Control by action representation and input selection (CARIS): a theoretical framework for task switching

A numerical example will clarify this point. In Model 1, response representation is perseverative. Let us assume for example that w _A = 0.51. A switch from SHAPE to SIZE would result in representing Response A and Response B as (.49 0 .51 0) and (0 .49 0 .51). (Note that in both the cases, the weight given to the relevant feature, size, is smaller than that given to the irrelevant feature, shape, reflecting the fact that shape was relevant in Trial n − 1). If the response used to indicate LARGE and CIRCLE is repeated w _A−REPEAT = w _A  + w _CR (where w _A−REPEAT represents w _A for the repeated response) and the selection vector for Response A becomes \( {\left( {\begin{array}{*{20}c} {{.49 - w_{{{\text{CR}}}} }} \\ {{.49 - w_{{{\text{CR}}}} }} \\ {{.51 + w_{{{\text{CR}}}} }} \\ {{.51 + w_{{{\text{CR}}}} }} \\ \end{array} } \right)} \) If w _CR = .03, the selection vector for Response A in the present example becomes \( {\left( {\begin{array}{*{20}c} {{.46}} \\ {{.46}} \\ {{.54}} \\ {{.54}} \\ \end{array} } \right)}. \) Note that the selection vector is translated into a response representation such as \( {\left( {\begin{array}{*{20}c} {{.46}} \\ {0} \\ {{.54}} \\ {0} \\ \end{array} } \right)}, \) for example. This implies that the bias in favor of the emphasized dimension in Response A is increased relative to Response B, for which the bias vector would be \( {\left( {\begin{array}{*{20}c} {{.49}} \\ {{.49}} \\ {{.51}} \\ {{.51}} \\ \end{array} } \right)} \) with a response representation that may be \( {\left( {\begin{array}{*{20}c} {0} \\ {{.49}} \\ {0} \\ {{.51}} \\ \end{array} } \right)} \) keeping in line with the example. Given that the emphasis is based on the n−1st task, it is counterproductive in switch trials, and becomes even more counterproductive if the response is repeated.

In comparison, the number of significant degrees of freedom (including the intercept term) in the standard 5-way ANOVA was 21 (actually almost 22 because of an additional parameter with p = 0.059). To determine these numbers we needed to break down the 2-df Switch variable into orthogonal contrast. The contrasts were switch cost: switch vs. repeat, and mixing cost: switch + repeat vs. single task. This was done for the main effect of Switch and all the interactions involving Switch).

Specifically, response selection is based on a competition between the correct response and the incorrect response. In congruent trials, no such competition exists because the irrelevant stimulus dimension activates the correct response. Competition arises in incongruent trials because the irrelevant stimulus dimension activates a competing response. Correct responding in these trials therefore depends on the fact that the correct response is more potent than the incorrect response. Without loss of generality, let us consider a special case in which the two response keys are: \( R_{{{\text{A(LARGE - CIRCLE)}}}} = {\left( {\begin{array}{*{20}c} {1} & {0} & {1} & {0} \\ \end{array} } \right)} \) and \( R_{{{\text{B(SMALL - SQUARE)}}}} = {\left( {\begin{array}{*{20}c} {0} & {1} & {0} & {1} \\ \end{array} } \right)} \) and an incongruent target, say, \( S_{{{\text{LARGE - SQUARE}}}} = {\left( {\begin{array}{*{20}c} {1} & {0} & {0} & {1} \\ \end{array} } \right)}. \) Let us assume further that the task is SIZE. In such a case, the correct response is R _A (the target is LARGE). The potency for that response according to Model 1 is either w _S · w _R in the case of a task repetition or w _S · (1 − w _R) in the case of a task switch. The potency of the competing response R _B is (1 − w _S) · (1 − w _R) in the case of a task repetition and (1 − w _S) · w _R in the case of a task switch. For R _A to be chosen, the following inequality must hold: P _Response-A  > P _Response-B. Replacing P _Response-A and P _Response-B with their values, as defined above shows that the inequality translates into w _S > (1 − w _R) for task repetitions and into w _S  > w _R for task switches. The competition between the two responses becomes even stronger in the case of a response repetition, where w _A is replaced by w _A + w _CR. Less formally, the processing strategy represented by Model 1 ensures correct responding by a strong-enough input selection that overcomes the counterproductive adjustment of action representation. Similarly, Model 2 strategy ensures correct responding by a strong enough bias in the action representation which counteracts the counterproductive adjustment in input selection.

Because the exponential distribution has only one parameter, Lambda, this parameter dictates both the variance (1/Lambda square) and the mean value (1/Lambda). Having decided to maintain the mean value the same as that estimated (Table 2) could have implicated that we could not vary the distribution to enable better fit to the data. To allow more freedom there, we changed Lambda and added a constant so that the variance could change without changing the mean.