Neurons as hierarchies of quantum reference frames

Information processing in biological systems has traditionally been represented as classical; despite theoretical arguments from a variety of perspectives [51, 52, 53, 54, 55] and experimental evidence for the functional relevance of quantum coherence in photoreception and magnetoreception [56, 57, 58], quantum biology remains in its infancy [59, 60, 61, 62]. Motivated in part by recent analyses showing that cellular bioenergetic resources fall orders of magnitude short of those needed for fully-classical computation at macromolecular scales [63], here we adopt a quantum-theoretic perspective from the start. This perspective allows full use of the quantum information toolkit, including the concept of a QRF; it offers the advantages of full generality and applicability at any scale [50].

Consider an isolated, finite, bipartite system $AB$ for which the interaction $H_{AB}$ is weak enough and the time interval of interest short enough that considering $AB$ to be separable, i.e. having a joint state $|AB\rangle$ that factors as $|AB\rangle=|A\rangle|B\rangle$, is a good approximation. In this case, it is always possible to choose a basis in which the interaction can be written:

$$H_{AB}=\beta^{k}k_{B}T^{k}\sum_{i}^{N}\alpha^{k}_{i}M^{k}_{i},$$

(1)

where $k=~{}A$ or $B$, the $M^{k}_{i}$ are $N$ Hermitian operators with eigenvalues in $\{-1,1\}$, the $\alpha^{k}_{i}\in[0,1]$ are such that $\sum^{N}_{i}\alpha^{k}_{i}=1$, and $\beta^{k}\geq$ ln 2 is an inverse measure of $k$’s thermodynamic efficiency that depends on the internal dynamics $H_{k}$ [40, 46, 49]. In the form given by Eq. (1), the interaction can, without loss of generality, be regarded as defined at a holographic screen $\mathscr{B}$ separating $A$ from $B$; the operators $M^{k}_{i}$ can in this case be regarded as “preparation” and “measurement” operators that alternately write and read bit values encoded on $\mathscr{B}$ [46, 64]. The screen $\mathscr{B}$ can be realized as an ancillary qubit array as in Fig. 1. The thermodynamic factor $\beta^{k}k_{B}T^{k}$ in Eq. (1) assures compliance with Landauer’s Principle [65, 66, 67], i.e. assures that the per-bit free-energy cost of classical bit erasure is paid on each cycle (see [44, 50] for discussion).

Now consider $A$ to be an “observer” that decomposes $B$ into a “system of interest” $S$ and its surrounding “environment” $E$. As a familiar example, $S$ may be some item of laboratory apparatus and $E$ the surrounding laboratory. We suppose further that $A$ is primarily interested in the state $|P\rangle$ of some proper component $P$ of $S$, conventionally called the “pointer” of $S$, that is separable from the remaining proper component $R$ (the “reference” component), i.e. $S=PR$ and $|PR\rangle=|P\rangle|R\rangle$ over the observation times of interest. Under these conditions, repeated observations of a fixed state $|R\rangle$ of $R$ allow the identification of $S$ as a system; in our example, repeated observations of the fixed size, shape, location, etc. of an item of laboratory apparatus allow its identification over time and hence unambiguous observations of its pointer state [47, 48]. These repeated observations effect decoherence of $S$, and hence enforce separability of $S$ from $E$ [46]. Nothing in the above changes when pure states $|P\rangle$ and $|R\rangle$ are replaced by densities $\rho_{P}$ and $\rho_{R}$, interpreted as distributions over time series of pure states, with the separability condition $\rho_{PR}=\rho_{P}\rho_{R}$.

Assigning mutually-exclusive subsets of bits encoded on $\mathscr{B}$ to $P$, $R$, and $E$ is clearly a computational process that must be implemented by the internal interaction $H_{A}$ of $A$. Indeed this process is completely independent of $H_{B}$ or, equivalently, of the decompositional structure of $B$ [47]. Tracking the states of $P$, $R$, and $E$ over time requires an adequate memory resource and a comparison function capable of detecting changes at some suitably coarse-grained resolution. As shown in [47, 48], these computations can be regarded as implementing QRFs for $P$, $R$, and $E$, the functions of which can, without loss of generality, be specified by networks (formally, cocones) of Barwise-Seligman classifiers as described in §2.3 below.

It is worth emphasizing here that the decomposition $B=PRE$ is not “objective” in any sense; the components $P$, $R$, and $E$ are defined, relative to and hence “for” $A$, by their respective QRFs. These QRFs can, therefore, themselves be labeled ‘$P$’, ‘$R$’, and ‘$E$’ without ambiguity. Similarly, the states $|P\rangle$, $|R\rangle$, and $|E\rangle$ encoded on $\mathscr{B}$ in the basis specified by Eq. (1) are not “objective” but rather defined as the domains, respectively, of the QRFs $P$, $R$, and $E$. They are, therefore, also strictly relative to $A$ (cf. the characterization of quantum states as personal in [68, 69] or as observer-relative in [70]). The computations implemented by $P$, $R$, and $E$ result in classical, i.e. irreversible encodings of state information to $A$’s memory; therefore they require free energy as discussed in [47]. This free energy can only come from $B$; hence some fraction of the bits encoded on $\mathscr{B}$ must be viewed as supplying the free energy required for computation. The values of these bits cannot, therefore, affect the computational outcomes of $P$, $R$, and $E$; they are effectively traced over. This obligate trace operation renders the outputs of $P$, $R$, and $E$ coarse-grained representations $\rho_{P}$, $\rho_{R}$, and $\rho_{E}$ that can be viewed as probability distributions, and effectively as weighted averages, in the basis specified by Eq. (1), or as pure states in a “computational basis” with reduced dimension. The dimension reduction or coarse-graining induced by the action of any QRF, solely in consequence of its free-energy requirements, enables the construction of QRF hierarchies with distinguishable layers as discussed in the case of neurons in §4 and §5 below.

A system $S$ is only useful, in practice, as a measurement apparatus if it can be identified over macroscopic time, i.e. has a stable coarse-grained reference state $\rho_{R}$. “Stability” requires, in particular, that observing the environment $E$ cannot affect $\rho_{R}$; hence the QRFs $R$ and $E$ must commute. The reference state $\rho_{R}$ that allows identification of $S$ must, moreover, remain stable when the pointer component $P$ is observed; hence $R$ and $P$ must commute. Similarly $P$ must commute with $E$. These commutativity conditions are, effectively, consequences of separability: they follow directly from the assumption that $P$, $R$, and $E$ are each distinguishable from the others.

These commutativity conditions on the QRFs $P$, $R$, and $E$ can be summarized in diagrammatic form by:

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(2)

where $|\mathscr{B}\rangle$ is the bit string encoded on $\mathscr{B}$ by the action of $H_{AB}$, $\mathcal{P}_{U}$ is the time propagator of the joint system $U=AB$, and $\mathbb{M}_{P}$, $\mathbb{M}_{R}$, and $\mathbb{M}_{E}$ are Markov kernels on the state spaces of $\rho_{P}$, $\rho_{R}$, and $\rho_{E}$, respectively. The boundary $\mathscr{B}$ functions as a Markov Blanket (MB) [71, 72] that renders the system state $|A\rangle$ conditionally independent of $|B\rangle$ at the “microscale” defined by $H_{AB}$ [73], while $\rho_{P}$, $\rho_{R}$, and $\rho_{E}$ are effective MB states at the coarse-grained “computational scale” at which $A$ writes observed state information to memory. The probabilities encoded by the $\mathbb{M}_{P}$, $\mathbb{M}_{R}$, and $\mathbb{M}_{E}$ are posterior probabilities for $A$ from a Bayesian perspective; the task of $A$ as an observer implementing active inference [20, 21, 22, 23, 24] is to learn priors, or equivalently, to implement actions on $B$, that predict $\mathbb{M}_{P}$, $\mathbb{M}_{R}$, and $\mathbb{M}_{E}$ as accurately as possible.

The operators $\mathbb{M}_{P}$, $\mathbb{M}_{R}$, and $\mathbb{M}_{E}$ are, clearly, Markovian if but only if the joint density $\rho_{PRE}$ factors as $\rho_{PRE}=\rho_{P}\rho_{R}\rho_{E}$, i.e. if but only if $P$, $R$, and $E$ can be assumed to be separable. As noted above, separability can be assumed only in the limit of weak interactions and/or short observation times. Hence Markovian evolution of $P$, $R$, and $E$ is an assumption valid in this limiting case, not a fact. Evolution being Markovian corresponds, by definition, to probabilities satisfying the Kolmogorov axioms and hence to the Bell [74] and Leggett-Garg [75] inequalities being satisfied, not violated. Hence physically, the Markovian evolution corresponds to phase correlations being negligible, i.e. the absence of quantum entanglement or changes of “intrinsic” measurement context [76, 77, 78] as discussed further below. These conditions can be summarized as observations being “sufficiently coarse-grained” at the computational scale to be treated as classical.

The commutativity conditions given by Eq. (2) are instances of the general condition required to interpret any physical process as computation [79]. We can, therefore, regard $\rho_{P}$, $\rho_{R}$, and $\rho_{E}$ as encoding a “semantic” representation of the microscale states $|P\rangle$, $|R\rangle$, and $|E\rangle$ encoded on $\mathscr{B}$. It is with respect to this completely-general notion of semantics that hierarchies of QRFs can be considered semantic, and hence virtual machine (VM) hierarchies [80], in §4 - 5 below.

In recent work [42, 43, 44, 45] we have drawn extensively upon the semantically enriched Channel Theory of information flow developed in [41], outlining many examples and applications (see also in particular [47, 48]). Here we summarize the basic ideas, starting with that of a (Barwise-Seligman) “classifier” (or “classification”), which categorically accommodates a “context” in terms of its constituent “tokens” in some language and the “types” to which they belong.

Note that this definition specifies a classifier/classification as an object in the category of Chu spaces [81, 82, 83] where ‘$\models_{\mathcal{A}}$’ is realizable by a satisfaction relation valued in some set $\mathbf{K}$ having no structure assumed.

Morphisms between classifiers are specified by the following:

This last definition can be represented schematically as the requirement that the following diagram commutes:

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(3)

Following these definitions, Channel Theory can be represented as a category comprising classifiers as objects and infomorphisms as arrows, which can be seen to be equivalent to the category comprising Chu spaces as objects and Chu morphisms as arrows [41].

Much of the formulism of [41] revolves around the idea of a distributed system in which the classifiers and infomorphisms between them function as ‘logical gates’ as further exemplified in [42, 43, 44, 45]. Significantly instrumental in this schemata is a finite, commuting cocone diagram (CCD) depicting a flow of infomorphisms sending inputs to a core $\mathbf{C^{\prime}}$ that is the colimit of the underlying classifiers if such exists:

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(4)

There is a dual construction to this CCD, namely a commuting finite cone diagram (CD) of infomorphisms on the same classifiers, where all arrows are reversed. In this case the core of the (dual) channel is the limit of all possible downward-going structure-preserving maps to the classifiers $\mathcal{A}_{i}$. Hence we can define the central idea of a finite, commuting cone-cocone diagram (CCCD) as consisting of both a cone and a cocone on a single finite set of classifiers $\mathcal{A}_{i}$ linked by infomorphisms as depicted below:

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(5)

Significantly, the CD functions as a “memory-write system” in Eq. (4); this is used to represent the time-stamped “write” operations of a QRF in [47, 48]. These diagrams can be extended to obtain a “bow tie” diagram as below in (6) which represents a coarse-graining of the semantics of $\mathcal{A}_{i}$ via $\mathbf{C^{\prime}}$, into a compressed representation $\mathcal{A}^{\prime}_{i}$.

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(6)

Diagrams such as (4) (6) can be further extended into hierarchical networks by adding intermediate layers of classifiers and infomorphisms. In this way they resemble artificial neural networks (ANNs), and in the “bowtie” form of Diagram (6), they resemble variational autoencoders (VAEs) [42]. The core $\mathbf{C^{\prime}}$ in (6) can be viewed as both an “answer” computed by the $f_{i}$ from inputs to the $\mathcal{A}_{i}$ and, dually, as an “instruction” propagated by the $h_{i}$ (or in Diagram (6), the $h^{\prime}_{i}$), to drive outputs from the $\mathcal{A}_{i}$ (or in Diagram (6), the $\mathcal{A}^{\prime}_{i}$). This kind of dual input/output is precisely that of a QRF. Crucially, a “bow tie” diagram within the distributed systems described above, is a descriptive mechanism for broadcasting control signals to multiple recipients. The Global Neuronal Workspace, as a massively parallel, distributed system, performs precisely this function for the mammalian central nervous system [84, 85, 86].

We refer the reader to the concepts of a (regular) theory and local logic as introduced in [41, Chs 9 12] (and extensively reviewed in [42, 45]) to specify the logical structure of a given situation. Basically, a local logic is a classifier with a theory, along with a subset of tokens as specified by a sequent; namely, a classification $M\models_{\mathcal{A}}N$ of a classifier given by a pair of subsets $M,N$ of $Typ(\mathcal{A})$, such that $\forall x\in Tok(\mathcal{A}),x\models_{\mathcal{A}}M\Rightarrow x\models_{\mathcal{A}}N$. In fact, any classifier generates its own natural local logic. Below, we will adopt the sequent as a ‘conditional’ when defining probabilities.

In general, given an information flow channel:

$$\longrightarrow\mathcal{A}_{\alpha-1}\longrightarrow\mathcal{A}_{\alpha}\longrightarrow\mathcal{A}_{\alpha+1}\longrightarrow\cdots$$

(7)

the semantic content can be extended by postulating local logics $\mathcal{L}_{\alpha}=\mathcal{L}(\mathcal{A}_{\alpha})$ generated by the corresponding classifiers $\mathcal{A}_{\alpha}$ (assumed, in principle, to be in relationship to a (regular) theory associated to the individual $\mathcal{A}_{\alpha}$, as specified in [41, Ch 9] and reviewed in [45, Appendix A]), thus postulating a flow of logic infomorphisms:

$$\cdots\longrightarrow\mathcal{L}_{\alpha-1}\longrightarrow\mathcal{L}_{\alpha}\longrightarrow\mathcal{L}_{\alpha+1}\longrightarrow\cdots$$

(8)

which may then comprise a CCD as in Eq. (4).

A probabilistic interpretation of information flow in Eq. (8) results when the sequent relation is weakened to require only that if $x\models_{\mathcal{A}}M$, there is some probability $P(N|M)$ that $x\models_{\mathcal{A}}N$. ²²2We use the upper case ‘P’ to denote a probability in relationship to classifiers, and a lower case ‘p’ for that pertaining to e.g. events or states as below. In effect, $\models_{\mathcal{A}}$ becomes a conditional probability that inherits the semantics of the local logic [87] (cf. [88]); that is, given a sequent in a classifier $\mathcal{A}$, along with its satisfaction relation satisfying:

$$M\models_{\mathcal{A}}N~{},\qquad\forall x(x\models_{\mathcal{A}}M\Rightarrow x\models_{\mathcal{A}}N),$$

(9)

we can define the conditional probability as:

$$M\models_{\mathcal{A}}^{P}N:=P(M|N).$$

(10)

In fundamental Bayesian terms, $M$ above can be regarded as an unobserved (or as below, unobservable) event, and $N$ an observable datum, in which case $P(M)$ becomes the prior, and $P(N)$ the evidence, that together generate a prediction. Given the likelihood $P(N|M)$ as the conditional obtained from weakening the sequent, Bayes’ theorem specifies this conditional as the posterior:

$$P(M|N)=\frac{P(N|M)P(M)}{P(N)}.$$

(11)

We introduce the idea of “contexts” of observation, following [45, §7.1] (see also [48, Appendix A.1]), by considering the following sets: i) $X$ is a set of events in a very general sense (e.g. a set of observed value combinations or atomic events); ii) $Y$ is a set of conditions (specifying objects/contents or influences; iii) $Z$ is a set of contexts (e.g. detectors, measurements, or methods); iv) $W:=Y\times Z$, where $X,Y$ and $Z$ are taken to be subsets of some (very) large probability space. Note that here $Y$ can be decomposed as $Y=Y^{+}\cup Y^{-}$ (disjoint union) where $Y^{+}$ consists of observed objects/contents/characteristics of the context $Y$, and $Y^{-}$ consists of what is not observed about $Y$. Hence $W:=Y\times Z=(Y^{+}\cup Y^{-})\times Z$. The classifier in question is then $\mathcal{A}=\langle X,W,\models_{\mathcal{A}}\rangle$ and represents an observable in context. Among several possible interpretations for the classification relation ‘$\models_{\mathcal{A}}$’ is valuation by the conditional probability $p(a|x)=p(a|\{b,c\})$, whenever defined, for $a\in X,b\in Y$ and $c\in Z$ [45] (see also below).

As a brief example, consider arbitrary classifiers $\mathcal{A}_{1}^{(a)},\ldots,\mathcal{A}_{5}^{(e)}$ in some part of an information channel where the classifiers correspond to events $a,b,c,d,e$, respectively, together with logic infomorphisms $f_{13},\ldots,f_{45}$ between them, and where the sequents are relaxed to become conditional probabilities as above:

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(12)

Following e.g. [89], this particular channel then generates a joint probability distribution given by:

$$p(abcde)=p(a)p(b)p(c|a)p(d|ab)p(e|d).$$

(13)

As developed in detail in [45], the above constructions provide a formal basis for classical Bayesian inference. Specifically, the diagrams (4) and (5) depict channels of logic infomorphisms, when on relaxing the sequent in each case, the classifications $\models_{\mathcal{A}_{\alpha}}$ are now explicitly realized as conditional probabilities $p_{\alpha}(\cdot|\cdot)$, whenever these are defined. Putting the above details concerning conditionals into the diagrammatic framework reveals a typical portion of a CCD computing a hierarchical Bayesian inference from a set of (posterior) observations $\mathcal{A}_{i}$ to an outcome $\mathbf{C}^{\prime}$, as having the form:

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(14)

The requirement that diagrams (4) and (14) commute amounts to the requirement that branching “upward” to an overall logic $\mathcal{L}$ along any one of the $f_{\alpha}$, is equivalent to following the inferential sequence (8) up to its termination, and then following the last of the $f_{\alpha}$ to $\mathcal{L}$. Thus the $f_{\alpha}$ can be seen as shortcuts to reaching $\mathcal{L}$: “insights” that allow the rest of the inferences in (8) to be bypassed. The local logic $\mathcal{L}$ can be seen as the “answer” to the problem (8) addresses: formally, it is the logic that solves the problem in one step. The probability that the answer is “right” is the product of the probabilities along (8). The commutativity requirement on (14) requires, in addition, that this overall probability is conserved; hence the probability associated with each “insight” $f_{\alpha}$ must be the combined probability of the inferential steps it replaces.

Commutativity of a CCD thus enforces inferential coherence, and the same clearly applies when the dual objects and maps of a CD are combined in constructing a CCCD as in diagram (5) to implement a recurrent Bayesian network (see also [43] where CCCDs are employed to model the interaction between bottom-up and top-down processing in visual object categorization). In contrast, non-commutativity of (4) implies that there is no consistently definable (conditional) probability distribution across the system in question, and characterises intrinsic contextuality [76, 77, 78] for that system [45, Th 7.1 and Coroll 7.1].

A QRF is only useful operationally, and hence only meaningful as a reference frame, if it can be deployed consistently over multiple episodes of observation. A meter stick or a voltmeter is, clearly, useful only if yields consistent measurements; the notion of a “standard” to which a device can be periodically recalibrated captures this consistency requirement operationally.

Recalling that a QRF is by definition a physical system, not merely an abstraction [39], the above condition can be stated as a requirement that a QRF have stable dynamics over the relevant time period, or if it is a subsystem of a larger system $A$, that it is an (or a component of an) attractor of the dynamics of $A$, i.e. of the self-interaction $H_{A}$. In computational language, a QRF must be a memory structure, one that is active (and hence executed as a computation) at all times, or one that is executable on demand via some higher-level control structure. We can interpret a CCD identifying $S=RP$ that is implemented by $A$ as encoding an expectation, and hence a prior probability for $A$, that bit patterns specifying $S$ will be encoded on the holographic screen $\mathscr{B}$ separating $A$ from its complement $B$. As emphasized earlier, neither the existence of this expectation not its satisfaction in practice imply that $S$ exists in any “ontic” sense as an observer-independent component of $B$ (cf. [90]).

Operational utility and hence meaningfulness also require that a QRF write its outcomes to a classical memory in a way that allows comparison of outcomes obtained at different times. Following [47, 48], we consider the transition from writing an observed pointer outcome $P_{i}$ with timestamp $\tau_{i}$ to writing $P_{k}$ with timestamp $\tau_{j}$ to be implemented by an operator $\mathcal{G}_{ij}$, the complete set of which forms a groupoid under composition. “Timestamping” here can be considered merely to be imposition of an order on the recorded data; a separate, observation independent clock is not assumed (cf. [50] for a general discussion of time QRFs). As compliance with Eq. (1) requires all classical data to be written on the holographic screen $\mathscr{B}$, the memory-write CD must write on $\mathscr{B}$ as shown in Fig. 2. Comparing previous with current data, in this case, requires reversing the arrows of the relevant memory-write CD to reconstruct the previously-written data. The comparison itself becomes a “meta” operation, i.e. one defined at the next-higher hierarchical level as described in the case of neurons below (cf. [43, 44, 45] for examples from human cognitive processing).

We can, therefore, consider QRFs to provide two distinct memory resources: 1) the stable QRF itself as an executable computation, and 2) the ordered sequence of classical data accumulated by deploying the QRF to make measurements. The executable QRF is, in general, a “quantum” memory, as it is a quantum computation implemented by the operator $H_{A}$. The classical data, in contrast, are written on $\mathscr{B}$ and subject to both space and free-energy constraints. As discussed in [50], these latter memories are effectively stigmergic. Any representation of the “self” that is readable as a classical memory, in particular, must be written on $\mathscr{B}$, and read from $\mathscr{B}$ when recalled [50]. These distinct memory resources support distinct expectations: QRFs as executable computations encode expectations about how the “world” faced by an agent is (or more properly, usefully can be) decomposed into identifiable “systems” while the classical data encode expectations about the states in which these systems will be encountered. These expectations can be represented as Markov kernels that are learned incrementally; the difference between these “expected” kernels and the “true” kernels given by Diagram (2) provides a representation of VFE to be minimized by active inference, i.e. by exploration (novelty seeking that enlarges the “training set”) and further learning. This definition of VFE for generic quantum systems allows a formulation of the FEP [20, 21, 22, 23, 24] for such systems that is developed in detail elsewhere [91].

As discussed in [49, 50], the simplest biological QRFs are switches that induce opposing behaviors whenever some parameter value is above or below some threshold value. The CheY system regulating bacterial chemotaxis provides a familiar example: local concentrations of the phosphorylated form CheY-P above or below a essentially fixed (at the relevant timescales) default concentration induce swimming or tumbling behavior, respectively [92, 93]. From a computational perspective, the CheY system implements a switch between depth-first and breadth-first search, and hence a generic mechanism for avoiding local maxima in a search space. More biologically, it implements both approach to resources and avoidance of toxins and other irritants, and hence serves as a homeostatic setpoint. Such setpoints are, from the present perspective, expectations consistent with continuing structural and functional integrity, i.e. continuing separability from the surrounding world. They can, therefore, be expected to arise in any system that maintains separability, i.e. any system to which Eq. (1) applies.

A second familiar QRF is the default or “resting” membrane voltage $V^{0}_{mem}$ across any local patch of biological membrane. Fluctuations above or below this setpoint induce actions, e.g. opening or closing voltage-gated ion channels. This QRF is obviously relevant to neuron function as discussed below; at larger scales, it becomes a critical regulator of multicellular morphology; see e.g. [94, 95, 96] and further discussion in §6.

Guided by these examples, it is straightforward to consider any system implementing linear threshhold or sigmoid kinetics as a QRF that switches between “negative” and “positive” responses as its input fluctuates below or above a default value (Fig. 3). With the advent of two-component signaling pathways, early enough in prokaryotic evolution that they appear in all analyzed lineages [97], the default value becomes adjustable, i.e. context-dependent via a classical “direct influence” mechanism [78] (but see [98] for evidence that lactose-glucose interference in E. coli [99] exhibits nonclassical contextuality). The multicomponent systems typical of eukaryotes, e.g. the Wnt [100], ERK [101], notch [102], BMP [103] pathways and the interactions between them, introduce further, multifactorial context dependence (see e.g. [104, 105] for general reviews). Such systems embody expectations about the relationships between multiple, simultaneously-measured values, each effectively calibrated to an underlying standard, e.g. a molecular concentration or local value of $V_{mem}$.

As noted earlier, cellular-scale biochemical and bioelectric signaling pathways have traditionally been conceptualized and depicted as fully classical, with each component occupying some determinate state at all relevant times. Cellular energy budgets cannot, however, support the thermodynamic cost of classicality, even at ms timescales, indicating that quantum coherence and hence quantum computation may be significant up to timescales of seconds [63]. Classical data can, in this case, only be encoded on intra- or intercellular boundaries as illustrated schematically in Fig. 1. “Chunking” the cell into functional components that process and exchange classical information or serve as classical memory structures (e.g. the genome, proteome, transcriptome, and “architectome” [106]) is, effectively, identifying the boundaries at which classical information is encoded, i.e. the boundaries that function as MBs. A signal transduction pathway that measures some environmental parameter at the cell membrane and transfers a context-dependent representation of this information to the genome is, in this picture, a single quantum computation, a QRF that detects, calibrates, and reports an observational outcome. It can be treated as a black box characterized by inputs and outputs, including free energy inputs and dissipated heat outputs. We adopt this approach to characterize the functional architecture of neurons in what follows.

As discussed in [107, 108, 109], the MB of a single neuron includes the pre- and postsynaptic membranes that mediate interactions with other neurons, as well as the overall cell membrane that mediates interactions with the neuron’s local microenvironment. Single neurons are, however, large, spatially-extended, computationally-complex structures (Fig. 4); it is the need for improved models of such structures that motivates this paper. Hence we will consider how the neuron’s MB is constructed from those of its components, identifying in this process sites at which intermediate steps in neuronal computations may be classically encoded.

Starting from the input side, the smallest cellular-scale component is the specialized postsynaptic area, typically located on a spine morphologically. Its computational role is to contribute a positive (excitatory) or negative (inhibitory) time-dependent $V_{mem}$ gradient to the overall electrical activity of its local dendritic branch. Performing this function requires energetic and molecular flows in addition to $\Delta V_{mem}(t)$, as illustrated in Fig. 5. At this level of abstraction, the presynaptic specialization at the axonal bouton is essentially equivalent, up to reversing the flow of small molecules from the cell membrane.

Making the reasonable assumption of balanced mass flows, i.e. assuming that the volume and density of the postsynaptic specialization, or in quantum terms, its Hilbert-space dimension remains constant, we can focus on information flows only, and treat the interaction between the postsynaptic specialization and its surroundings using Eq. (1). The boundary $\mathscr{B}$ in this case has two components, $\mathscr{B}=\mathcal{C}\mathcal{D}$, where $\mathcal{C}$ separates the postsynaptic specialization from the external microenvironment and $\mathcal{D}$ separates it from the bulk dendritic cytoplasm. Both components support both inward sensory flows and outward active flows. As these information flows are both physically implemented and referenced to default homeostatic/allostatic setpoints, they can be considered to be implemented by QRFs. Hence they can be represented as input – output or perception – action flows as in Fig. 6. In this representation, the boundary $\mathscr{B}$ is decomposed functionally into $\mathscr{B}=\mathcal{I}\mathcal{O}$, where $\mathcal{I}$ and $\mathcal{O}$ are “input” and “output” components respectively, each of which includes regions of both anatomical components $\mathcal{C}$ and $\mathcal{D}$. While each flow $Q$ has its own characteristic time constant $\tau^{Q}$ as in Fig. 2, the cross-modulatory couplings between pathways, and hence mutual dependence between setpoints, requires that they be mutually coherent. Hence we can represent the postsynaptic specialization as executing a single QRF with a vector output and a time constant $\tau^{PS}=max_{Q}\tau^{Q}$. Presynaptic specializations can be treated similarly.

As discussed in §2 above, all classical information in the system must be encoded on $\mathscr{B}$. Hence Fig. 6 depicts the ideal case in which the boundary component $\mathcal{O}$ encodes all output information, i.e. the computations implemented by the QRFs $X$, $Y$, and $Z$ have no classically-encoded intermediate states. In this ideal case, the free-energy cost of computation per unit time is given by the output bandwidth in bits, i.e. the area in bits of $\mathcal{O}$. Assuming a minimal thermal energy $\Delta E_{th}=\mathrm{ln2}k_{B}T$, the minimum dissipation timescale is given by the time-energy uncertainty relation [112] as $\Delta t_{diss}\geq\pi\hbar/(2\Delta E_{th})\approx$ 50 fs. Neuronal energy budgets, however, are insufficient for classical encoding at this timescale; assuming 30,000 synapses per cortical neuron [111] and a maximum power consumption of 250 Gbits/sec (where as a power unit 1 bit $=_{def}\mathrm{ln2}k_{B}T\approx 3\cdot 10^{-21}$ J at $T=310$ K) [26], each synapse could encode at most 8.3 Mbits/sec if the cell’s metabolic resources were devoted entirely to synaptic encoding. In fact the bulk of neuron energy usage is devoted to action potential (AP) generation and transmission, reducing the coding capacity of synapses by up to two orders of magnitude (see [63, 113, 114, 115] for further discussion of neuronal classical encoding costs and capacity). We can expect, therefore, that individual postsynaptic specializations can realistically encode at most about 100 kbits/sec, or 100 bits per unit time at the ms scales of dendritic postsynaptic potentials. Hence the idealization of Fig. 6 is not unrealistic from an energetic perspective, suggesting that consistent with the general considerations in [63], QRFs implemented at the scale of cellular compartments perform essentially pure quantum computations, i.e. employ quantum coherence as a short-term memory resource.

The form of Fig. 6 suggests a cobordism with $\mathcal{I}$ and $\mathcal{O}$ as boundaries. If the QRFs $X$, $Y$, and $Z$ perform pure quantum computation without classical intermediate steps, we can treat them in a path-integral formalism [116], and hence as defining a manifold of mappings from $\mathcal{I}$ to $\mathcal{O}$. This suggests that information processing in neurons is amenable to a treatment using topological quantum field theory [117] and the emerging theory of topological quantum neural networks [118]. We defer this possibility to future work.

While both dendritic arborization and spine density vary with neuron type and location [120, 121, 122, 123], both cortical and hippocampal pyramidal cells can exhibit dendritic branches with on the order of 1,000 spines and full dendritic trees with up to 30,000 spines [111]. Consider such a branch, neglecting information and resource flows through the non-spine membrane to focus on flows to and from the spines as modeled in Fig. 5. As the magnitudes of resource flows correlate with activity and hence with the magnitude of $\Delta V_{mem}$, we can further focus on the latter. In this case, we can think of the branch as “observing” on the order of 1,000 points of active signaling, analogous to 1,000 points of light in a visual field.

There are clearly two limiting cases for the activity observed by a branch. If every spine receives input from a distinct, independently-activated neuron, the correlation $C_{ij}$ between inputs $i$ and $j$ can be expected to be small, with $C_{ij}\rightarrow 0$ in the limit. If all spines receive input from a single neuron, $C_{ij}\rightarrow 1$ in the limit. Assuming roughly constant frequencies (though random phases) of presynaptic activity, the former limit yields noise with an input-frequency dependent amplitude as an output PSP from the branch; the latter yields an essentially digital signal encoding the activity of the single input neuron. An essentially constant noise signal is effectively a “dark room” from a VFE perspective, while a variable digital signal poses a well-defined, potentially high-information prediction problem.

Microconnectome methods [124] allow, in principle, counting the numbers of presynaptic (i.e. incoming) and postsynaptic (outgoing) partners for any neuron. Measured numbers of presynaptic partners range up to 50 in C. elegans [25] and from tens to several hundred in mammalian cortical cells [125, 126] to thousands for cerebellar Purkinje cells [124]. If as estimated [127] neocortical pyramidal cells typically receive 4 or 5 synaptic connections from each presynaptic partner, these cells may also have several thousand presynaptic partners. Such cells would be operating very close to the noise limit of $C_{ij}\rightarrow 0$ unless the activity of the presynaptic partners is already highly correlated. Perhaps significantly, most microscale connectome mapping has been achieved in sensory cortices in which highly correlated incoming information can be expected.

Following [110] and considering each dendritic branch to be a coincidence detector, the fundamental problem faced by a branch is distinguishing between i) correlated signals from a given presynaptic partner, ii) “true” correlates from multiple partners, and iii) “random” correlates. We can treat this uncertainty as a VFE function, and ask how it can be minimized. For a relatively short, distal, terminal branch exhibiting all three kinds of correlates, the answer is shown in Fig. 7: localizing partially-correlated regular inputs to distinct, non-branched branches and moving random inputs as far as possible from branch junctions disambiguates signals from different synaptic partners, allows “true” correlations to be identified specifically at branch junctions, and reduces the amplitude of noise signals.

A terminal branch in an initial state similar to Fig. 7a has synapse-level learning and spine remodeling available as tools for moving toward a state similar to Fig. 7b, with trophic reward as the imposed quality function. This is active inference, with spine remodeling as the “action” that alters environmental input, i.e. synaptic activity, independently of synapse-level priors. Success (given enough time) can be expected if trophic reward increases with the overall informativeness of the branch’s activity, i.e. if trophic reward to dendrites as functional units has the same activity-dependence as trophic reward to neurons as functional units [1, 128, 129], as studies of dendritic remodeling already suggest [3, 4]. This is mechanistically reasonable, as it transfers both the task of determining informativeness and that of adjusting trophic reward to the proximal part of the dendritic tree, and hence effectively to the soma itself. Both are, from the branch’s perspective, outside of its MB and hence part of the environment.

We can, therefore, state the following:

Prediction: Trophic reward to dendritic branches correlates with informativeness of the signal from the branch for more proximal dendritic branches, with informativeness to the soma as a limit.

What the soma needs to know is whether to fire an AP. A high-amplitude signal on a low-amplitude background, i.e. an effectively digital PSP, provides this information most efficiently. Hence we can predict that dendritic trees will remodel in the direction of maximizing digital processing.

A branch that has segregated its non-noise inputs into different compartments as in Fig. 7b is, effectively, an object detector. The object it detects is a presynaptic partner, or a spatially-localized collection of axonal processes from a presynaptic partner. It measures the detectable state of this object, its binned-average activity at some binning timescale, typically a few ms. Hence we can think of it on the model of Eq. (2) or Fig. 2, with the input correlation as the reference $R$, the binned-average activity as the pointer $P$, and any background noise as the environment $E$. Hence a compartmentalized branch is a canonical collection of QRFs.

From this perspective, we can see spine remodeling within dendritic branches as a process of assembling QRFs that identify objects and hence enable measurements of their states. Spine remodeling is thus an instance of the second, relatively neglected, form of active inference: the minimization of uncertainty about the decomposition of an agent’s world. A branch with maximally-ambiguous inputs as in Fig. 7a decomposes its world only into “points of light”; a branch that has remodeling to approach Fig. 7b sees instead two objects with well-defined states that can be compared in both amplitude and time dimensions. Spine remodeling is, as are the processes of object segregation and object persistence during infant development, learning what to see, and hence learning what coherent things populate the perceived world.

As the “cut” and hence the boundary imposed to render a branch “terminal” is moved toward the soma, the branch accumulates junctions separating sub-branches, and object-identity information is replaced by coincidence information. At each such sub-branch junction, the post-junction sector of the dendrite is faced with the task of reducing VFE by distinguishing “true” from “random” coincidences. Learning and remodeling driven by trophic reward remain the tools available for amplifying the former at the expense of the latter, thus increasing the signal-to-noise ratio and rendering signal processing effectively more digital. Hence one can expect neurons to remodel in the direction of segregating inputs to their dendritic trees in a way that maximizes true coincidences at branch junctions. To the extent that they do this, they increasingly operate as hierarchies of spatially-segregated QRFs

Dendritic geometry gives proximal synapses more influence over somatic activity than distal ones; the gating function of proximal inhibitory synapses, in particular, is well known [110]. Distal signals can only be processed with high (Bayesian) precision in periods of proximal silence. Hence it is natural to think of dendritic geometry as imposing a salience gradient from proximal to distal synapses. Stereotypical differences in arborization pattern between pyramidal cells from different cortical or hippocampal layers reflect different salience assignments, with cortical cells typically giving distal inputs, and hence distal presynaptic partners, higher salience than do hippocampal cells [110]; see [130, 131, 132] for specific differences between cortical cells from different layers. Higher distal salience is only useful from an information processing perspective if it is low-noise, i.e. if PSPs from distal dendritic branches, e.g. from the elaborate apical tufts of Layer V cortical pyramidal cells, are (primarily) time-convolutions of true coincidence signals. As remodeling takes time, one can expect neurons that compute the same function over long periods, i.e. that are required to exhibit only slow learning, to assign the greatest salience to distal inputs.

In the spiking neurons of interest here, somatic signal integration results, given sufficient amplitude, in AP generation. While APs are relatively high-amplitude, low-noise signals, outside of myelinated axonal segments differential AP degradation can be expected to contribute to differential presynaptic signal strength. Weak presynaptic signals are weak actions on the signaling neuron’s environment, and cannot be expected to yield strong confirmatory signals in return. Hence relatively weak synapses can be expected to be clustered, consistent with Fig. 7b.

As discussed in §2, a QRF corresponds to a specific choice of basis for a subset of the operators $M^{A}_{i}$ deployed by an agent $A$ to measure the state of its MB, or boundary $\mathscr{B}$. We can, therefore, think of the hierarchy of QRFs implemented by a neuron’s dendritic tree as a hierarchy of basis choices, with the “low-level” basis components corresponding to detections of correlated inputs from local clusters of synapses from single presynaptic partners and the “high-level” basis components corresponding to coincidences between increasingly more highly “chunked” aggregates of input signals. Each neuron can thus be thought of as deploying a specific hierarchy of basis vectors to measure the presynaptic activity of its environment.

Consider now a set N of neurons sampling the “same” environment, e.g. a set of Purkinje cells each sampling the axonal outputs of a set of cerebellar granule cells. The causal antecedents of the activity of the input neurons are clearly “unknowns” from the perspective of the neurons in N; the input activities can, therefore, be considered from this perspective an ensemble E of samples from an unknown, time-varying state. We can, in this case, think of N as making a set of measurements, each in a distinct basis, of the ensemble E of time-varying states.

If we think of the states in E as quantum states, then the set N of neurons can be regarded as performing quantum state tomography on E, and hence quantum process tomography on the process generating time variation in E [133]. Quantum state tomography generalizes classical tomography by generalizing the choice of basis away from “slices” of a three-dimensional geometry in which the system of interest is embedded. It is expensive in terms of basis dimensionality, requiring $d^{2}$ basis vectors for quantum states of (binary) dimension $d$, and $\sim d^{4}$ basis vectors for processes on states of dimension $d$. This dimensional cost translates well to the case of ensembles of neural activity, since as in the case of quantum states, it is phase correlation information that the tomographic measurement aims to extract.

Viewing neural computation as tomographic provides an immediate explanation for the enormously high input and output bandwidths and apparent redundancy of neuronal architectures, which are difficult to understand on the basis of simple logic-circuit models [127, 134]. An input space with 100-bit states ($d$ = 100) would require $\sim 100^{4}=10^{8}$ binary dimensions for process tomography, or roughly $10^{5}$ neurons at 1,000 distinct presynaptic partners per neuron. Following [135] and assuming $\sim 10^{8}$ minicolumns with 100 neurons each in human neocortex, analyzing a 100-bit state would require a minimum of $\sim 10^{3}$ minicolumns or $0.001\%$ of neocortical capacity. We can infer from this that tomographic computation is approximate, with substantial reduction of the input dimension by salience systems, e.g. active attention.

The stereotypically layered, only sparsely interconnected minicolumns of mammalian cortex, and in particular, human neocortex, are widely viewed as computational as well as neuroanatomical units, with modeling studies increasingly suggesting that minicolumns execute Bayesian predictive coding [30, 107, 109, 123]. Extending the model outlined above, we can think of minicolumns as functionally analogous to neurons, gathering input from and sending output to other minicolumns. As does a neuron, a minicolumn “sees” a spatially-distributed collection of (positive or negative) excitations, spatial and temporal correlations between which are informative to the extent that they are non-random. Hence a minicolumn is faced with a coarse-grained version of the VFE minimization problem faced by neuron: that of disambiguating “true” coincidences from random ones. Minicolumns can be expected to functionally remodel to increase signal-to-noise ratio for the same reasons neurons can, with trophic reward from the environment as the selective criterion.

Functional networks spanning multiple cortical regions, e.g. sensory pathways, impose a higher-level hierarchical organization. Predictive coding across layers of this higher-level hierarchy have been analyzed in terms of typical connections between minicolumns in adjacent layers; see e.g. [109, Fig. 4] or [123, Fig. 2] for interlayer connection maps. Top-down connections encoding likelihood at pathway layer $i$ arise mostly from minimcolumn Layer V pyramidal cells and target minicolumn Layer III cells at pathway layer $i-1$. Reciprocally, bottom-up connections arise mainly in minicolumn Layer II at pathway layer $i$, and project to networks of Layer IV cells at pathway layer $i+1$. Hence at any level of the pathway, empirical priors are localized in minicolumn Layer III, and likelihoods (or predictions) in minicolumn Layer V. This is further analyzed in [107] in terms of how MBs influence connectivity of microcircuits. Internal and external states are implemented by spiny stellate cells and interneurons of each minicolumn; via connections from and to these, respectively, superficial pyramidal cells of the next minicolumn become the (blanket) active states, while the deep pyramidal cells of the previous minicolumn become the (blanket) sensory states. Effectively, the minicolumns are the functional units comprising an MB of networks (either seen as e.g. a MB of MBs, or a Matryoshka of MBs). Once the MBs are functional, the overriding principle is that neurons, microcolumns and networks appear to dynamically self-organize thanks to the FEP. There are many interesting outcomes: consider for instance the visual network as composed of internal states influencing the dorsal and ventral attention networks, while the default-mode network plays the role of a sensory system mediating the influence between these former and external, sensorimotor states. Just as in the case of lower-level structures, such high-level, multi-layer processing pathways exhibit activity-dependent trophic reward and remodel as necessary to achieve it, as shown e.g. by studies of large-scale pathway remodeling following injuries that remove expected inputs [136] (cf. [109, 137]).

Since Friston’s proposal that the mammalian brain is an active-inference device [20], numerous studies have explored the implementation of active inference by large-scale networks, up to and including the global neuronal workspace [44, 45]. What we have shown here is that these principles extend downward to the scale of the individual synapse. Trophic rewards select for informative activity, i.e. high signal-to-noise ratios, at every scale. We can expect these considerations to generalize from neurons to non-neural cells, and from neural communication to communications between biological structures at every scale.