there is only
a stable limit cycle thereafter.
In the presence of noise, the neurons can be evoked to
discharge spikes, and there exists a noise-induced transition
in excitability with increasing D关19兴. To quantify such a
transition, we compute stochastic bifurcation diagram in the
same way as in Refs. 关19,20兴. Figures 7共b兲and 7共c兲show the
noise-evoked transition diagrams for R⫽0 and R⫽1, re-
spectively, providing a global view of how the stationary
distribution V99 of membrane potential changes with D. For
each fixed D, output membrane potential has been collected
for 100 s. Then the top and bottom limits of the distribution
are computed in the way so that 99% of the distribution is
below the upper line and 99% is above the lower one. Three
regions can be distinguished. For R⫽0, the distribution in-
creases linearly for low noise (D⭐0.5). As noise intensity
increases (0.5⬍D⭐2), the distribution evidently widens.
That is, a transition occurs around D⫽0.5. When Dis fur-
ther increased, the distribution changes slightly. For R⫽1,
however, the bifurcation point is shifted rightwards and the
second region also broadens. Tanabe et al. 关19兴demonstrated
that noise with intensities prior to the bifurcation point may
play an important role in enhancing spike timing precision.
Here we further show that the correlation of noise widens
this beneficial region, which can largely enhance the neu-
ronal synchronization and lead to high variability of the spik-
ing dynamics.
Comparing Figs. 2共b兲,5共a兲and 6共a兲with Figs. 7共b兲and
7共c兲, we see that the SNR, K, and Pare in step with the
distribution of membrane potential varying with D. Thus we
can interpret their dependence on Dand Rin terms of the
excitability of neurons. In the first regime, noise-induced
fluctuations improve the excitability of some units in the
ensemble, while their membrane potentials are located
around the resting potential. In the presence of input signal,
those neurons which are more excitable than at rest may fire
synchronously in some driving cycles. As noise is increased
within this range, more neurons evolve closer to the firing
threshold, and enhanced response is observed such as the rise
of the SNR and P. As the noise is further increased beyond
the first regime, the membrane potential fluctuations evi-
dently increase. A fraction of neurons that are evoked to fire
by noise alone may not respond to the input due to the re-
fractory period, while other neurons whose membrane poten-
tials are around the resting potential may be triggered to fire
simultaneously by the signal. This may reduce the overall
response of the ensemble. In the third regime with larger
noise intensity, the noise fluctuations become dominant, and
the neural coherence further decreases. Here the boundary
between attractive basins, which is related to the unstable
limit cycle, plays a crucial role in the noise-induced synchro-
nization, as reported in Ref. 关21兴wherein a saddle point em-
bedded in system dynamics is responsible for the noise-
evoked synchronization. It is worth noting that in the case of
correlated noise, the first region evidently expands, and this
gives rise to a prominent increment in Pand Kwith increas-
ing D.
Therefore, the correlated noise has two primary effects on
neuronal firing. The first is related to noise intensity. In the
absence of input signal, small noise disturbs the long-term
motion of the system and results in the dominance of con-
traction dynamics. The membrane potential is distributed in a
narrow region. In this case, the noise can trigger the neurons
to respond synchronously to a weak signal, leading to a
higher spike timing precision when noise intensity is in-
creased up to the transition point. For large noise intensity,
the membrane potential distribution is largely broadened. In
this case, when the weak signal is input to the system, noise-
induced firings become dominant while the effect of the sig-
nal on driving the ensemble in phase has been disturbed
heavily. This leads to a decrease of spike timing precision
with increasing D.
The second effect of noise is due to its correlation. For a
fixed noise intensity, neurons with independent noise can fire
spikes more independently, whereas correlation in noise
makes the neurons prone to act together, which increases the
inertia of ensemble neurons to be rest or to fire synchro-
nously. That is, the correlation drives neurons with different
initial conditions to converge to an identical response, being
inside the contraction or expansion region. This results in an
enhanced neural synchronization. Periodic signal plays a
similar synergic role to correlation in driving the neural dy-
namics in phase, improving the level of synchronization
among neurons.
Finally, it is worth noting that the correlation of noise not
only enhances spike timing precision 关see Fig. 6共a兲兴, but also
enlarges the spiking variability 关see Fig. 3共b兲兴. The increment
in ISI variability will improve the encoding capacity of neu-
rons. This is of functional significance when considering
how the nervous system tunes noise intensity to its optimal
values. Our results may give a reasonable mechanism for
why neural responses in the cerebral cortex often become
highly variable but precise in encoding input signals 关1兴.
IV. CONCLUSION
We have demonstrated that there exists precisely synchro-
nized activity in the presence of strong noise correlation. Our
results reproduce some of the typical firing characteristics
observed in cortical neurons. First, it is generally agreed that
the response variability originates in unreliable synaptic in-
puts 关22兴. We also showed that the spike sequences become
more variable in response to correlated noise. Second, large-
scale synchronized firings have been observed in a variety of
brain areas, especially the
␥
oscillations 共at frequencies of
30–70 Hz兲关23兴, which play functional roles such as pattern
segmentation and feature binding. The frequency sensitivity
shown in Figs. 4 and 6共b兲may give us an enlightenment why
the
␥
oscillations are so ubiquitous in the nervous system.
Third, it has been found that under some conditions the spike
timing can show a high precision and reproducibility with
the temporal resolution being 2–3 ms 关24兴. Thus precise
WANG et al. PHYSICAL REVIEW E 69, 011909 共2004兲
011909-6
temporal firing patterns can be exploited to encode the stimu-
lus.
As shown in Fig. 2共a兲, although the neurons exhibit syn-
chronized firings when R⫽0.7, they miss firing in many
driving cycles. But this does not necessarily mean that the
neurons are poorly processing information. In contrast, such
synchronized activity may subserve information processing.
For example, s(t) may just represent a modulation of neural
behavior providing the system with an effect of frequency
selection 关16兴, and the neurons may preferentially respond to
synchronized synaptic inputs. Alternatively, the neural net-
works can exploit precise temporal relations among neurons
to select responses for joint processing and to bind neurons
temporally into functionally coherent assemblies 关7兴. Fur-
thermore, the combination of synchronized firing of cortical
neurons and high temporal precision also makes it possible
for their downstream neurons to act as coincidence detectors
preferentially transferring synchronized activity 关8兴. For in-
stance, a coincidence-detection neuron can precisely deter-
mine whether two coupled neurons receive similar level of
sensory input 关25兴.
Although we did not directly model the random synaptic
input, our results suggest that the noise correlation is crucial
for cortical neurons to temporally process information and
that the cerebral cortex may convey more information via
temporal codes than exclusively using rate codes. In contrast,
it seems plausible to assume that sensory neurons in the pe-
ripheral nervous system are subject to more or less indepen-
dent noise. In that case, it is the averaging of neural re-
sponses that encodes a stimulus feature such as the signal
frequency.
In addition, it is noted that Rudolph and Destexhe inves-
tigated the case wherein a neuron is subjected to a periodic
weak signal plus large numbers of random synaptic inputs
关26兴. They reported that neuronal response can also be en-
hanced by the correlation among synaptic inputs, exhibiting
a SR-like behavior. Their results suggested that cortical neu-
rons are efficient in detecting such correlations within milli-
second time scales. This is consistent with the present work
that neurons may exploit correlated noise to encode and
transmit information.
In conclusion, in this paper we have explored the impact
of spatially correlated noise on neuronal firing. Noise can
play a constructive role in optimizing neuronal response to
subthreshold stimuli. The results illustrate how the presence
of correlated noise improves the degree of synchronization
among the neurons and the spike timing precision but mean-
while makes output spikes more variable. With a high timing
resolution, temporal structures of neural activity can be used
to convey more information. Thus correlated firings of neu-
rons play functional roles in signal processing. These results
are consistent with observations in cortical neurons and im-
ply that we should consider the correlation in noise or syn-
aptic input when we model the cortical dynamics. Finally, we
would like to point out that the present form of noise is a
simplification, and it is of interest to exploit more complex
configurations, such as the spatially decaying functions over
the population.
ACKNOWLEDGMENTS
This work was supported by the National Natural Sci-
ences Foundation of China 共under Grant No. 30070208兲and
the Nonlinear Science Project of NSM.
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