On the interaction of numerical and size information in digit comparison: a behavioral and event-related potential study

Abstract

To investigate effects of dimensional congruity in digit comparison processes we recorded response latencies, error rates and event-related potentials (ERPs) in a situation when the relevant and the irrelevant dimension of judgement, the digits physical size or its numerical value, varied unpredictably from trial to trial. Congruity effects may arise at early processing stages, where both numerical and size information are mapped onto a common, integrated representation. Alternatively, numerical and size information may be processed in functionally independent channels and interact only at a later stage of response activation.

For both, numerical and size comparisons, reliable distance effects on ERPs (starting 224 ms post-stimulus at frontal electrode sites) and on behavioral data were observed. Congruity effects follow a timecourse that mirror-images the temporal pattern found for the distance effects: while distance effects on difference potentials start about 64 ms earlier for numerical comparisons, congruity effects due to the irrelevant attribute are seen about 88 ms earlier for size comparisons. While Lateralized Readiness Potentials (LRPs) for incongruent trials are shifted in time, there was no indication of an activation for the incorrect response in incongruent trials, as an account of congruity effects in terms of response competition would predict. The most parsimonious explanation of our results is that numerical and size information are extracted in parallel, are both converted into an integrated representation and potentially interact during these early parallel stages.

Introduction

Humans are extremely efficient in processing and comparing symbolic information such as digits representing distinct numerical values. Judgements of numerical magnitude involve symbolic information because the relevant dimension—the abstract numerical value—cannot be deduced from the graphical appearance of the digits as such; rather, their numerical meaning relies on previously learned conventions and must be held in, and retrieved from, memory in order to successfully perform tasks such as digit comparison. Moyer and Landauer [19]first demonstrated that the time to determine the numerically larger out of two simultaneously presented digits systematically decreases with their numerical difference (the numerical distance effect; cf. 3, 4, 5, 6, 18). To account for this effect, Moyer and Landauer [19]proposed that the digits are automatically converted to percept-like analog quantities and that these analog representations are then in turn processed much like sensory representations of stimuli which differ along some extensive physical dimension.

One line of support for this hypothesis derives from Stroop-type interference experiments in which the digits are presented with varying physical (i.e. font) sizes. Given the hypothesis of an analog representation of numerical magnitude, one would expect facilitation or interference, depending on whether the relation of physical size and numerical value is congruent, or not. This size congruity effect predicted by the analog representation model was indeed first obtained by Besner and Coltheart [1]. In principle, their results are in line with a serial processing model in which the physical features of the digit (including its physical size) are first identified before the digits numerical value is retrieved from memory and is then compared to the numerical value of the other digit (or to a fixed standard) to activate the appropriate response. According to the serial view, the output of the earlier stage of physical feature encoding may influence subsequent processes during later stages at which numerical information is retrieved and compared, but not vice versa. In particular, the digits physical size may well influence processing of numerical information, but according to the serial model the numerical value of a digit should not in turn influence physical size comparisons.

Henik and Tzelgov [11]and Tzelgov et al. [23]first studied the complementary task of judging the digits physical size, independent of its numerical value. Contrary to the expectations from the serial model, they found that the (irrelevant) numerical value of the digit induced systematic congruity effects on the latency of physical size judgements. These findings of bidirectional, mutual interference of size and numerical information strongly argue against any strictly serial model, which we will, therefore, not consider further in the sequel. Even so, two broad interpretations in terms of parallel processing remain, which differ in their assumptions about the locus of the observed congruity effects.

Under one interpretation (Fig. 1a), extending Moyer and Landauers original proposal, both, the digits physical size and its numerical value are first mapped onto an integrated internal analog representation, which is then processed further to activate the appropriate response. According to this view, congruity effects with both, numerical and size comparisons occur at an early processing stage where both, size and numerical information are converted into an integrated representation and potentially interact with each other. Under this model, the digits attributes have no individual access to the stage of response activation; rather, they must first pass a processing stage at which the separate information from the two attributes is integrated into a single representation.

An alternative account (Fig. 1b) of size congruity effects, however, is that size and numerical information are first processed in parallel, functionally independent channels and that both of them can separately activate a specific subresponse. Therefore, no congruity effects are predicted for early processing stages such as digit identification. Under this interpretation, both digit attributes have independent access to the stage of response activation and they potentially interact only during these later processing stages.

To further investigate these alternative accounts of congruity effects we employed a variation of a paradigm introduced by Sudevan and Taylor [22](see also [20]). In this paradigm, a cue serves to define the task-relevant stimulus attribute for each particular trial. In the present study, the cue indicated to the subjects to either judge the physical size, or the numerical value, of a single digit presented shortly thereafter. For numerical comparisons, subjects had to decide whether the numerical value of the presented digit was smaller or larger than a fixed numerical standard, which was equal to 5. The test digit could be 3, 4, 6 or 7, corresponding to a numerical distance from the standard of 1 (digits 4, 6) vs 2 (digits 3, 7). Likewise, for physical size comparisons, subjects had to decide whether the physical size of the presented digit was smaller or larger than a standard size (called size c) to which they were practiced before, during training blocks. The size of the test digit could be (in ascending order of size) a, b, d, or e, corresponding to a size distance from the size standard c of 1 (sizes b, d) vs 2 (sizes a, e). Thus, each digit was characterized by (i) its numerical value, and (ii) its physical size. A specific digit is called congruent if both, its physical size and its numerical value, would yield the same (correct) response under the two types of comparison, otherwise it is incongruent. For example, the digit 7 would be a congruent stimulus if presented with size e, but an incongruent stimulus, if presented with size a. Conversely, the digit 3 is congruent if presented with size a, but incongruent with size e. Note, that the congruity of a specific combination of numerical value and physical size is independent of the task-relevant attribute, so that estimates of congruity effects for the two types of comparisons are based on the same subsets of physical stimuli.

For the paradigm described, let us now reconsider the hypothesis that processing of physical size and numerical value interferes at an early stage, when both, size and numerical information are mapped onto an integrated representation (Fig. 1a). Suppose, processing of one of the two stimulus attributes, say, of numerical value, shows, on average, a speed advantage over judgements of the other attribute, physical size. If the onsets of the induced effects could be suitably monitored, we should then expect, first, earlier differential effects of numerical distance 1 vs numerical distance 2 for numerical comparisons, as compared to differential effects of size distance 1 vs size distance 2 for size comparisons. This pattern would simply reflect our hypothetical assumption that numerical information is, on average, processed faster than is size information. At the same time, this assumption would imply that for size comparisons the information about the irrelevant digit attribute (i.e. numerical magnitude) is, on average, available earlier than information about the relevant attribute, physical size. Thus, congruity effects should show up earlier for size comparisons as compared to numerical comparisons: for size comparisons, the irrelevant, but faster, attribute (numerical magnitude) could already exert its influence while the relevant attribute (physical size) is typically still being processed, whereas for numerical comparisons it is the relevant attribute which (over repeated trials) would usually have shorter finishing times. A similar reasoning would apply to the case that size information is, on average, processed faster than is numerical information: earlier distance effects for size judgements should then go together with earlier congruity effects for numerical judgements. In summary, if size and numerical information interact at an early stage we would expect that the order of onsets of congruity effects mirror-images the order of onsets found for the distance effects of the respective irrelevant dimension.

Consider now the alternative hypothesis that numerical and size information are represented and processed in functionally independent channels (Fig. 1b). Under this assumption, no differential congruity effect should occur before the stage of response activation; in particular, no systematic relation is predicted for the relative onset of distance and congruity effects during earlier stages preceding response activation. Furthermore, suppose that the onset and timecourse of response activation could be suitably monitored and that the subresponse suggested by, say, the numerical information, on average, accesses this stage earlier than the subresponse suggested by the size information. For size comparisons we would then expect an initial activation towards the incorrect response with incongruent stimuli, due to the earlier activation in response to the numerical information. At the same time, for congruent stimuli the individual subresponses to size and numerical information suggest the same overt response. Also, both effects should be absent or attenuated for numerical judgements, because, under our hypothetical assumption of faster numerical processing, the slower subreponse suggested by the size information will, on average, arrive too late to contribute an initial correct (congruent trials) or incorrect (incongruent trials) response activation.

The present study tries to distinguish between these contrasting predictions using event-related potentials (ERPs). ERPs provide a continuous record of synchronous neural activity which can aid in tracing the onset and timecourse of the processing stages that intervene between stimulus and response (cf. [10]). In particular, differences between ERPs can be used to monitor the onset and timecourse of electrophysiological effects attributable to variations of the independent variables, such as type of judgement, distance-to-the-standard and congruity in the present case. Furthermore, specific components of the ERP can be related to stimulus evaluation processes, while other components are thought to reflect response-related processes.

Distance effects are systematically related to a particular positive-going component of the ERP centered around the central parietal electrode site (Pz), the P300 5, 8, 9, 16, 24. The P300 usually shows a clearly identifiable peak between 300 and 700 ms after stimulus onset and its latency varies systematically with factors which influence the duration of stimulus evaluation processes such as stimulus discriminability or numerical distance in the present case. A recent literature review [25]indicates that the usefulness of P300 latency as an index of stimulus evaluation processes (cf. 13, 14, 15) differs considerably across experiments, although for the present case of digit comparison previous work 5, 8, 9, 16, 24has clearly established that distance effects are well reflected by characteristics of the P300.

Effects of selective response activation can be monitored by the so-called Lateralized Readiness Potential (LRP), which is related to the preparation for, and execution of, a motor movement. The LRP is derived from the difference of potentials measured at the left and right central brain sites C3′ and C4′ above the motor cortex; it is based on the fact that a maximum negativity occurs contralateral to the responding hand. By comparing the difference potentials C3′–C4′ under right-hand responses and left-hand responses main lateralization effects not related to response processes are subtracted out, so that the LRP monitors selective response activation (for detailed accounts of the LRP, see 2, 7, 17, 21).

Let us now reconsider the models discussed previously in terms of difference potentials, P300 and LRP. Under the hypothesis of an early and common locus of size distance and numerical distance effects (Fig. 1a) we would expect that (a.i) numerical distance has a main effect on early difference potentials and P300 for numerical comparisons, (a.ii) size distance has a main effect on early difference potentials and P300 for size comparisons, (a.iii) congruity has a main effect on early difference potentials and P300 in both tasks, which should (a.iv) mirror-image the temporal order of effect onsets found in (a.i) and (a.ii); finally, (a.v) congruity effects may delay the LRP onset along the time axis, but should not lead to an initial deflection towards the incorrect response.

On the other hand, if numerical and size information is processed in functionally independent channels (Fig. 1b) and interact only via response competition, we expect that (b.i) congruity effects should not show up in early difference potentials and P300—even if size distance and numerical distance individually do induce main effects on difference potentials, or P300. Furthermore, (b.ii) the faster processed irrelevant attribute should induce an initial deflection of the LRP towards the incorrect response in incongruent trials. Conversely, (b.iii) the slower processed irrelevant attribute should not induce an initial deflection of the LRP towards the incorrect response in incongruent trials, although it may contribute incorrect response activation after the LRP onset for the correct response. Finally (b.iv), congruent trials should never induce an initial deflection of the LRP towards the incorrect response, but the LRP may onset earlier because both attributes co-activate the same subresponse. Clearly, because part of predictions (b.iii) and (b.iv) is shared by both models, special attention will be paid to predictions (b.i), and (b.ii). We also note that predictions (a.i) and (a.ii) of the early interaction model are also compatible with a response competition account and thus do not by themselves discriminate between the two models.