Introduction

Procrastination is increasingly becoming one of the most prevalent behavior problems, which reflects the irrational voluntary postponement of scheduled tasks albeit being worse off for such delays (Blake, 2019; Steel, 2007). In the cross-cultural investigations, more than 15% adults were identified as having chronic procrastination problems, and the situation for students was worse as 70-80% undergraduates engaged in procrastination (American College Health Association, 2022; Ferrari et al., 2005). Moreover, the behavioral genetic evidence indicated a certain heritability of procrastination in human beings as well (Gustavson et al., 2017; Gustavson et al., 2014, 2015). In addition to its prevalence, the deleterious consequences of procrastination behavior should warrant more caution. There was cumulative evidence to show that procrastination behavior endangers working performance, financial status, interpersonal relationships, and subjective well-being (Ferrari, 1994; Pychyl & Sirois, 2016; Steel et al., 2021). Further, many mental health problems have been found to be consequences of procrastination, such as sleep problems, depression, and anxiety (Hairston & Shpitalni, 2016; Johansson et al., 2023). Even worse, chronic procrastination behavior has been consistently observed in downgrading one’s life health as shown by the close association between procrastination and immune system disruption, gastrointestinal disturbance, as well as a high risk of hypertension and cardiovascular disease (Sirois, 2015; Sirois, 2016). Thus, given these critical consequences, many efforts have been spent to delve into why we procrastinate irrationally.

To probe into why we procrastinate irrationally, researchers have built upon theoretical bases of procrastination from different perspectives. For instance, Steel (2007) pioneered a promising temporal motivation theory (TMT) to explicate procrastination as the failure of self-regulation (Steel, 2007). This theory suggests that individuals would procrastinate the task as task utility is devalued in the far future (Steel & König, 2006). Based on insights into emotional regulation, the mood repair perspective provides another explanation to elucidate that procrastination is due to the failure of self-regulation to give priority to short-term mood repair caused by doing the task than long-term task reward (Sirois & Pychyl, 2013). Recently, the temporal decision model (TDM) of procrastination further provides an integrative framework to explain the procrastination decision, which highlights that procrastination would be contingent on the trade-off between task aversiveness and task-outcome value (Zhang & Feng, 2020; Zhang, Liu, et al., 2019). Task aversiveness reflects how unpleasant individuals perceive the tasks to be, with more unpleasant feelings making procrastination more likely (Zhang & Feng, 2020). Task outcome value indicates how much it is worth as we evaluate the benefits it provides us (e.g., keeping body health) once we complete the task before the deadline (e.g., doing scheduled exercise) (Zhang & Feng, 2020). If the task aversiveness is overvalued in this trade-off, the decision to postpone tasks would be made consistently.

It is worthwhile to note that this trade-off and resultant decisions to procrastinate were predominantly influenced by self-control capability, with higher self-control for less procrastination behavior (Blake, 2019; Ramzi & Saed, 2019; Zhao et al., 2021). Thus far, there have been three promising pathways attempting to clarify how self-control works for reducing procrastination: one for decreasing task aversiveness, the other one for increasing task-outcome value, and the last one for both (Zhang & Feng, 2020). As identified in both behavioral and neural evidence, procrastinators consistently reported high task aversiveness when receiving scheduled tasks, and are more likely to postpone tasks so as to devalue negative task aversiveness (Blunt & Pychyl, 2000; Zhang et al., 2021). Meanwhile, the self-control was observed as powerful for downstream regulation of negative emotional stimuli (Paschke et al., 2016; Tice & Bratslavsky, 2000). Therefore, the first possible pathway is to enhance self-control by facilitating emotional regulation towards task aversiveness (Eckert et al., 2016). On the other hand, as a value-based decision, procrastination behavior is contingent on the evaluation of future outcomes (Rebetez et al., 2016; Zhang, Becker, et al., 2019). Existing evidence has shown that procrastinators generally underestimate the task-outcome value (H. Wu et al., 2016; Zhang et al., 2021). In this vein, it is hard to generate the motivation to take immediate action (Taura et al., 2015). Notably, increasing the value of future rewards has been found effective in making individuals inclined to pursue future outcomes by strengthening self-control (Cho et al., 2015; Kelley et al., 2018). Thus, another pathway worth putting forward is that procrastination behavior could be shaped by increasing future task-outcome value. Also, given the multifarious roles of self-control, it leads us to hypothesize the third pathway that both decreased task aversiveness and increased task-outcome value contribute to determining procrastination simultaneously through the exertion of self-control.

Supporting that, the left dorsolateral prefrontal cortex (DLPFC), responsible for self-control, has been frequently shown to be closely associated with procrastination. As the biomarker of self-control, the neuroanatomical changes of the left DLPFC were revealed to predict procrastination robustly (Chen et al., 2020; Hu et al., 2018; Liu & Feng, 2017). In addition to structural brain hallmarks, the neurofunctional anomalies underlying procrastination were found in DLPFC-involved circuits (Y. Wu et al., 2016; Xu et al., 2021). Furthermore, the triple brain network theory provided network-based insights to highlight the neural signature of procrastination as the self-control neural network (e.g., left DLPFC; anterior cingulate cortex, ACC), emotional regulation network (e.g., insula and orbitofrontal cortex, OFC) and episodic prospection network (e.g., hippocampus and ventromedial prefrontal cortex, vmPFC, amygdala) (Chen & Feng, 2022; Chen et al., 2020; Schlüter et al., 2018; Wypych et al., 2019). Despite solid evidence indicating significant associations between self-control and the right DLPFC (Huang et al., 2025; Knoch & Fehr, 2007), Xu and colleagues demonstrated null effects of neuromodulating the right DPFC had no effects on either value evaluation or emotional regulation in reporting procrastination willingness. Moreover, a substantial amount of neural evidence supported this conclusion that DLPFC is involved in long-term reward evaluation and value encoding via top-bottom self-control circuits (Frost & McNaughton, 2017; Jimura et al., 2013; Smith et al., 2018). Using a neurocomputational model, Le Bouc & Pessiglione (2022) provided clear evidence indicating that dorsal PFC signaling expected effort values was significantly attenuated in procrastinators compared to healthy controls (Le Bouc & Pessiglione, 2022). In this vein, this evidence supports this conceptualization that the left DLPFC may be a domain-specific neural signature determining one’s procrastination. In light of technical advances, high-resolution transcranial direct current stimulation (HD-tDCS) has been widely used to reveal the causal neurocognitive mechanism of problematic behaviors by modulating cortical excitability, blood-brain barrier permeability and even neuroplasticity (Cirillo et al., 2017; Woods et al., 2016), which is regulated by the NMDA (N-methyl-D-aspartate) system to either bolster LTP (Long-term potentiation) or LTD (Long-term depression) processes (Chrysikou et al., 2022; Shin et al., 2020). For instance, anodic HD-tDCS applied to the left DLPFC was found effective in inhibiting problematic behaviors caused by the lack of self-control (Allenby et al., 2018), showing significantly amplified local neural oscillations (Chrysikou et al., 2022). Beyond regional neuromodulation in a dose-free protocol, cumulative evidence has well-documented that effects of tDCS for neuromodulation are highly dose-dependent and are involved in network-wise covariance (Sabé et al., 2024; Soleimani et al., 2023; Woodham et al., 2025). Thus, this study aims to provide causal evidence clarifying the brain-behavior association of procrastination and revealing how self-control works to shape one’s procrastination by manipulating left DLPFC activity in a multiple-session (dose) protocol.

Current Study

To clarify the causal cognitive mechanism of self-control on procrastination, we conducted a double-blind, randomized, multiple-session, placebo-controlled design, with a 2 (active HD-tDCS vs. sham control) × 2 (before first neural stimulation vs. after last neural stimulation) full factorial design (see Fig. 1). This HD-tDCS protocol consisted of 7 sessions spaced over 15 days, and each session was implemented every two days. To ensure sound ecological validity, each procrastinator was informed to report one REAL-LIFE task that he/she should complete the following day (e.g., Day 2) after the current neuromodulation session (e.g., Day 1), and was asked to report the ACTUAL performance for completing this task in that day (Day 2) (see Fig. 1). Based on the temporal decision model of procrastination (Zhang, Liu, et al., 2019), we drew on the experience sampling method (ESM) to estimate the real dynamics of task aversiveness and task-outcome value by using a parameter-free model in each session. More importantly, to clarify which pathway best explains the neurocognitive mechanism of procrastination, we built upon the mixed-effects general linear model and Quasi-Bayesian causal mediation model to test whether the changes in task aversiveness and task outcome value caused by HD-tDCS could predict decreased procrastination. Finally, the follow-up investigation for actual procrastination was conducted after the experiment for 6 months to attempt to test the long-term retention of this neural stimulation effect.

Figure 1

Materials and methods

To promote the reproducibility and transparency, all the data processes and statistics were recorded and made available for tracking by using the “ProjectTemplate” package (see Supplemental Materials, SM). This study fully adhered to CONSORT reporting guidelines, and was preregistered in the OSF repository a priori (DOI 10.17605/OSF.IO/Y3EDT).

Participants

Due to the lack of diagnostic criteria for clinical procrastinators, we recruited a large-scale sample (n = 1,682) to obtain a stable benchmark distribution. Thus, the procrastinators were captured once their procrastination scores were higher than 66, using General Procrastination Scale (GPS) (see Fig. 2A-B). Following this criterion, a total of 186 participants were included initially, which was in accordance with empirical evidence (i.e., 10 – 15% prevalence of procrastination) (Harriott & Ferrari, 1996). Subsequently, the semi-structured interview was performed to screen those suffering from problematic procrastination and volunteering for this study, thereby enrolling 53 participants (see SM Methods). Seven participants were eventually excluded for analyses because they voluntarily dropped out before experimental completion. All the included participants were screened for depression and anxiety symptoms (see Tab. 1).

Figure 2

Table 1

A full randomized block design was used to assign both groups (active neuromodulation group, NM; sham-control group, SC) (see Fig. 2C). Statistical power was predetermined by G* Power and Power contour estimation at medium effect size (1-β err prob = 0.80, f = 0.25), which indicated that total sample size at 18 can reach acceptable power (see SM Methods and Fig. S1). All the participants reported no history for HD-tDCS or neuromodulation. No significant differences were found between groups for any demographic characteristics (see Tab. 1). This study and protocol have been fully approved by the Institutional Review Board (IRB) of the School of Psychology, Southwest University (China, IRB200301108).

Measurement

The General Procrastination Scale (GPS) developed by Lay (1986) was used here to quantify one’s chronic procrastination symptom here (Lay, 1986). This scale was widely adopted in many cross-cultural contexts, and has been reported to have good psychometric properties (Klein et al., 2019). There were two additional items that we added for lie detection, including description of “the sky is red” and “I have never taken shower”. If either one was selected as “agree” by participant, his/her response would be discarded. Internal reliability of the GPS in the current study was found acceptable (Cronbach’s a = 0.890). No significant difference was found between groups for GPS scores (t = −1.08, 95% CI: −4.296 1.283, p = .283; Jeffreys-Zellner-Siow Bayesian factor, BF, BF₁₀ = 0.455, error = 0.020 %).

To quantify one’s daily emotions, the Positive and Negative Affect Schedule (PANAS) developed by Watson et al (1988) was adopted here. This scale consists of two subscales (i.e., positive affect, PA and negative affect, NA), each containing 10 terms, where higher scores indicate stronger affective states (Terracciano et al., 2003). To control for potential confounding effects of daily emotions, both subscale scores were included as covariates of no interest in the statistical model. Additionally, this tool demonstrated good internal consistency for both sub-scales (α = 0.801 for PA, α = 0.793 for NA).

Experimental design and procedure

Nested cross-sectional longitudinal design

This study used a nested cross-sectional longitudinal design to investigate whether the multiple-session anodal HD-tDCS targeting the left DLPFC could reduce actual procrastination behavior and to probe how this effect manifests (see Fig. 1). To achieve the first goal, in the first-level longitudinal model, we delivered a 2 (active HD-tDCS neuromodulation group vs. sham-control group) × 2 (before first HD-tDCS neuromodulation vs. after last HD-tDCS neuromodulation) full factorial design. In detail, the 20-min anodal HD-tDCS neuromodulation targeting the left DLPFC was performed for HD-tDCS group at intervals of 2 days, while the sham-control group received sham HD-tDCS training. This HD-tDCS training was repeated for a total of seven sessions, and lasted 15 days (see Fig. 1). To examine the long-term effect of tDCS on reducing procrastination, all the participants were required to reported task-execution willingness (TEW) (Zhang & Feng, 2020; Zhang, Liu, et al., 2019), for a real-life task 24 hours post-neuromodulation. Thus, procrastination willingness was quantified as 100-TEW score (see underneath for details). Furthermore, we asked participants to report the actual completion rate (CR) of the task at the deadline (e.g. participant A finished 90% homework at deadline and reported this situation to us at deadline). In this vein, the actual procrastination rate (PR) was quantified as 1-CR.

As for the second aim, in the second-level cross-sectional model, we developed a real-time interactive web-platform sampling system to implement experience sampling method (ESM) for tracking one’s real-time evaluation of task aversiveness and task outcome value at five moments within one day needing to do task (see Fig. 1, e.g., Day 1,3, 5). On the basis of the temporal decision model (TDM) of procrastination, the emotional changes in task should be dynamic along with nearing the deadline, and might be hyperbolic discounting manner (Zhang & Feng, 2020). In this vein, the five recording moments of ESM model were determined by using a hyperbolic function toward the deadline, such as moments of 10:00, 16:00, 18:00, 19:30, 20:00 (deadline). Thus, recording moments of tasks were task-specific and participant-specific. To obviate the confounds of daily emotions in task aversiveness evaluation, we used the averaged scores of PANAS at 10:00 (noon) and 16:00 (afternoon) as anchoring points to quantify one’s daily emotions by using this ESM system. Before each session of HD-tDCS training, each participant were required to report a real-life task whose deadline is tomorrow. To obtain the long-term effect of HD-tDCS (i.e., the interval between HD-tDCS and task completion is at least 24 hours), the task deadline that participants reported was required to be between 18:00 – 24:00.

Quantification of covariates of interests

Outcome variables of this study are twofold: one is task-execution willingness and another is procrastination rate (PR). Task-execution willingness is used to evaluate one’s subjective inclination to avoid procrastination (Zhang & Feng, 2020). In this vein, we used a 100-point scale to require participants to report their task-execution willingness (0 for “I will definitely procrastinate this task” and 100 for “I will take action to complete this task immediately”). This metric was recorded 24 hours after neuromodulation to examine its long-term effects. PR is used to quantify the extent to which one task has been procrastinated, and was calculated as 1 - CR (task completion rate). If the task was actually completed before the deadline, the CR would be 100% and the PR would be calculated as 0% (1-CR). PR was recorded at the actual task deadline for each participant. We are also interested in re-investigating their actual procrastination by using PR 6 months after the last neuromodulation to test the long-term retention of this neuromodulation effect.

From what has been mentioned above, task aversiveness and outcome value were considered key factors to explain the effect of neuromodulation on reducing procrastination in the current study. The task aversiveness describes how disagreeable one perceives when performing this real-life task (Zhang & Feng, 2020). To quantify one’s task aversiveness, participants are required to rate their feelings towards the task by using a 100-point scale (i.e., How do you feel in the current moment when you need to complete this task before deadline, with 0 for “extremely unpleasant”, 50 for “totally neutral” and 100 for “extremely pleasant”). Outcome value serves as the subjective benefits for the task outcome brought about by completing the task before the deadline (Zhang & Feng, 2020). Likewise, participants are also required to rate their outcome value by using a 100-point scale (i.e., How much do you desire to obtain incentive outcome of this task, with 0 for “extremely weak”, 50 for “medium” and 100 for “extremely strong”).

HD-tDCS protocol

The HD-tDCS suit (stimulater and 4 × 1 multichannel stimulation adapter, MSA) that this study used was produced by the Soterix Medical Inc., and has been widely verified safe, effective and reliable for public (Villamar et al., 2013). Based on advanced properties of 4 × 1 MSA, the targeted areas for current flow can be constrained within 2.5 cm² (Villamar et al., 2013).

To position electrodes into targeted areas (left DLPFC), the 10-20 international system for EEG was initially used to mark potential nodes, and determined Cz as reference point. There is compelling evidence to claim that the left F3 could be used as target node for modulating the left DLPFC (Seibt et al., 2015; Tsukuda et al., 2025). In this vein, the central anodal electrode was determined onto F3, and four return electrodes surrounded central electrode at outside of 7.5 cm, including F5, AF3, FC3 and F1 (see Fig. 2F). Ramp-up and ramp-down duration have been set to 30 seconds. To further locate the targeted areas, the high-definition neuronavigation system (ANT Neuro Inc., Welbergweg, Germany) was performed. Results indicated the accurate position that we pre-determined by showing a high overlap probability over the left DLPFC (MNI Coordinate: −51 40 18, 94.33 % overlapping probability) (see SM Method) (see Fig. 2F). In addition, the coordinates of this targeted area were retrieved from the Brede Database (http://neuro.imm.dtu.dk/services/), and showed highly pertinent functions related to DLPFC.

Before stimulation, participants were informed to clean the scalp to reduce resistance. Then, cotton swabs were used to separate hair until the scalp surface become visible. Subsequently, the electrically conductive gel (about 1.5 ml) was introduced into the plastic casing facilitating constraint of current flow. Next, the Ag/AgCI sintered ring electrodes were placed onto plastic casing and covered with a cap to lock them in the right positions. To reduce discomfort, the electrode cables were taped elsewhere. Further, the above processes would be re-adjusted if any electrode resistances were found larger than 1.5 units (Villamar et al., 2013). Once all the processes had been completed, the stimulator would be launched.

Participants in the HD-tDCS training group underwent constant electric current of 2.0 mA targeting the left DLPFC for 20 minutes. Results from the simulation of electric density showed a peak current of ~0.5mA/cm ² at the central electrode and of ~0.125 mA/cm² at the four return electrodes, thereby indicating the safety and effectiveness. As for the sham-controlled group, the stimulator would deliver current flow with 2.0 mA during the first and last 30 seconds to elicit sense of electric stimulation for blinding of them. To obtain the pure offline effect, these measures for task-execution willingness, task aversiveness, and outcome value were conducted after stimulation at least 12 hours (Bikson et al., 2016).

Statistics

All the statistics were implemented by R (https://www.rstudio.com/) and JASP software (https://jasp-stats.org/).

To clarify whether multiple-session HD-tDCS training can reduce actual procrastination, the general mixed-effect linear model (GLMM) was constructed using 2 (active vs. sham) × 2 (before first neuromodulation vs. after last neuromodulation) full factorial design for procrastination willingness and actual procrastination rates. Here, sex, age and socioeconomic status (SES) were modeled as covariates of no interest. As the National Bureau of Statistics (China) issued (https://www.stats.gov.cn/sj/tjbz/gjtjbz/), on the basis of per capita annual household income, the SES was divided into seven hierarchical tiers from 1 (poor) to 7 (rich). This analysis was conducted using the “lme4” package. Further, we capitalized on Markov Chain Monte Carlo Generalised linear mixed effects model (MCMCglmm) to overcome pitfalls of null-hypothesis significant test (NHST) that the above analysis might bring out (Hadfield 2009). Further, the post hoc analysis of the above statistics adopted Jeffreys-Zellner-Siow Bayesian examinations (see SM Methods) (Wagenmakers et al., 2018). Instead of continuous outcomes, we also conducted a between-group comparison for the number of tasks procrastination by using the x ² test with φ correction or Fisher exact test. Likewise, the Bayesian estimations were carried out as well. Regarding the 6-month follow-up investigation, this GLMM was also built to examine the long-term retention of neuromodulation on reducing procrastination. Also, these statistical models were validated using Bayesian factor analysis to provide more robust statistical evidence. In addition, we also capitalized on the stochastic gradient descent Cox model by using “BigSurvSGD” package to estimate the effect of neuromodulation on procrastination probability. To minimize the risk of a false-positive ratio, these statistical results have been validated by adjusting for the baseline effect from regressing pre-neuromodulation procrastination and other covariates to post-neuromodulation procrastination (see SI Results).

To reveal the potential mechanism of this effect, all statistics mentioned above were redone for the evaluation of task aversiveness and outcome values. On the basis of the temporal decision model of procrastination proposed by Zhang et al (2020), the task aversiveness evoked by executing a task was quite dynamic, and would increase dramatically once reaching the deadline, with the peak task aversiveness for the nearing deadline (and vice versa) (Zhang & Feng, 2020). In this vein, to quantify the task aversiveness with consideration for its dynamics, the model-free area under the curve (AUC) was estimated. We assumed that the task aversiveness scores that derived from continuous time points (being far away from the deadline to the nearing deadline) formed a closed interval, and thus allowed for the estimation for AUC by using a definite integral:

Here, by using ESM, task aversiveness was recruited at progressive time points close to the deadline, and thus formed an available function interval. Thus, the AUC represented the extent to which task aversiveness has changed in approaching the deadline, with a high AUC for low-level task aversiveness. As for the task outcome value, it was theoretically posited as a relatively stable evaluation of the task. In this vein, it was quantified by the self-reported 100-point scale after neuromodulation at least 12 hours later, to ensure no online effect.

Further, taking into account the longitudinal repeated measured data, the joint model of longitudinal and survival data (JM-LAD), in conjunction with a machine learning algorithm, was adopted, which has been reported to outperform mixed-effects models. As referred to in the existing study, we thus assumed that the procrastination probability (Pr(P)) could be determined by the joint statistic (m_i(t)) as follows:

where γ^T represented vector of regression coefficient, and a reported the fitting coefficient by longitudinal model. To determine the longitudinal component of this joint model, the vast majority of prevailing machine learning regression algorithms were initially placed into model, including decision tree regression, Bogging regression, random forest regression and support vector regression. The final model would be determined according to minimum normal mean square error (NMSE).

To ascertain the neurocognitive mechanism of tDCS in reducing procrastination, the Quasi-Bayesian causal mediation analysis was used to model the association between the effects of tDCS, task aversiveness/outcome and decreased procrastination. To build upon this model, the tDCS treatments were inputted as independent variables, and the task aversiveness/outcomes were modeled as causal mediating variables by using the “Mediation” package (https://cran.r-project.org/web/packages/mediation/) (Imai et al., 2010). We estimated these pathway effects (i.e., averaged causal mediation effects, δ; averaged direct effects, ζ; total effects, ρ) by using Markov Chain Monte Carlo (MCMC) sampling. To improve the statistical reliability, the sequential ignorability assumption was tested by using sensitivity analysis. Details for the statistical principals and basis could be found elsewhere (Imai et al., 2010).

Results

Blinding

In both groups, almost all participants reported perceiving acceptable pain stemming from current stimulation and believed they were receiving treatment (91.30% (21/23) for active neuromodulation group (NM), 86.95% (20/23) for sham control group (SC), x² = 0.224, p = .636). In addition, statistical models excluded Session 1 and Session 4 because participants reported additional side effects in both groups (see SI Result and Tab. S1).

Multiple-sessions HD-tDCS (ms-tDCS) can alleviate procrastination

To identify whether ms-tDCS targeting the left DLPFC can alleviate procrastination willingness and actual procrastination, a linear mixed-effects model with Scatterthwaite algorithm was built, with task-execution willingness and actual procrastination rates (PR) as outcomes, respectively. Procrastination willingness changes for each session have been described in the Fig. 3A. Results showed a significant two-way interaction (NM v.s. SC × before neuromodulation vs. after neuromodulation) for procrastination willingness (Task-execution willingness, F _{(1, 269)} = 43.86, p = 1.94 × 10 ^-8, = .21), and further demonstrated a significantly increased task-execution willingness (i.e., decreased procrastination willingness) after neuromodulation in the active neuromodulation group, but no such effects in the sham control group (Task-execution willingness, NM-before: 35.65 ± 30.20, NM-after: 80.43 ± 19.92, F _(1,44) = 76.46, p < .001; SC-before: 37.57 ± 26.46, SC-after: 47.35 ± 30.49, F _(1,44) = 1.35, p = .251) (Fig. 3B-C). A linear uptrend fir task-execution willingness was observed across multiple sessions (Fig. 3D; p < .01, Mann-Kendall test). Meanwhile, changes to procrastination rates across all the sessions have been detailed in the Fig. 3E. A statistically significant interaction effect was identified (F _{(1, 269)} = 35.54, p = 1.90 × 10 ^-10, = .17), and the post-hoc analysis showed decreased actual procrastination rates after ms-tDCS in the active neuromodulation group, but no such changes found in the sham-control group (actual PR, NM-before: 43.26 ± 39.09, NM-after: 0.00 ± 0.00, F _(1,44) = 58.06, p < .001; SC-before: 46.47 ± 40.75, NM-after: 33.34 ± 37.82, F _(1,44) = 1.28, p = .26) (Fig. 3F-G). Similarly, a significant downtrend for procrastination rates across all the sessions was identified (Fig. 3H; p < .01, Mann-Kendall test).

Figure 3

To obtain more solid statistical evidence, we further utilized a Bayesian mixed-effects model by using Markov Chain Monte Carlo (MCMC) sampling and revealed the same findings (see SI Results and Fig. S2–5). Furthermore, the x² test of R × C contingency table was conducted for the frequency of procrastination, as this outcome is an ordinal variable. A significant reduction in group-averaged procrastination frequency was found in the active neuromodulation group after HD-tDCS, but not in the sham-control group (NM-before: 69.56% (16/23 participants), NM-after: 0.00% (0/23 participants); SC-before: 69.56 % (16/23 participants), SC-after: 56.52% (13/23 participants), x² = 10.08, p < .001). In addition, using a stochastic gradient descent algorithm in the Cox model, procrastination probability (risks) for procrastinators in the neuromodulation group was found to be reduced by 5.12 times compared to the sham-control group after ms-tDCS neuromodulation (OR (Odds Rates) = 5.12, 95% CI: 1.97 – 13.26, p < .001). In brief, these findings provided evidence to support the claim that ms-tDCS neuromodulation targeting the left DLPFC can be an effective way to reduce both procrastination willingness and actual procrastination through its long-term after-effects.

Ms-tDCS changes task aversiveness and task-outcome value

Both task aversiveness and task outcome value serve as key pathways determining whether one would procrastinate. To this end, we further utilized a 2 (NM vs. SC) × 2 (before first neuromodulation vs. after last neuromodulation) mixed linear model to examine the effects of ms-tDCS on changes in task aversiveness and task outcome value. Task aversiveness changes across all the sessions have been shown in the Fig. 4A and 4C. Statistical analysis demonstrated significant decreases in task aversiveness and increases in task outcome value via ms-tDCS in the neuromodulation group but not in the sham-control group (Task aversiveness, F _{(1, 269)} = 45.92, p = 7.73 × 10^-11 = .13, NM-before _(AUC): 1.13 ± 0.53, NM-after _(AUC): 1.95 ± 0.85, F _(1,44) = 11.44, p < .005; outcome value, F _{(1, 269)} = 45.61, p = 8.87 × 10 ^-11 = .19, NM-before _(AUC): 35.86 ± 27.82, NM-after _(AUC): 73.08 ± 23.33, F_(1,44) = 18.07, p < .001) (see Fig. 4B and 4D; SI Results and Fig. S6–9). In the neuromodulation (NM) group, task aversiveness steadily decreased with the cumulative number of stimulation sessions, while perceived task outcome value increased significantly (see Fig. 4E-F, p < .05, , Mann-Kendall test). Thus, it provides causal evidence clarifying that neuromodulation to self-control capacity reduces task aversiveness and enhances task-outcome value simultaneously.

Figure 4

Increased task outcome value but not decreased task aversiveness predicts reduced procrastination

Given the dual neurocognitive pathways identified above—reduced task aversiveness and increased task-outcome value—we propose that these changes, driven by enhanced self-control, account for how neuromodulation reduces real-world procrastination behavior. We utilized a generalized linear model to relate decreased task aversiveness and increased task outcome value to changes in task-execution willingness. In this model, increased task outcome value (Δ _{Outcome value}) significantly predicted increased task-execution willingness (Δ _{task-execution willingness}) (x² = 15.95, p < .01, R² = .40; Δ _{Outcome value}: β = 0.61, S.E. = 0.12, p < .001, 95% CI: 0.373 – 0.860), whereas no significant effect was observed for predicting task-execution willingness through decreased task aversiveness (Δ _{Task aversiveness}: β = 0.10, S.E. = 0.12, p = .41,95% CI: −0.14 – 0.34).

Likewise, increased outcome value (Δ _{Outcome value}) significantly predicted one’s real-world procrastination behavior, whereas decreased task aversiveness showed no significant predictive effect (see Tab. 2). The Bayesian statistical evidence was further obtained to substantiate the above findings (Null model: BF(M|data) = 7.44 × 10 ^-4, BF₁₀ = 1.00; Model of including task outcome value: BF(M|data) = 0.77, BF₁₀ = 1.03 × 10 ⁴; Model of including task aversiveness: BF(M|data) = 9.42 × 10 ^-5, BF₁₀ = 1.26). In addition, by adopting the joint model of longitudinal and survival data in a machine learning scheme and linear probability model, we also revealed the predictive role of increased task outcome value to less procrastination when the longitudinal processes (sessions) were taken into account (see SI Results and Tab. S3–6). On balance, these findings provide evidence that reveals that the left DLPFC could strengthen one’s self-control to increase task outcome value so as to decrease procrastination.

Table 2

Increased task outcome value plays causal role to explain why ms-tDCS reduces procrastination

To clarify the causal neurocognitive mechanism of procrastination, the Quasi-Bayesian causal mediation analysis was undertaken by using White’s heteroskedasticity-consistent estimator, with increased task outcome value as a mediated variable. Results showed the significant causal mediated role of increased task outcome value in increasing the task-execution willingness (δ = 21.73, p < .01; ζ = 11.25, p = .07, ρ = 32.99, p < .01, simulation = 1,000; see Fig. 5A) and real-world procrastination (δ = 30.75, p < .01; ζ = 3.05, p = .52, ρ = 33.81, p < .01, simulation = 1,000; see Fig. 5B), as caused by ms-tDCS neuromodulation. To ensure the robustness and specificity of these findings, the sensitivity analysis was implemented by changing sampling parameters and outcome variables. Results showed that the above findings were replicated when the bootstrap sampling algorithm was used to change parameters, and thus confirm the robustness of this model (see SI Results and Tab. S6–7). Likewise, the results of the control analysis further validated the specificity of these findings by showing a null causal mediated effect of this model to predict one’s task aversiveness (see SI Results and Tab. S8). In summary, these findings revealed neurocognitive pathway underlying procrastination: the self-control governed by left DLPFC uniquely increases task-outcome value to reduce one’s procrastination willingness and real-world procrastination behavior.

Figure 5

Long-term effects of ms-tDCS

We have also attempted to conduct a follow-up investigation to test the long-term retention of ms-tDCS in reducing actual procrastination. Almost all the participants had undergone follow-up except one in the neuromodulation group after last neuromodulation for 6 months (N_NM = 22, N_SC = 23). Thus, the GLMM was constructed, with the PR before first neuromodulation vs. PR after last neuromodulation for 6 months as covariates of interest. Results showed the statistically significant group*time interaction effects (β = 16.5, SE = 9.9, p = .049). Simple-effect model demonstrated a decrease in actual procrastination rates in the active neuromodulation group after last stimulation for 6 months compared to baseline (β = −22.05, SE = 10.0, p = .038, Tukey correction; NM-before: 40.68 ± 37.96, NM-after_6-months: 18.63 ± 29.80), and revealed null effects in the SC group (β = 1.26, SE = 9.78, p = .99, Tukey correction; SC-before: 46.47 ± 40.75, SC-after_6-months: 47.73 ± 39.18) (see Fig. 6).

Figure 6

Likewise, the parallel Jeffreys-Zellner-Siow Bayesian factor model was constructed to provide solid statistical evidence. Results demonstrated the robustness of these findings by showing the acceptable Bayesian evidence strength to support the above statistical inferences (NM group: BF₁₀ = 2.88, error = 0.76 %, P_(inc|data) = 0.74; SC group: BF₁₀ = 0.29, error = 0.98 %, P_(inc|data) = 0.22). Furthermore, the difference of group-averaged procrastination frequency was found significant statistically between pre-neuromodulation and after-neuromodualtion for 6 months by using the x² test in the NM group (x² = 3.30, p = .035, NM-before: 68.19% (15/22), NM-after_6-months: 40.91% (9/22)), while observed no significant changes in the SC group (x² = 0.11, p = .74, SC-before: 69.56% (16/23), SC-after_6-months: 73.91% (17/23)). Taken together, this study revealed the long-term retention that was produced by ms-tDCS neuromodulation to reduce procrastination.

Discussion

In the current study, by performing anodal ms-tDCS neuromodulation on the left DLPFC, both procrastination willingness and actual procrastination behavior were significantly decreased in real life. Additionally, a 6-month follow-up investigation revealed the long-term retention of such effects. Furthermore, this neuromodulation was found to decrease task aversiveness and increase outcome values; notably, only increased task-outcome value could predict the decreased procrastination. On balance, our findings clarified the neurocognitive mechanism of procrastination by showing that self-control could increase task-outcome value so as to reduce procrastination. In addition, this study provided an effective way to reduce actual procrastination by using ms-tDCS neuromodulation.

One major contribution this study has made is to disentangle the neurocognitive mechanism of procrastination by demonstrating that self-control could increase task-outcome value so as to reduce procrastination. Neurobiological substrates of procrastination have been investigated in recent years, and demonstrated the crucial roles of the left DLPFC in predicting procrastination (Chen & Feng, 2022; Chen et al., 2021; Chen et al., 2022; Hu et al., 2018; Liu & Feng, 2017; Zhang et al., 2017; Zhang et al., 2016). Not only the brain functional anomalies of the left DLPFC but also the neuroanatomical disruptions of self-control brain network that was constituted by the left DLPFC were found to link with more procrastination behaviors (Chen & Feng, 2022; Zhang et al., 2016). Notwithstanding that, it still remains unclear to claim their causal brain-behavior relationship - that is - no known evidence existed to clarity whether changes of the left DLPFC lead to procrastination or vice versa. The current study demonstrated the causal role of the left DLPFC in procrastination by showing that the neuromodulation of the left DLPFC indeed manipulated procrastination, and thus provided straightforward and powerful evidence to fill this gap.

It has long been acknowledged that the left DLPFC is associated with many aspects of self-control communities, such as patience to wait long-term gratification for a delay, inhibition of impulsiveness, and control of game addiction (Cohen & Lieberman, 2010; Lin & Feng, 2024). Furthermore, the increased activation of the left DLPFC has been observed during the exertion of self-control which modulates value signals (Hare et al., 2009; Harris et al., 2013). Meanwhile, procrastination has been argued to be the consequence of self-control failures and self-regulation failure for a long time (Ariely & Wertenbroch, 2002; Rebetez et al., 2018; Rozental & Carlbring, 2014). Supporting this, both the brain morphological disruptions in the DLPFC and anomalies in the functional coupling of the DLPFC-based self-control network were interpreted as the phenotype of reduced self-control, making individuals more prone to procrastinate tasks, as reported in existing literature (Xu et al., 2021; Yang et al., 2021). Moreover, it was worth noting that the increased activation of the left DLPFC was found to be involved in outcome value evaluation through self-control regulation (Chen et al., 2018; Zha et al., 2019). There was more straightforward evidence to substantiate the role of manipulating the DLPFC in changing one’s subjective value evaluation (Huang et al., 2017; Minati et al., 2012). On the other hand, the theoretical explanations and empirical evidence have increasingly converged into one line for claiming that the increased task outcome value would prompt more motivation to drive one to take action immediately and thus reduce procrastination (Zhang, Liu, et al., 2019). In this vein, this study advanced our understanding of the neurocognitive mechanism of procrastination by showing that the cortical excitability of the DLPFC produced by active neuromodulation could boost self-control to increase task outcome value so as to reduce procrastination behavior.

In addition, another contribution of the current study is to provide an effective way to reduce both procrastination willingness and actual procrastination in real-life tasks. As mentioned above, despite the fact that multifarious behavioral interventions and evidence have been massively studied for overcoming procrastination, they have shared a common aim - that is - reducing the intention-action gap (Miao et al., 2024; van Hooft et al., 2005). Eerde and Klingsieck (2018) put forward an insightful standpoint established by meta-analytic evidence: procrastination is characterized as an intention-action gap rather than an intention to postpone (van Eerde & Klingsieck, 2018). This notion has been partly supported by showing that cognitive behavior therapy (CBT) for goal-directed behaviors may outperform other interventions focusing on time management (Rozental et al., 2018). Moreover, the trans-theoretical model of procrastination has shown that the behavioral intervention may be effective in changing one’s motivation to overcome procrastination but not in actual behaviors (Grunschel & Schopenhauer, 2015). Thus, the ms-tDCS neuromodulation the current study performed has a remarkable advantage in reducing procrastination - that is, the intention-action gap was attenuated so as to overcome procrastination behavior effectively. Furthermore, both 2-day-interval long-term effects and the 6-month long-term retention of the effects of ms-tDCS on reducing actual procrastination have been revealed as well. Thus far, the trends in adopting tDCS neuromodulation techniques in many aspects of behavioral therapies have emerged, but concern for a long-lasting effect of single session stimulation has continued (Brunoni et al., 2013; Brunoni et al., 2012). To tackle this concern well, instead of single-session tDCS, the current study adopted multiple-session stimulation to implement neuromodulation on the left DLPFC, which facilitates long-term effects (Au et al., 2017; Tedesco Triccas et al., 2016). Existing neurobiological theories and empirical evidence have demonstrated that multiple-session tDCS stimulation could boost cumulative effects of consolidation for activity-dependent LTP (long-term potentiation), which is crucial to neurobehavioral learning, and thus produce robust long-term after-effects (Agboada et al., 2020; Au et al., 2017). Intriguingly, the activity-dependent LTP process produced by multiple-session consolidation was found to contribute to long-term cortical plasticity, especially in the DLPFC (Jannati et al., 2023; Siebner & Rothwell, 2003). Thus, it led us to presume that the 6-month long-term retention of the ms-tDCS effect on reducing procrastination might be attributed to long-term neuroplastic changes in the dlPFC. On balance, this study provided an effective way to help procrastinators overcome actual procrastination in real-life.

Limitations

While the use of a multi-session design and the long-term assessment can be considered a strength of the present study, it also has several limitations. Even though various tDCS effects have been demonstrated so far, they also tend to be difficult to replicate and sensitive to not yet fully understood context conditions and interindividual differences, which also applies to transcranial magnetic stimulation (TMS) (Valle et al., 2009). To overcome this shortcoming, it will be necessary to establish an individually tailored tDCS protocol to improve the sensitivity of corresponding interventions (Chew et al., 2015). Thus, future research could further improve the effects of tDCS on reducing procrastination by adopting more individualized tDCS protocols. Another limitation is the lack of real-time functional neuroimaging measures to better monitor the impact of our intervention. In the absence of such measures, we had to rely on behavioral indicators to assess the success of the tDCS training. In addition to technical limitations, given the limited size of the sample and the underrepresented demographic characteristics (e.g., over 80% for females), it warrants caution in generalizing these findings elsewhere. Also, considering the lack of medical screening for psychiatric conditions (e.g., ADHD or depression) in this sample, it remains unclear whether these training effects are domain-specific for procrastination. Moreover, this study did not collect data on participants’ trait procrastination, obscuring the conclusion for whether effects of this neuromodulation were limited in those tasks or not. Finally, given the absence of cathodal HD-tDCS stimulation as a contrast condition in causal inference, it warrants caution that the increased DLPFC excitability may not be the exclusively neural mechanism for procrastination.

In conclusion, this study provides an effective way to reduce both procrastination willingness and actual procrastination behavior by using neuromodulation on the left DLPFC. Furthermore, such effects have been observed for 2-day-interval long-term after-effects, and were also found for 6-month long-term retention in part. More importantly, this study found that the ms-tDCS neuromodulation could decrease task aversiveness and increase task outcome value while, and further demonstrated that the increased task outcome value could predict decreased procrastination through enhancing self-control. In this vein, the current study moves forward our understanding of neurocognitive mechanism of procrastination by showing the prominent role of increased task outcome value in reducing procrastination. Also, it provides an effective method for intervening in human procrastination.

Data and Code Availability

We report how we determined our sample size, all data exclusions (if any), all manipulations, and all measures in the study, and the study follows JARS (Appelbaum, et al., 2018). Data, analysis code, and research materials for reproducing these findings are available at Open Science Framework (OSF) repository (10.17605/OSF.IO/Y3EDT). Data were analyzed using R, version 4.4.1 (R Core Team, 2020) and the package ggplot, version 3.5.2, as well as MATLAB (2021, MathWork Inc.) and GraphPad Prisma.

Supplementary information

SI Methods

To reach a higher reproducibility and transparency, “ProjectTemplate” package of R Studio (http://projecttemplate.net/index.html) was drawn upon for recording and tracking all of processes this study has done in a standardized document. This final record would be released once all the processes and statistics have been done. This study fully adhered to CONSORT reporting guideline for providing comprehensive reports for these findings. The CONSORT Checklist and CONSORT Diagram have been included below.

Randomised participants

To randomise participants into neuromodulation and sham-control group, the full randomised block design (FRBD) was conducted here. Furthermore, it could be beneficial from FRBD that the within-group variances can be minimized. In detail, the whole sample would be portioned into k blocks, which were manipulated for high homogeneity. For each block, the pairs of participants were then randomly designated to neuromodulation and sham-control group. Thus, adopting FRBD for randomised processes can reap huge fruits to keep within-group homogeneity for each group than do of that full randomised design (King et al., 2019). The randomised codes came from “www.random.org”, which produced random labels by using atmospheric.

Estimation for statistical power

To determine adequate statistical power beforehand, we capitalized on the G*Power software to estimate the minimum sample size obtaining the medium effect size (a err prob = 0.05, 1 -β err prob = 0.80, f = 0.25) by building a repeated measures, within-between, and interaction effect ANOVA model (Faul et al., 2007). In this vein, the medium effect size can be reached once total sample size n = 18, noncentrality λ = 15.75, critical F = 2.1945, numerator Df = 6, and denominator Df = 96. In addition, owing to the repeated measures for each participant, the 2-dimensional plot, so-called Power Contours estimation, was used to visualize and calculate the acceptable sample size by corresponding experimental design (Baker et al., 2020). Thus, according to the design this study made, the Power Contours estimation was done up front and indicated the total sample size of this study can attain the adequate statistical power (see Fig. S1). More detail for how to estimate and plot this map can be found elsewhere (Baker et al., 2020).

How to determine one’s socioeconomic status (SES)

In line with Human Connectome Project (HCP) project proposed by National Institutes of Health (NIH), this study also acquired demographic information for each participant, including gender, age, nationality, educational level (years), family’s economic status (poor, modest, middle and affluent class), family structure (FS), and status of birth (SB). In terms of criteria endorsed by the food and agriculture organization (FAO) of the United Nation (UN) and the financial statement published by National Bureau of Statistics (NBS) of China in 2019 (http://www.stats.gov.cn/tjsj/), the economic status of the family was rated as poor, modest, middle and affluent class according to family incomes, with < ¥ 45 thousands per year for poor class, ¥ 45-65 thousands per year for modest class, ¥ 65-105 thousands per year for middle class, and > ¥ 105 for affluent class.

To determine homogeneity between the HD-tDCS group and sham-controlled group, non-/parametric and Jeffreys-Zellner-Siow Bayesian examinations were adopted to identify significant differences between demographic variable.

Semi-structural interview questionnaire

To ensure the high ecological validity, we developed a semi-structural interview questionnaire to test whether participants were eligible for the current study. There were major four profiles to investigate them, including evaluation of how they perceived pain for procrastination, whether their social functions were disrupted by procrastination, whether their attitudes are to change procrastination and whether they accepted our protocol. The outlines of this questionnaire has been provided as below:

1. Please details your identity and investigation proposal; 2. Do you perceive you have procrastination symptoms, and how severe it is do you think; 3. Are you ever assumed you have “procrastination disorder”; 4. Do you feel pain due to procrastination and how it impacts your daily life; 5. What types of procrastination are do you think in yourself; 6. Do you think you would suffer from procrastination all the life; 7. Do you want to get rid of procrastination; 8. Have you try to stop procrastination; 9. Do you know neuromodulation technique, such as tDCS or TMS; 10. Do you feel panic or worrisome for medical technique, especially in electric medical technique; 11. Do you feel panic or worrisome for electric current; 12. Do you willing to receive electric medical technique for getting rid of procrastination despite limited skin pains.

High-definition neuronavigation system

To ensure the accuracy of targeting location (left DLPFC), this neuronavigation system was used to estimate the location from skull. This system reported the MNI coordinate of what we predetermined for the first, and would re-adjusted automatically twice to self-verify the estimation accuracy. Subsequently, the brain labels for these coordinates were reported by using AAL atlas and Broadman atlas, respectively. Results pronounced that the F3 node overlapped the left DLPFC with 94.33 % probability, which was cross-checked automatically.

Temporal difference-to-difference model

In the current study, in addition to the estimation for the model including all the repeated measures, we still aimed to do a random-sham pre-post tests for clarifying whether this HD-tDCS by using multiple sessions protocol was effective. In this vein, the difference coefficient (A) was computed as difference between post-test after last session and pre-test before first session, one-by-one for participants. In this vein, the pre-post within-participant differences upon outcome variables can be estimated as the HD-tDCS training effect.

Jeffreys-Zellner-Siow Bayesian examinations

Also, to make the claims for the null inferences more robust and reliable, the equivalence test (ET) and Bayesian estimations were jointly adopted. ET needs to be an estimator of the smallest effect size of interest (SESOI), describing whether the null effect falls within this interval, with overlapping for significant null effects. Thus, “TOSTER” package of R was used to test the null hypothesis by pre-determining the interval as ± 0.5 (default). Further, to confirm the null effect, Bayesian estimations were implemented in the BEST (Bayesian estimation supersedes the t-test) package of R. Building upon MCMC (Markov Chain Monte Carlo), subsequent distributions were defined as having normality with maximum variance, and determined whether the 90 % High Density Interval (HDI) overlapped in the region of practical equivalence (ROPE) for inferring the null effect. Likewise, the “BestCMCM” function was used for this analysis, adopting a default interval of ± 0.40.

There are no denying this fact that the conventional Null-hypothesis Significant Test (NHST) has ineluctable imperfections, particularly in the concerns on the misuse of p-value (Anderson et al., 2000). A growing number of studies have promoted this idea that the Bayesian model was the most attractive alternative for the statistical analysis (Wasserman, 1997, Montagnini et al., 2007). Compared to the NHST method, the Bayesian model can be profit from the apriori null hypothesis, thus balancing the fits between the apriori hypothesis and aposteriori cumulative evidence (Wasserman, 1997). In addition, it is of also attractive that the Bayesian model has the strength for the liberal precondition and sensitive detection power (Benavoli et al., 2016, Nathoo and Masson, 2016). In brief, we predefined the apriori distribution for the null hypothesis using the revised Cauchy profile (Wagenmakers et al., 2018); then, the aposterior evidence was obtained that relying on this Cauchy distribution; finally, canonical Bayes algorithm was utilized to estimate the Bayes factor (BF) for providing the power of evidence on the null-hypothesis or alternative-hypothesis:

As these functions showed, the left one is the algorithm for the apriori information and the right one for the estimation of the aposterior evidence. Thus, all the neural markers derived from these significant clusters and topological dynamics of brain network were progressively putted into this model to validate our findings. These processses were implemented in the JASP package (x64 v0.9.0.1; https://jasp-stats.org/; JASP team 2017).

SI Results

Side effects

In the Session 1, a significant unexpected side effect occurred by this fact that almost all the participants in both group do not procrastinate (Neuromodulation group, NM, Procrastination rate (PR): 4.34 % (1/23), Sham control group, SC, PR: 8.69 % (2/23)). In the follow-up investigation, all the participants reported a strong expectation for this neuromodulation as they received such treatment for the first time. In the Session 4, the PR was observed to be increased dramatically in both groups (Neuromodulation group, NM, Procrastination rate (PR): 86.95 % (20/23), Sham control group, SC, PR: 86.95 % (20/23)). In the follow-up investigation, all the participants reported that taking action immediately to complete task was hampered significantly due to weekend effect (Ryan et al., 2010, Stone et al., 2012). In this vein, data for these sessions were excluded in the analysis. Here, the follow-up investigation required participants to report whether their performance for completing task was influenced by additional effects. If in this case, they should report what additional effects occur here. The x² test was used to examine whether the additional side effect reports posed significantly differences between groups across all the sessions (see Tab. S1).

Table S1

Sensitivity analysis for interaction effects

To examine sensitivity of main findings regarding Group*Experiment interaction, we reanalyzed interaction effect by removing regressors of PANAS scores in the mixed-effect linear model. This replicated these findings by showing statistically significant interaction effect between Group (NM vs control) and training manipulation (active stimulation vs sham stimulation) on task-execution willingness changes (β = −25.8, SD = 14.7, p < .05) and procrastination rate changes (β = −25.8, SD = 14.7, p < .05).

Bayesian estimations for statistics upon the neuromodulation effects

Here, the Bayesian inference was used to reveal whether neuromodulation can alleviate one’s procrastination problem by “MCMCglmm” package. Likewise, the significant increments of task-execution willingness and actual PR were found in NM group after neuromodulation but it was found null in the SC group (task-execution willingness, post mean distribution = −21.56, 95 % CI: − 27.89 – −15.54, pMCMC < .001; actual PR, post mean distribution = 24.82 95 % CI: 16.38 – 33.23, pMCMC < .001). Also, the Jeffreys-Zellner-Siow Bayesian factors for the x² test was calculated to provide evidences for the changes of procrastination frequency before and after neuromodulation. Results demonstrated the significant effect to reduce procrastination by showing the significant decreasing procrastination frequency after neuromodulation in NM group but not SC group (BF_{10 hypergeometric} = 76.84; > 10 means a strong evidence to support alternative hypothesis).

Bayesian estimations for statistics upon the neuromodulation effects towards task aversiveness and outcome value

Likewise, the Bayesian inference was used to validate the neuromodulation effects towards the changes of task aversiveness and outcome value. Results showed the significant reduction of task aversiveness and increased outcome value in NM group after neuromodulation but null findings in the SC group (task aversiveness, post mean distribution = − 0.48, 95 % CI: −0.62 – −0.34, pMCMC < .001; outcome value, post mean distribution = − 20.89, 95 % CI : −26.98 – − 14.52, pMCMC < .001).

Results of joint model of longitudinal and survival data

taking into account for the longitudinal repeated measured data, the joint model of longitudinal and survival data (JM-LAD), in conjunction of machine learning algorithm, was adopted. Compared to standard JM, this one was found to outperform for analysis panel data by reaping huge fruits from machine learning (ML) algorithms in longitudinal sub-model (Wen, 2015). Thus, the normal mean standard error (NMSE) was used to determined which ML algorithm is best for this sub-model with lower NMSE value for better model fitting performance:

Results showed that K-Nearest Neighbor (KNN) was the best longitudinal sub-model as the minimum NMSE (see Tab. S3).

Table S3

By using KNN, the Outcome value of task was found to predict whether one would procrastinate significantly, whilst it showed null effects by using task aversiveness (see Tab. S4–5).

Table S4

Table S5

Given the longitudinal effects in multiple neuromodulation sessions, we capitalized on the joint model of longitudinal machine learning and survival data for the fits of both task aversiveness and outcome value to actual procrastination behavior across all the sessions. Outcome value of task was found to predict whether one would procrastinate significantly, whilst it showed null effects by using task aversiveness (Outcome value, β = 0.77, OR = 2.15, p < .05, method: pseudo-adaptive Gauss-Hermite; Task aversiveness, β = −0.001, OR = 0.99, p = .88, method: pseudo-adaptive Gauss-Hermite) (see Tab. S3–5).

Results of linear probability panel model

Given the panel data pattern (7 longitudinal sessions × task-execution willingness), this study has drawn upon the linear probability panel model (PLM) fitting the task aversiveness and outcome value to task-execution willingness. Specifically, the Fisher’s Augmented Dickey-Fuller Test (ADF) tests were performed to examine whether these data are suitable for this model. Results indicated the stationary properties for all the variables (task aversiveness, DF = −5.78, p < .05; overcome value, DF = −5.83, p < .05; task-execution willingness, DF = −5.83, p < .01). Further, the findings derived from Breusch-Godfrey/Wooldridge test maintained the decision to reject the stability of pool model (x² = 22.75, p < .0001). Also, the Hausman test was done for determining what types of models would be accepted. Results indicated that the random effect model is unstability (x² = 12.92, p < .0015). Thus, the fixed effect model controlling both time and individual variants was adopted as final one.

Results of sensitivity analysis

To examine the robustness and specificity of this causal medication model, we conducted several sensitive analyses. Thus, the we attempted to re-do this mediation model by replacing the sampling method from bootstrap-based bias-corrected and accelerated (BCa) intervals to “bca” sampling at 5000 simulations. Results demonstrated the robustness of this model by showing same findings (see Tab. S6–7).

Table S6

Table S7

Further, we inputted the age and gender as outcome variables into this model for testing whether this causal mediation model is specific to predict decreased procrastination. As hypothesized, no significant effects were found to predict task aversiveness by using this model, and thus supported the specificity of these findings (see Tab. S8).

Table S8

Results of GLMM adjusted for baseline

To minimize false-positive risks, rather to take time and group as predictors meantime, we reanalyzed these data by remodeling post-neuromodulation procrastination as dependent and remodeling pre-neuromodulation procrastination, group and other covariates as predictors. Results showed the significantly predictive of pre-neuromodulation procrastination willingness (β = 24.48, SE = 6.24, p = .0018) and procrastination rate (PR, β = 30.66, SE = 8.48, p = .0015) to post-neuromodulation ones. Moreover, for those covariates of interests on cognitive mechanisms, these findings are validated as well, showing statistically significant predictions from pre-neuromodulation cognitive processes to post-neuromodulation ones (task aversiveness, β = 0.58, SE = 0.23, p = .018; outcome value, β = 22.86, SE = 5.11, p = .0005). This replicated effect has been observed in predicting post-neuromodulation PR by pre-neuromodulation one for 6 month (β = 34.79, SE = 11.1, p = .005). Taken together, using stringent statistical constrain to adjust baseline, we revealed the same findings compared to traditional GLMM.

Figure S1

Figure S2

Figure S3

Figure S4

Figure S5

Figure S6

Figure S7

Figure S8

Figure S9

Table S9

Acknowledgements

We appreciate Zhibing Xiao (Beijing Normal University) for assistance with programming and coding; Chao Ran (Alibaba technical team) for assistance with developing experience sampling online platform; YanChen Tang (Peking University), Dr. Peikai Li (Utrecht University) and Dr. Hu Chuan-Peng (Nanjing Normal University) for assistance with statistics.

Additional information

Funding

This study was supported by the National Natural Science Foundation of China (32300907; 31971026; 31800959) and the Fundamental Research Funds for the Central Universities of China (SWU1509392, SWU1809357; SWU118091).

Author Contributions

Zhiyi Chen: Conceptualization, Methodology, Software, Writing - Original Draft and Visualization; ZhenZhen Huo, Chenyan Zhang, Bernhard Hommel: Writing - Original Draft, Formal analysis and Validation; ZhuangZheng Wang, Ye Liu, Bowen Hu, Wanting Chen: Writing - Review & Editing, Methodology, or Validation; Tingyong Feng, Conceptualization, Supervision, Project administration and Funding acquisition. CZY, RZL and LW contributed equally to this work.