Abstract Pigeons were trained to peck a certain numberof times on a key that displayed one of several possiblenumerical symbols. The particular symbol displayed indi-cated the number of times that the key had to be pecked.The pigeons signalled the completion of the requirementby operating a separate key. They received a food rewardfor correct response sequences and time-out penalties forincorrect response sequences. In the first experiment ninepigeons learned to allocate 1, 2, 3 or 4 pecks to the corre-sponding numerosity symbols s1, s2, s3 and s4 with levelsof accuracy well above chance. The second experimentexplored the maximum set of numerosities that the pi-geons were capable of handling concurrently. Six of thepigeons coped with an s1–s5 task and four pigeons evenmanaged an s1–s6 task with performances that were signif-icantly above chance. Analysis of response times sug-gested that the pigeons were mainly relying on a number-based rather than on a time-based strategy.

Key words Pigeons · Counting · Timing · Symbols ·Responses

Introduction

At the beginning of the last century animal behaviour stu-dents became interested in the experimental study of nu-merical abilities in animals, not least because they weregoaded by the apparent algebraical achievements of ahorse named Kluger Hans whose performance eventuallyturned out to be artefactual (Pfungst 1907). Later the nu-merical capacities of several avian and mammalianspecies were examined using a wide variety of better-con-

trolled numerical tasks (Rilling 1993). At present there islittle doubt that human numerical capabilities, at least atthe non-verbally-aided counting level, have an evolution-ary past and that animal studies can provide some com-parative insights into that phylogenetic history (Dehaene1997; Butterworth 1999). Beyond that the non-verbalproto-counting abilities of animals may play a role whenthey asses the novelty-familiarity of stimuli (Fersen andDelius 1989) and, more generally, when they quantify thebadness-goodness of some ecological variables (Deliusand Siemann 1997).

Two different general kinds of counting behaviour, i.e.“responsive” and “constructive” counting, may be distin-guished. The more obvious terms “receptive” and “pro-ductive” counting are avoided because they have alreadybeen used by Boysen (1992) with a different meaningfrom that intended here. In responsive counting the num-ber of some externally given set of items or events iscounted at a perceptual level and translated into a corre-sponding numeric-symbolic motor response. In humansmost often this response is the production of a numericword or a written numeral, e.g. “the 3 black sheep thatjumped the fence”. In constructive counting a perceivedsymbolic stimulus conveys a particular numerosity thathas to be converted into an equivalent number of discretemotor responses. In humans this could for example in-volve the instruction “×3” printed by a door-bell next to afriend’s name being converted into three button-pushes.The responsive form of counting has, in different guises,often been studied in animals, including birds and culmi-nating perhaps with the African grey parrot Alex who,upon being asked, for example, “how many blue cork?”were on a tray amongst other coloured bottle-corks andother items such as house-keys and toy cars, some of themalso blue, was capable of correctly answering for exam-ple, “three blue cork” (Pepperberg 1994). Indeed, in an-other paper we will ourselves be reporting a further studyconcerned with such responsive, though of course non-linguistic, counting in pigeons (L. Xia, J. Emmerton, M.Siemann, and J.D. Delius, unpublished work). The con-structive form of counting has received far less attention

Li Xia · Martina Siemann · Juan D. Delius

Matching of numerical symbols with number of responses by pigeons

Anim Cogn (2000) 3 :35–43 © Springer-Verlag 2000

Received: 11 October 1999 / Accepted after revision: 27 January 2000

ORIGINAL ARTICLE

L. Xia · M. Siemann · J. D. Delius (Y) Allgemeine Psychologie, Universität Konstanz, 78457 Konstanz, Germanye-mail: juan.delius@uni-konstanz-de, Tel.: +49-7531-883563, Fax: +49-7531-883184

in both animal and human research (reviews: Davis andPérusse 1988; Gallistel and Gelman 1992; Boysen andCapaldi 1993; Dehaene et al. 1998).

In an early study on constructive counting Mechner(1958) trained rats to press a lever a certain number oftimes in order to obtain reward. In half of the trials the re-ward was delivered as soon as they had pressed a requirednumber of times (n). In the other half of the trials theywere rewarded when they had pressed at least n times andhad then pressed a second lever. If the second lever waspressed before the first lever had been pressed n times, noreward was given and the trial was repeated. If they hadpressed the first lever less than n times before pressing thesecond one they received no reward. At different stagesthe animals were required to press the first lever up to 16 times (Mechner 1958). In the initial follow-up experi-ment of Platt and Johnson (1971) the rats signalled a re-sponse completion by putting their head into a reward-well equipped with a photo-electric sensor. In one groupof rats an erroneous premature head-poke ended the trial,while another group of rats could continue to press thelever after such an incorrect poke and head-poke again af-ter more presses. The rats received a reward if they head-poked after they had pressed the lever n or more times. Aninterval followed before the next trial began. With the lat-ter group the number of times the lever was pressedbunched around the smaller number of responses requiredbut, as this number increased, the distribution becamebroader, indicating a relative loss of precision with highernumerosities. With the former group the bunching aroundthe required numbers was less pronounced, and further-more at higher response requirements the response distri-bution peaks tended to be located around more pressesthan the minimum number of responses required. The lat-ter finding also applied to Mechner’s results, but here thenumber of surplus presses was fairly constant regardlessof the minimum number of responses required.

Zeier (1966) reported a study with pigeons using asomewhat similar, but more exacting procedure. He trainedthe birds to deliver a predetermined number of pecks toone of two pecking keys and to indicate the completion ofthe task with a peck to the other key in order to receive areward. The number of pecks required was increased instages. The issue of interest was the maximum responserequirement that pigeons could manage with some preci-sion. Zeier (1966) found that about a third of the pigeonsprogressed to the stage where they yielded an above-chance level of 75% correct trials with a requirement offive pecks on the first key. Some pigeons, however, couldonly manage a task requiring three pecks, while a fewbirds coped with a requirement of eight pecks. Note, how-ever, that at any given experimental stage Zeier’s pigeonswere only expected to cope with one, fixed numerical re-sponse requirement.

Koehler (1960) had earlier summarised several experi-ments by his students in which birds of various specieshad learned to grasp and eat a given number of food itemsdepending on either which numerosity array or which nu-merosity symbol had been presented to them previously.

Some avian species, notably crows and parrots (Lögler1959), were capable of learning to peck from one to sev-eral items, depending on which array or symbol had beenshown on particular trials, a more flexible form of con-structive counting than that described by Zeier (1966).Seibt (1982) criticised the inference from experimentswith numerosity arrays that the animals had converted thenumber of items seen in arrays into the number of re-sponses produced. She did an experiment in which pi-geons had to match sample arrays consisting of two andthree dots with either two pecks and three pecks respec-tively, or conversely had to match two and three dots withthree and two pecks, respectively. Her pigeons apparentlydid about equally well under both conditions, leading herto conclude that the birds were not transferring the nu-merosities seen but rather employing the item arrays asnumerical symbols that constructively elicited a specifiednumber of responses.

These studies show that rats and birds, pigeons in par-ticular, are capable of executing a particular number ofdiscrete responses, quite precisely when the requirednumber of responses is low but not so precisely when theresponse requirement is high. Except in the studies men-tioned in the previous paragraph, experimenters did notexplicitly test whether the animals could deal flexibly andconcurrently with varying response requirements, i.e.whether they could adjust the number of responses emit-ted depending on being cued with different stimuli by, forexample, delivering three pecks if a particular cue waspresent on one trial and delivering five pecks if anothercue was present on the next trial. In the present study pi-geons were trained to emit a certain number of pecks witheach of six different arbitrary symbols. They were firsttrained to respond to one of the symbols with a singlepeck, and then, one after another, additional symbols wereintroduced until the birds finally had to deal concurrentlywith six symbols that required one to six pecks each.

Experiment 1: matching up to four

This experiment was designed to find out whether pigeonswould be able to learn a variable constructive countingtask, and also explored which of two different trainingschedules might be the most efficient.

Method

Subjects

Nine experimentally naive pigeons (Columba livia) of lo-cal homing stock were used. They were housed in indi-vidual cages in a well-ventilated room with a 12 h light/12 h darkness cycle. Throughout the experiment theywere maintained deprived of food at 85% of their free-feeding body weight.

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Apparatus

A conventional conditioning chamber (30 × 34 × 34 cm)was used. The back wall of the chamber bore three side-by-side pecking keys (2.5 cm diameter, 9.5 cm apart, 16 cm above the floor). In-line micro-projectors allowedthe back-projection of stimuli onto the translucent keys.An automatic feeder opening was located 9 cm below themiddle pecking key. The chamber was illuminated by twohouse-lights (3 W) located 9 cm above the middle key.The upper half of the left side-wall was one-way transpar-ent and allowed observation of the pigeons. All experi-mental events were controlled and recorded by a personalcomputer (Commodore) furnished with interface cards(Computer Boards) and programmed in QuickBasic (Micro-soft).

Stimuli

The symbol stimuli used are shown in Fig.1. They wereselected as symbols because their surface areas and formcomplexities were more similar than those of the corre-sponding arabic numerals. The first four of the six pat-terns shown in Fig.1 were used in this first experiment.These stimuli could be individually projected onto the

right, symbol-key in white on a dark background. The keycould also be fully illuminated. The middle, enter-keycould only be fully illuminated. The left key was not usedand remained inactive and dark throughout.

Shaping

The pigeons were taught to peck the right pecking key us-ing an autoshaping procedure. A trial began with a 30-sinterval after which the key was illuminated for a maxi-mum period of 8 s. If the animal pecked the key withinthis period, mixed grain was offered immediately for 2 s,after which the key illumination was extinguished. If thebird did not peck the lit key the food was presented afterthe end of the period. When the pigeon pecked the key inmore than half of the trials during a 30-min period, thesame procedure was repeated with the middle key.

Training

The goal was to get the pigeons to respond to any sn num-ber stimulus presented on the symbol-key with n pecks onthat key and an n+1th peck on the enter-key. For this lastpeck they were rewarded with access to food. If the birdspecked the enter-key after fewer than n pecks on the sym-bol-key or if they pecked the symbol-key n+1 times theywere penalised with time-out. The first four entries ofTable 1 list the correct response sequences, the incorrectresponse patterns and the probabilities of a response se-quence being correct by chance correct with the stimuli s1to s4 used in this first experiment. The task was expectedto be difficult for the pigeons to learn because the higher-numerosity stimuli would initially yield rewards with onlylow probabilities. The training therefore proceeded instages using auxiliary procedures (assisted and consolida-tion sessions) to begin with. Alternative overall trainingdesigns were tried out with two different groups of pi-geons.

The pigeons were therefore randomly divided into twogroups, A and B. The five pigeons of group A were ini-tially trained to cope successively with only one sn stimu-lus at a time beginning with s1 and ending with s4. Theywere exposed to four successive series of daily assistedsessions, each session lasting 200 trials. A series of ses-sions with a given sn always started with assisted trials.These trials began with the display of that stimulus on the

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Table 1 Correct and incorrectresponse patterns associatedwith the various numerositystimuli and the chance proba-bilities of correct sequences(s1–s6 numerosity symbols,n number of s pecks required,s peck to the symbol-key,e peck to the enter-key). Thes5 and s6 rows only apply toexperiment 2

Stimuli n Correct response Incorrect response patterns Probability of correctpatterns response pattern

s1 1 se e, ss 33%s2 2 sse e, se, sss 25%s3 3 ssse e, se, sse, ssss 20%s4 4 sssse e, se, sse, ssse, sssss 17%s5 5 ssssse e, se, sse, ssse, sssse, ssssss 14%s6 6 sssssse e, se, sse, ssse, sssse, ssssse, sssssss 13%

Fig.1 Symbolic numerosity stimuli and experimental arrange-ment used. The first four stimuli were used in experiment 1 and allsix were used in experiment 2

symbol-key. Each peck on that key triggered a computerproduced 36 ms, 1800 Hz audible feedback sound in ad-dition to the mechanical clicking by the key itself. The en-ter-key remained unlit. When the animal had pecked the npecks corresponding to the sn, the symbol-key darkenedand the enter-key was lit. If the pigeon now pecked the en-ter-key the response pattern was counted as correct andyielded a reward consisting of 2 s access to mixed grain.If, however, the pigeon pecked the dark symbol-key oncemore or pecked the dark enter-key before completing then pecks on the symbol-key the response pattern wascounted as incorrect and yielded a 5-s houselights-offpenalty. During the reward and penalty periods the keyswere unlit and inactive. The next trial began immediatelyafter the reward or penalty period. After incorrect trials,the next trial was a correction trial, i.e. a repeat trial.These ceased after the pigeons produced a correct trial.Correction trials were disregarded for the purpose of trialcounts and performance scores.

Whether assisted trials were replaced by ordinary trialswas determined by the current level of performanceshown by the birds. A running weighted performance in-dex cn was updated after each trial n according to the for-mula

cn = cn–1 + w(1–cn–1)

when the pigeon had produced a correct response or ac-cording to the formula

cn = cn–1–wcn–1

when it had produced an incorrect response. The param-eter w, routinely set to w = 0.01, determined the impactwhich the outcome of the nth trial had on the cn–1 that hadaccumulated up to the preceding n–1th trial. If, after agiven trial, cn was below a criterion 0.3 the next n+1thtrial was an assisted trial, while if cn was equal to or abovethat criterion the n+1th trial was an ordinary trial. Theselatter trials began with both the symbol-key and the enter-key being lit and active. The response pattern definitionsand reinforcement outcomes, including correction trials,however, were the same as with assisted trials. As the pi-geons’ performance improved the proportion of assistedtrials decreased and that of ordinary trials increased. Thecurrent numerosity stimulus was replaced by the nexthigher numerosity stimulus as soon as the pigeons met acriterion of 75% correct responses within 20 consecutiveordinary trials.

After the pigeons had successively reached the learn-ing criterion for each and all of the four numerosity stim-uli, they were trained in consolidation sessions in whicheach and all the stimuli occurred at various times. All 200trials of these sessions were ordinary trials. To aid learn-ing, a given numerosity stimulus was presented repeat-edly in a run of trials. A run with that particular stimulusended when the pigeons produced three correct responseswithin five consecutive trials. The stimulus selected forthe next run was that exhibiting the worst, i.e. the lowestcn performance index at the time. As the pigeons’ perfor-mance improved the stimulus switch criterion was re-

duced to two correct trials out of three trials and then toone correct trial out of one. The training terminated whenthe animals achieved 75% correct out of 20 consecutivetrials within a single session under this last switch criterion.

The four pigeons in the B group were trained with thesame types of sessions as the A group pigeons but follow-ing a different overall design. They began by beingtrained on the s1 and s2 stimuli in exactly the same way asgroup A. However, when the birds had successivelyachieved the 75% correct criterion on both stimuli theywere trained in consolidation sessions where these twostimuli were presented in alternating runs of ordinary tri-als. When the pigeons met the 75% criterion with thesestimuli, they were trained with the numerosity stimulus s3in assisted sessions. Upon reaching the criterion withstimulus s3 they were exposed to consolidation sessionsinvolving stimuli s1, s2 and s3 until the 75% criterion wasmet again. The pigeons were then trained with assistedsessions on stimulus s4 until they reached criterion.Finally, they were trained in consolidation sessions in-volving all four stimuli presented in runs of ordinary tri-als. This terminal training was fully equivalent to the finalstage of group A and also ended when the pigeons met thesame final criterion of 75% correct trials in 20 consecu-tive trials.

Testing

To assess the pigeons’ response counting in the absence ofany auxiliary cues or procedures the birds of both groupswere finally exposed to a test session consisting of 80 or-dinary trials without correction trials. Each numerositystimulus was presented in 20 ordinary trials but in a quasi-random sequence incorporating the restriction that a givennumerosity stimulus could not appear in more than twoconsecutive trials. In contrast to the consolidation ses-sions, in this testing session incorrect trials did not lead torepeat correction trials.

Results and discussion

All nine pigeons completed all training stages by achiev-ing the criterion of 75% correct in 20 consecutive trials.The number of trials which group A and group B pigeonsneeded to reach this criterion with the different stages isshown in Fig.2. The groups differed significantly in thetotal number of trials they required to reach the final train-ing stage criterion when all four stimuli were concurrentlypresented (t7 = 2.42, P = 0.046). The group A pigeons re-quired an average of 13,180 trials (10,811–17,317) tocomplete the last stage and the group B pigeons an aver-age of 20,606 trials (15,527–27,781).

The results of the test session are shown in Fig.3. Thefive pigeons in group A yielded a mean of 80%, 59%,58% and 60% correct responses with stimuli s1, s2, s3 ands4, respectively. These accuracies are well above the cor-responding chance levels of 33%, 25%, 20% and 16%

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correct trials listed in Table 1 (t-tests; s1: t4 = 17.0, P =0.00007; s2: t4 = 7.0, P = 0.0022; s3: t4 = 7.0, P = 0.0022;s4: t4 = 10.3, P = 0.00050). The pigeons in group B alsorevealed an above-chance-level performance of 79%,72%, 64% and 46% correct responses with stimuli s1, s2,s3 and s4, respectively (t-tests; s1: t3 = 10.6, P = 0.0018; s2:t3 = 6.2, P = 0.0084; s3: t3 = 6.6, P = 0.0070; s4: t3 = 12.3,P = 0.0011). The accuracy differences between the twogroups were not significant except for the s4 numerosity(t7 = 2.65, P = 0.032).

Since there was only a small difference between thetest results of the two groups, the data of all nine pigeonswere pooled for the purposes of a finer analysis. The totalnumbers of correct and incorrect responses during the fi-nal consolidation stage and again during the test stage wascomputed separately for each of the numerosity stimulis1–s4. The number of correct and various kinds of incor-rect responses to each of these stimuli were expressed aspercentages of the corresponding totals and are shownplotted as response distributions of the responses in Fig.4.

Both diagrams demonstrate that the pigeons most oftenmade errors by pecking the symbol-key once more oronce less than the number of pecks defining the correctresponse patterns. This was more pronounced during thetest stage than during the consolidation stage. The latter ofcourse includes a phase when the birds were not yet show-ing their optimal performance. The levelling out of the re-sponse distributions may represent a loss of precision withhigher numerosities, perhaps because they had been lesstrained than the lower numerosities, and/or because theyallowed more kinds of error response patterns than thesmaller numerosities (Table 1).

The results show that pigeons are able to respond withsignificant accuracy to four different numerosity symbolswith a corresponding number of pecks, even when thevarious symbols were presented in a quasi-randomised se-quence during the test stage. That means that the pigeonswere able to match four different numerosity symbolswith the corresponding number of pecks.

Experiment 2: matching up to six

This experiment was designed to explore the upper boundsof the pigeons’ ability to learn to match different nu-merosity symbols with a corresponding number of pecks.

Method

Seven of the nine pigeons that had participated in the pre-vious experiment were used. Two pigeons were excluded

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Fig.2 Experiment 1. Cumulative number of trials that the individ-ual pigeons of A group A (5 pigeons) and B group B (4 pigeons)needed to complete the various training stages up to stage 1234

Fig.3 Experiment 1. Mean percent correct responses shown bygroup A and group B pigeons with the s1–s4 numerosity stimuliduring the test stage. Bars indicate chance-level performances

Fig.4 Experiment 2. Mean percent distributions of responses withthe s1–s5 numerosity stimuli during A the consolidation and B thetest stage

because they had required more than 20,000 trials to com-plete that experiment. The same apparatus and procedureswere used. All birds, regardless of which group they hadbelonged to, were now trained and tested in the same way.In addition to the four numerosity stimuli used in experi-ment 1, experiment 2 also involved the numerosity stimulis5 and s6 (Fig.1, Table 1). The birds were first trained onstimulus s5 using assisted sessions. When the animalsreached the criterion of 75% trials correct within 20 ordi-nary trials with that stimulus, all five stimuli s1–s5 wereused in consolidation sessions. The birds were trained un-til they yielded 75% correct trials out of 12 trials for eachstimulus within a single session, or until they had com-pleted more than 30,000 trials without reaching this crite-rion, in which case they were excluded from further par-ticipation. Then the remaining pigeons were tested in asingle testing session, within which all five stimuli werepresented 20 times in a quasi-randomised sequence as de-scribed earlier. The next stage involving stimulus s6 pro-ceeded in a similar manner.

Results and discussion

Six out of the seven pigeons reached the learning criterionwith the numerosity stimuli s1 to s5 after an average of11,800 training trials (3,800–27,500) additional to thosethat they had needed to complete experiment 1. Four ofthese six pigeons reached the criterion with the stimulis1–s6 after an average of 19,340 additional trials (11,500–27,500). Figure 5 shows the cumulative total number oftrials the six and four pigeons required to complete thevarious training stages.

The results of the test sessions are shown in Fig.6. Thesix pigeons that were successful with the s1–s5 trainingyielded 82%, 83%, 62%, 57% and 42% average correctresponses with the s1, s2, s3, s4 and s5 stimuli, respectively.All these scores were significantly above the chance lev-els (t-tests, s1: t5 = 12.7, P = 0.000054; s2: t5 = 14.5, P =0.000028; s3: t5 = 7.9, P = 0.00052; s4: t5 = 8.3, P =0.00041; s5: t5 = 7.2, P = 0.00080). Similarly, the four pi-

geons that also completed the training with the s1–s6 stim-uli yielded an average 79%, 84%, 74%, 68%, 53% and50% average correct trials with the s1, s2, s3, s4, s5 and s6stimuli, respectively. All these averages were again signif-icantly above the corresponding chance levels (t-tests, s1:t3 = 10.6, P = 0.0017; s2: t3 = 11.4, P = 0.0014; s3:t3 =12.5, P = 0.0011; s4: t3 = 15.7, P = 0.00056; s5: t3 = 6.4,P = 0.0077; s6: t3 = 10.6, P = 0.0017).

Figure 7 shows the response distributions, calculated inthe same manner as explained earlier, for the six stimulibased on the data of the final consolidation stage withstimuli s1–s6. There were not enough data for the teststage to yield meaningful response distributions. It is nev-ertheless apparent that, compared to the corresponding re-sults of the first experiment, the response distributionswith the numerosity stimuli s1–s4 were now steeper. Thismight be because by now the pigeons had acquired muchadditional experience with these lower numerosities, soerroneous responses occurred less frequently for thesestimuli than in experiment 1. As in experiment 1, how-ever, the spread of errors was less with the lower-nu-merosity stimuli than with the higher ones.

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Fig.5 Experiment 2. Cumulative number of training trials that in-dividual pigeons needed to reach the various stages up to stages12345 (6 pigeons) and 123456 (4 pigeons)

Fig.6 Experiment 2. Mean percent correct responses during thetest sessions with the s1–s5 (six pigeons; left columns) and the s1 tos6 stimuli (four pigeons; right columns). Bars indicate chance-levelperformances.

Fig.7 Experiment 2. Mean percent distributions of responses withstimuli s1–s6 during the final consolidation stage

General discussion

As far as we know this is the first demonstration in pi-geons of a flexible constructive-counting-like behaviourthat involved the production of up to six responses condi-tional upon the separate pre-presentation of up to six nu-merosity symbols. The counting behaviour was flexible inthe sense that the pigeons ended up responding in a con-current manner to four, five or six different arbitrary vi-sual symbols each standing for the different numerositiesbetween 1 and 6. The pigeons responded correctly to thesedifferent numerosity stimuli at levels that were signifi-cantly above chance by emitting the number of pecks thestimuli were meant to symbolise. This clearly goes be-yond the constructive counting ability that Zeier (1966)had previously demonstrated in pigeons. His birds hadonly to respond a fixed number of times at any stage of hisexperiment even though the particular numerical responsedemanded was progressively augmented. In our case thevarious symbolic stimuli signalled in a random trial-to-trial order the number of responses that the pigeons had toproduce to obtain reinforcement.

The pigeons undeniably required extensive training tomaster this variable constructive counting task. Theyneeded, for example, an average of some 15,000 trials be-fore they reliably counted up to four pecks at the end ofexperiment 1. But this training was nevertheless brieferthan the average 25,000 trials or so that other pigeons hadneeded to reliably count four visual stimuli in a respon-sive counting task (L. Xia, J. Emmerton, M. Siemann, andJ.D. Delius, unpublished work). Incidentally, four of theselatter pigeons that had managed to reliably count four oreven five stimuli in that study were then trained using thesame procedures as those employed with group A groupin experiment 1 of this study. These birds only took an av-erage of some 8,000 trials compared with the about13,000 trials that our present group A pigeons needed toreliably count up to four responses (Xia 1998). Althoughthe speedier learning of these experienced pigeons mighthave been due to their pre-exposure to conditioning pro-cedures generally, it is also possible that some of their ad-vantage might have been due to a more specific transfer ofnumerical skill from the responsive to the constructiveforms of counting.

Quite what information the pigeons utilised when theyresponded with the correct number of pecks beforeswitching over to the enter-key remains largely uncertain,however. A hypothetical cumulative peck-counting pro-cess could, for example, have relayed either on the regis-tration of the efferent neural peck commands or else of thereafferent sensory volleys arising from the pecking ac-tions. Among these latter signals one might expect thetactile and auditory stimuli arising from the beak’s impacton the key and from the mechanical motion of the key aswell as the computer produced feedback tone to possiblyplay a role. But besides these exteroceptive stimuli, inte-roceptive mechanical and vestibular stimuli consequenton the peck motions could also have intervened (Schall

and Delius 1991). It is, however, also possible that thecorrect response sequences might have been assessed onthe basis of the time that elapsed between the first and thelast peck, since repetitive pecks of pigeons are known tobe emitted at a rather constant rate of about 3–4 s–1

(Delius et al. 1986). We may thus assume that the produc-tion of six pecks, for example takes about three times aslong as the production of two pecks.

Roberts and Mitchell (1994) had pigeons learn to dis-criminate two different visual stimuli. One lasted 2 s andconsisted of two light pulses, another one lasted 8 s andconsisted of eight light pulses. To get a reward the birdshad to peck one key if the longer stimulus with morepulses had been presented and another key if the shorterstimulus with fewer pulses had been presented.Intermediate signals were then presented. They were ei-ther of a fixed duration (e.g. 4 s) while the number ofpulses varied from 2 s to 8 s or they consisted of a fixedintermediate number of pulses (e.g. 4) but their durationvaried from 2 s to 8 s. In these time-based and number-based tests the relative choice frequencies of thelonger/more key increased with both the number and theduration of the relevant test stimuli. This indicated that pi-geons were using both counts and times to achieve correctresponding during the test trials. Fetterman (1993) re-viewed older data and reported additional findings thatreplicated this ambivalent finding with rats. A currentlystill popular so-called mode control model assumes thatthere is a trade-off between a timing mechanism and acounting mechanism operating on the basis of the accu-mulation of either endogenous clock pulses or exogenousperceptual item pulses (Meck and Church 1983; but seeStaddon and Higa 1999). However, Emmerton et al. (1997)ran an experiment where pigeons were presented with ar-rays of dots. When one or two dots were presented, pi-geons had to peck one key, when six or seven dots werepresented they had to peck another key. Tests with inter-mediate arrays of three, four or five dots also resulted inan orderly graded choice of the two keys. In this instance,because of a simultaneous numerosity information pre-sentation, the pigeons are unlikely to have estimated thenumber of dots on a passage-of-time basis.

The timing of the pecks onto the symbol-key were infact recorded during the final test stages of experiments 1and 2. A summary of this data is presented in Table 2. Thetable ignores the intervals from the onset of the trial to thefirst peck of each sequence since these latencies were toovariable to convey truly meaningful information. Nodoubt their variability was due to episodes of inattentionwhich caused at least some of the birds to miss the begin-ning of the symbol-key illumination. The table suggeststhat the average interval between the pecks was in princi-ple regular enough for a timing-based production of cor-rect responses. Interestingly, the emission of the finalsymbol-key peck was quite consistently associated with alonger inter-peck interval than the other symbol-keypecks. This could mean that the required switch to the en-ter-key was proactively interfering with the end of thesymbol-key sequence. However, this effect is compatible

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with both a counting and a timing process. Beyond thisthere was no indication that the pigeons developed anysymbol-specific rhythmic response patterns which couldhave enabled them to generate the correct number ofpecks in a manner that bypassed counting- or timing-likeoperations.

To examine whether the pigeons had in fact used a tim-ing strategy we carried out an analysis of errors based onresponse times. First, the average times elapsed betweenthe first and the last numerosity key-peck were calculatedfor the correct responses in the experiment 2 test session.This was done separately for each pigeon and each nu-merosity from s3 to s6 (s2 was excluded because the corre-sponding single interval response times were too brief formeaningful comparisons). If the pigeons were relying ona timing mechanism the correct response times ct3 to ct6should have been memorized by the pigeons as a meancriterion time indicating when it would pay to switch frompecking the symbol key to pecking the enter key. It couldbe then expected that erroneous sequences that involved apeck too few (we ignored the few errors involving evenfewer pecks) might have arisen because the pigeons hadbeen exceptionally slow in pecking and had reached thecorresponding internalized criterion time before complet-ing the correct number of pecks. If so the response timesassociated with such errors etn–1 should, as a rule, haveequalled or even exceeded the corresponding ctn criteriontimes. It is also possible that erroneous sequences involv-ing a peck too many might have arisen because the pi-geons had been exceptionally fast in pecking and thus notreached the corresponding criterion time after the numberof pecks actually required, this causing them to produce asuperfluous, erroneous peck. If so the average cumulatedtimes etn associated with the nominally required numberof pecks (i.e. one less than that actually produced) associ-ated with such errors should as a rule been less than thecorresponding criterion time ctn. Table 3 shows that whilethe times associated with overcounting errors may have

partially supported the timing hypothesis, those associ-ated with undercounting errors definitely did not do so.The total number of cases supporting the timing hypothe-sis was statistically indistinguishable from the 50%chance level (binomial test). Inclusion of error cases in-volving fewer pecks than just one, of the s1 and s2 nu-merosity trials and of the trial-onset to first peck intervalsinto analogous evaluations did not alter the conclusionthat a timing strategy played at most a very partial role indetermining the counting performance of our pigeons

Returning again to the counting aspect, it is apparentthat, as shown in Figs. 4 and 7, the distribution of error re-sponses varied systematically with the number of pecksrequired. In this respect the results compare with thoseobtained in rats by Platt and Johnson (1971). Errorsmostly concerned numbers of pecks that were near ratherthan distant from the correct response pattern, that is, ifanimals were required to peck four times then they tendedto erroneously peck three or five times rather than erro-neously peck two or six times. Moreover, the proportionsof correct response sequences decreased as the number ofelementary responses required increased. In other words,the response distributions became flatter and broaderwhen the required number of lever presses/pecks was in-creased. At first sight this seems to roughly obey Weber’slaw, i.e. the bigger the absolute number of responses de-manded, the larger the just-discriminable numerical dif-ference that could be achieved. But in our study the errorswere not only downwards-censored by necessarily ex-cluding less than zero responses, but also upwards cen-sored by allowing no more than n+1 responses. Thismeant that, for example, with s2 there were only two erro-neous response sequences and that with s5 there were nofewer than five such erroneous sequences among whichthe animals could distribute their errors (Table 1). Thiswould necessarily tend to generate Weber-like data arte-factually based on a procedural constraint rather than onany sensory mechanism. One must also consider that thepigeons had more training experience with the part-tasksinvolving lower numerosities. Due to these proceduralconstraints it is difficult to be sure whether there was asignificant additional perceptual effect even though it islikely that there was. This is regrettable because much hasbeen theoretically made of the Weber effect in responsivecounting contexts where it was valid to assume that it wastruly based on perceptual factors (e.g. Meck and Church1983; Emmerton 1998). Note that applied to our countingtask Weber’s law would suggest that although pigeonsmight well have difficulties in accurately producing say,seven and eight responses to a pair of corresponding sym-bols, they might still be capable of accurately producing,

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Stimulus s2 s3 s4 s5 s6Sequence sse ssse sssse ssssse sssssse

Experiment 1 334 278 349 274 274 324 – –Experiment 2 258 227 234 215 213 251 244 245 251 289 271 232 228 239 312

Table 2 Average intervals (ms) between pecks to the numerositystimuli s2, s3 and s4 during the correct response sequences indi-cated. Intervals in (ms) between 1st-2nd, 2nd-3rd, 3rd-4th, 4th-5th

and 5th-6th pecks of these sequences for final test sessions of ex-periment 1 (9 pigeons) and experiment 2 (4 pigeons). The closingmean intervals for each sequence are printed in bold numbers

Table 3 Proportion of error response times connected with under-or overproduction of number of pecks that were in agreement witha timing strategy. Based on the test session of experiment 2 (4 pi-geons)

Numerosity stimulus s3 s4 s5 s6 Total

Proportion cases 0/11 1/12 4/17 2/20 7/60where etn–1 ≥ ctn

Proportion cases 7/10 7/12 10/17 14/15 38/54where etn < ctn

Total agreeing with 7/21 8/24 14/34 16/35 45/114timing hypothesis

say, 15–17 and 18–22 responses to a pair of correspondingsymbols. That is, the count resolution becomes coarserwith increasing numerosities (cf. Rilling 1965). Whetherthis is actually so still needs to be investigated.

Dehaene and Changeux (1993) have presented a neuralnet model capable of a counting-like performance of thiskind. Its attraction is that it offers a rigorously mechanisticaccount of a responsive kind of non-verbal counting. Thenet consists of several layers: dots of different sizes and lo-cation can be presented to an input layer, which is con-nected to an array of neural Gaussian filters. Through thisfiltering, the sizes and locations of the dots become irrele-vant. The output of this so-called topographical mappinglayer is transferred to a summation layer, containing anumber of units that have increasing thresholds. The sum-mated and threshold-exceeding activation of a certain unitin this layer is roughly proportional to the number of dotspresent in the stimulus array. These units are connected toan output layer of units within which, through lateral con-nections, units responsible for higher numerosities can in-hibit units responsible for lower ones. This ensures that thenet output layer only responds to the highest numerositythat is activated, because units responsible for lower nu-merosity, which are activated by subsets of the input stim-ulus, are suppressed. The activity distributions of the out-put layer units show Weber-like properties insofar as withincreasing numerosity arrays the peak activities declineand the activity distributions become broader (implyingthat the overlap between adjacent active units increases).Clearly, as presented the model cannot produce a specifiednumber of responses upon recognising a given numericalsymbol, i.e. behave in a constructive manner, but it wouldbe feasible to add such a module. Dehaene and Changeux(1993) have in any case sketched how the model may beeasily modified to count successive events, such as thereafferent stimuli upon pecking or indeed the peck com-mands themselves, as demanded in our particular case.The development of a fully explicit neural model of con-structive counting is desirable because it would generatemore precise predictions about the neurophysiological ba-sis of such counting (cf. Thompson et al. 1970) than purelyformally descriptive models of counting.

Acknowledgements The research was supported by the DeutscheForschungsgemeinschaft, Bonn. We are grateful to Prof. JackyEmmerton for excellent advice while she was on sabbatical leavefrom Purdue University and a visiting professor in Konstanz. Shealso helped much by improving an earlier draft. We also thankJessica Grante for revising the language of the final manuscript.We dedicate this paper to the memory of Otto Koehler whose pio-neering work on averbal counting in animals has in the past beenfrequently doubted but which when replicated has nearly invari-ably proven right.

References

Boysen ST (1992) Counting as the chimpanzee views it. In:Fetterman G, Honig WK (eds) Cognitive aspects of stimuluscontrol. Erlbaum, Hillsdale, pp 367–383

Boysen ST, Capaldi EJ (eds) (1993) The development of numeri-cal competence: animal and human models. Erlbaum, Hillsdale

Butterworth B (1999) The mathematical brain. Macmillan, London

Davis H, Pérusse R (1988) Numerical competence in animals:Definitional issues, current evidence, and a new research agenda.Behav Brain Sci 11:561–615

Dehaene S (1997) The number sense: how the mind creates math-ematics. Oxford University Press, Oxford

Dehaene S, Changeux JP (1993) Development of elementary nu-merical abilities: a neuronal model. J Cogn Neurosci 5:390–407

Dehaene S, Dahaene-Lambertz G, Cohen L (1998) Abstract repre-sentations of numbers in the animal and human brain. TrendsNeurosci 2:355–361

Delius JD, Siemann M (1997) Transitive responding in animalsand humans: exaptation rather than adaptation? Behav Proc 42:107–137

Delius JD, Lindenblatt U, Lombardi C (1986) Coupling betweenpecking and heartbeat in pigeons. J Interdis Cycle Res 17:243–251

Emmerton J (1998) Numerosity differences and effect of stimulusdensity in pigeons’ discrimination performance. Anim LearnBehav 26:243–256

Emmerton J, Lohmann A, Niemann J (1997) Pigeons’ serial order-ing of numerosity with visual arrays. Anim Learn Behav 25:234–244

Fersen L von, Delius JD (1989) Long term retention of many vi-sual patterns by pigeons. Ethology 82:141–155

Fetterman JG (1993) Numerosity discrimination: both time andnumber matter. J Exp Psychol Anim Behav Proc 19:149–164

Gallistel CR, Gelman R (1992) Preverbal and verbal counting andcomputation. Cognition 44:43–74

Koehler O (1960) Le dénombrement chez les animaux. J PsycholNorm Pathol 1:45–58

Lögler P (1959) Versuche zur Frage des “Zähl”-Vermögens aneinem Graupapagei und Vergleichsversuche an Menschen. ZTierpsychol 16:179–217

Mechner F (1958) Probability relations within response sequencesunder ratio reinforcement. J Exp Anal Behav 1:109–122

Meck WH, Church RM (1983) A mode-control model of countingand timing processes. J Exp Psychol Anim Behav Proc9:320–334

Pepperberg IM (1994) Numerical competence in an African greyparrot (Psittacus erithacus). J Comp Psychol 108:36–44

Pfungst O (1907) Das Pferd von Herrn Osten. Leipzig, reprinted(1977) as: Der kluge Hans. Ein Beitrag zur nicht-verbalenKommunikation. Fachbuchhandlung für Psychologie, Frankfurt

Platt JR, Johnson DM (1971) Localization of position within a ho-mogeneous behavior chain: effects of error contingencies.Learn Motiv 2:386–414

Rilling M (1993) Invisible counting animals: a history of contribu-tions from comparative psychology, ethology, and learning the-ory. In: Boysen ST, Capaldi EJ (eds) The development of nu-merical competence: animal and human models. Erlbaum,Hillsdale, pp 3–37

Rilling M, McDiarmid C (1965) Signal detection in fixed-ratioschedules. Science 148:526–527

Roberts WA, Mitchell S (1994) Can a pigeon simultaneouslyprocess temporal and numerical information? J Exp PsycholAnim Behav Proc 20:66–78

Schall U, Delius JD (1991) Grasping in the pigeon: controlthrough sound and vibration feedback mediated by the nucleusbasalis. Physiol Behav 50:983–988

Seibt U (1982) Zahlbegriff und Zählverhalten bei Tieren. NeueVersuche und Deutungen. Z Tierpsychol 60:325–341

Staddon JER, Higa JJ (1999) Time and memory: towards a pace-maker-free theory of interval timing. J Exp Anal Behav 71:215–251

Thompson RF, Mayers KS, Robertson RT, Patterson CJ (1970)Number coding in association cortex of the cat. Science 168:271–273

Xia L (1998) Numerische Fähigkeiten bei Tauben. Tectum, Mar-burg

Zeier H (1966) Über sequentielles Lernen bei Tauben, mitspezieller Berücksichtigung des “Zähl”-Verhaltens. Z Tier-psychol 23:161–189

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