Different strategies in solving series completion inductive reasoning problems: an fMRI and computational study.

INTPSY-10796; No of Pages 8 International Journal of Psychophysiology xxx (2014) xxx–xxx

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International Journal of Psychophysiology journal homepage: www.elsevier.com/locate/ijpsycho

Different strategies in solving series completion inductive reasoning problems: An fMRI and computational study Peipeng Liang a,b,c, Xiuqin Jia a,b,c, Niels A. Taatgen d, Ning Zhong b,e, Kuncheng Li a,b,c,⁎ a

Department of Radiology, Xuanwu Hospital, Capital Medical University, Beijing, China Beijing Key Lab of Magnetic Resonance Imaging and Brain Informatics, Beijing, China Key Laboratory for Neurodegenerative Diseases, Ministry of Education, China d Institute of Artificial Intelligence and Cognitive Engineering, University of Groningen, Groningen, The Netherlands e Department of Life Science and Informatics, Maebashi Institute of Technology, Maebashi, Japan b c

a r t i c l e

i n f o

Article history: Received 30 October 2013 Received in revised form 6 May 2014 Accepted 7 May 2014 Available online xxxx Keywords: Number series Letter series Inductive reasoning Adaptive control of thought-rational (ACT-R)

a b s t r a c t Neural correlate of human inductive reasoning process is still unclear. Number series and letter series completion are two typical inductive reasoning tasks, and with a common core component of rule induction. Previous studies have demonstrated that different strategies are adopted in number series and letter series completion tasks; even the underlying rules are identical. In the present study, we examined cortical activation as a function of two different reasoning strategies for solving series completion tasks. The retrieval strategy, used in number series completion tasks, involves direct retrieving of arithmetic knowledge to get the relations between items. The procedural strategy, used in letter series completion tasks, requires counting a certain number of times to detect the relations linking two items. The two strategies require essentially the equivalent cognitive processes, but have different working memory demands (the procedural strategy incurs greater demands). The procedural strategy produced significant greater activity in areas involved in memory retrieval (dorsolateral prefrontal cortex, DLPFC) and mental representation/maintenance (posterior parietal cortex, PPC). An ACT-R model of the tasks successfully predicted behavioral performance and BOLD responses. The present findings support a general-purpose dual-process theory of inductive reasoning regarding the cognitive architecture. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Inductive reasoning, defined as inferring a general rule or relation from specific elements, is traditionally considered as one of the most important constitutes of human intelligence (Spearman, 1923). Several studies were performed to investigate the neural underpinnings of human inductive reasoning using different types of tasks, including sentential (e.g., House cats have 32 teeth; lions have 32 teeth; therefore, all felines have 32 teeth. Was the given conclusion plausible given the premises? Goel et al., 1997; Goel and Dolan, 2004), figural (e.g., infer the rule underlying the figural stimuli consisting of novel animals. Goel and Dolan, 2000) and numerical (e.g., number series completion tasks Liang et al., 2007; Yang et al., 2009; Zhong et al., 2011; Jia et al., 2011) tasks. The recruitment of fronto-parietal regions together with their left lateralization are convergent reported in most of these studies, however, the detailed activation patterns are modulated by the

⁎ Corresponding author at: No. 45 Changchun Street, Xicheng District, Xuanwu Hospital, Capital Medical University, Beijing 100053, China. Tel./fax: +86 10 83198376. E-mail address: [email protected] (K. Li).

heterogeneousness of the experimental task. The neural correlate of human inductive reasoning process is still unclear, and needs more experimental studies. Some psychological contemporary theories have been proposed to characterize the cognitive architecture underlying reasoning. Regarding the cognitive architecture, two competitive theories were put forward (Barrett and Kurzban, 2006; Goel and Dolan, 2000; Jia et al., 2011). The modular theory views the mind as containing specialized reasoning modules, and predicts that the neural systems for reasoning should be relatively localized, implementing modules that are cognitively impenetrable and informational encapsulated. Alternatively, the general-purpose theory views the mind as containing general reasoning systems. In particular, dual-process theory rose that reasoning is based on two general-purpose systems (Evans, 2003; De Neys, 2006; Tsujii and Sakatani, 2012): an associative system and a rule-based system. The dual-process model predicts that reasoning recruits different neural systems, depending on cognitive demand. When tasks are easy, the associative system is sufficient for correct reasoning and primarily recruits left inferior frontal gyrus, PPC and the temporal lobes. As tasks become more difficult, the rule-based system becomes required for correct reasoning and recruits the prefrontal cortex, especially the

http://dx.doi.org/10.1016/j.ijpsycho.2014.05.006 0167-8760/© 2014 Elsevier B.V. All rights reserved.

Please cite this article as: Liang, P., et al., Different strategies in solving series completion inductive reasoning problems: An fMRI and computational study, Int. J. Psychophysiol. (2014), http://dx.doi.org/10.1016/j.ijpsycho.2014.05.006

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P. Liang et al. / International Journal of Psychophysiology xxx (2014) xxx–xxx

ventrolateral sub-region, which has been implicated in rule maintenance (Barbey and Barsalou, 2009). Many more neuroimaging experimental studies are still required to test these theories. 1.1. Series completion tasks The series completion problem, including letter series (e.g., c e g ?) and number series (e.g., 3 5 7 ?), is a kind of typical inductive reasoning task (Pelligrino, 1985; Thurstone, 1938; Thorndike et al., 1986) and always used in a general fluid intelligence (Gf) test (Cattell, 1963; Hayslip et al., 1995; Johnston et al., 2010; Redick et al., 2012). However, number series and letter series completion tasks are solved differently. Evidences demonstrated that different strategies (by definition, a strategy is a goal-directed procedure under the deliberate control of the participant (Rosenberg-Lee and Anderson, 2009)) were employed in number series and letter series completion tasks, in which each item in one has a same-rule counterpart in the other (Quereshi, 2001), and number series tasks are easier and more familiar than letter series tasks (Quereshi and Seitz, 1993; Quereshi and Smith, 1998). This has been confirmed by a recent pilot study with post-test oral report in our group, in which subjects are required to solve the two kinds of series completion tasks comprising items based on identical rules. A “retrieval” strategy is used in solving number series tasks, in which the relation between two adjacent items can be directly attained by retrieving the corresponding arithmetic fact from long-term memory (e.g., 11 13: 11 + 2 = 13, thus, the rule is + 2). As to letter series tasks, a “procedural” strategy is adopted, in which participants require to step-wise count the corresponding adjacent letter of an item in order to find the relation linking two items (e.g., k n: k … l … m … n; thus, the rule is + 3). The investigation of strategies using imaging has the potential to enrich our understanding of the neural substrates of inductive reasoning, in terms of locus, level, and duration of activity. Correspondingly, the first goal of the present study is to examine “strategy-independent” and “strategy-dependent” brain regions in a series completion inductive reasoning process (Note: The “strategyindependent” brain regions represent the regions commonly recruited by the two strategies; while the “strategy-dependent” regions represent those specifically involved in one of the two strategies). The retrieval and procedural strategy involve performing equivalent cognitive processes (retrieving of declarative memory to detect the relation between the adjacent two items, internal representation and maintenance, and response output), thus engage the same brain areas. However, the working memory demands differ between the two strategies. In the retrieval strategy (e.g., 11 13), participants directly get the relation (e.g., +2) between the adjacent two items by retrieving once the corresponding arithmetic fact (e.g., 11 + 2 = 13); while in the procedural strategy (e.g., k m), participants require to perform twice retrieval step by step (e.g., l is next to k, and m is next to l) and twice internal maintaining/updating a counter, and then the rule can be determined according to the counter. In this way, the procedural strategy incurs many more working memory demands than the retrieval strategy. Thus, the two strategies should differentially engage brain areas that are sensitive to working memory load. It is predicted that the two strategies can be differentiated by the extent of activity within the same brain areas, including the left prefrontal cortex, recruited in memory retrieval, and posterior parietal cortex, involved in mental representation (see a summarization in Anderson et al., 2008). 1.2. ACT-R (adaptive control of thought-rational) The second goal of the current study was to employ computational cognitive modeling to make specific predictions about the strategy differences. Specifically, based on the aforementioned specification, we expect to demonstrate that the strategy difference can be distinguished neurally by differential engagements of the same brain areas. To make our predictions more precise (in terms of the timing and

level of activity), we plan to build computational models of the experimental tasks using the ACT-R (Anderson, 2007). According to the ACT-R theory, cognitive process is the result of the independent activity of distinct peripheral/central modules which are coordinated by a central production system (Fig. S1). The peripheral modules include the visual and aural modules for perception, and the motor and vocal module for interacting with the external world. The central cognitive modules contain the procedural module for implementing the central production system, the declarative memory module for retrieving information, the goal module for state control, and the imaginal module for problem state representation. The interaction among modules depends on buffers corresponding to each module. There are two types of knowledge representation in ACT-R, i.e., declarative knowledge and procedural knowledge. Declarative knowledge corresponds to things that we are aware we know and can usually describe to others. Procedural knowledge is knowledge which we display in our behavior but which we are not conscious of. In ACT-R, declarative knowledge is represented in structures called chunks and procedural knowledge is represented as rules called productions. Thus chunks and productions are the basic building blocks of an ACT-R model. There are several parameters that can be estimated in ACT-R, including the parameters of the time scale moving visual attention from the screen, the time to retrieve a declarative memory fact, the time to modify the contents of the imaginal module, etc. In addition, in order to a make prediction for the exact time course of the BOLD response, ACT-R convolves the module activity with a gamma function. If the module is engaged, it will produce a BOLD response t time units later according to the function (Boyton et al., 1996; Anderson et al., 2003): α −ðt=sÞ

H ðt Þ ¼ mðt=sÞ e

where m is the magnitude parameter and determines the height of the function; the parameter s is the scale parameter and determines the time scale, and α is the shape parameter and determines the narrowness of the function. The cumulative BOLD response in a particular module is the sum of the individual BOLD responses driven by a module's activities. This can be modeled by convolving the hemodynamic response, H(t), with a demand function, D(t), which has a value of 1 when the module associated with that region is active, and 0 otherwise: t

BðtÞ ¼ ∫ DðxÞHðt−xÞdx: 0

Once the timings of buffer actions are set, we can predict the BOLD functions by estimating the magnitude parameter m, the shape parameter α, and the latency scale s for each brain region. The methodology is to find an ACT-R model that can account for the behavioral performance and then see whether it can predict the pattern of brain activity in the related brain regions. For the tasks in the present study, the differential engagement was mainly due to the differences in retrieval and maintenance demands between the two strategies (Quereshi and Seitz, 1993; Quereshi, 2001; Liang, 2009). Two modules were thus of particular interest in this study: the retrieval module, which is responsible for the retrieval of declarative memories and linked to the lateral prefrontal cortex (LPFC) and the imaginal module, which is responsible for the encoding and maintenance of internal representation of the problems and linked to the posterior parietal cortex (PPC) (Anderson, 2007; Qin et al., 2004). As the procedural strategy incurs more demands of retrieval and maintenance, thus, we expected LPFC and PPC to have a greater response to the procedural strategy (used in letter series tasks) than the retrieval



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strategy (used in number series tasks). Together, we will test the two competitive theories empirically and computationally. 2. Material and method Fig. 1. Paradigm of stimuli presentation. Reaction time (RT) is recorded by the first response and accuracy is attained by the second response.

2.1. Subjects Twenty-three paid healthy undergraduate and postgraduate students (11 females; 24.1 ± 3.7 years old) participated in the experiment. All subjects were right-handed and had the normal or corrected-to-normal vision. None of the subjects reported any history of neurological or psychiatric diseases. Written informed consent was obtained from each participant and this study was approved by the Ethics committee of Xuanwu Hospital, Capital Medical University. 2.2. Stimuli and experimental design Thirty number series induction tasks, thirty letter series induction tasks, twenty-four number judgment baseline tasks and twenty-four letter judgment baseline tasks were organized into a 2 × 2 factorial design (see Table 1). The first factor was Content, consisting of two levels, number-related (24 inductions and 24 baselines) and letterrelated (24 inductions and 24 baselines) tasks. The second factor was Task in which the first level was the induction condition consisting of series completion tasks (24 number series inductions and 24 letter series inductions) and the second level was the baseline condition (24 number judgment baselines and 24 letter judgment baselines). This yielded four types of tasks: number series induction (NumIR), letter series induction (LetIR), number judgment baseline (Is10) and letter judgment baseline (IsJ). In particular, interferential tasks, which are identical in pattern to inductions but without common rules (e.g., “1 3 8” or “a c f”), were included into NumIR and LetIR tasks based on a pilot study. There were twelve interferential tasks within 60 induction tasks in the current study, with six in NumIR tasks and another six in LetIR tasks. For all kinds of tasks, there were three sequentially presented numbers or letters (e.g., “1 3 5” or “a c e”). Number-related tasks and letter-related tasks were matched for magnitudes and operations. All the letters involved were lowercase and within the range of a–z. Correspondingly, all the numbers involved were within the range of 1–26. Half of NumIR and LetIR tasks was forward (e.g., “1 3 5” or “a c e”; the rule is: + 2) while another half was backward (e.g., “13 9 5” or “m i e”; the rule is: − 4). In the answer options of NumIR and LetIR tasks, the distances between the correct and the false answers were less than 3. Half of Is10 and IsJ tasks were with the answer of “yes” while another half were with the answer of “no”.

3.4 s. After the appearance of the third number/letter, subjects were instructed to press the left or right button (counterbalanced among subjects) when the answer was acquired. Three numbers/letters would remain on the screen for at most 6.6 s (since the presentation of the third number/letter), or the button-pressing response within 6.6 s would stop it. Subsequently, two answer options were displayed and subjects were asked to choose the correct answer by pressing the corresponding buttons (left button for “A”). Subjects were instructed to respond as accurately and quickly as possible and move to the next trial if the stimuli advanced before they could respond. The interstimulus interval (ISI) was 8 s and quick responses would leave more rest time within the trial. Thus, reaction times (RT) were recorded based on the first button response and accuracies were acquired based on the second button response. For NumIR and LetIR tasks, subjects were required to identify the underlying rule governing the number or letter sequence, and to complete the following number or letter (e.g., for “1 3 5”, the rule is “+2” and the answer is “7”). For Is10 or IsJ tasks, subjects were required to judge whether there was a number of “10” or a letter of “j” in the presented three numbers or letters. 2.4. MR data acquisition Scanning was performed on a 3.0 T MRI system (Siemens Trio Tim; Siemens Medical System, Erlangen, Germany) with a 12-channel phased array head coil. Foam paddings and headphones were used to limit head motion and reduce scanning noise. Structural images were obtained by using a three-dimensional magnetization prepared rapid acquisition gradient echo (3-D MPRAGE) sequence with the following parameters: repetition time (TR)/echo time (TE) = 1600/2.25 ms, flip angle = 9°, 192 sagittal slices, voxel size = 1 × 1 × 1 mm3. Functional data were acquired using a T2* gradient echo-planar imaging (EPI) pulse sequence (TR/TE = 2000/31 ms, 30 axial slices, voxel size = 3.75 × 3.75 × 4.0 mm3, 0.8 mm inter-slice gap, 90° flip angle, 64 × 64 matrix size in 240 × 240 mm2 field of view [FOV]). This acquisition sequence generated 364 volumes for each session. The presentation of every trial was synchronized with the scanner.

2.3. Stimuli presentation

2.5. Data analysis

Stimuli from all conditions were organized into three sessions and presented randomly in an event related design (Fig. 1). The order of sessions was counterbalanced among subjects. The beginning of a trial was signaled by a cue of the task type (“Finding a rule” for NumIR and LetIR tasks, “Is there ‘10’” for Is10 tasks and “Is there ‘J’” for IsJ tasks) for 2 s. The numbers/letters then appeared on the screen with the first number/letter appearing at 2 s, the second at 2.6 s, and the last at

2.5.1. Data preprocessing Data were analyzed using SPM8 software (http://www.fil.ion.ucl.ac. uk). The first four images for each session were discarded to allow for T1 equilibration effects. The remaining fMRI images were first corrected for within-scan acquisition time differences between slices and then realigned to the first volume to correct for inter-scan head motions (Head movement was b 2 mm and b 2° in all cases). The structural image was co-registered to the mean functional image created from the realigned images using a linear transformation. The transformed structural images were then segmented into gray matter (GM), white matter (WM) and cerebrospinal fluid (CSF) by using a unified segmentation algorithm (Ashburner and Friston, 2005). The realigned functional volumes were spatially normalized to the Montreal Neurological Institute (MNI) space and re-sampled to 3 mm isotropic voxels using the normalization parameters estimated during unified segmentation. The registration of the functional data to the template was checked for each individual subject. Subsequently, the functional images were

Table 1 Examples of experimental tasks.

NumIR LetIR Is10 IsJ

Task

Options

11 13 15 tsr 14 23 10 nwj

A. 18 A. q A. Yes A. No

Answer B. 17 B. p B. No B. Yes

B A A B


4


spatially smoothed with a Gaussian kernel of 8 × 8 × 8 mm3 full width at half maximum (FWHM) to decrease spatial noise. 2.6. FMRI analysis Only correct response trials were included in data analysis (wrong response trials and interferential trials were not included). The epoch of interest is the duration between the presentations of the first number/letter to the first button response. The BOLD signal was modeled using the canonical HRF with time derivative and the RT as duration trial by trial. Condition effects at each voxel were estimated according to the general linear model and regionally specific effects were compared using linear contrasts. Each contrast produced a statistical parametric map of the t-statistic, which was subsequently transformed to a unit normal z-distribution. The contrast images of each subject were then used in random effect analysis to determine what regions were the most consistently activated across subjects using a onesample t-test. For confirmatory analysis, region of interest (ROI) analyses focused on two predefined functional regions in ACT-R, LPFC (TAL., − 43, 23, 24) and PPC (TAL., − 23, − 63, 40) (Anderson, 2007). The statistical results were based on the beta-values (of general linear model implemented in SPM5) averaged within the two ROIs. In order to relate our results to previous studies, we reported the contrast associated with two simple main effects of Task (NumIR– Is10) and (LetIR–IsJ), but are primarily interested in the activations common to both strategies [(NumIR–Is10) and (LetRI–IsJ)]. The common activation map was achieved by the following two steps: The map of (NumIR–Is10) was first outputted as a mask, and then (LetIR– IsJ) was examined within the mask. The Task by Content interaction comparisons ([(NumIR–Is10)–(LetIR–IsJ)] and [(LetIR–IsJ)–(NumIR– Is10)]) were also executed to reveal the areas specific to each strategy. The activation reported survived a voxel-level intensity threshold of p b 0.05 corrected for false discovery rate (FDR) for the whole brain volume with a minimum cluster size of 10 contiguous voxels. 2.7. ACT-R modeling Having now reviewed the result of the experiment, we come to the question whether we can understand them in the framework of the ACT-R theory. The model we constructed primarily depends on the visual module to perceive the stimuli, the manual module to respond, the retrieval module to retrieve a fact from memory, and the imaginal module to encode and update its stored representation. Communication among these modules is achieved through a procedural module (see Fig. S1 in the supplementary material). For example, the following might be a production rule for producing the of number series induction: IF the goal is to solve the number series problem and the problem is of the form “Number1 Number2 Number3” and “Number3 + Rule = Number4” has been retrieved THEN the answer of the problem is Number4. As the fronto-parietal network plays an important role in inductive reasoning, the retrieval and imaginal module are of special interest in the present study. In the following, we would illustrate the prediction of BOLD responses in the two modules. Fig. 2 illustrates the sequences of activities in the four relevant buffers of ACT-R, by taking the number series completion task (“11 13 15”) and letter series completion task (“t s r”) as examples. Each box reflects a period of time during which that buffer is active, and the vertical length of the box reflects the duration that the buffer is active. With the processes in Fig. 2 and their durations set, we will test whether the BOLD responses in the frontoparietal regions can be predicted.

Fig. 2. A schematic representation of the ACT-R model's solution to the number series induction of “11 13 15”, and letter series induction of “t s r”.

3. Results 3.1. Behavioral performance We carried out analyses of variance for two factors: Content (number vs. letter) and Task (induction vs. baseline) on both RT and accuracy. Behavioral scores indicated that subjects performed the task in the expected manner (Table 2). The main effects of Content and Task were significant for RT and accuracy: Response to induction tasks was significantly longer (F(1,22) = 70.10, p b 0.001) and less accurate (F(1,22) = 61.56, p b 0.001) than that of baseline tasks; Response to Table 2 Behavioral scores. RT (ms)

Induction Baseline

Accuracy (%)

Number (SD)

Letter (SD)

Number (SD)

Letter (SD)

586.39 (148.45) 530.45 (104.37)

2221.43 (1001.60) 606.09 (165.94)

98.37 (3.01) 99.09 (2.16)

72.64 (15.07) 97.64 (3.03)



letter tasks was significantly longer (F(1,22) = 68.38, p b 0.001) and less accurate (F(1,22) = 69.94, p b 0.001) than that of number tasks. The interaction effect between Task and Content was significant for RT (F(1,22) = 69.24, p b 0.001) and accuracy (F(1,22) = 66.75, p b 0.001): the difference between LetIR and IsJ (i.e., LetIR–IsJ) (RT: F(1,22) = 30.59, p b 0.001; accuracy: F(1,22) = 73.81, p b 0.001) and the difference between NumIR and Is10 (i.e., NumIR–Is10) (RT: F(1,22) = 6.39, p b 0.05; accuracy: F(1,22) = 2.89, p = 0.10). This indicated that the significant interaction effect is primarily driven by the strategy difference, i.e., the difference between LetIR and NumIR. Additionally, we also checked the RT derived from the second response (from which we attained the accuracies), and found that all effects were not significant.

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Table 3 Regions activated commonly to both strategies. The results were revealed by the common activations of the two contrasts, i.e., (NumIR–Is10) and (LetIR–IsJ), using the gray matter as a within mask. Loci of maxima are in MNI coordinates in mm. The activation was thresholded at a False Discovery Rate (FDR) b 0.05 with a minimum cluster size of 10 contiguous voxels. Lt, left; Rt, right. Regions

BA

Cluster size

Lt. middle frontal gyrus Lt. inferior frontal gyrus Lt. middle frontal gyrus Lt. superior parietal lobule

46 9 6 7

67

Rt. superior parietal lobule

7

59

24

Coordinate

T

x

y

z

−48 −51 −51 −27 −24 30 27

27 12 3 −63 −69 −72 −63

21 30 45 45 51 51 48

11.54 10.26 7.12 7.85 7.16 6.02 5.81

3.2. Imaging results 3.2.1. Confirmatory analysis We performed repeated measures Task by Content for the two ROIs (as shown in Fig. 3). The two ROIs showed the identical patterns. The main effect of Content (F(1,22) = 35.46, p b 0.001 for LPFC and F(1,22) = 14.23, p b 0.001 for PPC), Task (F(1,22) = 67.10, p b 0.001 for LPFC and F(1,22) = 35.45, p b 0.001 for PPC) and the Task by Content interaction effect (F(1,22) = 29.64, p b 0.001 for LPFC and F(1,22) = 4.60, p b 0.001 for PPC) were all significant. The post-hoc multiple comparisons showed that the interaction effect was driven by the strategy difference, i.e., the difference between letter series reasoning tasks and number series reasoning tasks (F(1,22) = 46.58, p b 0.001 for LPFC and F(1,22) = 13.51, p b 0.001 for PPC). There was no significant difference between the two kinds of baselines (i.e., IsJ versus Is10; F(1,22) = 3.98, p = 0.06 for LPFC and F(1,22) = 1.85, p = 0.19 for PPC). 3.2.2. Exploratory analysis The two simple main effects of Task, i.e., (NumIR–Is10) and (LetIR–IsJ), showed the similar activation pattern (see Fig. S2 in the supplementary material). The common activations of (NumIR–Is10) and (LetIR–IsJ) included the left dorsolateral prefrontal cortex (DLPFC, BA 46, 9) and bilateral superior parietal lobe (SPL, BA 7) extending to the left inferior parietal lobe (IPL, BA 40) (see Table 3 and Fig. 4). Regions specific to LetIR [(LetIR–IsJ)–(NumIR–Is10)] were found in the bilateral superior frontal gyrus (SFG, BA 6, 8) extending into the insula (BA 13) and anterior cingulate cortex/medial frontal gyrus (ACC/MeFG, BA 9, 32), bilateral inferior frontal gyrus (IFG, BA 46, 47), and bilateral superior parietal lobule (SPL, BA 7) extending into the precuneus (BA 7), left thalamus extending into the lateral globus pallidus, right caudate and putamen (see Table 4). No significant activations were found specific to NumIR [(NumIR–Is10)–(LetIR–IsJ)]. We further used the reaction time (RT) as covariates to controlling for the task difficulty effect as reflected by RT. It was found that the

Fig. 3. Results of confirmatory analysis. Parameter estimates of two ACT-R ROIs located at LPFC (TAL., −43 23 24) and PPC (TAL., −23 −63 40).

left DLPFC and bilateral posterior cingulate cortex (PPC) still survived but with less voxel number. Additionally, many frontal regions, including bilateral MeFG and IFG, survived in [(LetIR–IsJ)–(NumIR– Is10)], but no activations were found in [(NumIR–Is10)–(LetIR–IsJ)]. 3.3. Results of ACT-R modeling The fit of the predictions of the model to the RT and ACC data are presented in Fig. 5. The parameters were estimated to fit the behavioral data: a factor that scaled the time to retrieve a declarative memory fact (0.4 s) and the time to modify the contents of the imaginal module (i.e., 0.2 s). This leads to a predicted effect size (648.0 ms for NumIR, 2332.8 ms for LetIR, and 645.0 ms for baseline conditions) that is highly similar to the observed effect size (586.4 ms for NumIR, 2221.4 ms for LetIR, 530.4 ms for Is10, and 606.1 ms for IsJ). Only the deviation of data is presented because the model makes identical predictions. It can be seen that the model produced a reasonable fit to the behavioral data. We fixed α to be 2 and s to be 2 s. Once the timings of buffer actions are all set, we can predict the BOLD functions by estimating the magnitude parameter m from the BOLD response in each brain region (Anderson et al., 2009; Rosenberg-Lee and Anderson, 2009). The estimates of these parameters and measurement of the quality of the prediction are given in Table 5. Fig. 6 displays percent change of BOLD response (relative to the baseline defined by the average BOLD response of the first two scans and the last two scans), along with the prediction of the ACT-R model to be presented. 4. Discussion The primary purpose of the current study was to investigate the neural correlates of the two cognitive strategies in series completion

Fig. 4. Activations common to both strategies. Result revealed by the two contrasts, i.e., (LetIR–IsJ) and (NumIR–Is10). Color bar indicates the t-score. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)


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Table 4 Regions specific to LetIR procedural strategy. The results were revealed by the contrast of [(LetIR–IsJ)–(NumIR–Is10)]. Loci of maxima are in MNI coordinates in mm. The activation was thresholded at a False Discovery Rate (FDR) b 0.05 with a minimum cluster size of 10 contiguous voxels. Lt, left; Rt, right. Regions

BA

Lt. superior frontal gyrus

6

Lt. insula Rt. inferior frontal gyrus

13 47 46 8 32 9 47 47 7 7 7

Rt. superior frontal gyrus Rt. cingulate gyrus Rt. medial frontal gyrus Lt. inferior frontal gyrus Rt. inferior frontal gyrus Lt. superior parietal lobule Rt. superior parietal lobule Rt. precuneus Lt. thalamus Lt. lateral globus pallidus Rt. caudate Rt. putamen

Cluster size

1356

374 507

16 13 344 250 256 178

Coordinate (MNI)

T

x

y

z

−3 −9 −30 30 45 3 9 3 −33 33 −12 36 12 −9 −12 9 15

12 3 24 21 18 15 24 39 30 30 −57 −75 −66 −12 3 6 0

54 60 3 −6 27 54 33 36 −3 −3 48 45 51 3 6 6 9

9.85 8.09 7.75 8.06 4.61 7.45 6.48 6.11 7.43 6.02 4.08 4.38 4.05 4.88 4.53 5.08 4.87

inductive reasoning tasks, i.e., the retrieval strategy in number series tasks and the procedural strategy in letter series tasks. Both confirmatory and exploratory analyses indicated that DLPFC (L. N R.) and PPC (bilaterally) were commonly recruited in the two kinds of tasks, but with different extent. Additionally, the exploratory analyses identified additional regions in the dorsomedial prefrontal cortex (DMPFC) and ventrolateral prefrontal cortex (VLPFC) which were active for the procedural strategy but not for the retrieval one (see Table 4). These results suggest that some aspects of the behavioral signatures of strategies may be recovered from imaging data. We constructed the computational model to simulate participants' behavior. The result showed

Fig. 5. Data (solid lines) and model fits (dashed lines) for NumIR, LetIR, Is10, and IsJ. Error bars, shown only for subject data, represent standard error.

Table 5 Parameters and the quality of the BOLD function prediction.

Exponent (α) Scale (s) Magnitude MΓ(α + 1)⁎ Correlation (r)

Imaginal

Retrieval

2 2

2 2

6.10 0.97

4.26 0.95

⁎ This is a more meaningful measure because the height of the function is determined by the exponent as well as M.

that there was reasonable fitness between the model prediction and the empirical data (see Fig. 6). In the current study, the significant co-activation of the left DLPFC and bilateral PPC is identified to be strategy-independent. These results replicated a recent fMRI study of number series completion task in our group (Jia et al., 2011), and were consistent with previous studies of sentential inductive reasoning task (Goel and Dolan, 2004) and figural inductive reasoning tasks (Induction minus Perceptual baseline; Goel and Dolan, 2000). This implies that these two regions may be the key regions involved in inductive reasoning. Moreover, the left DLPFC is more specific to sentential induction, in contrast to deduction (Goel and Dolan, 2004). The activity of the DLPFC is right lateralized in Goel and Dolan (2000), but is left lateralized in the other studies (Jia et al., 2011; Goel and Dolan, 2004) and the current study, which may be ascribed to the different kinds of experimental materials (figure versus number, sentences). Inductive reasoning is the core of human intelligence (Cattell, 1963; Hayslip et al., 1995) and series completion tasks are widely used in measuring general intelligence. We thus are also interested in comparing the current results with the neuroimaging studies of other intelligence

Fig. 6. The BOLD functions obtained for the prefrontal and parietal regions for NumIR and LetIR conditions and the prediction of the ACT-R model of the task.



test tasks. It was found that our findings, the strategy-independent co-activation of fronto-parietal regions, were congruent with those in fMRI experiments using Raven Progressive Matrix (RPM) (Christoff et al., 2001), fluid analogizing (Masunaga et al., 2008), three-back working memory task (Gray et al., 2003) and Topology Test (Geake and Hansen, 2010), and thus add new supports to the Parieto-Frontal Theory (P-FIT) of intelligence (Jung and Haier, 2007). As to the functional role of DLPFC and PPC, different interpretations were proposed. In the domain of inductive reasoning, the left DLPFC was related to the use of world knowledge in the generation and evaluation of hypotheses (Goel and Dolan, 2004), rule identification and extrapolations (Jia et al., 2011), and the fronto-parietal network was more specific to rule identification (Jia et al., 2011). (Note: In Goel and Dolan (2000), the authors are interested in the bilateral hippocampus, which is specifically associated with rule inference, and the right lateral orbital prefrontal cortex, which is associated with the task by difficulty interaction. The former is interpreted in terms of semantic encoding of novel stimuli, and the latter in terms of hypothesis selection. While the DLPFC (R. N L.) and bilateral PPC, identified in Rule Induction minus Perceptual Baseline, are not explained of their functional roles.) While in the domain of intelligence, PPC is related to symbolism, abstraction and elaboration, and DLPFC as well as other frontal regions are associated with hypothesis testing (Jung and Haier, 2007). It is not surprise of these different explanations, as the previous studies were situated in different domains and contexts. In the current study, with the help of computational modeling, we made a more general account. The DLPFC and PPC regions identified in the exploratory analysis, are largely overlapped by the regions involved in the ACT-R LPFC and ACT-R PPC. Therefore, consistent with the explanation in the ACT-R theory, it was inferred that the left DLPFC is associated with memory retrieval of semantic information/knowledge and PPC is associated with mental representation of problem states (Anderson, 2007; Danker and Anderson, 2007). In the current study, although the procedural strategy and the retrieval strategy commonly recruited the left DLPFC and bilateral PPC, the strategy variation can be distinguished by the extent of cortical activity within these regions. The procedural strategy incurs more working memory demands (both the retrieval and maintenance demands) than the retrieval strategy. The ACT-R model fitted well with the experimental behavioral and imaging data, which demonstrated the validity of this general account. The current results may inform the theories of human reasoning that address the cognitive architecture underlying reasoning. Although the current study together with previous studies reported the important roles of DLPFC and PPC in inductive reasoning, the modular theory is not advocated, as these frontoparietal regions are also widely coactivated in deductive reasoning (see a meta-analysis: Prado et al., 2011), working memory (Chein et al., 2011) and fluid intelligence tasks (Jung and Haier, 2007). On the contrary, the current results are in agreement with the cognitive demand hypothesis motivated by the dual-process theory, which predicts that reasoning recruits different neural systems depending on cognitive demand. In the current study, it was found that the more difficult tasks (letter series tasks with the procedural strategy) exclusively activated more areas in bilateral VLPFC and DMPFC, in addition to stronger recruitment of frontoparietal regions. Thus, the present findings have lent support to the dual-process theory of inductive reasoning. Moreover, ACT-R, with which the predicted behavioral and imaging data fit well with the participants' performance, is a kind of general cognitive theory. This implies that the general-purpose cognitive architecture can explain the experimental facts well. Actually, dual-process mechanism has been empirically demonstrated in category-based induction (Feeney et al., 2007). In summary, by coupling this empirical work with computational modeling, the neural underlying of the strategy variation (the retrieval strategy vs. the procedural strategy) in series completion tasks was studied. These results support the general-purpose dual-process theory of inductive reasoning.

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Acknowledgments This work was supported by the grant from the National Natural Science Foundation of China (61105118), Beijing Nova Program (No. Z12111000250000, Z131107000413120), and the Open Research Fund of the State Key Laboratory of Cognitive Neuroscience and Learning (CNLZD1302). ACT-R code can be accessed by logging in ccmureasoninglab@gmail. com (userid: ccmureasoninglab; password: peipengliang), and open the email entitled with “ACT-R code for ‘Different Strategies in Solving Series Completion Inductive Reasoning Problems: An fMRI and Computational Study’”.

Appendix A. Supplementary data Supplementary data to this article can be found online at http://dx. doi.org/10.1016/j.ijpsycho.2014.05.006.

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