Exaggerated redundancy gain in the split brain: A hemispheric coactivation account

Abstract

Recent studies of redundancy gain indicate that it is especially large when redundant stimuli are presented to different hemispheres of an individual without a functioning corpus callosum. This suggests the hypothesis that responses to redundant stimuli are speeded partly because both hemispheres are involved in the activation of the response. A simple formal model incorporating this idea is developed and then elaborated to account for additional related findings. Predictions of the latter model are in good qualitative agreement with data from a number of sources, and there is neuroanatomic and psychophysiological support for its underlying structure.

Introduction

A fundamental phenomenon in the area of divided attention is that responses to redundant stimuli are faster than responses to single stimuli. In a simple reaction time (RT) task, for example, an observer may be asked to press a button as quickly as possible to the onset of any visual stimulus. In that task, responses will be much faster if two LEDs are illuminated simultaneously in both the left and right visual fields than if just one of these LEDs is illuminated. Analogous redundancy gains have been observed with bimodal visual and auditory stimuli (e.g., Diederich & Colonius, 1987), with stimuli in three modalities (e.g., Diederich, 1992; Todd, 1912), and with go/no-go tasks (e.g., Mordkoff & Yantis, 1991) and choice RT tasks (e.g., Grice, Canham, & Boroughs, 1984).

As reviewed briefly below, numerous studies have investigated the mechanisms responsible for redundancy gain, and a number of mathematical models for it have been proposed (for additional reviews, see Diederich, 1992; Luce, 1986; Miller & Reynolds, 2003; Townsend and Nozawa, 1995, Townsend and Nozawa, 1997). Recent studies of redundancy gain in split-brain individuals, however, have uncovered an intriguing fact that seems quite counterintuitive in terms of the existing models. Specifically, when redundant stimuli are presented several degrees to the left and right of fixation, redundancy gain is much larger in individuals without a functioning corpus callosum than in control individuals in whom this structure functions normally. This article develops an hypothesis to account for the striking increase in redundancy gain in split-brain individuals. The hypothesis is used to develop a formal mathematical model, and its predictions are in good qualitative agreement with experimental results. Before presenting the new hypothesis and developing the model, I briefly review background on models of redundancy gain and on the results of studies using split-brain individuals.

It is common in RT research to suppose that observed RTs are the sum of a decision time, $\text{D}$, and a motor time, $\text{M}$, (i.e., $\text{RT}\text{=}\text{D}\text{+}\text{M}$; cf. Luce, 1986).¹ Within this framework, it is natural to consider race models as a possible explanation of observed redundancy gains (Raab, 1962). According to race models, the time needed for deciding to respond to stimulus i is a random variable, $\text{D}{}_{\mathrm{i}}$. Across many trials, the average RT to stimulus i is simply the mean decision time, $\text{E}\text{[}\text{D}{}_{\mathrm{<mtext>i</mtext>}}\text{]}$, plus the average motor time $\text{E}\text{[}\text{M}\text{]}$. When two redundant stimuli are presented, the decision can be triggered by the first stimulus to be processed, so the decision time is $\text{D}{}_{\mathrm{r}}\text{=}\text{min}\text{(}\text{D}{}_1\text{,}\text{D}{}_2\text{)}$. In this case, the average RT is $\text{E}\text{[}\text{min}\text{(}\text{D}{}_1\text{,}\text{D}{}_2\text{)]+}\text{E}\text{[}\text{M}\text{]}$. If the distributions of $\text{D}{}_1$ and $\text{D}{}_2$ overlap—i.e., the same racer does not always win the race—then the mean decision time in redundant trials will be less than both of the single-stimulus means. Intuitively, the average time of the winner of a race is faster than the average time of either racer. Hence, race models predict redundancy gain. In addition, several more detailed predictions about mean RTs in redundant trials can be derived from such models (e.g., Ulrich & Miller, 1997).

There is evidence, however, that race models cannot fully account for the observed speedup in trials with redundant stimuli. Assuming that the decision time needed with a given stimulus is independent of whether the other stimulus is presented (i.e., context independence; cf. Luce, 1986), race models predict that the decision time for redundant stimuli, $\text{D}{}_{\mathrm{r}}$, is related as follows to the decision times with single stimuli:

$$\text{Pr}\text{(}\text{D}{}_{\mathrm{r}}\text{⩽t)=}\text{Pr}\text{(}\text{D}{}_1\text{⩽t)+}\text{Pr}\text{(}\text{D}{}_2\text{⩽t)−}\text{Pr}\text{(}\text{D}{}_1\text{⩽t∩}\text{D}{}_2\text{⩽t)}$$

for every value of t. This predictive equation cannot be evaluated in practice, because the final term $\text{Pr}\text{(}\text{D}{}_1\text{⩽t∩}\text{D}{}_2\text{⩽t)}$ depends on the correlation between $\text{D}{}_1$ and $\text{D}{}_2$, which is unobservable. Nonetheless, given that this final term is a probability and therefore nonnegative, it follows from race models (Miller, 1982) that

$$\text{Pr}\text{(}\text{D}{}_{\mathrm{r}}\text{⩽t)⩽}\text{Pr}\text{(}\text{D}{}_1\text{⩽t)+}\text{Pr}\text{(}\text{D}{}_2\text{⩽t).}$$

This predicted inequality can be tested by comparing the cumulative probability density functions (CDFs) of the RTs in the two single-stimulus conditions and the redundant-stimulus condition, F₁, F₂, and F_r respectively. Specifically, if race models are correct, it is expected that

$$\text{F}{}_{\mathrm{r}}\text{(t)⩽F}{}_1\text{(t)+F}{}_2\text{(t)}$$

for every value of t. Eq. (3) is sometimes referred to as the race model inequality. It follows immediately from inequality (2) if the motor time $\text{M}$ is assumed to be a constant, and it can also be derived if the motor time is a random variable (Ulrich & Giray, 1986).²

RT distributions observed in divided attention experiments often violate the race model inequality for relatively small values of t (e.g., Diederich, 1992; Egeth & Mordkoff, 1991; Grice et al., 1984; Miller, 1982; Mordkoff, Miller, & Roch, 1996). Specifically, there are often more very fast responses in the redundant condition than can be predicted by race models. Apparently, then, processing in redundant-stimulus trials is not simply a race between two separate processes of the same sort seen in single-stimulus trials. This fact has led a number of theorists to develop alternative models.

Coactivation models have received the most attention as alternatives to race models. In these models, redundant stimuli combine their activations rather than being processed separately (e.g., Miller, 1982; Schwarz, 1989, Schwarz, 1994). Clearly, response initiation can occur more rapidly if both stimuli contribute activation (redundant trials) than if only one does (single-stimulus trials), so coactivation models also predict redundancy gain. Moreover, coactivation models need not obey the race model inequality, so they are at least indirectly supported by observed violations of it.

Several other classes of models violating the race model inequality have also been described. For example, Mordkoff and Yantis (1991) described a modified race model with between-stimulus interactions that could produce violations of the race model inequality when there were certain contingencies in the stimulus set or task, as there often are in studies of redundancy gain (e.g., Mordkoff, 1996). In addition, Townsend and Nozawa (1997) have pointed out that serial exhaustive models can also violate the race model inequality in go/no-go and target-detection tasks if target stimuli are processed faster than nontargets.

Recent studies of redundancy gain with split-brain individuals have revealed a finding that was quite unexpected in terms of the models for redundancy gain just described.³ Specifically, when these individuals are presented with bilateral redundant visual stimuli (i.e., redundant stimuli presented to different visual hemifields and thus different hemispheres), their RTs reveal unusually large redundancy gains and unusually large violations of the race model inequality (e.g., Iacoboni, Ptito, Weekes, & Zaidel, 2000; Pollmann & Zaidel, 1999; Reuter-Lorenz, Nozawa, Gazzaniga, & Hughes, 1995). Moreover, these enhanced effects of redundancy are not observed when two redundant stimuli are presented to the same visual hemifield (i.e., same hemisphere; Iacoboni et al., 2000; Pollmann & Zaidel, 1999; Reuter-Lorenz et al., 1995). It is only with bilateral presentation that redundancy gains and violations of the race model inequality are enhanced for split-brain individuals.

The finding of enhanced redundancy gains and race model violations for split-brain individuals is quite surprising in terms of coactivation models. If redundancy has its effects because information about the two redundant stimuli is combined together to activate the response, such effects should be reduced or eliminated when such information combination is prevented. Surely, it should be especially difficult to combine information about bilateral stimuli when the connections between the two hemispheres are disrupted, as they are in split-brain individuals. Therefore, it would seem that the effects of redundancy would be much smaller than usual with these individuals—not much larger. Indeed, the actual observed enhancement of redundancy effects in split-brain individuals has been described as “astonishing” (Pollmann & Zaidel, 1999, p. 246) and “paradoxical” (Corballis, 1998, p. 1795).

Other models for redundancy gain fare no better than coactivation models in explaining its enhancement with split-brain individuals. The exhaustive serial search model of Townsend and Nozawa (1997) is not directly applicable to the simple RT in which this enhancement has been observed, because these tasks do not include nontarget stimuli. Within the contingency-based model of Mordkoff and Yantis (1991), violations of the race model inequality depend on stimulus–stimulus and stimulus–response contingencies, but it is not clear why these would be any different for split-brain individuals than for normals. Moreover, to the extent that disconnected hemispheres eliminate the between-channel interactions responsible for stimulus–stimulus contingency effects, it could be argued that this model also predicts smaller violations of the race model inequality in split-brain individuals.

To date, no completely satisfactory explanation of the split-brain results has emerged. Reuter-Lorenz et al. (1995), who were the first to report it, suggested that redundancy gain with bilateral stimuli must somehow arise during response processes. They suggested an AND-OR model in which each hemisphere produces response activation along two output pathways. One output pathway leads to an OR gate, and the response is initiated when either hemisphere generates a motor command at this gate. The other output pathway leads to an AND gate. This pathway produces response inhibition until both hemispheres are ready to generate a response (i.e., until the AND gate is satisfied), at which time the inhibition ceases. When a single lateralized stimulus is presented to a split-brain individual, only one hemisphere produces input into the AND gate, so the OR gate is inhibited throughout the trial, producing an abnormally slow response. When both hemispheres receive stimuli, though, the AND gate is satisfied, inhibition of the OR gate terminates, and a much faster response is produced. According to this model, then, redundancy gain is enhanced in split-brain individuals because their responses to single stimuli are especially slow, not because their responses to redundant stimuli are especially fast. This prediction of the model corresponds well to observed findings with split-brain individuals (i.e., substantial slowing of responses to single stimuli and approximately normal-speed responses to redundant stimuli; e.g., Pollmann & Zaidel, 1999; Reuter-Lorenz et al., 1995).

The model of Reuter-Lorenz et al. (1995) has been quite influential, because it captures well the basic phenomenon that both hemispheres must be independently stimulated to produce fast responses in split-brain individuals (cf. Corballis, Hamm, Barnett, & Corballis, 2002; Roser, 2002; Roser & Corballis, 2002). On the other hand, the model is not entirely satisfactory for three reasons. First, the model is impossible to evaluate quantitatively, because no details are given about how the stimuli activate the AND and OR gates or about how the inhibition generated from the AND gate influences the RT. Without such details it is impossible to tell, for example, whether the model predicts violations of the race model inequality in normal individuals. Second, it does not seem very parsimonious to postulate both activation and inhibition processes. RT modelers generally assume that a simple RT response is initiated when enough activation is produced (e.g., Luce, 1986). Alternatively, one could imagine a model in which a response was initiated when enough inhibition had been removed. To the extent that activation and inhibition appear to be opposites of one another, however, it seems theoretically extravagant to postulate them both. Third, the model does not address certain other important phenomena observed with split-brain individuals in redundancy gain paradigms, especially the crossed-uncrossed difference that will be discussed later.

The new model presented here is both a simplification and an extension to that of Reuter-Lorenz et al. (1995). Although the new model uses conceptual building blocks that are similar to those of the Reuter-Lorenz et al. model (i.e., AND and OR gates), these building blocks are arranged in a simpler structure involving only response activation and no response inhibition. Importantly, the simplifications make it possible to extend the model by deriving exact predictions from it for quantitative analysis of both normal and split-brain results. In addition, as will be seen later, the new model can conveniently be extended to handle other phenomena not addressed by the model of Reuter-Lorenz et al.

Before proceeding to the new model, it should be noted that several other suggestions have been put forward to explain enhanced redundancy gain in split-brain individuals. For example, Corballis (1998) suggested that the neural summation in split-brain individuals could arise in the superior colliculus, which receives sensory projections from both visual hemifields despite the absence of a corpus callosum. Involvement of the superior colliculus seemed indicated by the finding that the exaggerated coactivation for split-brain individuals was drastically reduced or eliminated when bilateral stimuli were rendered invisible to this structure by presenting them equiluminant with the background (Corballis, 1998). Although this hypothesis could clearly explain how redundancy gain might be preserved in split-brain individuals because it arose in a normal, intact neural structure, it is less clear how the hypothesis would explain why redundancy gain is increased in these individuals relative to normals. Moreover, this hypothesis has been tested by comparing the redundancy gains for bilateral stimuli in locations that were symmetric versus asymmetric with respect to fixation (Corballis et al., 2002; Roser & Corballis, 2002). Contrary to what would be predicted from the retinotopic organization of the superior colliculus, redundancy gains were not dependent on symmetry (cf. Roser, 2002). This led Corballis et al. (2002) to suggest that perhaps redundancy gains depend on summation of sensory inputs by the reticular formation, which are also preserved in split-brain individuals. Again, however, it is not clear why redundancy gain would increase in split-brain individuals relative to normals if it simply arises within a structure that remains intact.

Another possible explanation of the enhanced redundancy gain in split-brain individuals is that the corpus callosum transmits interhemispheric inhibition in normal individuals (Corballis et al., 2002). When stimuli are presented simultaneously to both visual fields (i.e., both hemispheres), the activation within each hemisphere might be reduced somewhat by inhibition from the opposite hemisphere transmitted over the corpus callosum. Thus, normals’ responses to redundant stimuli would be slower with the corpus callosum than they would have been without it, because of inhibition. Unlike the model of Reuter-Lorenz et al. (1995), however—and contrary to fact—this model seems to suggest that responses to redundant stimuli should be slower in normals (where they are slowed by inhibition) than in split-brain individuals (where they are not). As noted earlier, redundancy gain seems to be enhanced in split-brain individuals because responses to single stimuli are slowed, not because responses to redundant stimuli are speeded.

Finally, another possibility for explaining enhanced redundancy gain in split-brain individuals is based on a model suggested by Mordkoff and Yantis (1993). In brief, they proposed that multiple sources of activation within different modules (e.g., color and shape targets) produce coactivation, whereas those within a single module (e.g., two shape targets) simply race, producing no coactivation.⁴ Although Mordkoff and Yantis did not apply their model to split-brain individuals, one might argue that in these individuals the two hemispheres represent two separate modules, and thereby produce coactivation, whereas in normals the two hemispheres process visual information as a single module and produce no coactivation. In principle, this could explain the finding of enhanced redundancy gain in split-brain individuals relative to normals.

Unfortunately, there seem to be two major problems with an account based on this extension of the Mordkoff and Yantis (1993) model. First, it suggests that split-brain individuals differ from normals mainly in what happens in redundant trials, not in what happens in single-stimulus trials. Thus, it predicts that split-brain individuals should produce normal RTs in single-stimulus trials and especially fast responses in redundant trials, contrary to what has been observed (i.e., especially slow responses in single-stimulus trials and relatively normal RTs in redundant trials). Second, what is known about neural information transmission in split-brain individuals makes it hard to see how the model could work in this situation. In these individuals, the two hemispheres or modules are not connected by any fast pathways, so they cannot communicate with one another as would seem to be needed to coactivate the response.

In this article I develop a new model of the enhanced redundancy effects observed with split-brain individuals. I consider two versions of the new model. One is a simple all-or-none version from which it is possible to derive relatively convenient equations for predicting results. Although this model is not very realistic and cannot account for all of the results, it provides a relatively transparent illustration of the crucial mechanism proposed to explain the enhanced redundancy effects observed with split-brain individuals. The second is a more complex, continuous model based on the same idea, from which predictions must be obtained by computer simulation. This model can account not only for the enhanced redundancy gain but also for additional results beyond the scope of the all-or-none model and the model of Reuter-Lorenz et al. (1995).

It is possible to state simply the essential hypothesis on which both versions of the new model are based. Suppose that both cerebral hemispheres normally contribute to the initiation of a response, even with unilateral stimuli and responses (cf. Roser, 2002; Saron, Foxe, Schroeder, & Vaughan, 2003). One possibility, for example, is that the motor areas of both hemispheres influence the activation of a single keypress response, possibly summating at a spinal level. Another possibility is that the sensory areas of both hemispheres normally contribute activation to the response, possibly because of cortico–cortical interactions between homotopic sensory areas (e.g., Payne, 1986). It is not really essential to the present model whether the two hemispheres coactivate within sensory structures, motor structures, other structures (e.g., basal ganglia), or some combination of structures. Instead, the crucial feature of the new model is that both hemispheres jointly contribute in some fashion to the activation of the response.

To present a concrete example of the hemispheric coactivation hypothesis, I will temporarily make the subsidiary assumption that hemispheric coactivation occurs at a motor level—that is, that the motor areas of both hemispheres contribute to response activation. As will be considered further in Section 5, there is actually very good evidence for this assumption (see Saron et al., 2003; for a recent review), contrary to the classical position that control of a unilateral manual response resides exclusively in the motor area of the contralateral hemisphere (e.g., Poffenberger, 1912). As will also be considered in Section 5, however, the essential idea that both hemispheres contribute to the response may be correct even if the idea that their effects summate at a motor level is not.

The key idea of the new model is this: If both hemispheres normally contribute to the initiation of a response, split-brain individuals should show especially large benefits when redundant stimuli are presented to both hemispheres precisely because their interhemispheric communication is poor. For these individuals, information about the presence of a unilateral stimulus in either visual hemifield will be detected very quickly in the contralateral hemisphere to which that hemifield projects, but it will be detected very slowly by the other hemisphere, which can only get information about it via slow subcortical pathways. If a stimulus were presented to one hemisphere, then, it would take a relatively long time for information about that stimulus to become available to both hemispheres. If both hemispheres influence responses, RTs would therefore have to be especially long with unilateral stimuli, as indeed they are (e.g., Corballis, Corballis, & Fabri, 2003; Pollmann & Zaidel, 1999; Reuter-Lorenz et al., 1995). With presentation of bilateral stimuli, however, both hemispheres would quickly receive information about stimulus presentation via fast within-hemisphere pathways, without the need to wait for transmission along the slow subcortical pathways. In that case, both hemispheres could contribute to the initiation of the response relatively quickly, and responses would be fast. Under this hypothesis, then, split-brain individuals benefit greatly from simultaneous presentation of bilateral stimuli, because this condition obviates the need to use slow subcortical routes for transferring information between hemispheres. Thus, these individuals would be expected to show especially large redundancy effects.

In contrast, the hypothesis predicts that normal individuals would show much smaller redundancy gains because of the good communication between the hemispheres. When a unilateral stimulus is presented to either visual field of a normal individual, information received by one hemisphere can be transmitted quickly to the opposite hemisphere via the intact corpus callosum. Responses might be slightly faster with bilateral stimuli than with single stimuli for various reasons, as elaborated below, but the advantage for bilateral stimuli would be small because bilateral stimuli only eliminate the need for a fast callosal transmission, not for a slow subcortical one as in the split-brain individuals.