# Neurons as hierarchies of quantum reference frames

Chris Fieldsa111Corresponding author at: 23 Rue des Lavandières, 11160 Caunes
Minervois, FRANCE; E-mail address<EMAIL_ADDRESS>James F.
Glazebrook${}^{b},^{c}$ and Michael Levind

a 23 Rue des Lavandières, 11160 Caunes Minervois, FRANCE
ORCID: 0000-0002-4812-0744
b Department of Mathematics and Computer Science,
Eastern Illinois University, Charleston, IL 61920 USA
c Adjunct Faculty, Department of Mathematics,
University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA
d Allen Discovery Center at Tufts University, Medford, MA 02155 USA
ORCID: 0000-0001-7292-8084

Abstract
Conceptual and mathematical models of neurons have lagged behind empirical
understanding for decades. Here we extend previous work in modeling biological
systems with fully scale-independent quantum information-theoretic tools to
develop a uniform, scalable representation of synapses, dendritic and axonal
processes, neurons, and local networks of neurons. In this representation,
hierarchies of quantum reference frames act as hierarchical active-inference
systems. The resulting model enables specific predictions of correlations
between synaptic activity, dendritic remodeling, and trophic reward. We
summarize how the model may be generalized to nonneural cells and tissues in
developmental and regenerative contexts.

Keywords
Activity-dependent remodeling; Bayesian inference; Bioelectricity;
Computation; Learning; Memory

##  1 Introduction

Neurons are canonical biological information processors. Theoretical and,
particularly, conceptual models of neural information processing, however, lag
increasingly far behind our developing empirical understanding of neurons as
electrically-excitable cells. Experimental work over the past two decades has,
for example, firmly established that dendrites undergo activity-dependent
remodeling [1, 2, 3], particularly alterations of spine location, density, and
function [4], even in adult humans. This ontogenic process is functionally
analogous to the evolution of structural and positional diversity of dendrites
as they have adapted to a spectrum of functional roles [5], e.g. the
implementation of deep learning via synaptic plasticity [6, 7]. Neurons are
not, therefore, static structures, but rather can be regarded as undergoing
continuous development throughout the life cycle. This dynamic process has
significant consequences for both neuron-level and organism-level function.
For example, in organisms (such as caterpillars transitioning into butterflies
or moths) that exhibit drastic remodeling and reconstruction of their brains,
some of their learned memories remain and survive the process [8]. In other
contexts, memories can be imprinted on newly regenerating brains from other
tissues [9, 10], underscoring the plasticity of large-scale neural structure
and of the information stored within it. These effects of remodeling are not
just an issue for so-called lower animals, as applications in regenerative
medicine are likely to soon produce human patients in whom some portion of the
brain has been replaced by the progeny of naïve stem cells to treat
degenerative disease or brain damage.

It has also been shown that subnetworks of dendrites can compute local logical
operations including exclusive-or (XOR) [11]. Nonetheless, the idea that
dendritic trees effectively compute weighted sums, dating back to [12],
continues not only to dominate artificial neural network (ANN) development,
but also to guide biological thinking. Explicit cable-theory modeling can
reproduce time-dependent signal propagation and processing in simplified
representations of dendritic trees [13], but rapidly becomes intractable with
increasing geometric complexity. Increasingly sophisticated hardware models of
neurons allow exploration of phase coding, frequency modulation, and other
forms of hybrid analog-digital computing [14, 15], but are not readily mapped
to biological neural networks and are not standard components of a theoretical
neuroscientist’s toolkit.

Conceptualizing neurons as biological wires – even functionally sophisticated
wires – provides, moreover, no insight into the fundamental question of
cognitive neuroscience: the question of how information present in the
external world is encoded, via some combination of evolution, development,
remodeling, and learning, into the functional architecture of a nervous
system. Brains are not constructed, as ANNs are, in the absense of functional
activity, nor do they “begin learning” from a default initial state of uniform
connectivity. Brains are not “wired up” into some starting configuration,
after which they are turned on and exposed to the world. Brains instead
develop from neural primordia that are already functionally complex
multicellular microenvironments. The function of the nervous system, from its
earliest phylogenetic and ontogenetic role in directing morphogenesis [16] to
adult cognition, depends on the ability of individual neurons to negotiate
this microenvironment, both structurally and functionally. The local
microenvironment, with its diverse cell types, biochemical and
(micro)anatomical complexity, and network of biomolecular and bioelectric
signaling, is the “world” with which each individual neuron interacts, and is
each neuron’s sole source of nongenomic information. There are many
interesting scenarios supporting this viewpoint; some are of the
‘socialization’ and decision making type. For instance, assortative neural
networks demonstrate collective resilience, whereby nodes of a certain degree
have an affinity to team up with those of the same topological type, and thus
often contribute to formation of a small world network structure [17, 18].
Neural network modeling of independent tasks in the prefrontal cortex in [19],
for example, reveals how seemingly separate neuronal groups engaged in their
respective tasks may sometimes be drawn into a coalition, casting votes for a
stimulus, and then enacting a committee-like decision.

In line with the Free Energy Principle (FEP) and the idea of Bayesian active
inference [20, 21, 22, 23, 24], individual neurons can be considered cognitive
agents that minimize aggregate uncertainty about their future states
(variational free energy or VFE) by actively exploring and developing
predictive models of their local microenvironments. It is the joint activity
of hundreds (in C. elegans [25]) to tens of billions (in primates [26]) of
neurons within this jointly-constructed environment that encodes information
sourced from the external world. Hence an adequate conceptual model of neurons
as biological systems must explain how each neuron’s construction of a
predictive model of its local microenvironment results in the joint encoding
of a predictive model of the external environment at the scale of the entire
brain. The utility of FEP based models of cell sorting in morphogenesis [27,
28], cortical minicolumn and local-circuit organization [29, 30, 31, 32, 33],
and network and whole-brain function [21, 34, 35, 36, 37] suggests that a
scale-free conceptual framework is the right way to approach this question of
functional coherence between microenvironment models constructed by individual
neurons and external environment models constructed by brains. The FEP is,
however, fundamentally a statement about error minimization; it is an
implementation-independent principle of thermodynamics [24]. It does not tell
us anything specific about neuronal functional architecture, or about how
neurons interact with either each other or the non-neuronal components of
their microenvironments.

Here we suggest that the idea of a quantum reference frame (QRF) developed
within quantum information theory [38, 39] is usefully applied to characterize
neurons both architecturally and functionally. While quantum information
theory employs the formalism of standard quantum theory, it makes no scale-
dependent assumptions and is applicable from sub-microscopic to cosmological
scales. The formalism of QRFs, in particular, is applicable at any scale. A
QRF is a physical system that assigns units of measurement, and hence an
operational semantics, to each observational outcome, enabling outcomes
obtained at different times or places, or with different QRFs, to be compared.
Macroscopic systems such as meter sticks, clocks, gyroscopes, and the Earth
with its gravitational and magnetic fields are canonical examples of QRFs.
While reference frames are typically thought of as fully-specifiable
abstractions in classical physics, this is not the case in quantum theory:
QRFs are physical systems that encode unmeasurable but functionally relevant
quantum phase information. Every QRF is, therefore, in an important sense
unique. A QRF cannot, in particular, be fully specified either structurally or
functionally by any finite bit string; it is “nonfungible” in the terminology
of [39]. Alice can, in this case, share a QRF with a distant observer Bob only
by physically transferring the QRF to Bob; sending any finite description is
provably insufficient. Alice can, moreover, transfer her QRF to Bob
successfully only if Bob already possesses a functionally equivalent QRF [40];
Alice cannot, for example, successfully communicate the meaning of “1 meter”
to Bob unless Bob already has a QRF that effectively functions as a meter
stick. From an information processing point of view, a QRF is a quantum
computer: it outputs an outcome value that is reproducible within some fixed
resolution, and that is assigned standardized units that confer an operational
semantics, when given a “raw” measurement, or simply a physical interaction,
as input.

We propose, in particular, that neurons are usefully regarded as hierarchies
of QRFs. Neurons in this representation are hierarchical measurement devices
that sample their surrounding cellular microenvironments at multiple scales
and encode scale-specific expectations about how those microenvironments will
behave. The output of a particular neuron encodes, for each cell that receives
it, a specific, nonfungible representation of the transmitting neuron’s local
measurements. These representations are encoded in the “units of measurement”
defined by the sending neuron’s particular combination of QRFs. They impose
operational semantic constraints on all downstream processing. These
constraints are inherited by, and give a particular operational meaning to,
the activity patterns of the neural system as a whole.

We develop this proposal using a general formalism, that of Barwise-Seligman
classifiers [41] organized into networks with well-defined limits and colimits
(cone-cocone diagrams or CCCDs) developed in [42] and applied to human
cognitive architecture in [43, 44, 45]. Barwise-Seligman classifiers are
naturally interpreted as representing either logical or probabilistic
constraints; hence CCCDs represent maximal collections of mutually-consistent
constraints, and indeed generalize hierarchical Bayesian inference [45]. To
model neurons as hierarchies of QRFs is to represent them as 1) quantum
computers that 2) implement hierarchical Bayesian inferences determined by 3)
their specific three-dimensional (3d) geometries and 4) the differential
signal-transduction velocities that their geometries impose. This formalism
extends naturally to networks of neurons, at any scale from a local circuit –
e.g. a minicolumn – up to an entire brain.

We begin in §2 by briefly reviewing the implementation of QRFs by generic
quantum systems [46, 47, 48] and their representations by CCCDs. We then turn
to biological systems, reviewing the representation of simple homeostatic
setpoints as QRFs [49, 50] and extending this treatment to perisynaptic
processes in neurons in §3. This allows us in §4 to represent the hierarchical
structures of dendrites as hierarchical QRFs that detect the states – activity
patterns – of particular “objects” – spatially-organized aggregates of
presynaptic axon terminals – in their microenvironments. We then extend this
representation to neurons in §5, showing how the integration of signals from
multiple dendritic branches implements an effectively tomographic computation.
This view of neurons as state identifiers scales upwards to functional
networks, the global activity of which provide coarse-grained representations
of particular components of the external environment. We then briefly review
in §6 results from a variety of systems showing that with suitable spatial and
temporal scaling, this model applies to electrically-excitable cells
generally. We consider simulation modeling and experimental approaches
suggested by this framework in §7.

##  2 QRFs and their representation by CCCDs

###  2.1 QRFs as generic representations of measurement

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).

Figure 1: A holographic screen $\mathscr{B}$ separating systems $A$ and $B$
with an interaction $H_{AB}$ given by Eq. (1) can be realized by an ancillary
array of noninteracting qubits that are alternately prepared by $A$ ($B$) and
then measured by $B$ ($A$). There is no requirement that $A$ and $B$ share
preparation and measurement bases, i.e. QRFs. Adapted from [45] Fig. 1, CC-BY
license.

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.

###  2.2 Commutativity constraints on QRFs

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.

###  2.3 QRFs as constraint networks

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.

######  Definition 1.

A classifier $\mathcal{A}$ is a triple $\langle
Tok(\mathcal{A}),Typ(\mathcal{A}),\models_{\mathcal{A}}\rangle$ where
$Tok(\mathcal{A})$ is a set of “tokens”, $Typ(\mathcal{A})$ is a set of
“types”, and $\models_{\mathcal{A}}$ is a “classification” relation between
tokens and types.

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:

######  Definition 2.

Given two classifiers $\mathcal{A}=\langle
Tok(\mathcal{A}),Typ(\mathcal{A}),\models_{\mathcal{A}}\rangle$ and
$\mathscr{B}=\langle
Tok(\mathscr{B}),Typ(\mathscr{B}),\models_{\mathscr{B}}\rangle$, an
infomorphism $f:\mathcal{A}\rightarrow\mathscr{B}$ is a pair of maps
$\overrightarrow{f}:Tok(\mathscr{B})\rightarrow Tok(\mathcal{A})$ and
$\overleftarrow{f}:Typ(\mathcal{A})\rightarrow Typ(\mathscr{B})$ such that
$\forall b\in Tok(\mathscr{B})$ and $\forall a\in Typ(\mathcal{A})$,
$\overrightarrow{f}(b)\models_{\mathcal{A}}a$ if and only if
$b\models_{\mathscr{B}}\overleftarrow{f}(a)$.

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.

###  2.4 CCCDs implement Bayesian inferences in defined contexts

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].

###  2.5 QRFs as memory resources

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).

Figure 2: A CD $W_{kj}$ (green triangle) specifies a memory-write operation of
the time-stamped state $(\rho^{RP_{k}},\tau_{j})$ to $\mathscr{B}$. The
timestamp $\tau_{j}$ is generated by the groupoid action $\mathcal{G}_{ij}$ on
the previous (at $\tau_{i}$) output from the CCD $PR$. Adapted from [48] Fig.
9, CC-BY license.

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].

##  3 QRFs at the macromolecular scale

We now turn from generalities to the specific case of mammalian cortical
neurons, and show how these can be conceptualized as hierarchies of QRFs, i.e.
hierarchies of physically-implemented computations, each of which can be
regarded as a measurement, at a defined scale and with respect to defined
units, of some aspect of its local environment. We then generalize this
treatment to arbitrary cells in §6.

###  3.1 QRFs as homeostatic setpoints

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 $V_{mem}$.

Figure 3: a) A generic system executing sigmoid kinetics. b) Reflecting the
response curve at its inflection point redescribes the system as a QRF that
switches behavior at a default value.

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.

###  3.2 The postsynaptic complex as a QRF

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.

Figure 4: Cartoon structure of a mammalian cortical neuron; cf. [110, 111] for
anatomical details. a) Voltage gated channels and pumps are the primary input
(sensory) and output (active) MB components; b) Specialized postsynaptic
structures, e.g. on dendritic spines, collect incoming information from other
neurons; c) dendritic subtrees integrate and process this incoming
information; d) dendritic information is integrated at the soma and
distributed to outgoing axonal channels. Postsynaptic receptors may also
appear on the soma or on axons; this level of detail is neglected here.

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.

Figure 5: Cartoon structure of a mammalian postsynaptic specialization; cf [4]
for anatomical and functional details. Primary energy (red), molecular (blue)
and $V_{mem}$ gradient (orange) flows are shown. Considered as a system, the
spine’s MB includes the spine membrane (solid line) and the
cytoplasmic/cytoskeletal interface between the spine and the bulk dendritic
cytoplasm (dashed line).

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.

Figure 6: Information flows abstracted to QRFs ($X$, $Y$, $Z$; colored
triangles) that read from an input boundary component $\mathcal{I}$ (right
hand blue ellipse) and write to a output boundary component $\mathcal{O}$
(left hand blue ellipse), where the overall system boundary
$\mathscr{B}=\mathcal{I}\mathcal{O}$. Each QRF $Q$ has an associated time
constant $\tau^{Q}$ and a groupoid-structured set of operators
$\mathcal{G}^{Q}_{ij}$ as depicted in Fig. 2.

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.

##  4 Dendrites as spatially-organized QRFs

Dendritic branches are traditionally viewed as lossy but otherwise passive
integrators of relatively slow (10s of ms) postsynaptic potentials (PSPs),
though faster spike-like activations and backwards propagation of APs are also
possible [110]; see [119] for detailed dendritic spike models. This passive
view of dendritic activity renders the execution of complex logical operations
like XOR [11] surprising and activity-dependent remodeling [2, 3, 4]
mysterious. It decouples the processes required to maintain a successful
synapse: learning is largely localized to the postsynaptic side, while target
search and synapse formation are localized to the presynaptic side. Finally,
it decouples dendritic function from the dendrite’s trophic requirements.
Branches that transmit only low-amplitude noise signals cannot, on the passive
model, be regarded as less successful or less worthy of continued metabolic
maintenance in comparison to those branches that consistently transmit high-
amplitude, highly informative signals.

Dendritic branches satisfy the physical requirements for implementing QRFs and
hence engaging in active inference. Viewing dendrites in this way raises two
immediate questions:

  1. 1.

What is a dendritic branch measuring?

  2. 2.

What is the VFE function that a dendritic branch is minimizing?

However, while dendrites can be described as coincidence and hence activity-
correlation detectors [110] even on the passive view, the second of these
questions is difficult to even formulate from a passive perspective. A partial
answer to both questions is discussed in [29, 30]: basically, dendrites can
self-organize to minimize the VFE on surprise of their presynaptic inputs,
showing that postsynaptic gain is itself optimized with respect to VFE.
Extending the modeling approach of the previous section to dendritic branches
suggests answers to these questions that can be summarized by the hypothesis
that dendritic branches identify “objects” with relatively small numbers of
states, and execute approximately binary logic on the measured state vectors.

###  4.1 Colocating consistently correlated synapses minimizes VFE

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.

Figure 7: a) Least and b) most efficient arrangements of synapses with
partially-correlated (red and blue) and random (green) activity on a two-sided
symmetrical, terminal dendritic branch. By minimizing the ambiguity of
correlation, arrangement b) allows maximally digital processing at the branch
junction.

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.

###  4.2 Branches as object detectors

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.

##  5 Neurons as measurement devices

###  5.1 Dendritic geometry imposes salience

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.

###  5.2 Neurons as tomographic computers

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.

###  5.3 Scaling upwards: minicolumns to functional networks

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.

##  6 Generalizing neural computation to cellular computation

The prior discussion was framed in the context of neurons; however, it becomes
more generally applicable when we note that neurons did not appear de novo but
in fact evolved slowly from other cell types which already shared many of
their features (reviewed in [16]). Not only are the basic molecular components
of neurons (ion channels, electrical synapses, neurotransmitter machinery)
already present in most cells including unicellulars, but all cells produce
bioelectric gradients and comprise tissues with propagating changes in resting
potential [95]. It has been suggested that developmental bioelectricity is the
evolutionary precursor to neural dynamics, and indeed that evolution speed-
optimized the slow bioelectrical signaling that was first used to solve
problems in morphospace (exerting anatomical control over body shape by
regulating cell behaviors) before it was exapted to solve problems in
3-dimensional behavior space (by regulating muscle function) [138, 139].

Consistent with this hypothesis, bioelectric signaling in non-neural cells has
been implicated in control of morphogenetic decisions in embryogenesis,
regeneration, and cancer [140, 141, 142]. Recent work targeting the native ion
channels and gap junctions in tissue has shown that bioelectric states can be
readily modulated to predictably alter organ identity, induce regeneration of
limbs, control the axial patterning of whole bodies, and repair complex
structures such as craniofacial birth defects [143]. Indeed, it has been
suggested that parallel to the architecture of brains, non-neural
bioelectricity is the medium implementing the information processing that
enables collective intelligence of cellular swarms during body construction
and remodeling [144].

Morphological change is a deeply computational process that relies on
calibrated measurement, and hence QRFs, at multiple organizational levels.
While the DNA determines the micro-level hardware that each cell gets to have
(e.g., its complement of ion channels), genomes do not directly encode
morphology. Instead, they specify machines that have to deploy considerable
intelligence, using William James’ definition of intelligence as the ability
to reach the same outcome from different starting conditions and despite
perturbations. The physiology of cellular collectives implements very robust
problem-solving capacities by making important decisions about collective
macrostates that are not defined at the level of individual cells. For
example, mammalian embryos cut in half do not result in two half-embryos – the
system regulates to make complete, normal bodies of monozygotic twins
(regulative development). Tadpoles that are developmentally disrupted to have
all of their craniofacial organs in the wrong positions still result in normal
frogs after metamorphosis because the eyes, nostrils, mouth, etc. move through
novel paths to get to the same invariant outcome – a correct frog face [145,
146, 147]. Salamanders whose arms are amputated at the shoulder or wrist
regenerate precisely what is needed and then stop when a correct limb is
complete. These are just a few examples of a nearly ubiquitous property of
morphogenetic systems: anatomical homeostasis toward a specific pattern
memory, and the ability to reach that state despite sometimes drastic,
unpredictable perturbations. This degree of anatomical control requites
fundamentally computational functions by cell collectives: they need to be
able to measure the current state (e.g. the length of a limb, configuration of
the face, etc.), remember the correct state (represent aspects of the correct
target morphology), and execute a kind of means-ends analysis to reduce error
by controlling cell proliferation, migration, and differentiation. Trophic
reward plays a critical role in such processes: structures or bodies that are
functionally insufficient to obtain nutrients and other resources from their
environments do not survive.

Consistent with a functional continuity between neural and non-neural
dynamics, morphogenesis shares a number of major features with behavior in
addition to the reliance on bioelectric networks to implement large-scale
coordination. The first is the hardware/software distinction: genomes do not
encode final outcomes, they encode the structure of a system with plasticity,
context-sensitivity, and the flexibility to produce different outcomes from
the exact same hardware. This is why genetically wild-type flatworm cells can
generate head structures appropriate to other species when their bioelectric
signaling is shifted to different attractors [148], why normal skin cells
liberated from frogs spontaneously self-assemble into different, motile proto-
organisms (“Xenobots”) without any genomic editing [149, 150], and why embryos
with severe genetic defects can be bioelectrically coaxed to normal brain
morphogenesis by reinforcing specific voltage patterns [151, 152].
Developmental bioelectric circuits also feature a kind of re-writable memory.
In addition to the default voltage patterns (like the “electric face” [153,
154]), new ones can be written into tissues and maintained by the circuit,
such as the two-headed planaria that result in genetically wild-type worms
when a different $V_{mem}$ pattern memory is temporarily incepted into the
tissue [155, 156]. These two-headed worms continue to regenerate as two-headed
in perpetuity (without additional treatment) or can be bioelectrically
switched back to one-headed [155, 157], illustrating the stable but re-
writable aspects of the information in the tissue that guides behavior.

Bioelectric signaling in all tissues, like in nervous systems, integrates
information across distance to enable decisions and behavior toward adaptive
outcomes in novel circumstances. This basic scheme was already discovered by
evolution as far back as the time of bacterial biofilms [158, 159], and is
exploited very widely across the web of life from microbes to humans [96].
Many aspects of cognitive neuroscience have clear parallels in developmental
biology [160], suggesting that much of the reasoning in §3–5 above applies not
just to nervous systems but in fact to all cells solving problems in various
spaces. Indeed, all living systems are deeply hierarchical, exhibiting aspects
of basal cognition and decision-making at levels including molecular pathways
[161, 162], physiological problem-solving by cells [163, 164], and tissue and
organ memory [165, 166, 167] among others.

##  7 Conclusions and future directions

Neurons, networks of neurons, and biological structures generally exchange
information with their environments via physical interactions interpretable in
terms of measurement (or the gathering of sensory input) and manipulative
action. This interpretation of biological activity forms the basis of the
active-inference principle [20, 21] and has achieved wide currency in the
biophysics, evo-devo, and neuroscience communities. We have shown here how to
formulate this view of biological activity in the very general yet powerful
language of quantum information theory, employing in particular the idea of a
QRF as a calibrated measurement, and in the output direction, a calibrated
action. These results extend the analysis of QRFs as hierarchical Bayesian
systems developed in [43, 44, 45, 47]. They allow us to model processes
implemented by neurons and networks of neurons, from the biomolecular pathway
scale upward, as hierarchies of QRFs. This representation makes explicit where
and how classical information is encoded on successive layers of MBs, enabling
models that explicitly comply with energy-budget considerations. Such models
require that cells employ quantum coherence as a resource for bulk computation
[63]. Hence the current framework represents neurons as explicitly quantum
devices.

Our goal here has been to develop a framework for building detailed models of
specific neuronal cell types in specific environments; these will be pursued
in future work. Even at the current abstract level, however, we are able to
predict generically that dendritic trees will remodel in the direction of
maximal informativeness, with trophic reward as the selection criterion. While
this prediction is generally supported by the phenomenology of dendritic
remodeling [3, 4], it is specifically testable by correlating measures of
branch level activity and branch level active transport. As branch level
remodeling is now known to be involved in neurodegenerative conditions [3, 4],
an understanding of the relations between architecture, computation, and
trophic reward at this scale may be useful in ameliorating these conditions.

From a more general perspective, treating neurons in explicitly quantum-
theoretic terms introduces new possibilities for mathematical modeling, e.g.
with topological field theory as discussed in §3. Neuroscience and computer
science have both benefitted from the abstraction of neurons to sum-threshold
units connected in layered ANNs. As interest in and tractable architectures
for quantum computing continue to develop, we may expect a comparable, but
considerably deeper, connection between biological and artificial computing
systems.

## Acknowledgements

ML gratefully acknowledges funding from the Guy Foundation and the Finding
Genius Foundation.

## Conflict of interest

The authors declare no competing, financial, or commercial interests in this
research.

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