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e c o l o g i c a l m o d e l l i n g 2 1 3 ( 2 0 0 8 ) 105–118

avai lab le at www.sc iencedi rec t .com

journa l homepage: www.e lsev ier .com/ locate /eco lmodel

horus model of the synchronizing bushcricketpecies Mecopoda elongata

anfred Hartbauer ∗

arl-Franzens University Graz, Department of Zoology, 8010 Graz, Universitatsplatz 2, Austria

r t i c l e i n f o

rticle history:

eceived 13 July 2007

eceived in revised form

5 November 2007

ccepted 20 November 2007

ublished on line 3 January 2008

eywords:horusing

oupled oscillators

a b s t r a c t

Males of the Malaysian bushcricket species Mecopoda elongata synchronize or alternate their

cyclically occurring song elements (chirps) in a duet. The acoustic interaction of males inter-

acting in a duet was successfully simulated by means of mutually coupled song oscillators,

which respond to a disturbance by a phase shift which is known from the phase response

curves (PRCs) of real males. However, little is known about the acoustic interaction of males

in a complex chorus situation. Therefore, the aim of the current study was to extend the

duet model to a chorus taking into account an inhomogeneous spacing of agents and a

natural variability of oscillator properties. This chorus model was used to study oscillator

coupling in a chorus consisting of 15 agents. Since such a computer model allows one to

simulate chorus manipulations that far exceed the possibilities of behaviour experiments,

cological modeling

ulti-agent simulation

nsect swarm

ynchronization

the following scenarios were simulated: modification of chorus density, sensory bias dur-

ing sound production, selective attention to only a subset of neighbors and males joining

or leaving a chorus. Simulation results allow one to draw conclusions about the chorus-

ing behavior of males in a real chorus and about signaler and receiver aspects influencing

and

chorusing formation

. Introduction

he synchronization of communication signals in aggrega-ions of many individuals has attracted the interest of manyesearchers throughout the last century. Near perfect syn-hrony of light signals can be found in some species ofast-Asian fireflies (Buck, 1938; Buck and Buck, 1976; Buck,988). But also acoustic advertisement signals often show aigh degree of synchrony. Such synchrony has been seen inhe cricket Oecanthus fultoni, in the bushcricket species Neo-onocephalus spiza, Mecopoda elongata, Pterophylla camelliflora,latycleis intermedia, Neoconocephalus caudellianus in the genusawanaphila (Tettigoniidae: Zaprochilinae) and in the period-

cally occurring cicada Magicicada cassini and in the anuran

pecies Smilisca sila (Alexander and Moore, 1958; Greenfield,994; Hartbauer et al., 2005; Mason and Bailey, 1998; Sismondo,990; Walker, 1969).

∗ Tel.: +43 316 380 8751; fax: +43 316 380 9875.E-mail address: Manfred.Hartbauer@uni-graz.at.

304-3800/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.ecolmodel.2007.11.010

mate choice.

© 2007 Elsevier B.V. All rights reserved.

Synchrony within a population of biological oscillatorsis achieved by mutual entrainment. Similar mechanisms,responsible for the establishment of synchrony, were found inthe pacemaker cells of the heart, circadian pacemakers, thehippocampus, insulin-secreting cells of the pancreas and themenstrual periods of women (for an overview see: (Winfree,1967; Winfree, 1980)). Much theoretical work is based on thePeskin model (Peskin, 1975) of “integrate and fire” oscillatorsin which the interaction between two oscillators are eithersmooth or pulse like (Mirollo and Strogatz, 1990; Goel andErmentrout, 2002). As Peskin (1975) already conjectured andMirollo and Strogatz (1990) proved, for almost all initialconditions a steady-state evolves among a population of

homogeneous “all to all” coupled oscillators in which alloscillators fire in synchrony. However, little work has beendone so far considering the behavior of a population of biolog-ical oscillators with different natural frequencies and other

i n g

106 e c o l o g i c a l m o d e l l

geometries different from “all to all” coupling or ring-likeformations.

In a natural chorus situation the distances between sig-nalers show a high degree of variability (e.g. Nityananda etal., 2007; Romer and Bailey, 1986; Thiele and Bailey, 1980) andthe coupling strength of song oscillators strongly dependson the distance between individuals. This physical constrainttogether with an intrinsic variability of signalers with regardto solo chirp periods (solo CPs), signal level, cycle-to-cyclefluctuations and variability of phase change in response toa stimulus might result in an oscillator coupling which neverresults in global stable synchrony.

Males of the Malaysian bushcricket species M. elongataperiodically display advertisement signals (called chirps)at a period of ∼2 s. Males in a duet mutually couple theirsong oscillators by signals either displayed in synchronyor alternation. So far almost all investigations addressingthe establishment of synchronization in this species wereperformed in real or simulated male duets (Sismondo, 1990;Hartbauer et al., 2005) neglecting the complexity that existsin a chorus situation. Multi-agent based simulations thatconsider transmission effects and a natural variability ofsignalers therefore constitute a useful approach for theinvestigation of chorusing in this species.

A so-called “inhibitory-resetting oscillator” (see Section4) is often used to describe the underlying neural oscillatorresponsible for song oscillator coupling (Greenfield, 1994).However, the acoustic interaction of M. elongata males cannotbe successfully described by this model, because a distur-bance only affects the disturbed cycle but not the cyclesubsequent to a disturbance in this species (Hartbauer et al.,2005). In addition, depending on stimulus phase the oscillatorcycle is either shortened or prolonged. In order to accountfor these special oscillator properties in the current chorusmodel, agents mutually couple their song oscillators basedon known phase response curves (PRCs) obtained from realmales. This constitutes a more realistic modeling approachin comparison to an “inhibitory-resetting oscillator” model,which relies on several unknown variables. This modelingapproach was successful in simulating firefly synchrony(Ermentrout, 1991) and in the simulation of the acousticinteraction of a male duet of a chirping Indian M. elongataspecies (Nityananda and Balakrishnan, 2007).

A chorus model not only allows one to study oscillator cou-pling in a reasonably realistic simulation of a chorus but italso allows one to investigate the influence of several receiveraspects on chorusing. Furthermore, the robustness of chorus-ing to influences arising from different chorus densities andagents that acoustically join or leave a chorus can also bestudied.

Males of a chirping Indian Mecopoda species (Nityananda etal., 2007) and males of the bushcricket species Neoconocephalusnebrascensis (Meixner and Shaw, 1979) space themselves out ina way which preserves an average inter-male distance of about5–6 m. The mean inter-male distance for M. elongata in the fieldis currently unknown and may vary in aggregations of differ-

ent bushcricket species substantially (e.g. Thiele and Bailey,1980; Romer and Bailey, 1986). Therefore, agents in the cho-rus model exhibit an uneven inter-male spacing. Simulationsperformed with different chorus densities offer the possibil-

2 1 3 ( 2 0 0 8 ) 105–118

ity to elucidate a minimum inter-male distance necessary forthe establishment of steady-state oscillator coupling in a M.elongata chorus.

Apart from male density, chorus composition may alsoaffect chorusing in a M. elongata chorus. Chorus attendance ofa single male in a chorus of M. elongata is of limited duration(about 30 min) and individual males acoustically join or leavethe chorus from time to time. This can be easily simulated inthe computer model and allows one to draw inferences aboutthe robustness of steady-state oscillator coupling to changesof chorus composition.

Some continuously signaling insects are known to hearwhile they sing. However, the rate of syllables in a chirpof M. elongata is quite high (125 Hz) and it is still unknownwhether males hear while they produce a sound. This mayresult in a sensory bias that affects oscillator coupling and inconsequence chorusing behavior. Simulations were thereforeperformed modeling a shift of the hearing threshold to a levelwhich exceeds the current chirp level by 3 dB.

The attention of a male in a chorus may be influenced notonly by self-generated sound but also by selective attentionto a subset of neighbors in close proximity. In such a situa-tion selective attention to agents close to a focal agent resultsin a higher coupling strength of their song oscillators com-pared to more distant agents. This however may neglect theprobable existence of a selective response to only a few near-est neighbors as was found in insect choruses of Ligurotettixplanum, Ligurotettix coquilletti, Ephippiger ephippiger and Neocono-cephalus spiza (Snedden et al., 1998; Greenfield and Snedden,2003) and also in an anuran chorus (Physalaemus pustulosus(Greenfield and Rand, 2000)). In the current chorus model,a selective response of agents can be simulated by forcingagents to ignore all signals except those of nearest neighbors.

All these chorus simulations may lead to a betterunderstanding of signaler and receiver aspects influencingchorusing in M. elongata. Results will be discussed in the con-text of insect chorusing and female choice.

2. Methods

2.1. Biological background

Males of the species Mecopoda elongata (M. elongata) exhibita chirp period (CP) of 1.75 to 2.3 s in songs displayed inisolation. Within such a song bout only little chirp-to-chirpvariability is found and therefore fast and slow singingmales can be discerned. Each chirp consists of syllables ofincreasing amplitude and are regarded as single oscillatoryevents displayed once in each cycle. In the current modelagents couple their oscillators using acoustic signals whichexhibit an amplitude profile that is similar to the increaseof syllable amplitudes within a conspecific chirp (Fig. 1A).Individual agents were modeled on the basis of signal oscil-lators exhibiting the same properties as was found in 11 realM. elongata males. The phase response curves (PRCs) of these

individual males were obtained from playback experimentsusing a conspecific stimulus, which was broadcast at 50, 60and 70 dB SPL at random phases in the cycle of the songoscillator. All PRCs were already published in Hartbauer et al.

e c o l o g i c a l m o d e l l i n g 2 1

Fig. 1 – Chirp signal of a M. elongata male. The oscillogramof a male chirp recorded at a distance of 1 m consists ofsyllables of increasing amplitude (A). (B) Represents thesame chirp as shown in (A) but after echo processingperformed in CoolEdit (Syntrillium Inc.) using a delay timeof 8 ms and 70% decay. The amplitude profile of a simulatedchirp is shown in (C) (black line). In response to theperception of a stimulus in the final phase of the oscillatorcl

(asaSam

2

TmaI3asi

ycle, the simulated chirp was found to be shortened andoud syllables are brought forward in time (grey line).

2005). Most simulation parameters as well as their variabilityre known from behavioral experiments. These include theolo CP of individual males, cycle change after presenting ancoustic stimulus, chirp level, chirp duration and the drop ofPL over distance. In order to develop a realistic chorus modelll these parameters were fully implemented in the chorusodel considering their natural variability.

.2. Model description

he acoustic interaction between males (agents), whichutually couple their song oscillators on the basis of

coustic signals (chirps), was modeled in the JAVA (Sunnc.) based multi-agent simulation environment Netlogo

.1 (http://ccl.northwestern.edu/netlogo). Although Netlogollows one to increase the number of agents up to 1000, aimulated chorus in the current study always consisted of 15ndividuals. This size covers a male aggregation large enough

3 ( 2 0 0 8 ) 105–118 107

to study principal chorusing effects. Within a simulation runof at least 5000–7000 simulation steps in duration agents kept afixed position in the chorus. Each simulation step in the modelrefers to a 10 ms time period.

2.3. Simulated chirp signals

Conspecific chirps of M. elongata are characterized by a steadyincrease of syllable level including brief pauses between sub-sequent syllables (Fig. 1A). Due to reverberations occurring inthe acoustic transmission channel, chirps loose their charac-teristic temporal pattern (Fig. 1B). The average chirp durationof 12 individual males was found to be 273 ± 28.6 ms. In themodel, a longer average chirp duration of 31 ± 2 simulationsteps (1 step = 10 ms) was chosen. This compensates for a pro-longation of a chirp signal as a result of echoes added to theoriginal signal in the transmission channel.

The average amplitude of loud syllables in a chirp of asinging male recorded at a distance of 1 m corresponds tothe amplitude of a 4 kHz continuous sine wave of approxi-mately 86 dB SPL. The loudest syllable of simulated chirps inthe model reached the same level. In the final phase of theoscillator cycle the remaining cycle length (cl) becomes equalor smaller than the duration of a chirp (chirp dur). Then eachagent starts to produce a chirp of increasing amplitude mod-eled after Eq. (1). The amplitude of a chirp starts at 40 dB SPLand ends at 86 dB SPL as a function of the remaining cyclelength.

s level =(

86 − 40chirp dur

)∗ (chirp dur − cl) + 40 (1)

Chirp dur denotes the current chirp duration of an agent andfluctuates on a chirp-to-chirp basis (31 ± 2 steps). The SPL atsource is given as s level. Loud syllables at the end of a malechirp are of about 20 ms in duration corresponding to two timesteps in the model. Therefore, after every third simulation stepsignal level was decreased by 7 dB. The resulting amplitudeprofile of a simulated chirp (Fig. 1C, black line) roughly cov-ers the envelope of a chirp considering transmission effectsobliterating the characteristic temporal syllable structure.

The average maximum chirp level was on average 86 dB SPLand chirp-to-chirp level variability was taken from a Gaussiandistribution with a S.D. of 3 dB.

2.4. Signal oscillator properties

Agents in the model mutually coupled their oscillators bychirp signals that are generated in the final phase of the songoscillator. The relation between stimulus phase (time periodbetween the last signal and the stimulus/solo CP) and the nor-malized response phase (length of the disturbed cycle/soloCP) are displayed in a PRC. In the model, the resulting phaseshift after perception of a stimulus was calculated from PRCsthat were obtained in playback experiments of 11 individualmales.

The left branch of a PRC refers to responses after stimula-tions with a conspecific chirp occurring shortly after the focalmale’s chirp and was modeled using second- or third-orderpolynomials. Stimuli up to a phase of about 0.7 (transition

108 e c o l o g i c a l m o d e l l i n g

Fig. 2 – Phase response curve and distribution of solo chirpperiods. (A) Shows an example of a PRC (male #1) obtainedin playback experiments with a conspecific chirp presentedat a stimulus level of 70 dB SPL. In such a plot the stimulusphase was plotted against the normalized response phase(length of the disturbed cycle/solo CP). The data shown inPRCs were fitted with polynomials or linear functions(curves in (A)). These equations (see appendix) were usedto calculate the change in phase after the perception ofstimulations. The distribution of solo CP (mean: 1.97 s)within a sample of real males is shown in (B). A similardistribution of solo CP of agents was simulated in the

s

and the maximum level of the perceived stimulus. The modelallowed that more than one suprathreshold stimulus could

model chorus (C). A total of 57 males/agents contributed tothe data shown in (B) and in (C).

phase) resulted in a prolongation of the disturbed cycle. Incontrast, a stimulation that occurred after the transition phaseresulted in a shortening of the disturbed cycle length (anexample is shown in Fig. 2A). Linear equations or first-order

2 1 3 ( 2 0 0 8 ) 105–118

polynomials were used to model the right branch of PRCs refer-ring to responses to stimulations occurring late in the cycle.The data for PRCs were obtained by measuring stimulus timesand response times at the end of chirps. In the appendix allfitting equations describing the PRCs of all 11 males are listedtogether with individual transition phases.

2.4.1. Simulation of oscillatory behaviorEach agent in the model exhibits its own solo chirp period(solo CP) in addition to an individual song oscillator defined bya PRC. If not stated differently all of these individual oscillatorproperties were assigned to an agent at the beginning ofeach simulation run, whereby the PRCs and the appropriatetransition phases were randomly chosen from a total of 11PRCs. This was done in order to investigate the influenceon chorusing arising from different chorus compositions.Individual solo CPs were taken from a normal distributionwith a mean of 2 s (200 simulation steps) and a S.D. of 70 ms(7 simulation steps) (see example in Fig. 2C). This results in adistribution of solo CPs similar to that obtained from a sampleof real males (Fig. 2B).

The cycle length of real males entrained to a conspecificstimulus with a period of 2 s exhibits some variability. Agentsin the model exhibited a similar degree of cycle-to-cycle vari-ability that was simulated according to Eq. (2).

Tc = int(T0 + (T0 ∗ rand(0.02))) (2)

T0 denotes the average solo CP of an agent and Tc rep-resents the solo CP of the current cycle. “rand” refers to aGaussian distributed variable with a S.D. of 0.02 simulationsteps. According to Eq. (2) agents with a solo CP length of 200steps will therefore exhibit a cycle-to-cycle variability of ±4simulation steps.

2.4.2. Simulation logicIn every simulation step (corresponding to 10 ms), each agentexecutes signaler and receiver rules (summarized in Fig. 3).In the first step the remaining oscillator cycle length (cl) isdecremented by one. In the terminal phase of the oscilla-tor cycle (if cl ≤ chirp dur) each agent starts to generate achirp signal, which will only be detected by agents (in theactive space of a signaler) if the perceived stimulus level(level) exceeds hearing threshold (thresh = 48 dB SPL). Signalsoverlapping in time will be summed up at the receiver site(sum level). As soon as the perceived level of the stimulussignal drops below hearing threshold and the stimulus wascontinuously present within at least 5 simulation steps,a perturbation of the oscillator cycle was calculated. Theend of a stimulus was automatically recognized when totalstimulus duration exceeded 35 simulation steps. The changeof oscillator phase depends on the phase of perturbation (˚ )

perturb the oscillator in a single cycle. This was necessarybecause playback experiments revealed that both conspecificsignals falling within a single oscillator cycle contributed tothe perturbation of the oscillator.

e c o l o g i c a l m o d e l l i n g 2 1 3 ( 2 0 0 8 ) 105–118 109

Fig. 3 – Signaler and receiver rules. In every simulation step all agents follow signaler and receiver rules independentlyfrom each other. For details see Section 2. cl = remaining cycle length; chip dur = chirp duration; s level = stimulus level;s = cu

2

Aiaa

a

T4as

um level = perceived stimulus level; T0 = average solo CP; Tc

.5. Signal propagation

signal will exceed hearing threshold (thresh) of receiversn a circle around the signaler with radius aspace calculatedccording to Eq. (3). The area of this circle is defined as thective space of a signal.

space = 10(s level−thresh)/20 (3)

hresh refers to the hearing threshold, which was fixed at8 dB SPL. Almost all agents were found in the active space ofsignaler when the chorus was modeled with the parameter

et which is given in Table 1.

Table 1 – Standard simulation parameters

Parameter

Chorus sizeHearing thresholdMean chirp levelS.D. chirp levelMean free-run chirp periodS.D. free-run periodMean chirp durationS.D. chirp durationS.D. left branch of the PRCS.D. right branch of the PRCMinimum distance between agents

Agents start at a random phase in their oscillator cycleAgents hear during chirp productionSelective attention to local neighbors

rrent solo CP.

The level of a signal attenuates with distance because ofspherical spreading and was calculated according to Eq. (4).

level = s level − (20 ∗ log10 distance) (4)

The delay of a signal traveling from one agent to anotherwas calculated assuming a transmission velocity of 340 m/s.Each agent services a list of delays and appropriate levelsof detected signals (level ) and considers these data for sig-

i

nal level summation. Signals overlapping in time amplify thesummed stimulus level (sum level) at the receiver. This wascalculated according to Eq. (5) whereby the signals belong-ing to different signalers were treated as incoherent sound

Value Unit

15 N48 dB SPL86 dB SPL3 dB200 Steps4 Steps30 Steps2 Steps6 Degrees3.2 Degrees6 or 9 m

YesYesNo

i n g

110 e c o l o g i c a l m o d e l l

sources.

sum level =n∑

i=1

10 ∗ log10(10sum level/10 + 10leveli/10) (5)

2.6. Signal perception

The phase of perturbation of the oscillator cycle (�s) afterdetection of a suprathreshold stimulus was calculated accord-ing to Eq. (6).

�s =(

Tc − clT0

)(6)

Tc refers to the cycle length of the current cycle. By multi-plication of �s by 360 the phase of perturbation in degrees canbe obtained.

The resulting phase shift following an oscillator perturba-tion was calculated from different PRCs, which were derivedfrom playback experiments with real males performed atthree different stimulus levels (50, 60 and 70 dB SPL). In orderto obtain phase shifts other than these stimulus levels lin-ear interpolations and extrapolations of response phases werecalculated for the left branch exclusively. See appendix for adetailed description of the interpolation method. The rightbranch of PRCs is not much different between PRCs obtained at50 and 60 dB SPL and, respectively, 60 and 70 dB SPL (Hartbaueret al., 2005). For this reason and because of a simulatedresponse variability (see below), it was not necessary to inter-polate or extrapolate responses to stimulus levels for the rightbranch of PRCs.

In order to account for a naturally observed variability ofthe resulting phase change following an oscillator perturba-tion, Gaussian noise with a S.D. of 0.017 was added to theresponse phase (�r) representing the left branch of the PRC.The right branch exhibits less variability with a S.D. of 0.009.The size of this response variability was drawn from devi-ations of real response phases from the response phasespredicted by equations fitting the data representing PRCs. Allagents in the model exhibited the same degree of responsevariability.

The resulting response phase (�r) following an oscillatorperturbation was transformed into the length of the disturbedcycle (Pcycle) according to Eq. (7).

Pcycle = int(T0 ∗ �r) (7)

After detection of a stimulus, the remaining oscillator cyclelength (cl) was calculated by subtracting the already passedcycle length before a stimulus occurred (Tc − cl) from the totallength of the disturbed cycle (see Eq. (8)).

cl = Pcycle − (Tc − cl) (8)

When cl became negative its value was set to zero.

2.7. Male spacing

Spacing of males observed in nature was found to be more orless clustered. In the model clustering was achieved by a sort

2 1 3 ( 2 0 0 8 ) 105–118

of random walk performed by each agent starting in the centerof the simulated world. The heading of 15 agents after theirsequential creation in Netlogo follows a systematic scheme.Each agent heads towards a direction which is 24◦ higher com-pared to the agent which was created before. Agents keptwalking by heading towards a randomly chosen direction inthe range of 0–19◦ (calculated from the current heading) untilthere were no other agents in a user defined radius. This radiuscorresponds to a user defined nearest neighbor distance. Areasonably realistic spacing of agents was achieved whenagents covered a randomly chosen distance in each walkingstep in the range of 0–3 patches. After agents spaced them-selves in the simulated world, agents did not move within asimulation run.

2.8. Variables left unconsidered in the model

All agents shared the same average chirp duration andthe same average maximum chirp level. Nevertheless, bothparameters were found to show some variability among malesin a population.

Agents did not face a certain direction: as a result soundpropagation and sound perception did not suffer from thecurrent heading of agents. The sound field of an agent issomewhat ideal because it ignores obstacles present in everyhabitat and a directional sound output of signalers. Confusionwith heterospecific signals was not considered in the model.

2.9. Evaluation of chorus synchrony

If not stated differently all simulations were repeated at least12 times without changing the parameter setting. This wasnecessary to simulate different chorus compositions butalso to account for the implemented variability of oscillatorproperties. The standard chorus situation refers to simulationruns performed with the parameter set listed in Table 1. Ineach run agents started at a random phase in their oscillatorcycle. In the current study, synchrony in a chorus consistingof 15 agents was defined to have been established as soonas eight agents overlapped their signals in time. The numberof simulation steps before this kind of imperfect chorussynchrony was established was divided by the average solo CPand represents the periods of asynchronous interactions. Thedegree of chorus synchrony was quantified from cycle to cycleby calculating the maximum number of agents overlappingtheir signals in time.

In order to measure the accuracy of oscillator synchronyalso between periods of common signaling a synchronizationindex (SI) was calculated according to Eq. (9) (suggested by(Goel and Ermentrout, 2002)).

SI =

√√√√[∑N

i=1(sin(2 ∗ Pi ∗ ˚i))

N

]2

+[∑N

i=1(cos(2 ∗ Pi ∗ ˚i))

N

]2

(9)

N denotes the total number of oscillators (agents) and ˚

represents the current oscillator phase of an oscillator. Thisindex is 1.0 as soon as all oscillators are in phase and dropstowards zero when all oscillators run out of phase.

e c o l o g i c a l m o d e l l i n g 2 1 3 ( 2 0 0 8 ) 105–118 111

Fig. 4 – Establishment of chorus synchrony. In each simulation run agents start at a random phase in their oscillator cycle.After about 2600 simulation steps (13 cycles) synchrony was established in a standard chorus model (for simulationparameters see Table 1). This becomes obvious by the number of agents overlapping their chirps in time (black oscillatingcurve in (A)). In an asynchronous chorus the number of agents that signaled in synchrony was much smaller and lacked acertain periodicity (A, grey line). Those oscillator phases in which stimuli were perceived in a synchronous chorus areshown in a polar plot in (B). Each dot refers to the phase of disturbance of the oscillator cycle obtained in four subsequentchirp interactions. The outer circle in (B) represents the phase of disturbance in a chorus in which agents hear while theyproduce a chirp (standard chorus situation). In contrast the inner circle was obtained from a chorus in which the hearingthreshold of a receiver was 3 dB above the current level of syllables in a chirp. Zero degree refers to that oscillator phase inwhich the end of a chirp is reached. The synchronization index was calculated in three synchronous choruses differing intheir degree of oscillator variability (C). In a chorus in which all agents share the same oscillator properties and cycle noisewas absent the synchronization index was high (C, curve #1). This index dropped between periods of common signaling(asterisks) in a chorus in which all agents were assigned to different oscillators but share the same solo CP (cycle noise wasabsent) (C, curve # 2). In a standard chorus in which agents exhibit their own solo CP and cycle noise is present a dramaticr ). No

2b

3

3

Asof

eduction of the synchronization index was found (curve #3

All statistical calculations were performed with Sigmastat.03 (SPSS Inc.). All data were evaluated for normal distributionefore applying non-parametric tests.

. Results

.1. Oscillator coupling in a simulated chorus

gents start at a random phase in their oscillator cycle and asoon as agents start to synchronize their signals the numberf agents that overlap their signals in time gradually increasesrom cycle to cycle (Fig. 4A, black line). Using the standard

te that curve #2 was shifted to the right.

parameter set shown in Table 1 synchrony was establishedwithin only a few cycles. This kind of imperfect synchronyturned out to be the only global stable oscillator coupling.Chorus synchrony could be temporarily lost due to the noiseadded to the oscillator cycle. In 30 simulation runs of a stan-dard chorus (nearest neighbor distance = 9 m) more than eightagents overlapped their signals in 88% of the simulation time(calculated after 2000 time steps).

Twenty percent of all simulation runs resulted in asyn-chronous choruses when a nearest neighbor distance of 6 mwas simulated and about 30% at a nearest neighbor distanceof 9 m. The CP of agents participating in a synchronous chorus

112 e c o l o g i c a l m o d e l l i n g

Fig. 5 – Faster signaling agents are more likely leaders.Correlation of the solo CP of agents in a synchronizedstandard chorus with oscillator phase (A) and with therelative time difference between chirps of males (B) (nearestneighbor distance: 6 m). In (A), the solo CP of an agent thathas terminated his chirp was plotted against the currentoscillator phase (in degrees) of all other agents present inthe active space. The relative timing of chirps (�t) observedwithin four successive chirp interactions was plottedagainst solo CP in (B). ((A) p < 0.001; correlationcoefficient = −0.60; n = 90; (B) p < 0.001; correlationcoefficient = 0.614; n = 60; Spearman rank order correlation).

was similar to the mean solo CP of all agents (∼200 steps). Dur-ing asynchronous interactions only a small number of agentsoverlapped their signals in time (Fig. 4A, grey line) and theaverage CP of agents was higher compared to a synchronizedchorus (asynchronous chorus: ∼220 steps, synchronous cho-rus: ∼200 steps).

Once synchrony was established in the standard chorus sit-uation, 85–95% of all agents in the active space of a signaleroverlapped their chirps. About 50% of all perceived stimulioccurred in a phase range of 30◦ before and 30◦ after the endof a chirp (±17 steps of the average CP) (Fig. 4B, outer circle).In such an imperfectly synchronized chorus, agents exhibit-ing a longer solo CP perceived stimuli in the final part of theiroscillator cycle. In contrast, agents exhibiting a shorter soloCP perceived slower agents in the first part of their oscillatorcycle. In the example shown in Fig. 5A a significant corre-lation exists between the solo CP of individual agents andtheir oscillator phase in a synchronous interaction (p < 0.001,correlation coefficient = −0.60, n = 90, Spearman rank order

correlation). Among all agents intrinsically faster ones initi-ated chirping first in a synchronous chorus. The significantcorrelation between solo CP and the timing of chirps obtainedin four subsequent synchronous interactions is shown in

2 1 3 ( 2 0 0 8 ) 105–118

Fig. 5B (p < 0.001, correlation coefficient = 0.614, n = 60, Spear-man rank order correlation).

3.2. Synchronization index

The synchronization index was high (∼0.9) in a chorus sim-ulation in which all 15 agents shared the same oscillatorproperties and cycle-to-cycle fluctuations were not simulated(Fig. 4C, curve # 1). When agents were assigned to differentoscillators but share the same solo CP a high synchroniza-tion index was restricted to phases when agents signaled insynchrony (Fig. 4C, curve # 2). In the standard chorus modelagents additionally exhibit individual solo CPs and cycle-to-cycle fluctuations of this period. This dramatically reducedthe maximum synchronization index to 0.5–0.6. In periodsbetween synchronized signaling this index gradually droppedtowards zero (Fig. 4C, curve #3). This result emphasizes thatchorus synchrony in M. elongata is established on a chirp-to-chirp basis by propelling song oscillators, which tend tofire asynchronously, forward and backward in their cycle. Fur-thermore, the results summarized in Fig. 4C demonstrate theinfluence of individual oscillator properties on the accuracy ofoverall chorus synchrony.

3.3. Alternation

In the standard chorus stable alternating choruses were some-times found in which agents belong to either of two chorusessignaling in alternation. When this happened agents belong-ing to one chorus need not necessarily be close to each other.However, this type of chorusing was not stable and often inter-rupted by asynchronous signal interactions. The likelihood forthe establishment of alternating choruses increased when atleast 5 out of 15 agents shared either PRC #5 or PRC #9. ThesePRCs are characterized by a late transition phase at a stimuluslevel of 70 dB SPL and/or a very steep left branch.

3.4. Simulation of a receiver bias

In simulations in which all agents were simulated with anincreased hearing threshold, that is 3 dB above their currentchirp level, almost no oscillator perturbations occurred at aphase later than 330◦ in the oscillator cycle (Fig. 4B, inner cir-cle). However, the number of perturbations were doubled inthe first 60◦ of the oscillator cycle. Interestingly this simulatedreceiver bias had no significant influence on the maximumnumber of agents that synchronized their signals in a syn-chronous chorus (p > 0.05, N = 30 runs each, t-test). In contrast,when agents were able to hear while they produce a chirp,alternating choruses were found in a significantly higher pro-portion of simulation runs (26% versus 8% asynchronous cho-ruses, p < 0.05, N = 50 runs each group, z-test). In all followingsimulations agents were able to hear while producing chirps.

3.5. Duration of chirps in a simulated chorus

Interestingly, stimuli perceived in the final phase of the oscil-lator cycle often resulted in a shortening of the chirp durationand loud syllables were brought forward in time. An exampleof this phenomenon is shown in Fig. 1C (grey line). Conse-

e c o l o g i c a l m o d e l l i n g 2 1

Fig. 6 – Influence on chorus density on the establishmentof synchrony. In a simulated standard chorus inter-maledistance was increased without changing the spatialarrangement of agents (A). With increasing nearestneighbor distance it took a higher number of oscillatorperiods before at least eight agents in the active spacesignaled in synchrony (grey bars in B). The average numberof synchronously chirping agents obtained at differentnR

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earest neighbor distances is shown in (B) (open bars). (**)efers to p < 0.001, N = 12, Mann–Whitney U-test.

uently, the average chirp duration in the standard chorusodel was found to be 23.7 ± 6.9 steps, which is signifi-

antly lower compared to the average chirp duration of solohirping agents in the model (31 ± 2 steps) (p < 0.001, N = 300,ann–Whitney U-test). This unexpected result is a direct con-

equence of the property of the implemented PRCs. A stimuluserceived in the final phase of the oscillator cycle results in ahortening of the cycle. There the remaining oscillator cycleay already be shorter than solo chirp duration.When agents where simulated with a higher hearing

hreshold during their own sound production (+3 dB aboveound level), the average chirp duration was found to be5.7 ± 6.9 steps. This chirp duration was significantly higherompared to the chirp duration obtained in a standard cho-us situation simulating a fixed hearing threshold (p < 0.001,= 300, Mann–Whitney U-test).

.6. Influence of inter-agent distance on chorusingehavior

he influence of inter-agent distance (chorus density) wasnvestigated in simulations in which the minimum inter-maleistance was varied between 3 and 18 m without affecting thepatial arrangement of agents. At a nearest neighbor distance

ower than 9 m chorus synchrony (at least 8 agents signaling inynchrony) was found after only a few oscillator cycles (Fig. 6,rey bars). In simulations performed at inter-agent distancesarger than 9 m the establishment of synchrony was some-

3 ( 2 0 0 8 ) 105–118 113

times heavily delayed. The average number of agents signalingin synchrony significantly dropped after exceeding a nearestneighbor distance of 12 m (9 m versus 15 m: p < 0.001, N = 12,t-test) (Fig. 6, open bars). At a distance of 15 m only 50% ofall simulation runs resulted in the establishment of chorussynchrony.

3.7. Agents joining or leaving a synchronous chorus

Adding either two or three agents to a synchronous chorusoriginally consisting of 15 agents significantly reduced thepercentage of agents signaling in synchrony after 10 cyclespassed (p < 0.001, n = 24, Mann–Whitney U-test) (Fig. 7A, com-pare open bars with grey bars). This chorus manipulationresulted in a 30% loss of synchronously signaling agents. Thiseffect indicates a rearrangement of mutual oscillator couplingafter introducing new agents that started at a random phase intheir signal oscillator. A decreased proportion of synchronizedagents was not found after removing two or three agents ran-domly chosen among agents of a synchronized chorus (Fig. 7B)(p > 0.05, n = 24, Mann–Whitney U-test).

3.8. Selective attention to nearest neighbors

The number of agents chirping in synchrony quickly droppedafter selective attention to the nearest three agents (Fig. 8A,arrow at 12,000 steps). This effect was reversible and over-all synchrony was re-established quickly after switching offthe selective response to nearest neighbors (Fig. 8A, arrow at16,000). Throughout this manipulation of receiver attention,synchrony within local neighbors was quite constant (aboveline in Fig. 8A) and waves of synchronous signaling spreadthroughout the chorus.

Interestingly, in a synchronous chorus in which agentsinteracted with only three nearest neighbors solo CPs sig-nificantly correlated with the phase of song oscillatorsof nearest neighbors (p < 0.05, N = 38, correlation coeffi-cient = −0.43, Spearman rank order correlation) (Fig. 8B). Thisresult is similar to what was found in the standard choruswithout this receiver attention (Fig. 5A). However, a significantcorrelation of solo CPs with the oscillator phases of neigh-bors was restricted to periods in which local synchronizationexceeded 85%. When agents attended to only two nearestneighbors it happened quite frequently that synchronizationamong agents in close proximity dropped below 70%.

4. Discussion

4.1. Establishment of chorus synchrony

Even with an irregular arrangement of agents and despiteparameter variability, simulations of a M. elongata chorus mostlikely resulted in a steady-state oscillator coupling in whichalmost all agents signaled in synchrony. Synchrony is anemergent property of agents simulated after song oscillatorproperties observed in real males. Nevertheless, synchrony

was not perfect and was maintained on a beat-to-beat basis.In this respect, the simulation yields a similar result as wasfound in an “all to all” coupled oscillator model in whichthe natural frequencies (solo CP) of sinusoidal oscillators var-

114 e c o l o g i c a l m o d e l l i n g 2 1 3 ( 2 0 0 8 ) 105–118

Fig. 7 – Changing the size of a chorus. After 5000 simulation steps (open bars) synchrony was established in the standardchorus model. Subsequently, two or three new agents either joined (A) or left (B) the chorus. The percentage ofsynchronously signaling agents found 2000 simulation steps later are shown as grey bars. Each bar represents theaverage ± S.D. of the proportion of synchronously signaling agensignificant difference between 5000 and 7000 simulation steps (p

Fig. 8 – Selective attention to only three local neighbors.After synchrony was established in the standard chorusmodel, selective attention to only three neighbors wasturned on (small arrow at 12,000 steps). This resulted in arapid decay of agents overlapping their chirps in time(lower line in A) without affecting the average synchronyamong local agents much (upper line in A). After turning offthis kind of selective attention (small arrow at 16,000 steps)chorus synchrony was re-established quite fast. Despite theloss of global synchrony a significant correlation betweenthe solo CP of agents and the oscillator phase of closeneighbors was found (B) (p < 0.05, correlation coefficient:−0.427, N = 38, Spearman rank order correlation). This

result is similar to what was found in a synchronouschorus simulated without selective attention (see Fig. 5A).

ied (Ermentrout, 1985). In this mathematical model and inthe current chorus model perfect synchrony could only beestablished after assigning all oscillators the same natural fre-quency (Fig. 4C).

ts calculated from 24 simulation runs. (**) Indicates a< 0.001, N = 24, Mann–Whitney-U test).

The behavior of many different biological oscillators canbe modeled by a saw-toothed pacemaker that slowly buildsup an oscillator level (membrane potential) and triggers anoscillatory event after attaining a certain threshold. There-after the level returns quickly to a basal level and the oscillatorlevel starts to build up again. The simplest type of this oscil-lator model is known as the “integrate and fire” oscillator(Peskin, 1975; Mirollo and Strogatz, 1990) and was modifiedseveral times in order to be able to describe the observed cyclechange following a stimulation. These modifications includea resetting of the oscillator level after stimulus perception,the introduction of an “effector delay” retarding the produc-tion of an oscillatory event and the addition of a “refractoryperiod” which avoids a resetting of the oscillator level duringgeneration of the oscillatory event. Because the song oscilla-tors of agents in the current chorus model exhibited the sameproperties as real males, it was not necessary to know anyof these underlying neuronal oscillator variables. Therefore,using PRCs obtained from real signalers constitutes a morerealistic modeling approach.

Most chorus participants in the model perceived signals ofneighbors in the first phase of their oscillator cycles as soonas synchrony was established. At this phase the shape of thePRC (an example is shown in Fig. 2A) is shallow and cyclelength will be affected only by noise added to the disturbedcycle length. This results in a stabilization of mutual oscillatorcoupling and finally leads to the establishment of chorussynchrony. In a real male chorus in which males are presentin high density, periods of synchrony are often interrupted byalternating choruses. The chorus model was able to reproducethis behavior but for a proper validation simulation resultswill be compared with a natural chorus in the field in a futureproject.

An increased hearing threshold during chirp productionseems to have a stabilizing effect on the establishment of cho-rus synchrony. This simulation result gives rise to the notion

that M. elongata males do not hear during the production ofloud syllables. This may protect ears from being deafenedby the own sound and may result from a corollary dischargewhich is known to desensitize the ears of field crickets dur-

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ng the production of their own chirp (Poulet and Hedwig,002).

.2. Chirp plasticity

he chorus model revealed a shortening of simulated chirpsf a stimulus was perceived shortly before the end of angent’s cycle (Fig. 1C, grey line). This effect was even presenthen agents increased their hearing threshold during soundroduction. Since this was not explicitly programmed, thisehavior seems to be an emergent property of song oscillators.nterestingly a similar result was found in real males entrainedo a periodic stimulus that was played back every 2 s. In suchxperiments, a maximum chirp shortening of 90.0 ± 20.5 msas found when individuals initiated their chirps as follow-

rs. In contrast, the chirp duration of leader chirps was nothanged by the presence of follower chirps.

A shortening of follower chirps and especially the gener-tion of louder syllables earlier in time (see Fig. 1C, grey line)ay be interpreted as an attempt of the follower to mask

he leader chirp. This could represent a countermeasure ofollowers and may explain why followers sing at all despiteheir unattractive role (see below). On the other hand, suchehavior focuses the signals displayed by males in a chorus

n a certain time period. In consequence, a higher maximumound level of the chorus sound have to be expected. Thisnteresting behavior will be investigated in future experimentssing real males.

.3. Leader—follower roles in a chorus

horus synchrony in Necoconocephalus spiza was found to ben evolutionary stable outcome of female choice selecting theeader of a pair of imperfectly synchronized males (Greenfieldnd Roizen, 1993). In M. elongata, the preference of a leader in awo-choice setting (Fertschai et al., 2007) was found to be basedn a sensory bias that diminishes the neural response of fol-

ower signals in receivers (Romer et al., 2002). This bias forcesales to gain the leader role thereby establishing imperfect

horus synchrony as a by-product.The chorus model has shown that oscillator properties sim-

lated after M. elongata males enabled agents to overtake theeader role in a synchronous interaction (Fig. 5B). A similarorrelation of solo CP and the establishment of the leaderole was found in real male duets of an Indian and Malaysianpecies of M. elongata (Hartbauer et al., 2005; Nityananda andalakrishnan, 2007) as well as in Neoconocephalus nebrascensis

Meixner and Shaw, 1986), Necoconocephalus spiza (Greenfieldnd Roizen, 1993) but also in the firefly Pteroptyx cribellata (Buckt al., 1981). Interestingly, in the model such a correlation wasbsent as long as asynchronous interactions were going on.

The results of this simulation suggest that chorus syn-hrony is a prerequisite for the establishment of the leader

ole of intrinsically faster chirping agents and emphasizeshe possibility of females selecting males exhibiting a shorterolo CP. These males are displaying a trait which is ener-etically demanding (Hartbauer et al., 2006; Prestwich, 1994)nd according to the handicap principle (Johnstone, 1995), ahorter solo CP may indicate a higher male quality.

3 ( 2 0 0 8 ) 105–118 115

4.4. Rhythm adjustment

In M. elongata synchrony was found to be maintained on abeat-to-beat basis which is principally different from otherbiological oscillators in which a slow adaptation of the intrin-sic CP to the period of a repetitive signal was found (Pteroptyxmalaccae: (Hanson, 1982); electric organ in fish: (Zelick, 1986)).In a chirping Indian Mecopoda species, an adjustment of theintrinsic CP to the joint CP observed in a male duet wasnecessary for a successful simulation of the proportion ofleading chirps and the proportion of chirps displayed insynchrony (Nityananda and Balakrishnan, 2007). This IndianMecopoda species produce chirps at a rate four times higherthan the Malaysian M. elongata species and lacks a phaseadvance mechanism which enables males of the Malaysianspecies to lock as follower to the chirps displayed by a leader.In a male duet of the Malaysian M. elongata in which bothmales exhibited different solo CPs (male A: 1.98 s, male B:2.17 s) a clear distinction between leader and follower roleswas possible throughout their complete song bout, despite aconsiderable variability of the joint CP (Hartbauer et al., 2005).This suggests that the possible existence of the adjustmentof the intrinsic CP to the CP of the faster male in this speciesseems to be not sufficient to reduce the difference betweenthe chirp onsets of males to zero or even to reverse leaderand follower roles. Neglecting a possible adjustment of theintrinsic CP to faster stimulus periods in the current modeltherefore constitutes a valid approach.

4.5. Chorus density

The average distance between agents had a strong effect onthe percentage of agents overlapping their chirps in time(Fig. 6, open bars) whereby the likelihood of establishing asyn-chronous choruses increased dramatically above a nearestneighbor distance of 12 m. This has two reasons: (1) Increas-ing nearest neighbor distance reduces the number of agentspresent in the active space. (2) PRCs obtained at 50 dB SPLexhibit a shallower slope and a higher transition phase. Suchoscillator properties were found to favor alternation in a maleduet at an inter-male distance of more than 4–5 m in M. elon-gata (Sismondo, 1990).

In the chorus model, however, steady-state synchrony canbe expected to be about 80% as long as agents maintain aminimum nearest neighbor distance of less than 12 m. Thisresult suggests a higher probability for the establishment ofsynchrony in a chorus situation compared to male duets.

In some bushcricket species males space themselves suchthat a focal male perceives signals of close neighbors at a levelof about 65–70 dB SPL (an Indian Mecopoda species: Nityanandaet al., 2007) Mygalopsis marki (Romer and Bailey, 1986) and Tet-tigonia viridissima (Romer and Krusch, 2000). Using standardchorus parameters, this sound level corresponds to an inter-agent distance of ∼6 m in the chorus model. This averageinter-male distance was frequently observed in male chorusesof an Indian Mecopoda species (Nityananda et al., 2007), and

in the synchronizing bushcricket Neoconocephalus nebrascen-sis (Meixner and Shaw, 1979). Simulation results predict thatmales of the Malaysian Mecopoda species keep an average near-est neighbor distance of about 6 m in a natural chorus. In a

i n g 2 1 3 ( 2 0 0 8 ) 105–118

LB50

RB

50T

P50

LB60

RB

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380.

685

y=

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1.99

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0.48

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116 e c o l o g i c a l m o d e l l

planned field study this simulation result will be comparedwith the spacing of real males in their natural habitat.

4.6. Adding agents to a synchronous chorus

Agents joining a synchronous chorus by starting at a randomphase in their oscillator cycle had a strong effect on cho-rus synchrony that was obvious even 10 cycles later (Fig. 7A).This suggests that chorusing needs to be re-established afterthis manipulation and males joining a synchronous chorusdo better in already synchronizing their oscillator cycle beforegenerating their first chirp. This let me investigate the ini-tiation of male songs in response to a conspecific stimulusthat was already on for at least a few periods. Astonishingly, asynchronous initiation of songs was found in all investigatedplayback experiments (a total of 11 males were investigatedin at least two playback experiments per male). These resultsobtained from real males question the way agents were addedto a simulated chorus in the current model. A loss of synchronyinitiated by males joining a real synchronous chorus there-fore need not be expected. In contrast removing agents fromthe simulated chorus did not disturb steady-state synchronyevaluated 10 cycles later (Fig. 7B).

4.7. Selective attention

Selective attention to only three agents in the simulatedchorus disrupted overall chorus synchrony quickly withoutaffecting the synchronization of agents in close proximitymuch (Fig. 8). By doing so waves of synchronized signalsspread throughout the chorus similar to the flashing in aggre-gations of some firefly species (Buck, 1988). The result ofthe current chorus model suggests that the strategy of somebushcricket females to attend only to a subset of males, andchoosing the leader among them (Greenfield, 1994; Greenfieldet al., 1997; Fertschai et al., 2007), is also possible in a chorusin which agents attend to only two or three neighbors in closeproximity (Fig. 8, lower plot). This assumption holds true aslong as males synchronize their signals with the same subsetof neighbors as females pay attention to.

For this point of view simulation results suggest that thereis no need for M. elongata males to selectively attend to only asubset of nearest neighbors in a synchronized chorus. Thismight be different in choruses in which individuals alter-nate their signals like in Ligurotettix planum, and Ephippigerephippiger and Physalaemus pustulosus (Snedden et al., 1998;Greenfield and Snedden, 2003). In such a chorus situationselective attention seems to be a easier task since advertise-ment signals are not overlapping to such a high degree.

Recently, in an Indian species of M. elongata it was shownthat selective attention is achieved by active spacing of malesrather than by neurophysiological filtering (Nityananda et al.,2007). Whether at all males of the Malaysian species of M.elongata only pay attention to nearest neighbors is currentlyunknown. However, the existence of waves of synchronizedsignaling in a real chorus would indicate selective attention to

only a subset of spatially close neighbors.

In summary, the results obtained from simulations of aM. elongata chorus resulted in a better understanding of theestablishment and the robustness of steady-state oscillator # 1

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oupling. In particular, the establishment of chorus synchronyespite a simulated natural variability of inter-male distancesnd variability of song oscillator properties is remarkable. Inhe future, the results of the current study will be comparedith the behavior of a natural chorus. Additionally, choice

xperiments will be performed in a chorus situation with theim of proving the relevance for mate choice of the inferencesrawn from the current study.

cknowledgements

am grateful to Vivek Nityananda for proofreading thisanuscript.

ppendix A

he following polynomials fit the data of PRCs obtainedt three different stimulus levels. Equations were usedo model the change in cycle length after perception of

stimulus at a certain phase (x = phase of perturbation,= normalized response phase). LB = left branch; RB = rightranch; TP = transition phase.

.1. Linear interpolation of response phases

he phase shift following a stimulus with a maximum stim-lus level between 50 and 60 dB SPL (at the receiver) wasalculated according to Eq. (10). If the maximum stimulus levelas in the range of 60–70 dB SPL Eq. (11) was used to interpo-

ate the resulting response phase. Division by 10 accounts forhe range of interpolation (10 dB).

r = �50 + (max(sum level) − 50) ∗ (�60 − �50)10

(10)

r = �60 + (max(sum level) − 60) ∗ (�70 − �60)10

(11)

Signals below 50 dB SPL and above 70 dB SPL were linearlyxtrapolated as a linear function of stimulus level.

e f e r e n c e s

lexander, R.D., Moore, T.E., 1958. Studies on the acousticalheavier of seventeen-year cicadas (Homoptera: Cicadidae:Magicicada). Ohio J. Sci. 58, 102–107.

uck, J., 1938. Synchronous rhythmic flashing of fireflies. Quart.Rev. Biol. 13, 301–314.

uck, J., 1988. Synchronous rhythmic flashing of fireflies. II.Quart. Rev. Biol. 63, 265–289.

uck, J., Buck, E., 1976. Synchronous fireflies. Sci. Am. 234, 74–85.uck, J., Buck, E., Case, F., Hanson, F.E., 1981. Control of flashing in

fireflies. V. Pacemaker synchronisation in Pteroptyx cribellata. JComp. Physiol. 144, 287–298.

rmentrout, B., 1985. Synchronization in a pool of mutuallycoupled oscillators with random frequencies. J. Math. Biol. 22,1–9.

rmentrout, B., 1991. An adaptive model for synchrony in thefirefly Pteroptyx malaccae. J. Math. Biol. 29, 571–585.

3 ( 2 0 0 8 ) 105–118 117

Fertschai, I., Stradner, J., Romer, H., 2007. Neuroethology offemale preference in the synchronously singing bushcricketMecopoda elongata (Tettigoniidae; Orthoptera): why dofollowers call at all? J. Exp. Biol. 210, 465–476.

Goel, P., Ermentrout, B., 2002. Synchrony stability, and firingpatterns in pulse-coupled oscillators. Physica D 163,191–216.

Greenfield, M.D., 1994. Synchronous and alternating choruses ininsects and anurans: common mechanisms and diversefunctions. Annu. Rev. Ecol. Syst. 25, 97–126.

Greenfield, M.D., Roizen, I., 1993. Katydid synchronous chorusingis an evolutionary stable outcome of female choice. Nature364, 618–620.

Greenfield, M.D., Rand, A.S., 2000. Frogs have rules: selectiveattention algorithms regulate chorusing in Physalaemuspustulosus (Leptodaytylidae). Ethology 106, 331–347.

Greenfield, M.D., Snedden, W.A., 2003. Selective attention and thespatio-temporal structure of Orthopteran choruses. Behaviour140, 1–24.

Greenfield, M.D., Tourtellot, M.K., Snedden, W.A., 1997.Precedence effects and the evolution of chorusing. Proc. RoyalSoc. B. 264, 1355–1361.

Hanson, F.E., 1982. Pacemaker control of rhythmic flashing offireflies. In: Carpenter, D. (Ed.), Cellular Pacemakers. JohnWiley, New York, pp. 81–100.

Hartbauer, M., Kratzer, S., Steiner, K., Romer, H., 2005.Mechanisms for synchrony and alternation in songinteractions of the bushcricket Mecopoda elongata(Tettigoniidae: Orthoptera). J. Comp. Physiol. A 191, 175–188.

Hartbauer, M., Kratzer, S., Romer, H., 2006. Chirp rate isindependent of male condition in a synchronizing bushcricketspecies. J. Insect. Physiol. 52, 221–230.

Johnstone, R.A., 1995. Sexual selection, honest advertisementand the handicap principle: reviewing the evidence. Biol. Rev.70, 1–65.

Mason, A.C., Bailey, W.J., 1998. Ultrasound hearing and male-malecommunication in Australian katydids(Tettigoniidae:Zaprochilinae) with sexually dimorphic ears.Physiol. Entomol. 23, 139–149.

Meixner, A.J., Shaw, K.C., 1979. Spacing and movement of singingNeoconocephalus nebrascensis males (Tettigoniidae:Copiphorinae). Ann. Entomol. Soc. Am. 72, 602–606.

Meixner, A.J., Shaw, K.C., 1986. Acoustic and associated behaviorof the coneheaded katydid Neoconocephalus nebrascensis(Orthoptera Tettigoniidae). Ann. Entom. Soc. Am. 79, 554–565.

Mirollo, R.E., Strogatz, S.H., 1990. Synchronization ofpulse-coupled biological oscillators. SIAM J. Appl. Math. 50,1645–1662.

Nityananda, V., Balakrishnan, R., 2007. Synchrony during acousticinteractions in the bushcricket Mecopoda ‘Chirper’(Tettigoniidae:Orthoptera) is generated by a combination ofchirp-by-chirp resetting and change in intrinsic chirp rate. J.Comp. Physiol. A 193, 51–65.

Nityananda, V., Stradner, J., Balakrishnan, R., Romer, H., 2007.Selective attention in a synchronising bushcricket: physiology,behaviour and ecology. J. Comp. Physiol. A 193, 983–991.

Peskin, C.S., 1975. Mathematical Aspects of Heart Physiology.New York University, New York, p. 268–278.

Poulet, J.F.A., Hedwig, B., 2002. A corollary discharge maintainsauditory sensitivity during sound production. Nature 418,872–876.

Prestwich, K.N., 1994. The energetics of acoustic signaling inanurans and insects. Am. Zool. 34, 625–643.

Romer, H., Bailey, W.J., 1986. Insect hearing in the field II. Malesapacing behaviour and correlated acoustical cues in thebushcricket Mygalopsis marki. J. Comp. Physiol. A 159, 627–638.

i n g

118 e c o l o g i c a l m o d e l l

Romer, H., Krusch, M., 2000. A gain-control mechanism forprocessing of chorus sounds in the afferent auditory pathwayof the bushcricket Tettigonia viridissima (Orthoptera;Tettigoniidae). J. Comp. Physiol. A 186, 181–191.

Romer, H., Hedwig, B., Ott, S.R., 2002. Contralateral inhibition as asensory bias: the neural basis for a female preference in asynchronously calling bushcricket, Mecopoda elongata. Eur. J.Neurosci. 15, 1655–1662.

Sismondo, E., 1990. Synchronous, alternating, and phase-lockedstridulation by a tropical katydid. Science 249, 55–58.

Snedden, W.A., Greenfield, M.D., Jang, Y., 1998. Mechanisms ofselective attention in grasshopper choruses: who listens towhom? Behav. Ecol. Sociobiol. 43, 59–66.

2 1 3 ( 2 0 0 8 ) 105–118

Thiele, D., Bailey, W.J., 1980. The function of sound in malespacing behaviour in bush-crickets (Tettigoniidae,Orthoptera). Aust. J. Ecol. 5, 275–286.

Walker, T.J., 1969. Acoustic synchrony: two mechanisms in thesnowy tree cricket. Science 166, 891–894.

Winfree, A.T., 1967. Biological rhythms and the behavior ofpopulations of coupled oscillators. J. Theor. Biol. 16, 15–42.

Winfree, A.T., 1980. The Geometry of Biological Time Springer.Verlag, New York.

Zelick, R., 1986. Jamming avoidance in electric fish and frogs:strategies of singal oscillator timing. Brain Behav. 28, 60–69.