A good proxy measure for spatial associations of symbolic information is available in the Spatial-Numerical Association of Response Codes (SNARC) effect, which is evident in cognitive performance during very simple two-choice reaction tasks. When healthy participants classify features of sequential or numerical stimuli by key presses on the left- or right-hand side, the central finding of the SNARC effect consists in relatively faster left-hand over right-hand responses for small over large magnitudes (and initial over posterior sequence positions), and vice versa (Dehaene et al., 1993; Gevers et al., 2003; Wood et al., 2008; van Dijck and Fias, 2011). Regarding the neurocognitive processes beyond the SNARC effect, several theoretical positions are currently available in the literature and the exact mechanisms may be more multifaceted than initially declared, including the context of mental number representations in long-term memory (Hubbard et al., 2005), polarity correspondence (Proctor and Cho, 2006), working memory and / or spatial attention (van Dijck and Fias, 2011; Ginsburg et al., 2014; Abrahamse et al., 2016).

Regarding the underlying functional neuroanatomy, studies in numerical cognition have increasingly acknowledged the role of prefrontal-parietal circuits in intracortical recordings (Nieder and Dehaene, 2009; Nieder, 2016) or diffusion-tensor-imaging of white matter connectivity in human cortex (Klein et al., 2016). Thus, prefrontal regions appear to complement the established role of parietal regions in number representation (Dehaene et al., 2003) and spatial-numerical associations (Cutini et al., 2014). These recent results dovetail with theoretical accounts of the SNARC effects that predict prefrontal involvement in the form of verbal working memory. Using subthreshold neuromodulation of brain activity in prefrontal areas concurrent to respective tasks, studies with transcranial direct current stimulation (tDCS) can investigate and causally bolster these supposed linkages between prefrontal networks and subtle activation of implicit spatial associations of numbers and non-numerical sequences.

By testing the effect of left-hemispheric prefrontal tDCS on SNARC effects, we recently observed that inhibitory cathodal tDCS specifically blocked the generation of spatial associations in case of numerical symbols, but the stimulation did not affect performance when visuospatial distraction was directly available in the spatial displacement of stimuli (Schroeder et al., 2016). Precisely, in three sham-controlled cross-over experiments with either 1 mA anodal or cathodal tDCS, healthy participants were asked to classify centrally presented numbers. The task included left-hand or right-hand classifications according to features that were irrelevant to the spatial dimension, namely: number parity or magnitude. During a sham tDCS condition (which elicits comparable sensations, but no changes in brain activity), we obtained relatively faster response times for the congruent combinations of small (large) numbers and left-side (right-side) responding than for the incongruent combinations (e.g., small number and right-side responding), but this SNARC effect for single digits (1–9, without 5) was specifically abolished during another session of concurrent 1 mA cathodal tDCS (target cathode: left PFC, return anode: extracephalic placement on right upper arm). The effect of cathodal tDCS on SNARC effects was reproduced in two different paradigms (parity judgment and magnitude judgment, (Schroeder et al., 2016), experiments 1 and 2), but there were no general performance modulations in a control task, and polarity-specificity could be substantiated by a descriptive increase in SNARC effects during a separate experiment with excitatory anodal tDCS (Schroeder et al., 2016, experiment 3).

In a follow-up study, we investigated whether stimulation effects were specific for numbers or whether the polarity-specific tDCS effects would generalize to non-numerical sequence items. Having established a valid sham-control for modulations of spatial associations of numbers in the first study, we now tested different SNARC effects in a more economic parallel design with two groups of participants who performed baseline assessments (without tDCS) immediately followed by 1 mA anodal or cathodal tDCS applied concurrent to a second assessment of task performance (Schroeder et al., 2017b). The cardinal number sequence included number symbols 1, 2, 4, and 5, and the ordinal non-numerical sequence included written weekday names Monday, Tuesday, Thursday, and Friday (in German). In separate tasks, participants classified whether a number was smaller/greater than 3, and whether a weekday name came before/after Wednesday, by means of a left-side or right-side button press in congruent and incongruent blocks of trials. In line with the literature (Gevers et al., 2004), we observed left-to-right oriented SNARC effects during the baseline condition in the order-relevant magnitude judgment tasks for both sets of stimuli, with some variations across participant groups. The tDCS results, however, showed polarity-dependent dissociations of the directional arrangement (spatial associations) of the numerical vs. non-numerical stimuli. Specifically, in this previous study, participants displayed the reverse orientation for a SNARC effect for weekday stimuli (that is, Monday and Tuesday were faster responded to with their right hands, while Thursday and Friday were responded faster to with their left hands) when stimulated with anodal tDCS, but not with cathodal tDCS. For the group stimulated with anodal tDCS, we thus observed a striking reversal of the spatial association of a non-numerical sequence in the same two-choice reaction time testing paradigm, highlighting a dissociation between numerical and non-numerical sequential (weekday) spatial associations (Schroeder et al., 2017b).

However, weekdays are only one instance of non-numerical sequential stimuli. To conclude that the diverging stimulation results hold for non-numerical sequential stimuli in general, and not only specifically for weekdays, a conceptual replication with at least one other non-numerical sequence was required.

In contrast to the stable and replicated effect of cathodal tDCS (Schroeder et al., 2016), the overall effect of anodal tDCS was less clear. For instance, although the reversal of spatial associations of weekday stimuli during anodal tDCS was theoretically meaningful and statistically clear-cut in our previous analysis (Schroeder et al., 2017b), results were based on a single tDCS study in parallel design without sham control condition. Thus, in the current study, we set out to challenge the robustness of our previous finding and to test the generalizability to another set of sequential stimuli. Moreover, by testing month stimuli as another non-numerical sequence, we also tested the presented multiple-coding framework which would predict dissociations between number and month stimuli by anodal tDCS, unlike potential unifying accounts.

The full publication of such replication attempts is particularly important in the domain of tDCS to transparently address the contemporary skepticism among researchers and practitioners. Specifically, the conception of the neuromodulation technology is rather reserved and some recent articles have sparked doubt on its potential effects in quantitative review (Horvath et al., 2015) or have outlined large variability in physiological measures of motor cortex excitability modulations (Strube et al., 2016). Due to publication biases, it is possible that an even larger number of negative results had not been reported on, but underlying reasons remain elusive. For instance, we found in our previous results that the efficacy of modulation depended on the level of cognitive activity induced by a task (Zwissler et al., 2014) or on the timing of stimulation in a first, second, or third session of repeated task performance (Dockery et al., 2009). Moreover, if a task classification rule incorporated a stimulus dimension not necessarily recruiting the respective (prefrontal) network (Fias et al., 2001), stimulation was not effective (Schroeder et al., 2017b). These findings line up with the general theoretical premise of state dependency of transcranial brain stimulation (Silvanto and Pascual-Leone, 2008; Fertonani and Miniussi, 2016).

The aim of the current study was therefore threefold. First, following the striking reversal of the weekday sequence spatial associations, we sought to replicate the reversal of a non-numerical ordinal sequence with anodal tDCS in a sham-controlled design. Our primary outcome was defined as differences in the unstandardized regression coefficients capturing the SNARC effects (Fias et al., 1996; see Data Treatment in the next section) between sham and anodal tDCS condition, which should resemble the observed difference between baseline and anodal tDCS in the same parameter in our previous study (Schroeder et al., 2017b, p. 43, paired t-test on unstandardized regression coefficients). We decided to change the experimental design in order to establish a valid sham-control that elicits comparable sensations and thus to control for any motivation-related changes (although it has to be noted that we could not observe significantly different sensations between anodal and cathodal tDCS before). In the present study, participants were tested on separate days with a minimum wash-out period of 48 h to circumvent any possible long-term neuroplastic (after-) effects of the stimulation. Second, to explore whether the observed directionality switching was directed by the verbal presentation format of weekday stimuli (which is somewhat related to the theoretical markedness correspondence account), we now also presented single-digits in their verbal, i.e., written, form (i.e., German word “eins” for “1,” and so on). Finally, we included another non-numerical sequence task with month names to examine the generalizability of our original results. With the same month names used in the new task (albeit in Dutch language), previous experiments demonstrated a spatial mapping from left-to-right akin to the SNARC effect for numbers (Gevers et al., 2003). Thus, the third experiment on month names was specifically designed to conceptually replicate our finding with another non-numerical sequence and to test the predictions of the presented multiple-coding account. Previously, we had suggested that the dissociative tDCS effect in weekdays was driven by the markedness features of their ordinal item structure (i.e., as weekdays constitute a non-numerical sequence). If these conclusions hold, similar effects should be obtained with other non-numerical sequences as well, i.e., also with month names. We hypothesized that 1 mA anodal tDCS would reverse the spatial associations of both weekdays and month names, but that stimulation would rather yield a small enhancement in opposite direction of regular left-to-right SNARC effects for numbers.

Based on a-priori power calculation (see respective section on the next page), healthy volunteers (N = 24, mean age: 21.8 year, range: 18–26 year, 5 males) were recruited from the general and student population. All participants attended a sham and a stimulation session with 1 mA anodal tDCS, in counterbalanced order (mean inter-session interval: 5.1 days, range: 2–9 days). This study was carried out in accordance with the recommendations of University Hospital Tuebingen Ethical Commision with written informed consent from all subjects. All subjects gave written informed consent in accordance with the Declaration of Helsinki. The protocol was approved by the University Hospital Tuebingen Ethical Commision (Number of approval: 701/2015BO2). Eligibility for the study was assessed by self-reports collected prior to the first experimental session. Right-handed volunteers (according to the questionnaire by Oldfield, 1971) were eligible for participation if they were native German speakers without neurological or psychiatric impairments and if they fulfilled tDCS safety requirements (no metallic implants, cardiac pacemaker, pregnancy, medication, or use of recreational drugs). Participation was compensated with money or student credits. The experiments were performed in Tübingen between June-November 2016.

The study followed a sham-controlled cross-over design. Transcranial direct current stimulation (tDCS) was administered in either active anodal or sham configuration on separate testing days in counterbalanced order across all participants. Direct current was generated by a CE-certified stimulator (neuroConn GmbH, Illmenau, Germany). Sham stimulation was realized by fading out the direct current after only 40 s of stimulation, yet participants started the first task in both conditions always after 5 min following the tDCS fade-in. In contrast, active anodal tDCS was administered continuously at 1 mA intensity for a total duration of 25 min (“online” to the task). In contrast to offline tDCS effects (i.e., behavioral effects are assessed after the termination of stimulation), online tDCS effects can be linked directly to resting membrane threshold changes in cortical excitability as opposed to longer-lasting neuroplastic responses. A minimum wash-out period of 48 h was imposed between sessions. The target anode (5 × 7 cm) was fixed over F3, the return cathode (5 × 7 cm) was fixed over the right upper arm. Impedances were below 10 kΩ. Participants rated adverse effects (Brunoni et al., 2011) after completion of the stimulation protocol.

This specific anodal tDCS protocol was motivated by our previous observation that the reversal of SNARC effects for non-numerical ordinal words was induced by 1 mA anodal, but not by 1 mA cathodal tDCS over the left PFC with extracephalic return electrode. In contrast to the previous study, the cross-over study design was implemented here to establish a valid sham control.

Participants completed all three experimental tasks during both sessions either following a 4:20 min stimulation-free rest phase in the sham session or concurrent to the anodal tDCS following 5 min of at-rest stimulation. All tasks were completed online in the active anodal tDCS session.

Since we observed significant modulations of spatial associations exclusively during the most active tasks with explicit comparison instructions (Schroeder et al., 2017b), the testing sessions in this study included only the order-relevant magnitude comparisons in all three stimulus set variants.

The tasks closely followed previous order-relevant implementations (Gevers et al., 2003, 2004). In all three tasks, participants had to repeatedly classify stimuli by following a response-mapping rule and by operating two keyboard buttons with their right-hand and left-hand index fingers (covert “s” and “l” on a QWERTZ-keyboard). Stimuli included a numerical sequence in German verbal notation (“eins,” “zwei,” “vier,” “fünf”; English translation: “one,” “two,” “four,” “five,” respectively), the weekday sequence (“Montag,” “Dienstag,” “Donnerstag,” “Freitag”; English translation: “Monday,” “Tuesday,” “Thursday,” “Friday,” respectively), and the month series (“Januar,” “Februar,” “März,” “April,” “September,” “Oktober,” “November,” “Dezember”; English translation: “January,” “February,” “March,” “April,” “September,” “October,” “November,” “December,” respectively). The task instruction was to classify whether a number was smaller or larger than “drei” (three), whether a weekday was before or after “Mittwoch” (Wednesday), and whether a month was before or after “Juli” (July).

Response assignments to the classification decisions (e.g., right = large or right = before) were alternated within the tasks in counterbalanced order across participants. Thus, participants always solved two blocks with “compatible” and “incompatible” response assignments (e.g., a left-hand assignment to small numbers/before positions would be considered to be compatible following the standard results of relatively faster responses in this block as compared to the opposite right-hand assignment).

All single tasks were preceded by a brief training block (16 trials) and an error count emphasized correct responding. The experimental procedure comprised 40 target repetitions in each task, with 300 ms fixation hash (#), 2 s target presentation (or until response), and conditional 300 ms error feedback (German words “Fehler” [error] or “Bitte schneller antworten” [please respond faster]) or a blank inter-trial interval. Figure 1 displays a schematic of a trial. Note that consequently, because the month task included twice as many stimuli as both other tasks, experimental blocks took longer for this task. There was the possibility to take self-paced breaks between tasks and blocks.

The order of the three experimental tasks (number words, weekdays, months) was balanced across participants in a Latin Square Design, resulting in three conditions (WMN, MNW, NWM). Half of all participants started each task with the incongruent response mapping.

A power calculation was performed to estimate sample size based on the previously observed reversal effect (Schroeder et al., 2017b) using the program MorePower 601 (Campbell and Thompson, 2012). In our previous experiment, the observed effect size of anodal tDCS on weekday stimuli in the within-group comparison was d = 0.95. Assuming this effect size, the sample size for a replication (at α = 0.05 and a priori power of 1–ß = 0.99) was calculated to require N = 24 participants (dependent samples t-test of unstandardized regression coefficients).

Only correct trials were considered for analyses of response times (RTs, 95.4%). Additional post-error trials (3.6%, to reject systematic RT variation in post-error slowing; Notebaert et al., 2009) as well as stimulus-repetition trials (17.9%) were rejected from analyses (SNARC effects are reduced in stimulus repetition trials due to episodic memory; Tan and Dixon, 2011; Pfister et al., 2013). Next, response latencies that deviated more than 3.0 SD from the respective cell mean (computed separately for each target-response combination) were rejected (1.0%). These criteria left 73.0% of all trials available for subsequent analysis. The rationale of this data treatment was to reject trials that included systematic RT variation due to other known cognitive effects. Following a reviewer's comment, and acknowledging the fact that different analysis strategies could in principle yield different outcomes (Silberzahn et al., 2017), we also demonstrate in the Supplementary Materials (SA1) that the main results did not change substantially in an alternative analysis based on correct mean RT without further trial rejection.

We used unstandardized regression coefficient analyses¹ to extract the relative advantage of right-hand over left-hand responses with increasing numerical magnitude or sequence position, as it is usually implemented to quantify SNARC effects (Lorch and Myers, 1990; Fias et al., 1996) for numerical and non-numerical sequence stimuli (Gevers et al., 2003, 2004). The computation involves several steps: First, RT right-hand—left-hand differences are determined separately for each participant, stimulation condition, task set, and target stimulus combination from median RTs. Next, RT hand differences are predicted by numerical magnitude (1-5) or sequence position bins (numerically coded as 1-5; see Figure 2). A negative regression coefficient (tested with one sample t-tests against zero) indicates the typical behavioral pattern of SNARC effects, that is: Participants give relatively faster left-hand than right-hand responses to small/before stimuli and participants give relatively faster right-hand than left-hand responses to large/after stimuli. The resulting regression coefficients were submitted to separate paired-samples t-tests for each task (number, weekdays, months) to outline the effect of anodal tDCS.

Adverse effect ratings for both sham and active anodal tDCS are summarized in Table 1. Participants reported slightly more “tingling elsewhere” [t₍₂₃₎ = 2.01, p = 0.057], but all values were relatively close to the lower boundary of the self-report scale (see Table 1).

To address the shortcoming of reduced power due to smaller effects than anticipated, we decided to resubmit the previously obtained regression coefficients from earlier data sets to joint analyses. The respective experiments and their main findings are fully reported in Schroeder et al. (2016, experiment 3; S16) and Schroeder et al. (2017b; S17). Here, we re-analyzed only those datasets which were collected during the exact same anodal tDCS configuration and the respective sham or baseline control conditions.

In recent years, it has come to public awareness that Type I error rates (false positives) are particularly pronounced in psychological research under the umbrella of reproducibility crisis (Open Science Collaboration, 2015), supported by publication biases, questionable research practices, and other factors that lead researchers to publish and report only on successful (“significant”) manipulations in their studies (Nosek et al., 2015). Relative to the impact of the current research topic, and also the controversy around tDCS effects on cognition, it seems reasonable to assume that some publications on the efficacy of tDCS could include Type I errors. False positives rates can inflate with inclusion of more measurement factors or tasks, which is particularly critical when only positive procedures are reported on.

Over the current set of studies, we believe that the pattern of results and especially the successful partial replication in the month sequence does not point toward a false positive in the original results (Schroeder et al., 2017b). Actually, in series of experiments, the probability that one replication would not work is statistically plausible and hinges on the power of the test. For a simplified example, given a power of 73% (i.e., the power to detect the effect of anodal tDCS on the month series in the current experiment), the chance for 3 out of 3 experiments to turn positive is relatively low (0.73³ = 38%). In contrast, the probability to obtain 2 out of 3 significant results (i.e., as it was the case here) would be even slightly higher (43%), although this example assumes the case that the effect is actually true (see Francis, 2012 for discussion of this too-good-to-be-true approach).

Following up on the observed power of 0.73 (related to the reversal of spatial associations for the month sequence) it could be plausible that a tDCS effect was simply not observed in the weekday sequence due to a Type II error. Actually, running low-powered studies has a long tradition in clinical psychology and estimates were below 50% chance to detect a medium-sized effect (i.e., d = 0.5 or larger), but power was acknowledged only in most seldom cases (Sedlmeier and Gigerenzer, 1989). Importantly, the failure to detect an effect in null hypothesis significance testing does not provide evidence for absence of the effect in the first place.

On this occasion, it is revealing to notice that effect sizes diverged tremendously between studies. A priori sample size estimation initially suggested sufficient power with the included sample size, but the post-hoc power analysis deviated from the a priori sample size estimation due to the diminished effect size. Although it is acknowledged that initial discoveries of psychological effects may report larger effect sizes than replications (with a standardized mean difference of 0.21, Open Science Collaboration, 2015, p. 5), the reduction in effect size in the present case may have been also supported by other systematic differences between studies.

However, while power issues could in principle account for our results pattern, there are also reasonable alternative accounts suggesting that the obtained results are due to systematic underlying differences between our previous study and the current results. For example, the facts that other stimuli and control tasks were tested in different study designs (i.e., parallel vs. cross-over design) could play into the obtained results. Although all stimuli comprised sequence items, the month series includes a larger number of items than both number and weekday sequences, which could have influenced also the mapping of the relatively smaller item sets onto spatial templates. Actually, our joined analysis showed larger SNARC effects for extended number ranges, which dovetails with the differences in SNARC effects for different stimuli of the present study during the sham condition (showing the largest effect in the extended month range).

Next, since the markedness correspondence account attests different possible strategies to produce spatial associations from verbal markedness, but also from other mechanisms such as visuospatial simulation or serial-order processing, it may be possible that detachment from a previous task set (e.g., classification of month stimuli) to perform on a new, yet comparable task (e.g., classification of weekday stimuli) involved a subtle switch in the spatial association strategy used to allow for more effective task set representations. Thus, we speculate that the presence of different tasks in the same session could lead to contagion to another spatial association mechanism in changing item sets.

Furthermore, the possibility of individual differences could be relevant for both the modulation with tDCS and also for the effect inclined by our tasks. Regarding the SNARC effect, variation is remarkable in general and only ~70% of the general population present regular spatial-numerical associations (Wood et al., 2006; Cipora and Nuerk, 2013). We also noted that different inclinations of SNARC effects for weekdays during sham/baseline were present in the current study and the previous study, which could suggest a lack of modulation effects due to the lack of sham effects in the present sample (see Figure 4). Specifically, the SNARC effect in the sham tDCS condition trended toward being negative only in the month order classification task, but not in the numerical magnitude or weekday order classification tasks. Moreover, there is also noticeable variability in the responses to tDCS, already when motor cortex stimulations and physiological measurements are performed (Wiethoff et al., 2014; Strube et al., 2016), which could result in different effectivities as also captured by the effect size of the tDCS modulation in the month task.

In any case, all presented possibilities remain post-hoc hypotheses generated from the data and thus they need to be submitted to respective confirmatory testing in the future, since they also raise potential relevance to the systematic investigation of tDCS effects in general.

We assume that spatial associations can result from different mechanisms and we propose several verbal and non-verbal simulation processes (see Figure 5). In this multiple-coding framework, anodal tDCS of left-hemispheric prefrontal circuits could facilitate the processing of verbal markedness, render correspondence effects between right-before and left-after classifications as task-relevant, and thus result in the observed reversal of SNARC effects for the ordinal sequence (Schroeder et al., 2017b). For the numerical sequence, the same mechanism would enhance correspondence effects between large-right and small-left classifications as task-relevant and thus result in the regular left-right direction of the SNARC effect (Schroeder and Pfister, 2015). However, in this theoretical model, it is possible that also other verbally mediated strategies (such as working memory mechanisms of sequential order) or visuospatial simulations produce spatial associations that are resistant to a certain stimulation, especially considering the possibilities of individual differences or task contagion. Such a multiple-coding framework also agrees with the inconsistent outcomes for non-numerical stimuli, as it was actually observed in the weekday stimuli in the present study.

For a conceptualization of the neurocognitive processes involved in mentally aligning (properties of) objects with physical space, the results collected here basically reiterate the connotations drawn from the previous dissociation of spatial associations of number and sequence (Schroeder et al., 2017b, in press). Further support for the different coding strategies comes also from a recent dissociation of parity-space and number-space associations during tDCS over left and right parietal cortices, where only the former were modulated by cathodal tDCS (Di Rosa et al., 2017). Complementary to these results, the current study underscores that the spatial alignments of both numerical and non-numerical sequences are guided by prefrontal activity, albeit with orthogonal responses to tDCS polarities. Furthermore, the data highlight a critical dissociation between numbers and months, which opposes a unified theoretical account and provokes a multiple-coding framework.

The systematic reasons outlined above can be easily arranged along the proposed model by emphasizing the flexible and situated nature of a spatial association multiple coding framework. For example, findings from a principle component analysis showed that the SNARC effects in the parity judgment and magnitude classification tasks were placed in two separate components, suggesting unrelated spatial coding mechanisms depending on the task (van Dijck et al., 2012). Moreover, the spatial coding processes underlying spatial associations of numerical magnitudes were found to change depending on task instructions (Georges et al., 2015). In general, considerable heterogeneity across stimulus, task, and participant attributes was documented in a meta-analysis (Wood et al., 2008). Moreover, the replicated dissociation between SNARC effects for number and order information is in line with the taxonomy proposed to disentangle the multitude of mechanisms involved in spatial associations (Patro et al., 2014; Cipora et al., 2015a,b). Interestingly, in right-to-left reading native Hebrew participants, reversed SNARC effects akin to the month-performance during anodal tDCS were previously obtained for sequential stimuli (but not when the magnitude dimension was emphasized in the task instruction; Shaki and Gevers, 2011). However, also in native Hebrew participants, a regular left-to-right SNARC effect for numbers was finally observed when parity-space response mappings were tested on separate days (Zohar-Shai et al., 2017). At large, this result could imply mutual interactions between markedness-based codes (parity-space association) and the active coding strategy for spatial-metric associations. Moreover, the findings by Zohar-Shai et al. (2017) also reiterate the crucial influence of seemingly trivial task design parameters, i.e., counterbalanced key assignments.

By drawing on the linguistic structure of the sequence comparison task, a relatively plausible (and testable) account was proposed by first assuming that multiple mechanisms can be dominant when generating spatial association for any concept. Loosely, this proposal also reflects the earliest models of numerical magnitude representations (e.g., the triple-code model) that assumed a verbal, visual spatial, and analog representation (Dehaene et al., 2003) and the proposal is also consistent with the traditional dual-code assumption for mental representations in general (Paivio, 1986). In greater detail, the markedness property describes the formal requirements for identification of the default member of verbal opposite pair (Nuerk et al., 2004; Proctor and Cho, 2006). A general compatibility effect then consists in better task performance in cases where either default or non-default members are present in both dimensions of a task.

The combined results also demonstrate the asymmetry of polarity-specific tDCS effects in the cognitive domain (Jacobson et al., 2012): For numerical stimuli, we found that the effect size for anodal stimulation was low (d = 0.2), and the effect turned significant only for the least conservative test across all available data from N = 72 healthy participants. Here it should be noted that the three included studies used different number ranges and classification tasks (1–9 in parity judgment or 1–5 in magnitude comparison tasks) and number notations (1–5 or “eins”-“fünf,” magnitude comparison tasks). The mechanisms beyond producing spatial associations for these different number stimuli and tasks are likely to differ (Georges et al., in press). The results of our joint analysis can only suggest that the systematic interactions of these range- and task-effects are (at best) only marginally pronounced in modulating the effect of anodal tDCS on spatial-numerical associations and they are thus less decisive given the statistically non-significant interaction effects. By analysing data drawn from three experiments, the current result shows that anodal tDCS of the left prefrontal cortex can increase the inclination of SNARC effects for numerical stimuli, but the effect size of this stimulation effect is small.

In contrast, the effect of cathodal tDCS on numerical stimuli was relatively effective in previous observations (d = 0.5; Schroeder et al., 2016). This pattern of results is remarkably opposite to the typical observation that cathodal tDCS was less effective for modulating cognition (Jacobson et al., 2012; Pirulli et al., 2014). Furthermore, DC polarity-specific effects appeared rather linear and symmetric at physiological level in motor cortex studies showing excitability changes (Priori et al., 1998; Nitsche and Paulus, 2000) and also in in vitro studies (e.g., Bikson et al., 2004). The observed patterns suggest that polarity asymmetries are more likely in cognitive tasks and that they can also render the cathodal stimulation more effective than the anodal stimulation (Schroeder and Plewnia, 2016). However, the sources of polarity asymmetry and other non-linear effects of tDCS such as current intensity and individual differences (Batsikadze et al., 2013; Benwell et al., 2015) are currently unknown and must be scrutinized systematically in future research. Finally, the results nicely illustrate that there is no binary outcome of tDCS, but that rather different mechanisms are available for producing spatial associations, which makes the stimulation more efficacious in certain cases.

The conclusions of this study must be accompanied by some caveats as some further limitations exist. First, given the non-significant positive coefficient in the month sequence during anodal tDCS, it is not clear whether anodal tDCS reverses or abolishes spatial associations of non-numerical ordinal sequences generally⁴. We believe that a reversal better describes the mechanism due to the markedness correspondence account, the significantly positive coefficient in our previous study, the positive sign of the coefficient in this study, and the dissociation with numerical symbols. In future research, it may prove fruitful to study larger stimulus ranges (e.g., longer ranges of sequence (months, letters) during anodal tDCS or single-digit numbers (1–9) during cathodal tDCS).

Secondly, as stimulus attributes and ranges may affect SNARC results, future study designs should ensure that different stimulus ranges in within-subject (within-session) experiments comprise the same numbers of items. This was not the case for the present experiments with different weekday, number, and month name ranges. Given that switching between number ranges may induce different cognitive strategies (Abrahamse et al., 2016; Huber et al., 2016), such manipulations may be studied with care.

Discrepancies in the present results may have been influenced as well by the results during the sham condition. Unlike other reports, we did not observe significantly negative SNARC effects for numbers (and for weekdays) with the 1–5 range without any stimulation (e.g., Dehaene et al., 1993; Fias et al., 1996). It should be acknowledged as well that internal consistency and reliability of the SNARC effect can be medium to low (Cipora and Wood, 2017, for simulations; Cipora and Nuerk, 2013; Viarouge et al., 2014; Georges et al., 2016; for estimates of reliability from 0.27 to 0.70), which may also influence differences between two tDCS conditions.

Several interesting tDCS parameters have not been explored with the present cognitive effects (see Schroeder et al., 2017a, for a review of different tDCS parameters in stimulation studies). It is currently unknown whether changes in intensity or electrode configuration (such as a bilateral vs. extracephalic return electrode configuration) would lead to comparable stimulation outcomes. Moreover, there are no definite data on right-hemispheric tDCS. Also here, the data of Di Rosa et al. (2017) are interesting, because they show dissociations between parity-space and number-space associations in a parietal-supraorbital configuration, independent of cortical hemisphere. However, a possible effect of laterality was not significant in their study and the electrode configuration targeted entirely different areas than the prefrontal-extracephalic configuration used in the current and in our previous studies.

In sum, we view our study as a starting point, which shows at least that the neurocognitive mechanisms underlying the SNARC seem to be not as simple as often assumed. However, because these mechanisms and the codes involved may depend on task, stimuli, participants, and stimulation parameter, we wish to acknowledge that much more research is needed to explore the generality and specificities of associations of space and different cardinal or ordinal metrics.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.