If we are to interpret quantum probability (QP) theory as a mechanistic cognitive theory, there must be some method whereby the operations postulated by QP are implemented within the human brain. Whereas this initially seems like it would require some sort of large-scale quantum effect in the brain, the target article notes that “the relevant mathematics is simple and mostly based on geometry and linear algebra” (sect. 1.2). No special quantum physics effects are needed. If we can show how neurons can compute these operations, then we can interpret QP as making strong claims about how brains reason, rather than merely acting as a novel behavioral description of the results of cognitive processing.

Interestingly, there is already a family of cognitive models that make use of geometry and linear algebra to describe cognitive mechanisms, and these share many features with QP. Vector symbolic architectures (VSAs; Gayler Reference Gayler and Slezak2003) use high-dimensional vectors to store structured information, and use algebraic operations to manipulate these representations. The closest match to QP among VSAs is holographic reduced representations (HRRs; Plate Reference Plate2003). As with QP, vectors are added to combine information and the dot product is used to evaluate similarity. For example, if there is one vector for HAPPY and another for UNHAPPY, the current mental state representation might be 0.86HAPPY+0.5UNHAPPY, representing a state more similar to HAPPY than to UNHAPPY. While the notation is different, this is identical to the |Ψ> = a|happy> + b|unhappy> example in the target article (sect. 2.1; Fig. 1a). If EMPLOYED and UNEMPLOYED are other vectors that are similar to HAPPY and UNHAPPY, respectively, then we get Figure 1b.

For the case of Figure 1c, HRRs and QP differ. Rather than using tensor products like |happy>⊗|employed> (sect 2.2.2), HRRs use circular convolution (HAPPY⊛ EMPLOYED). This was specifically introduced by Plate as a compressed tensor product: an operation that gives the same effects as a tensor product, but that does not lead to an increase in dimensionality. To explain this, consider the transition from Figure 1b to 1c in the target article. In the first case, we are dealing with a two-dimensional space, and in the second it is a four-dimensional space. How is this represented in the brain? How do neurons dynamically cope with changing dimensionality? Are different neurons used for different cases? For HRRs, these concerns are addressed by having a fixed (but large) dimensionality for all representations. If HAPPY is a particular 500-dimensional vector, and EMPLOYED is a different 500-dimensional vector, then HAPPY⊛EMPLOYED gives a new 500-dimensional vector (a tensor product would give a 250,000-dimensional vector). Importantly, in high-dimensional spaces, the newly created vector is highly likely to be (almost) orthogonal to the original vectors. This gives a close approximation to all of the required orthogonality requirements mentioned in the target article, but does not lead to an unlimited explosion of dimensions as representations get more complicated. As we have shown elsewhere, adult human vocabularies fit well within 500 dimensional spaces (Eliasmith, in press).

Given cognitive theories expressed in terms of vector symbolic architectures, we have created large-scale neural models that implement those theories. In particular, we use the neural engineering framework (NEF; Eliasmith & Anderson Reference Eliasmith and Anderson2003), which gives a principled method for determining how realistic spiking neurons can represent vectors, how connections between neurons can implement computations on those vectors, and how recurrent connections can be used to provide memory and other dynamics. This allows us to turn abstract descriptions of cognitive processing into specific brain mechanisms, connecting a plethora of neural data (functional magnetic resonance imaging [fMRI], electroencephalograms [EEG], single cell recordings) to cognitive function.

In the NEF, distributed representations of vectors are made by generalizing the standard notion of each neuron having a particular preferred direction vector (e.g., Georgopoulos et al. Reference Georgopoulos, Schwartz and Kettner1986). Whereas Hebbian learning rules can be used to adjust connection weights, we can also directly solve for the desired connection weights, as this kind of distributed representation allows a much larger range of functions to be computed in a single layer of connections than is seen in typical connectionist models. This makes it straightforward to create models that accurately compute linear operations (such as the dot product), and even more complex functions such as a full 500-dimensional circular convolution. These models are robust to neuron death and exhibit realistic variability in spiking behavior, tuning curves, and other neural properties.

Although these techniques have not yet been used on the specific tasks and theories presented in the target article, all of the operations mentioned in the article have been implemented and scaled up to human-sized vocabularies (e.g., Eliasmith Reference Eliasmith, Bara, Barsalou and Bucciarelli2005; Stewart et al. Reference Stewart, Bekolay and Eliasmith2011). Furthermore, we have shown how to organize a neural control structure around these components (based on the cortex–basal ganglia–thalamus loop) so as to control the use of these components (e.g., Eliasmith, in press; Stewart & Eliasmith Reference Stewart, Eliasmith, Carlson, Hölscher and Shipley2011). This architecture can be used to control the process of first projecting the current state onto one vector (HAPPY) and then on to another (EMPLOYED), before sending the result to the motor system to produce an output. These neural models generate response timing predictions with no parameter tuning (e.g., Stewart et al. Reference Stewart, Choo, Eliasmith, Salvucci and Gunzelmann2010), and show how the neural implementation affects overall behavior. For example, the neural approximation of vector normalization explains human behavior on list memory tasks better than the ideal mathematical normalization (Choo & Eliasmith Reference Choo, Eliasmith, Ohlsson and Cattrambone2010).

Although the NEF provides a neural mechanism for all of the models discussed in the target article, it should be noted that this approach does not require Gleason's theorem, a core assumption of QP (sect. 4.3). That is, in our neural implementations, the probability of deciding one is HAPPY can be dependent not only on the length of the projection of the internal state and the ideal HAPPY vector, but also on the lengths of the other competing vectors, the number of neurons involved in the representations, and their neural properties, all while maintaining the core behavioral results. Resolving this ambiguity will be a key test of QP.