suppression of jamming in excitable systems by aperiodic...

October 27, 2004 9:42 01145

International Journal of Bifurcation and Chaos, Vol. 14, No. 10 (2004) 3519–3539c© World Scientific Publishing Company

SUPPRESSION OF JAMMING IN

EXCITABLE SYSTEMS BY APERIODIC

STOCHASTIC RESONANCE

YING-CHENG LAIDepartment of Mathematics and Statistics,

Department of Electrical Engineering,

Arizona State University,

Tempe, AZ 85287, USA

ZONGHUA LIUDepartment of Mathematics and Statistics,

Arizona State University, Tempe, AZ 85287, USA

ARJE NACHMANAir Force Office of Scientific Research,

801 North Randolph Road,

Arlington, VA 22203, USA

LIQIANG ZHUDepartment of Electrical Engineering,

Arizona State University, Tempe, AZ 85287, USA

Received August 29, 2003; Revised September 29, 2003

To suppress undesirable noise (jamming) associated with signals is important for many appli-cations. Here we explore the idea of jamming suppression with realistic, aperiodic signals bystochastic resonance. In particular, we consider weak amplitude-modulated (AM), frequency-modulated (FM), and chaotic signals with strong, broad-band or narrow-band jamming, andshow that aperiodic stochastic resonance occurring in an array of excitable dynamical systemscan be effective to counter jamming. We provide formulas for quantitative measures character-izing the resonance. As excitability is ubiquitous in biological systems, our work suggests thataperiodic stochastic resonance may be a universal and effective mechanism for reducing noiseassociated with input signals for transmitting and processing information.

Keywords : Aperiodic signal; stochastic resonance; noise reduction; excitable system.

1. Introduction

The presence of undesirable noise is a common

problem in many scientific and engineering appli-

cations. To suppress noise, particularly when it is

present in input signals, is naturally of great inter-

est. The issue becomes critical in applications such

as communication, where externally imposed noise,

or jamming, may be present and strong. Tradi-tional methods of antijamming are based mainlyon filtering. For instance, if the frequency spectraof the signal and jamming are relatively well sep-arated, a bandpass filter can be useful for reduc-ing jamming so as to enhance the signal. In-bandnoise, whose spectrum overlaps significantly withthat of the signal, is more difficult to deal with.

3519

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3520 Y.-C. Lai et al.

Often, sophisticated filtering procedure is necessary.In some cases the underlying dynamics generatingthe signal can be exploited. For instance, if the sig-nal comes from a deterministic chaotic process, thenthe intrinsic redundancy of the chaotic dynamicscan be used to suppress in-band noise [Rosa et al.,1997; Mario et al., 2000].

A closely related problem is how natural sys-tems deal with noisy signals. In a biological or neu-ral network, for instance, input signals that containcertain information to be processed or transmittedare often contaminated by noise. To understand themechanisms by which noise is suppressed and use-ful information retrieved in biological systems is offundamental importance.

In a recent work [Liu et al., 2002, 2004], weproposed an approach to jamming suppression. Theidea is to make use of the principle of stochas-tic resonance [Benzi et al., 1981, 1983; Jung, 1993;Moss et al., 1994; Wiesenfeld & Moss, 1995;Gammaitoni et al., 1998; McNamara & Wiesenfeld,1989; Longtin et al., 1991; Jung et al., 1992; Dou-glass et al., 1993; Misono et al., 1998; Chapeau-Blondeau & Godivier, 1997; Neiman et al., 1997;Russell & Moss, 1999; Goychuk & Hanggi, 2000;Greenwood et al., 2000; Collins et al., 1995b;Heneghan et al., 1996; Gailey et al., 1997; Nozakiet al., 1999; Collins et al., 1995a, 1996; Inchiosa& Bulsara, 1995; Bulsara & Zador, 1996; Inchiosaet al., 1998; Hanggi et al., 2000; Inchiosa et al.,2000; Stocks, 2000; Goychuk, 2001; Stocks &Mannella, 2001; Stocks, 2001a, 2001b], where ajammed signal is taken as the input to a nonlinearsystem and the signal is enhanced by deliberatelyapplying adjustable noise to the system. While thisseems counterintuitive, the mechanism lies in thenonlinear system’s ability to respond, resonantly, tothe input signal at some optimal noise level. Thisis particularly so when the system is excitable andsub-threshold, as in many biological oscillators. Wehave shown how this can be accomplished for theparticular case where the input signal is dominantlyperiodic with a well-defined peak in its Fourier spec-trum [Liu et al., 2002, 2004]. For noisy periodic sig-nals, stochastic resonance can be quantified by thesignal-to-noise ratio (SNR) defined with respect tothe spectral peak. In this case, SNR is a properand convenient measure to characterize the degreeof signal enhancement.

This paper addresses the problem of jam-ming suppression for aperiodic signals. In partic-

ular, we consider amplitude-modulated (AM),frequeny-modulated (FM), and chaotic signals,which are used in many applications ranging fromradio broadcasting to more advanced communica-tion systems such as the global positioning sys-tem (GPS) that is key to the infrastructure ofa modern society. A common feature associatedwith these signals is that they have broad Fourierspectra, which renders the SNR measure improper.It is necessary to explore more general measures.In this paper we will use the correlation measureproposed by Collins et al. [Collins et al., 1995b;Heneghan et al., 1996; Gailey et al., 1997; Nozakiet al., 1999; Collins et al., 1995a, 1996] for char-acterization of aperiodic stochastic resonance. Ourquestion is whether stochastic resonance can helpsuppress the jamming so as to enhance the correla-tion measure. Here we will show this is feasible. Animplication is that stochastic resonance provides anatural way for biological systems to retrieve in-formation from weak input signals contaminatedby noise.

An issue requiring a special consideration forjamming suppression is the range of noise amplitudein which a stochastic resonance can occur. For a sin-gle nonlinear oscillator, resonance usually occurs ina narrow range of the noise amplitude. If the jam-ming is weak, applying additional noise gives rise toa combined noise level that results in a stochasticresonance. However, for strong jamming, additionalnoise will bring the total noise level in the system farbeyond the resonant point. In order to achieve theresonance, an array of nonlinear oscillators shouldbe used [Liu et al., 2002, 2004], in which case a widenoise range for resonance can be achieved. This isthe phenomenon of extended stochastic resonance[Collins et al., 1995b; Heneghan et al., 1996; Gaileyet al., 1997; Nozaki et al., 1999; Collins et al., 1995a,1996]. Thus, we will use an array of nonlinear oscil-lators to generate extended stochastic resonance tosuppress jamming.

In Sec. 2, we briefly describe the phenomenonof stochastic resonance and define the correlationmeasures. In Sec. 3, we introduce our numericalmodel and present extensive computational resultsfor jamming suppression for AM, FM and chaoticsignals. In Sec. 4, we derive formulas for the correla-tion measure under two independent noise sourcesto explain the features of our numerical results. Adiscussion is presented in Sec. 5.

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Suppression of Jamming in Excitable Systems by Aperiodic Stochastic Resonance 3521

2. Stochastic Resonance and

Characterization

2.1. Stochastic resonance and our

antijamming scheme

The following simple mechanical system has been aparadigm for studying stochastic resonance [Jung,1993; Moss et al., 1994; Wiesenfeld & Moss, 1995;Gammaitoni et al., 1998]. Consider the motion of aparticle of mass m in a one-dimensional, symmetric,double-well potential U(x), under heavy dampingcharacterized by viscous friction. Suppose a weakperiodic forcing is applied and the output signalof the system is proportional to the hopping rateof the particle between the two potential wells. Be-cause the forcing is weak, in the absence of noise theparticle cannot overcome the potential barrier. Thepresence of noise, in combination with the weak pe-riodic forcing, can give rise to a nonzero probabilityfor hopping of the particle between the wells. Here,the two distinct time scales are apparent: one is theinverse of the rate of transition between the poten-tial wells and another is the period of the externalforcing. The rate depends on noise as it is caused bythe fluctuational force due to noise and it is given bythe Kramers formula [Jung, 1993; Moss et al., 1994;Wiesenfeld & Moss, 1995; Gammaitoni et al., 1998]:rK ∼ exp (−∆U/D), where ∆U is the height ofthe potential barrier. The intrinsic time scale τi inthis case is then the average first-passage time forthe particle to cross the potential barrier. A matchin the time scales occurs when this time is half ofthe period of the forcing, leading to a stochasticresonance.

In the above setting of a single, relatively low-dimensional nonlinear oscillator, stochastic reso-nance occurs at the optimal noise level D. It wasfound later that for an array of nonlinear oscilla-tors, stochastic resonance can occur in a wide rangeof the noise level [Collins et al., 1995b; Heneghanet al., 1996; Gailey et al., 1997; Nozaki et al., 1999;Collins et al., 1995a, 1996]. Such a system is typ-ically high-dimensional and in principle providesmany more time scales that can potentially lead toresonance. When noise varies in a range, the dom-inant intrinsic time scale varies, but if the externalsignal is aperiodic with many time scales, it is pos-sible that time-scale match can occur in a rangeof noise levels, leading to an extended stochasticresonance.

The mechanism of stochastic resonance wasfirst proposed by Benzi et al. to explain the ap-

proximately periodic occurrence of the earth’s iceages [Benzi et al., 1981, 1983]. They regard theearth’s climatic system as nonlinear and subjectto constant random fluctuations. The earth’s wob-ble acts as the small periodic perturbations thatprovide the external time scale. A stochastic res-onance can lead to large-scale climatic changessuch as the ice age. Since then the phenomenonhas been identified in many situations and it be-comes one of the most active areas in nonlinear andstatistical physics [Jung, 1993; Moss et al., 1994;Wiesenfeld & Moss, 1995; Gammaitoni et al.,1998; McNamara & Wiesenfeld, 1989; Longtinet al., 1991; Jung et al., 1992; Douglass et al.,1993; Misono et al., 1998; Chapeau-Blondeau &Godivier, 1997; Neiman et al., 1997; Russell &Moss, 1999; Goychuk & Hanggi, 2000; Greenwoodet al., 2000; Collins et al., 1995b; Heneghan et al.,1996; Gailey et al., 1997; Nozaki et al., 1999;Collins et al., 1995a, 1996; Inchiosa & Bulsara,1995; Bulsara & Zador, 1996; Inchiosa et al., 1998;Hanggi et al., 2000; Inchiosa et al., 2000; Stocks,2000; Goychuk, 2001; Stocks & Mannella, 2001;Stocks, 2001a, 2001b].

Our interest is in antijamming and signal en-hancement by stochastic resonance, so we considera nonlinear system as a signal-processing unit. Theinput consists of the desirable signal and jammingof amplitude DJ . As we have demonstrated pre-viously [Liu et al., 2002, 2004], in the presence ofjamming, to have a stochastic resonance it is nec-essary to use an array of N nonlinear elements, asshown in Fig. 1. Each element can be a nonlinearoscillator described by a set of differential equationsdxi/dt = fi(xi) or it can simply be a threshold de-vice [Stocks, 2000; Goychuk, 2001; Stocks & Man-nella, 2001; Stocks, 2001a, 2001b]. The input con-sists of the desirable signal S(t) to be enhanced andjamming. Independent Gaussian noise of adjustableamplitude D is applied to each element. The out-put xi from a single element is one of the dynami-cal variables of this element, and the output signalX(t) of the system is the average of outputs fromall elements.

2.2. Characterization of stochastic

resonance

When a periodic signal is under jamming of am-plitude DJ , its spectrum is broad but nonethelessthere are dominant peaks at the corresponding fre-quency f0 and its harmonics. In this case, it is

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f

f

f

fN−1

N

1

2x

x 1

2

N−1x

x N

< >

Dξ

Dξ 2

1

DξN−1

DξN

(t)

(t)

(t)

(t)

K

K

K

K

X(t)JD

S(t) +

(t)η

Fig. 1. Our proposed system of an array of nonlinear ele-ments to suppress jamming by using adjustable noise to in-duce a stochastic resonance. The input consists of the desir-able signal and jamming, and the output is the average ofoutputs from all elements.

proper to use the SNR to characterize stochasticresonance, which is the ratio of the height H(f0)of the dominant peak in the spectrum to the noisypower B(f0) at the same frequency,

βf =H(f0)

B(f0). (1)

The value of the SNR depends on the amplitudeD of the adjustable noise. A stochastic resonanceoccurs if the plot of βf versus D exhibits a max-

imum at D. The amplitude D is thus a tunableparameter that can be adjusted in applications toachieve a maximum amount of jamming suppres-sion. Strictly speaking, the notion of SNR is mean-ingful only when the system is linear and noise isGaussian. For weakly nonlinear systems and pe-riodic signals, which are characterized roughly bythe persistence of a set of dominant spectral peaks,the SNR measure can still be utilized to quantifystochastic resonance.

For aperiodic signals in excitable systems,Collins et al. suggested the cross-correlation mea-sure to characterize stochastic resonance [Collinset al., 1995b; Heneghan et al., 1996; Gailey et al.,1997; Nozaki et al., 1999; Collins et al., 1995a, 1996].Specifically, let S(t) be a zero-mean input signal,and let R(t) be the response function determined bythe bursting or firing rate of the excitable system.For output signal consisting of spikes, the instan-taneous firing rate R(t) can be computed by using

a moving-window filter, such as the Hanning filter[Collins et al., 1995b; Heneghan et al., 1996; Gai-ley et al., 1997; Nozaki et al., 1999; Collins et al.,1995a, 1996]. The power norm is defined to be

C0 = max{S(t)〈R(t + τ)〉} , (2)

where the overbar denotes time average, 〈·〉 is theensemble average, τ accounts for the time delay(usually small) for the system to respond to aninput impulse, the max{·} is with respect to τ .Clearly, C0 measures the coherence between the in-put and output signals. This definition of C0 is mo-tivated by the fact that in an excitable system, in-formation is transmitted and represented in its av-erage firing rate 〈R(t)〉. Depending on the relativestrength of the input signal and the average firingrate, the numerical value of C0 can be rather arbi-trary. The following normalized power norm is thenuseful:

C1 =C0

[S2(t)]1/2[〈R2(t)〉 − (〈R(t)〉)2]1/2

, (3)

where 0 ≤ C1 ≤ 1. For the special case of periodicdriving, C1 is proportional to SNR.

Recently, information theoretic measures havebeen proposed to characterize stochastic resonancewith aperiodic signals [Inchiosa & Bulsara, 1995;Bulsara & Zador, 1996; Inchiosa et al., 1998; Hanggiet al., 2000; Inchiosa et al., 2000; Stocks, 2000;Goychuk, 2001; Stocks & Mannella, 2001; Stocks,2001a, 2001b]. It was shown that the average mutualinformation (or the information transfer), whichmeasures the amount of information transmittedthrough the system, can characterize stochastic res-onance for discrete-value dynamical systems such asa threshold device. For our antijamming scheme inFig. 1, the information transfer can be defined as

I = H(X) − H(X|S)

= −N∑

n=0

PX(n) log2 PX(n)

−[

−∫

∞

−∞

p(S)dS

N∑

n=0

P (n|S) log2 P (n|S)

]

,

(4)where H(X) is the information (or entropy) con-tained in the output signal X(t) and H(X|S) is theinformation content of the output signal providedthat the information about S(t), the desirable sig-nal to be enhanced, is known. Here the output as-sumes a set of (N +1) discrete values, PX(n) is the

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probability distribution function of the discrete ran-dom variable X, P (n|S) is the conditional probabil-ity distribution function, and p(s) is the probabilitydensity function associated with the continuous in-put signal S(t). In the ideal situation where thereis no jamming or other noise and the system is lin-ear, knowing S(t) implies a full knowledge aboutthe output signal X(t) so that it provides no newinformation, giving H(X|S) = 0. In this case, in-formation transfer I is maximum, as it should be.In the opposite case where the noise is overwhelm-ingly strong, knowing S(t) gives no improvement inthe prediction of the output and, hence, we haveH(X|S) ≈ H(X). In this case, the informationtransfer is approximately zero, indicating that noinformation about the desirable signal S(t) can berevealed in the output. We see that the quantityH(X|S) can be regarded as the amount of encodedinformation lost in the transmission of the signalthrough the system. Due to jamming, informationabout the desirable signal S(t) will be reduced in theinput signal. For discrete-value signals the informa-tion measure is advantageous and can be computedrelatively easily. However, to compute the informa-tion for continuous-time input and output signalsis difficult. For this reason we will not consider theinformation measure in this paper.

3. Numerical Experiments

3.1. Model and signal description

The following FitzHugh–Nagumo (FHN) model[FitzHugh, 1961; Scott, 1975] represents a simpleexcitable system to simulate the dynamics of neu-rons, which has been a paradigm to study stochasticresonance:

εdx

dt= x(x − 1/2)(1 − x) − y ,

dy

dt= x − y − b ,

(5)

where ε and b are parameters. The value of ε ischosen to be much smaller than unity so that thex-variable is faster than y in time. The typical dy-namics consists of slow motion near a fixed pointand rapid excursions away from it. We use b = 0.15and ε = 0.005 in our numerical experiments. Nowconsider a network of N excitable systems mod-eled by an array of FHN oscillators, under noise of

adjustable amplitude D,

εdxi

dt= xi(xi − 1/2)(1 − xi) − yi + SI(t) ,

dyi

dt= xi − yi − b + Dξi(t) , for i = 1, . . . , N ,

(6)

where ξi(t)’s are Gaussian random variables of zeromean and unit variance 〈ξi(t)ξj(t

′)〉 = δijδ(t−t′) (soD2 is the noise variance), SI(t) is the jammed inputsignal. We focus on additive jamming and write

SI(t) = S(t) + DJη(t) , (7)

where S(t) is the information-carrying signal, η(t) isa random process simulating the additive jamming,and D2

J is the variance of the jamming. The outputis

X(t) =1

N

N∑

i=1

xi(t) .

We integrate the stochastic differential equationsEq. (6) by using the standard, second-order Mil-shtein routine [Kloeden & Platen, 1992]. In our nu-merical experiments we consider various combina-tions of types of signal and noise including AM,FM and chaotic signals, and both broad-band andnarrow-band jamming. For AM signals we choose

S(t) = 0.06[1 + 0.15 cos (0.5t)] sin (ω0t) , (8)

where ω0 = 0.15. This choice of ω0 is made suchthat it is much smaller than the firing frequency ofthe FHN oscillator. This is necessary because, tosuppress jamming, the system must be able to re-spond to the signal so that the bursting frequencychanges with the signal. This cannot be achieved ifthe signal frequency is comparable with the intrinsicfrequency of the excitable system.

For FM signals, we use

S(t) = 0.06 sin [ω0t + 0.15 cos (ω1t)] , (9)

where the frequency ratio is incommensurate:ω1/ω0 = (1 +

√5)/2.

For chaotic signals, we use the standard Lorenzsystem [Lorenz, 1963]

du

dt= 10(v − u) ,

dv

dt= 28u − v − xw ,

dw

dt= uv − 8

3w ,

(10)

and choose S(t) = 0.004u(t/10) so that its ampli-tude and frequency are comparable to those of theAM and FM signals.

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For broad-band jamming we choose η(t) to bea Gaussian random variable of zero mean and unitvariance, and DJ = 0.05 so that the signal and jam-ming amplitudes are comparable. In fact, the stan-dard deviation of the jamming is 0.05 but those ofthe AM, FM, and chaotic signals are about 0.03.In this sense, the signals are somewhat immersed inthe jamming. In spite of this, we will demonstratethat the system can still pick up the signal effec-tively through the firing mechanism by stochasticresonance.

For narrow-band jamming, the time domainnoisy signal is constructed by choosing a narrowsubset from a Gaussian spectrum, which covers thatof the signals, and then performing inverse Fouriertransform (detailed in Secs. 3.2–3.4). This way weensure a significant overlap between the jammingand signal spectra.

3.2. Stochastic resonance with

AM signals

3.2.1. Broad-band jamming

We consider the case where an AM signal (thicksolid line) is immersed in a broad-band jammingsignal, as shown in Fig. 2. Figure 3(a) shows a

Fig. 2. AM signal S(t) (thick solid line) and jammed signalSI (t) (dots).

typical firing pattern of an FHN oscillator in re-sponse to this jammed signal, where N = 40 andno adjustable noise is applied (D = 0). We ob-serve spikes concentrated in some narrow time in-tervals and a noisy background. The correspondinginstantaneous firing rate R(t) computed from thenetwork output signal X(t) is shown in Fig. 3(b)(the thick solid line), where the original AM signal

0 50 100 150

0

0.5

1

1.5

t

x i

0 50 100 150

0

0.5

1

t

S(t

),R

(t)

0 50 100 150

0

0.5

1

1.5

t

x i

0 50 100 150

0

0.5

1

t

S(t

),R

(t)

(a) (b)

(c) (d)

Fig. 3. For jammed AM signal and an array of N = 40 oscillators, (a) typical firing pattern of an individual oscillatorfor D = 0, (b) the corresponding average firing rate R(t) (the thick solid line), (c) firing pattern under adjustable noise ofamplitude D = 0.2, and (d) average firing rate R(t), which resembles the original AM signal (the thin solid line).

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0 0.1 0.2 0.3 0.4 0.5

0.6

0.8

1

D

C1

0 20 40 60 80

0.6

0.8

1

N

C1

(a)

(b)

0 0.1 0.2 0.3 0.4 0.5

0.6

0.8

1

DC

1

0 20 40 60 80

0.6

0.8

1

N

C1

(a)

(b)

Fig. 4. (a) For N = 40, the normalized power norm C1 versus D, the amplitude of the adjustable noise. (b) For D = 0.04,C1 versus the system size N .

S(t) is shown as the thin solid line. While it can beseen that firing tends to occur when the AM sig-nal reaches its local maxima, overall there appearsto be no resemblance between R(t) and S(t). Thus,information carried by the signal cannot be codedproperly by the network of oscillators through thefiring rate. As we turn on the adjustable noise, theoscillators fire more frequently, as shown by the fir-ing pattern of a typical oscillator in Fig. 3(c) forD = 0.2. The overall firing rate R(t) is shown inFig. 3(d) (the thick solid line), which appears to fol-low closely the original signal S(t). Thus, noise canhelp improve the network ability to code the infor-mation through its firing rate, despite the presenceof strong jamming.

Figure 4(a) shows the normalized power normC1 versus D for N = 40. There is apparently a rangeof D values for which C1 is close to its maximallypossible value. For any choice of D in this range,C1 is close to unity, indicating that jamming hasbeen effectively suppressed as the AM signal canbe extracted from the firing rate almost perfectly.Figure 4(b) shows, for D = 0.04, C1 versus N .We see that the correlation increases toward unityrapidly as N is increased, and saturates for N > 10.This indicates a wide range of system sizes forstochastic resonance and jamming suppression, in-sofar as the number of elements is more than a few.The biological implication is that the ability of aneural network to suppress jamming seems to beindependent of its size, making stochastic resonance

a robust and effective mechanism for informationcoding in noisy environment.

3.2.2. Narrow-band jamming

The frequency spectrum of the narrow-band jam-ming signal is shown in Fig. 5(a), which coversthe frequencies of the AM signal (indicated by thevertical thick dashed lines). The time-domain jam-ming signal is shown in Fig. 5(b), together with theAM signal S(t) (thick solid line). Figures 6(a) and6(b) show, respectively, the typical firing patternof an oscillator and the average firing rate R(t) ofthe whole system (N = 40) in the absence of ad-justable noise, where the correlation between R(t)and S(t) is poor. As we turn on the adjustablenoise, the oscillators fire more frequently, as shownin Fig. 6(c) and, the correlation is improved, as canbe seen even visually in Fig. 6(d) for D = 0.03.Figures 7(a) and 7(b) show, the normalized powernorm C1 versus D for N = 40 and versus N forD = 0.04, respectively. An extended stochastic res-onance is present because C1 achieves maximumfor a finite range of the noise level. Comparingwith the case of broad-band jamming [Figs. 4(a)and 4(b)], we see that the maximum power normfor narrow-band jamming is slightly smaller. Thiscan be understood by noticing that the jamming-suppression process is essentially an averaging pro-cess. Broad-band noise is relatively more uni-form in time and can therefore be averaged outmore easily.

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−0.5 0 0.50

4000

8000

f

H(f

)

0 50 100 150−0.5

0

0.5

t

S(t

),η(

t)

(a)

(b)

Fig. 5. (a) Frequency spectra of AM signal (thick line) and narrow-band jamming signal (thin line). (b) Time-domain signals(AM signal — thick line; jamming noise — dots).

0 50 100 150

0

0.5

1

1.5

t

x i

0 50 100 150−0.2

0

0.2

0.4

t

S(t

),R

(t)

0 50 100 150

0

0.5

1

1.5

t

x i

0 50 100 150−0.2

0

0.2

0.4

t

S(t

),R

(t)

(a) (b)

(c) (d)

Fig. 6. For AM signal, narrow-band jamming, and N = 40, (a) the typical firing pattern of an individual oscillator for D = 0,(b) the corresponding average firing rate R(t) (the thick solid line), (c) the firing pattern under adjustable noise of amplitudeD = 0.03, and (d) the average firing rate R(t), where the thin solid line represents the AM signal.

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0 0.1 0.2 0.3 0.4 0.5

0.6

0.8

1

D

C1

0 20 40 60 80

0.6

0.8

1

N

C1

(a)

(b)

0 0.1 0.2 0.3 0.4 0.5

0.6

0.8

1

DC

1

0 20 40 60 80

0.6

0.8

1

N

C1

(a)

(b)

Fig. 7. For AM signal and narrow-band jamming, (a) the normalized power norm C1 versus D for N = 40, and (b) C1 versusthe system size N for D = 0.03.

0 50 100 150−0.5

0

0.5

t

Sig

nals

Fig. 8. FM signal S(t) (solid line) and jammed signal SI (t) (dots).


FM signals


Figure 8 shows an FM signal with jamming, wherewe see that the signal is immersed in jamming.Figures 9(a) and 9(b) show, respectively, the fir-ing pattern and the average firing rate R(t) forD = 0, where we see that R(t) does not seem to

be able to code the signal (thin solid line) properly,as most oscillators do not fire most of the time.As we turn on the adjustable noise to increase thelevel of excitation of the system, the firing is en-hanced [as shown in Fig. 9(c)] and the system’s abil-ity to code the original FM signal is improved, ascan be seen from Fig. 9(d). Figures 10(a) and 10(b)show, respectively, the normalized power norm C1

versus D for N = 40 and versus N for D = 0.04.

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0 50 100 150

0

0.5

1

1.5

t

x i0 50 100 150

−0.2

0

0.2

0.4

t

S(t

),R

(t)

0 50 100 150

0

0.5

1

1.5

t

x i

0 50 100 150−0.2

0

0.2

0.4

tS

(t),

R(t

)

(a) (b)

(c) (d)

Fig. 9. For FM signal, broad-band jamming, and N = 40, (a) the typical firing pattern of an individual oscillator for D = 0,(b) the corresponding average firing rate R(t) (the thick solid line), (c) the firing pattern under adjustable noise of amplitudeD = 0.2, and (d) the corresponding average firing rate R(t), where the thin solid line is the FM signal.

0 0.1 0.2 0.3 0.4 0.5

0.6

0.8

1

D

C1

0 20 40 60 800.8

0.9

1

N

C1

(a)

(b)

0 0.1 0.2 0.3 0.4 0.5

0.6

0.8

1

D

C1

0 20 40 60 800.8

0.9

1

N

C1

(a)

(b)

Fig. 10. For FM signal and broad-band jamming, (a) normalized power norm C1 versus D for N = 40, and (b) C1 versusthe system size N for D = 0.04.

Stochastic resonance similar to that in Figs. 4(a)and 4(b) is observed.


The Fourier spectrum of the FM signal is shownin Fig. 11(a). Narrow-band jamming signal is con-

structed by limiting the spectrum of a Gaussian ran-

dom variable to a range that covers that of the FM

signal. In the time domain, the FM signal is im-

mersed in the jamming, as shown in Fig. 11(b).

Figures 12(a) and 12(b) show, respectively,

the typical firing pattern of an oscillator and the

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−0.5 0 0.50

4000

8000

f

H(f

)

0 50 100 150−0.5

0

0.5

t

S(t

),η(

t)

(a)

(b)

Fig. 11. (a) Fourier spectrum of FM signal, (b) time-domain signals (FM signal — thick line; jamming — dots).

0 50 100 150

0

0.5

1

1.5

t

x i

0 50 100 150−0.2

0

0.2

0.4

t

S(t

),R

(t)

0 50 100 150

0

0.5

1

1.5

t

x i

0 50 100 150−0.2

0

0.2

0.4

t

S(t

),R

(t)

(a) (b)

(c) (d)

Fig. 12. For FM signal, narrow-band jamming, and N = 40, (a) typical firing pattern of an individual oscillator for D = 0,(b) the corresponding average firing rate R(t) (the thick solid line), (c) the firing pattern under adjustable noise of amplitudeD = 0.03, and (d) the average firing rate R(t), where the thin solid line is the FM signal.

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average firing rate R(t) of the system (N = 40)for D = 0. We see that R(t) does not appear tofollow S(t). As the adjustable noise is turned on,the oscillators fire more often and the system’s fir-ing rate follows relatively more closely the original

signal S(t), as shown in Figs. 12(c) and 12(d), re-spectively, for D = 0.03. The behavior of stochas-tic resonance is shown in Fig. 13(a), the normal-ized power norm C1 versus the noise amplitude D.Figure 13(b) shows C1 versus the system size N .

0 0.1 0.2 0.3 0.4 0.5

0.6

0.8

1

D

C1

0 20 40 60 80

0.6

0.8

1

N

C1

(a)

(b)

0 0.1 0.2 0.3 0.4 0.5

0.6

0.8

1

D

C1

0 20 40 60 80

0.6

0.8

1

N

C1

(a)

(b)

Fig. 13. For FM signal and narrow-band jamming, (a) C1 versus D for N = 40, and (b) C1 versus the system size N forD = 0.03.

0 50 100 150−0.2

0

0.2

t

S(t

)

−2 −1 0 1 20

1000

2000

f

H(f

)

0 50 100 150−0.5

0

0.5

t

Sig

nals

(a)

(b)

(c)

Fig. 14. (a) Chaotic signal S(t) from the Lorenz system; (b) the corresponding Fourier spectrum; (c) broad-band jammingsignal (dots) that covers the original signal S(t) (thick line).

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chaotic signals


We now consider a chaotic signal from the Lorenzsystem Eq. (10), as shown in Fig. 14(a). Its Fourier

spectrum is shown in Fig. 14(b). The broad-bandjamming signal is shown in Fig. 14(c), which coversthe chaotic signal S(t) almost entirely in the timedomain.

Without adjustable noise, the firing patternof an individual oscillator consists of relatively

0 50 100 150

0

0.5

1

1.5

t

x i

0 50 100 150−0.2

0

0.2

0.4

t

S(t

),R

(t)

0 50 100 150

0

0.5

1

1.5

t

x i

0 50 100 150−0.2

0

0.2

0.4

t

S(t

),R

(t)

(a) (b)

(c) (d)

Fig. 15. For chaotic signal, broad-band jamming, and N = 40, (a) typical firing pattern of an individual oscillator for D = 0,(b) the corresponding average firing rate R(t) (the thick solid line), (c) the firing pattern under adjustable noise of amplitudeD = 0.05, and (d) the average firing rate R(t), where the thin solid line is the original chaotic signal.

0 0.1 0.2 0.3 0.4 0.50.4

0.6

0.8

1

D

C1

0 20 40 60 800.4

0.6

0.8

1

N

C1

(a)

(b)

0 0.1 0.2 0.3 0.4 0.50.4

0.6

0.8

1

D

C1

0 20 40 60 800.4

0.6

0.8

1

N

C1

(a)

(b)

Fig. 16. For chaotic signal and broad-band jamming, (a) C1 versus D for N = 40, and (b) C1 versus the system size N forD = 0.05.

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−2 −1 0 1 20

500

1000

1500

f

H(f

)

0 50 100 150−0.5

0

0.5

t

η(t)

,S(t

)

(a)

(b)

Fig. 17. (a) Spectrum of narrow-band jamming signal; (b) jamming (dots) that covers the chaotic signal S(t) (solid line).

0 50 100 150

0

0.5

1

1.5

t

x i

0 50 100 150−0.2

0

0.2

0.4

t

S(t

),R

(t)

0 50 100 150

0

0.5

1

1.5

t

x i

0 50 100 150−0.2

0

0.2

0.4

t

S(t

),R

(t)

(a) (b)

(c) (d)

Fig. 18. For chaotic signal, narrow-band jamming, and N = 40, (a) typical firing pattern of an individual oscillator for D = 0,(b) the corresponding average firing rate R(t) (the thick solid line), (c) the firing pattern under adjustable noise of amplitudeD = 0.05, and (d) the average firing rate R(t), where the thin solid line is the original chaotic signal.

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0 0.1 0.2 0.3 0.4 0.50.4

0.6

0.8

1

DC

1

0 20 40 60 800.4

0.6

0.8

1

N

C1

(a)

(b)

0 0.1 0.2 0.3 0.4 0.50.4

0.6

0.8

1

DC

1

0 20 40 60 800.4

0.6

0.8

1

N

C1

(a)

(b)

Fig. 19. For chaotic signal and narrow-band jamming, (a) C1 versus D for N = 40, and (b) C1 versus the system size N forD = 0.05.

infrequent spikes, as shown in Fig. 15(a). There is asubstantial fraction of time during which the firingrate is zero, as shown in Fig. 15(b). The instan-taneous firing rate R(t) thus cannot represent thesignal S(t) properly in this case. With the presenceof adjustable noise, the firing pattern of the sys-tem tends to follow more closely the behavior ofthe signal S(t), as shown in Figs. 15(c) and 15(d)for D = 0.05. An extended stochastic resonance isobserved, as shown by the plot of C1 versus D inFig. 16(a) for N = 40. Figure 16(b) shows C1 versusN for D = 0.05.


The spectrum of the narrow-band jamming is shownin Fig. 17(a), and its time-domain signal is shownin Fig. 17(b). Again, our choice of the jamming issuch that it covers the signal S(t). The firing pat-tern of an individual oscillator and the average firingrate R(t) of the system are shown in Figs. 18(a)–18(d), respectively, for D = 0 and D = 0.05, whereR(t) tends to follow S(t) more closely under ad-justable noise. The behavior of extended stochasticresonance is shown in Fig. 19(a), where C1 versus Dis plotted for N = 40. Figure 19(b) shows C1 versusthe system size N for D = 0.05.

4. Theory

4.1. Jamming suppression

The ability of an excitable system in suppressingjamming lies in its firing mechanism. A burst oc-curs only when the input signal is strong enoughso that the “excitation” level is above a threshold.For above-threshold signals, the number of burstsper unit time, or the firing rate, is approximatelyproportional to the difference between the signal in-tensity and the threshold. The role of the adjustablenoise is to make the system fire more often so thata sufficient number of spikes are generated all thetime. If the jamming and the adjustable noise areuniform in time, as when they are Gaussian andbroad-band, they contribute to roughly a constantincrease in the firing rate. Variations in the firingrate is due mainly to the temporal variation of theoriginal signal, as shown schematically in Fig. 20(a),where the dashed line denotes jamming and theupper solid line represents the instantaneous firingrate in the presence of jamming. We see that jam-ming is effectively eliminated in the sense that thetemporal variation of the firing rate does not dependon the details of the jamming, even if it is strong.This is why we observe close-to-unity values of thecorrelation measure C1 for all cases of broad-bandjamming, regardless of the nature of the signal.

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S(t)

t

t

S(t)

R(t)

R(t)

nonuniform jamming

"uniform" jamming

(a)

S(t)

t

t

S(t)

R(t)

R(t)

nonuniform jamming

"uniform" jamming

(b)

Fig. 20. Schematic illustrations of the firing rate R(t) withreference to the signal S(t) in the presence of jamming.(a) For Gaussian, broad-band jamming that tends to in-crease the firing rate by approximately a constant, jammingis effectively suppressed in the sense that the correlation be-tween the output signal R(t) and input S(t) is generally high.(b) For narrow-band jamming correlated with the signal S(t),there can be severe distortations in R(t) with respect to S(t).

Narrow-band jamming is less uniform in time.Because of this, it can happen that the jammingcan influence the firing rate in a way that is notcompletely independent of that due to the signal.For instance, in time intervals where the signal isstrong, the system is likely to fire more as a re-sult of the “correlated” jamming. Likewise, rela-tively fewer bursts occur when the signal is weak,as shown schematically in Fig. 20(b). As a result,there can be a severe amount of distortion in thefiring rate with reference to the signal, resulting inlower values of C1, as we have observed in numericalexperiments.

4.2. Characterization of aperiodic

stochastic resonance under

doubly additive noise

4.2.1. Single FHN oscillator

General expressions for the dependences of thepower norm measures C0 and C1 on the amplitudes

of adjustable noise and on jamming can be derivedbased on the work of Collins et al. [Collins et al.,1995b; Heneghan et al., 1996; Gailey et al., 1997;Nozaki et al., 1999; Collins et al., 1995a, 1996]. Theidea is that bursting dynamics can in general beunderstood by a universal model for stochastic res-onance: particle motion in a potential well undernoise. Consider, for instance, a double-well poten-tial with a barrier. In the absence of noise, parti-cles with energy less than the barrier height (“sub-threshold state”) are confined within one well. Theycan move across the barrier when noise is present.The switches between the wells correspond to burstsor spikes in an excitable system. Given an excitablesystem, the task is then to find an equivalent po-tential configuration.

For the FHN model, the starting point is touse the change of variables: v(t) = x(t) − 1/2 andw(t) = y(t) + b − 1/2 to convert Eq. (5) in the ab-sence of signal and noise into

εdv

dt= −v(v2 − 1/4) − w + A ,

dw

dt= v − w ,

(11)

where A is the threshold parameter and for theparticular parameter setting in Eq. (5), A = A ≡b − 1/2. The fixed points of Eq. (11) are the inter-secting points of two nullclines: w = A−v(v2−1/4)and w = v. Threshold stability of the fixed pointsis achieved when the linear nullcline passes throughthe minimum of the cubic nullcline determined byw′(v) = 0, which occurs at v− = −1/(2

√3) and

w− = A − 1/(12√

3). Setting w− = v− yields thethreshold value for A: AT = −5/(12

√3), where

the fixed points are stable (unstable) for A < AT

(A > AT ). For the parameter setting used in ournumerical experiments, we have A = −0.35 < AT ,so the system is sub-threshold. The threshold fixedpoint (v−, w−), nonetheless, provides a convenientreference point for a perturbative treatment forfinding the fixed points under signals and noiseand, consequently, the equivalent potential model.In particular, let vf (t) be the fixed point of the FHNequation under signal S(t). It is determined by

−vf (v2

f − 1/4) − vf +A + S(t)

≡ −vf (v2

f − 1/4) − vf + AT − γ(t)

= 0 , (12)

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where γ(t) ≡ (AT −A)−S(t), and B ≡ (AT −A) isthe “distance” of the excitation level to the thresh-old. For weak signal, Collins et al. suggested thefollowing perturbative assumption [Collins et al.,1995a, 1996]:

vf (t) = v− + α1γ(t) + β1γ2(t) , (13)

which, when substituted into Eq. (12), yields α1 =−(3/4 + 3v2

−)−1 = −1 and β1 = −3α1v−/(3/4 +

3v2−) =

√3/2.

Now consider the FHN model under signal S(t)and doubly additive noise, written as

εdv

dt= −v(v2 − 1/4) − w + AT

− γ(t) + DJη(t) ,

dw

dt= v − w + Dξ(t) .

(14)

For the system to be excitable, the parameter εis chosen to be small: ε � 1. Since the signal isweak and the system is sub-threshold, in the ab-sence of noise the fixed point [vf (t), wf (t)] is sta-ble, where wf (t) = vf (t). Asymptotically, a trajec-tory approaches this time-dependent stable state.Under noise, the dynamics consists of staying nearthe fixed point for relatively long time and rapidbursting away from it, creating a spike in v(t). Be-cause ε is small, the time rate of change of v(t) ismuch larger than that of w(t), so relatively, v(t) is“fast” while w(t) is “slow”. Approximately we canassume dw/dt ≈ 0, which gives

w(t) ≈ vf (t) + Dξ(t) .

Substituting this approximation into the v-equationin Eq. (14) yields

εdv

dt≈ −v(v2 − 1/4) − vf (t) + AT

− γ(t) + DJη(t) − Dξ(t)

≡ −∂U(v)

∂v+ DJη(t) − Dξ(t) , (15)

which describes the motion of a heavily dampedparticle in a potential under the influence of com-bined noise DJη(t) − Dξ(t), where the potentialfunction U(v) is given by

U(v) ≡ v4

4− v2

8+ [vf (t) − AT + γ(t)]v

=v4

4− v2

8+

[

v− − AT +

√3

2γ2(t)

]

v . (16)

We see that U(v) is a time-dependent, tilted double-well potential. Since v(t) = vf (t) is a solution of∂U(v)/∂v = 0, one well is located at vf (t). Wewrite vw(t) = vf (t). To find the location vb(t) ofthe barrier, a perturbative approach can again beused in which we assume, to first order in γ(t),vb(t) = v− + α2γ(t). Substituting this assumptioninto ∂U(v)/∂v = 0 yields α2 = 1. The height ofthe potential barrier, which the noise-driven parti-cle must cross to generate a spike, is

∆U(t) = U(vb) − U(vw)

≈ U(v− + γ) − U(v− − γ)

≈ 2√3

γ3(t) . (17)

Since we are interested in weak signals, we canassume |S(t)| � B and, hence, we have γ3(t) =[B − S(t)]3 ≈ B3 − 3B2S(t). The barrier height isthus given by

∆U(t) ≈ 2√3

B3 − 2√

3B2S(t) . (18)

The analysis proceeds by utilizing the standardKramers’ formula [Kramers, 1940; Hanggi et al.,1990], which gives the rate of escape of particlesfrom a potential well driven by noise. In particular,if the noise variance D2

T is small compared with thebarrier height, the ensemble average of the escape(switching) rate is given by

〈R(t)〉 ∼ exp

(

−∆U/ε

D2

T

)

, (19)

where the normalizing parameter ε has been takeninto account. From Eq. (15), the noise variance isgiven by

⟨

1

ε2[DJη(t) − Dξ(t)][DJη(t′) − Dξ(t′)]

⟩

= D2

T δ(t − t′) . (20)

Since the jamming and adjustable noise are inde-pendent Gaussian white noise, we obtain

D2

T =D2

J + D2

ε2. (21)

Substituting Eqs. (18) and (21) into Eq. (19) gives

〈R(t)〉 ∼ exp

[

− 2ε√3

B3 − 3B2S(t)

D2

J + D2

]

. (22)

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The ensemble-averaged power norm is thus given by

〈C0〉 = S(t)〈R(t)〉

≈ exp

[

− 2εB3

√3(D2

J + D2)

]

×[

1 +2√

3εB2S(t)

D2

J + D2

]

S(t)

∼ 1

D2

J + D2exp

[

− 2εB3

√3(D2

J + D2)

]

S2(t) .

(23)

To compute 〈C1〉, we write [Collins et al., 1995a,

1996]: R(t) ≡ 〈R(t)〉 + ζ(t), where ζ(t) = 0 and

ζ2(t) ≡ Γ(DJ , D). The variance of the switchingrate is thus

(∆R(t))2 = 〈R(t)〉2 − (〈R(t)〉)2 + Γ(DJ , D) .

If |S(t)| � (D2

J + D2)/(εB2), we can expand the

exponential dependence on S(t) of 〈R(t)〉 to obtain

〈R(t)〉 ∼ exp

[

− 2εB3

√3(D2

J + D2)

]

×{

1 +2√

3εB2S(t)

(D2

J + D2)+

6ε2B4S2(t)

[(D2

J + D2)]2

}

,

and

(〈R(t)〉)2 ∼ exp

(

− 4εB3

√3(D2

J + D2)

)

×{

1+4√

3εB2S(t)

(D2

J + D2)+

24ε2B4S2(t)

[(D2

J + D2)]2

}

.

Taking the time average of 〈R(t)〉 and (〈R(t)〉)2 and

noting that S(t) = 0, we obtain

(∆R(t))2 ∼ exp

[

− 4εB3

√3(D2

J + D2)

]

12ε2B4S2(t)

[(D2

J + D2)]2

+ Γ(DJ , D) , (24)

which yields the following expression for 〈C1〉:

〈C1〉 ∼exp

(

− 2εB3

√3(D2

J + D2)

)

√

S2(t)

(D2

J + D2)

√

12ε2B4S2(t)

(D2

J + D2)2exp

(

− 4εB3

√3(D2

J + D2)

)

+ Γ(DJ , D)

. (25)

The following general conclusions can be drawnfrom the formulas of 〈C0〉 and 〈C1〉 in Eqs. (23)and (25).

(1) For a single FHN oscillator, the jamming andadjustable noise play the same dynamical rolein generating the resonance because the depen-dencies on the jamming variance D2

J and ad-justable noise variance D2 are the same. This issomewhat expected because both noise sourcesare additive.

(2) As in [Collins et al., 1995a, 1996] for aperi-odic stochastic resonance under a single ad-ditive noise source, in our case the ensemble-averaged power norms 〈C0〉 and 〈C1〉 both ex-hibit a hump as a function of the combinednoise variance (D2

J + D2) and attain maximum

for D2

J + D2 = K0εB3/√

3, where K0 = 2 forC0 and K0 ≈ 2 for C1. This dependence sug-gests that if jamming is strong such that its

variance is greater than K0εB3/√

3, additiveadjustable noise cannot induce a stochasticresonance in a single FHN unit.

4.2.2. Array of FHN oscillators

For strong jamming, in order to induce a stochasticresonance to suppress it, an array of N FHN oscilla-tors is necessary. For such an array, we consider themean-field variable V (t) =

∑Ni=1

vi(t)/N . Writingvi(t) = V (t) + δi(t), where δi(t) is the deviation ofthe fast variable of each individual oscillator fromthe mean field which satisfies

∑Ni=1

δi(t) = 0, wehave

1

N

N∑

i=1

v3

i (t) = V 3 + 3〈δ2〉V + 〈δ3〉 ,

where 〈δ2〉 ≡ (1/N)∑N

i=1δ2

i and 〈δ3〉 ≡ (1/N) ×∑N

i=1δ3

i . The equation of motion in a potential thus

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becomes

εdV

dt≈ −V 3 + (1/4 − 3〈δ2〉)V − 〈δ3〉

− vf (t) + AT − γ(t) + DJη(t)

− D

N

N∑

i=1

ξi(t) . (26)

For an excitable system, most of the time it is nearthe fixed point so that the spike amplitude is nearzero. Thus, we have |〈δ3〉| � 〈δ2〉 � 1. Thesetwo terms can thus be neglected in Eq. (26). Let-

ting DSχ(t) ≡ DJη(t) − (D/N)∑N

i=1ξi(t), where

D2

S = D2

J + D2/N and χ(t) is a Gaussian ran-dom variable of zero mean and unit variance, wesee that Eq. (26) is completely equivalent in formto Eq. (15). Power-norm formulas for a single FHNoscillator, namely Eqs. (23) and (25), are thus ap-plicable for the mean-field variable from the array,with D2 replaced by D2/N and Γ(DJ , D) replacedby Γ(DJ , D)/

√N .

As N is increased from one, 〈C1〉 increases, butif N is sufficiently large such that Γ(D)/

√N is neg-

ligible compared with the additive term in the de-nominator of 〈C1〉, the normalized power norm isessentially independent of N . Strikingly, in this case〈C1〉 appears to depend on neither DJ nor D, indi-cating an extended aperiodic stochastic resonance.These are features we observe in numerical exper-iments. The array system thus has the capabilityto make use of the noise, regardless of whether itis associated with the incoming signal, internal, ordeliberately added, in such a way that a stochas-tic resonance is induced by which the signal is en-hanced. This may be a natural mechanism for abiological network to deal with, and more impor-tantly, to take advantage of various noise sources inits environment for signal processing.

5. Discussion

Aperiodic stochastic resonance in excitable dynam-ical systems was discovered by Collins et al. and be-lieved to be important for information processing inbiological networks [Collins et al., 1995b; Heneghanet al., 1996; Gailey et al., 1997; Nozaki et al., 1999;Collins et al., 1995a, 1996]. In the typical setting in-vestigated in previous works, a weak signal is passedthrough an array of excitable units and stochas-tic resonance is induced by external or internalnoise [Collins et al., 1995b; Heneghan et al., 1996;

Gailey et al., 1997; Nozaki et al., 1999; Collins et al.,1995a, 1996]. Our interest here is whether stochas-tic resonance can be used to suppress strong noiseassociated with a relatively weak signal, by deliber-ately introducing independent noise sources in thesystem dynamics. The signals considered are thosecommonly encountered in many applications: AM,FM and chaotic signals. The results reported inthis paper indicate that stochastic resonance can beachieved in wide ranges of the noise levels, with a va-riety of system configurations. Aperiodic stochasticresonance with doubly additive noise sources thusappears to be a quite general phenomenon for sub-threshold, excitable systems. This extension fromthe situation of a single noise source is the maincontribution of this paper [Zaikin et al., 2000, 2003;Singh et al., 2003].

While our work is motivated by the problemof antijamming, we wish to point out that with theconfiguration considered in this paper, the approachof stochastic resonance does not appear to be ad-vantageous compared with the traditional methodof filtering. The key reason lies in the thresholdmechanism in the excitable system on which we relyto generate stochastic resonance. Because of this,the output signal consists of spikes and all infor-mation associated with the signal is encoded in theinstantaneous firing rate R(t). It is thus necessaryto use a weighted linear filter (e.g. the Hanning win-dow in our computations) to extract the firing ratefunction. It is quite conceivable that a carefully de-signed linear filter, applied directly to the contam-inated signal, would reduce the jamming. Indeed,in all numerical examples we examined, applying amoving Hanning window to the jammed input sig-nal yields normalized power norm with comparablevalues to those that can be achieved by aperiodicstochastic resonance. Whether this would be truein general is not known. Particularly, the specificchoice of the excitable system to generate stochas-tic resonance may not be a practical way to counterjamming. Nonetheless, we feel that the idea of us-ing stochastic resonance to counter jamming andthe underlying philosophy of using noise to suppressnoise is immensely interesting and worth further in-vestigation, possibly with other choices of nonlineardynamical systems.

At a fundamental level, our work provides aplausible explanation for how biological or neuralnetworks deal with weak signals immersed in noise.Conceivably, input signals to a wide variety of bio-logical networks are noisy. A threshold mechanism

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to generate spike trains, with the help of inter-nal noise that is independent of the input noise(jamming), can result in stochastic resonance andconsequently, efficient extraction and transmissionof the information through the network. We haveshown that this can indeed be achieved for com-mon types of aperiodic signals and for broad-bandor narrow-band input noise.

Acknowledgments

This work was sponsored by AFOSR under GrantsNo. F49620-98-1-0400 and No. F49620-03-1-0290.

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