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Fig. 2. A double bisection task is a conditional dis- crimination in which the animal learns two mappings, {S1, S4j}—{Red, Green} on ‘“‘Short’’ trials, and {S4, S;¢}—>{Blue, Yellow} on ‘“‘Long”’ trials. The subscripts indicate the sample duration; the arrow indicates that the first and second responses in each pair are correct following the first and second samples, respectively.

Figure 2 A double bisection task is a conditional dis- crimination in which the animal learns two mappings, {S1, S4j}—{Red, Green} on ‘“‘Short’’ trials, and {S4, S;¢}—>{Blue, Yellow} on ‘“‘Long”’ trials. The subscripts indicate the sample duration; the arrow indicates that the first and second responses in each pair are correct following the first and second samples, respectively.

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Fig. 1. The left panels show the structure of SET. A pacemaker generates pulses which are added in an accumulator and stored at the end of the to-be-timed interval in one or more long-term memories. To decide when to respond, the animal compares the number currently in the accumulator with samples extracted from the memories. In FI schedules (top left) only one memory is formed; in the bisection procedure (bottom left) two memories are formed. The right panels show the structure of LeT. After a time marker, a set of states (top circles) is activated in series. The states may be coupled to various degrees (associative links) with one or more operant responses (bottom circles). The strength of each response is determined by the dot product between the vectors of state activation and coupling. In FI schedules (top right) only one vector of couplings is formed; in the bisection procedure (bottom right) two vectors are formed.
Fig. 1. The left panels show the structure of SET. A pacemaker generates pulses which are added in an accumulator and stored at the end of the to-be-timed interval in one or more long-term memories. To decide when to respond, the animal compares the number currently in the accumulator with samples extracted from the memories. In FI schedules (top left) only one memory is formed; in the bisection procedure (bottom left) two memories are formed. The right panels show the structure of LeT. After a time marker, a set of states (top circles) is activated in series. The states may be coupled to various degrees (associative links) with one or more operant responses (bottom circles). The strength of each response is determined by the dot product between the vectors of state activation and coupling. In FI schedules (top right) only one vector of couplings is formed; in the bisection procedure (bottom right) two vectors are formed.
Similarities and differences between SET and LeT. and a more complex decision rule (see bottom right panel of Figure 1; Machado, 1997). The states become active at sample onset, the time marker, and each state is now coupled with two responses. The strength of the link connecting state n with response r, W,(n), changes only after the animal chooses. If the choice response is reinforced, the links between the states and that response increase, whereas the links be- tween the states and the other response decrease, always in proportion to each state’s activation. Conversely, if the choice response is extinguished, the links between the states and that response decrease, whereas the links between the states and the other response increase. In other words, the model assumes that, in bisection tasks, when the link between a state and one response changes, the link between the same state and the other response also changes, albeit in the opposite direction. Hence, the model’s learning rule implements a strong form of response competition’. —< = oe Green will be weak—hence the preference for Red after short samples. However, after the 4-s samples, later states will be the most active and because of the reinforcement contingencies their link with Red will be weak whereas their ink with Green will be strong—hence the preference for Green after the long samples. LeT predicts that preference for Red decreases as sample duration ranges from 1 to 4s. Moreover, it also predicts (see Machado, 997) a PSE close to, but slightly greater than, the geometric mean of the training stimuli and, when A is proportional to the overall reinforcement rate during the trials, that psychometric functions obtained with samples holding the same ratio will superimpose when plotted on a common axis.
Similarities and differences between SET and LeT. and a more complex decision rule (see bottom right panel of Figure 1; Machado, 1997). The states become active at sample onset, the time marker, and each state is now coupled with two responses. The strength of the link connecting state n with response r, W,(n), changes only after the animal chooses. If the choice response is reinforced, the links between the states and that response increase, whereas the links be- tween the states and the other response decrease, always in proportion to each state’s activation. Conversely, if the choice response is extinguished, the links between the states and that response decrease, whereas the links between the states and the other response increase. In other words, the model assumes that, in bisection tasks, when the link between a state and one response changes, the link between the same state and the other response also changes, albeit in the opposite direction. Hence, the model’s learning rule implements a strong form of response competition’. —< = oe Green will be weak—hence the preference for Red after short samples. However, after the 4-s samples, later states will be the most active and because of the reinforcement contingencies their link with Red will be weak whereas their ink with Green will be strong—hence the preference for Green after the long samples. LeT predicts that preference for Red decreases as sample duration ranges from 1 to 4s. Moreover, it also predicts (see Machado, 997) a PSE close to, but slightly greater than, the geometric mean of the training stimuli and, when A is proportional to the overall reinforcement rate during the trials, that psychometric functions obtained with samples holding the same ratio will superimpose when plotted on a common axis.
Note. ‘‘Early’’, ‘“Middle’”’ and ‘‘Late”’ represent the states most active after 1-, 4, and 16-s samples, respectively. Initially, all links equal 0.5. The arrows show the effects of training in the double discrimination {S;, S4} — {Red, Green} and {S4, S;¢6} — {Blue, Yellow}. Strength of the links (W) between the states and the choice responses.
Note. ‘‘Early’’, ‘“Middle’”’ and ‘‘Late”’ represent the states most active after 1-, 4, and 16-s samples, respectively. Initially, all links equal 0.5. The arrows show the effects of training in the double discrimination {S;, S4} — {Red, Green} and {S4, S;¢6} — {Blue, Yellow}. Strength of the links (W) between the states and the choice responses.
Fig. 4. The left and middle panels show the predic- tions of SET and LeT, respectively, for the test trials of two groups exposed to a double bisection task. Both groups learned the mapping {S;, S4}-{Red, Green} on ‘‘Short”’ trials, but, on “‘Long”’ trials, Group 8 learned the mapping {S4, Sg}-{Blue, Yellow}, whereas Group 16 learned the mapping {S4, Sj¢}—>{Blue, Yellow}. The right panels show the data from Machado & Pata (2005).
Fig. 4. The left and middle panels show the predic- tions of SET and LeT, respectively, for the test trials of two groups exposed to a double bisection task. Both groups learned the mapping {S;, S4}-{Red, Green} on ‘‘Short”’ trials, but, on “‘Long”’ trials, Group 8 learned the mapping {S4, Sg}-{Blue, Yellow}, whereas Group 16 learned the mapping {S4, Sj¢}—>{Blue, Yellow}. The right panels show the data from Machado & Pata (2005).
Fig. 5. The top panels show the predictions of LeT for Groups Consistent and Inconsistent. Each curve shows the probability of choosing Green over Blue as a function of sample duration. The number on each curve identifies the session for which the curve applies (e.g., curve 0 = immediately after double bisection training, curve 1 = after one session with the new discrimination training, etc.). The bottom panels show the data from Machado & Arantes (2006). Choices following the 1-s and 16-s samples were reinforced provided they were correct, but choices following the 2-, 4- and 8-s samples were not reinforced. Green was correct following the 16-s samples for Group Consistent and the 1-s samples for Group Inconsistent.
Fig. 5. The top panels show the predictions of LeT for Groups Consistent and Inconsistent. Each curve shows the probability of choosing Green over Blue as a function of sample duration. The number on each curve identifies the session for which the curve applies (e.g., curve 0 = immediately after double bisection training, curve 1 = after one session with the new discrimination training, etc.). The bottom panels show the data from Machado & Arantes (2006). Choices following the 1-s and 16-s samples were reinforced provided they were correct, but choices following the 2-, 4- and 8-s samples were not reinforced. Green was correct following the 16-s samples for Group Consistent and the 1-s samples for Group Inconsistent.
Fig. 6. Psychometric functions obtained with the Free Operant Psychophysical Procedure—pecks to one key, say L, are reinforced (according to one or more VIs) only during the first half of the trial, and pecks to the other key, R, are reinforced (also according to one or more VIs) only during the second half of the trial. In the top panels, when the overall reinforcement rate favored the L key (first one of the two VI schedules), the psychometric function shifted to the right; when it favored the R key (last one of the two VI schedules), it shifted to the left. The middle panels show that when the overall reinforcement rates differ, the psychometric function shifts only if the local reinforcement rates differ around the middle of the trial; the bottom right panels show that when the overall reinforcement rates are equal, the functions shift provided the local reinforcement rates differ in the middle of the trial. The data are from Bizo & White (1995a) (top left panel) and Machado & Guilhardi (2000) (remaining panels). The curves show the fit of the LeT model.
Fig. 6. Psychometric functions obtained with the Free Operant Psychophysical Procedure—pecks to one key, say L, are reinforced (according to one or more VIs) only during the first half of the trial, and pecks to the other key, R, are reinforced (also according to one or more VIs) only during the second half of the trial. In the top panels, when the overall reinforcement rate favored the L key (first one of the two VI schedules), the psychometric function shifted to the right; when it favored the R key (last one of the two VI schedules), it shifted to the left. The middle panels show that when the overall reinforcement rates differ, the psychometric function shifts only if the local reinforcement rates differ around the middle of the trial; the bottom right panels show that when the overall reinforcement rates are equal, the functions shift provided the local reinforcement rates differ in the middle of the trial. The data are from Bizo & White (1995a) (top left panel) and Machado & Guilhardi (2000) (remaining panels). The curves show the fit of the LeT model.
Fig. 7. Average data from one pigeon exposed to a mixed FI 30s—-FI 120s schedule (points) and the fit of the LeT model (curve). Data from Catania & Reynolds (1968).
Fig. 7. Average data from one pigeon exposed to a mixed FI 30s—-FI 120s schedule (points) and the fit of the LeT model (curve). Data from Catania & Reynolds (1968).
Fig. 10. Model output for two mixed-FI schedules, mixed FI 15s-FI 120s and mixed FI 30s-FI 240s. Panel A: distribution of W(n). The smooth lines plot approximate closed-form solutions derived in the Appendix. Panel B: response distribution for one stat-pigeon during the last trial. Panel C: average response rate. The smooth curves plot the approximations derived in the Appendix. Panel D: the scalar property.
Fig. 10. Model output for two mixed-FI schedules, mixed FI 15s-FI 120s and mixed FI 30s-FI 240s. Panel A: distribution of W(n). The smooth lines plot approximate closed-form solutions derived in the Appendix. Panel B: response distribution for one stat-pigeon during the last trial. Panel C: average response rate. The smooth curves plot the approximations derived in the Appendix. Panel D: the scalar property.
Learning rules in the bisection task. corresponding peaks. The scalar property holds in two ways. First, the two average rate curves overlap when plotted in relative time (panel D—scalar property across mixed-FIs). And second, although not shown in the figure, the two ascending limbs of each response rate curve also overlap when plotted in relative time (scalar property within a mixed-FI).
Learning rules in the bisection task. corresponding peaks. The scalar property holds in two ways. First, the two average rate curves overlap when plotted in relative time (panel D—scalar property across mixed-FIs). And second, although not shown in the figure, the two ascending limbs of each response rate curve also overlap when plotted in relative time (scalar property within a mixed-FI).
Fig. 11. Model output for the temporal bisection task. Panel A: distribution of the associative links after training with the simple bisection task {S4, S;¢}{Red, Green}. The filled and empty circles show the associative links with the Red and Green responses, respectively. Panel B: psycho- metric functions plotting the probability of choosing “Short”’ during test trials in four simple bisection tasks. From left to right, the training sample durations were 1 vs. 4, 2 vs. 8, 4 vs. 16, and 8 vs. 32. Panel C: the scalar property with bisection at the geometric mean in the four simple bisection tasks. Panel D: results from the tests with novel key pairings after training on the double bisection task {S}, S4}-{Red, Green} and {S4, S;¢}—{Blue, Yellow}. Simulation details: training sessions comprised 400 trials with each sample; the generalization sessions comprised 64 trials for each test sample plus 384 trials for each training sample; in the double bisection task, the test sessions with the novel key pairings comprised 64 test trials for each test sample and 3200 trials for each training sample.
Fig. 11. Model output for the temporal bisection task. Panel A: distribution of the associative links after training with the simple bisection task {S4, S;¢}{Red, Green}. The filled and empty circles show the associative links with the Red and Green responses, respectively. Panel B: psycho- metric functions plotting the probability of choosing “Short”’ during test trials in four simple bisection tasks. From left to right, the training sample durations were 1 vs. 4, 2 vs. 8, 4 vs. 16, and 8 vs. 32. Panel C: the scalar property with bisection at the geometric mean in the four simple bisection tasks. Panel D: results from the tests with novel key pairings after training on the double bisection task {S}, S4}-{Red, Green} and {S4, S;¢}—{Blue, Yellow}. Simulation details: training sessions comprised 400 trials with each sample; the generalization sessions comprised 64 trials for each test sample plus 384 trials for each training sample; in the double bisection task, the test sessions with the novel key pairings comprised 64 test trials for each test sample and 3200 trials for each training sample.
Fig. 12. Model predictions for the temporal general- ization task (see Eq. 22 in the Appendix). Top panels: Generalization gradients when the reinforced signal was 4- s long and the nonreinforced signals were spaced linearly or logarithmically for a shorter (A) or longer (D) range. Middle panels: the reinforced signal was 2-, 4-, or 8-s long; the right panel shows the scalar property. Bottom left panel: effect of changing the reinforcement probability following the 4s, target signal. Bottom right panel: effect of changing the probability of presenting the 4-s, target signal on each trial.
Fig. 12. Model predictions for the temporal general- ization task (see Eq. 22 in the Appendix). Top panels: Generalization gradients when the reinforced signal was 4- s long and the nonreinforced signals were spaced linearly or logarithmically for a shorter (A) or longer (D) range. Middle panels: the reinforced signal was 2-, 4-, or 8-s long; the right panel shows the scalar property. Bottom left panel: effect of changing the reinforcement probability following the 4s, target signal. Bottom right panel: effect of changing the probability of presenting the 4-s, target signal on each trial.
Fig. Al. The top panel shows two samples paths of N(t), the active state at time t, in an FI T = 15 s. The activation speed 4 came from a Gaussian distribution with mean t= | and standard deviation c= 0.2. When A= 0.8, the last active (and reinforced) state, N(T), equalled 13; when A= 1.2, N(T) = 19. The bottom panel illustrates, for n = 12, how q(n,T), p(n,T) and r(n,T) relate to the Gaussian density function for A. respectively, with t = 0 andn = 1, @... State dynamics, N(t). On each trial, the states are activated serially, starting with state | at trial onset. The states are activated at a rate of 1 states per second, with A> 0 a Gaussian random variable with mean pt and standard deviation o. We represent by N(x, pt, o) the density function, and by ®(x, pt, o) the distribution function, of that Gaussian variable evaluated at x. Sn a a ee ee ae
Fig. Al. The top panel shows two samples paths of N(t), the active state at time t, in an FI T = 15 s. The activation speed 4 came from a Gaussian distribution with mean t= | and standard deviation c= 0.2. When A= 0.8, the last active (and reinforced) state, N(T), equalled 13; when A= 1.2, N(T) = 19. The bottom panel illustrates, for n = 12, how q(n,T), p(n,T) and r(n,T) relate to the Gaussian density function for A. respectively, with t = 0 andn = 1, @... State dynamics, N(t). On each trial, the states are activated serially, starting with state | at trial onset. The states are activated at a rate of 1 states per second, with A> 0 a Gaussian random variable with mean pt and standard deviation o. We represent by N(x, pt, o) the density function, and by ®(x, pt, o) the distribution function, of that Gaussian variable evaluated at x. Sn a a ee ee ae
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