Behavioural Processes 85 (2010) 28–35 Contents lists available at ScienceDirect Behavioural Processes journal homepage: www.elsevier.com/locate/behavproc Categorical counting J Gregor Fetterman a,b,∗ , P Richard Killeen a,b a b Indiana University Purdue University, Indianapolis, United States Arizona State University, United States a r t i c l e i n f o Article history: Received 24 March 2010 Received in revised form 20 May 2010 Accepted June 2010 Keywords: Count Time Pacemaker a b s t r a c t Pigeons pecked on three keys, responses to one of which could be reinforced after a few pecks, to a second key after a somewhat larger number of pecks, and to a third key after the maximum pecking requirement The values of the pecking requirements and the proportion of trials ending with reinforcement were varied Transits among the keys were an orderly function of peck number, and showed approximately proportional changes with changes in the pecking requirements, consistent with Weber’s law Standard deviations of the switch points between successive keys increased more slowly within a condition than across conditions Changes in reinforcement probability produced changes in the location of the psychometric functions that were consistent with models of timing Analyses of the number of pecks emitted and the duration of the pecking sequences demonstrated that peck number was the primary determinant of choice, but that passage of time also played some role We capture the basic results with a standard model of counting, which we qualify to account for the secondary experiments © 2010 Elsevier B.V All rights reserved Wearden (1995) and Fetterman and Killeen (1995) introduced a technique for the study of timing in animals called categorical scaling (Guilford, 1954) Trials began with the illumination of three pecking keys (right, center, left), corresponding to temporal epochs; their illumination “started the clock.” Responses to the right key were sometimes reinforced after a short interval (e.g 8-s), to the center key after a longer interval (e.g., 16-s) and to the left key after the maximum interval (e.g., 32-s) On any trial the reinforcer was arranged for only one of the categorical responses at the prearranged time of reinforcement The resulting pattern of behavior was orderly: Each bird began by pecking on the right key; if food was not obtained at the prescribed time it switched to the center key, and if not at the prescribed time for that key, they switched to the left key Graphs of this behavioral pattern showed a decreasing rate on the “short” key as trial time passed, an increasing then decreasing rate on the “intermediate” key and, finally, an increasing rate on the “long” key (e.g., see Fig in Fetterman and Killeen, 1995) The pigeons experienced different triplets of intervals (e.g., 4–8–16-s, 8–16–32-s, 16–32–64-s) and the regularities in performance, normed to the middle value, were consistent with scalar timing (e.g., Gibbon, 1977), and thus with Weber’s law After exposure to the different conditions described above, in which a reinforcer was primed on every trial, the pigeons experienced a condition in which only one fourth of the trials ended ∗ Corresponding author at: Department of Psychology, IUPUI, 402 N Blackford St., Indianapolis, IN 46202, United States Tel.: +1 317 274 6768; fax: +1 317 274 6756 E-mail address: gfetter@iupui.edu (J.G Fetterman) 0376-6357/$ – see front matter © 2010 Elsevier B.V All rights reserved doi:10.1016/j.beproc.2010.06.001 with food; the other trials ended with the inter-trial interval (ITI) after the longest interval expired Subsequently, the pigeons were returned to baseline conditions where food was available on every trial This manipulation was motivated by a theory of timing proposed by Killeen and Fetterman (1988) which posits a pacemaker whose rate changes with changes in the rate of reinforcement According to the theory, reducing reinforcement rate reduces pacemaker speed and increasing reinforcement rate increases pacemaker speed The resulting effect on behavior should be a shift to the right in a timing function when reinforcement rate is reduced and a shift to the left in the function when reinforcement rate is increased The results of Fetterman and Killeen (1995) confirmed this prediction (see, e.g., their Fig 10) In the experiment reported here the categorical scaling method was adapted to the dimension of number, specifically number of pecks emitted to three pecking keys The payoffs were structured such that responses to one key were reinforced after a small number of pecks had been emitted, to a second key after a somewhat larger number of pecks were emitted, and to a third key after a large number of pecks were emitted (e.g., 4–8–16) Otherwise, the contingencies were the same as described for the categorical scaling of time By comparison with the temporal dimension, less is known about the empirical principles of counting and the mechanisms that may explain the principles An exception to this generalization may be found in the data and theory published by Gibbon, Church, and Meck (e.g., Meck and Church, 1983), who proposed an extension of their timing model to accommodate situations in which rats were taught to discriminate between different numbers of events J.G Fetterman, P.R Killeen / Behavioural Processes 85 (2010) 28–35 dispersed through time (e.g., sequences of tones) For instance a rat may hear a sequence of or tones and learn to make one response after the smaller number and a different response after the larger number Their proposed extension of the timing model stipulates that the timing mechanism may function in either one of two modes: a time mode or a count mode In either mode, the “functional stimulus” consisted of pulses gated to an accumulator, such that both time and number are represented as pulse number values This extension has subsequently been explored and further developed by other researchers (e.g., Roberts, 1995, 1997), who conclude that the dual-mode approach, having the virtue of economy of mechanism, suffices More recent work by Brannon and Terrace (2000) sets a new standard for abilities in monkeys, but does not challenge the dual mode hypothesis Feigenson et al (2004) document another bipolarity in core number competencies, that between precise control by small numbers of objects, and approximate control by larger magnitudes It is the latter system that is engaged by the present experiments Fetterman (1993, 2006) trained pigeons to discriminate between different fixed-ratio (FR) schedules in which the pigeons emitted different numbers of responses to the center key of a threekey operant chamber After completion of the FR one judgment was reinforced after a small number of pecks was emitted and another after a larger number was emitted Different FR pairs were used and probe FRs intermediate to the extreme values were introduced in every condition The psychophysical data obeyed Weber’s law: The scalar principle holds in the numerical as well as the temporal domain Fetterman also recorded the times taken to emit the different FRs Using multiple regression analyses he found that both time and number contributed to the pigeon’s choice behaviors (see Fetterman, 1993, for further considerations of this result) The present research explores the characteristics of numerical processing using the categorical scaling paradigm We used the paradigm to explore the statics and dynamics of timing, and to discern whether changes in the rate of reinforcement affect numerosity discriminations as they affect temporal discriminations Because the duration of the pecking sequences was correlated with the times taken to emit the sequences we took care to record their duration and time as well as number in our analyses and modeling of the data Methods 1.1 Subjects Four adult male Silver King pigeons (Columba livia) maintained at 85% of their free-feeding weights served as subjects The pigeons had free access to water and grit and were housed individually in a room with a 12-h day/night cycle with dawn at am The pigeons were experimentally naïve at the beginning of the experiment Key pecking was established through an autoshaping procedure by illuminating each of pecking keys 30 times per session (total of 90 trials per session), followed by s access to grain All birds reliably pecked all keys after sessions of autoshaping 1.2 Apparatus The experimental enclosure was a standard BRS-LVE threekey operant chamber (32 cm high × 34 cm wide × 34 cm deep) The pecking keys were accessible through 2-cm circular openings in the work panel on the front wall, with the center of the openings 6.3 cm apart, 25 cm above the chamber floor A force of approximately 0.15 N was required to operate the keys The feeder opening was located directly below the center response key and measured cm on all dimensions; the bottom of the feeder opening was 10 cm 29 above the chamber floor When activated, the food hopper provided 3-s of access to mixed grain White noise served to mask extraneous sounds; additional masking and ventilation were provided by an exhaust fan attached to the chamber wall Experimental events were scheduled and recorded by an IBM PC 1.3 Procedure 1.3.1 Pretraining Trials began with the illumination of the right pecking key with red light; a single peck to the key produced 3-s access to mixed grain and turned off the light and illuminated the center key with red light A peck to this key produced mixed grain, turned off the light and illuminated the left key with red light, a peck was followed by reinforcement and the trial ended A 15-s inter-trial interval (ITI) was followed by the next trial There were 90 trials per session in this phase of pretraining, which lasted sessions In the next phase of pretraining the requirements were increased to pecks on the right key, pecks on the center key, and 16 pecks on the left key, where they were held for sessions Finally, as a preliminary to exposing the pigeons to the regular task structure, two kinds of trials were presented, with two keys (right and center or center and left) lit on each trial Reinforcement was primed on just one of the keys For instance, if the trial involved illumination of the right and center keys, and food was primed on the center key, a pigeon had to switch to the center key before it emitted a cumulative total of eight pecks; otherwise food was not delivered Similarly, if food was primed on the right key and the pigeon switched to the center key after emitting just pecks, food was not delivered 1.3.2 Baseline After pretraining, trials began with red illumination of all of the keys, with food primed for just one of them; if food for the designated key was missed, the trial ended without reinforcement and a new 15-s ITI, with all lights in the chamber off, was initiated A switch from one key to the next turned off the light on the key that the pigeon had just left; “illegal” switches from Key to Key terminated the trial After a week of baseline training, very few illegal switches occurred Each peck, regardless of its location, contributed to the cumulative total for the trial Thus, in the case where the requirements were 8, 16, and 32 responses, if the pigeon left the first key after responses, an additional 10 responses on the center key would satisfy the requirement for that key All trials were non-correction: Success or failure in obtaining reinforcement on one trial had no effect on the requirements on the next trial By the end of training the birds missed very few reinforcers (typically