Bidirectional Modulation of Numerical Magnitude

Abstract

Numerical cognition is critical for modern life; however, the precise neural mechanisms underpinning numerical magnitude allocation in humans remain obscure. Based upon previous reports demonstrating the close behavioral and neuro-anatomical relationship between number allocation and spatial attention, we hypothesized that these systems would be subject to similar control mechanisms, namely dynamic interhemispheric competition. We employed a physiological paradigm, combining visual and vestibular stimulation, to induce interhemispheric conflict and subsequent unihemispheric inhibition, as confirmed by transcranial direct current stimulation (tDCS). This allowed us to demonstrate the first systematic bidirectional modulation of numerical magnitude toward either higher or lower numbers, independently of either eye movements or spatial attention mediated biases. We incorporated both our findings and those from the most widely accepted theoretical framework for numerical cognition to present a novel unifying computational model that describes how numerical magnitude allocation is subject to dynamic interhemispheric competition. That is, numerical allocation is continually updated in a contextual manner based upon relative magnitude, with the right hemisphere responsible for smaller magnitudes and the left hemisphere for larger magnitudes.

Keywords: VOR; dynamic interhemispheric competition; mental number line; numerical magnitude; vestibular cognition.

Figure 1.

Experimental setup for number pair bisection and clock drawing (i.e., motor transformation) tasks. (A) Subjects lay supine with the head tilted up by 30° and with the knees flexed at 45°. The BR (“RIV”) was delivered using afterimages. A board was rested on the subject's thighs to provide writing support for the clock drawings. (B) Caloric irrigation (either cold 30°C or warm 44°C water irrigations) were applied to either the right (R) or left (L) ear for a duration of 40 s. Immediately at the end of the caloric irrigation, subjects performed either the mental number pair bisection task or clock drawings (Experiment 1). The vestibular activation in response to a caloric evokes nystagmus at around 20 s as represented by slow phase velocity (SPV) eye movement trace (dashed line; schematically drawn based on our normative data) (Experiment 2). In the CALORIC + RIV condition, the BR (“RIV”) was applied before the onset of the caloric and lasted for the entire duration of the task.

Figure 2.

Results from mental number pair bisection experiments following physiological manipulations. We present the mean % bisection error from the midpoint of the numerical interval. (A) “Caloric + RIV” condition (gray diamonds) resulted in subjects significantly underestimating the midpoint (i.e., shift to the left as indicated by red arrow) compared with “Caloric-only” (black diamonds) condition following RIGHTCOLD + RIV (LOWER PANEL), but no effect was found during LEFTCOLD + RIV (upper panel). (B) During LEFTWARM + RIV, the subjects demonstrated a significant shift toward larger numbers (i.e., rightward shift as indicated by red arrow), suggesting overestimation of the midpoint (upper panel). No effect of RIGHTWARM + RIV was observed (lower panel). Gray-shaded area in panels indicates 95% confidence limits calculated from baseline measures (i.e., no caloric or BR stimulation). Dashed line at 0 corresponds to 0% error, i.e., accurate bisection. Data marked ** are significant at P < 0.01; data marked * are significant at P < 0.05. Error bars indicate standard errors.

Figure 3.

Heat maps (upper panel) illustrate numerical clock-drawing performance in caloric-only conditions (RIGHTCOLD on top left; LEFTWARM on top right) with corresponding ‘+RIV’ conditions below. Center of mass results are displayed in the lower panel: (A) (left panel) Following RIGHTCOLD + RIV when subjects were asked to draw the clocks clockwise (CW), a significant shift to the right (indicated by red arrow) is seen in the “Caloric + RIV” condition (gray diamonds) compared with “caloric alone” (black diamond) condition. (B) (right panel) Following LEFTWARM + RIV when subjects were asked to draw the clock anticlockwise (ACW), a significant shift to the left (indicated by red arrow) was observed in “Caloric + RIV” (grey diamonds) compared with “caloric alone” (black diamond) condition. Gray-shaded area in lower panels indicates 95% confidence limits calculated from baseline measures (i.e., no caloric or BR stimulation). Dashed line at 0.5 indicates the midline of a perfectly symmetrical clock. Data marked ** are significant at P < 0.01. Error bars indicate standard errors.

Figure 4.

Schematic model illustrating proposed hemispheric activation in the Caloric + RIV condition. The perceptual switching in BR (RIV) is proposed to activate the right hemisphere (gray circle). Hemispheric activations following caloric stimulation are shown by the red circle following warm irrigations or by blue circles following cold irrigations. The labyrinth represents the side of the caloric irrigation. The size of the circles illustrates the relative degree of the activation. (A) In the RIGHTCOLD + RIV condition, the hemispheres are in conflict; however, the right hemisphere exerts a predominant effect (as shown by the relative thickness of the arrows). The interhemispheric conflict is not present during the RIGHTWARM + RIV condition as the right hemisphere is preferentially activated by both the visual and vestibular stimuli. (B) Similarly, no conflict is present in LEFTCOLD + RIV condition, whereas during the LEFTWARM + RIV condition conflict presents, but critically here the left hemisphere exerts a greater influence during the interhemispheric conflict.

Figure 5.

Asymmetrical modulation of the VOR during combined caloric irrigation and rivalry-viewing. On the y-axis, we represent the mean % change in peak SPV when comparing the CALORIC alone condition with the corresponding CALORIC + RIV condition. On the x-axis, we have represented the different conditions, namely cold or warm water irrigations of either the right (dark gray bar) or left (light gray bar) ear. Note that we observe a marked suppression of the VOR for the following conditions, RIGHTCOLD + RIV compared with RIGHTCOLD irrigations alone and LEFTWARM + RIV compare with LEFTWARM irrigations alone. No suppression of the VOR was observed when comparing LEFTCOLD + RIV with LEFTCOLD irrigations alone or RIGHTWARM + RIV to RIGHTWARM irrigations alone. Data marked *** are significant at P < 0.001. Error bars indicate standard error.

Figure 6.

Probing the neural correlates of the asymmetrical VOR modulation following combined CALORIC + RIV stimulation using tDCS. (A) Top panel represents the results from the cold water irrigations (i.e., Group 1). On the y-axis, we represent the mean % change in peak SPV when comparing the CALORIC alone condition with the corresponding CALORIC + RIV condition. On the x-axis, we have represented the different conditions of either the right (dark gray bar) or left (light gray bar) ear cold water irrigations following unipolar left anodal, left cathodal, right anodal, or right cathodal stimulation. Note that for RIGHTCOLD + RIV, we only observed asymmetries of the VOR following unipolar right hemisphere anodal stimulation and unipolar left hemisphere cathodal stimulation. Note that the asymmetries in the VOR during RIGHTCOLD + RIV were attenuated following either unipolar anodal stimulation of the left hemisphere or unipolar cathodal stimulation of the right hemisphere. (B) Lower panel represents the results from the warm water irrigations (i.e., Group 2). Again on the y-axis, we represent the mean % change in peak SPV when comparing the CALORIC alone condition with the corresponding CALORIC + RIV condition. On the x-axis, we have represented the different conditions of either the right (dark gray bar) or left (light gray bar) ear warm water irrigations following unipolar left anodal, left cathodal, right anodal, or right cathodal stimulation. Note that for LEFTWARM + RIV, we only observed asymmetries of the VOR following either unipolar left hemisphere anodal stimulation or unipolar right hemisphere cathodal stimulation. Note that the asymmetries in the VOR during LEFTWARM + RIV were attenuated following either unipolar anodal stimulation of the right hemisphere or unipolar cathodal stimulation of the left hemisphere. Data marked ** are significant at P < 0.01. Error bars indicate standard errors.

Figure 7.

Relationship between numerical perceptual biases and degree of VOR suppression. (A) On the x-axis, we present the degree of vestibular nystagmus suppression (i.e., % change in SPV) between right cold caloric alone and RIGHTCOLD + RIV. On the y-axis, we represent the number pair bisection error (%). We observed a significant negative correlation between the number pair bisection error (i.e., bias toward smaller numbers) and the degree of vestibular nystagmus suppression. That is, those individuals who exhibited a larger bias toward smaller numbers during RIGHTCOLD + RIV also demonstrated a larger degree of vestibular nystagmus suppression. (B) On the x-axis, we present the degree of vestibular nystagmus suppression (i.e., % change in SPV) between left warm caloric alone and LEFTWARM + RIV. On the y-axis, we represent the number pair bisection error (%).We observed a significant positive correlation between the number pair bisection error (i.e., bias toward larger numbers) and the degree of vestibular nystagmus suppression. That is, those individuals who exhibited a more pronounced bias toward larger numbers demonstrated greater vestibular nystagmus suppression.

Figure 8.

Relationship between lateral shifts observed during numerical clock drawings and degree of VOR suppression. On the x-axis, we represent the degree of vestibular nystagmus suppression and on the y-axis we represent the relative shift in the center of mass (arbitrary units). (A) For RIGHTCOLD + RIV, we observed a positive correlation, in that those individual who exhibited greater VOR asymmetries had larger shifts in the center of mass to the right. (B) For LEFTWARM + RIV, we also observed a positive correlation, in that those individual who exhibited greater VOR asymmetries had larger shifts in the center of mass to the left.

Figure 9.

Summary of the results for the effects of frontal tDCS upon numerical magnitude allocation. On the y-axis, we represent the mean change in bisection error (%) when comparing caloric alone with the corresponding CALORIC + RIV condition, either before (dark gray bars) or after (light gray bars) application of tDCS. On the x-axis, we represent the 4 different tDCS stimulation paradigms implemented. (A) No effect of tDCS upon number pair bisection was observed in any of the 4 stimulation conditions during RIGHTWARM + RIV. (B) For LEFTWARM + RIV, there was a bias toward larger numbers before application of tDCS, which was abolished following unipolar right anodal and left cathodal stimulation. Notably, this bias toward larger numbers was augmented following either unipolar left anodal or right cathodal stimulation. (C) During RIGHTCOLD + RIV, there was a bias toward smaller numbers before tDCS; however, this bias was abolished following either unipolar left anodal stimulation or right cathodal stimulation and augmented following either unipolar right anodal or left cathodal stimulation, respectively. (D) No effect of tDCS was observed upon number pair bisection in any of the 4 stimulation conditions during LEFTCOLD + RIV. Data marked * significant at P < 0.001. Error bars indicate standard errors.

Figure 10.

Computational modelling. The figure illustrates the probability distribution p(x; l,r) that occurs for several different values of l where the following parameters were implemented in the model r = 3.0 and β = 1.