Participants

We recruited 72 healthy, right-handed participants (36 males, mean age = 21.75, SD = 3.21, range = 18–30), and 36 females (mean age = 22.1, SD = 2.63, range = 19–28). Participants were randomly assigned into three stimulation groups (i.e., sham tRNS versus active dlPFC-tRNS versus active PPC-tRNS), and each of the three stimulation groups consisted of 12 females and 12 males. All the participants provided written consent before the experiment started. The study was approved by the Berkshire Research Ethics Committee (National Research Ethics Service Committee South Central—Berkshire, approval number: 10/H0505/72) and was conducted according to the principles expressed in the Declaration of Helsinki.

Learning task

In the current study, we used mathematical learning as a model for studying academic learning. The mathematical learning paradigm used here was adopted from Delazer and colleagues [53]. Participants learned to solve equations either by memorizing solutions to given problems (“drill learning,” denoted by “#”) or by using an algorithm (“calculation learning,” denoted by “§”) (Fig 1 and S1 Text).

tRNS

tRNS uses a rapidly alternating current at a fixed range of frequencies to induce cortical excitation [27] and provides adequate participant-blinding in sham-controlled research [57,58]; the current was applied via electrodes on the scalp (see below for details). Electrodes were placed above the bilateral dlPFC (F3 and F4) or PPC (P3 and P4), as defined by the international 10–20 system for EEG recording (for additional information, see S1 Text).

MRS

Single VOI ¹H-MRS data were acquired using a 3T Verio system (Siemens Healthcare, Erlangen, Germany) at the Wellcome Centre for Integrative Neuroimaging (WIN) at the University of Oxford. The system body coil was used for signal transmission and a 32-channel head receive array (HEA-HEP; Siemens Healthcare, Erlangen, Germany) was used for signal reception. T1-weighted MR images with a slice thickness of 1 mm were acquired (MPRAGE; magnetization prepared rapid gradient echo) with repetition time (TR) = 2,040 ms, echo time (TE) = 4.68 ms, TI = 900 ms (inversion time), and a flip angle of 8°. The localized MRS sequence SPECIAL (spin-echo, full-intensity, acquired, localized) technique was used to acquire 128 averages with a TR = 4,000 ms, a TE = 8.5 ms, a bandwidth of 4,000 Hz, and 4,096 points [59,60]. Water suppression was performed using VAPOR (variable power radio frequency pulses with optimized relaxation delays) [61]. Three 2 × 2 × 2 cm voxels of interest were manually localized on axial and coronal slices and placed over the left PPC (S1 FigA) or the left dlPFC (S1 FigB), depending on the tRNS condition, and a control region in the primary visual cortex (V1) of each participant. For additional information, including MRS data analyses, see S1 Text.

Resting fMRI

Functional images were acquired with an interleaved sequence (TR = 2410 ms, TE = 30 ms, flip angle 90°, number of slices = 44, voxel dimensions = 3 × 3 × 3 mm, number of volumes = 128). fMRI data were preprocessed and analyzed using the CONN toolbox (www.nitrc.org/projects/conn, RRID: SCR_009550) in SPM12 (Wellcome Department of Imaging Neuroscience, Institute of Neurology, London, UK) using a standard preprocessing pipeline that involves MNI-space normalization [62]. For additional information, including the calculation of the dlPFC-PPC (henceforth, frontoparietal) functional connectivity, see S1 Text.

Procedure

Participants were screened for safety over the phone and invited to the WIN at the University of Oxford (for the scanning sessions). A scan (i.e., structural MRI, ¹H-MRS, and resting-state fMRI) was performed, and each participant was subsequently invited to the Department of Experimental Psychology for 5 days of tRNS and mathematical learning (for the behavioral sessions). On the first day before the learning sessions started, mathematical attainment was assessed using the Wechsler Individual Achievement Test (WIAT) [63], as has been done previously [64]. This standardized test (Mean score = 100, SD = 15) includes Numerical Operations, which assesses written calculation abilities, and Mathematical Reasoning, which assesses verbal word problem mathematical abilities. We used the composite score (henceforth, the mathematical attainment) to match the participants in each group according to their mathematical abilities. Computerized drill learning and calculation learning, paired with active tRNS condition or sham tRNS condition, were provided for 30 min. After the last learning session, participants underwent another scanning session involving the same sequence of events, including repeated safety screening.

Statistical analyses

We performed linear, mixed-effects analyses with learning as the dependent variable (mean RT of correct trials only) and using several time points (i.e., day [5 as there were 5 days] and learning type [2 levels: drill and calculation]). For additional information, see S1 Text.

Baseline frontoparietal positive connectivity predicted improved learning

First, we examined whether mathematical (i.e., “calculation” or “drill”) learning was influenced by the baseline (pre-tRNS) frontoparietal functional connectivity (Fig 2A). To control for the confounding effect of tRNS, we focused on the sham tRNS condition only. We predicted learning using the variables day, learning type (drill versus calculation), and baseline frontoparietal connectivity. We observed an interaction between frontoparietal connectivity*learning type both for the left (Fig 2B, B = 2,462, SE = 839, T₂₁₀ = 2.9, P = 0.004, CI = [808, 4,116]) and for the right connectivity (Fig 2C, B = 1,960, SE = 619.0, T₂₁₀ = 3.17, P = 0.002, CI = [740, 3,180]). In both cases, the effect was restricted to the calculation learning; individuals with stronger baseline positive connectivity showed a better performance.

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Fig 2. (A) A 3D volume, generated manually in CONN [62], depicting the four frontoparietal seeds (left dlPFC, right dlPFC, left PPC, right PPC) as well as the right and left frontoparietal connectivity that was used in the functional connectivity analyses.

Predicting RT in the calculation learning task (for the corresponding analyses depicting the drill learning task, see S3 Fig) using the left and right frontoparietal connectivity (Panels B and C, respectively). Predicting RT in the calculation learning task using the left and right frontoparietal connectivity (Panels D and E, respectively) as a function of tRNS condition. Following Aiken and colleagues’ [65] suggestion for plotting continuous variables, we plotted the results here and in the other figures for one SD above the mean (i.e., mean + 1 SD), the mean, and one SD below the mean (i.e., mean − 1 SD). Error bars correspond to the 95% standard error of the mean. The data underlying the results in the Panels (B–E) can be found in S2 Data. dlPFC, dorsolateral prefrontal cortex; PPC, posterior parietal cortex.

https://doi.org/10.1371/journal.pbio.3003200.g002

Based on previous work showing that cortico-hippocampal connectivity is involved in mathematical learning [36], we assessed in a separate analysis whether dlPFC- or PPC-hippocampus connectivity explains variance in calculation or drill learning (see S1 Table). We found two significant effects: the left dlPFC–hippocampus connectivity (B = 2,650, SE = 988, T₂₂ = 2.68, P = 0.01, CI = [601, 4,699]) and the left dlPFC–hippocampus connectivity*learning type interaction (B = −2,720, SE = 1,045, T₂₁₀ = −2.6, P = 0.01, CI=[−4,780, −661]), where more positive connectivity was associated with poorer calculation learning, again with no significant differences for drill learning (S2 Fig).

We subsequently assessed whether left dlPFC–hippocampus connectivity would explain the additional variance in calculation learning above and beyond left frontoparietal connectivity. However, the left dlPFC–hippocampus baseline connectivity was no longer significant, whereas the frontoparietal baseline connectivity predictors were still significant (S2 Table).

Active tRNS over the dlPFC improved learning

Second, we examined whether tRNS induced changes in mathematical learning. As can be seen in Table 1, active tRNS over the dlPFC improved calculation learning.

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Table 1. Behavioral results when predicting RT from stimulation condition (compared to sham tRNS, which serves as the reference), day, and learning type.

Statistics: Value = regression coefficient, SE = standard error, DF = degrees of freedom, T = t-value, p = p-value, dlPFC = dorsolateral prefrontal cortex, PPC = posterior parietal cortex, tRNS, transcranial random noise stimulation. Interactions are denoted by the * symbol. The data underlying the results in this table can be found in S1 Data.

https://doi.org/10.1371/journal.pbio.3003200.t001

Active tRNS over the dlPFC in individuals with weaker positive baseline frontoparietal connectivity improved learning

To examine whether the frontoparietal connectivity plays a causal role in mathematical learning we modulated its activity by using active tRNS. By targeting one node (dlPFC) or another (PPC) in different individuals, we tested whether we could enhance learning. We predicted learning using the tRNS condition (sham tRNS versus dlPFC-tRNS versus PPC-tRNS), day, (1–5), learning type (drill versus calculation), and baseline frontoparietal connectivity (for additional information, see S1 Text). We observed a tRNS condition*frontoparietal connectivity*learning type interaction for the left (Fig 2D, B = −3,312, SE = 1,007, T₆₀₃ = −3.3, P = 0.001, CI = [−5,289, −1,335]) and the right hemisphere (Fig 2E, B = −2,291, SE = 838, T₆₀₃ = −2.7, P = 0.006, CI = [−3,937, −646]). In both cases, we observed that the effect was restricted to calculation learning, showing a difference only between the sham and dlPFC-tRNS conditions. Participants who were predicted to perform poorly due to weaker positive baseline frontoparietal connectivity, as observed in their peers in the sham tRNS condition, performed as well as or even better than those who were expected to outperform them when they received dlPFC-tRNS. Those who received PPC-tRNS showed a similar pattern to those in the sham tRNS condition (Fig 2D–2E). These findings were specific to frontoparietal connectivity, indicating that the effect of dlPFC-tRNS on learning depends on the baseline levels of frontoparietal connectivity, but not occipitofrontal or occipitoparietal functional connectivity (for statistical details, see S3 Table).

To assess whether the effect of active tRNS is confounded by baseline mathematical ability we performed an additional analysis with the inclusion of three covariates (i.e., mathematical attainment, first block drill RT, and first block calculation RT). The effect of active tRNS was not confounded by baseline mathematical ability (S4 Table).

In contrast to our predictions, active tRNS did not reliably alter GABA or glutamate levels (S5 Table). We examine their moderating role on the effect of dlPFC-tRNS on learning as indicated by the interaction between Δ dlPFC neurochemical level (i.e., post-tRNS – baseline neurochemical concentrations) and Δ frontoparietal functional connectivity (Materials and methods, S1 text, and S6 Table). We observed a tRNS condition*Δ dlPFC GABA*Δ right frontoparietal connectivity*learning type interaction (Fig 3, B = 18,110, SE = 6,651, T₁₂₀ = 2.7, P = 0.007, CI = [4,942, 31,278]). This interaction remained significant after controlling for Δ dlPFC glutamate (B = 29,571, SE = 8,523, T₁₁₁ = 3.47, P = 0.0007, CI = [12,682, 46,460]). Compared to the sham tRNS, dlPFC-tRNS induced the largest improvement in calculation learning for participants who showed reductions in dlPFC GABA and positive frontoparietal connectivity (P = 0.0148, Fig 3A). However, if an increased, rather than decreased, frontoparietal connectivity was observed, the performance was detrimental (P < 0.0001, Fig 3A).

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Fig 3. Plotting the simple three-way interaction of tRNS*right frontoparietal connectivity*dlPFC GABA.

This is based on the four-way interaction between learning type*tRNS condition*right frontoparietal connectivity across three levels of the dlPFC GABA. For simplicity, we display the calculation learning results. For the corresponding drill learning results, see S4 Fig. Participants who showed reductions in positive frontoparietal connectivity and reductions in dlPFC GABA showed calculation learning improvement in the dlPFC-tRNS (vs. sham tRNS) condition. The effect of the dlPFC-tRNS (vs. sham tRNS) condition is reversed for participants who showed increases in positive frontoparietal connectivity and reductions in dlPFC GABA. dlPFC-tRNS benefited those participants who performed the worst, namely, the participants in the sham tRNS condition who showed the most pronounced reductions in positive frontoparietal connectivity and the most pronounced reductions in dlPFC GABA (Fig 3A, leftmost subpanel). Following Aiken and colleagues’ [65] suggestion for plotting continuous variables, we plotted the results for one SD above the mean (i.e., mean + 1 SD), the mean, and one SD below the mean (i.e., mean – 1 SD). Error bars correspond to the 95% standard error of the mean. The data underlying Fig 3 can be found in S3 Data. dlPFC, dorsolateral prefrontal cortex; tRNS, transcranial random noise stimulation.

https://doi.org/10.1371/journal.pbio.3003200.g003

A closer examination of the sham tRNS group (red color in Fig 3) suggests that greater positive connectivity increase was beneficial for those in the plasticity state as indicated by reduced GABA (−1SD change in GABA, P = 0.0001, Fig 3A and S7 Table A) but was detrimental for those in the stability state (+1SD change in GABA, P = .013, Fig 3C and S7 Table A).

In contrast, for the active tRNS group (blue color in Fig 3), compared to those with a smaller positive connectivity increase, a greater positive connectivity increase was detrimental both in the plasticity (P = 0.0014, Fig 3A and S7 Table B) and the stability state (P = 0.009, Fig 3C and S7 Table B).

The role of baseline frontoparietal connectivity in predicting learning and tRNS efficacy

Our connectivity analyses revealed that baseline frontoparietal connectivity interacts with learning type in predicting learning: individuals with a stronger positive baseline frontoparietal connectivity showed better learning that was specific to calculation.

Furthermore, individuals with weaker positive baseline connectivity, who would have otherwise shown poorer calculation learning, improved their calculation learning when they received active tRNS over the dlPFC compared to the sham tRNS condition. Again, we obtained this effect specifically for calculation learning. By contrast, stimulating the other node in this network, the PPC did not yield a similar effect. Taken together, these findings suggest that the key causal effect of connectivity on learning is rooted in the dlPFC. The enhancement effect of those with predicted poorer calculation learning raises the possibility that tRNS has a “scaling up” effect that can release the brakes due to suboptimal brain activity to support learning [66,41]. The fact that individuals with stronger positive baseline connectivity showed better calculation learning suggests a neurobiological equivalent of the Matthew effect. The Matthew effect [4] has been shown in different types of learning, including mathematical learning [67], language learning [68,69], and memory-related processes such as vocabulary growth [69], whereby a baseline cognitive gap between high and low achievers leads to an even greater increase in inequality between these two groups.

Our results, in line with previous results [7–9], support the view that such differences may stem from biological origin. However, such differences may be independently influenced by socioeconomic status and polygenetic factors [70]. By using a learning task and modulating the neural corelates involved in mathematical learning, we were able to show that the Matthew effect can be explained and mitigated at the biological level. Our results do not undermine the environmental factors (e.g., socioeconomic status) that could also draw a causal effect but are less likely to play a role in this study due to the random allocation of participants and the use of a learning task. Our results highlight the possibility of reversing such biological disadvantage using an excitatory form of neurostimulation to target the neural substrates involved in calculation learning. This is in line with a recent study that identified genes that correlate with mathematical abilities, as crucial for neuronal excitability [9]. Future studies can build on our findings by assessing how long this effect lasts and whether such an approach can have practical relevance outside the controlled lab environment. Answering these questions would have important implications in the fields of education and the science of learning.

Furthermore, the application of active tRNS, our excitatory neurostimulation protocol, sheds light on the involvement of the frontoparietal network in skill acquisition in calculation learning but not by drill learning. Unlike in calculation learning, in drill learning, active tRNS did not produce a robust enhancement effect. These behavioral results are in contrast to a previous study [33], but in line with another [71]. It might indicate that the effect of tRNS on mathematical drill learning is less robust. This might be attributed to the ease of acquisition of such learning, which required shallower processing during the encoding stage, and therefore faster learning [72]. This, in turn, may lead to an early shift from plasticity to stabilization [40], and therefore reduce the beneficial excitability effect of tRNS.

Active tRNS effect is mitigated by changes in dlPFC GABA

Subsequently, we revealed a multimodal interaction whereby, compared to the sham tRNS, the dlPFC-tRNS induced an improvement in mathematical learning for participants who showed reduced dlPFC GABA and reduced positive frontoparietal connectivity. This beneficial effect was reversed if the participants showed increased positive frontoparietal connectivity. In the sham condition, the worst-performing individuals were those who exhibited the most pronounced reductions in positive frontoparietal connectivity and dlPFC GABA (Fig 3A, leftmost subpanel). However, dlPFC-tRNS benefited the participants with the same neurobiological profile.

How do these results fit with the putative mechanisms of tRNS [26,66,73]? Different studies have attributed the effect of tRNS to angiogenesis [33], stochastic resonance [26,66,71], E/I [41], a reduction in GABA precursor [43], and sodium channels [44]. It would be simplistic to assume that a single mechanism can explain the effect of tRNS. Rather, factors such as the timescale in which tRNS was applied, the stimulated region, the cognitive domain, the level of skill acquisition, and the neural state during stimulation, which vary across studies, may play an important role in the effect of tRNS and the underlying mechanisms on learning [74]. The beneficial effect of tRNS on those who would otherwise underperform is in line with the principle of stochastic resonance [26,66,71], a phenomenon by which the addition of random noise to a weak signal enhances the detectability of that signal [75]. Previous studies have demonstrated performance enhancements both when the random noise was added directly to the weak perceptual stimuli or via tRNS over cortical regions [27,52,66,76]. The current results suggest that the tRNS effect may have been achieved by shifting subthreshold receptor potentials within the dlPFC toward the firing threshold, increasing the likelihood of action potentials in neurons that would otherwise not reach the threshold [66]. This, in turn, is likely to increase the E/I level, as indicated by decreased GABA concentration. However, the null result in our MRS measurement create challenge to this explanation. The reason for this null result could be multiple, including the length of our training that span over 5 days and might have led to a shift from plasticity to stability, resulting in a spurious E/I effect [40], and the hours that passed from the end of the learning and tRNS to the scanning session, which can show fluctuations in E/I dynamics [77].

The importance of stability and plasticity states during mathematical learning and the interplay between functional connectivity and GABAergic modulation in the efficacy of brain-based interventions to augment learning outcomes

Previous studies on visual procedural learning demonstrated that neural E/I regulates the transition between stability and plasticity states (for a review, see [40]). Here we extend these previous studies by demonstrating how the states of plasticity and stability, as indicated by the relative change in dlPFC GABA concentration, impact mathematical learning. Initially, we showed that overall performance (i.e., across stimulation conditions and connectivity levels) is better in those in the stability state (+1SD GABA, Fig 3C) than those in the plasticity stage (−1SD GABA, Fig 3A). This set of findings suggests that more stability is associated with better calculation learning.

Importantly, however, by employing a multimodal experiment, we showed that the mechanisms that regulate learning improvement are more complex than described before, and co-depend on both functional connectivity and GABA concentration. Normally, as exhibited by the sham tRNS group, a greater increase in positive connectivity is beneficial for those individuals in the plasticity state (−1SD change in GABA, Fig 3A) and for those neither in the plasticity nor stability state (Mean change in GABA, Fig 3B) but detrimental for those in the stability state (+1SD change in GABA, Fig 3C). A potential explanation is that when the reorganization of frontoparietal connectivity occurs under plasticity (versus stability) state, such reorganization allows optimal synchronization between frontal and parietal regions to support mathematical learning. This can allow effective progression from a stage of skill acquisition that is dominated by executive control and intentional processing subserved by the PFC to more automatic processing subserved by the PPC [20,21,25]. In contrast, increased stability, which signified an advanced skill acquisition stage that should be signaled by less controlled and more automatic processing [77], with increased positive connectivity between the PFC and PPC, might indicate difficulty in moving to automatic processing.

Furthermore, a closer examination of the active dlPFC-tRNS group (blue color in Fig 3) suggests that increased positive connectivity is detrimental both in the stability state (+1SD change in GABA, Fig 3C) and also in the plasticity state (−1SD change in GABA, Fig 3A). These findings tie in with our aforementioned findings that participants with weaker positive baseline frontoparietal connectivity experienced enhanced learning outcomes following dlPFC-tRNS. Therefore, dlPFC-tRNS seem to be beneficial to those with weaker changes in positive baseline frontoparietal connectivity and may be detrimental to those with stronger changes in positive baseline connectivity, regardless of the state (plasticity or stability) (for details see S7 Table). Relatedly, we also report that across changes in GABA concentration (Fig 3B), changes in connectivity predict learning. Specifically, in the sham condition (red panels in Fig 3B), those with a greater increase in positive connectivity exhibited better learning. However, the administration of tRNS reverses this pattern by benefiting those with the least increase in positive frontoparietal connectivity while degrading the performance of those with the greatest change in positive frontoparietal connectivity. At the same time, tRNS did not significantly alter functional connectivity, suggesting that while its effect depends on baseline functional connectivity and its changes, if it causes changes, it is via other mechanisms.

Moreover, our findings expand previous studies that have shown that the effect of brain stimulation varies among individuals [76,41,78–82]. We highlight a key determinant of brain stimulation outcome on neurophysiology, namely, the initial levels of frontoparietal connectivity. Some previous studies in children showed a detrimental influence of stronger functional connectivity in participants with lower mathematical abilities [83,84]. However, the relationship between functional connectivity and math performance is more complex than originally thought and it is influenced by several factors including developmental stage (children versus adults), concurrent math measures versus learning studies, and in response to interventions (e.g., before versus after math tutoring). For example, greater IPS connectivity with the premotor cortex and supramarginal gyrus has been associated with better performance in adulthood [85]. However, IPS connectivity was negatively associated with math performance in adolescence [86] as in children. Moreover, frontohippocampal connectivity was shown to be negatively correlated with concurrent math measures in adolescence [86], which contrasts with results from learning studies. For example, tutoring-induced strengthened connectivity between IPS with lateral PFC (as well as VTOC, and hippocampus) was also associated with improved mathematical learning in children [87]. In previous studies, we have reported that the association between different brain measures (e.g., GABA, glutamate, aperiodic exponent) and cognition (e.g., math) can be reversed as a function of the developmental stage [15,88]. Future studies should examine whether and how the different brain indices (GABA, glutamate, functional connectivity, aperiodic exponent) are linked to each other and may reflect the changes in the association between functional connectivity and math as a function of the developmental stage. Our results provide important considerations for future individual-specific brain stimulation interventions aimed at enhancing academic learning in typically developing adults or those who suffer from learning difficulties.

The causal role of dlPFC in calculation but not drill learning in adults

This study provides novel insights into the neural mechanisms underlying mathematical learning, specifically in terms of the role of dlPFC in calculation but not drill learning in adults. Namely, in contrast to previous fMRI studies that highlighted the role of dlPFC also in drill learning, our findings suggest that dlPFC may instead be a correlate and may indicate redundancy in brain functions [89] with regards to drill learning in the adult brain. While we included a mathematical control task, future studies should include a non-mathematical learning control task that also heavily depends on executive functions to help distinguish between specific executive function components. This will allow the examination of whether the observed neurostimulation effect is specific to mathematical learning or due to a domain-general modulation of executive functions. Nevertheless, our findings are theoretically consistent with current meta-analyses and models of arithmetic processing [32,35,90–94]. Our two-time point (i.e., after versus before tRNS) additional analyses did not provide evidence that tRNS modulated working memory/executive functions (S8 Table). Recent tRNS experiments that used a similar prefrontal montage as in this study, improved sustained attention performance or symptoms that are associated with sustained attention impairments [42,49,95]. Therefore, we hypothesize that our results are due to sustained attention enhancement that has been linked with improved academic learning [96]. In this case, the lack of effect on drill learning may be explained by task difficulty. This is in line with previous tRNS studies that have shown that in line with the stochastic resonance framework, task difficulty moderates the effect of tRNS [71,97]. Namely, the presence of a dlPFC-tRNS effect for calculation learning might not necessarily be due to the fact the dlPFC is less involved in drill learning but because drill is an easy task and less demanding compared to calculation. Highly task engagement is thought to boost the effectiveness of non-invasive brain stimulation [98]. Based on the stochastic resonance framework [97] it is expected that the more difficult the task the more the benefit from tRNS. Our findings, which are in line with previous work in the field of cognitive learning [41,71], match these predictions.