Vibrational resonance in neuron populations
Bin Deng, Jiang Wang, Xile Wei, K. M. Tsang, and W. L. Chan
Citation: Chaos 20, 013113 (2010); doi: 10.1063/1.3324700
View online: http://dx.doi.org/10.1063/1.3324700
View Table of Contents: http://chaos.aip.org/resource/1/CHAOEH/v20/i1
Published by the American Institute of Physics.
Related Articles
Multiscale characterization of recurrence-based phase space networks constructed from time series
Chaos 22, 013107 (2012)
Optimal pinning synchronization on directed complex network
Chaos 21, 043131 (2011)
Variability of contact process in complex networks
Chaos 21, 043130 (2011)
Detecting the topologies of complex networks with stochastic perturbations
Chaos 21, 043129 (2011)
Vibrational resonance in excitable neuronal systems
Chaos 21, 043101 (2011)
Additional information on Chaos
Journal Homepage: http://chaos.aip.org/
Journal Information: http://chaos.aip.org/about/about_the_journal
Top downloads: http://chaos.aip.org/features/most_downloaded
Information for Authors: http://chaos.aip.org/authors
Downloaded 27 Feb 2012 to 158.132.161.52. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/about/rights_and_permissions
Vibrational resonance in neuron populations
Bin Deng,
1
Jiang Wang,
1
Xile Wei,
1
K. M. Tsang,
2
and W. L. Chan
2
1
School of Electrical and Automation Engineering, Tianjin University, Tianjin 300072, China
2
Department of Electrical Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong
共Received 26 August 2009; accepted 26 January 2010; published online 9 March 2010兲
In this paper different topologies of populations of FitzHugh–Nagumo neurons have been introduce
to investigate the effect of high-frequency driving on the response of neuron populations to a
subthreshold low-frequency signal. We show that optimal amplitude of high-frequency driving
enhances the response of neuron populations to a subthreshold low-frequency input and the optimal
amplitude dependences on the connection among the neurons. By analyzing several kinds of topol-
ogy 共i.e., random and small world兲 different behaviors have been observed. Several topologies
behave in an optimal way with respect to the range of low-frequency amplitude leading to an
improvement in the stimulus response coherence, while others with respect to the maximum values
of the performance index. However, the best results in terms of both the suitable amplitude of
high-frequency driving and high stimulus response coherence have been obtained when the neurons
have been connected in a small-world topology. © 2010 American Institute of Physics.
关doi:10.1063/1.3324700兴
In bistable systems, it has been shown that the role of
noise in improving the quality of signal detection can be
played by other types of driving, such as a chaotic signal
or a high-frequency periodic force. In the latter case,
known as vibrational resonance (VR), the system is under
the action of a two frequency signal. Such bichromatic
signals are pervasive in different fields, including brain
dynamics, where, for instance, bursting neurons may ex-
hibit two widely different time scales, and telecommuni-
cations, where information carriers are usually high-
frequency waves modulated by a low-frequency signal
that encodes the data. The dynamics of neurons is often
modeled by using differential models. Networks of many
neurons can be studied by connecting many of these mod-
els and studying the global behavior of the system. The
importance of these studies is related to a deep under-
standing of the neural mechanisms underlying neural
systems. This paper focuses on several topological struc-
tures of networks of these models and studies the dy-
namical response of coupled neurons to bichromatic sig-
nal with two very different frequencies from the
viewpoint of VR.
I. INTRODUCTION
The external influence can considerably affect the signal
detection by nonlinear system. Stochastic resonance 共SR兲,
where the response of a nonlinear system to a weak deter-
ministic signal is enhanced by external random
fluctuation,
1–3
is the most relevant example of this fact. Re-
cently, Ullner et al. gave a detailed description of several
new noise-induced phenomena in the FitzHugh–Nagumo
共FHN兲 neuron in Ref. 4. They have investigated the Canard-
enhanced SR,
5
the effect of noise-induced signal processing
in systems with complex attractors,
6
and a new noise-induce
phase transition from a self-sustained oscillatory regime to
an excitable behavior.
7
They also showed that optimal am-
plitude of high-frequency driving enhances the response of
an excitable system to a low-frequency subthreshold signal.
8
In the latter case, known as VR, the system is under the
action of two frequency signal.
9–11
Such bichromatic signals
are pervasive in many different fields, including brain
dynamics,
12
where, for instance, bursting neurons may ex-
hibit two widely different time scales.
However, most of the relevant studies only considered
the single neuron.
5,7,8
Recently, Ref. 13 has investigated SR
on excitable small-world networks so the focus of this paper
is on the investigation of the role of topology in neuron
networks in the presence of high-frequency driving. We will
refer to the enhancement in the stimulus response due to the
presence of high-frequency driving as a generalized VR ef-
fect. Several network topologies have been investigated fo-
cusing on the positive effects of connections in networks of
nonlinear FHN neurons affected by high-frequency driving.
Structures, such as chains or fully connected graphs, random
graphs, and small-world networks,
14
have been simulated by
connecting FHN neurons excited by subthreshold low-
frequency signal. The Fourier coefficients have been evalu-
ated to point out the VR features of extended neuron popu-
lations versus the topology configuration. The phenomenon
studied in this paper is also related to noise-enhanced SR in
coupled oscillation.
15,16
The contents of this paper are arranged as follows. In
Sec. II, VR in single FHN neuron is introduced briefly. The
analysis of the VR in different models of neuron population
is given in Sec. III. Finally, conclusions and discussions are
made in Sec. IV.
II. VIBRATIONAL RESONANCE IN SINGLE
FHN MODEL
In the presence of two harmonic signals, the FHN
model
17
is defined by the equations
dx
dt
= x −
x
3
3
− y, 共1兲
CHAOS 20, 013113 共2010兲
1054-1500/2010/20共1兲/013113/7/$30.00 © 2010 American Institute of Physics20, 013113-1
Downloaded 27 Feb 2012 to 158.132.161.52. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/about/rights_and_permissions
dy
dt
= x + a + A cos共
t兲 + B cos共⍀t兲, 共2兲
where x共t兲 represents the membrane potential of neuron and
y共t兲 is related to the conductivity of the potassium channels
existing in the neuron membrane. The value of the time scale
ratio =0.01 is chosen so that the activator x共t兲 evolves
much faster than the inhibitor y共t兲. The terms A cos共
t兲 and
B共⍀t兲 stand for the low- and high-frequency components of
the external signal, respectively. In what follows we will
choose A =0.01, so that the system is below the excitation
threshold, and ⍀Ⰷ
, in particular, ⍀=5 and
=0.1. In Eq.
共2兲 we have considered no phase shift between the two driv-
ing signals, but it can be checked that the existence of an
arbitrary phase shift does not alter the results that follow. The
parameter a determines the behavior of the system. For a
⬎1.0 the FHN model is excitable, and for a ⬍1.0 it shows an
oscillatory behavior. At the bifurcation a=1.0 the stability of
the only fix point will be changed.
18
Between these two cases
an intermediate behavior can appear. For values of the pa-
rameter a slightly beyond the bifurcation point, small oscil-
lations near the unstable fix point exist instead of large spike
and these are the so-called canard oscillations.
19
An impor-
tant fact of the treatment of canard oscillation is that a very
small change in the parameter a leads to a large different in
the trajectories. This change in the parameter a can be caused
by some instantaneous influence of noise as investigated in
Refs. 5 and 18 or external high-frequency driving as inves-
tigated in Ref. 8. In this paper, the parameter of a is fixed to
be 1.05. We fix the amplitude of the low-frequency signal
component and increase the high-frequency amplitude. In
FIG. 1. Firing of a single FHN neuron subjected to a subthreshold low-frequency signal for increasing the amplitude of high-frequency driving: 共a兲
subthreshold low-frequency signal; 共b兲 x共t兲 for weak high-frequency driving 共no spikes are emitted兲; 共c兲 x共t兲 for optimal high-frequency driving; 共d兲 x共t兲 for
too strong high-frequency driving. 共e兲 The ratio of the time duration of spikes and subthreshold oscillation vs the amplitude of high-frequency driving.
013113-2 Deng et al. Chaos 20, 013113 共2010兲
Downloaded 27 Feb 2012 to 158.132.161.52. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/about/rights_and_permissions
Fig. 1 the behavior of the system with respect to increasing
the amplitude of high-frequency driving is reported. The in-
put is a periodic, underthreshold signal, as shown in Fig.
1共a兲. As can be noticed when the amplitude of high-
frequency driving is low 关Fig. 1共b兲兴, the FHN neuron does
not fire, but when the amplitude is too high 关Fig. 1共d兲兴, the
firing is not coherence with the input. The optimal case is
represented by Fig. 1共c兲. As the information is carried
through neuron spikes, for a sinusoidal input signal, the more
the ratio of the time duration of spikes and subthreshold
oscillation is close to 1, the better the information is carried
through spike trains. From Fig. 1共e兲, it can be seen that while
the amplitude of high-frequency driving is about 0.06, the
information will be carried best.
To evaluate the amplitude of the input frequency in the
output signal, we calculate the Fourier coefficient Q for the
input frequency
. We use the Q parameter instead of the
power spectrum because we are interested in the transport of
the information encoded in the frequency
. For this task the
Q parameter is a much more compact tool than the power
spectrum
1,20
Q
sin
=
2n
冕
0
2
n/
2x共t兲sin共
t兲dt,
Q
cos
=
2n
冕
0
2
n/
2x共t兲cos共
t兲dt,
Q =
冑
Q
sin
2
+ Q
cos
2
,
where n is the number of period 2
/
covered by the inte-
gration time. The maximum of Q shows the best phase syn-
chronization between input signal and output firing. It is to
be noted that in the case of phase synchronization one could
expect a response of the Q measure but not vice versa, and
the phase synchronization between the input signal and the
output of neuron can be seen from the following figures of
time series. Also, as information in neuron system is carried
through large spikes instead of subthreshold oscillations, we
are more interested in the frequency of spikes. So following
Ref. 6, we set the threshold V
s
=0 in the calculation of Q.If
V⬍ V
s
, V is replaced by the value of the fixed point V
f
共here
V
f
=−1兲; otherwise, V remains the same.
Figure 2 shows VR in the single FHN neuron with value
of parameter a = 1.05. The dependence of neuron’s response
on the amplitude of the high-frequency driving displays a
resonant form with clearly defined maxima at the optimal
values of B, similar to what happens in SR. The staircase
form of this dependence is caused by the abrupt discrete
appearance of new spikes in the spike train as the forcing
amplitude changes.
III. VIBRATIONAL RESONANCE IN NEURON
POPULATIONS
Now we consider the coupled FHN neuron populations
subject to series of high-frequency driving with the common
amplitude and frequency but different phases described by
dx
i
dt
= x
i
−
x
i
3
3
− y
i
− I
i
syn
, 共3兲
dy
i
dt
= x
i
+ a + A
i
cos共
t兲 + B cos共⍀t +
i
兲, 共4兲
where i =1,2,...,N index of the neurons. Without loss of
generality, let
i
be uniformly distributed in 关0
兴. = 0.01,
A
i
=0.01,
=0.1 and ⍀= 5 as used in Sec. II. A
i
cos共
t兲 and
B cos共⍀t+
i
兲 are low- and high-frequency driving, respec-
tively. I
i
syn
is the synaptic current through neuron i.
For the electrical coupling,
I
i
syn
=
兺
j=i−r
j=i+r
g
syn
共x
i
− x
j
兲, 共5兲
where g
syn
is the conductance of synaptic channel. g
syn
=0.01, which is large enough to synchronize the coupled
neurons. r is the radius of the neighborhood and the value of
it will be given in each case followed. By varying r the
network architecture changes from local connection to all-to-
all global coupling.
In neural systems with a large amount of neurons, it is
unnecessary and impossible to add external signals to all
involved individuals. Only weak and local input is reason-
able and guarantees the low energy consumption in large
neural networks, so in all cases in this paper, the same low-
frequency signal has been applied to a fraction f 共0⬍ f ⬍1兲
of elements in the neuron populations, whereas high-
frequency driving with different phases for each neuron has
been taken into account. The response of the entire network
has been monitored by considering the average of each mem-
brane variable x
i
, namely, V共t兲= 共1/ N兲兺x
i
共t兲.
FIG. 2. Response Q of the FHN neuron at the low frequency
vs the
amplitude B of the high-frequency input signal.
013113-3 VR in neuron populations Chaos 20, 013113 共2010兲
Downloaded 27 Feb 2012 to 158.132.161.52. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/about/rights_and_permissions
A. Random connections
First of all, the case in which the neurons are random
coupled has been investigated. The random networks are
characterized by asymmetric structures and capture in an ide-
alized way the features of many real systems. These net-
works have a short average path length which drives from
the fact that, starting from each node, any other node of the
network can be reached in a small number of links.
14
Starting
from a population of N FHN neurons the random network
has been built by using the following rule. Given a probabil-
ity p of connections, each pair of neurons is connected by a
link with probability p so the topology structure of a random
network is determined by the probability, as shown in Fig. 3.
First, we fix f = 50% and p= 0.5 which means that half of
the neurons are subjected to the low-frequency signal and
each pair of neurons is connected by a link with probability
0.5. The effects of high-frequency driving in a random net-
work of FHN have been characterized with respect to differ-
ent values of the number of FHN units. The result is shown
in Fig. 4共a兲, where it is evident that increasing the number of
FHN units, the response dependence is shifted to the right.
The larger the neuron population is, the more energy is
needed for the emergence of VR in a randomly coupled neu-
ron network, and while the neuron population is large
enough the corresponding high-frequency driving amplitude
B
VR
of VR will not increase notably with the size of the
neuron network, as shown in Fig. 4共b兲.
For an Erdös–Rényi 共Ref. 21兲 random graph with N
nodes, if the connection probability p is greater than a certain
threshold p
t
⬃共ln N兲/ N, then almost every random graph is
connected, so for a randomly coupled neuron network with
50 FHN units, if connection probability is less than
ln共50兲 / 50⬇0.08, there may be some isolated neurons in the
network and the value of Q increases with the increase in p,
as shown in Fig. 5共a兲.Ifp is greater than 0.08, almost every
graph is connected, so the value of Q will not increase with
the increase in p, as shown in Fig. 5共b兲.Ifp is greater than
0.08, the optimal amplitude of high-frequency driving B
VR
will increase with the increase in p, as shown in Fig. 5共d兲.
We suspect that it may be because that while almost every
random graph is connected the additional links will lead to
more energy cost needed for VR in a randomly coupled neu-
ron population. If p is less than 0.08, B
VR
will not increase
with the increase in p, as shown in Fig. 5共c兲.
B. Small-world networks
Many real networks have a short average path length,
but at the same time show a high clustering degree due to the
presence of both short-range and long-range links. In order
to model these systems Strogatz and Watts introduced the
concept of small-world networks that successfully captures
the essential features of the neuronal systems of the C.
elegans.
14
Small-world networks can be built starting from a
network of locally coupled neurons, i.e., each neuron is
FIG. 3. Example of considered random network topologies. Given 50 iso-
lated nodes, one connects every pair of nodes with probability 共a兲 0.01,
共b兲 0.1.
FIG. 4. Effect of the size of neuron network on the VR in a randomly
coupled neuron network. 共a兲 Response Q of three randomly coupled neuron
networks with different amounts of elements vs the amplitude B of high-
frequency driving. 共b兲 The corresponding high-frequency driving amplitude
B
VR
vs the amount of neuron population. The parameters are p= 0.5, f
=50%. This figure is the average result of ten trials.
013113-4 Deng et al. Chaos 20, 013113 共2010兲
Downloaded 27 Feb 2012 to 158.132.161.52. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/about/rights_and_permissions
