We agree in general with the views expressed in the target article by Pothos & Busemeyer (P&B) about the application of quantum probability (QP) to cognitive science. Moreover, we believe that the mathematical framework of QP can be used as well to interpret brain network interactions underlying cognitive tasks, and that this should be further explored in the future. Here, we emphasize that the concept of functional brain network embodies notions similar to those of QP, such as entanglement, interference, and incompatibility.
Modern neuroimaging experimental designs are often based on the idea of studying a complex cognitive task as a sum of multiple sensory/cognitive factors that have corresponding processing modules in the brain (Sternberg Reference Sternberg2011). For example, to study auditory–visual integration, one would design unimodal control tasks (factors) employing visual and auditory stimuli separately, and examine the change of brain responses during presentation of combined visual–auditory stimuli (Molholm et al. Reference Molholm, Ritter, Javitt and Foxe2004). Another example comes from investigating the neural correlates of bimanual coordination: execution of movements with both limbs, for example, two hands or two feet. Here, unimanual movements (movement with one limb at a time) serve as factors for studying the neural correlates of bimanual movements (Banerjee et al. Reference Banerjee, Pillai, Sperling, Smith and Horwitz2012a; Debaere et al. Reference Debaere, Swinnen, Beatse, Sunaert, Van Hecke and Duysens2001).
A fundamental problem with factor-based approaches is that they implicitly assume that each cognitive component triggers an additional neural correlate that is the same, irrespective of the cognitive or physiological context (Banerjee et al. Reference Banerjee, Tognoli, Assisi, Kelso and Jirsa2008). For example, while studying multisensory integration, one often assumes that the same brain response during an auditory stimulus presentation is expected to be triggered in auditory areas when a visual stimulus is tagged with the auditory stimulus presentation as a combined auditory–visual stimulus (Molholm et al. Reference Molholm, Ritter, Javitt and Foxe2004). The reader of P&B will note that this assumption is identical to the concept of “compatibility,” and is not appropriate for studying the neural basis of human sensory and cognitive processes that are adaptive to even subtle changes of context (McIntosh Reference McIntosh2004).
Furthermore, an implicit assumption in factor-based neuoimaging designs is that any residual (e.g., bimanual: sum of left and right unimanual) brain activity comes from recruitment of additional brain areas related only to the missing factor not addressed by the control tasks (Banerjee et al. Reference Banerjee, Tognoli, Assisi, Kelso and Jirsa2008). Hence, the possibility of “interference” of control-related networks during the active cognitive task is ignored by factor-based designs. This approach thus disregards the “degenerate” nature of cognitive neural systems; different networks can coordinate to perform the same tasks (Price & Friston Reference Price and Friston2002; Tononi et al. Reference Tononi, Sporns and Edelman1999). Degeneracy is an omnipresent phenomenon of complex biological systems, and the search for measures to characterize degenerate neural systems is an ongoing area of research (Edelman & Gally Reference Edelman and Gally2001; Tononi et al. Reference Tononi, Sporns and Edelman1999). Current neuroscience research indicates that there are two distinct conceptualizations of neural degeneracy: structural and functional. The same neural structures can give rise to different functions (structural) and different neural structures can generate the same function (functional) (McIntosh Reference McIntosh2004; Price & Friston Reference Price and Friston2002). An example of the former was observed in an electroencephalographic (EEG) study of bimanual coordination (Banerjee et al. Reference Banerjee, Pillai, Sperling, Smith and Horwitz2012a) in which participants were asked to move their left or right fingers independently and simultaneously in different trials at specific frequencies by syncing with a rhythmic visual stimulus. During execution of stable bimanual coordination patterns, neural dynamics were dominated by temporal modulation of unimanual networks. An example of functional degeneracy occurred in an investigation of the temporal microstructure of a long-term memory retrieval process (Banerjee et al. Reference Banerjee, Tognoli, Kelso and Jirsa2012b). Here, magnetoencephalographic (MEG) data were used to examine the network recruitment of auditory areas during a visual–auditory paired associate task compared with a visual–visual paired associate task. It was found that visual–visual and visual–auditory memory recollection involved equivalent network components without any additional recruitment during an initial period of the sensory processing stage, which was then followed by recruitment of additional network components for modality-specific memory recollection.
P&B point out that the QP concept of “entanglement” plays a crucial role in understanding cognitive behavior. The mathematical framework of expressing “entanglement” involves quantifying the evolution of correlated state variables, and hence is pertinent for the study of functional brain networks. Data acquired by functional neuroimaging (i.e., functional magnetic resonance imaging [fMRI] and EEG/MEG) are being investigated by cognitive neuroscientists using a variety of network-based mathematical tools, such as functional connectivity (Beckmann et al. Reference Beckmann, DeLuca, Devlin and Smith2005; Horwitz et al. Reference Horwitz, Grady, Haxby, Ungerleider, Schapiro and Mishkin1992), effective connectivity (Friston et al. Reference Friston, Harrison and Penny2003; McIntosh et al. Reference McIntosh, Grady, Ungerleider, Haxby, Rapoport and Horwitz1994), and graph theory (Bullmore & Sporns Reference Bullmore and Sporns2009). A critical point with respect to functional networks is that as all nodes are functionally linked to one another, any alteration in even a single link results in alterations throughout the network (Kim & Horwitz Reference Kim and Horwitz2009). In essence, one can say that brain network nodes (regions) that may not be anatomically connected and are physically separated by large distances can become “entangled” with one another to facilitate task execution. This aspect of network behavior has important ramifications for the way in which functional brain imaging data are interpreted in comparing tasks, or patients and healthy subjects (Kim & Horwitz Reference Kim and Horwitz2009).
To conclude, we think that the insights that QP brings to cognitive science, as reviewed by P&B, are likely to be as significant when used for interpretation of brain network analysis. An examination of the recent brain imaging literature suggests that such an approach already is under way (Banerjee et al. Reference Banerjee, Pillai, Sperling, Smith and Horwitz2012a; Reference Banerjee, Tognoli, Kelso and Jirsa2012b; Noppeney et al. Reference Noppeney, Friston and Price2004). Although the words used by functional brain imagers to describe their methods differ from the language of QP, we have attempted to point out that there are strong similarities in the fundamental mathematical problem each strives to solve.
We agree in general with the views expressed in the target article by Pothos & Busemeyer (P&B) about the application of quantum probability (QP) to cognitive science. Moreover, we believe that the mathematical framework of QP can be used as well to interpret brain network interactions underlying cognitive tasks, and that this should be further explored in the future. Here, we emphasize that the concept of functional brain network embodies notions similar to those of QP, such as entanglement, interference, and incompatibility.
Modern neuroimaging experimental designs are often based on the idea of studying a complex cognitive task as a sum of multiple sensory/cognitive factors that have corresponding processing modules in the brain (Sternberg Reference Sternberg2011). For example, to study auditory–visual integration, one would design unimodal control tasks (factors) employing visual and auditory stimuli separately, and examine the change of brain responses during presentation of combined visual–auditory stimuli (Molholm et al. Reference Molholm, Ritter, Javitt and Foxe2004). Another example comes from investigating the neural correlates of bimanual coordination: execution of movements with both limbs, for example, two hands or two feet. Here, unimanual movements (movement with one limb at a time) serve as factors for studying the neural correlates of bimanual movements (Banerjee et al. Reference Banerjee, Pillai, Sperling, Smith and Horwitz2012a; Debaere et al. Reference Debaere, Swinnen, Beatse, Sunaert, Van Hecke and Duysens2001).
A fundamental problem with factor-based approaches is that they implicitly assume that each cognitive component triggers an additional neural correlate that is the same, irrespective of the cognitive or physiological context (Banerjee et al. Reference Banerjee, Tognoli, Assisi, Kelso and Jirsa2008). For example, while studying multisensory integration, one often assumes that the same brain response during an auditory stimulus presentation is expected to be triggered in auditory areas when a visual stimulus is tagged with the auditory stimulus presentation as a combined auditory–visual stimulus (Molholm et al. Reference Molholm, Ritter, Javitt and Foxe2004). The reader of P&B will note that this assumption is identical to the concept of “compatibility,” and is not appropriate for studying the neural basis of human sensory and cognitive processes that are adaptive to even subtle changes of context (McIntosh Reference McIntosh2004).
Furthermore, an implicit assumption in factor-based neuoimaging designs is that any residual (e.g., bimanual: sum of left and right unimanual) brain activity comes from recruitment of additional brain areas related only to the missing factor not addressed by the control tasks (Banerjee et al. Reference Banerjee, Tognoli, Assisi, Kelso and Jirsa2008). Hence, the possibility of “interference” of control-related networks during the active cognitive task is ignored by factor-based designs. This approach thus disregards the “degenerate” nature of cognitive neural systems; different networks can coordinate to perform the same tasks (Price & Friston Reference Price and Friston2002; Tononi et al. Reference Tononi, Sporns and Edelman1999). Degeneracy is an omnipresent phenomenon of complex biological systems, and the search for measures to characterize degenerate neural systems is an ongoing area of research (Edelman & Gally Reference Edelman and Gally2001; Tononi et al. Reference Tononi, Sporns and Edelman1999). Current neuroscience research indicates that there are two distinct conceptualizations of neural degeneracy: structural and functional. The same neural structures can give rise to different functions (structural) and different neural structures can generate the same function (functional) (McIntosh Reference McIntosh2004; Price & Friston Reference Price and Friston2002). An example of the former was observed in an electroencephalographic (EEG) study of bimanual coordination (Banerjee et al. Reference Banerjee, Pillai, Sperling, Smith and Horwitz2012a) in which participants were asked to move their left or right fingers independently and simultaneously in different trials at specific frequencies by syncing with a rhythmic visual stimulus. During execution of stable bimanual coordination patterns, neural dynamics were dominated by temporal modulation of unimanual networks. An example of functional degeneracy occurred in an investigation of the temporal microstructure of a long-term memory retrieval process (Banerjee et al. Reference Banerjee, Tognoli, Kelso and Jirsa2012b). Here, magnetoencephalographic (MEG) data were used to examine the network recruitment of auditory areas during a visual–auditory paired associate task compared with a visual–visual paired associate task. It was found that visual–visual and visual–auditory memory recollection involved equivalent network components without any additional recruitment during an initial period of the sensory processing stage, which was then followed by recruitment of additional network components for modality-specific memory recollection.
P&B point out that the QP concept of “entanglement” plays a crucial role in understanding cognitive behavior. The mathematical framework of expressing “entanglement” involves quantifying the evolution of correlated state variables, and hence is pertinent for the study of functional brain networks. Data acquired by functional neuroimaging (i.e., functional magnetic resonance imaging [fMRI] and EEG/MEG) are being investigated by cognitive neuroscientists using a variety of network-based mathematical tools, such as functional connectivity (Beckmann et al. Reference Beckmann, DeLuca, Devlin and Smith2005; Horwitz et al. Reference Horwitz, Grady, Haxby, Ungerleider, Schapiro and Mishkin1992), effective connectivity (Friston et al. Reference Friston, Harrison and Penny2003; McIntosh et al. Reference McIntosh, Grady, Ungerleider, Haxby, Rapoport and Horwitz1994), and graph theory (Bullmore & Sporns Reference Bullmore and Sporns2009). A critical point with respect to functional networks is that as all nodes are functionally linked to one another, any alteration in even a single link results in alterations throughout the network (Kim & Horwitz Reference Kim and Horwitz2009). In essence, one can say that brain network nodes (regions) that may not be anatomically connected and are physically separated by large distances can become “entangled” with one another to facilitate task execution. This aspect of network behavior has important ramifications for the way in which functional brain imaging data are interpreted in comparing tasks, or patients and healthy subjects (Kim & Horwitz Reference Kim and Horwitz2009).
To conclude, we think that the insights that QP brings to cognitive science, as reviewed by P&B, are likely to be as significant when used for interpretation of brain network analysis. An examination of the recent brain imaging literature suggests that such an approach already is under way (Banerjee et al. Reference Banerjee, Pillai, Sperling, Smith and Horwitz2012a; Reference Banerjee, Tognoli, Kelso and Jirsa2012b; Noppeney et al. Reference Noppeney, Friston and Price2004). Although the words used by functional brain imagers to describe their methods differ from the language of QP, we have attempted to point out that there are strong similarities in the fundamental mathematical problem each strives to solve.
ACKNOWLEDGMENT
This work was supported by the Intramural Research Program of the National Institute on Deafness and Other Communications Disorders.