Predictive brain represents an alternative to the orthodox framework based on the general assumption that sensory inputs “ascend” by integrating the complexity of perceptual experience. Predictive brain proposes an inverse explanation as “the brain became an organ of inference, actively constructing explanations for what’s going on ‘out there’, beyond its sensory epithelia” (Friston, 2018, p. 1019). Predictive brain challenges the general assumption that the brain is a device that extracts knowledge from sensations, and proposes a unifying theory of cortical function postulating that the core function of the brain is the minimization of the prediction error, which represents the mismatch between the predicted input and the actual evidence (Rao and Ballard, 1999; Friston, 2009; Clark, 2013). In this view, the brain is composed of a hierarchy of layers, where each layer makes predictions about the layers below in the hierarchy. A prediction error—the information that, at each layer, is not successfully predicted—is formed through a comparison between the downward descending predictions and the activity of the layer. Prediction errors serve as input to higher levels and can be used to update the mental model of the world. Generally speaking, by minimizing the prediction error, the brain performs perceptual inference and learning, as predictions are instantiated at multiple scales and hierarchy levels, from the low-level variation of sensory data to high-level models of causes of sensory data (Clark, 2015). Prediction error minimization is affected by precision, related to the “signal to noise ratio” in updating mechanisms. Attention can be considered as the inference of precision in hierarchical perception (Feldman and Friston, 2010; see also Hohwy, 2012; Yon and Frith, 2021). In more detail, prediction error minimization passes through the inference of hidden external states (perception) through updating the model of the world (learning) or through the actions employed to conform to predictions. Despite the elegance of the predictive brain framework, its neurobiological dimension is nowadays subject to increasing investigation, as the evidence is still mixed (Walsh et al., 2020).

The principles of predictive brain present important theoretical antecedents—in particular, Kant’s a priori knowledge, Helmholtz’s unconscious inference, and the notion of feedback control in cybernetics. The modern mathematical formalization of predictive brain—also known as predictive coding (Huang and Rao, 2011)—is relatively recent. After the seminal article of Rao and Ballard (1999), such formalization passed through the recognition that predictive coding could be approximated by Bayesian inference based upon Gaussian generative models (e.g., Friston, 2003, 2008). As discussed by Friston (2012, p. 1230): “Bayesian brain is a corollary of the free energy principle, which says that any self organizing system (like a brain or neuroimaging community) must maximize the evidence for its own existence, which means it must minimize its free energy using a model of its world.” The current mathematical formalization of predictive brain is not secondary to its theoretical tenets and its well-developed formalism often represents a barrier for the outsiders. Maximizing Bayesian evidence (or minimizing variational free energy) has been described as a gradient descent minimization of variational free energy (Friston et al., 2017), able to account for several phenomena such as “repetition suppression, omission responses, violation responses, place cell activity, phase precession, theta sequences, theta-gamma coupling, evidence accumulation, race-to-bound dynamics, and transfer of dopamine responses” (Friston et al., 2017, p. 37). Generally speaking, the minimization of variational free energy allows a parsimonious but accurate description of observable outcomes of different types.

Bayesian inference does not necessarily need to be implemented by predictive coding (Aitchison and Lengyel, 2017). Millidge et al. (2021) propose a review of the mathematical components of predictive coding. Predictive coding can be considered, in its foundational principle, as a form of variational inference (under Gaussian assumptions) where multi-layered topology, dynamical properties, the multiplicity of stimulus, and precision are essential factors. Interestingly, Millidge et al. (2021) discuss the future directions of research on predictive coding, identifying the following domains: the plausible existence of non-Gaussian but discrete generative models at the highest cognitive levels, the necessity of formalizing the basis of long term memory involved in cortical working, the opportunity of considering the actual architecture of the cortex (which could not be fully connected through hierarchical layers, but presents sparse connectivity and “columnar” structures), and the implementation of precision consistent with known neurobiological mechanisms. Furthermore, they identify several machine learning algorithms that could be comparatively used to shed light on updating mechanisms involved in predictive coding.

Current literature on predictive coding (extensively discussed by Millidge et al., 2021) never raises the more foundational issues about the very nature of probability, assumed in predictive coding. In particular, current research in predictive brain does not consider the existence of alternative probabilistic frameworks, which are not based on Kolmogorov’s probability axioms and Boolean logic, but informed by quantum principles. Among the exceptions, Fields et al. (2021) discuss quantum systems as observers assigning semantics to observational outcomes; the authors emphasize that context-switching challenges the classical formulation of the free-energy principle so as to justify a quantum formulation. Such a quantum approach represents a novel and promising perspective for future research on the free energy principle. Generally speaking, quantum mechanics can be used to solve the so-called “Bayesian blur” problem—related to the contrast between the univocal nature and the probabilistic roots of human experience (Clark, 2018)—where consciousness is considered a “collapsing agent” (Safron, 2021), which is consistent with von Neumann & Wigner’s interpretation, according to which, consciousness is necessary for quantum measurement (von Neumann, 1932/1955; Wigner and Margenau, 1967).

The QPB—presented in the next section through a stylized model—aims to update the current view of the predictive brain through a quantum probabilistic framework.

In each instant, an organism is in a state |S⟩, which is a unit-length vector, expressed in bra-ket notation, in an N-dimensional Hilbert space, representing the states of the system. A Hilbert space is a generalization of Euclidean space and was originally used to formalize quantum mechanics (for a detailed formulation of the nature of events and their properties in a Hilbert space, see Busemeyer and Bruza, 2012, Sections 2.1 and 2.5). The N dimensions of this space represent potential states that are possible for an organism and the unit-length vector state represents the state of the system. In our model, we do not consider that the dimensions could be infinite and include real or complex coefficients; neither do we specify additional constraints. This deliberate simplification is consistent with the geometric approach (proposed by Busemeyer and Bruza, 2012, Section 2.1) and is suitable to introduce quantum cognition to non-experts (as we assume that predictive coding scholars are not versed in quantum formalism).

The dimensions of the system must be conceptualized as the possible values of the neurobiological degrees of freedom, where events can be of two general types: top-down predictions or bottom-up evidence. As we explain in the following sections, the fundamental idea of a quantum view of predictive brain is that this state admits incompatible—complementary—representations depending on the fact that we are considering top-down or bottom-up processes.

Predictive coding is based on the hypothesis that the brain minimizes the prediction errors with respect to a generative model of the world. Both classical and quantum theories provide a tool for model updating the state conditioned to some observations. In a classical Bayesian framework, when an event is observed, the original probability function is changed into a new conditional function. In particular, the joint probability is normalized and this guarantees that the new probability function sums to one. Quantum theory proposes, mutatis mutandis, a similar model: the original state vector is transformed into a new state vector through a projection. In particular, the original state is projected onto the subspace of the observed event and then such projection is divided by its length. This transformation of the state vector onto a subspace corresponding to the observed event is called “projective measurement.” In our stylized model, we call it collapse to generally indicate any state that eventuates, after some processing, in a definite output. In the words of Busemeyer and Bruza (2012, p. 23), collapse refers to “the transition from a superposition state to a definite state associated with the observed outcome.”

The difference between a classical and a quantum framework is marginal when the systems are compatible (the probability of the union of independent events is simply their sum, and the probabilities of all the events sum to unity). However, things change when we consider complementary systems, that is to say, predictions and evidence are incompatible. The certainty about predictions may lead to uncertainty about evidence (and vice versa), as incompatibility is, in the words of Busemeyer and Bruza (2012, p. 23), “mathematically implemented by the non-commutativity of quantum measurements.” Indeed in case of incompatibility, predictions and evidence do not share the same basis, but require separate spaces. In particular, each one (predictions and evidence) considered separately admits a Boolean framework, but they cannot be treated in a Boolean way once they are “pasted together.” A quantum probabilistic approach presents more flexibility than the Bayesian, as it does not require specifying the joint probabilities. More specifically, as we explain in section “The Role of Quantum Contextuality”, the notion of “quantum contextuality”—whose role is nowadays debated with reference to quantum evidence in the behavioral domain (e.g., Dzhafarov et al., 2016)—represents a fundamental theoretical device able to provide alternative explanations to incompatibility.

Incompatibility is central in the QPB and represents a substantial innovation compared to existing theories of Bayesian predictive coding, assuming a unique space for both predictions and evidence. We know from the previous two sections that S—the vector representing the state of the system—can be represented in terms of either the top-down system (that is {|P1⟩,|P2⟩} basis) or the bottom-up system (that is {|E1⟩,|E2⟩} basis). Indeed, we know that:

and

so

where

represents the coordinates of the state |S⟩ with respect to {|P1⟩,|P2⟩}, and

represents the coordinates of the state |S⟩ with respect to {|E1⟩,|E2⟩}. Now we know that the same state |S⟩ can be represented using two incompatible bases, that is to say, we have different coordinates depending on the basis, as in Figure 3.

We can represent the basis vectors of the top-down system in terms of the bottom-up system basis:

And, vice versa, we can represent the basis vectors of the bottom-up system in terms of the top-down system basis:

We can express each system with respect to the other. If we want to express the coordinates of the bottom-up system with {|E1⟩,|E2⟩} basis in terms of the top-down system with {|P1⟩,|P2⟩} basis, we have:

And if we want to express the coordinates of the top-down system with {|P1⟩,|P2⟩} basis in terms of bottom-up system with {|E1⟩,|E2⟩} basis, we have:

where † indicates the transpose-conjugate operation, as UP⁢E⋅UP⁢E†=I, where I is an identity matrix.

The incompatibility between the prediction system and evidence system, in Figure 3, represents a form of fundamental uncertainty, related to the famous Heisenberg uncertainty principle. Indeed, if we are certain in the evidence system—we observe a specific sensory input—we must accept some uncertainty in the prediction system—different predictions are still potential—and vice versa. In other words, once the state vector collapses in a system, it remains in a superposed state in the complementary system. This point is a tenet of the QPB, positing that we can determine the state of the top-down system only if we remain uncertain about the state of the bottom-up system, and vice versa.

Let us consider, again, the incipit of this contribution, “It’s not the deer that crosses the road, it’s the road that crosses the forest.” If we remain bounded in a Bayesian framework, we will consider the deer as an unexpected event and we will update our model of the forest: deer are not so much rare, so the next time we cross a forest, we should pay attention. Unfortunately, this Bayesian mechanism is as much plausible as it is trivial. The citation is not about deer living in a forest, but lies in a different cognitive order, unveiled by a metaphor (about a deer and a forest). We can conceive this metaphorical meaning only if we conceive the sensory evidence on a totally different and incompatible order of meanings. Indeed, the most interesting surprise is instantiated by the discovery of the ecological embeddedness, occurring when we realize that the road is nothing more than a small line in a huge forest. But, there is more. Once we access this higher-order meaning and we re-formulate the metaphor through an internally consistent and accurate description (the metaphor reveals our ecological embeddedness, disclosing the incapacity of conceiving our life as a part of a bigger picture), its poetical meaning suddenly disappears and becomes vague and uncertain. Notice also, when we try to express the meaning through a consistent and accurate description, new metaphors (such as the “bigger picture”) will enter again into the game of sense-making. Yet, again, there is more. There is no actual experience of a forest and a deer, our narrative is not part of a poetry book, nor are we interested in ecological embeddedness. The metaphor is a part of a scientific publication and it is instrumental to a totally different purpose and topic. Again, we found a further, higher-level order of sense-making. In short, when we are certain about a specific order of sense-making, we remain vague on the others that instantiate the sense. So, it seems to us that we can be certain about one thing precisely because of the vagueness of the complementary things.

We think that the Bayesian brain is not able to model such kinds of higher-order dynamics among different levels of metaphorical mapping, sensory evidence, poetical value, and logical consistency, and we also think that QPB presents some advantages. A fundamental implicit assumption of the Bayesian brain is that predictions and evidence are modeled as orthogonal sub-spaces. This fundamental property represents a fundamental limitation if analyzed through the quantum lens. Indeed, QPB assumes that prediction and evidence could not be independent (not orthogonal), as they can be expressed, one with respect to the other, with a precision trade-off (they are conjugate systems). The orthogonality between the prediction and evidence systems (in the Bayesian brain) is only an extreme case, and QPB explores all the other infinite possibilities of non-orthogonal but complementary interactions between prediction and evidence. Such types of incompatible interactions are central in QPB.

Speculatively, the human brain presents the “ability” of adjusting the predictive system with respect to the sensory evidence, and vice versa. Such adjustment is consistent with the well-known “adaptive resonance theory” (e.g., Grossberg, 2013), which considers the brain is organized into complementary parallel processing systems—respectively related to top-down expectations and bottom-up sensory information—whose interactions generate intelligent behaviors (Grossberg, 2000). Contrary to adaptive resonance theory—emphasizing the role of resonance among the processes—QPB considers complementarity as a phenomenon that works precisely because top-down and bottom-up processes cannot always be “reduced” one to the other, but must preserve their incompatibility to generate higher-order cognitive processes. In QPB, the adjustment between predictions and evidence—corresponding in our model to a rotation of one system with respect to the other (as shown in Figure 3)—instantiates the complementarity of the prediction, endogenizing error minimization and precision, both constructs being central in predictive brain theory. Furthermore, complementarity preserves the history of the dynamic interplay (through order effects) and the ability to reinvent the interaction (through interference) in a non-trivial manner. In the QPB framework, we speculatively posit that incompatibility is necessary for higher-order surprisal. Specific forms of surprisal (such as the deer metaphor, which makes sense precisely when one oversteps the veil of sensory obviousness) occur when the interaction between prediction and evidence presents a form of non-reducible uncertainty, requiring some indeterminism in the predictive brain (cf. De Ridder and Vanneste, 2013). Such uncertainty preserves the degrees of freedom necessary to instantiate such incommensurable-but-meaningful wild flights, which are a mark of the human being.

The possibility of conceiving prediction and evidence as complementary systems is not a hypothesis that is meaningful only on a cognitive and computational level, but aims to find evidence in the areas of the brain involved in predicting and coding the sensory evidence. Despite the mounting (but also mixed) neuroimaging research on predictive coding (Walsh et al., 2020), and postulating distinct brain areas for prediction and sensory evidence, we posit that QPB is more consistent with neural reuse, hypothesizing that the same area of the brain can be used for different purposes (Gallese, 2008; Anderson, 2010; cf., Gatti et al., 2021). In particular, the possibility of conceiving the same state vector through incompatible descriptions is a way of modeling neural reuse, where the same neural endowments admit a representation both in the predictive system and in the evidence system. Hence, the possibility of adjusting the prediction and the sensory-motor systems, preserving their incompatibility, is strictly related to the idea of exploiting the same neural resources for both predicting and coding the sensory evidence. Incompatibility is what preserves the specificities of prediction and evidence systems within the same neural resources; speculatively, when a brain area is involved in prediction, its role in coding evidence is uncertain, and vice versa, that is to say, we hypothesize that the same neural endowment enables partial determinations in the form of prediction or evidence.

In QPB, predictions and evidence are not necessarily distinct brain areas, but they are analytical distinctions that identify specific functions, which share some neural resources. This fundamental mechanism—the same neural resources for different functions—challenges a fundamental property assumed in existing predictive brain theories, that is, the existence of a strictly hierarchical organization of the brain. QPB proposes an alternative topological model that shifts the theoretical focus from layers to orders. If we admit that the same neural resources enable different functions, we are substituting the necessity for layers with the possibility of instantiating specific functions, without a strict functional topology but through complementarity. Indeed, predictions and evidence can be determined only by assuming that the same neural resources collapse in a system, while preserving the indeterminacy of the complementary system in a dynamic, non-commutative fashion. It can be noticed, again speculatively, that by avoiding a strictly hierarchical organization and allowing neural “shortcuts”, complementarity can be also useful to shed new light on neuroplasticity. More specifically, a Bayesian brain is not able to fully account for such predictions that, being phylogenetically-determined adaptations, are resistant to contextual changes (Yon et al., 2019). A QPB, being consistent with neural reuse, preserves both the adaptive functions and the degree of freedom for the instantiation of high-level processes, to be intended as exaptations of the sensory-motor system (Anderson, 2007; Mastrogiorgio et al., 2022).

Hence, we do not deny that the human brain presents hierarchies in the topology of neural networks. However, we think that the strict hierarchical organization assumed in current theories of predictive brain seems to be more related to the necessity of formulating a theory consistent with the Bayesian framework, than to the opportunity of interpreting the flourishing neuroimaging evidence through the Bayesian lens. Put differently, it seems to us that a Bayesian framework imposes specific and restrictive topological conditions to preserve its theoretical consistency. This limitation is one that QPB tries to overcome, endorsing the view that specific old brain structures are, evolutionarily speaking, exapted to enable high-level faculties employed in civilized niches.