Synchronized Rhythmic Oscillation in a Noisy Neural Network

Yuguo YU, Feng LIU and Wei WANG�

National Laboratory of Solid State Microstructure and Department of Physics, Nanjing University, Nanjing 210093, P. R. China

(Received June 19, 2003)

The occurrence of synchronized oscillation and its critical behavior in a globally coupled stochasticHodgkin–Huxley (HH) neuronal network are studied in this paper. It is found that there is a critical curvefor the coupling strength versus noise intensity, which shows a V-shaped structure and divides thenetwork behavior into an asynchronous firing state and a synchronous one. Analysis of the scalingbehavior near the bifurcation point reveals that this transition is analogous to a second-order phasetransition. The frequency of synchronized oscillations is within the range of 40–80Hz, and its physicalorigin is explored by studying single HH neuron’s impedance. The intrinsic property of single neuronmay account for the generation and the frequency characteristics of synchronized rhythmic oscillations.

KEYWORDS: synchronization, non-equilibrium phase transition, coherence resonanceDOI: 10.1143/JPSJ.72.3291

1. Introduction

Synchronized rhythmic oscillations are ubiquitous phe-nomena in the brain and have been studied intensively inrecent years.1) These oscillations, resulting from the syn-chronous firings of neurons, play functional roles such aspattern segmentation, feature binding, selective attention andmemory.1,2) Especially, much attention has centered on thesynchronized 40Hz oscillations (with frequencies in therange of 30–80Hz). As realized by many researchers,however, two focal issues remain open: 1) what are themechanisms that bind single neuronal activity into rhythmiccoherent modes? 2) what are the dominant features thatdetermine the characteristic frequency of each brain rhythm?Many computational analyses using different models sug-gested that it is the network structure and the mutualinteractions between neurons that enable the generation ofvarious rhythmic activity (see refs. 1 and 3 for review). Thefrequency characteristic can be derived from the reverberat-ing activity within re-entrant neural circuits and theinteraction based on synaptic connectivity.3–5) Nevertheless,the spatiotemporal scales of various oscillations may also becorrelated with the intrinsic dynamics of single neuronitself.3,5) Among many theoretical studies, we have previ-ously discussed this topic using different neuronal models.6)

Moreover, noise-induced ordered states in nonlinear ornonequilibrium systems have recently attracted generalattention. A wide variety of interesting phenomena havebeen reported, such as noise-enhanced weak signal detectionvia stochastic resonance (SR) (see ref. 7 and referencestherein), noise-induced spiral waves,8) and noise-enhancedwave propagation.9) It was also shown that even in theabsence of external forcing, a nonlinear excitable unit withnoise can display SR-like behavior as well. This phenom-enon has been termed autonomous SR or coherenceresonance (CR).10) A common feature of such nonlinearphenomena is that an optimal activity occurs at some noiseintensity, displaying periodic (or quasi-periodic) oscillation.Thus this system can be viewed as a self-sustained CR-oscillator. When these CR-oscillators are coupled together,spontaneous rhythmic oscillation will be generated as the

coupling strength exceeds some critical value.11) [Synchron-ized firing behavior has also been studied in nervous systemswithout noise (for example see ref. 12 and referencestherein).] This phenomenon enriches the possible mecha-nisms underlying the origin of various rhythmic oscillationsin nature. In this paper, we assume that neurons are onlysubjected to Gaussian noise and focus on the followingaspects. What is the mechanism underlying the transition ofthe network activity from an asynchronous state to asynchronous one? What physical features does the transitionpresent? How does it relate to information transduction?How do the coupling and noise affect the frequencycharacteristic of the rhythmic oscillations?

To address the aforementioned questions, we explore aglobally coupled Hodgkin–Huxley (HH) neuronal network.We investigate the critical dynamic conditions for thegeneration of synchronized oscillations. It is shown thatthere exists a critical curve for the coupling strength versusnoise intensity, which divides the network activity into anasynchronous state and a synchronous one. The criticalcurve is V-shaped and presents a global minimum. Theanalysis of the scaling behavior near the boundary revealsthat such a transition can be regarded as a second-orderphase transition. Both the coupling and noise can induce thetransition. But only for the coupling strength above aminimal critical value can noise enhance the networkactivity via the CR mechanism. The frequency of synchron-ized firings is within the range of 45–80Hz. We also findthat the impedance of single HH neuron is frequencydependent and has a resonant peak in the range of 40–100Hz. It is this intrinsic property that affects the synchro-nous firings of neurons and the frequency characteristics ofrhythmic oscillations. Clearly, these provide a physicalinterpretation for the generation of the 40Hz oscillations.The correlations of dynamic behaviors with neural signalprocessing are also discussed.

This paper is organized as follows. The network model isdescribed in §2. In §3, the results and discussion for thephysical features relating to the generation of synchronizedoscillations are presented. An argument for signal processingbased on the dynamic behavior is also made. Finally, aconclusion is given in §4.

�E-mail: wangwei@nju.edu.cn

Journal of the Physical Society of Japan

Vol. 72, No. 12, December, 2003, pp. 3291–3296

#2003 The Physical Society of Japan

3291

2. Model

Let us consider a network composed of HH neurons whichare globally coupled via excitatory synapses. The dynamicequations for the network are presented as follows:13,14)

Cm

dVi

dt¼ �gNa

m3i hiðVi � ENa

Þ � gKn4i ðVi � EKÞ

� glðVi � ElÞ þ I0 þ �iðtÞ þ Isyni ðtÞ; ð1Þ

dmi

dt¼ �mðViÞð1� miÞ � �mðViÞmi; ð2Þ

dhi

dt¼ �hðViÞð1� hiÞ � �hðViÞhi; ð3Þ

dni

dt¼ �nðViÞð1� niÞ � �nðViÞni; i ¼ 1; . . . ;N: ð4Þ

Here Cm ¼ 1 mF/cm2, ENa¼ 50mV, EK ¼ �77mV, El ¼

�54:4mV, gNa¼ 120mS/cm2, gK ¼ 36mS/cm2, gl ¼

0:3mS/cm2, and �mðVÞ ¼ 0:1ðV þ 40Þ=ð1� e�ðVþ40Þ=10Þ,�mðVÞ ¼ 4e�ðVþ65Þ=18, �hðVÞ ¼ 0:07e�ðVþ65Þ=20, �hðVÞ ¼1=ð1þ e�ðVþ35Þ=10Þ, �nðVÞ ¼ 0:01ðVþ55Þ=ð1� e�ðVþ55Þ=10Þ,and �nðVÞ ¼ 0:125e�ðVþ65Þ=80. The number of neurons istakes as N ¼ 1000. I0 is a constant bias which can adjust theintrinsic state of neurons.13–15) Each neuron is subjected to anindependent Gaussian white noise with the same intensity:

h�iðtÞi ¼ 0; h�iðt1Þ�jðt2Þi ¼ 2D�ij�ðt1 � t2Þ: ð5Þ

Here D is referred to as noise intensity and is in units ofmA2/cm4. The synaptic current is described as

Isyni ðtÞ ¼ �

XN

j¼1;j 6¼ i

gsyn

N�ðt � tjÞðViðtÞ � VEÞ; ð6Þ

where �ðt � tjÞ ¼ �ðt0Þ ¼ ðt0=�sÞe�t0=�s , tj is the firing time ofthe j-th neuron; �s ¼ 2ms is the characteristic time of thesynaptic interaction. VE is the synaptic reversal potential setto 30mV, and gsyn is the coupling strength (in units of mS/cm2).14) All the currents are in units of mA/cm2.

The output of the network is defined as

V�ðtÞ ¼1

N

XN

i¼1

ViðtÞ; ð7Þ

which is considered the averaged activity of the neurons.Numerical integration is performed by a second-orderalgorithm,16) and the integration step is 500/32768ms.

3. Results and Discussion

Owing to the CR effect10) and the inherent excitability,2,14)

the HH neuron with noisy input can be regarded as a self-sustained oscillator whose firings are around a mainfrequency depending largely on the values of D and I0.

17)

When many such oscillators with an identical D and I0 arecoupled together, the network activity exhibits different timecourses with increasing gsyn. For weak coupling, V�ðtÞfluctuates randomly around mean value [see Fig. 1(a)],implying that the neural firings exhibit weak correlation andstochastic phases [see Fig. 1(b)]. That is, the neuronsdischarge spikes nearly independently. The phase plot ofdV�=dt versus V� also reveals that the asymptotical stablestate is around a global attractor [see Fig. 1(c)]. The powerspectrum density (PSD) of V�ðtÞ is shown in Fig. 1(d), wherethere is a peak around 50Hz. This peak frequency fpeak

denotes the average oscillatory frequency. However, thepeak is rather low since the firing phases of neurons arerandomly distributed.

Differently, as gsyn increases to a critical value, gcsyn ¼ 0:6,

V�ðtÞ presents a distinct periodic feature [see Fig. 2(a)],indicating the formation of synchronized rhythm. Theoscillatory activities of neurons are driven in phase andshow a well-defined order [see Fig. 2(b)]. The phase plotdisplays that the network activity goes along a limit cycle[see Fig. 2(c)], which corresponds to periodic firing ofspikes. In comparison with Fig. 1(c), the network behaviorvarying with the coupling strength indeed undergoes a Hopf

2000 2100 2200

-63

-60

-57

-54

-65 -60 -55

-5

0

5

10

2000 2100 22000

500

1000

0 100 200 300 400

0.00

0.01

0.02

(a)

t (ms)

V* (

mV

)

(d)

(c)

dV* /d

t (m

V/m

s)

V* (mV)

Neu

ron

Inde

x

t (ms)

(b)

P(f)

f (Hz)

Fig. 1. Under the conditions of gsyn ¼ 0:1, D ¼ 5, and I0 ¼ 2, (a) average

activity V�ðtÞ of the neurons vs time; (b) spike time rastergram; (c) phase

plot of dV�=dt vs V�; (d) the PSD of V�ðtÞ.

Fig. 2. Under the conditions of gsyn ¼ 0:6, D ¼ 5, and I0 ¼ 2, (a) average

activity V�ðtÞ of the neurons vs time; (b) spike time rastergram; (c) phase

plot of dV�=dt vs V�; (d) the PSD of V�ðtÞ.

3292 J. Phys. Soc. Jpn., Vol. 72, No. 12, December, 2003 Y. YU et al.

bifurcation. That is, a transition from the global attractor tothe limit cycle occurs at gcsyn. This critical coupling strengthdetermines the bifurcation point in the dynamic phase space.At gcsyn ¼ 0:6, the PSD for V�ðtÞ presents a high peak aroundfpeak ¼ 55Hz, implying coherent firings of the neurons [seeFig. 2(d)]. Clearly, this results from the excitation of theintrinsic oscillations of neurons via the CR. The couplingstrength really affects the formation and evolution of thesynchronized rhythm. From the point view of dynamics, thetransition of the collective dynamics from an asynchronousmode to a synchronous one is due to the Hopf bifurcation.The birth of the limit cycle indicates the occurrence ofsynchronized rhythm.

To quantify the synchronization between the neurons, weuse a coherence measure based on the normalized cross-correlations of spike trains.18) Supposing that a long timeinterval T is divided into small bins of � and that two spiketrains are given by XiðlÞ ¼ 0 or 1 and YjðlÞ ¼ 0 or 1, withl ¼ 1; 2; . . . ;m (here m ¼ T=�). The coherence measure forthe pair is then defined as

kijð�Þ ¼Pm

l¼1 XiðlÞYjðlÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPml¼1 XiðlÞ

Pml¼1 YjðlÞ

p : ð8Þ

A population coherence measure K is defined by an averageof kijð�Þ over all pairs in the network, that is,

K ¼1

NðN � 1Þ

XN

i¼1

XN

j¼1;j 6¼i

kijð�Þ: ð9Þ

Thus K is used to measure the level of synchronizationamong the neurons. Here � is set to 1ms.

For each noise intensity D, the population coherencemeasure K goes through a sharp transition near the transitionpoint when gsyn is increased [see Fig. 3(a)]. We define thecritical coupling strength, gcsyn, corresponding to the max-imal slope in the K–gsyn curve. Figure 3b depicts gcsyn versusD. The curve is V-shaped and divides the parameter space ofgsyn and D into two regions corresponding to asynchronousand synchronous firing, respectively. There exists an optimalnoise intensity at which the network needs a minimal valueof gsyn to generate synchronized oscillation. The criticalcurve also depends on the constant bias I0. For I0 ¼ 2, theminimum critical coupling strength is gcsyn ¼ 0:6 and thecorresponding optimal noise intensity is Do ¼ 5, comparedto gcsyn ¼ 0:35 and Do ¼ 2 for I0 ¼ 5. Obviously, large I0can set the neurons in a more active state so that it isrelatively easier to excite them to fire. Thus, a lower noiseintensity and weaker coupling strength can still evoke thetransition. However, when D becomes too small or too large,strong coupling strength is needed to induce synchronousfirings. This clearly implies a re-entrance to the asynchro-nous state as D increases or decreases (at a fixed gsyn). Wewill discuss this in detail below. As I0 decreases, the concavearea becomes small and nearly disappears when I0 < 2. Butthe concave part becomes sharper if I0 increases further.

The existence of such a concave curve results from theneural excitability and is closely related to informationprocessing, since synchronous firing has been argued toimprove the spike timing precision.19) For the constantstimulus I0 ¼ 5, when gsyn is too small (gsyn < 0:35), theneurons fire asynchronously at any level of noise. When

gsyn > 0:35, however, the neurons discharge synchronouslyprovided that D is within the concave region. Thus, thestimulus will be encoded if the synchronization presumablyprovides a mechanism for the coding. This is indeed the caseas mentioned at the beginning of this paper. As a matter offact, the noise intensity in the brain is always dynamicallyvariable, so that the stimuli distributed on neurons may beprocessed selectively by different pathways associated withthese variables. More specifically, as proposed in ref. 19,when a total effective stimulus composed of a constant biasplus Gaussian white noise is input to the neurons, there aresynchronous firings if the input variance (or the noise level)is just located in the concave area. Such synchronous firingis argued to improve the reliability of spike timing. But thespike timing reliability becomes weak (meanwhile thesynchronous state cannot be established) when the meanstimulus I0 is decreased. From our simulations, it is true thatthe network activity is in an asynchronous state when thevalues of I0, gsyn and D do not match each other. Thus, thefiring behaviors of the neurons are closely correlated withinformation processing. The critical curve or the boundarybetween the synchronous and asynchronous states doseparate these different dynamic behaviors.

When viewed as a critical phenomenon, the abovetransition may be analogous to a phase transition20) in thenonequilibrium system, which has been extensively studied(for example, see ref. 21 and references therein). Broeck et

0

1

10

10-1 100 101

10-2

10-1

100

I0=2

I0=5

asynchronous state

synchronous state

gc syn(m

S/cm

2 )

D (µA2/cm

4)

D=1D=10D=40

(b)

(a)

K (

arb.

uni

ts)

gsyn

(mS/cm2)

4020

Fig. 3. (a) For I0 ¼ 2, the population coherence measure K vs the

coupling strength gsyn for D ¼ 1, 10 and 40, respectively. (b) The critical

curves of gcsyn vs D for I0 ¼ 2 and 5, respectively.

J. Phys. Soc. Jpn., Vol. 72, No. 12, December, 2003 Y. YU et al. 3293

al. reported a noise-induced nonequilibrium phase transitionin a nonlinearly coupled network with multiplicativenoise.22) But they argued that the system did not undergo aphase transition in the presence of additive noise. Since thenmany efforts have been devoted to clarifying the nature ofnonequilibrium phase transition in different dynamic sys-tems such as the simple overdamped nonlinear system,23) thevan der Pol oscillators,24) and the representative Kuramotomodel.25) These abounding studies showed that for differentmodels, the coupling and additive or multiplicative noise caninduce a first- or second-order phase transition and thetransition can be characterized roughly by standard tools inequilibrium statistical mechanics.20)

Now let us present a clear characterization of neuronalbehavior near the critical point. Phenomenologically, for aphase transition to occur there is in general a symmetry-breaking in the collective dynamics of the network. Let ussee whether there is such a change here. We define the firingphase of the i-th neuron via a time difference �ti between afiring and a referential period T , i.e.,

’i ¼ 2��ti=T: ð10Þ

Here T is the inverse of the main frequency fpeak in the PSD,which characterizes the collective dynamics of the system.Figures 4(a) and 4(b) show the phase distributions for D ¼ 5

and gsyn ¼ 0:1, and 0.6, respectively. For weak couplingwith gsyn ¼ 0:1, the network generally has a random anduniform phase distribution with ’i 2 ½0; 2�Þ, which corre-sponds to a symmetry of continuous time-translation. In thiscase, the network is settled in an asynchronous state.Differently, when the coupling strength increases to thecritical value, gcsyn ¼ 0:6, the symmetry is broken, and all thefiring phases become in order and are localized within anarrow range with ’i 2 ½0; �=2�. The stronger the coupling

is, the narrower the range is. Therefore, such a transitionmay be regarded as a dynamic phase transition, and thesynchronized firing is a phase synchronization due to thestrong couplings between the neurons. It is the symmetrybreaking that results in not only a phase transition, but alsothe long-range order, or synchronized firing. Furthermore,the critical behavior near the transition point can bedescribed by a scaling relationship between K � Kc andðgsyn � gcsynÞ=gcsyn [see Fig. 4(c)] (Kc is the coherencemeasure at gcsyn), that is,

K � Kc � ½ðgsyn � gcsynÞ=gcsyn�

� ð11Þ

with a scaling exponent �. This suggests a resemblancebetween this dynamic transition and the well-studied Landauphase-transition in equilibrium systems,20) providing aninsight into the mechanism of the synchronization transition.The critical curve in Fig. 3(b) is indeed the phase transitionboundary. Interestingly, we find that � is between 0.5 and0.8 and depends slightly on D. When D becomes too large(D � 40), � increases to a large value around 0.8. It is notedthat for other values of I0, the scaling exponent is basicallylocated in the same range. Furthermore, for the fixedcoupling strength and noise intensity, nearly the samescaling behavior is observed when the number of neurons inthe network is increased. In fact, it may be the interplay ofthe coupling, noise-induced oscillation, and noise that affectsthe critical exponent. This needs a further examination.

Now we fix the coupling strength and explore theinfluence of noise on the transition. In the previousstudies,10,11) it was shown that for a system near a bifurcationpoint, an optimal noise intensity can evoke the mostcoherent activity via the CR mechanism. Here we study indetail the critical dynamic behavior for this phenomenon.Figure 5 plots the population coherence measure K against Dfor different values of gsyn. For I0 ¼ 2, as we already knewfrom Fig. 3(b), the minimum critical coupling strength isgcsyn ¼ 0:6 together with the optimal noise intensity Do ¼ 5.Indeed, when gsyn < gcsyn, there is no synchronized rhythm atany noise level. K increases monotonously with D due to thestatistic effect. For gsyn � gcsyn, however, the profile of K

versus D is no longer monotonic, and there is a global

-3 -2 -1 0

-4

-3

-2

-1

3π/23π/2

π0 0

π/2π/2

(c)

D=1: α =0.50D=10:α =0.59D=40:α =0.62

ln[(gsyn

-gc

syn)/g

c

syn]

ln(K

-Kc)

(b) gsyn

= 0.6(a) gsyn

= 0.1

π

1

Fig. 4. For D ¼ 5 and I0 ¼ 2, the firing phase distributions for gsyn ¼ 0:1

(a) and 0.6 (b), respectively. (c) Scaling relationship between lnðK � KcÞand ln½ðgsyn � gcsynÞ=gcsyn� for D ¼ 1, 10 and 40, respectively.

0.1 10 100

0.0

0.2

0.4

0.6g

syn=0.1

0.50.6135

10

K (

arb.

uni

ts)

D (µA2/cm

4)

1

Fig. 5. For I0 ¼ 2, the population coherence measure K vs noise intensity

D for different coupling strengths.

3294 J. Phys. Soc. Jpn., Vol. 72, No. 12, December, 2003 Y. YU et al.

maximum, indicating a CR-like behavior.10) Here noiseindeed induces the symmetry-breaking which results in agenuine nonequilibrium phase transition, for no such atransition is observed without noise. For 1 � gsyn < 3, thetransition is found to be re-entrant: the synchronized stateappears above a critical noise intensity but disappears at ahigher noise level. Both the critical noise intensities can befound in Fig. 3(b) by the intersection points of the criticalcurve and the straight line gcsyn ¼ gsyn. For gsyn � 3, thenoise-induced transition is rather sharp, i.e., a very smallincrease in noise intensity can turn on the transition. K firstincreases dramatically with D and then is saturated. Thismeans that the self-sustained synchronized rhythm is stableagainst a range of noise intensities due to the strongcoupling. Finally, when D becomes large, the stochasticnature of noise dominates the collective dynamics and thusK decreases evidently.

Therefore, the coherent mode can be evoked andenhanced by both the coupling and noise. They are themain factors controlling the nonequilibrium phase transition.Our numerical analysis clearly demonstrated that thetransition from asynchronous to synchronous state isanalogous to a second-order phase transition. Importantly,we found that the scaling exponent � is basically fixed in therange of 0.5–0.8 and depends slightly on D. This has notbeen reported before and may be the characteristic of thenonequilibrium phase transition in the stochastic HH neuro-nal network. However, due to the complexity and non-linearity inherent in the system, the mechanism underlyingthis phase transition has not been explored rigorously yet,and largely theoretical analysis is needed.

The noise intensity D and coupling strength gsyn also playa critical role in adjusting the frequencies of the rhythmicoscillations. Figure 6 plots the frequency versus noiseintensity. When D varies from 1 to 60, fpeak increases from45 to 66Hz for gsyn ¼ 0:1, while fpeak increases from 46 to77Hz for gsyn ¼ 20. fpeak tends to saturate with furtherincreasing D. Thus the frequency of oscillations is mainly inthe range of 40–80Hz. It is noted that even for differentvalues of I0, this frequency range is more or less the same.

This robust behavior implies that the 40Hz synchronizedfirings are indeed closely correlated with the features of thecoupled neurons, especially their intrinsic properties. Theseinclude the spatiotemporal summation of the stimulus, therecovery effect or adaption after the firing of spikes.Therefore, the reciprocal coupling and (internal or external)noise can modulate the collective dynamics of the network.

To clearly see the intrinsic frequency feature of neurons,we analyze the impedance feature of single neuron. Theinput signal with frequency sweeping the range of 0–200Hzis described by sðtÞ ¼ A cos½2�f ðtÞt� and f ðtÞ ¼ f0 þ ðfm �f0Þt=T for t 2 ½0;TÞ, with A ¼ 1 mA/cm2, f0 ¼ 0Hz, fm ¼200Hz and T ¼ 5000ms. Input the signal sðtÞ to a noiselessHH neuron and the impedance is found by dividing thepower spectrum (computed by the Fast Fourier Transform,FFT) of the output (i.e., the membrane potential) by that ofthe input.26) From Fig. 7 we can see that the impedance ofsingle neuron is frequency-dependent and has a resonantpeak in the range of 40–100Hz. Therefore, the neuron maybehave as a bandpass filter and exhibits frequency prefer-ence in response to stimuli. That is, the neural responseexhibits the most coherence to the signals with specifiedfrequencies. This resonant phenomenon reflects the intrinsicoscillation of the HH neuron. It is noted that the summarymade in ref. 5 based on experimental observations andtheoretical argument is more about the rhythms withfrequency below 20Hz. Our results further demonstrate theexistence of the 40Hz oscillations.

Such an intrinsic dynamic feature of the HH neuron doesplay important roles in determining the collective dynamicsor the rhythmic oscillations. In the evolution of the noise-induced oscillations, activities with frequencies close to thenatural frequency of neurons will be selected by theresonance, while those with frequencies far from theresonant frequency will be depressed and even disappearfinally. Resonance and frequency preference have significantimplications for signal processing in the brain, such as weaksignal detection.27) With these features the neurons mayserve as a substrate for adjusting the rhythmic activityaround a particular frequency. These rhythms then playcritical roles in information processing and higher brain

040

50

60

70

80

gsyn

=0.1g

syn=1

gsyn

=10g

syn=20

f peak (

Hz)

D (µA2/cm4)

10 20 30 40 50 60

Fig. 6. For I0 ¼ 2, the peak frequency fpeak in the PSD vs noise intensity

D for different values of gsyn.

-1

0

1(a)

S (µ

A/c

m2 )

100 1000-68

-64

(b)

V (

mV

)

t (ms)100 200 300 400

0

3

6

9(c) Impedance=

FFTout

/FFTin

Mag

nitu

de

f (Hz)

Fig. 7. (a) The input current. (b) The membrane potential of single neuron

vs the time. (c) Impedance analysis in the frequency domain.

J. Phys. Soc. Jpn., Vol. 72, No. 12, December, 2003 Y. YU et al. 3295

functions.1–3) Recently, it has been shown that the synchro-nous firings with some specified frequencies can bepropagated in neural circuits.28) Our results may providean enlightenment why the working brain is characterized byvarious coherent rhythms with characteristic temporalscales.

4. Conclusions

When neurons are only subjected to Gaussian white noiseand reciprocal interactions, synchronized rhythmic oscilla-tions can be generated if the coupling strength and noiseintensity exceed certain critical values. The transition fromasynchronous state to synchronous one corresponds to aHopf bifurcation in the dynamic phase space. When viewedas a critical phenomenon, such a dynamic transition can beregarded as an analogue of the second-order phase transitionand can be induced by both the coupling and noise. Weobtained the critical curve describing the transition boundaryin the gsyn–D parameter space, which is V-shaped. Itsconcave part has neurophysiological implications. In addi-tion, only for the coupling strength above the minimalcritical value can noise enhance the network activity via theCR mechanism. The analysis of the scaling behavior near thetransition point indicates that the transition is related to thesymmetry-breaking of the firing phase differences. We foundthat the impedance of single HH neuron is frequencydependent and has a resonant peak in the range of 40–100Hz. It is this intrinsic characteristic that largelyinfluences the collective dynamics of the neurons. Thefrequency of the synchronized rhythmic oscillation is in the�-band (30–80Hz). Thus, our study provides a possiblemechanism for the generation of the �-band (or 40Hz)synchronized oscillations.

Finally, as our point of view, although some physicalfeatures of the synchronization transition have been reportedin other systems such as the simplified neuronal models, thepresent work is to explore the mechanism underlyingsynchronized oscillations based on a relatively realisticneuronal model. It is known that for ‘‘real neuron’’, there arerecovery and adaption effects and the spatiotemporalsummation of inputs to the neuron. The time scales relatingto the depolarization/recovery effects also have an impacton the synaptic interactions when noise is present. Thedetailed neuronal models, like the HH model, may enable usto obtain more realistic interspike intervals and to giveprecise frequency characteristics in the analysis of thefrequency spectrum. But these could not be obtained fromsome simplified neuronal models. In addition, the dynamicnatures such as the concave part in Fig. 3(b) and thefrequency range of synchronized oscillations provide a morequantitative interpretation of the neurophysiological obser-vations. Of course, different modeling studies may reachdifferent conclusions. Here our goal is to investigate thesynchronized oscillations in a realistic neural network.

Acknowledgement

This work is supported by the NNSF of China (GrantNo. 19625409 and No. 30070208) and the Nonlinear ScienceProject of the NSM.

The authors are grateful to the Institute of Pure and

Applied Physics for financial support in publication.

1) C. M. Gray, P. Konig, A. K. Engel and W. Singer: Nature 338 (1989)

334; R. R. Llinas, A. A. Grace and Y. Yarom: Proc. Natl. Acad. Sci.

U.S.A. 88 (1991) 897; M. A. Whittington et al.: Nature 373 (1995)

612; P. Konig, A. K. Engel and W. Singer: Proc. Natl. Acad. Sci.

U.S.A. 92 (1995) 290; W. Singer and C. M. Gray: Annu. Rev.

Neurosci. 18 (1995) 555; P. Konig and A. K. Engel: Curr. Opin.

Neurosci. 5 (1995) 511; W. Singer: Neuron 24 (1999) 49; W. M. Usrey

and R.C. Reid: Annu. Rev. Physiol. 61 (1999) 435.

2) C. M. Gray and W. Singer: Proc. Natl. Acad. Sci. U.S.A. 86 (1989)

1698; W. Singer: Annu. Rev. Physiol. 55 (1993) 349; T. J. Sejnowski:

Nature 376 (1995) 21.

3) J. G. R. Jefferys, R. D. Traub and M. A. Whittington: Trends Neurosci.

19 (1996) 202; E. Marder and R. L. Calabrese: Physiol. Rev. 76 (1996)

687.

4) R. D. Traub and R. Miles: Neuronal Networks of the Hippocampus

(Cambridge University Press, 1991).

5) R. Llinas: Science 242 (1988) 1654; B. Hutcheon and Y. Yarom:

Trends Neurosci. 23 (2000) 216.

6) W. Wang: J. Phys. A 22 (1989) L627; W. Wang, G. Perez and H. A.

Cerderia: Phys. Rev. E 47 (1993) 2893; W. Wang, G. Chen and Z. D.

Wang: ibid. 56 (1997) 3728.

7) J. K. Douglass, L. Wilkens, E. Pantazelou and F. Moss: Nature 365

(1993) 337; K. Wiesenfeld and F. Moss: ibid. 373 (1995) 33; L.

Gammaitoni, P. Hanggi, P. Jung and F. Marchesoni: Rev. Mod. Phys.

70 (1998) 223.

8) P. Jung and G. Mayer-Krers: Phys. Rev. Lett. 74 (1995) 2130.

9) M. Locher, D. Cigna and E. R. Hunt: Phys. Rev. Lett. 80 (1998) 5212;

J. F. Lindner, S. Chandramouli, A. R. Bulsara, M. Locher and W. L.

Ditto: ibid. 81 (1999) 5048.

10) W.-J. Rappel and S. H. Strogatz: Phys. Rev. E 50 (1994) 3249; A.

Neiman, P. I. Saparin and L. Stone: ibid. 56 (1997) 270; A. S.

Pikovsky and J. Kurths: Phys. Rev. Lett. 78 (1997) 775; A. Longtin:

Phys. Rev. E 55 (1997) 868; S. G. Lee, A. Neiman and S. Kim: ibid.

57 (1998) 3292.

11) W.-J. Rappel and A. Karma: Phys. Rev. Lett. 77 (1996) 3256; M. G.

Rosenblum, A. S. Pikovsky and J. Kurths: ibid. 76 (1996) 1804; J.

Pham, K. Pakdaman and J.-F. Vibert: Phys. Rev. E 58 (1998) 3610.

12) D. Hansel and H. Sompolinsky: Phys. Rev. Lett. 68 (1992) 718; C. van

Vreeswijk: Phys. Rev. E 54 (1996) 5522; M. Tsodyks, I. Mitkov and

H. Sompolinsky: Phys. Rev. Lett. 71 (1993) 1280.

13) A. L. Hodgkin and A. F. Huxley: J. Physiol. 117 (1952) 500.

14) D. Hansel, G. Mato and C. Meunier: Europhys. Lett. 23 (1993) 367.

15) W. Wang, Y. Wang and Z. D. Wang: Phys. Rev. E 57 (1998) R2527.

16) R. F. Fox: Phys. Rev. A 43 (1991) 2649.

17) Y. Yu, W. Wang, J. Wang and F. Liu: Phys. Rev. E 63 (2001) 021907.

18) X. J. Wang and G. Buzsaki: J. Neurosci. 16 (1996) 6402.

19) Z. F. Mainen and T. J. Sejnowski: Science 268 (1995) 1503.

20) S. K. Ma: Modern Theory of Critical Phenomena (Benjamin, London,

1976).

21) W. Horsthemke and R. Lefever: Noise-induced Transitions (Springer-

Verlag, Berlin, 1984); Noise in Nonlinear Dynamical Systems, ed. F.

Moss and P. V. E. McClintock (Cambridge University Press, Cam-

bridge, 1989); Instabilities and Nonequilbrium Structures, ed. E.

Tirapegui and W. Zeller (Kluwer, Amsterdam, 1993); Modeling

complex phenomena, ed. L. Lam and V. Narroditsky (Springer, New

York, 1992); S. H. Strogatz: Nonlinear dynamics and chaos (Addison-

Wesley, Reading, MA, 1994).

22) C. Van den Broeck, J. M. R. Parrondo and R. Toral: Phys. Rev. Lett.

73 (1994) 3395.

23) P. Landa and A. Zaikin: Phys. Rev. E 54 (1996) 3535.

24) for example see, H. K. Leung: Phys. Rev. E 58 (1998) 5704.

25) for example see, J. D. Crawford and K. T. R. Davies: Physica D 125

(1999) 1.

26) E. Puil et al.: J. Neurophysiol. 55 (1986) 995.

27) F. Liu, J. F. Wang and W. Wang: Phys. Rev. E 59 (1999) 3453.

28) M. Diesmann, M. O. Genwaltig and A. Aertsen: Nature 402 (1999)

529.

3296 J. Phys. Soc. Jpn., Vol. 72, No. 12, December, 2003 Y. YU et al.