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A connectionist model of stimulus class formation with a yes/no procedure and compound stimuli.

Stimulus classes or categories have been defined as stimulus sets that occasion common responses in a specific context; such sets include stimuli involved in an explicit learning history and new stimuli to which behavioral control functions acquired by such history can be transferred (Zentall, Galizio, & Critchfield, 2002).

Research on stimulus equivalence classes in humans--as a case of category formation--has been done principally with the matching-to-sample (MTS) procedure, in which conditional relations between some stimuli of a potential class can be established. For example, in order to establish a class formed by stimuli A1, B1, and Cl, relations A1rB1 and B1rC1 are trained. In order to evaluate class formation, the relations that were not directly trained, known as emergent relations, are tested through symmetry (i.e., BlrAl and C1rB1), transitivity (i.e., A1rC1), and equivalence trials (i.e., ClrA1; see Sidman, 1992, 1994, 2000; Sidman & Tailby, 1982).

Research on equivalence classes through the use of connectionist models (CMs) has been done. These models, also known as artificial neural networks, can simulate complex behavior by learning to produce response patterns that are similar to the responses observed in humans when a similar stimulation is presented.

CMs are considered research tools that enable behavioral scientists to explore, analyze, and question data obtained by other strategies (for a review of connectionism, see McClelland, 2009; Plunkett & Elman, 1997). It has been proposed that CMs offer advantages such as the complete control of stimuli introduced to the network; therefore, phenomena are explored without the complication of an unspecified preexperimental learning history (Lyddy & Barnes-Holmes, 2007).

CMs consist of a number of interconnected processing units. Each unit receives, computes, and propagates information, which is represented by different activation values. The connections between the units are defined by a weight value. Units are organized in different layers, simulating either specific brain structures or behavioral processes, by a system with input/stimulation, hidden/representation, and output/response layers.

The activation values of the input layer are specified directly by the experimenter and represent different stimulation environments; the activation values of the hidden and output layers are generated by the functioning of the model. CMs are able to learn complex tasks by modifying their connection weights by means of a learning algorithm (see, e.g., McClelland, 2009; O'Reilly & Munakata, 2000; Thomas & McClelland, 2008).

These models have been used in the study of diverse psychological phenomena, such as pattern recognition (Rumelhart, McClelland, & PDP Research Group, 1986), conditional discriminations (Maki & Abunawass, 1991), generalization (Gluck, 1991), and early word learning (Mayor & Plunkett, 2010). CMs have been used also for the study of equivalence class formation. Barnes and Hampson (1993) proposed a model called RELNET--there are other RELNET-derived models (e.g., Cullinan, Barnes, Hampson & Lyddy, 1994; Lyddy & Barnes-Holmes, 2007; Lyddy, Barnes-Holmes, & Hampson, 2001). These authors pointed out that one contribution of RELNET models has been to provide evidence showing that CMs are able to respond correctly to relations between stimuli that have not been directly trained, just like human participants do.

The network developed by Barnes and Hampson (1993) has three principal elements in the input pattern: The first element represents the actual stimuli that function as samples and comparisons; the second element (the "sample-marking duplicator") marks the stimulus that functions as the sample; and the third element represents the contextual stimulus (same, different, and opposite). Barnes and Hampson (1993) pointed out that
  the second element of the input ... basically copies, or
  mirrors, the activation from each task ... and marks one
  of the stimuli as a sample from the task (e.g., when A1
  is activated as a sample with BI and B2 as comparisons, then
  Z1, Z1/s, Z2, and Z3 are activated ...). The sample-marking
  duplicator mirrors activation in exactly the same way for
  each individual task across each of the eight stimulus sets
  (e.g., when W1 is activated as a sample with X1 and X2 as
  comparisons then Z1, Z1/s, Z2, and Z3 are activated in the
  sample marking duplicator). (p. 625)


During probes for emergent relations, the stimulation patterns are identical regarding the second element of the input (sample-marking duplicator) to some of the training trials (look at Z input units in the previous paragraph). Thus, even when the first element changes depending on the particular trial, the marking element of the sample stimulus remains constant. This has a determinant influence on the activation values obtained in the output layer; as a consequence, RELNET models do not give an "emergent" response in the test trials. The outputs generated by these networks result from previously learned information about the sample marking.

In order to avoid stimuli marking problems, one alternative is to design a simulation that will not require the representation of sample and comparison stimuli functions. In this context, the go/no-go and yes/no procedures with compound stimuli have been suggested as alternatives to the MTS procedure for the study of stimulus class formation (Debert, Huziwara, Faggiani, Simoes de Mathis, & McIlvane, 2009; Debert, Matos & McIlvane, 2007; Fields, Doran, & Marroquin, 2009). In these procedures, compound stimuli are formed by two components (e.g., abstract figures) that belong either to the same class or to different classes, arbitrarily defined by the researcher.

In the go/no-go procedure, participants are required to emit a response (go) in the presence of compound stimuli with components that belong to the same class and not emit the response (no-go) in the presence of compound stimuli with components that belong to different classes (Debert et al., 2007).

In the yes/no procedure, participants emit two different responses; when the two components of a compound stimulus belong to the same class, participants should respond with the yes option; when the two components do not belong to the same class, participants should respond with the no option (Fields et al., 2009).

Using go/no-go responses, Debert et al. (2007) established three 3-member stimulus classes. The participants were taught to respond (go) in the presence of compound stimuli formed by the pairs A1B1, B1C1, A2B2, B2C2, A3B3, and B3C3 and not to respond (no-go) when the stimulus pairs A1B2, A1B3, B1C2, B1C3, A2B1, A2B3, B2C1, B2C3, A3B1, A3B2, B3C1, and B3C2 are presented. During emergent relation probes, four out of six participants responded (go) when facing new stimulus pairs whose components belonged to the same class (i.e., A1C1, A2C2, and A3C3); and they did not respond (no-go) in the presence of stimulus pairs whose components belonged to different classes (i.e., A1C2, A1C3, A2C1, A2C3, A3C1, and A3C2).

Fields and colleagues (2009) showed the formation of stimulus classes through compound stimuli with yes/no and same/diff responses. In their second experiment, they found that 100% of the participants in the preliminary training condition (i.e., generalized transitivity repertoire) formed all equivalence classes; such preliminary training consisted of teaching all the stimulus relations in different equivalence classes. Transitivity trials were presented with direct reinforcement instead of testing under extinction.

The studies with compound stimulus procedures showed that it is not essential to consider different functions for stimuli, which, in an MTS arrangement, work as conditional/sample or discriminative/comparison (Debert et al., 2007; Debert et al., 2009). We consider that these procedures are preferable when working with computational simulations because they allow the design of parsimonious CMs; these models can be centered on the analysis of stimulus relations without considering special stimulus functions.

Therefore, our main purpose was to develop a CM to evaluate stimulus class formation, avoiding the problems generated by the marking of stimulus functions and allowing an appropriate evaluation of CMs capability to respond correctly to emergent relations between stimuli.

In this context, we carried out a stimulus class formation study with human participants using compound stimuli and yes/no responses. This was made in order to obtain data to contrast the performance of the CM that simulates stimulus class formation.

Method

Participants

Six female undergraduate students, ages 19 to 22 years, participated in this study. None of them had previous experience with class formation experimental tasks. They received partial course credit for their participation, regardless of performance. Informed consent was obtained from all participants.

Apparatus

Sessions were conducted in a 4 m x 6 m room; each participant was seated facing a touch-screen monitor connected to a Dell computer. A custom software developed in Visual Basic controlled all stimulus presentations and recorded all responses. The stimuli were six abstract figures such as the ones used by Debert et al. (2007), originally developed by Markham and Dougher (1993); they were designated as Al, Bl, Cl, A2, B2, and C2 (see Figure 1).

[FIGURE 1 OMITTED]

Procedure

The procedure was similar to the one used by Debert et al. (2007), but instead of working with go/go-no we used yes/no responses, the training phase was divided into six stages, and we worked with two experimental classes instead of three. The experiment was carried out in two sessions conducted in two consecutive days.

Session 1, Phase 1: Training of baseline relations. During this phase, the compound stimuli were introduced and participants chose between the yes and no responses for each stimulus pair. The yes response was reinforced in the presence of the pairs A1B1, B1C1, A2B2, and B2C2; the no response was reinforced in the presence of the pairs A1B2, B1C2, A2B1, and B2C1. The designation of each stimulus pair makes reference to the spatial location of the abstract figures: When A1B1 is read, Al appears on the left side and B1 on the right side (see Figure 2).

[FIGURE 2 OMITTED]

At the beginning of Session 1, participants were asked to read the following instructions (translated from the Spanish).
  Thank you for your participation.

  The following study is not an intelligence test
  and will not evaluate any aspect of your intellectual
  abilities. At the end of the study you will receive
  a complete explanation. A researcher will remain inside
  the room in order to help you if a technical problem
  arises, but he will not provide you with assistance
  to answer the test.

  Your goal is to obtain as many points as possible;
  these points will be shown on a counter placed in
  the upper left corner of the screen.

  At the center of the screen, two figures will be
  displayed. Your task is to choose yes by touching the
  screen on the yes option whenever you consider that
  figures are related, and you will touch the no option
  whenever you consider figures are not related.

  You will be able to learn which figures are related
  and which are not by checking the points on the counter.

  The task seems simple, but it will get more difficult
  as it goes on, so pay attention.

  Please repeat these instructions to the experimenter,
  and when he approves, touch in the "Go on" box.


After touching the "Go on" box, the trials began. On each trial, a compound stimulus appeared at the center of the screen, and the yes and no responses were at the bottom. There was a counter on the upper left corner of the screen with the total points displayed in black (see Figure 2).

When participants made a correct response, the counter increased its value by 5 points and flickered in green for 2 seconds; at the same time, the participant heard an "approval" sound. When the responses were wrong, the counter decreased its value by 5 points and flickered in red for 2 seconds, accompanied by an "error" sound.

During the 2 seconds of the flickering, the compound stimulus and the response options remained visible on the screen; however, no consequences were programmed for responding during that period. The trials were separated by 1 s intertrial intervals. The next trial started with the value on the counter in black.

Phase 1 was divided into six stages. Table 1 shows the sequence of stages and learning criteria used during this phase. The trained relations in each stage were presented in a random order; there were no more than ft ree consecutive trials of the same type. Each stage was repeated until participants reached the mastery criterion.
Table 1
Sequence of Stages During Phases 1 and 2

Stage  Compound  Correct   Reinforcement    Phase 1 Training
       Stimuli   Response     Percentage  Number of  Criteria
                                             Trials

1      A1B1      YES                100%         12     21/24

       A1B2      NO                 100%         12

2      B1C1      YES                100%         12     21/24

       B1C2      NO                 100%         12

3      A2B2      YES                100%         12     21/24

       A2B1      NO                 100%         12

4      B2C2      YES                100%         12     21/24

       B2C1      NO                 100%         12

5      A1B1      YES                100%          3     23/24

       B1C1      YES                100%          3

       A2B2      YES                100%          3

       B2C2      YES                100%          3

       A1B2      NO                 100%          3

       B1C2      NO                 100%          3

       A2B1      NO                 100%          3

       B2C1      NO                 100%          3

6      A1B1      YES                  0%          3     23/24

       B1C1      YES                  0%          3

       A2B2      YES                  0%          3

       B2C2      YES                  0%          3

       A1B2      NO                   0%          3

       B1C2      NO                   0%          3

       A2B1      NO                   0%          3

       B2C1      NO                   0%          3

Stage    Compound          Phase 2 Retraining
          Stimuli        Number of   Criteria
                            Trials

1            A1B1                6      11/12

             A1B2                6

2            B1C1                6      11/12

             B1C2                6

3            A2B2                6      11/12

             A2B1                6

4            B2C2                6      11/12

             B2C1                6

5            A1B1                2      15/16

             B1C1                2

             A2B2                2

             B2C2                2

             A1B2                2

             B1C2                2

             A2B1                2

             B2C1                2

6            A1B1                2       15/16

             B1C1                2

             A2B2                2

             B2C2                2

             A1B2                2

             B1C2                2

             A2B1                2

             B2C1                2

Note. The compound stimuli, percentage of reinforcement,
total number of trials, and criteria are shown for each
stage.


During Stages 1 to 5, all responses generated the programmed consequences. At the beginning of Stage 6 the participants were informed about the lack of feedback, and they read the next instructions in the computer screen: "In the next trials, the counter and the sounds will be removed. However, your correct and wrong responses will be considered. Keep doing your best in accordance with what you have learned."

Session 1 was extended until participants met the learning criteria of Phase 1; no more than 30 minutes were required for each participant.

Session 2, Phase 2: Retraining of baseline relations. During this phase, the baseline relations were retrained. The six stages of Phase 1 were presented with half of the trials. Table 1 shows information about the total number of trials and the learning criteria for each stage in Phase 2.

After completion of Phase 2, participants moved to emergent relations tests. Phases 3 and 4 tested for symmetry-like and transitivity-like relations, respectively. We use these terms because, traditionally, symmetry and transitivity are used in MTS procedures.

Phase 3: Symmetry-like tests. This phase tested for symmetry-like relations; components of the stimulus pairs were presented in a reversed spatial location. For example, if A1 was located at the left side of Bi during training (A1B1), in the symmetrylike test, Bi was located at the left side of A1 (B1A1). There were eight symmetry-like trials, one for each trained relation (see Table 2). Each relation appeared four times in a random order; thus, a total of 32 trials were presented. At the beginning of this phase, participants were informed about the lack of feedback. At the end of the 32 trials, participants moved to Phase 4.

Phase 4: Transitivity-like tests. During this phase we tested the emergence of the yes and no responses in the presence of new compound stimuli. A total of 96 trials were presented, 48 of transitivity-like relations and 48 of equivalence-like relations. Each relation was presented 12 times in a random order (see Table 2). At the beginning of this phase, participants were informed again about the lack of feedback. Session 2 lasted no more than 25 minutes.
Table 2
Compound Stimuli Presented During Phases 3 and 4

           Phase                       Phase
               3                           4
Compound  Number   Correct  Compound  Number Correct
              of                          of
Stimuli   Trials  Response   Stimuli  Trials  Response

B1A1           4       YES      A1C1      12       YES

B2A2           4       YES      A2C2      12       YES

B1A2           4       NO       A1C2      12       NO

B2A1           4       NO       A2C1      12       NO

C1B1           4       YES      C1A1      12       YES

C2B2           4       YES      C2A2      12       YES

C1B2           4       NO       C1A2      12       NO

C2B1           4       NO       C2A1      12       NO


Results and Discussion

All six participants completed the two experimental sessions. Four out of six participants formed the two stimulus classes (A1B1C1 and A2B2C2); in Phases 3 and 4, they responded in accordance with the stimulus class memberships established during training with accuracy levels of 85% or higher.

Table 3 presents the number of stage repetitions and number of trials required to complete the training and retraining phases as well as the number and percentage of correct trials during symmetry-like and transitivity-like tests. In most cases, participants required only one stage repetition in order to achieve the criterion. An outstanding exception was Participant 101, who required five repetitions of Stage 5 to meet the criterion. Her errors were not made in a particular relation; however, during the retraining phase, she made only one error in Stage 4 and four errors in Stage 5. The other participants showed a minimal number of errors during retraining stages. Participant 106, for example, showed an errorless performance in all retraining stages.
Table 3
Performance of the six participants during Sessions 1 and 2

             Training and Retraining (Phases 1 and 2)

Participant  Session  Stage 1  Stage 2  Stage 3  Stage 4      Stage5
                                                                 All
                                                           relations
                                                              reinf.
                                                                100%

101          1              1        1        1        1  5 (86/120)
                      (21/24)  (23/24)  (24/24)  (24/24)

             2              1        1        1        1   2 (28/32)
                      (12/12)  (12/12)  (12/12)  (11/12)

102          1              2        1        1        1   2 (44/48)
                      (39/48)  (23/24)  (21/24)  (22/24)

             2              1        1        1        1   1 (15/16)
                      (11/12)  (12/12)  (12/12)  (12/12)

103          1              1        1        1        1   1 (22/24)
                      (22/24)  (23/24)  (24/24)  (23/24)

             2              1        1        1        1   1 (16/16)
                      (12/12)  (12/12)  (12/12)  (11/12)

104          1              1        1        1        1   1 (23/24)
                      (22/24)  (22/24)  (23/24)  (24/24)

             2              1        1        1        1   1 (15/16)
                      (12/12)  (11/12)  (12/12)  (12/12)

105          1              2        1        1        1   1 (23/24)
                      (44/48)  (22/24)  (23/24)  (24/24)

             2              1        1        1        1   1 (15/16)
                      (12/12)  (12/12)  (12/12)  (12/12)

106          1              1        1        1        1   1 (24/24)
                      (21/24)  (23/24)  (24/24)  (24/24)

             2              1        1        1        1   1 (16/16)
                      (12/12)  (12/12)  (12/12)  (12/12)


                                    Tests (Phases 3 and 4)
Participant  Session     Stage6  Symmetry-like  Transitivity-like
                            All    trials             trials
                      relations
                      reinf. 0%

101          1        3 (59/72)

             2        1 (16/16)   (32/32) 100%      (94/96) 97.9%

102          1        1 (23/24)

             2        2 (30/32)   (32/32) 100%      (60/96) 62.0%

103          1        1 (24/24)

             2        1 (16/16)  (30/32) 93.7%      (93/96) 96.8%

104          1        1 (24/24)

             2        1 (16/16)   (32/32) 100%      (54/96) 56.0%

105          1        1 (24/24)

             2        1 (16/16)  (31/32) 96.8%       (96/96) 100%

106          1        1 (24/24)

             2        1 (16/16)   (32/32) 100%       (96/96) 100%

Note. The number of stage repetitions is indicated in bold type.
The number of correct responses and total of trials are in
parentheses. Percentage of correct performance during testing
is also shown.


During tests for emergent relations (Phases 3 and 4), Participants 101, 105, and 106 showed the highest performance by responding correctly in at least 96% of the trials. Participant 103 achieved accuracy levels of 93.7% correct and 96.8% correct in Phases 3 and 4, respectively. These four participants were the ones who formed classes.

Participant 102 responded correctly to all the symmetry-like trials; however, she achieved 60% correct in the transitivity-like trials. She made 36 errors. The trials in which she made most errors were A1C1 (12 errors) and C1A1 (9 errors). For these two stimulus pairs, the yes response was expected and she responded with no. The 15 remaining errors were made in the presence of the other compound stimuli.

Participant 104 responded correctly to all symmetry-like trials, but in the transitivity-like tests she achieved 56% correct. At the beginning of Phase 4, her errors were distributed across all compound stimuli--the ones whose components were related and the ones whose components were not related--but throughout the course of the trials she began to show more errors in the presence of compounds whose components were related while she showed correct responses in the presence of compounds whose components were not related; this was because she responded no in the last 25 trials. This suggests that Participant 104 could use the no response by default; this type of performance has been described previously by Fields et al. (2009).

As suggested by Debert et al. (2009), components of the compound stimuli do not have a specific function as conditional or discriminative stimuli. However, we can consider a conditionality between the compound stimuli and the response options because the display used in this study can also be considered as a MTS format, where the compound stimulus could be viewed as a conditional or sample stimulus and the yes and no responses could be viewed as discriminative or comparison stimuli. Therefore, the compound stimuli A1B1, B1C1, A2B2, B2C2 and the yes response can be integrated in a class, and the compound stimuli A1B2, B1C2, A2B1, B2C1 and the no response can be integrated in another class. However, we did not administer the necessary probes in order to evaluate this hypothesis.

Another aspect of the present results that requires discussion is the equal likelihood of the yes and no responses to emerge in the presence of new compound stimuli. As Fields et al. (2009) pointed out, during baseline training the stimulus pairs A1B1 and B1C1, associated with the yes response, established the basis to evoke the yes response in the presence of A1C1. Similarly, the stimulus pairs A1B2 and B2C1, associated with the no response, also established the basis to evoke the no response in the presence of A1C1. Likewise, the stimulus pair A2C2 could evoke the yes and no responses with equal likelihood. However, in the present study, four out of six participants used the yes response when the pairs A1C1 and A2C2 were presented. We think that their performance could be biased by the semantic-inclusion role that the yes response has acquired extraexperimentally, probably enhanced by the instructions on the use of the yes/no responses (see Method section). Future research should examine this hypothesis by using arbitrary figures as response options.

In the next section, we show the architecture of the CM, the experimental task used, and the obtained results.

Connectionist Model

We developed a three-layer feed-forward artificial neuronal network to simulate the human study (see Figure 3). The input layer was composed of nine units. The activation (i.e., 1) or no activation (i.e., 0) of input units represented the presence or absence of each component of the compound stimuli (e.g., the compound stimulus A1B1 was represented with the input pattern 100000010). Table 4 shows all input patterns used during training. The position of the components of the compound stimulus in the input layer was determined in a way that did not allow two components of the same class to be adjacent. This was made in order to avoid teaching spatial or sequential relations between stimuli.

[FIGURE 3 OMITTED]

The hidden layer consisted of four units. The activation values on this layer were considered as the representation developed by the network for each trial presented in the input layer; the hidden layer was fully connected to the input and the output layers. The output layer consisted of two units that represent the yes and rio responses, respectively (see Figure 3).

Each input pattern activated the units in the input layer, and then the activation passed on to the hidden layer and from this to the output layer. This kind of propagation is known as feed-forward (see Thomas & McClelland, 2008). The spread of activation levels was determined by the connection weights, initially randomized.

In the training phase, baseline relations were established. The network learned to activate the yes response when the compound stimuli A1B1, B1C1, A2B2, and B2C2 were presented as inputs. It also learned to respond to the compound stimuli A1B2, B1C2, A2B1, and B2C1 by activating the no response (see Table 4). Training used with humans was simulated in this way.
Table 4
Input Patterns Presented During Training and Correct
Output Patterns for Each Compound Stimulus

          Components of the Compound Stimuli in the
                Input Layer

Compound             A1  Z  B2         X  CI  Y  C2  B1  A2      YES
Stimuli                            Input                     Correct
                                Patterns                      Output

A1B1                  1  0   0         0   0  0   0   1   0        1

A1B2                  1  0   1         0   0  0   0   0   0        0

B1C1                  0  0   0         0   1  0   0   1   0        1

B1C2                  0  0   0         0   0  0   1   1   0        0

A2B2                  0  0   1         0   0  0   0   0   1        1

A2B1                  0  0   0         0   0  0   0   1   1        0

B2C2                  0  0   1         0   0  0   1   0   0        1

B2C1                  0  0   1         0   1  0   0   0   0        0

XY                    0  0   0         1   0  1   0   0   0        1

YZ                    0  1   0         0   0  1   0   0   0        1

XZ                    0  1   0         1   0  0   0   0   0        1

           Response Units in the Output Layer

Compound     NO Patterns (trained)
Stimuli

A1B1                            0

A1B2                            1

B1C1                            0

B1C2                            1

A2B2                            0

A2B1                            1

B2C2                            0

B2C1                            1

XY                              0

YZ                              0

XZ                              0


During this phase, the model also learned all the relations between the components of a third class designated as XYZ; that is to say, the model learned to activate the yes response when the pairs XY, YZ, and XZ were presented as inputs. This constitutes a generalized transitivity repertoire (Fields et al., 2009). This was made with the purpose of simulating the previous knowledge acquired extraexperimentally by human participants about class formation.

Other connectionist models have suggested that the complete training of additional classes facilitates the formation of new equivalence classes (Barnes & Hampson, 1993; Lyddy & Barnes-Holmes, 2007). The complete XYZ class training also allows the model to learn to use differentially the units of the output layer as yes and no responses. As the units are functionally similar, the model must learn that the activation of the yes response unit in the output layer follows the presentation of any stimulus pair whose components belong to the same class.

In each trial, the connectionist model learned the correct response by comparing the activation values generated in the output layer with the ones expected in such layer (see right side of Table 4). in this way, the network found an error value to which the connection weights were adjusted. As a consequence, in the following trials, error decreased. The model learned by error correction, modifying the connection weights through the back-propagation algorithm. For the adjustment of the weights, a learning rate of 0.3 was used. The learning rate is a constant value between 0 and 1 that determines the magnitude of change in the weights (McLeod, Plunkett, & Rolls, 1998; Rumelhart, McClelland, et al., 1986).

Training phase ended when the root mean square error (RMS) reached a value [greater than or equal to] 0.05, then we proceeded with the tests for emergent relations.

During testing, the learning algorithm was deactivated so that there were no more changes in the network due to learning processes, and the connection weights achieved at the end of the training phase were fixed. In this phase, the compound stimuli A1C1, A2C2, A1C2, and A2C1 were presented to the network. Each compound stimulus was presented only once (see Table 5). Since this model is inspired in a compound stimuli arrangement that do not consider spatial location, no symmetry-like trials were tested.
Table 5
Input Patterns Presented During Testing

             Components of the compound stimuli

Compound  A1  z          B2  X  C1  Y  C2  B1  A2
Stimuli

             Activation pattern in the input layer

A1C1       1  0           0  0   1  0   0   0   0

A1C2       1  0           0  0   0  0   1   0   0

A2C2       0  0           0  0   0  0   1   0   1

A2C1       0  0           0  0   1  0   0   0   1


The training and testing phases were repeated in six runs of the CM, which were equivalent to the six participants of the human study; each of the six runs started at different initial random weights (between 0 and 1).

Results and Discussion

During training, the six runs achieved an RMS error value 0.05 in an average of 13,898 iterations. This indicates that the model acquired the baseline relations properly. During testing, we considered the model execution as indicative of class formation if the yes unit reached an activation value [greater than or equal to] 0.85 when the pairs A1C1 and A2C2 were presented and also if the no unit reached that value when the pairs A1C2 and A2C1 were presented in the input layer.

Based on the above criteria, Runs 1 to 5 showed class formation, in the sixth run activation values were 0.788 in the yes unit for the stimulus pair A1C1 and 0.841 in the no unit for the stimulus pair A1C2. For stimulus pairs A2C2 and A2C1, the correct output unit exceeded 0.85 activation value. However, when considering all trials, this run did not form the two stimulus classes (see Figure 4).

[FIGURE 4 OMITTED]

Cluster Analysis of Hidden Layer

In the hidden layer, the first transformation of the input patterns is done so that an intermediate representation between the input and the output patterns is developed (see Thomas & McClelland, 2008). With a cluster analysis we can observe which stimulation patterns were transformed to generate similar representations in the hidden layer. This analysis grouped similar patterns by computing the Euclidean distance between them. Figure 5 shows the graphical representation of the cluster analysis of the second run. Cluster analyses conducted on the other five runs generated similar groups; these are not presented here but can be obtained from the authors.

Three main clusters can be observed in Figure 5: the first with the stimulus pairs A1B1, B1C1, and A1C1; the second cluster with the pairs A2B2, B2C2, and A2C2; and the third with the pairs A2C1, B1C2, A2B1, A1B2, B2C1, and A1C2. The stimulus pairs of the first two clusters were associated with the yes response; the pairs of the third one were associated with the no response.

[FIGURE 5 OMITTED]

This clustering indicates that during training in the hidden layer the network transformed the input patterns into similar activation values, depending on their class membership. For example, stimulus pair A2B2 was represented in the input layer with the pattern 001000001, and stimulus pair B2C2 was represented with the pattern 001000100. However, both pairs had an activation value of 0.9, 0.0, 0.0, and 0.9 in the four units of the hidden layer, respectively. The network established the connection weights that allowed the stimulus pair A2C2 (represented with the input pattern 000000101) to produce the values 0.9, 0.0, 0.0, and 0.9 in the four units of the hidden layer during testing.

This could be one strategy for responding correctly during arbitrary class formation tests that humans can also use; no matter how different the stimulus pairs are, all of them can evoke a similar representation.

Cluster analysis revealed that the model classifies the compound stimuli before producing the "final" yes or no response. This final response can be considered as a general classification; multiple class memberships can be seen for the different compound stimuli. For example, pairs A1B1, B1C1, and A1C1 formed Class 1; pairs A2B2, B2C2, and A2C2 formed Class 2. These two classes are included in the yes class. The model is able to work with multiple class memberships. It can generate the same response for different classes and keep the integrity of each class in the hidden layer representation.

A similar process could occur with the human participants but in a behavioral level that is difficult to analyze because we can just observe the general classification response. However, the data of a postexperimental interview applied to the human participants point out that they can also distinguish between Classes 1 and 2; thus, the data obtained from the model and from the postexperimental interview seem to match.

Further Implications of the Model

In order to test the semantic bias supposition (see Results and Discussion section of the human study), we performed six new runs (7-12) of the CM. We removed the training of the XYZ class to suppress the complementary learning history associated with the response options so that they can be considered just "Response-1" and "Response-2"; this is similar to using arbitrary response options with human participants. The CM only learned to respond to the pairs A1B1, B1C1, A2B2, and B2C2 with Response-1 and to respond to the pairs A1B2, B1C2, A2B1, and B2C1 with Response-2.

Probes for emergent relations under this condition revealed as expected: that the model cannot respond consistently along all runs. Only in Runs 7 and 8, Response-1 unit was activated to the test pairs AlC1 and A2C2 and Response-2 unit was activated to the test pairs A1C2 and A2C1; in Runs 9,11, and 12, the model inverted this activation pattern; finally, in the Run 10 the output, activation was undefined.

These results support our hypothesis and suggest that when working with two stimulus classes in a compound stimuli procedure, the use of arbitrary response options will generate a heterogeneous pattern of responses during testing.

General Discussion

The characteristics of the yes/no procedure with compound stimuli enabled the simulation of stimulus class formation in a CM. The model was capable of acquiring all baseline relations and responding correctly to the emergent relations. This model focused on the relations between the components of the compound stimuli. The simulation did not require the inclusion of sample and comparison functions. This represents a different approach to class formation connectionist modeling (see Barnes & Hampson, 1993; Cullinan et al., 1994; Lyddy & Barnes-Holmes, 2007; and Lyddy et al., 2001, for other approaches). These results showed that CMs are capable of responding correctly to nondirectly trained relations between stimuli.

The present results also extend the previous findings on arbitrary stimulus class formation in humans using yes/no procedures. This supports the idea that class formation can be analyzed with procedures other than MTS (e.g., Delbert et al., 2007; Debert et al. 2009; Fields et al., 2009). Future research should focus on detailing the role of the semantic bias of the response options and the instructions.

An advantage of the CM is that it allows us to analyze the behavioral process in an observable system. In the next paragraphs, we summarize what we consider the main contributions of the present model.

By working with compound stimuli and yes/no responses, we were able to use simple stimulation patterns. Using cluster analysis, we could observe how the model classifies the compound stimuli in the hidden layer. With this classification process, the model is able to produce the correct response; the compound stimuli that belong to the same class generated the same activation values in the hidden layer. We hypothesized that this could be one strategy that humans can also use, no matter how different the stimuli are. Once the class is already formed, all stimuli evoke a similar representation.

This model allowed us to manipulate the effect of the complete training of a third class (XYZ). When it was trained, five out of six runs exhibited stimulus class formation, and when training of XYZ was removed, the model did not form the stimulus classes along six new runs consistently. This indicates that the XYZ training established the necessary experience in order to acquire a "generalized transitivity repertoire" (e.g., Fields et al., 2009).

As mentioned in the Results and Discussion section of the human study, the experimental procedure established the basis to evoke the yes and the no responses with equal likelihood when A1C1 and A2C2 trials were presented. Nevertheless, the preferred election of the yes response by both the human participants and the model leads us to think that there is a functional similarity between the semantic bias and the generalized transitivity repertoire trained to the model. Future research should analyze stimulus class formation with compound stimuli and arbitrary responses. The present model indicates that class formation will be difficult to demonstrate under these conditions.

In general, both the human study and the CM showed the relevance of analyzing arbitrary stimulus class formation with different research strategies. Both strategies give each other feedback and generate a reciprocal heuristic value. The use of different experimental and simulation procedures extends our knowledge on the determinant factors involved in arbitrary categorization and symbolic processing.

This study was conducted in partial fulfillment of the requirements of the PhD degree by the first author at Universidad Nacional Autonoma de Mexico. The first author was supported by a fellowship from the Mexican National Council for Science and Technology (CONACYT). We thank Dr. Florente Lopez for his thoughtful suggestions on the manuscript.

Correspondence concerning this article should be addressed to Angel E. Tovar or Alvaro Torres Chavez, both at Departamento de Psicologia Experimental, Universidad Nacional Autonoma de Mexico, Av. Universidad #3004, Mexico, Distrito Federal, Coyoacan, Codigo postal 04510. E-mail: eugeniot@comunidad.unam.mx or alvarot@unam.mx

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Angel E. Tovar and Alvaro Torres Chavez

Universidad Nacional Autonoma de Mexico
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Date:Sep 22, 2012
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