Fireflies as Role Modelsfor Synchronization in Ad Hoc NetworksAlexander Tyrrell Gunther Auer Christian Bettstetter

DoCoMo Euro-Labs University of Klagenfurt, Mobile Systems Group80687 Munich, Germany 9020 Klagenfurt, Austria

Email: lastname@ docomolab-euro.com Email: Firstname.Lastname@ uni-klu.ac.at

Abstract- Fireflies exhibit a fascinating phenomenon of spon- menstrual periods [6], and people walking next to each othertaneous synchronization that occurs in nature: at dawn, they on the street tend to walk in synchrony.gather on trees and synchronize progressively without relying In Section II, we look at experiments that have been madeon a central entity. The present article' reviews this process . 'by looking at experiments that were made on fireflies and the ian order to comp rehend the synchronizaon mechanism. Inmathematical model of Mirollo and Strogatz [1], which provides particular, we focus on experiments made on fireflies andkey rules to obtaining a synchronized network in a decentralized how this affected their flashing instants, providing interestingmanner. This model is then applied to wireless ad hoc networks. insights to compare nature with mathematical models.To properly apply this model with an accuracy limited only to In biological systems distributed synchronization is com-the propagation delay, a novel synchronization scheme, whichis derived from the original firefly synchronization principle, is monly mo singathe teory of coupl clors[7]. Forpresented, and simulation results are given. fireflies, an oscillator represents the internal clock dictating

when to flash, and upon reception of a pulse from otherI. INTRODUCTION oscillators, this clock is adjusted. Over time, synchronization

In certain parts of South-East Asia alongside riverbanks, emerges, i.e. pulses of different oscillators are transmittedsimultaneously. Synchronization in populations of coupled

male fireflies gather on trees at dawn, and start emitting flashes sillatorsle withinieldof cs.regularly. Over time synchronization emerges from a random A theoretical framework for the convergence to synchronysituation, which makes it seem as though the whole tree isflashing in perfect synchrony. This phenomenon [2] forms illy-meshed networ was p id in [1]. Ts modelan amazing spectacle, and has intrigued scientists for several wil be prsentaeinSchtionI and wil sssshundred years. Over the years, two fundamental questions have otrderobeen studied: Why do fireflies synchronize'? And how do they sYse.been studied: Why do fireflies synchronize?And how do they The Mirollo and Strogatz model of [1] has already been

applied to wireless networks. One of the first papers toThe first question led to many discussions among biologists. apply the firefly synchronization model to wireless networks

In all species of fireflies, emissions of light serves as a meansw.'.' ~~~~~~~~~was[8]. It utilized the characteristic pulse of Ultra-Wide-

of communication that helps female fireflies distinguish males Band (UWB) to emulate the synchronization process of pulse-of its own species: the response of male fireflies to emissions coupled oscillators, and included more realistic effects such asfrom females is different in each species. However it is not channel attenuation and noise.clear why in certain species of fireflies, males synchronize. To lift the restriction of using UWB pulses and apply theSeveral hypothesis exist: Either it could accentuate the males model of [1] to wireless systems, delays need to be taken intorhythm or serve as a noise-reduction mechanism that helps account. Both models of [1] and [8] assume that fireflies formthem identify females [3]. This phenomenon could also enable a fully-meshed network and communicate through pulses.small groups of males to attract more females, and act as a However pulses are hardly considered for communications incooperative scheme [3]. a wireless environment, because they are difficult to detect.

Although the reason behind synchronization is not fully un- To reflect more realistic effects such as message delay andderstood, fireflies are not the only biological system displaying loss, [9] proposed to synchronize using a low-level timestampa synchronized behavior. This emergent pattern is present in on the MAC layer. The principle is similar to the originalheart cells [4], where it provides robustness against the death firefly synchronization scheme, in the way that each nodeof one or more cells, and in neurons, where it enables rapid adjusts its clock when receiving such a timestamp. Becausecomputation [5]. Among humans synchronizatitn the ap cof9titFor example, women living together tend to synchronize their aodteielcs fteMrloadSrgt oe hr

'A preliminary version of this paper was presented at 3rd workshop of the al noe. rnmtsmlaeul.Ti aecetstomnESE-funded scientific program MiNEMA, Leuven, Belgium, February 2006. collisions, which prevails nodes from synchronizing. How-The authors would like to acknowledge MiNEMA for their support. ever, from a physical layer perspective, all nodes transmitting

I1-4244-0463-0/06/$20.OO ©2006 IEEE

synchronously a common word can help a faraway receiver A K 94m102synchronize and communicate with the rest of the network,because it receives the sum of all transmitted powers (known BK ;as the reachback problem). Hence, unlike data transmission,a synchronization process where all nodes transmit the same 82 6 --

word is not affected by collisions in a similar way to flooding.Taking advantage of a common synchronization word also c 950helps the network synchronizing quicker.

Therefore our approach, which also integrates realistic ef-fects such as transmission delays, bases the synchronizationalgorithm entirely on the physical layer. This has several Fig. 1. Experiments by Buck et al. (from [13] with kind permission ofadvantages over [9] such as the fact that collisions are in fact (®)Springer Science and Business Media). Delays are expressed in ms.

a benefit to the scheme and the time to reach a synchronizedstate is shorter because there is no random backoff. Thissynchronization strategy combats transmission and processing response from the firefly occurs at a normal time ofdelays by modifying the intrinsic behavior of a node. 950 is, the signal does not seem to have modified the

II. EXPERIMENTS ON FIREFLIES natural response. This corresponds to a refractory period:Early hypotheses had difficulties explaining the firefly syn- during this time, the potential of the flash regains the

ca"resting" position and no modification of the internalchronization phenomenon. For example, Laurent in 1917 dis-clock ispossiblec

missed what he saw and attributed the phenomenon to the clock iS possible.blinking of his eyelids [10]. Others argued that synchrony . In responsesfBly andsB2, themsignalinhibitsthe responsewas provoked by a single stimulus received by all fireflies ofthe firefly: instead ofemitting lightafter about 960mis,it delays its response until 920ms and 940ms after re-on the tree [11]. However the presence of a leading firefly ora single external factor is easily dismissed by the fact that not cng themsignal Ts succssie flashes ocura 1355omall fireflies can see each other and fireflies gather on trees and an 1635onse, whi aisfia moetan tenural pefri. In response C, the artificial signal occurs 150ms beforeprogressively synchronize. The lack of a proper explanation X

n

until the 1960s is mostly due to a lack of experimental data. othe t flash hin an doe not have day incdenAmongearlhypthese, Rihmon [12]statd in1930 on this flash. This is due to a processing delay in the

whAtameg veary clposetothes actualprchmocess:2 "suped tha central nervous system of a firefly, which is equal to aboutwhat came very close to the actual process: "Suppose that8Os[3.Teeoeteetra inlifune h

in each insect there is an equipment that functions thus: when 8the normal time to flash is nearly attained, incident light on following flash, which is advanced by 150ms. Thus thethe insect hastens the occurrence of the event. In other words, external signal has an excitatory effect on the response

and brings the firefly to flash earlier.if one of the insects is almost ready to flash and sees other g yinsects flash, then it flashes sooner than otherwise. On the The modified behavior of the firefly depended only onforegoing hypothesis, it follows that there may be a tendency the instant of arrival of the external signal. The responsesfor the insects to fall in step and flash synchronously." display both inhibitory (responses B1 and B2) and excitatory

This statement identifies that synchronization among fire- (response C) couplings depending on the instant the externalflies is a self-organized process, and fireflies influence each flash is perceived. Furthermore a refractory period placed afterother: they emit flashes periodically, and in return are receptive emission is also present (response A).to the flashes of other males.

To understand this process, a set of experiments was con- In all cases, the external flash only altered the emissionducted by Buck et al. [13]. These experiments concentrated of one following flash, and in the following period, nodeson the reaction of a firefly to an external signal depending regained their natural period of about one second. Variatingon when this light is received. Naturally a firefly emits light the amplitude of the input signal also yielded similar results.

periodically every 965 ± 90 ms [13], and the external signal For more insights into the phenomenon or firefly synchro-changes this natural period. nization, Chapter 10 in [11] provides a history of studies on

For the experiments, the firefly was put in a dark room fireflies, including early interpretations, and analyzes differentand was restrained from seeing its own flashes. Stimuli were experiments including the one presented in this section.made by guiding 40 ms signals of light from a glow modulator These experiments have helped mathematicians to modellamp into the firefly's eye via a fiber optics. Responses were fireflies. However proving that synchrony occurs when bothrecorded and are shown on Fig. 1. inhibitory and excitatory coupling are present has, to ourFrom Fig. 1, when an external signal is emitted, three knowledge, not been done yet. Therefore we will concentrate

different responses are identified: on the existing model of [1], which considers exhitatory. In response A, the artificial signal occurs only 20ms coupling and no delays, in order to derive a synchronization

after the firefly's spontaneous flash. As the following algorithm suited for wireless systems.

III. FIREFLY SYNCHRONIZATION jfire fire

The internal clock of a firefly, which dictates when a Iflash is emitted, is modeled as an oscillator, and the phase Pt ----------------------------- -----

of this oscillator is modified upon reception of an externalflash. In general this type of oscillator is termed relaxationoscillators, which are not represented by a typical sinusoidalform but rather by a series of pulses. There is no general modeldescribing this class of oscillators, and some examples includeVan der Pol oscillators and integrate-and-fire oscillators [1]. A (a) (b)review of this class of oscillators and their implications canbe found in Chapter 1 in [14]. Fig. 2. Time evolution of the phase function

In the remainder of this paper, we will focus on integrate-and-fire oscillators, which are also termed "pulse-coupledoscillators". Pulse-coupled oscillators interact through discrete The phase function encodes the remaining time until theevents each time they complete an oscillation. The interaction next firing, which corresponds to an emission of light for atakes the form of a pulse that is perceived by neighboring firefly. The goal of the synchronization algorithm is to align alloscillators. This model is used to study biological systems internal counters, so that all nodes agree on a common firingsuch as heart cells, neurons and earthquakes [5]. This section instant. To do so, the phase function needs to be adjusted.describes how time synchronization is achieved in a decentral- The phase function dictates when a pulse or a flash isized fashion in a system of N oscillators. emitted. For a firefly, the natural period T is equal to about

one second. In order to simplify the analysis, the MirolloA. Mathematical Model and Strogatz model does not encompass the clock jitter of

Each oscillatori 1 < i< N, is described by a state variable ±90 ms that is experimentally observed (Fig. 1). Thereforexi, similar to a voltage-like variable in a RC-circuit, and its in the following, we consider that all nodes have the sameevolution and interactions are described by a set of differential dynamics, i.e. clock jitter is considered negligible.equations [15]: For considerations about clock jitter and frequency adjust-

dxi (t) N ment, a different oscillator model was proposed by Ermentrout= d-Xi + Io + S Ji,j Pi (t) (1) in [16], where oscillators have different frequencies.dt

j=1j:Ai B. Synchronization of Pulse-Coupled Oscillators

where 1o controls the period of an uncoupled oscillator andJij determines the coupling strength between oscillators. Mirollo and Strogatz analyzed spontaneous synchronizationWhen xi = xth, where xth is the state variable threshold, an phenomena and also derived a theoretical framework based onoscillator is said to 'fire': at this instant, its state is reset to pulse-coupled oscillators for the convergence of synchrony [1].zero and it emits a pulse, which modifies the state of other When coupled to others, an oscillator is receptive to thecoupled oscillators. The coupling function Pj is defined as a pulses of its neighbors. Coupling between nodes is consideredtrain of emitted pulses: instantaneous, and when a node j (1. j < N) fires at t =Tj,

Pj (t)= 0(t 7 J '"l)(2) i.e. oj (Tj) = 4th, all nodes adjust their phase as follows:m OiQ(Tr))0(4

where T'm represents the nmth firing time of oscillator j and l5i(Tj) =5(Tj) + Aq(5i(Tj)) for i j)d(t) is the Dirac delta function.As (1) is not solvable in closed-form for arbitrary N, [1] Fig. 2(b) plots the time evolution of the phase when receiv-

relies on a discrete approach. To demonstrate that synchrony ing a pulse. To simplify notations, the parameter m in Eq. (2)is always achieved independently of initial conditions, each is dropped, which is coherent with Fig. 1 where the externaloscillator is described by a phase function Oi which linearly signal affected only one following flashing.increments from 0 to a phase threshold Oth and periodically By appropriate selection of AO, a system of N identicalfires every T seconds: oscillators forming a fully-meshed network is able to syn-

chronize their firing instants within a few periods [1]. Thed (t) = X&h (3) phase increment AO is determined by the Phase Response

dt T Curve (PRC). For their mathematical demonstration, MirolloWhen Xi~(t) = bth, a node resets its phase to 0. If not and Strogatz derive that synchronization is obtained whenever

coupled to any other oscillator, it naturally oscillates and fires the firing map xi(t) fQJ(i(t)) is concave up and the returnwith a period equal to T. Fig. 2(a) plots the evolution of map Xi~(t) + Ai\Q75(t)) =gQri(t) + e), where e is thethe phase function during one period when the oscillator is amplitude increment, is its inverse [1]. The resulting operationisolated. q5 (t) + Aq5(q5(t)) =g(f(q5(t))) yields the PRC, and is a

piecewise linear function: instantly as before, but after T('j). If this causes j to reach0the threshold and transmit a pulse, then i also increments itsphase after To ' which can cause it to fire, and so on. If more0

a o= exp(b -E) (5) than two nodes are present in the system, nodes continuouslywith /3 exp(b.e)-1 fire.

exp(b) -1 To avoid this unstable behavior, a refractory period ofwhere b is the dissipation factor. Both factors a and Q duration Trefr is added after firing: after transmitting its pulse,determine the coupling between oscillators, and are identical a node i stays in a refractory state, where $i (t) = 0 andfor all. The threshold ¢bth is normalized to 1. no phase increment is possible, and then goes back into the

It was shown in [1] that if the network is fully-meshed listening state where its phase follows (3). This period is alsoas well as a > 1 and Q > 0 (b > 0, e > 0), the observed in case A in the experimental data of Fig. 1.system always converges, i.e. all oscillators will fire as one, Stability is maintained if echoes are not acknowledged,independently of initial conditions. The time to synchrony is which translates to a condition on Trefr:inversely proportional to a.As it can be observed on Fig. 2(b), detection of a pulse Trefr > 2 Tornax (6)

shortens the current period and causes an oscillator to fire where Tormax] is the maximum propagation delay between twoearly, because Ao(Qi(t)) > O,Voi(t). Compared to the nodes in the network. With the introduction of the refractoryexperimental data of Fig. 1, the Mirollo and Strogatz model state, the accuracy of the synchronization scheme is equal orexhibits only excitatory coupling, and no refractory period is smaller to the maximum propagation delay.present.

However the main features of the experiments are present: B. Transmission Delays in Wireless Systemsnodes do not need to distinguish the source of the synchro- The previous scheme implies that nodes communicatenization pulse, and adjust their current phase upon reception through pulses and that a receiver is able to immediatelyof a pulse. The synchronization scheme relies on the instant detect a single pulse of infinitely small width, and no decodingof arrival of a pulse and receivers adjusting their phases when is done by the receiver. In a wireless environment solitarydetecting this pulse.Applctied tois wirelses systems, this hastheadvantagethatpulses are hardly used alone as they are virtually impossibleApplied to wireless systems, this has the advantage that

to deet.oeraitclyasycrnzto odisd.. . .....................to detect. More realistically a synchronization word iS used.interference and collision is not observed, because a receiver In the original firefly synchronization scheme, nodes do notdoes not need to identify the source of emission. Furthermore need to distinguish between emitters. Therefore a commontwo pulses emitted simultaneously can superimpose construc- synchronization word is broadcasted by all nodes when firing.tively, which helps a faraway receiver synchronize. This type The synchronization word can be chosen from a variety ofof spatial averaging has been shown to beneficially bound the schemes: it can correspond to a sequence of pulses, a Pseudo-synchronization accuracy to a constant, making the algorithm Noise (PN) sequence [19] or the 802.11 preamble [20]. In allscalable with respect to the number of nodes [17]. these cases, the synchronization word has a certain duration

IV. APPLICATION TO WIRELESS SYSTEMS TTX.

In wireless systems, different delays need to be taken into During the transmission of this word, a node is unable to

account. The algorithm needs to be modified to account receive. This constraint is due to limitations on the RadioFrequency (RF) part of transceivers.for propagation delays, so that the system remains stable. qu y (R) p

Moreover long synchronization words need to be considered After the message has propagated and been received by nodef, some processing time is required to correctly declare that a

decoding delay to properly identify that a synchronization synchronization message has been received. This results in amessage was transmitted. As these delays affect the achievable decoding delay Tdeclaccuracy, we will modify the intrinsic behavior of a node, so Altogether, four delays need to be taken into account tothat high accuracy is regained, and evaluate the novel scheme model the synchronization strategy to a wireless network:through simulations. . T(')j: Propagation delay - time taken for a burst to

propagate from the emitting to the receiving node. Thistime is proportional to the distance between two nodes.

Within the field of nonlinear oscillators, it is known that . TTX: Transmitting delay - length of the synchronizationwhen a delay occurs, even if it is constant between all nodes, word. A node cannot receive during this time.then a system of pulse-coupled oscillators becomes unstable, . Tdec: Decoding delay - time taken by the receiver to prop-and is never able to synchronize [18]. In wireless systems, even erly identify the emissions of a synchronization word.when considering communication through pulses, a propaga- This time needs to be overestimated to account for thetion delay dependent on the distance between nodes occurs. slowest receiver.

If considering a propagation delay between two nodes i . Trefr: Refractory delay -time necessary after transmittingand j, denoted by T(t'2, then the pulse of i influences j not to maintain stability.

In this paper, all propagation delays are considered neg- the receiver. The output of the correlator produces a large peakligible in order to focus our analysis on the effects of the that occurs at exactly Oj =Tj + TTX [19]. In order to properlytransmission time and the decoding delay on the original identify this peak, node i increments its phase Tdec= 0.12 Tscheme. This assumption is valid when considering Wireless after the peak, i.e. at O. + Tdec.LAN settings in an ad hoc scenario. Typically the maximum These delays are the most significant difference from theoperation range is 50 m, which limits the propagation delay Mirollo and Strogatz model, which assumes no propagationto Tonax] = 50m 0.17,us. In comparison, the preamble of delay, an infinitely short transmission time and no decodingan 802.11 frame, which can be used as the synchronization delay [1]. The total delay is defined by:word, has a duration of TTX= 8 ,us [20]. Noise at the receiverand channel gain are also neglected to emphasize the effect of Tdel TTX + Tdec (8)delays on the original scheme. When several nodes transmit, if the PN sequence has low

For simplicity, transmission and decoding delays TTX and auto-correlation properties such as a Gold sequence [21], thenTdec are the same for all nodes. To assume that the decoding distinct peaks appear at the output of the correlator Ai exactlydelay is the same for all nodes, slow receivers need to Tdel after a node has fired [19]. Thus several transmissionsbe accounted for. Therefore, in practice, this delay should are distinguishable and several phase increments occur. Ifbe overestimated, so that all nodes increment their phases nodes fire and transmit synchronously, peaks superimposesimultaneously upon proper reception of a synchronization constructively.word. The total delay Tdel represents the inherent time difference

To illustrate the impact of these delays, we choose the between the beginning of the transmission of a synchroniza-synchronization word to be a PN sequence. When node j tion burst and its successful reception. From the theory offires, it enters a transmit state: a synchronization word x(t) coupled oscillators, it is known that delays impact heavily onpasses through a shaping filter and starts being emitted. It then the synchronization process and its stability [22]. For pulse-propagates through a channel h(t) before being received by coupled oscillators, it has been shown that the system becomesnode i, which collects the incoming signal through a matched unstable in the presence of delays [23], but this model doesfilter and samples it [19]. As x(t) is the same for all nodes, not account for the transmission and refractory periods, wherenode i detects it by correlating the received signal with the no coupling is possible. In our model, as coupling is onlyknown message. The output of the correlation detector is given possible when nodes are in a listening state, stability issuesby [19]: are prevented by adjusting Trefr, and proper choice for Trefr is

Ai(t) = x(-t) * yi(t) (7) done through simulations.

where y. (t) is the incoming signal at node i' and * denotes the .In any case, due to Tdel and to the fact that during transmis-sion it is not possible to receive, a deafness of duration Tdel

convoltio sypteratorel.described previously, Fig. 3 plotsappears in which nodes cannot listen to the network. WithinFrm hesyte mde dscibd reiosl, ig 3plt this deafness, no mutual coupling between nodes can occur,the output of the correlation detector Ati (t) and the correspond-

ti efes omta opigbtenndscnocrthe output ofuncthe correlaionde tectoriAg (t) anriod.tecrsod which implies that the attainable synchronization accuracy is

lower bounded by Tdel. For transmission techniques where the100 time for one symbol block, TTX, cannot be assimilated as a80 - T pulse, such an accuracy is clearly unacceptable. Therefore,

T de there is a need to modify the synchronization strategy.60 - Tx

40- C. Compensating Delays20 - In order to regain high accuracy, we propose to combato_ _ ll transmission delays by modifying the intrinsic behavior of0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 a node: after firing, a node delays its transmission of theIt. timel/T

J 01j+Tdec synchronization word. This approach is similar to the one1_ observed in the experiments of fireflies: in response C of

0.8 * IA4O(Oi(O.+T Fig. 1, the advance in flashing is not effective immediately0.6 iJ dec'1 upon reception of a signal, but occurs in the following period.

0.4o1s1X___de 2The waiting delay is chosen to be:0.2 --____ _Twajt = T-Tdel = T -(TTx +Tdec) (9)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 where T denotes the synchronization period. With this ap-time / T

proach, receivers increment their phases exactly T secondsFig. 3. Example of the output of the correlator when considering noise after a transmitter fired.

This scheme modifies the natural oscillatory period of aOn Fig. 3 a synchronization sequence of duration TTX node, which is now equal to 2 .T. Nodes are coupled only

0.2. T is transmitted by node j at t =Tr, and is collected by if they can hear each other during TRX, which is the time

during which the phase function will linearly increments over D. Simulation Resultstime. This time is reduced by the waiting, transmitting, and Deriving a thorough mathematical demonstration that syn-refractory delays, and is now equal to: chrony is reached when each node follows the simple rules of

TRX 2 T (Twwt + TTx + Trefr) T + Tdec Trefr I10) waiting before transmitting is a task of formidable difficulty,TRx= 2 .T - (T wait + TTx + Trefr) T + Tdec - Trefr (10) as proving synchrony when no delays are present is already

difficult. The behavior of the system lies within the field ofToss the modife beh of a node , Frepre- nonlinear dynamics, and a complete description and analysissents the four successive states of a node: wait, TransmiT, does not seem easil reachable. Therefore we rel on simula-refr and listen when lWi 0.75 - T, lTT= 0.2 - TlyTdec = 0.05 . T, Trefr = 0.2 . T and TRX = 0.85 T. A tion results to evaluate our synchronization scheme.

node i r e mTo verify the validity of the synchronization scheme, MonteCarlo simulations are carried out. Fig. 5 plots the synchroniza-diagram linearly over time and counterclockwise. Using this tion rate for several values of TTX, Trefr and a. For simulations,diagram, N nodes can be represented on the same circle, each period T is decomposed into 1500 steps, and at each

which helps analyzing the dynamic evolution of the system. step, state and interactions of each node are evaluated. TheOne full rotation of a marker corresponds to a period 2 T. decoding delay is fixed to Tdec .0. I T. Nodes are able to

perfectly distinguish each transmitted synchronization word,e.g. by using a long Gold sequence as the synchronizationword. The initial conditions correspond to the case whereall nodes have randomly distributed state variable, i.e. eachnode is assumed to be active starting with a random phase

Tdec pfire oi (0). Successful synchronization is declared if two groups<-l g | l | | l_@gF 01~o oscillators firing T seconcis apart form, anci the synchrony

rate iS defined as the number of successful synchronizationsafter 50 -T over the number of realizations of initial conditions,

listen which iS set to lUUU. Initial conditions correspond to the worstcase scenario where initially all state variables are randomlydistributed around the phase diagram of Fig. 4.

t lO-Aiii ~~~~~~~~~~~~ ~~100 ...................

Fig. 4. Phase diagram of a system of N nodes following the novel 80 ..................,synchronization strategy. Two groups of oscillators form, spaced exactly T \apart. a)

..

For a system of N oscillators, all firings instants are initially ° 40 \ /;.jrandomly distributed over a period of 2 T, i.e. all markers a = 13 Trefr = 0.2 Tare randomly uniformly distributed around the circular state OC=1.3,Trefr=0.4Tmachine representation. Each oscillator follows the same rules 2 =1.4, Trefr =0.2 Tof waiting before transmitting. When a message is successfully a =.4,Tf =04Treceived during listen, the marker representing this node 0.1 0.2 0.3 T /T 0.6 0.7 0.8

abruptly shifts its position towards the diametrically opposedstate on the representation of Fig. 4. Over time the oscillators Fig. 5. Synchronization rate when TTX variessplit into two groups diametrically opposed on the statemachine representation, each group firing T seconds apart and For Trefr = 0.2 T, a = 1.4 (high coupling), and low valueshelping each other to synchronize. Therefore T is still used as of TTX, the system is unstable and synchrony is not alwaysthe reference synchronization period. reached. In this case, clusters of oscillators form and oscillateThe formation of two groups is a necessary requirement for more rapidly (phase-locking mechanism). This phenomenon

maintaining high accuracy, because nodes that transmit almost tends to disappear when the transmitting time increases, but itsimultaneously cannot hear each other (deafness while trans- is never completely resolved and the synchrony rate is nevermitting). Therefore each group helps the other to synchronize higher than 90%.by transmitting T seconds after the other. When increasing the refractory time Trefr to 0.4 -T, synchro-

With this new transmitting strategy the accuracy of syn- nization is always obtained for a large range of TTX. Thus,chronization is no longer limited by Tdel. Successful synchro- a relatively long refractory time is preferable. For TTX >nization is therefore declared when firing instants are spread 0.6 .T, the synchrony rate becomes lower than 90% andover a time interval that is equal or smaller than the maximum drops very rapidly. This can be explained by the proportionpropagation delay. of the listening time, which becomes small compared to the

transmitting time. This makes it difficult for nodes to hear may result in a "deafness effect", where a local group ofanother and reach a consensus. nodes all transmit at similar time instants, which implies

Fig. 6 plots the mean time taken by a system of 30 that these nodes cannot hear each other. While for a fully-oscillators to synchronize. The duration of a time slot T is meshed network the probability that all local nodes are withinused as reference. one group tends to zero, the deafness effect causes severe

problems for meshed networks, and is a suitable topic for24 - = 1.3, Tf =0.2T further research.

LC22 = 1.3, Tr0 =0.4TREFERENCES

u=l1.4,T =0.2TC 20'refr20 1.4,Trefr =04T [1] R. Mirollo and S. Strogatz, "Synchronization of pulse-coupled biological

0 18 oscillators," SIAM J. Appl. Math, vol. 50, no. 6, pp. 1645-1662, Dec.cn 1990.

16 [2] D. Attenborough, "BBC Trials of Life, "Talking to Strangers"," 1990.E 14 [3] D. Otte and J. Smiley, "Synchrony in texas fireflies with a consideration

12 t4k*ofmale interaction models," Biology of Behaviour, vol. 2, pp. 143-158,

[4] C. Peskin, Mathematical Aspects of Heart Physiology. New York:10 i Courant Institute of Mathematical Sciences, 1975.

8 [5] J. Hopfield and A. Herz, "Rapid local synchronization of action po-0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 tentials: Toward computation with coupled integrate-and-fire neurons,"Tx T Proc. National Academy of Sciences USA 92, vol. 92, pp. 6655-6662,

Fig. 6. Mean time to synchrony when TTX varies July 1995.[6] S. Strogatz and I. Stewart, "Coupled oscillators and biological synchro-

nization," Scientific American, vol. 269, no. 6, pp. 68-74, Dec. 1993.For Trefr = 0.2 T, the mean time to synchrony T is lower [7] A. Winfree, "Biological rhythms and the behavior of populations of

than for Tref = 0.4 * T. However synchrony rates are lower coupled oscillators," J. Theor Biol., vol. 16, no. 1, pp. 15 - 42, Julyand it is preferable to ensure that synchrony will be reached 1967.

[8] Y-W. Hong and A. Scaglione, "Time synchronization and reach-backalthough it may take one or two periods more in average. communications with pulse-coupled oscillators for UWB wireless ad

For Trefr = 0.4 T, T is lower when ag = 1.4 than hoc networks," in Proc. IEEE Conference on Ultra Wideband Systemswith a

e1. .4 T,

This Tlies lowit theo nal M .4rtoan and Technologies 2003, Nov. 2003, pp. 190-194.with ag = 1.2. This complies with the original M[rollo and 19] G. Werner-Allen, G. Tewari, A. Patel, M. Welsh, and R. Nagpal, "Firefly-Strogatz model where the mean time to synchrony is inversely inspired sensor network synchronicity with realistic radio effects," inproportional to the coupling factor a. Proc. SenSys '05, Nov. 2005.

[10] S. Strogatz, Sync: The Emerging Science of Spontaneous Order, 1st ed.

V. CONCLUSION Hyperion, 2003.[11] S. Camazine, J.-L. Deneubourg, N. R. Franks, J. Sneyd, G. Theraulaz,

Fireflies provide an amazing spectable with their ability and E. Bonabeau, Self-Organization in Biological Systems. Princeton,2001.

to synchronize using simple rules: each node maintains an201

[12] C. Richmond, "Fireflies flashing in unison," Science, vol. 71, pp. 537-internal clock dictating when to emit, and in return, this clock 538, 1930.is adjusted when receiving. These synchronization rules are [13] J. Buck, E. Buck, J. Case, and F. Hanson, "Control of flashing in

partiull sfireflies. V. Pacemaker synchronization in pteroptyx cribellata," Journalparticularly simple and well suited for a deployment in ad of Comparative Physiology A, vol. 144, pp. 630-633, 1981.hoc networks. However they are not directly applicable when [14] G. M. R. Avila, "Synchronization phenomena in light-controlled oscil-accounting for transmission delays and the fact that a node lators," Ph.D. dissertation, Universit6 Libre de Bruxelles, 2004.cannotreceive and transmit simultaneously. [15] S. R. Campbell, D. L. Wang, and C. Jayaprakash, "Synchrony and de-

cannot synchrony in integrate-and-fire oscillators," Neural Computation, vol. 11,To regain a level of accuracy that is upper bounded by no. 7, pp. 1595-1619, 1999.

the propagation delay, the intrinsic behavior of nodes was [16] B. Ermentrout, "An adaptive model for synchrony in the firefly pteroptyxmalaccae," Journal of Mathematical Biology, vol. 29, no. 6, pp. 571-modified to compensate for transmission and decoding delay. 585, 1991.

Thanks to this modification, communication through pulses is [17] A. Hu and S. Servetto, "On the scalability of cooperative time synchro-no longer required, and accurate synchrony of oscillators is nization in pulse-connected networks," IEEE Transactions on Informa-

possible.The simplicity and generality of the synchronization [ tion Theory, vol. 52, pp. 2725-2748, June 2006.possible. The simplicity and generality of the synchronization [8] U. Ernst, K. Pawelzik, and T. Geisel, "Synchronization induced byscheme makes its implementation very appealing. If all nodes temporal delays in pulse-coupled oscillators," Phys. Rev. Lett., vol. 74,cooperate, synchrony can be reached within 15 periods. Once no. 9, pp. 1570-1573, Feb. 1995.nodes have agreed on a common time scale, they are then [19] J. Proakis, Digital Communications, 3rd ed. McGraw-Hill, 1995.

[20] Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY)able to use the full time slot to communicate in a synchronous Specifications, IEEE Std. 802.1la, 1999.manner. [21] R. Gold, "Optimal binary sequences for spread spectrum multiplexing,"

While the rules of the novel synchronization strategy are Information Theory, IEEE Transactions on, vol. 13, no. 4, pp. 619-621,Oct. 1967.

simple, the waiting time imposes additional delays, raising [22] M. Yeung and S. Strogatz, "Time delay in the Kuramoto model ofthe constraints to achieving convergence and stability. In a coupled oscillators," Phys. Rev. Lett., vol. 82, no. 3, pp. 648 -651,not fully-meshed multi-hop network, however, the stain [23] M. Timme, F. Wolf, and T. Geisel, "Prevalence of unstable attractors inis more complicated. Due to the fact that two groups are networks of pulse-coupled oscillators," Phys. Rev. Lett., vol. 89, no. 15,established, formations might occur where one node is sur- Oct. 2002.rounded by nodes which are all in the same group. This