On the role of prefrontal and parietal cortices in mind wandering and dynamic thought

Mind wandering is a common phenomenon in our daily lives and can have both an adaptive and detrimental impact. Recently, a dynamic framework has been proposed to characterise the heterogeneity of internal thoughts, suggesting there are three distinct thought types which can change over time – freely moving, deliberately constrained, and automatically constrained (thoughts). There is very little evidence on how different types of dynamic thought map onto the brain. Previous research has applied non-invasive transcranial direct current stimulation (tDCS) to causally implicate the prefrontal cortex and inferior parietal lobule in general mind wandering. However, a more recently developed and nuanced technique, high-definition tDCS (HD-tDCS), delivers more focal stimulation able to target specific brain regions. Therefore, the current study investigated the effect of anodal HD-tDCS applied to the left prefrontal and right inferior parietal cortices (with the occipital cortex included as an active control) on mind wandering, and specifically, the causal neural substrates of the three internal dynamic thought types. This was a single session study using a novel task which allows investigation into how dynamic thoughts are associated with behavioural variability and the recruitment of executive control operations across the three brain regions. Anodal stimulation to the prefrontal cortex decreased freely moving thought and anodal stimulation to the parietal lobule decreased deliberately constrained thought, with preliminary evidence for an increase in freely moving thought in the occipital cortex as well. These findings support the heterogenous nature of mind wandering, revealing that different brain regions are implicated in distinct dynamic thought types.

We spend up to half our waking hours directing our thoughts towards self-generated, internally orientated representations: a phenomenon known as mind wandering (Kane et al., 2007;Seli, Beaty, et al., 2018;Smallwood & Schooler, 2006).Whilst mind wandering can be detrimental in contexts where a high level of sustained attention is required e such as air traffic control (Gouraud, Delorme, & Berberian, 2017) and driving (Baldwin et al., 2017;Yanko & Spalek, 2014) e it has also been hypothesised that there are also associated benefits such as eliciting creativity (Baird et al., 2012) and facilitating future planning (Pachai, Acai, LoGiudice, & Kim, 2016).Given the broad range of impacts mind wandering has on our daily lives, and the extent to which we engage in it (Killingsworth & Gilbert, 2010), it is important to understand the neural, cognitive, and behavioural aspects which underly this phenomenon.
Mind wandering encompasses a broad range of experiences (Callard, Smallwood, Golchert, & Margulies, 2013;Seli, Kane, et al., 2018), yet there has been limited exploration into the underlying complexities and heterogeneity of the associated internal thought processes (Kam et al., 2021;Martel, Arvaneh, Robertson, Smallwood, & Dockree, 2019).The "dynamic framework" has been proposed to account for this heterogeneity, suggesting that internal mental states arise and shift over time and can be distinguished as different thought types (Poerio et al., 2017;Seli, Kane, et al., 2018).There have been three key types of dynamic thoughts identified in the literature: 1) deliberately constrained thoughts, 2) automatically constrained thoughts and 3) freely moving thoughts (Christoff, Irving, Fox, Spreng, & Andrews-Hanna, 2016;Martel et al., 2019).Deliberately constrained thoughts require cognitive control and are directed towards goal-orientated information; automatically constrained thoughts focus largely on affective or sensory salient information which is difficult to disengage from and typically does not require cognitive control (Christoff et al., 2016;Irving, 2015;Seli, Risko, & Smilek, 2016); whereas, in comparison, freely moving thoughts arise without strong constraints or overarching direction and are not focused on particular topics for any period of time (Martel et al., 2019;Mills, Raffaelli, Irving, Stan, & Christoff, 2018).
In terms of the neural substrates underlying mind wandering generally, an extensive literature has employed functional magnetic resonance imagining (fMRI) to implicate a large-scale collection of brain regions, known as the default mode network, in self-reported task unrelated thought (Christoff, Gordon, Smallwood, Smith, & Schooler, 2009;Fox, Spreng, Ellamil, Andrews-Hanna, & Christoff, 2015;Groot et al., 2021).Interestingly, this network shows increased metabolic activity when the mind is at rest and during internal thought processes, and decreased activity during cognitively demanding tasks (Mason et al., 2007;Poerio et al., 2017).The default mode network consists of a broad range of areas, with prominent regions including the medial prefrontal cortex (PFC), posterior cingulate cortex and posterior temporoparietal cortex (Fox et al., 2015;Groot et al., 2021).Researchers have also found mind wandering recruits the dorsolateral prefrontal cortex (DLPFC) and the anterior PFC, which are part of the frontoparietal control network (Christoff et al., 2009).These regions are associated with executive control processes, which hints at a negative correlation between executive functioning and mind wandering (Christoff et al., 2009;Fox et al., 2015).Executive functions are associated with actions including response selection, inhibition, attentional control, and multitasking (Diamond, 2013).These functions may be used to direct attention away from the current task, to allow the continuity of internal task unrelated thoughts and to guide and select between different types of dynamic thoughts (Christoff et al., 2016;Kam & Handy, 2014;Smallwood & Schooler, 2006).
Recently, cognitive neuroscientific investigations have begun to examine the neural processes associated with dynamic thought, with hints that different types may be linked with distinct neurocognitive operations (Martel et al., 2019;Wang et al., 2018).Initial research by Martel et al. (2019) used electroencephalogram (EEG) to investigate the neural correlates of the intentionality of task unrelated thought, reporting differences in the alpha oscillations and evoked sensory responses for deliberate and freely moving task unrelated thoughts.Kam et al. (2021) conducted further research on this topic and observed the parietal event related potential P3 was greater for task-related compared to unrelated thoughts whereas the frontal P3 was larger for deliberate compared to automatic thoughts.This research provides correlational evidence that different types of dynamic thought may have distinct neural substrates, however, to date, there have been no attempts to causally implicate specific brain regions in dynamic thought.
One approach commonly used to investigate the casual neural substrates of mind wandering is transcranial direct current stimulation (tDCS).This is a form of non-invasive brain stimulation which passes low-intensity currents (typically 0.5 mAe3 mA) between electrodes attached to the scalp (see Fig. 1; Filmer, Dux, & Mattingley, 2014, 2020;Mosayebi Samani, Agboada, Jamil, Kuo, & Nitsche, 2019).The effects of tDCS on modulating resting membrane potentials were initially thought to be polarity dependent, with anodal stimulation increasing the likelihood of cortical excitation and cathodal stimulation decreasing the likelihood of cortical excitation (Nitsche & Paulus, 2000).However, recent research suggests that the polarity of tDCS effects is far more complex and is affected by factors such as stimulation intensity, duration, and task type (Filmer, Griffin, & Dux, 2019, 2020;Mosayebi Samani et al., 2019).While the field of tDCS has faced criticism over variability in the outcomes and the use of poor methodological practices (Horvath, Carter, & Forte, 2016), many of these criticisms have more recently been addressed.Specifically concerns about the reproducibility of tDCS have been discussed in a more recent review by Filmer, Mattingley, and Dux (2020), which identified various fields with consistent findings across studies, alongside highlighting the importance of methodological and scientific rigour for developing reproducible findings using tDCS.Thus, this form of non-invasive brain stimulation offers novel insights into our understanding of the brain and has many potential benefits within the field of cognitive neuroscience and for society (Filmer et al., 2020).
There have been several studies investigating the effect of bipolar tDCS on mind wandering in the prefrontal cortex (Axelrod, Rees, Lavidor, & Bar, 2015, 2018;Filmer et al., 2019).Initial studies observed that low intensity stimulation, specifically 1 mA anodal tDCS applied to the left DLPFC (but was likely more broadly the left PFC given the bipolar tDCS montage, see Fig. 1), increased participants mind wandering propensity (Axelrod et al., 2015(Axelrod et al., , 2018)).However, Boayue et al. (2020) completed a high-powered replication of Axelrod et al. (2015) and found strong evidence for a failure to replicate the effect of anodal stimulation over the left PFC.Bertossi, Peccenini, Solmi, Avenanti, and Ciaramelli (2017) also investigated mind wandering and the medial PFC and found a decrease in the propensity to mind wander following cathodal stimulation, however the effect was only found in male subjects.Given evidence hinting at the importance of tDCS intensity, a recent large scale pre-registered study explored the effect of this variable on mind wandering propensity (Filmer et al., 2019).This study targeted the left PFC, applying 1 or 2 mA of anodal or cathodal stimulation.The experiment found that only 2 mA cathodal stimulation to the left PFC influenced mind wandering, such that the frequency of task unrelated thoughts was increased (Filmer et al., 2019).The inconsistencies in findings and failure to replicate early results has led to questions over the causal role of the prefrontal cortex in mind wandering and the mechanisms which may explain these underlying differences.
An additional region which has been implicated in modulating mind wandering propensity is the right inferior parietal lobule (IPL; Filmer, Marcus, & Dux, 2021;Kajimura, Kochiyama, Abe, & Nomura, 2019;Kajimura & Nomura, 2015).Research applying 1.5 mA anodal tDCS to the right IPL found stimulation reduced the propensity to mind wander (Kajimura et al., 2019;Kajimura & Nomura, 2015).Interestingly, Kajimura et al. (2019) uniquely stimulated the IPL region, finding the effect of anodal tDCS on mind wandering was specific to the right IPL alone, as opposed to finding effects in the IPL while concurrently stimulating the PFC.Filmer et al. (2021) investigated the role of both the PFC and IPL in mind wandering using 1 mA and 2 mA stimulation intensities and inversed polarity montages.These researchers found that anodal stimulation to the left PFC and cathodal stimulation to right IPL increased mind wandering propensity (Filmer et al., 2021).This research aligns with previous literature which, collectively, provides evidence for a causal role of both the PFC and IPL in mind wandering.However, these results display inconsistencies in the directionality of the effects and the interplay between the different types of dynamic thought and activity in these regions is still unknown (Chaieb, Antal, Derner, Leszczy nski, & Fell, 2019).
There is debate that the effects of traditional bipolar tDCS are diffuse, with the electrodes stimulating relatively broad brain regions and thus not targeting a focal cortical area (Datta et al., 2009;Pisoni et al., 2018).To address this issue, a newly developed approach -high definition tDCS (HD-tDCS) -utilises multiple smaller electrodes (typically a 4 x 1 ring montage) to deliver more targeted stimulation (Nikolin, Lauf, Loo, & Martin, 2019;Villamar et al., 2013).As seen in Fig. 2, the electrodes used in the HD-tDCS configuration cover a smaller cortical area than the larger electrodes used for bipolar tDCS, and are also more focal, with the 4 Â 1 ring configuration designed to restrict the area of cortical excitability modulation to the region within the ring perimeter (Villamar et al., 2013).
A recent study conducted by Chou, Hooley, and Camprodon (2020) investigated the effects of bilateral HD-tDCS to the posterior IPL on maladaptive forms of mind wandering.This study manipulated participants mood and probed them on the contents of their thoughts in between completing a multi-source inference task.The study utilised a double 3 x 1 electrode montage, applying 1 mA to both the right and left IPL and included an anodal, cathodal, and sham group.Interestingly, this study found no effect of either anodal or cathodal stimulation on overall mind wandering frequency.However, cathodal stimulation was found to significantly decreased the frequency of negative mindwandering thoughts about the past relative to sham.These findings offer initial support for distinctions in the causal brain regions of internal thought types; however, the research focused on negatively valanced thoughts for a single brain region.
An additional large-scale study conducted by Boayue et al. (2021) is the only study to employ HD-tDCS to the DLPFC to investigate the causal neural substrates of mind wandering.This study applied 2 mA anodal HD-tDCS to the left DLPFC and tested performance on a novel task which consisted of generating random sequences in time with a metronome (Boayue et al., 2021).Throughout the task a thought-probe was presented asking participants to rate how focused they were on the task using a 4-point Likert scale.These researchers employed hierarchical order probit modelling and found anodal stimulation decreased the propensity to mind wander relative to sham.They argued this analysis technique improved sensitivity as the thought probe data is treated as an   ordinal variable and assessed against a number of predictors including measures of task performance and the trial number (Boayue et al., 2020(Boayue et al., , 2021)).This research provides preliminary evidence for the feasibility of anodal HD-tDCS modulating self-reported mind wandering in the PFC, using a simple thought probe measuring participants focus on the task (Boayue et al., 2021).However, HD-tDCS has not yet been applied to understand the causal neural substrates of different types of dynamic thought across brain regions.

The present study
Considering the importance of mind wandering for both theoretical and applied settings it is crucial to understand the complex heterogeneity of dynamic thoughts.The current research employed a similar protocol to Boayue et al. (2021), however anodal HD-tDCS was used to explore the causal neural substrates of the dynamic thought types across three brain regions.Thus, our study aimed to use HD-tDCS to investigate whether task unrelated, freely moving, deliberately constrained, and automatically constrained thoughts were associated with distinct causal neural substrates, specifically in the DLPFC and IPL, with the occipital cortex included as an active control region.We employed a double-blind protocol with six equal groups of participants who received 2 mA anodal or sham HD-tDCS.The study also aimed to understand the effect of HD-tDCS on the frequency of different dynamic thoughts and task performance across these brain regions.We predicted that there would be differences in the frequency of task unrelated thoughts between the active and sham HD-tDCS groups, consistent with the effect found by Boayue et al. (2021).Specifically, active anodal stimulation to the left DLPFC and right IPL would decrease the frequency of task unrelated thoughts, compared to sham and the occipital cortex (H 1 ).Furthermore, we hypothesised that there would be a decrease in the frequency of the novel types of dynamic thought (i.e., freely moving, deliberately constrained, and automatically constrained thoughts) for groups receiving anodal HD-tDCS to the left DLPFC and the right IPL, relative to sham and the occipital cortex (H 2 ).Finally, we hypothesised that HD-tDCS would affect task performance, such that active stimulation to the left DLPFC and right IPL would increase participants randomness (approximate entropy) and reduce behavioural variability, relative to sham and the occipital cortex (H 3 ).Consistent with Boayue et al. (2021), we also predicted there would be a relationship between behavioural variability and approximate entropy.

Overview
We report how we determined our sample size, all data exclusions, all inclusion/exclusion criteria, whether inclusion/ exclusion criteria were established prior to data analysis, all manipulations, and all measures in the study.All participants completed one session, which consisted of HD-tDCS to one of three brain regions in conjunction with the Finger-Tapping Random-Sequence Generation Task (FT-RSGT).This task was designed for participants to generate random number  sequences while listening to a repetitive metronome.Participants were pseudo randomly assigned to one of six equal groups, with half the participants for each brain region receiving active anodal HD-tDCS and the other half receiving sham HD-tDCS.The specific groups were: (1) anodal HD-tDCS over the left DLPFC; (2) sham HD-tDCS over the left DLPFC; (3) anodal HD-tDCS over the right IPL; (4) sham HD-tDCS over the right IPL; (5) anodal HD-tDCS over the occipital cortex; and (6) sham HD-tDCS over the occipital cortex.At the beginning of the session, participants completed a questionnaire assessing their experience with musical instruments and video games.They were then taught about how to generate random sequences with a short practice block; given explanations on the four types of thought probes which were included in the experiment and, following this, tested on their ability to identify each dynamic thought type, alongside completing a short training block on the task, and thought probes combined.All participants then completed a baseline block of the FT-RSGT, followed by 30 min s of online active or sham stimulation in conjunction with the FT-RSGT.After concluding the experiment, participants completed an end of session questionnaire and were debriefed on their involvement in the study.See Fig. 3 for the full outline of the experimental procedure.The full Stage 1 registered report manuscript can also be accessed on the OSF (https://doi.org/10.17605/OSF.IO/XZ4RN) Q2 (see Fig. 4).
Fig. 3 e Conceptual diagram of the experimental procedure beginning with the pre-screening and pseudo random allocation to one of the 6 stimulation groups and followed by training on both the dynamic thought probes and the task, before participants complete a 10-min baseline block and the 30-min online stimulation and task block.The session then concludes with an end of session questionnaire and debriefing on the experiment.Following recruitment and screening, participants were pseudo-randomly allocated to demographically balanced groups, using a script which factored in age, sex, time spent playing video games and musical instruments and time of day for testing (see Table 1 for demographic information).We also included a question assessing participants hours of sleep from the night before and this variable was included in the script to ensure sleep was equivalent across the groups.To be included in the analysis, participants were also required to comply with all the instructions throughout the experiment and the stimulation had to remain functional across the entire session.The University of Queensland Human Research Ethics Committee approved this study.

Sample size e Bayesian sampling plan
A Bayesian sampling approach was used, and thus, the sample size was not predetermined for this study (Fox, Filmer, & Dux, 2022;Horne et al., 2020;Leow et al., 2023).A total of 251 participants were recruited.During testing, 11 of these participants were removed from the sample and not analysed as they did not complete their session successfully.In line with our Stage 1 exclusion criteria, five of these participants were removed because they did not understand the thought probes or the task correctly.Two did not meet the eligibility criteria and an additional four were removed due to issues relating to the stimulation (e.g., the impendences were too high prior to starting the stimulation block).These datasets were then replaced with additional participants, resulting in a useable sample of 240 participants at the completion of testing.A further 12 participants were removed after testing was completed, as per our post-study data exclusion criteria outline in the Analysis Overview below, resulting in a final sample of 228 participants.These 12 excluded participants included eight participants removed in the baseline block, as per our Stage 1 behavioural data exclusion criteria.An additional exclusion criterion, which was added for the stimulation block, also removed four participants, who were more than 3 standard deviations away from the mean for the task performance or thought probe measures (see Analysis Overview for a detailed justification as to why this criterion was added).The demographic details for the sample including only participants who were excluded as per our original Stage 1 criteria (i.e., 232 participants) can be found in the supplementary results (see Table S4).
It was stated that participants would continue to be recruited until a Bayes Factor (BF) 10 > 6 or BF 01 > 6 was established for the critical hypothesis test (see below).However, recruitment was stopped when the maximum sample size of 240 (40 participants per group) was reached or if testing continues until the specified end date of December 2022, 1 as we believed inconclusive results at this sample size would still contribute meaningfully to the field.The stopping rule was first checked once 10 participants in each group were tested and for every 5 participants thereafter.The critical hypothesis test which was selected was the effect of stimulation on task unrelated thought in the DLPFC, which was the thought probe that broadly asked participants whether they were focused on or off task.Specifically, we predicted that there would be a difference in the frequency of task unrelated thoughts between participants who received active anodal HD-tDCS to the left DLPFC and those who received sham stimulation.We selected this test as Boayue et al. (2021) previously found that HD-tDCS to the DLFPC significantly decreased the frequency of task unrelated thought.Thus, we believed finding this effect using the same stimulation montage would be a meaningful result.However, as we did not find evidence for this effect in the t-test analysis, the maximum sample size was reached.

2.3.
Behavioural assessments 2.3.1.Finger-Tapping Random-Sequence Generation Task (FT-RSGT) In the FT-RSGT task, adopted from Boayue et al. (2021), participants were required to respond in a random sequence to an ongoing metronome tone using two response-buttons (see 1 The end date went beyond the December threshold, due to difficulties with the original HD-tDCS electrode set up using a conductive paste, which we have explained in the HD-tDCS Montage section below.Fig. 3).The button presses were mapped to two separate keys which participants used their index fingers to press ('z' for left-hand and 'm' for right-hand).Participants were instructed to time their responses as accurately as possible to a metronome tone which was presented at 440 Hz for 75 msec duration, with an inter-stimulus interval of 750 msec.They were also instructed to maintain their focus on a white (RGB 255 255 255) fixation cross in the centre of the screen with a grey (RGB 128 128 128) background for the duration of the experiment.
Performance on this task was determined by the randomness of the sequence, which was measured through approximate entropy (see Analysis Overview for detailed explanation).This measure was used as there is an expected relationship between executive functioning and randomness, such that diverting more executive resources towards the task will result in greater randomness which can be used to imply participants are more focused on the task (Boayue et al., 2021).Behavioural variability was also used to measure task performance and was calculated by the deviation in response times from the metronome tone.For the 20 trials before each set of mind wandering probes, the difference between participants key response and the tone was calculated and the standard deviation of the difference scores for each set of 20 trials was included in the analysis.This measure was designed to directly assess participants focus on the task and is also associated with executive functioning (Boayue et al., 2021).
The task was presented on a 24-inch LED monitor, with a refresh rate of 100 Hz.Participants sat approximately 70 cm away from the monitor and used a standard Macintosh keyboard and mouse to respond.The auditory tone was presented through the computer speakers.Initially, participants completed 20 practice trials to familiarise themselves with the task alone.Following training on the thought probes, an additional 20 trials were completed which included the task and the four thought probes.Participants then completed a baseline block of 720 trials which will ran for 10 min, followed by the stimulation block which consisted of 2160 trials 2 over approximately 30 min, with a 30 s break after 15 min.The stimulation block was completed while participants received online active or sham stimulation to one of the three targeted brain regions.

Mind wandering probes
Throughout the experiment, participants were also presented with four thought probes which were designed to assess the contents of their thoughts.The probes were presented pseudo randomly, every 45e75 s during the baseline and stimulation blocks.Thus, there were 10 probes in the baseline block and 30 probes in the stimulation block.The questions assessed participants thoughts in the 10e15 s before the probes were presented.The four questions were: (1) Before the probe, were you thinking about something other than the random sequence generation task; (2) Before the probe, was your mind wandering around freely; (3) Were you actively directing your thoughts; and (4) Was your mind stuck on something.These questions were presented sequentially, with participants rating their responses on a 7-point Likert scale ranging from "Not at all" (1) to "Very much" (7), and the middle point (4) was labelled "Moderately".Participants responded using the number keys labelled 1 through to 7 and they had an unlimited amount of time to categorise their thoughts before moving onto the next probe question.
Before participants completed the task, they were given detailed explanations on the four types of thought probes, alongside example scenarios where these thoughts may occur.The four dynamic thought probes, alongside explanations and questions given to participants replicated those used in research by Kam et al. (2021).Participants were then tested on their knowledge by categorising example scenarios, and the experimenter explained and corrected any incorrect answers.See supplementary methods for the full description of the probes and the examples which were shown to participants, alongside the four Likert scales which were shown during the experiment.All instructions and questions were presented in the centre of a grey background (RGB 128 128 128) in white Arial font (visual angle ¼ 1.1 ).

Self-report questionnaires
Prior to the task participants indicated how much time per week they spent playing musical instruments or video games.This was included as the FT-RSGT requires fast responses to the metronome tone and thus it was important to account for any influence of musical or video game training on Table 1 e The demographic information which was used to balance the six groups of participants.This includes age, sex (Female ¼ 1, Male ¼ 2), time spent on video games, time spent playing musical instruments and hours of sleep from the night before.performance (Boayue et al., 2021).After the FT-RSGT task was completed, participants completed the Mindful Attention and Awareness Scale (MAAS), which was designed to assess participants disposition to focus on the present.The MAAS has 15 items such as: "I find myself doing things without paying attention" and "I find myself listening to someone with one ear, doing something else at the same time".Participants rated how frequently they currently had each experience on a 6-point scale ranging from "Almost Always" (1) to "Almost Never" (6).The mean of the 15 items was used to calculate participants scores, with higher scores indicating higher levels of dispositional mindfulness.There were two additional measures which were designed to assess participants distractibility and rumination, to account for group differences across these domains.The Adult ADHD Self-Report Scale was designed to identify the frequency participants exhibited symptoms associated with attention-deficit/hyperactivity disorder (ADHD) and consisted of eighteen questions.The first six questions have been found to be the most predictive symptoms of ADHD, such as "When you have a task that requires a lot of thought, how often do you avoid or delay getting started?".The remaining twelve questions probed additional areas which the participant may have exhibited symptoms, such as "How often are you distracted by activity or noise around you?".The questions were rated on a 5-point scale ranging from "Never" (1) to "Very Often" (5) and the total score for each section was calculated and added together to develop a total score, with higher scores indicating a greater number of symptoms consistent with ADHD.The Rumination Responses Scale was also included and consisted of 22 items that assessed how often participants engaged in these thoughts when they are feeling down, sad, or depressed.Participants rated responses on a 4-point scale ranging from "Almost never" (1) to "Almost always" (4) and the items included thoughts such as: "What am I doing to deserve this?" and "Why can't I handle things better?".The total score for participants was calculated, with higher scores indicating greater ruminative symptoms.
There was also a detailed questionnaire assessing participants involvement with musical instruments and video games, including the types of games or instruments they played, alongside their engagement in other forms of entertainment.Finally, participants completed an end of session questionnaire including a motivation assessment, asking participants "How motivated were you to perform well in this task?"(Seli, Cheyne, Xu, Purdon, & Smilek, 2015).This question was rated on a 7-point scale ranging from "Not at all motivated" (1) to "Extremely motivated" (7).This questionnaire also assessed participants perspective on their task performance and stimulation experience.The full end of session questionnaire details can be found in the supplementary methods.

Stimulation protocol
Each participant received one of six stimulation protocols, which were delivered online during the FT-RSGT task: (1) anodal HD-tDCS over the left DLPFC; (2) sham HD-tDCS over the left DLPFC; (3) anodal HD-tDCS over the right IPL; (4) sham HD-tDCS over the right IPL; (5) anodal HD-tDCS over the occipital cortex; and (6) sham HD-tDCS over the occipital cortex (see Fig. 5).This was a between-groups design, such that there was only one session for each group of participants.The stimulation conditions for each brain region were doubleblinded, such that the experimenter and participant were both unaware whether participants were receiving active or sham stimulation.It was not feasible to blind participants to the location of the targeted brain region, as the electrodes were visible to the participants, however the participants were not informed of the study's hypotheses.At the end of each participant's session, they were asked which experimental condition they thought they were in e active or sham e to assess the effectiveness of the blinding.To ensure that the double-blinding was effective, the same question was also included for the experimenter to respond at the end of each testing session.In addition, a confidence rating was also included for the experimenter and participants asking, "How confident are you in your judgement of your experimental condition?"(see supplementary methods.This question consisted of a 7-point likert scale, ranging from "Not at all confident" (1) to "Extremely confident" (7), with the middle value as "Moderately confident" (4) (see Fig. 6).

HD-tDCS montage
The HD-tDCS was administered using a Nurostym stimulator, with a 4 x 1 electrode ring arrangement.The stimulator was manufactured in the United Kingdom, by the Neuro Device Group.The electrodes were 12 mm Ag/AgCl electrodes which  were attached to the scalp using a conductive gel.3All electrode placements were determined using the International 10e20 EEG system.For the left DLPCF stimulation, the anode was placed over F3, and the four reference cathodes were placed on F7, Fp1, C3, and Fz.This configuration was chosen to replicate the HD-tDCS montage used by Boayue et al. (2021).
The right IPL stimulation groups had the anode placed on P4 and the surrounding cathodes placed at T6, O2, Pz and C4.
There is evidence anodal bipolar tDCS to this region also has an effect on mind wandering propensity (Kajimura et al., 2019;Kajimura & Nomura, 2015), thus it was included to explore the effects using the anodal HD-tDCS.Finally, the occipital cortex groups had the anode placed at Oz with the upper cathode at Pz and three additional cathodes measured to each point, one below Oz and one straight out to the left and one to the right of Oz, at an equal radius as the distance from Oz to Pz.The occipital cortex was included as an active control region (Filmer et al., 2020).It is important to note that there has been research which utilises more focal HD-tDCS designs, with the electrode placements in more proximal locations such as the 3 Â 1 montage used by Chou et al. (2020).However, the current study was designed to replicate the findings of Boayue et al. (2021) in the DLPFC using the same task and to expand on this research into the IPL and occipital cortex, with the inclusion of the additional dynamic thought probes, thus the more distal 4 Â 1 montage used by Boayue et al. (2021) was most applicable for the three regions being investigate in this research.
All stimulation groups used 2.0 mA current density with 0.5 mA to each of the four reference electrodes (Boayue et al., 2021).The anodal stimulation lasted 30 min, including a 30 s ramp up and ramp down period.Participants in the sham conditions experienced 75 s of stimulation, with the same ramp up and ramp down times as the active stimulation groups.All participants initially completed the baseline block of the FT-RSGT, before then completing the 30-min task block with stimulation (active or sham).If participants experienced any discomfort, the stimulation was discontinued, and alternatively if there are any technical issues that affected the intensity or duration of the stimulation, or if the electrode impedance increased above the programmed impedance cutoff4 , then the session did not proceed.

Overview
The raw data files for each participant were immediately uploaded to The University of Queensland Research Data Manager cloud storage at the conclusion of their session or onto the lab servers managed by the School of Psychology.All analyses were completed in RStudio using the brms (Bayesian Regression Models using Stan; Bu ¨rkner, 2017) and BayesFactor packages (Mulder et al., 2021), or using JASP.All analyses employed a default Jeffreys-Zellner-Siow prior of r ¼ .707,centred around 0 (Rouder, Speckman, Sun, Morey, & Iverson, 2009).We chose to use a default prior as there was no previous research that we were aware of in this field which investigated the causal neural substrates of dynamic thoughts, thus it was not possible to use informed priors for this study.The results assessed which model the observed results are more likely under, as indicated by BF 10 and BF 01 .
We also reran the analyses for all four dynamic thought types using frequentist methods for completeness and to allow researchers who are more familiar with p-values to fully comprehend the results.All the raw data, study materials, supplementary methods, and analysis scripts from our study can be located at https://doi.org/10.48610/74fcc20.

Post-study data exclusion
After participants completed the study, the data was preprocessed and individuals who scored 3 standard deviations above or below the mean for approximate entropy and behavioural variability across all participants in the baseline block were excluded from the study.This resulted in eight participants being removed.Furthermore, we assessed participants responses to the end of session questionnaire and if the answers suggested that the participant did not generate random number sequences for the task correctly, we excluded their data before the analyses were conducted.An example which would suggest the task has not been completed correctly would be if the participant cites a specific pattern measure the online impedance, as it was not provided by the machine in the blinding mode.However, we did the nearest thing we could: we removed participants who had a high starting impedance.Overall, this should be comparable as the impedance checks are conducted at a much lower intensity, which means this was a more conservative criteria to begin stimulation.Q3 .An additional exclusion criterion, which was not included in the Stage 1 manuscript, was that if any participants scored more than 3 standard deviations above or below the mean for approximate entropy and behavioural variability or the four thought probe responses, for each group in the stimulation block, they were also removed.Given the critical analyses in this study were focused on the stimulation block, this criterion was included to ensure that participants who did not respond correctly during this block were also removed from the dataset.Furthermore, this cut off was the same as the Stage 1 baseline exclusion criteria for the two task performance measures, and it was only possible to apply it to the thought probes in this block because there were 20 more probes included in the stimulation block than the baseline block.This resulted in an additional four people being removed, creating a final sample of 228 participants.Finally, to avoid extreme behavioural variability and randomness scores skewing the time-on-trial analyses in the hierarchical order probit modelling, trials which were above or below 3 standard deviations for each brain region were also removed post-hoc.This resulted in 38 trials (1.25% of trials) being removed for the DLPFC groups, 53 trials (1.74%) being removed for the IPL groups and 42 trials (1.38%) for the occipital cortex groups being removed, with a final trial sample of 3002, 2987 and 2998 for each region, respectively.To ensure consistency with the Stage 1 submission, we reran all analyses without these additional stimulation block and trial exclusion criteria and there were no meaningful differences for the key stimulation effects.The supplementary materials contain a full summary of results based only on the Stage 1 exclusions (i.e., the sample consisting of 232 participants).

3.3.
The effect of HD-tDCS on dynamic thought types

Task unrelated thought
We predicted that there would be meaningful evidence of a difference between active anodal HD-tDCS and the frequency of task unrelated thoughts for the left DLPFC and right IPL (H 1 ).
To determine the effect of stimulation on task unrelated thought, we ran three Bayesian independent samples t-tests.
These tests compared the anodal HD-tDCS group to sham in each brain region (i.e., DLPFC anodal HD-tDCS compared to sham; IPL anodal HD-tDCS compared to sham; and Occipital anodal HD-tDCS compared to sham).Each analysis assessed whether the observed results were more likely under the "Alternate" or the "Null" (i.e., that the difference between the two groups equals 0), as indicated by BF 10 and BF 01 values.While these analyses were conducted across the three brain regions, the results from the Bayesian t-test for stimulation specifically in the DLPFC was the critical test for the study, as there was preliminary evidence for this effect in the PFC found by Boayue et al. (2021).In addition, we ran three 2 (stimulation: active, sham) x 2 (region: DLPFC, IPL, Occipital Cortex) Bayesian between-subjects ANOVAs to assess the overall effect of stimulation across the three brain regions on the frequency of task unrelated thought (i.e., through investigating the effects of active and sham stimulation across each possible comparison of the three brain regions).

Novel dynamic thought types
As there was evidence to suggest different types of dynamic thought may be occurring in distinct regions of the brain (Kam et al., 2021;Martel et al., 2019), the data for three novel types of dynamic thought was also analysed (H 2 ).This consisted of independently analysing the thought probe data from the freely moving, deliberately constrained and automatically constrained probes.To determine the effect of stimulation on the frequency of freely moving thought across the three brain regions, we re-ran the same analyses as for task unrelated thought, including the independent samples t-tests and the between-subjects ANOVAs.The same t-tests and ANOVAs were also applied to assess the effect of stimulation on differences in the frequency of deliberately constrained thought and the automatically constrained thought, respectively, across the three different brain regions.

3.4.
The effect of HD-tDCS on task performance 5 We were also interested in investigating the effect of stimulation on task performance (H 3 ), using hierarchical order probit modelling for each type of thought probe (Boayue et al., 2021;Filmer et al., 2019).For each dynamic thought type, we ran 23 models including the predictor variables of behavioural variability, approximate entropy, trial, part (baseline vs stimulation) and stimulation group (active vs sham), alongside their interactions.This analysis technique allowed the thought probe responses to be treated as an ordinal response variable, which has been argued to be a more accurate approach to investigate the response variability within an individual's performance during the task and to account for the time-on-task effect (Boayue et al., 2021;Filmer et al., 2019).
The 23 models increased in complexity and the model weights were interpreted using two different methods.The model weights will first be calculated using pareto smoothed importance sampling leave-one-out cross-validation scores (PSIS-LOO; Vehtari, Gelman, & Gabry, 2017,2022), whereby the leave-one-out cross-validation information criterion values (LOOIC; Vehtari et al., 2017;Wagenmakers & Farrell, 2004), which is based on Akaike weights, were compared using a stacking procedure (Vehtari & Gabry, 2018;Yao, Vehtari, Simpson, & Gelman, 2018).The model weights were also assessed using pseudo-Bayesian model-averaging (pseudo-BMA; (Vehtari & Gabry, 2018;Yao et al., 2018 Where there were discrepancies between the LOOIC and pseudo-BMA winning models, we chose to select the preferred LOOIC model.This is because it has more reliable predictive accuracy by selecting the optimal predictive distribution, rather than being biased towards the model closest to the Kullback-Leibler divergence (as the pseudo-BMA method is) when the true model is not included in the model list (Vehtari & Gabry, 2018;Yao et al., 2018).When these discrepancies occurred, we ran additional post-hoc exploratory analyses to confirm our findings.Given the main analyses included the baseline data which had relatively low numbers of trials (720), the model comparisons were re-run on the stimulation block alone, to ensure any potential confounds due to the differences in probe and trial numbers between the baseline and stimulation blocks were removed.The additional analyses on the stimulation block data can be seen in the exploratory analyses section.
We also included a simplified analysis to investigate the relationship between task performance and the thought probe responses.This was conducted by calculating the approximate entropy and behavioural variability mean values for the 15 s of trials (n ¼ 20 trials) before the four thought probes.For the task unrelated thought probes, the mean values for approximate entropy and behavioural variability was then compared for on task and off task responses.A score of 1e3 was considered on task and a score of 4e7 was considered off task, with 4 considered off-task for this experiment as the labelling of the Likert scale and phrasing of the questions indicated 4 may be considered as "moderately" off-task by participants.The comparison of the mean approximate entropy and behavioural variability scores between periods "on" and "off" task was conducted for each dynamic thought type, with the same categorisation of response scores for all four thought probes.This analysis was repeated to compare all trials which preceded each presentation of the four thought probes to also assess the relationship between task performance and selfreported dynamic thoughts across the entire experiment.

Calculating approximate entropy
An approximate entropy (ApEn) value was used to calculate the randomness of a participant's generated sequence, as it is mathematically impossible to calculate the entropy of a finite sequence.ApEn is designed to evaluate the randomness in a sequence of m numbers, by calculating the predictability of the next item in a sequence, given the previous sequence of m numbers (Pincus, 1991;Pincus & Singer, 1996;Yentes et al., 2013).A value of m ¼ 2 was used, which replicated the m value used by Boayue et al. (2021), as this value was calculated to be the most sensitive for detecting differences in the randomness of sequences for the FT-RSGT.A smaller ApEn value indicated a data set contains many repetitive patterns, whereas a higher ApEn score indicated less predictable response patterns in the data.

Testing for baseline differences
We also reran the primary 12 Bayesian independent samples t-tests on the baseline data alone.This was to ensure there were no significant differences between the groups in their baseline performance.A BF 01 ! 3 indicated there was moderate evidence for no meaningful differences between the groups, and thus the null hypothesis was accepted.This analysis was repeated for all four thought probes.Further to this, to investigate any group differences in responses to the three self-report measures we used three oneway independent groups ANOVAs, including the MAAS, the Adult ADHD Self-Report Scale, the Rumination Responses Scale as the dependent variable, respectively, and the six stimulation groups as the independent variable (anodal over left DLPFC, sham over left DLPFC, anodal over right IPL, sham over right IPL, anodal over occipital cortex, sham over occipital cortex).Any significant differences found between the groups were followed up using t-tests, and any self-report measures with significant group differences were to be included in the primary analyses as a covariate.

Assessing blinding
To ensure the HD-tDCS blinding was effective across the groups, we also assessed the proportion of correct guesses for people in the active compared to sham conditions.In addition, to understand the effect of subjective belief on mind wandering frequency, we implemented two Bayesian ANOVAs for each thought probe, employing objective intervention and subjective intervention as between-subject factors and the outcome measure of the average ratings for each thought probe, respectively.The first ANOVA compared all stimulation groups; however, the second ANOVA excluded conditions which showed limited evidence for differences between stimulation and sham according to the behavioural data, thus only evaluating conditions that demonstrate meaningful results.

The effect of HD-tDCS on dynamic thought types using Bayesian t-tests and ANOVAs
To investigate the direct relationship between stimulation and participants mean thought probe responses we used a series of Bayesian independent samples t-tests, comparing the anodal HD-tDCS and sham groups for each thought probe and region.We also ran three 2 (Stimulation: active, sham) x 2 (Region: DLPFC, IPL, or occipital cortex) Bayesian betweensubjects ANOVAs for each probe to assess the overall effect of stimulation across the three brain regions on the frequency of the dynamic thought responses.In line with Cortex's guidelines, we interpreted meaningful effects when BF 10 > 6 or BF 01 > 6, for the alterative and null hypotheses, respectively.

Task unrelated thought
There was no evidence for a meaningful effect of HD-tDCS on participants task unrelated thought in any brain region.To wit, there was no meaningful evidence for any effects found in Please cite this article as: Rasmussen, T., et al., On the role of prefrontal and parietal cortices in mind wandering and dynamic thought, Cortex, https://doi.org/10.1016/j.cortex.2024.06.017 the three Bayesian ANOVAs (BF excl > 1.95 for all).The Bayesian independent sample t-tests also found anecdotal to moderate evidence against a difference between the active and sham groups across the three regions (BF 01 > 1.35 for all).However, this outcome was only meaningful in the IPL region (BF 01 ¼ 6.73).These findings do not align with our predicted effect of active HD-tDCS reducing task unrelated thought in the DLPFC and IPL regions, relative to the sham and occipital cortex groups.

Freely moving thought
There was a meaningful interaction between the active and sham conditions in the DLPFC and occipital cortex for freely moving thought.This was revealed in the 2 (Stimulation: active, sham) x 2 (Region: DLPFC, Occipital Cortex) ANOVA which showed meaningful evidence for the inclusion of the interaction term [BF incl ¼ 8. ).Overall, these findings suggest there was evidence for a meaningful interaction between the effects of stimulation in the DLPFC and the occipital cortex, however there was no evidence for an interaction between the IPL and occipital cortex.Furthermore, our hypothesised reduction in freely moving thought with active stimulation to the DLPFC and IPL, relative to the sham conditions, was not supported by these analyses as none of the t-test results meet our threshold to interpret the differences, or a lack thereof, between the two groups.

Deliberately constrained thought
There was no evidence for an effect of HD-tDCS on deliberately constrained thought, with the t-tests finding moderate evidence against differences between the active and sham conditions for the three brain regions (BF 01 > 3.08 for all).However, only the DLPFC revealed a meaningful result (BF 01 ¼ 7.29).
Together, these results provide evidence against the hypothesised reduction in the active stimulation groups for the DLPFC and IPL.

Automatically constrained thought
There was no evidence for differences between the active and sham conditions found in any of the three brain regions for automatically constrained thought (BF 01 > 1.56 for all).Importantly, the evidence against a difference between the two groups was only meaningful in the IPL region for this thought type (BF 01 ¼ 7.27).These findings suggest the null hypothesis was supported in the IPL, and there were no meaningful effects found for the DLPFC, which does not support our predicted reduction in automatically constrained thought for the active group in these regions.

Testing for baseline differences
Collectively there was little evidence that any differences at baseline had an impact on the above findings.To ensure there were no meaningful differences between the groups in their baseline performance, we reran the primary 12 Bayesian independent samples t-tests on the baseline data alone (see Table 2).A BF 01 ! 3 was used to indicate there were no meaningful differences between the groups, and thus the null hypothesis was accepted.All baseline comparisons were in favour of the null with one exception.For occipital cortex stimulation, there was anecdotal evidence for an effect (BF 10 ¼ 1.89).
Further to this, to investigate any group differences in responses to the three self-report measures we used three oneway independent groups ANOVAs.These included the MAAS, the Adult ADHD Self-Report Scale, the Rumination Responses Scale as the dependent variable, respectively, and the six stimulation groups as the independent variable (anodal and sham over the DLPFC, IPL, and OC, respectively).These analyses revealed strong evidence for excluding the group variable in all three questionnaires (BF excl > 34.79 for all), suggesting there were no differences between the stimulation groups.

Assessing blinding
It was also important to ensure that the HD-tDCS blinding was effective across the groups and did not contribute to the pattern of results observed.Thus, we assessed the proportion of correct guesses for people in the active compared to sham conditions.Overall, the sham blinding was effective in blinding people to their stimulation condition.Specifically, this analysis revealed that 115 out of 228 participants correctly guessed their group (50%).Generally, there was a bias for participants to select active stimulation regardless of group.In the active group, 99 participants out of 114 correctly guessed their group (87%) and 15 guessed incorrectly (13%).However, in the sham group only 16 participants, out of 114, correctly guessed their group (14%) and 98 participants guessed incorrectly (86%).
To further understand the effect of subjective belief on the frequency of the dynamic thought types, we implemented two Bayesian ANOVAs for each thought probe, employing objective intervention and subjective intervention as betweensubject factors and the outcome measure of the average ratings for each thought probe (Gordon et al., 2022).The first set of ANOVAs compared all stimulation groups for each region (see Table S1); however, the second set of ANOVAs excluded conditions and probes which showed limited evidence for differences between stimulation and sham according to the modelling analyses, thus only evaluating conditions that demonstrate meaningful results (see Table 3).For the freely moving thought results, in the DLPFC which found evidence for stimulation effects in the modelling analyses, the results showed the strongest predictor was the objective stimulation.The IPL result for the deliberately constrained thought found no evidence for both objective and subjective condition.
We also reran the proportion analyses on the experimenter blinding responses.Overall, blinding was effective.The experimenter made 128 correct guesses out of 228 sessions (56%).In the active group, 48 out of 114 guesses were correct (42%) and in the sham group 80 out of 114 guesses were correct (70%).

4.4.
Exploratory analyses 4.4.1.Using modelling to investigate the effect of HD-tDCS on task performance and the dynamic thought types As outlined in our introduction and methods, we aimed to employ the same approach as Boayue et al. (2021) and to expand on this research by investigating the effects of stimulation on the novel dynamic thought types across three brain regions.At Stage 1, we proposed employing hierarchical order probit modelling to investigate the effect of stimulation on both the dynamic thought types and behavioural performance (using our measures of approximate entropy and behavioural variability).However, at the Stage 2 submission, it became clear that no explicit criteria for making inferences from the probit modelling had been defined at Stage 1.Despite receiving in-principle acceptance at Stage 1, this analysis did not align with Cortex's guidelines for a registered report, specifically a Bayes value of 6 or p value of p < .02.Thus, the probit modelling is now reported as an exploratory analysis, subsequent to the main planned hypothesis testing analyses.
In line with the interpretation used by Boayue et al. (2021), we determined meaningful results from the probit modelling as regression coefficients with a posterior probability !95%.These values were determined from hypothesis testing on the winning model predictors for each region, whereby a onetailed test for each predictor of interest was employed, based on the direction of the hypothesised effect for that predictor and their interactions.Furthermore, the evidence ratios for each effect were also included, and these odds ratios represent the likelihood of the regression coefficient being in the hypothesised direction (e.g., a negative effect), as opposed to the opposite direction (e.g., a positive effect) or to the effect being zero.When there were discrepancies between the LOOIC and Pseudo-BMA winning models, an additional analysis was implemented that expanded on the approach used by Boayue et al. (2021), and that which was specified at Stage 1.In these cases, the same model comparisons were re-run on the stimulation block data alone, which resulted in 16 models being compared for relevant thought probes and regions.The same model weighting techniques were applied and to be consistent with the complete modelling analyses, if there were conflicting findings, the winning LOOIC model was selected (Vehtari & Gabry, 2018;Yao et al., 2018).The full list of models and model comparisons for the complete models and stimulation block only models can be seen in Figure S1 and S2, respectively.

Stimulation did not modulate propensity for task unrelated thoughts
The above reported t-tests found no effect of DLPFC HD-tDCS on task unrelated thought.In sum, this was also consistent in the modelling analyses.Specifically, if a stimulation effect had been present the key predictor would be the stimulation Â block interaction.A meaningful negative coefficient for this interaction would replicate the reduction in task unrelated thought with active stimulation found by Boayue et al. (2021).However, in our data the model favoured  S3 for complete model winning model predictors).The LOOIC and Pseudo-BMA weights did not agree on the preferred model in this region (see Table S2 for the complete model top model selection weights), and thus the probit modelling was also run on the stimulation block data alone.The key predictor to indicate an effect of stimulation on the dynamic thought type for this block alone was a meaningful main effect of stimulation.However, the preferred LOOIC model for the stimulation block (p PSIS-LOO ¼ .48;see Table S3 for the stimulation block top model selection weights) also showed no evidence for an effect of stimulation in the DLPFC (b ¼ À.23, 95% credible interval (CI) [À.76, .28],ER-¼ 4.09, posterior p ¼ .42).See Figure S4 for the stimulation block winning model predictors.
Similarly, for IPL stimulation, the preferred LOOIC model included only a main effect of behavioural variability, randomness, and block, alongside a behavioural variability Â randomness interaction (p PSIS-LOO ¼ .24,R 2 ¼ .47[.46, .49]).Given the LOOIC and Pseudo-BMA weights did not agree on the preferred model, the analysis was conducted on the stimulation block data alone and the winning LOOIC model (p PSIS-LOO ¼ .39)also showed no evidence of a stimulation effect.Thus, these findings suggest HD-tDCS did not influence task unrelated thoughts for these two regions.There was also no evidence for the predicted meaningful interaction between behavioural variability and randomness in the DLPFC and IPL, (b ¼ .06,95% CI [À.21, .32],ERþ ¼ 2.23, posterior p ¼ .69)and (b ¼ .07,95% CI) [À.17, .30],ERþ ¼ 2.41, posterior p ¼ .71),respectively.Furthermore, there was no support for our hypothesised effect of stimulation increasing randomness and reducing behavioural variability, as these interaction effects were not selected in the winning models for both regions.
No influence of stimulation on task unrelated thought or our two measures of task performance was observed for the occipital cortex.However, there was a positive relationship between behavioural variability and randomness, such that when participants were more random (i.e., they had greater executive control), greater variability in responses was predictive of more task unrelated thought, than when participants were less random.The preferred model did not agree between the selection procedures, but the winning LOOIC model (p PSIS-LOO ¼ .25,R 2 ¼ .45[.44, .47])only included a meaningful effect of behavioural variability, block, and a positive behavioural variability Â randomness interaction (b ¼ .55,95% CI [.28, .83],ERþ ¼ ∞, posterior p ¼ 1.00).There was no evidence for a main effect of stimulation or a block Â stimulation interaction.Nor was there evidence for a stimulation effect in the winning LOOIC model for the stimulation block data alone (p PSIS-LOO ¼ .60).In the complete model, there was a meaningful randomness Â stimulation interaction (b ¼ .13,95% CI [.03, .22],ERþ ¼ 221.22,posterior p ¼ 1.00), however as the stimulation variable was averaged across the task blocks, this was not a direct effect of stimulation on randomness.Thus, these results do not support our hypothesised effect of stimulation increasing randomness or reducing behavioural variability in this region, however they do support the predicted positive relationship between randomness and behavioural variability.These results are consistent with the t-test analyses showing that HD-tDCS did not influence task unrelated thought in the occipital cortex.

Freely moving thoughts were influenced by DLPFC stimulation
The modelling revealed DLPFC stimulation reduced freely moving thought.Specifically, the critical block Â stimulation interaction showed that in the stimulation block, relative to the baseline block, freely moving thought was reduced in the active group compared to the sham group (b ¼ À.31, 95% CI [À.69, .07],ER-¼ 17.26, posterior p ¼ .95).There was also a meaningful relationship between behavioural variability and randomness, suggesting that during periods of greater randomness, greater variability was more predictive of freely moving thought compared to during periods with more predictable sequences.The winning Pseudo-BMA and LOOIC models did not align, however the preferred LOOIC model (p PSIS-LOO ¼ .49,R 2 ¼ .41[.39, .43])also included main effects of behavioural variability, randomness, block, trial, and stimulation, alongside a positive behavioural variability Â randomness interaction (b ¼ .22,95% CI [À.03, .48],ERþ ¼ 21.10, posterior p ¼ .95;see Fig. 7).This supports the hypothesised relationship between the two measures of task performance, showing that greater response variability was predictive of greater freely moving thought when participants were more random.However, there was no evidence for an effect of stimulation on either behavioural variability or randomness, as these predictors were not selected in the winning model.There was also no main effect of stimulation alone (b ¼ À.03, 95% CI [À.50, .43],ER-¼ .81,posterior p ¼ .45),as would be expected as this included pre-stimulation (baseline) data.In the stimulation block alone, there was also evidence in the direction of anodal stimulation reducing freely moving thoughts via a main effect of stimulation (b ¼ À.34, 95% CI [À.81, .08],ER-¼ 14.87, posterior p ¼ .94),however this finding did not meet the threshold to be considered meaningful.The preferred LOOIC model for the stimulation data also found evidence for an effect of behavioural variability and trial (p PSIS-LOO ¼ .35).Overall, these findings provide preliminary support for freely moving thought being impacted anodal stimulation, however this is specific to the DLPFC alone (see General Discussion).
For participants receiving stimulation to the occipital cortex, the winning LOOIC model (p PSIS-LOO ¼ .32,R 2 ¼ .40[.38, .42])was consistent with the DLPFC findings and included main effects of behavioural variability, randomness, block, trial, and stimulation, alongside a behavioural variability x randomness and block Â stimulation interaction (see Fig. 8).However, there was no evidence for a block Â stimulation interaction for this model (b ¼ À.08, 95% CI [À.39, .23],ER-¼ 2.23, posterior p ¼ .69),nor for the interaction between the behavioural measures (b ¼ .09,95% CI [À.17, .35],ERþ ¼ 3.04, posterior p ¼ .75).Furthermore, there was no evidence for an effect of stimulation on the two measures of task performance, given these predictors were not selected in the winning model.In contrast, there was evidence for a main effect of stimulation (b ¼ .51,95% CI [.07, .96],ERþ ¼ 94.24, posterior p ¼ .99),averaged across the two task blocks.This suggests there was a difference between the stimulation groups, however as this is not a function of task block, the full c o r t e x x x x ( x x x x ) x x x modelling suggests there is no effect of stimulation in the occipital cortex.Interestingly, the modelling for the stimulation block alone revealed a meaningful main effect of stimulation, whereby freely moving thoughts were increased in the active group relative to sham (b ¼ .43,95% CI [.01, .83],ERþ ¼ 41.11, posterior p ¼ .98).The preferred LOOIC model for the stimulation data also found evidence for an effect of behavioural variability and trial (p PSIS-LOO ¼ .46).Despite not finding evidence for a block Â stimulation interaction in the complete model analysis, this main effect of stimulation suggests anodal HD-tDCS may increase freely moving thought in the occipital cortex, relative to sham.
For the IPL, the preferred model did not align between the model selection procedures, and the winning LOOIC model (p PSIS-LOO ¼ .34,R 2 ¼ .39[.37, .41])found no evidence for effects of stimulation on freely moving thought nor on either measure of task performance.This model only included a main effect of behavioural variability, randomness, block, trial, and stimulation, alongside a behavioural variability Â randomness interaction, in which the CIs crossed 0. The stimulation block modelling results for the winning IPL model (p PSIS-LOO ¼ .34)also showed no effect of stimulation (b ¼ À.29, 95% CI [À.71, .11],ER-¼ 10.11, posterior p ¼ .91).Thus, there was no evidence for an effect of stimulation on behavioural task performance or the thought probe responses in this region.

Deliberately constrained thoughts were reduced by IPL stimulation
Anodal stimulation of IPL reduced deliberately constrained thought, as was indicated by a negative block Â stimulation interaction (b ¼ À.52, 95% CI [À1.02,À.01], ER-¼ 42.48, posterior p ¼ .98).Here, the Pseudo-BMA and LOOIC weights agreed upon winning model (p PSIS-LOO ¼ .38,R 2 ¼ .45[.43, .46]),which also included weak evidence for an effect of trial and a block x trial Â stimulation interaction (for both active and sham as the reference condition), and no evidence for an effect of trial or a Fig. 7 e The winning complete model for freely moving thought in the prefrontal cortex, which shows active stimulation reduced these thoughts in the stimulation block relative to baseline.
Fig. 8 e The occipital cortex winning complete model for freely moving thought, which found a main effect of stimulation, but no interaction across the two blocks.trial Â stimulation interaction (see Fig. 9).There was also no evidence for a main effect of stimulation, averaged across the two blocks (b ¼ .43,95% CI [À.13, 1.02], ERþ ¼ 12.47, posterior p ¼ .93)which supports the changes in deliberately constrained thought reported in the active condition, being specific to the differences found between the baseline and stimulation blocks.This does not align with the t-tests, which found anecdotal evidence against a difference between the two groups, however, this may be due to the improved sensitivity of the hierarchical order probit modelling to detect time-ontask effects across the task, rather than taking an average for each participant's probe responses.The winning model did not select any behavioural predictors, thus the expected relationship of stimulation reducing behavioural variability and increasing randomness was not supported.Furthermore, there was no evidence for a relationship between the two measures of performance.
Stimulating the DLPFC or the occipital cortex did not modulate deliberately constrained thoughts, nor participants behavioural variability or randomness.For the DLPFC, the winning LOOIC model (p PSIS-LOO ¼ .20,R 2 ¼ .39[.37, .40])only included main effects of behavioural variability, randomness, and block, alongside a behavioural variability Â randomness interaction, where the CIs crossed 0. However, this model was not selected by the Pseudo-BMA procedure in this region, thus the probit modelling was conducted on the stimulation block data alone (p PSIS-LOO ¼ .37)and the findings were consistent as there was also no evidence for a main effect of stimulation in this dataset (b ¼ .18,95% CI [À.24, .61],ERþ ¼ 3.65, posterior p ¼ .78).While the model selections agreed for the occipital cortex (p PSIS-LOO ¼ .25,R 2 ¼ .42[.40, .44]), the winning model did not include any evidence for a main effect or interaction including stimulation.

Stimulation did not modulate automatically constrained thoughts
For the automatically constrained thought probe, there was no evidence of stimulation effects in any region, with all three winning models not including the stimulation predictor in any main effects or interactions.In addition, the winning models did not agree between the Pseudo-BMA and LOOIC procedures for any region.In the DLPFC, the winning LOOIC model (p Consistent with the overall modelling, there were no effects of stimulation across the three brain regions on automatically constrained thoughts for the stimulation data.For the DLPFC and IPL, the winning LOOIC models did not include the stimulation predictor (p PSIS-LOO ¼ .41 and .28,respectively).Finally, in the occipital cortex, the agreed on preferred model (p PSIS- LOO ¼ .41)showed no evidence the main effect of stimulation (b ¼ .33,95% CI [À.16, .76],ERþ ¼ 12.79, posterior p ¼ .93).

The relationship between task performance and dynamic thought probe responses
In addition to the probit modelling, we also conducted a simplified analysis investigating the relationship between task performance and the dynamic thought responses.The mean values for approximate entropy and behavioural variability were compared for on-task and off-task responses, averaged across conditions.As anticipated, and across all dynamic thought types, randomness was lower when off-task and higher when on-task (see Fig. 10).Also, in line with our predictions, ratings of being off-task for freely moving thoughts led to higher, and on-task lower, behavioural variability.However, for the other three thought types, behavioural variability showed the opposite pattern: off-task leading to reduced variability compared to on-task.This may reflect that due to the nature of our paradigm and tasks requirements participants prioritised generating random sequences over accurately timing their responses to the tone.

Discussion
Here, we applied HD-tDCS to three brain regions, to investigate the causal neural substrates of distinct internal thought types in a sham-blinded, active control study.Our hypothesis driven results found no evidence for an effect of stimulation on task unrelated thought and no evidence for an effect on our dynamic thought types, nor on task performance.However, replicating the analysis approach used by Boayue et al. (2021) in an exploratory analysis, we found preliminary evidence that anodal stimulation had differential effects across the brain regions examined.Specifically, freely moving thought was reduced by stimulating the DLPFC, while deliberately constrained thought was decreased by IPL stimulation.There was also preliminary evidence for an effect of stimulation increasing freely moving thought in the occipital cortex.While we cannot draw definitive conclusions from these exploratory findings, they provide insight into the importance of continuing to explore mind wandering as a heterogeneous construct which may recruit a range of neural regions according to the current direction of an individuals' thoughts (Kam et al., 2021;Martel et al., 2019).
The evidence against stimulation affecting task unrelated thought using the Bayesian t-tests it did not support our hypothesised effect of stimulation reducing generalised mind-wandering.Our exploratory probit modelling was also consistent in finding no evidence for this effect.However, the evidence against an effect aligns with the recent failed attempt to replicate Boayue et al.'s (2021) reported modulations to task unrelated thought using HD-tDCS by Alexandersen, Csifcs ak, Groot, and Mittner (2022), who observed no evidence for the original effect of 2 mA HD-tDCS anodal stimulation applied to DLPFC reducing mind wandering.Given the inconsistencies in mind wandering literature to date, it is important to understand the methodological practices which may affect these results (Alexandersen et al., 2022;Boayue et al., 2020;Chaieb et al., 2019).In this study, the groups were demographically balanced with large sample sizes in each group, which reduces the possibility of individual differences causing differences between the groups.In addition to the large sample size, the sham-blinding was effective  for both the subjects and experimenter which suggests the current findings were not driven by a placebo effect.Furthermore, the improved focality of the HD-tDCS may provide a more nuanced understanding of the role of each region for distinct dynamic thought types, compared to studies using traditional tDCS.This is because it allows for more targeted stimulation covering a smaller cortical area (Nikolin et al., 2019;Villamar et al., 2013), rather than stimulating across brain regions.Thus, our findings suggest a measure of task unrelated thought may not always be sensitive to detecting shifts to internal thought processes.
We also hypothesised that anodal stimulation would reduce freely moving thought in both the IPL and DLPFC, relative to the sham and occipital conditions.Thus, this hypothesis was not supported.The planned hypothesis testing analyses found no meaningful evidence of a stimulation effect in the DLPFC.However, the exploratory probit modelling found preliminary evidence for a reduction in freely moving thought induced via DLPFC stimulation alone.This inconsistency in the results may be due to the improved sensitivity of the hierarchical order probit modelling to detect changes in an individual's performance over the duration of the task (Boayue et al., 2021).Our exploratory findings converge with EEG research by Kam et al. (2021), who found frontal alpha power was associated with freely moving thoughts.Furthermore, this result highlights the potential for freely moving thought to act as a more sensitive measure for detecting mind wandering.To wit, freely moving thought has been found to be independent of task-relatedness (Kam et al., 2021;Mills et al., 2018), and it has also been linked to creative and spontaneous thought (Christoff et al., 2016), which may represent a more valid assessment of mind wandering.
The ANOVA results highlighted a meaningful interaction between the DLPFC and occipital cortex stimulation, illustrating there was a difference in the stimulation effects between these two regions.However, the t-test comparisons found no evidence for a meaningful effect in the occipital cortex.There was, however, weak evidence from the exploratory probit modelling on the stimulation block data alone for an increase in freely moving thought.While the weak evidence for a baseline difference in the occipital cortex group for freely moving thought could explain why there was only an effect in the stimulation block, this is unlikely because there was significantly more data in the stimulation block for detecting differences between the stimulation groups.Overall, these findings do not support our hypothesised effect for differences in the DLPFC and IPL, relative to the sham and occipital groups.However, the exploratory findings provide interesting insight into the potentially differential effects of HD-tDCS across regions for freely moving thought.
In relation to deliberately constrained thought, we also hypothesised anodal stimulation would reduce these thoughts in both the IPL and DLPFC, relative to the sham and occipital conditions, which was not supported by the results.As with the freely moving thought data, there was no evidence for a meaningful effect using the t-tests.However, again, this analysis may not have detected the IPL effect as the t-tests only assessed the mean probe responses for each group, which does not allow a sensitive exploration into the effect of stimulation on the reporting of these thoughts throughout the task.There was preliminary evidence for anodal stimulation reducing deliberately constrained thought in the IPL alone using the exploratory probit modelling.This is consistent with Kam et al. (2021) finding distinct electrophysiological signatures for different type thoughts.However, interestingly the exploratory results differed from Kam et al. (2021), as they found differences for the frontal ERPs and in this study the preliminary effects were found in the parietal lobule.This contrasting result is likely due to the use of distinct methods.For example, EEG has been recognised as having poor spatial resolution, which makes the localisation of the electrical activation highlighted by the frontal effects difficult (Michel et al., 2004).Overall, these findings suggest there was no support for our hypothesised effect of stimulation on deliberately constrained thought, however there may be preliminary evidence for distinct effect of stimulation on these thoughts across brain regions.
The changes in randomness scores and behavioural variability between on-and off-task thoughts of freely moving thought also suggests a relationship between internal thoughts and the recruitment of executive functioning resources.Specifically, shifts towards periods of freely moving thought reduce randomness and increase individuals' behavioural variability, thus limiting the executive resources required to stay on task.This is consistent with the relationship found by Boayue et al. (2021), and supports evidence for the overlap in the brain regions recruited during executive control processes and mind wandering (Christoff et al., 2009;Fox et al., 2015).However, the exploratory analyses found no effect of stimulation on the measures of task performance across the thought probes, which did not align with the hypothesised reduction in behavioural variability and increase in randomness for the anodal stimulation groups in the DLPFC and IPL.There is previous literature which has also found limited evidence of stimulation on task performance measures in the context of mind wandering, with some task effects being dependent on the stimulation intensity (Filmer et al., 2021).This suggests the null effect may also be due to methodological factors which were not manipulated in the current research.
In summary, this study found no clear evidence for stimulation modulating the dynamic thought types, however there was preliminary exploratory evidence which suggests distinct neural substrates may be implicated in different internal thought processes.The predicted effect of stimulation on task unrelated thought found by Boayue et al. (2021) failed to replicate here, however, this aligns with a more recent highpowered replication of the original study by Alexandersen et al. (2022).This suggests that task unrelated thought may not always be sensitive to detecting shifts to internal thought processes.Furthermore, there was no evidence to support our hypothesised effects of stimulation on the dynamic thought types, as we expected a reduction for each thought type to be found for both the DLPFC and IPL groups.Instead, we have shown in an exploratory context, using sensitive behavioural and analysis techniques, that there is evidence anodal stimulation to the left DLPFC may reduce freely moving thought and increase freely moving in the occipital cortex, alongside right IPL stimulation potentially reducing deliberately constrained thought.By understanding the importance of distinct types of dynamic thought, across the brain, this will enable c o r t e x x x x ( x x x x ) x x x Please cite this article as: Rasmussen, T., et al., On the role of prefrontal and parietal cortices in mind wandering and dynamic thought, Cortex, https://doi.org/10.1016/j.cortex.2024.06.017 researchers to target key areas in a context dependent manner, to improve executive functioning and cognitive control processes.These preliminary findings may also have clinical applications in being able to target pervasive negative thoughts generated in distinct brain regions, expanding on research such as that by Chou et al. (2020) who observed that cathodal tDCS applied to the IPL reduced negative thoughts about the past.While there was no evidence for stimulation affecting participants performance on the task, there was a relationship between participants behavioural variability and randomness when they reported freely moving thought.Thus, this research may also be used to improve performance in contexts which require sustained attention.Collectively, this research provides preliminary evidence for the potential effectiveness of HD-tDCS modulating dynamic thought types across the brain.

Open practices
The study in this article has earned Open Data, Open Materials and Preregistered badges for transparent practices.The data, materials and preregistered studies are available at: https:// doi.org/10.17605/OSF.IO/XZ4RN.

Fig. 1 e
Fig. 1 e An illustration of a bipolar tDCS montage showing the target electrode (red) over the prefrontal cortex (F3) and reference electrode (blue) over the parietal cortex (P4).
c o r t e x x x x ( x x x x ) x x x

Fig. 2 e
Fig. 2 e Current modelling comparing the 4 x 1 ring HD-tDCS montage over the left prefrontal cortex (on the upper left), with the anode at F3 and cathodes placed at F7, C3, Fz and Fp1 and the right parietal cortex (on the upper right), with the anode at P4 and cathodes placed at T6, O2, Pz and C4.This is compared to a bipolar tDCS montage (lower image), with the active electrode over the left prefrontal cortex and reference electrode positioned over the right parietal cortex.

Fig. 4 e
Fig. 4 e An illustration of the Finger-Tapping Random-Sequence Generation Task, showing the two keys' participants used to generate the random sequences (on the left) and the fixation cross displayed during the task (on the right).

Fig. 5 e
Fig. 5 e An illustration of the three HD-tDCS montages, showing the set up for the left dorsolateral prefrontal cortex (far left), the right inferior parietal cortex (middle) and the occipital cortex (far right) with the anode (in red; positioned at F3, P4 and Oz, respectively) and four reference cathodes for each montage.

Fig. 6 e
Fig. 6 e The effect of anodal HD-tDCS on freely moving thought in the three target brain regions.

Fig. 9 e
Fig.9e The parietal lobule winning full model for deliberately constrained thought, which showed active stimulation reduced these thoughts in the stimulation block relative to the baseline block.

Fig. 10 e
Fig. 10 e The relationship between behavioural variability and randomness for off-and on-task thoughts across (a) task unrelated thought, (b) freely moving thought, (c) deliberately constrained thought, (d) automatically constrained thought.

©
Available online at www.sciencedirect.comScienceDirect Journal homepage: www.elsevier.com/locate/cortex2024 The Author(s).Published by Elsevier Ltd.This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
used to approach the task (e.g., they repetitively used z,z,m,m,m,z,z,m,m,m to generate the sequences) 5 c o r t e x x x x ( x x x x ) x x x CORTEX3961_proof ■ 12 July 2024 ■ 9/21 Please cite this article as: Rasmussen, T., et al., On the role of prefrontal and parietal cortices in mind wandering and dynamic thought, Cortex, https://doi.org/10.1016/j.cortex.2024.06.017 that they used to report the most representative model of the relationship between each thought probe and the predictor variables.It is important to note the R 2 values for each winning model are reported throughout, in line with our Stage 1 analysis plan; however, as the thought probe responses are treated as an ordinal response variable, it is difficult to meaningfully interpret these values.
Boayue et al. (2021)Although the probit analysis was included in the accepted Stage 1 plan, at the time of the Stage 2 submission, it was realised by both a new editor and the authors that the threshold being employed for making inferences from the probit modelling, while being based on the approach ofBoayue et al. (2021), and being accepted at Stage 1, was not explicit and did not conform to the threshold required at Cortex. T the findings have been reported as an exploratory analysis subsequent to the main planned hypothesis testing analyses. spec methods used to calculate the model weights, however these calculations have not been altered from the methods used byBoayue et al. (2021), or those specified in the Stage 1 manuscript.Please cite this article as: Rasmussen, T., et al., On the role of prefrontal and parietal cortices in mind wandering and dynamic thought, Cortex, https://doi.org/10.1016/j.cortex.2024.06.017 value will be 55, F(1, 148) ¼ 8.145, p ¼ .005,h p 2 ¼ .052].The t-tests also found anecdotal evidence for a difference between the active and sham conditions in the DLPFC [BF 10 ¼ 1.93 (M Active ¼ 3.51, M Sham ¼ 4.08), t(74) ¼ À1.84, p ¼ .035,d ¼ À.422)] and in the occipital cortex [BF 10 ¼ 1.89 (M Active ¼ 4.11, M Sham ¼ 3.46, t(74) ¼ 2.21, p ¼ .030,d ¼ .507)].In the IPL, there was moderate evidence against a difference between the active and sham groups [BF 01 ¼ 3.1 (M Active ¼ 3.99, M Sham ¼ 4.10, t(74) ¼ À.38, p ¼ .705,d ¼ À.087)], and no evidence for an effect in this region in the Bayesian between subject ANOVAs (BF excl > .88

Table 2 e
Descriptive statistics, sample size, and independent samples t-tests for all baseline tests.