The free energy principle induces neuromorphic development

Abstract

We show how any finite physical system with morphological, i.e. three-dimensional embedding or shape, degrees of freedom and locally limited free energy will, under the constraints of the free energy principle, evolve over time towards a neuromorphic morphology that supports hierarchical computations in which each “level” of the hierarchy enacts a coarse-graining of its inputs, and dually, a fine-graining of its outputs. Such hierarchies occur throughout biology, from the architectures of intracellular signal transduction pathways to the large-scale organization of perception and action cycles in the mammalian brain. Formally, the close formal connections between cone-cocone diagrams (CCCD) as models of quantum reference frames on the one hand, and between CCCDs and topological quantum field theories on the other, allow the representation of such computations in the fully-general quantum-computational framework of topological quantum neural networks.

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Neuromorph. Comput. Eng. 2(2022) 042002 https://doi.org/10.1088/2634-4386/aca7de
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TOPICAL REVIEW
The free energy principle induces neuromorphic development
Chris Fields1,2,∗, Karl Friston3, James F Glazebrook4,5, Michael Levin2,6
and Antonino Marcian`
o7,8,9
123 Rue des Lavandières, 11160 Caunes Minervois, France
2Allen Discovery Center at Tufts University, Medford, MA 02155, United States of America
3Wellcome Centre for Human Neuroimaging, University College London, London WC1N 3AR, United Kingdom
4Department of Mathematics and Computer Science, Eastern Illinois University, Charleston, IL 61920 United States of America
5Adjunct Faculty, Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801 United States of America
6Wyss Institute for Biologically Inspired Engineering at Harvard University, Boston, MA 02115, United States of America
7Center for Field Theory and Particle Physics & Department of Physics, Fudan University, Shanghai, People’s Republic of China
8Laboratori Nazionali di Frascati INFN, Frascati (Rome), Italy
9INFN sezione Roma ‘Tor Vergata’, I-00133 Rome, Italy
∗Author to whom any correspondence should be addressed.
E-mail: fieldsres@gmail.com
Keywords: Bayesian active inference, generative model, quantum reference frame, tomographic measurement,
topological quantum neural network
Supplementary material for this article is available online
Abstract
We show how any finite physical system with morphological, i.e. three-dimensional embedding or
shape, degrees of freedom and locally limited free energy will, under the constraints of the free
energy principle, evolve over time towards a neuromorphic morphology that supports hierarchical
computations in which each ‘level’ of the hierarchy enacts a coarse-graining of its inputs, and
dually, a fine-graining of its outputs. Such hierarchies occur throughout biology, from the
architectures of intracellular signal transduction pathways to the large-scale organization of
perception and action cycles in the mammalian brain. The close formal connections between
cone-cocone diagrams (CCCD) as models of quantum reference frames on the one hand, and
between CCCDs and topological quantum field theories on the other, allow the representation of
such computations in the fully-general quantum-computational framework of topological
quantum neural networks.
1. Introduction
The quest to understand how collections of cells form nervous systems that give rise to cognitive capacities
has driven research into computational systems using architectures observed in neural tissues. The
fundamentals of neuromorphic computing, i.e. computing with systems having functional architectures
similar or analogous to those of biological neurons, can be traced back to the work of Mead [1], who
pioneered the implementation of very large-scale integration (VLSI) methods. These kinds of functional
(neuromimetic) architectures use analog components that approximately mimic neurobiological systems,
and were conducive to solving real-world problems with high efficiency and low cost. Hybrid analog-digital
systems emulating spiking neurons were also developed as an alternative to purely analog models [2]. Since
then, neuromorphic computers have evolved to further emulate the computational architectures of neurons
and of functional networks of neurons (for recent reviews, see [3–10]).
As living systems, both neurons and networks of neurons implement computation, in part, using
morphology; differential delays between signals, for example, can be implemented by dendritic or axonal
processes of different lengths and widths. Changes in morphology also contribute to the implementation of
learning; for example, growing or regressing dendritic spines facilitates or inhibits synapse formation and
hence location-specific interneural communication [11–14]. Spike-based and structural plasticity together
implement memory-write circuits amenable to neuromorphic design [15] (and references therein). At the
network scale, activity-dependent pruning during neural development shapes both short- and long-range
© 2022 The Author(s). Published by IOP Publishing Ltd

Neuromorph. Comput. Eng. 2(2022) 042002 C Fields et al
cortical connectivity [16–18]. Hence from a biological perspective, a key feature of neuromorphic computing
is that it is dynamic: changes in morphology implement changes in computation and vice-versa. This is
exemplified in applications of hybrid analog/digital VLSI devices implemented as neuromorphic vision
sensors that model concept-learning in relatively simple biological neural networks (BNNs), such as
described in [19]10 Neuromorphic computing foregrounds a separation of temporal scales implicit in natural
computation; namely, the distinction between fast inference and slow learning; sometimes considered in the
light of ‘dynamics on structure’ [21]. However, on the neuromorphic view, structure itself is dynamic,
inheriting from fast inference (e.g. activity-dependent plasticity) and scaffolding inference (e.g. cortical
hierarchies and other aspects of functional brain architectures that rest upon synaptic connections) [22]. The
use of morphology as a computing resource is not, moreover, unique to neurons. It is an ancient biological
strategy, employed by plants, fungi, ameboid cells of diverse lineages, and even microbial biofilms [23–30].
All of these systems could, therefore, be considered ‘neuromorphic’ computers.
Here, we review and extend previous theoretical work suggesting that any finite physical system capable
of employing morphology as a computational resource will, if given a sufficiently informative environment
but locally-limited free energy, develop—i.e. evolve in time toward—a ‘neuron-like’ morphology. We show,
in particular, that this outcome can be expected on the basis of the free energy principle (FEP, [31–33])
applied to systems with morphological degrees of freedom but locally-limited free energy. Locally-limited
free energy restricts local measurements to a few degrees of freedom. Predictive power in this case is
maximized if the environment is addressed tomographically. Morphological plasticity is a key enabler of and,
when suitably abstracted, a requirement for tomographic measurements. As the FEP is completely scale-free,
this result applies at any spatiotemporal scale, and is indeed confirmed by systems from the µm scale of
intracellular signaling pathways to the planetary scale of the internet. We suggest on this basis that
morphological plasticity, even if merely simulated, is an important resource for neuromorphic computing.
In what follows, we first outline, in section 2, the fundamental role of morphology as a computational
resource. We then review, in section 3, the basic ideas underlying the FEP, including the definition of
variational free energy (VFE) and its interpretation as Bayesian surprisal, and the key concept of a Markov
blanket (MB, [34,35]) separating a time-persistent system from its environment. Here and in later sections
we employ the formalisms of both quantum and classical information theories; however, we make no claims
about the role, if any, of long-range coherence in biological computation and neither support nor challenge
any particular ‘quantum brain’ models. We show in section 4how defining the MB of a system defines its
environment, and consider the thermodynamics of the MB. We note a critical difference between current
artificial computing systems and organisms: the rigid segregation in the former between free-energy
exchange with the environment (via a power supply and heat exchangers) and data exchange with the
environment (via I/O interfaces or APIs). We then consider MBs as measurement surfaces with
locally-limited free energy resources in section 5, and show how morphological degrees of freedom enable
varying the correlations between measurement sites in nonuniform environments. This enables us to
consider, in section 6, how the FEP drives morphologically-plastic systems—with locally-limited free energy
resources—to measure the environment’s state tomographically, using measurements made at different
locations to reconstruct a best predictive model of the state. We provide a fully-general physical model of this
process in section 7, employing the formalism of topological quantum neural networks (TQNNs, [36]). This
shows that TQNNs provide general models of neuromorphic systems. We conclude with implications,
predictions, and next steps in section 8. As we employ formalism and concepts from several disciplines, a
glossary of terms is provided in supplementary material, appendix A.
2. Morphology as a computational resource
Abstract models of computation, e.g. the lambda calculus [37] or the Turing machine [38], make no mention
of morphology. In conventional computing systems, the three-dimensional (3d) layout or ‘shape’ of the
computer is chosen to minimize transmission delays and maximize heat dissipation capacity. Biological
systems, in contrast, possess evolved morphologies that determine, in part, how they are able to interact with
their worlds. The morphological structures of even single, free-living cells such as amoeba or paramecia
encode both intergenerational memory and immediate capabilities for action [39]. The key role of
morphology as a computational resource in robotics has been emphasized by Brooks [40] and by the
‘embodied cognition’ movement more generally (see [41] for review).
10 As reported in [19], the common honeybee stands out as an exemplar having remarkable capabilities for conceptualizing and categor-
izing (the bees having ≈106neurons compared to ≈1011 neurons in the human brain); in particular, their ability to distinguish between
‘odd’ and ‘even’ numeric quantities [20].
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Implementation in 3d space, and hence morphology, imposes two fundamental constraints on
computing systems that abstract models ignore. The first is thermodynamic: any irreversible encoding of
classical information—writing a bit value to a memory for later retrieval—has a finite free energy cost, at
least ln2kBT, where kBis Boltzmann’s constant and Tis ambient temperature [42–44]. This free energy must
be obtained from the environment, via a power supply, photosynthesis, or metabolism. The second
constraint is on informational coupling: the implementation determines how inputs are obtained from the
environment and how outputs are transferred to it. It determines what features or aspects of the
environment the computing system can detect—or in psychological language, perceive—and similarly, what
features or aspects of the environment it can directly affect by its actions. It is the response to this second
constraint that most strongly distinguishes ordinary computers from robots.
The effects of morphology become obvious when comparing conventional artificial neural networks
(ANNs) to networks of biological neurons. Setting thermodynamics aside, the ‘environment’ with which an
ANN interacts comprises sets of training and test data. It is commonplace to think of an ANN as interacting
with, for example, images that have a spatial (here 2d) structure. This, however, is an anthropomorphism;
the ANN in fact interacts with sets of finitely-encoded and therefore rational numbers. We can consider a
node in a layered, feedforward ANN to have the following structure:
where here {xi}is the set of input values from upstream nodes, {∆i}is the set of training (backpropagated
error) values, and the rational number ois the output. The ‘sensed environment’ of this node is the ordered
pair ({xi},{∆i}); the ‘acted-upon environment’ of the node is the rational number o. We will see in
section 3below that these sensed and acted-upon sectors of the environment can be represented formally as
sectors of the node’s MB.
Note that drawing the node as diagram (1) imposes on it a ‘morphological’ degree of freedom, namely its
layout on the 2d Euclidean surface of the page. This, in turn, imposes orders onto the sets {xi}and {∆i},
making them vectors with the obvious metric. This morphological degree of freedom is not, however,
intrinsic to the node; it appears nowhere in a mathematical specification of the function that the node
computes, nor does it characterize the (completely abstract) sets {xi}or {∆i}or the number o. This absence
of morphology is, more than absence of hierarchical structure or spiking (which biological neurons can lack),
what renders a node in an ANN non-neuromorphic. Nodes in ANNs are non-neuromorphic because they
are amorphic; they have no morphology.
Implementing an ANN in hardware gives it a morphology: the 3d morphology of the hardware. It also
confers a resource requirement for thermodynamic free energy; hence it adds a thermodynamic sector to its
environment, which in section 3below will become a thermodynamic sector of its MB. This exposure to
energetic exchange with the environment renders the implemented ANN a ‘thing’ in the language of the FEP.
How the implemented ANN behaves, i.e. how it regulates its energetic exchange with its environment,
determines whether it will persist over time. This regulation of energy exchange in service of persistence, or
survival, is the core meaning of embodiment. The ever-present possibility of dysregulation is what renders
embodiment ‘precarious’ [45].
Let us now consider a biological neuron, which is by definition embodied and therefore has a
morphology. A neuron’s sensed environment is, like the sensed environment of any other system, defined by
the sensory structures that it deploys. In the case of a neuron, these are mostly post-synaptic specializations,
including clusters of post-synaptic receptors and channels as depicted in [46], figures 4(a) and (b); we will
focus on these at the expense of more uniformly distributed biochemical and bioelectric sensors. The
neuron’s sensed environment is then the set {si}of activations detected by these post-synaptic
specializations. The neuron’s acted-upon environment is, similarly, the set {ai}of activations generated by
its pre-synaptic specializations, again ignoring more uniformly-distributed pumps, secretory systems, etc.
These sets {si}and {ai}comprise the neuron’s MB. Perception and action are linked together by the
dynamics on the internal states, which are supported by all internal degrees of freedom of the cell, including
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genome, mitochondria and other organelles, cytoskeletal network, etc; these internal dynamics implement
the cell’s generative model. Hence while it is commonplace to think of a neuron as ‘detecting pressure’ or
‘exciting a muscle’ these descriptions are possible only from a larger, tissue-scale perspective. From the
neuron’s own perspective, it is acting to regulate the bioelectrochemical gradients it detects as state variations
of {si}. See [47] for a worked example of dendritic self organization, in terms of structure learning, using the
minimization of VFE to implement model selection in terms of dendritic spines. In this example, the
morphology of the dendritic tree aligns itself with the temporal sequence of presynaptic inputs that itself
depends upon morphology of the neuropil [48]. However, at no point does the (synthetic) neuron ‘know’ its
morphology.
Both inputs to and outputs from a neuron are organized spatially by its morphology. However, the
neuron itself cannot detect or represent its morphology, though local changes in morphology are locally
detectable, e.g. by differential strain on the cytoskeleton. Hence the neuron’s inputs and outputs remain, for
the neuron itself, only sets without structure. The overall function of the neuron, and hence the values of its
outputs, depend however on its computational (i.e. message-passing) architecture and hence on its
morphology. It is the role of morphology in determining function—and hence action on the
environment—that the FEP explains [46].
The thermodynamic constraints faced by neurons—and indeed on any implemented computing
system—force them to trade off the energetic requirements of obtaining data against those of processing
data. As discussed in section 6below, biological systems respond to this tradeoff by coarse-graining, i.e. by
compressing their outputs into fewer bits than employed for their inputs. Diagram (1) illustrates this in an
extreme case. Coarse-graining leads naturally to a hierarchical computational architecture, and hence to a
hierarchical morphology. The combination of high fan-in and hierarchical morphology is spectacularly
evident in mammalian cortical neurons, with their elaborate, layer-specific dendritic trees and upwards of
10 000 input synapses [49,50].
Our goal in what follows is to understand the role of morphology—and in particular, of neuromorphic
morphology—as a resource for computation from first principles. To do this, we adopt the formal
framework of the FEP, which is applicable, in principle, to any physical system at any scale. After introducing
the FEP in the next section, we focus in sections 4–6on how the structure and functions of the MB
surrounding any system define its interaction with its environment and hence shape its morphology. We then
show in section 7how these effects of morphology can be captured in terms of fundamental physical theory.
3. The FEP as a general physical principle
Since its application to brain function [31,51–53], the variational FEP has been extended into an explanatory
framework for living systems at all scales [32,54–57], along with an extensive scope of related clinical studies
(e.g. [58–61]). When formulated as a general principle of classical physics, it characterizes the behavior of all
random dynamical systems that remain measurable, and hence identifiable as distinct, persistent entities,
over macroscopic times [33]. To summarize, it is shown in [33] that any system that has a non-equilibrium
steady state (NESS) solution to its density dynamics (a) possesses an internal dynamics that is conditionally
independent of the dynamics of its environment, and (b) will continuously ‘self-evidence’ by returning its
state to (the vicinity of) its NESS. Condition (a) can be thought of as a precondition for any system to have a
‘state’ that is clearly distinct from the state of its environment; it is effectively the requirement that system
and environment are weakly coupled. Given weak coupling and local interactions, the joint
system–environment state space can be partitioned into internal (i.e. system), external (i.e. environment)
and intermediary MB states. The MB surrounding a set {µ}of internal states comprises all states that are
parents of states in {µ}, children of states in {µ}, or non-{µ}parents of the children of states in {µ}, where
as usual, the ‘parents’ and ‘children’ of a state µare states with causal arrows into and out of µ, respectively
[34,35]. The MB states can, in turn, be partitioned into sensory states that mediate the influence of external
states on internal states and active states that mediate the influence of internal states on external states. In the
language of perceptual psychology, the MB functions as an ‘interface’ [62] that encodes perceptions and
actions. It is worth emphasizing that the MB states are elements of the joint system-environment state space;
while the MB states are embedded in a physically-continuous spatial boundary in canonical examples such as
biological cells, this is not required by the definition of an MB. With this partitioning, Condition (b) then
requires that the system behaves so as to preserve the functional integrity of its MB, i.e. that its dynamics does
not diverge following a perturbation. The FEP is the statement that any measurable, i.e. bounded and
macroscopically persistent, system will behave so as to satisfy these requirements.
More formally, the FEP is a variational or least-action principle stating that a system enclosed by an MB,
and therefore having internal states µ(t)that are conditionally independent of the states η(t)of its
environment, will evolve in a way that tends to minimize an VFE that is an upper bound on (Bayesian)
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surprisal. This free energy is effectively the divergence between the variational density encoded by internal
states and the density over external states conditioned on the MB states. If πis a ‘particular’ state π= (b,µ),
where b(t) is the state of the MB, the VFE F(π)can be written [33, equation (2.3)],
F(π) = Eq(η)[lnqµ(η)−ln p(η, b)]
| {z }
Variational free energy
=Eq[−ln p(b|η)−ln p(η)]
| {z }
Energy constraint (likelihood&prior)
−Eq[−lnqµ(η)]
| {z }
Entropy
=DKL[qµ(η)|p(η)]
| {z }
Complexity
−Eq[ln p(b|η)]
| {z }
Accuracy
=DKL[qµ(η)|| p(η|b)]
| {z }
Divergence
−ln p(b)
| {z }
Log evidence
⩾−ln p(b).
(2)
The VFE functional F(π)is an upper bound on surprisal (a.k.a. self-information)
I(π) = −logP(π) = −ln p(b)because the Kullback–Leibler divergence term (DKL) is always non-negative.
This KL divergence is between the density over external states η, given the MB state b, and a variational
density Qµ(η)over external states parameterized by the internal state µ. If we view the internal state µas
encoding a posterior over the external state η, minimizing VFE is, effectively, minimizing a prediction error,
under a generative model supplied by the NESS density. In this treatment, the NESS density becomes a
probabilistic specification of the relationship between external or environmental states and particular
(i.e. ‘self’) states. We can interpret the internal and active states in terms of active inference, i.e. a Bayesian
mechanics [63], in which their expected flow can be read as perception and action, respectively. In other
words, active inference is a process of Bayesian belief updating that incorporates active exploration of the
environment. It is one way of interpreting a generalized synchrony between two random dynamical systems
that are coupled via an MB.
We have recently reformulated the FEP within a scale-free, spacetime background-free quantum
information theory [64]. Quantum information theory provides a particularly simple and convenient
representation of physical interaction as information exchange; supplementary material, appendix B
provides a brief review. In this representation, the MB is implemented by a decompositional boundary in the
joint system-environment Hilbert space that functions as a holographic screen, a topological generalization
[65,66] of the original geometric construction [67–69]. The criterion of conditional independence is
implemented by the quantum-theoretic notion of joint-state separability, i.e. absence of entanglement across
the holographic screen. The action of the internal system dynamics implements a quantum computation,
which can be decomposed as a hierarchy of quantum reference frames (QRFs, [70,71]). Decomposition into
QRFs has the advantage of assigning an explicit semantics, interpretable as a system of units of measurement,
to each ‘thread’ of the computation. It is this semantic information that renders measurement outcomes
comparable across instances of measurement, and hence renders them ‘differences that make a difference’
[72,73], that is, differences that are actionable. Each QRF can, in turn, be given a functional specification as a
category-theoretic structure, a ‘cone-cocone diagram’ (CCCD) of Barwise–Seligman [74] classifiers. Such
CCCDs specify semantically-interpreted information flows, where the semantics are given by the satisfaction
conditions of the classifiers, within distributed systems (reviewed in [75,76]; see also supplementary
material, appendix C). In informational/logical terms a CCCD specifies ‘measurement’ and ‘preparation’ as
dual memory read/write operations. We have employed this representation to characterize neurons as
hierarchical measurement devices [46] as discussed further in section 4below. Didactically, this allows one to
think of perception and action in terms of measurement (read) and preparation (write) operators that stand
in for answers (outputs) and questions (inputs), about or from the environment as discussed below.
In either classical or quantum formulations, the FEP provides a generic theory of self-organization for
physical systems with sufficient dynamical stability to be identified over time and subjected to multiple
measurements, i.e. systems that can be considered ‘things’ that are distinct from their surrounding
environments (see especially the discussion of this point in [33]). The MB of any such ‘thing’ underwrites its
conditional independence—between its internal states and the external states of its environment—by
localizing and thereby restricting information exchange between them. Hence the FEP provides a generic
characterization of physical interaction as information exchange, and a generic characterization of internal
system dynamics as (Bayesian) inference [33,63,64].
This reading of self organization—sometimes referred to as self evidencing [77]—rests, in a foundational
way, on the notion of a generative model. Technically, this generative model can be associated with the NESS
density over the particular partition of systemic states described above. This density can be factorized into a
likelihood (the density over particular states, given their causes; i.e. external states beyond the MB) and a
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prior density over particular states that are characteristic of the particle or ‘thing’ in question. The states
constitute the attracting set that underwrites the NESS solution to density dynamics. In short, if there exists
an MB—defined in terms of conditional dependencies under an NESS density—then there is a lawful
description of systemic dynamics that can be cast as gradient flow, asymptotically toward the NESS [33], on a
free energy functional of a generative model. Teleologically, the generative model specifies the states to which
self-organization (i.e. evidencing) are attracted; namely, the characteristic or preferred states of the ‘thing’ in
question. The role of a generative model will be foregrounded in what follows; simply because the structure
of a generative model underwrites the dynamics and message-passing we associate with self organization.
4. Defining the MB defines the environment
4.1. Informative versus uninformative sectors
The partitioning of ‘everything’ into ‘system’ and ‘environment’ (where in equation (2) the MB is considered
part of the system) built into the FEP formalism has the immediate consequence that every system, by
definition, interacts with exactly one other system, its environment. The formalism is, moreover, completely
symmetric: the system maintains a well-defined, conditionally-independent state if and only if its
environment does as well. We can, indeed, think of system and environment as comprising a generative
adversarial network, with each side adapting, as its resources allow, to the other’s actions [78]. This
symmetry is particularly manifest in the quantum formalism, which is a completely general representation of
two systems (i.e. components of a bipartite Hilbert-space decomposition) open to interaction exclusively
with each other.
This exclusive coupling of system to environment has two consequences, both of which have been
explored more explicitly within the quantum formalism [64]; see also [65,79,80] for further discussion.
First, all (thermodynamic) free energy acquired by the system from, and all waste heat dissipated by the
system to, its environment must traverse the MB. The MB (or in the quantum formulation, the holographic
screen B), is thus partitioned into ‘informative’ (or ‘observed’) and ‘uninformative’ (or ‘unobserved’)
sectors as shown in figure 1. The function of the uninformative sector is purely thermodynamic; formally, it
exchanges the free energy required to support irreversible classical computation [42–44].
The second consequence of the system–environment decomposition is that the system of interest Shas
no access to the decompositional, or in quantum terms entanglement, structure of its environment E. Any
‘objects’ detected by Sin Eare in fact sectors of mutually correlated components of the state of the MB, or in
the quantum formulation, sectors of mutually correlated bits encoded on the screen B[64,65,79,80]. The
informative sector of the MB can, therefore, be thought of as implementing an applications programming
interface (API) between Sand E. Read and write operations to this API are implemented by the internal
dynamics of S(respectively, E). In the quantum formulation, these are implemented by QRFs that effectively
define the ‘data structures’ encoded on each face of B.
4.2. Learning is learning a message-passing structure
On the classical view of the Bayesian mechanics entailed by the FEP, the minimization of VFE can be usefully
considered at different timescales. For example, optimizing the states or activities of a BNN or ANN is
distinct from optimizing the connections or weights; which is distinct from optimising the structure of the
neural network per se. These three aspects of VFE minimization map neatly to the distinction between
inference, learning and model selection (a.k.a., structure learning), respectively. We start with this
observation because, to anticipate the discussion in section 4below, the structure just is the computational
architecture in question and thereby specifies the nature of the message-passing entailed by inference and
learning. Morphology in 3d physical space is an implementation resource for computational architecture, as
3d layout in VLSI exemplifies.
On this view, the structure or morphology of any ‘thing’ is subject to the same imperatives as the
message-passing; namely to maximize morphological or model evidence (or minimize the associated VFE
bound). This can be cast as structure learning in radical constructivism [81–83], or Bayesian model selection
in statistics [84]. The implication here is that any morphology must be a ‘good’ model of how its sensory
states are caused by external states. This is just an expression of the good regulator theorem from early
formulations of self organization in cybernetics [85,86]. In other words, statistical correlations beyond the
MB must be installed in the generative model, in terms of sparse coupling (i.e. message-passing) among
internal states (which themselves are ‘things’ equipped with MBs). So what kind of structures, architectures
or morphology might one expect to find in things that are good models of their external milieu?
If morphology maximizes model evidence, then any morphology effectively encodes a model specifying
expectations about the environment. Cell membranes, for example, can be viewed as encoding expectations
about viscosity and ambient chemical potentials, and skeletal systems can be viewed as encoding
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Figure 1. (a) A system Sis separated from its environment Eby a holographic screen Bthat implements an MB. Note that this
depiction is purely topological; no geometry is assumed for either the joint system SE or the boundary B. (b) Both sensation (s)
and action (a) states on the screen Bare divided into informative (i.e. data I/O) and uninformative (i.e. thermodynamic I/O)
sectors (clear versus hatched areas). Reproduced from [78]. CC BY 4.0.
expectations about gravity and the buoyancy of media such as air or water. These morphology-encoded
models should comply with Occam’s principle—or Jaynes maximum entropy principle [87,88]—in virtue of
having minimal complexity. This follows from the fact that log evidence (i.e. negative surprisal) is accuracy
minus complexity. Equation (2) shows that complexity is the degree of belief updating incurred by
message-passing. Technically, complexity is the KL divergence between posterior and prior, before and after
belief updating. In short, a ‘good’ model is that which provides an accurate account but is as simple as
possible. In turn, this requires the right kind of ‘coarse graining’ or compression [89], to provide an accurate
explanation for impressions on the sensory part of the holographic screen implemented by the MB. So, what
kind of coarse graining might emerge in a Universe that features probabilistic structure?
This explanation can only be in terms of ‘things’ and their lawful relationships as described below. At this
point, one can conjecture that things—and the (space-time) background that describes their relationships in
a parsimonious fashion—would feature in the structure of generative models or morphology. This is evinced
in a compelling way by neuroanatomy, which speaks to a distinction between ‘what’, ‘where’ and ‘when’ in
carving the sensorium at its joints. For example, one of the most celebrated aspects of brain connectivity is
the separation of dorsal and ventral streams that are thought to encode ‘where’ and ‘what’ attributes of visual
objects, respectively [90]. The argument here is that knowing ‘what’ something is does not tell you ‘where’ it
is and vice versa. This statistical independence translates into a morphological separation between the dorsal
and ventral streams. This separation minimizes complexity and thereby maximizes the efficiency of
(variational) measure-message passing and belief updating in terms of statistical, algorithmic and
thermodynamic complexity costs [54,91]. Similar arguments can be made for a separation of ‘what’ and
‘when’ [92]; in the sense that knowing ‘what’ something is, does not tell you ‘when’ it was ‘there’.
This kind of coarse graining (c.f., carving nature at its joints) is ubiquitous in statistics and physics, where
it emerges in the guise of mean field approximations; namely, factorizing a probability density into
conditionally independent factors [22,93–96]. Indeed, VFE and message passing are defined under a mean
field approximation to a posterior density [91,97]. Another important structural or morphological feature
of ‘good’ generative models is their deep or hierarchical structure, with an implicit separation of scales in the
genesis of—or explanation for—sensory impressions.
The common theme here is a morphology underwritten by the sparsity or absence of message-passing on
some factor graph. This foregrounds the imperatives for shielding or sequestering various internal states
from other internal states, which brings us back to MBs; however, these are internal MBs that define an
internal morphology or message-passing structure. One might conjecture that much of biological
self-organization is concerned with isolation and shielding, as a necessary part of internal autopoiesis
(e.g. the role of enzymes and catalysts, gap junctions, and many other highly controllable mechanisms for
setting up signaling paths and boundaries [98–101]. This occurs at all scales, from subcellular organelles that
partition biophysical and chemical reactions to nascent organ compartment boundaries, to the dynamics
that guide which members of a swarm pass messages to which others [102–106].
In this respect, morphogenetic self-organization, seen as a pattern formation, requires each individual
cell (and/or its progeny) to occupy its own place in the final morphology, and autopoietic self-assembly
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results only when each cell successfully detects local patterning signals as predicted by its own generative
model [57,107]. Morphological development thus implies a pre-determined patterning to which a cell
ensemble converges—the so-called Target Morphology [108,109]. Note that this is an essentially classical
statement; it assumes the existence of effectively-classical boundaries and hence distinctions between cells. If
each cell minimizes VFE then it infers its correct location and its function within the ensemble [57,107,110].
Examples in the case of neurons include assortative neuronal migration towards groups with very close or
identical node degrees [111,112] and amalgamation of groups with a common stimulus, following which
they ‘cast a vote’ to decide on how to proceed collectively [113]. More generally, the Good Regulator
Theorem again applies when each cell, by evidencing its own existence, can vouch inferentially for the same
model as the one of the local group in which it is accommodated. In this way, the cell contributes to the
eventual release of effective signaling by the ensemble to other formations. This is a basis for a theoretical
framework of autopoiesis expressed in terms of VFE minimization, and hence active inference [107].
A final consideration—afforded by the classical FEP—is that the same Bayesian mechanics must apply in
a scale-free fashion [33,110,114,115]. In other words, MBs of MBs (i.e. things composed of things) must
evince the same kind of message-passing. For example, the intracellular components of a single cell must
have the right morphology to maintain the cell’s MB (e.g. a cell surface). Similarly, the ensemble of cells that
constitute a multicellular structure must be so structured to maintain the MB of the tissue or organ in
question (e.g. a somatic cell on its endothelial surface) [116,117]. In a similar vein, this implies that the
message-passing between MBs (e.g. cells and organs through to conspecifics and cultures) must (look from
the outside as if they) comply with the same free energy minimizing imperatives. This translates into efficient
communication at the level of intracellular communication, through to languages with minimal algorithmic
complexity. In short, message-passing between ‘things’ should incur the minimum amount of belief
updating, while communicating as accurately as possible. In what follows, we will see these themes
re-emerge, both in terms of biological intelligence and quantum information theory. The semi-classical limit
of a TQNN model, in particular, constructs generalizations with the shortest possible trajectories and the
maximum topological information as discussed in section 7.
4.3. Object identification by QRFs
From the perspective of an observer S, ‘things’ are located in the environment E. As is obvious from the
definition of an MB, however, Scannot ‘see’ E;Scan only detect encodings on its MB B. A ‘thing’ for Sis,
therefore, a cluster of bits on Bwith high mutual information and hence high joint predictability.
Recognizing a ‘thing’—determining that some bits have high mutual information—requires multiple
measurements. In particular, any ‘thing’ Xcan be considered to have two components, a ‘reference’
component Rthat maintains a constant (up to measurement resolution and relevant coarse-graining) state
(or state density or expectation value), and a ‘pointer’ component Pwith a time-varying state that Sconsiders
‘the state of interest’ of X[64,65,79,80]. Ordinary items of laboratory apparatus provide a canonical
example; one can only identify a voltmeter or an oscilloscope if most of their state variables—size, shape,
brand name, etc—remain fixed while the ‘pointer’ variables vary to indicate some measured value [118].
Measurements of the states of Rand Pcan, without loss of generality, be regarded as implemented by
QRFs [64,65,79,80]. A QRF is simply a physical system with which a measurement is enacted; such a system
is a quantum reference frame because, being physical, it must at some suitable scale be regarded as a quantum
system, and at that scale it encodes unmeasurable, and hence unencodable or ‘nonfungible’ [71] quantum
phase information. Such systems are intrinsically semantic: they report not just values, but also units of
measurement that render such values mutually comparable. Even a non-standardized QRF such as the length
of one’s arm defines a unit of measurement, although an idiosyncratic one. Hence repeated observations,
which must determine at minimum the state of Rand are therefore measurements, are intrinsically semantic:
they are actions on the world that yield mutually-comparable, and hence actionable observational outcomes.
In a quantum theoretic formulation, measurement and its dual, state preparation, have the same formal
representation; a ‘preparation’ process is just a measurement reversed in time. A QRF is, therefore, a
preparation device as well as a measurement device: one can prepare a 0.75 m board with a meter stick, just as
one can measure a 0.75 m board. Preparation is an action on the environment; preparation and measurement
together constitute interaction. Indeed any physical interaction can be considered a sequence of alternating
preparation and measurement steps, as shown in detail in [64]. This duality is preserved in the classical
formulation, but remains implicit (i.e. perception as time-reversed action and vice versa). As we have pointed
out, the dual character of preparation and measurement as enacted by QRFs allows their representation, in
full generality, by category-theoretic structures, namely, the CCCDs Barwise–Seligman classifiers [74], as
constructed in [75,76]; formal definitions and examples are given in supplementary material, appendix C.
This representation has been extensively applied in computer science as reviewed in [75]; we prove its
generality in the present setting in [76], to which we refer for details. Such structures have the form:
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where the Aiare Barwise–Seligman classifiers and C′, also a classifier, is the category-theoretic limit of the
outgoing maps hiand the colimit of the incoming maps fi. The diagram shown in equation (3) is required to
commute, i.e. all directed sequences of maps from any node to any other node are equivalent. The
construction developed in [76] further places these diagrams within the context of general graph (e.g. ANN)
networks; in particular, the form of diagram (3) clearly suggests a variational auto-encoder.
Structures of the form of diagram (3), provided that they all mutually commute, can be assembled into
hierarchies of the form:
where here we have suppressed the outgoing arrows from C, and hence the mirror-image ‘upper half’ of the
diagram that is shown explicitly in diagram (3), for ease of illustration. Such diagrams represent
simultaneous actions by multiple QRFs, or alternatively, the construction of a functionally more complex
QRF from simpler QRFs. Failure of commutativity prevents such assembly, and can be interpreted as
indicating quantum (or ‘true’) contextuality; we do not pursue this here, but refer to [64,119] for extensive
discussion.
Diagram (4) resembles a dendritic tree in combining high fan-in with a hierarchical structure. It is this
generic functional form that allows the representation of neurons as hierarchies of QRFs [46]. We will show
in what follows that ‘neuromorphic’ structures of this form follow as a consequence of the FEP whenever two
conditions are met: the existence of morphological degrees of freedom and the constraint of locally-limited
(thermodynamic) free energy.
4.4. The environment in practice: spaces and contexts
The idea that any system interacts with ‘its environment’ as a whole follows immediately from the concept of
an MB or a holographic screen, which renders the environment a ‘black box’ of indeterminate internal
structure [120,121]. This is counter-intuitive, as we tend to regard our own interactions as interactions with
specific, identified objects. In fact, our interactions are with, or more properly via, QRF-identified sectors of
our MBs as described above. This is the case for all finite physical systems that interact only weakly with their
environments, i.e. for all systems that possess MBs.
Our human intuition and ordinary language similarly identify ‘the environment’ as our perceived
environment, which we tend to regard as ‘objective’ or observer-independent. When we consider systems at
different scales and with different QRFs, however, it becomes clear that their ‘environments’ are very
different from ours. Robbins et al [122] have emphasized, for example, that microbes interact with the world
primarily biochemically: the microbial environment is one of varying chemical potentials. Most mammals
retain much more sensitivity to the chemical potentials of the environment, via olfaction, than we do. In
general, both living systems and physical systems more generally can be characterized as sensing and acting
in a variety of ‘spaces’ instead of or in addition to the 3d space that we intuitively regard as a ‘container’ for
all others. These include (bio)chemical, (bio)electric, and morphological spaces at scales from the molecular
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to the macroscopic [123]. Systems can be characterized by ‘cognitive light cones’ that specify their sensory,
action, and memory capacities in each such ‘space’ that they inhabit [117,124,125].
When the ‘perceived environment’ of a system is understood in terms of QRF-identified sectors of the
system’s MB, it becomes obvious that the entire state of the MB, including its thermodynamic sector,
provides the ‘context’ of any observation. The ubiquity of context effects has been emphasized by Dzhafarov
et al, where it forms the basis of the ‘contextuality by default’ approach to statistical analysis of empirical data
[126–129]. Conceptually similar, but employing different methods, is the work of Abramsky et al [130–133].
Classical and quantum (i.e. entanglement-based) context effects, both of which can be observed even at the
macroscopic scale of human behavior [134,135], have been rigorously distinguished to a degree of generality
by commutativity relations between QRFs [76,119]. Context effects cross the boundaries between ‘spaces’
detected by distinct QRFs; for example, biochemical and bioelectric effects cross-modulate each other not
only in neurons, but in all cell types across phylogeny [136–140].
Systems with morphological degrees of freedom, including macromolecules, cells, multicellular
organisms, and multi-organism communities, as well as many artifacts, can employ these degrees of freedom
to segregate QRFs from each other and hence to regulate context effects. Folded proteins, for example, can
separate distinct binding sites in 3d space, and bacteria can segregate chemotactic receptors from
flagellar-motor proteins. Such segregation allows interacting with different parts of the environment in
different ways. We focus below on characterizing morphological degrees of freedom, and examining the
consequences of QRF segregation for computational architecture.
5. MBs with morphological degrees of freedom
The state bof the MB, or of the screen B, and in particular the state (s,a) of its informative sector, has thus
far been considered a state in some arbitrary (e.g. Hilbert) state space. In particular, no positional (e.g.
ordinary Euclidean 3d spatial) degrees of freedom have been assumed. We now add to the MB states, as a
parameter, an ancillary ‘morphological’ degree of freedom ξthat, as we will see, in naturally interpretable as
a spatial degree of freedom. This ancillary degree of freedom is ancillary in the sense of having no effect on
the total system–environment information exchange across the boundary B; in the purely-topological
notation of figure 1, the interaction HSE does not depend on ξ. As we show below, however, S’s QRFs
partition Binto sectors that are ‘localized’ in the space defined by ξ. Hence ξusefully parameterizes HSE in a
way that a neuron can take advantage of by varying its morphology to selectively deploy its QRFs to specific
sectors of B. We can, therefore, regard HSE as depending locally on ξ; this will be made explicit in section 7
below when we assign spatial coordinates to the input states of a TQNN.
We further assume that the states |ξ⟩of ξ(here adopting the Dirac notation for states) are vectors and
hence provide a distance measure ⟨ξ|ξ′⟩. This effectively ‘geometrizes’ the states bby assigning to each a
‘location’ ξand allowing ‘distances’ between states to be calculated. In this way ξplays the role of the 2d
geometry of the page in diagram (1); it allows the states bto be placed in an ordered array with the
dimension of ξ. From a physical perspective, the simplest geometrization of B(embedded in a 3d space)
represents the space of MB states bas a 2d array of qubits (it hence considers a minimal binary encoding of
the states band implements each bit with a quantum bit, e.g. a spin degree of freedom), and positions each
qubit in a voxel of volume 2∆x×2∆x×2c∆tas shown in figure 2, where ∆xis the minimal ‘grain size’ of
space, ∆tis the minimal time to encode one bit, and cis the maximal speed of ‘causal’ classical information
transfer. If ∆xand ∆tare the Planck length and time, respectively, this reproduces the idea of a ‘stretched
horizon’ subject to the original, geometric holographic principle, which encodes information at the
maximum density given by the Bekenstein bound [67–69]. For biological systems at temperature T∼310 K,
∆t∼50 fs and ∆x∼1 Å, and cis the speed of bond-vibration waves in macromolecules [141], a scale
roughly 25 orders of magnitude larger than the minimum set by quantum theory.
In order to model neurons, we will assume that ξhas an ‘embedding’ dimension in addition to the ‘2 +1’
space +time structure shown in figure 2. As we will see in section 6.3 below, the interpretation of this extra
dimension depends on the QRFs available to measure it.
Our two principal assumptions can now be stated:
(a) The state (s,a) of the informative sector of the MB/screen Bis non-uniform in ξ. Parameterizing HSE
with ξtherefore reveals local structure in HSE.
(b) The free energy available via the uninformative sector of the MB/screen Bis sufficiently limited so that
only a ‘few’ cycles of classical computation can be performed on each bit in the informative sector.
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Figure 2. One qubit degree of freedom (represented as a Bloch sphere), e.g. a spin, embedded in a 3d voxel at some minimal scale
∆x,∆t. Here cis the maximum speed of (classical) information transfer.
We will, for simplicity, also assume that the state of the uninformative sector has two components, each
with a uniform state. This allows us to treat thermodynamic exchange with the environment as an
interaction with ‘hot’ and ‘cold’ heat baths that supply free energy and exhaust waste heat, respectively.
Qualitatively, assumption (a) assures that the informative sector of Eis potentially interesting to S,
i.e. not perceived merely as noise, while assumption (b) limits S’s ability to ‘make sense’ of Eby performing
predictive computations. These assumptions thus keep Sin a regime of finite VFE, avoiding the ‘prefect
prediction’ limit of VFE →0 that, as we show in [64], corresponds to loss of separability (i.e. to an approach
to quantum entanglement) between Sand E.
On a classical view, these assumptions express the FEP in terms of maximizing accuracy (assumption (a)),
while minimizing the complexity cost of belief updating (assumption (b)). In machine learning, a failure to
minimize complexity leads to overfitting [142] that can be read as the perfect prediction limit (i.e. quantum
entanglement)—topological features of TQNNs, connected to topological quantum field theory (TQFT),
have been advocated [143] to explain generalization by DDNs, avoiding both underfitting and overfitting.
We can now develop our main result, showing in section 6below that given limited free energy, the FEP
imposes a hierarchy on the structure of computation over the informative state (s,a) that, when
parameterized by the morphological degree of freedom ξ, becomes a tomographical computation defined
over effectively ‘spatial’ measurements and producing effectively ‘spatial’ actions. This computation
represents the informative sector of Eas comprising ‘objects’ that interact ‘causally’ against a
spatially-extended background ˜
Eof non-objects. We then show in section 7that this action of the FEP can be
captured in full generality in the formalism of TQNNs, making explicit that ‘space’ is emergent from
connection topology. In this formalism, the role of the spatial embedding (i.e. of ξ) is to enforce a
coarse-graining in which the ‘objects’ detected by Sare separable and hence statistically conditionally
independent. This is exactly the role of ‘space’ in quantum field theories [66]. It allows the mean-field
assumption that allows us, as discussed above (section 4.2) in the classical setting, to talk about objects as
persistent entities with their own MBs.
6. Tomographic measurements minimize VFE
6.1. ‘Objects’ as sectors in E
We have previously demonstrated the converse of our desired result: that if the bits encoded on a sector
X=RP of Bhave sufficient mutual information to satisfy the logical criteria (e.g. as encoded by a CCCD)
implemented by some QRF X=RP over some macroscopic time interval τ—and in partcular, if the
measured state density ρRof Rremains fixed throughout τ—then Xwill appear to the observer Sto be an
‘object’ or ‘persistent thing’ during τ. In particular, the state |RP⟩(or density ρRP) will appear to be
decoherent from (i.e. not entangled with and hence conditionally independent from) the remainder of B
[64,65,79,80]. Decoherence corresponds, classically, to conditional independence [144,145]. It is what
makes ‘thingness’ and behavioral predictability possible.
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Assumption (a) above states that Econtains potentially-detectable objects; stated more carefully,
assumption (a) states that Bincludes sectors that encode bits with significant mutual information. These
sectors present Swith opportunities for predictability, i.e. for local (on B) reduction of VFE. Our question is
then: what computational (i.e. message-passing) structure can take advantage of ‘islands’ of predictability to
minimize VFE while remaining within the free-energy constraint imposed by assumption (b)?
6.2. Hierarchical measurements optimize the accuracy/complexity cost tradeoff
While pure quantum (i.e. unitary) computation costs no free energy, Landauer’s Principle imposes a finite
cost on classical bit erasure and hence on classical memory updating [42–44] as noted above. Assumption
(b), therefore, effectively limits the writing of classical memories. Recording previous measurement
outcomes for comparison with future ones is, therefore, the energetically-limited step in computing
predictions. Markov kernels with rational matrix elements provide, for a given (finite) measurement
resolution, the most efficient representation of prior measurement outcomes and hence of prior probability
distributions [64]. Hence the fundamental energetic tradeoff faced by any VFE-minimizing system—that is,
any system compliant with the FEP—is a tradeoff between the resolution with which both prior and
posterior probabilities are encoded and the predictive power that they provide11.
The optimal solution to the above tradeoff is, obviously, to only encode probabilities that actually
contribute to predictive power. Hence we can expect the FEP to drive systems toward identifying and
processing input data from only those sectors of their MBs that encode bits with high mutual information,
i.e. high redundancy or high error-correction capacity; we will see this also for TQNNs below. Biologically,
this corresponds to the (phylogenetic) evolution or (ontogenetic) development of systems with sensory
structures and processing pathways specialized to the detectable affordances of their ecological niches
[146,147]. Indeed, many authors have cast natural selection as, implicitly or explicitly, a VFE minimizing
process in terms of natural Bayesian model selection or structure learning [148–153].
Sectors of high mutual information induce a connection topology on B, with these sectors as the open
sets. This topological structure breaks the exchange symmetry of bit ‘positions’ (i.e., values of ξ) on B. This
symmetry breaking corresponds to a choice of basis for the Hamiltonian HSE; again see [64,65,79,80] for
details. Under the FEP, the local action of the internal dynamics HSon each such sector implements a QRF
that alternately measures and acts on the bits encoded by that sector. The action of this QRF can, without loss
of generality [76], be specified by a CCCD as shown in figure 3.
A limit/colimit and infomorphisms from/to it exist over any mutually-commuting subset of the CCCDs
specifying actions of QRFs on B[141, theorem 7.1]. Hence any mutually-commuting subset of the CCCDs
can be hierarchically composed as components of a larger CCCD that processes their combined outputs, as
shown in diagram (4). This larger CCCD specifies the action of a QRF that can be thought of as alternately
measuring and preparing the states of the component QRFs in the hierarchy. Indeed, this larger QRF induces
a single TQFT ([154]) on the collection of sectors measured/prepared by the component QRFs [76]; we will
pursue the consequences of this in section 7below.
Whenever any of the component CCCDs specify (the component QRFs implement) nonlinear processes,
e.g., logical AND or XOR, hierarchical decomposition implements coarse-graining. The FEP will drive any
system toward such coarse-graining provided the loss of predictive power at the component level is
compensated for by a gain of predictive power at the higher level. This will be the case whenever the sectors
spanned by the combined CCCD/QRF encode significant mutual information about each other, i.e., in any
situation in which there are information relations between sectors and hence apparent ‘interactions’ between
the ‘objects’ that the sectors represent to S. The FEP, in other words, will drive any system to discover ‘macro
variables’ that characterize and ‘emergent causality’ [155,156] between its identified sectors.
Teleologically speaking, while scientists have only recently developed reliable tools with which to
quantitatively track the causal power of different levels of a system [157–164], biological life forms emerging
under realistic temporal and energy (metabolic) constraints have always faced selection pressure to estimate
causality of meso-scale ‘objects’ in their Umwelten. It is essential for survival that an agent spends its precious
resources attempting to affect or communicate with (or track) the features of its environment that make a
difference, which requires them to coarse-grain their experience into models in which the massive stream of
11 Basically, for any sector Xdefined by a QRF X, a generic (k-)time-stamped quantum system Aconfronts the task of minimizing pre-
diction error ErX(k)given by
ErX(k) = d(MA
X(k),MX(k))
where MA
X(k)and MX(k)denote Markov kernels derived from observables, and dis the metric distance between kernels. The FEP in this
case asserts that a generic quantum system will act so as to minimize ErXfor each deployable QRF X(for details, see [64, sections 3 and
4]).
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Figure 3. (a) A CCCD associated with a sector of the boundary B, depicted as an array of qubits as in figure 2. The operators MS
i
and ME
jare Hermitian components, in the bases chosen by Sand Erespectively, of the interaction Hamiltonian HSE ; see [64,80]
for details. Adapted from [80] figure 3, CC-BY license. (b) Cartoon representation of kQRFs s1,...,sk(red triangles, with the
apex the limit/colimit of the corresponding CCCD) acting on high mutual-information sectors of B.
sensory information and potential activities (at all scales) is cut up into convenient ‘objects that do things’ for
the purposes of efficient action. Living beings cannot afford a ‘Laplace’s daemon’ (micro-reductionist) view
of cause and effect, and the pressure to form models that acknowledge causal potency of higher levels is
baked in from the very beginning of the evolution of life.
Biological spiking (e.g., mammalian cortical) neurons are canonical examples of such hierarchical,
coarse-graining measurement/preparation systems, with dendritic trees implementing hierarchical
measurement and axonal branches implementing hierarchical preparation, namely, action on the
surrounding environment [46]. Convolution of post-synaptic potentials at dendritic branch points
implement nonlinearities including logical AND and XOR [165]. Gating of action potentials implements
similar nonlinearities. These functions are, indeed, precisely the features that most distinguish neurons from
simplified models such as diagram (1), and precisely the features that most neuromorphic computing
models seek to replicate or at least emulate [3,4].
6.3. Hierarchical QRFs as tomographic computers
As discussed in section 4.2 above, the relationship between computational structure and morphology
exemplified by neurons should characterize any system subject to the FEP. We are now in a position to see
why. When the morphological degree of freedom ξis given the structure of a vector space, it provides a
distance measure ⟨ξ|ξ′⟩as discussed in section 4above; however, we have given no interpretation of this
distance. Hierarchical decomposition provides such an interpretation: if sectors of high mutual information
are regarded as ‘locations’ on B—and hence open sets in the connection topology are regarded as
simply-connected ‘areas’ in the induced geometry—then mutual information between sectors provides a
natural distance measure. A hierarchy of QRFs, in this case, induces a hierarchy of distances on Bthat are
encoded by the values of the morphological degree of freedom ξ.
Note that the distance measure ⟨ξ|ξ′⟩is a formal description of Bapplicable to any interaction HSE in
which at least one of the interacting systems, which we by convention label S, has an internal dynamics HSof
sufficient complexity to implement a QRF hierarchy. This does not imply that Sitself is capable of measuring
the distance ⟨ξ|ξ′⟩. Indeed we have explicitly assumed that HSE does not depend on ξ. The system Swill be
capable of perceiving ‘space’ and thus of measuring distance only if it implements a QRF for spatial
measurements. While vertebrates, cephalopods, and some arthropods appear capable of perceiving space,
this is not necessary for acting in space (as perceived by us), and the vast majority of organisms, including
perhaps all unicells, may lack spatial QRFs altogether [116].
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Figure 4. Cartoon showing S’s QRFs s1,s2,...,skmeasuring ‘slices’ of the effective Hilbert space HB; each slice corresponds to a
sector of B.
Let us consider what a system Simplementing hierarchies of QRFs, but having no spatial QRFs perceives.
The bits encoded by a high mutual-information sector siof the informative sector (s,a) of Bare written by
the action of Eon that sector. We can think of them, therefore, as outcomes of measurements of basis vectors
of the effective Hilbert space HBof the MB B; this is depicted in figure 3, with the MS
ias single-qubit
(e.g., z-spin sz) measurement operators. Hence, we can view each of S’s QRFs as measuring a projection or
‘slice’ of HBas shown in figure 4.
Measurements that (partially) reconstruct the state of some system by measuring independent
projections of that system are tomographic measurements; the (typically hierarchical) process of
reconstructing the total state is tomographic reconstruction. Hence QRFs acting on sectors are implementing
tomographic measurements of the state of B, and hierarchies of QRFs are computing a tomographic
reconstruction of the state of B.
By analogy with tomographic reconstruction from image planes as implemented by Positron Emission
Tomography in medical imaging, we can think of these slices as having two spatial dimensions as depicted in
figure 4. The ‘depth’ dimension (horizontal in figure 4) is, in this case, notionally perpendicular to the 2d
‘surface’ of B. This depth dimension can, therefore, be identified with the embedding dimension of ξ
assumed in section 4above.
The notion of ‘depth’ and hence the embedding dimension of ξhas, in addition, a second interpretation
in terms of time; it can be identified with the total time required to process the inputs from a particular
sector through the multiple layers of a hierarchical QRF. This means we can think of the embedded
morphology of Sin two complementary ways: the morphology both ‘extends into’ the state space being
measured (i.e., into HB) and ‘wraps around’ the hierarchical computational structure, conferring a spatial
(or space-like) structure on computational (or message-passing) time. This dual aspect of embedding is
evident in neurons, which both extend into their (3d) environments and require more time to process distal
inputs than proximal ones. In consequence, distal signals are degraded in time resolution and lower in
amplitude, rendering time resolution (relative to some proximal standard) and amplitude (relative to some
proximal standard) alternative interpretations of the embedding dimension. As discussed in [46], neurons
also perform state tomography in time, measuring multiple temporal replicates of input activity patterns to
reconstruct the relatively slowly-varying state of the (individual neuron’s) environment as a whole.
Perhaps the most celebrated examples of spatiotemporal encoding in the brain are the characteristic
responses of the hippocampus; variously read in terms of encoding space—with place and (hierarchically
superordinate) grid cells—or, tellingly, as having a key role in memory and the encoding of time [166–182].
From the current perspective, the very existence of place cells—and perhaps receptive fields more
generally—speak exactly to the coarse graining of the brain’s implicit explanations for its sensorium. And,
crucially, how neuronal computations leverage, or are scaffolded by, the morphology of neurons and the
connectomes in which they are embedded. This is manifest at many levels empirically; ranging from the
emergence of late waveforms in event-related potential research attributed to deep hierarchical processing
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[183]; through to temporal gradients as we pass from the back (visual) brain to the front ascending both
hierarchal depth and temporal scales [184–187], and indeed the functional anatomy of the hippocampus
[188]. The dénouement of this analysis suggests that space and time are perceptual constructs that supervene
on the ordinal structure and scheduling of message-passing on neuronal architectures with a certain
morphology—a morphology that is apt to explain sensory exchange with the environment, as it is actively
queried through (what we perceive as) navigation [61,189].
What we have said so far here suggests a rather consequential hypothesis: that the translational,
rotational, and boost symmetries of spacetime—hence the Poincaré group and Special Relativity—are
consequences, for any observer S, of the independence of the Hilbert space of Bfrom the ancillary coordinate
ξ. Observing these symmetries requires identifying, via specific QRFs, objects as sectors on B. Symmetries of
motion are, therefore, consequences of the conservation by Sof object identity through time, something that
follows immediately from defining, for each object, a reference component Rwith fixed ρR. Hence they are,
effectively, consequences of the structure of HSas a quantum computation. We will pursue this remarkable
hypothesis elsewhere; we turn now to the construction of TQNNs as models of neuromorphic computation.
7. TQNNs as general neuromorphic systems
We show in [76] how any sequence of measurements by some fixed QRF on some fixed sector(s) of a
boundary Binduces a TQFT on that sector (or those sectors). The proof is completely general and procedes
by two category-theoretic constructions. First, we show that any QRF can be represented by some CCCD,
and construct a category CCCD of CCCDs that represent QRFs. The morphisms in this category represent
transitions between QRFs and thus represent sequential measurements. We then construct a functor from
CCCD to the category Cob of finite cobordisms. A TQFT is a functor from Cob to the category Hilb of (in
this case finite-dimensional) Hilbert spaces, which interprets each cobordism as a manifold of linear maps
between Hilbert spaces that serve as boundaries. Transitions between QRFs correspond, therefore, to
manifolds of maps between Hilbert spaces, i.e. to TQFTs; see supplementary material, appendix D for a more
detailed sketch of this proof.
A TQFT on a boundary Bcan be thought of as encoding all possible smooth transformations (e.g. all
possible Feynman paths) from some initial configuration Bin of Bto some final configuration Bout.
Similarly, a TQFT induced on some sector of Bby sequential actions of one or more QRFs encodes all
possible Feynman paths of that sector. One immediate physical consequence is that all effective field theories
on observed sectors must be gauge invariant [66,76,190]; see [76,190] for further physical implications of
this construction. With a suitable choice of basis, such TQFTs can be implemented by TQNNs [36,76,143].
These generalize conventional quantum ANNs [191,192] by allowing the number and organization of
‘layers’ to be indeterminate. We start by clarifying this peculiar, novel quantum feature of TQNNs, which
stands at the forefront of the implementation of TQFTs in machine learning.
TQNNs implement computations on quantum states of the Hilbert space associated to the boundary
sectors B, and can be expanded on the spin-network basis [36,76,143]. Spin-networks are in turn
supported on 1-complexes (graphs or loops) embedded on the boundary sectors B, and are colored by
certain irreducible representations (irreps) of whatever symmetry describes the system. Note that this
embedding requires Bto have, or be extended to have, spatial (i.e., ξ) degrees of freedom. The relevant
symmetry is then either individuated by some Lie group or by some quantum group, namely a non-trivial
Hopf-algebra [193,194] (see e.g. [195] for a review of some general extensions of these topics). TQNNs are
then represented as superpositions of the basis elements on the boundary sector B. Spin-networks provide
an orthonormal basis, but is worth reminding that loop states as well span the Hilbert space of quantum
states over B, and that a unitary transform exists [196] that connects the two classes of states. The dynamical
evolution of the TQNN is then described by TQFT transition amplitudes between Bin and Bout, which are
supported on 2-complexes interpolating the TQNN’s 1-complexes [36,76]. The role of symmetry, as
customary in any (effective) quantum field theory, is crucial in recovering the dynamics, as it dictates the
interaction among different TQNNs/quantum states embedded on the boundary sectors Bin and Bout. The
superposition principle induces a summation over an infinite set of interpolating 2-complexes, supporting
virtually fluctuating TQNNs, namely the sum over all the possible evolutions of colored quantum states. The
superposition principle also forces to sum over all the possible colours, i.e., all possible assignments of irreps
of the symmetry group appropriate to describe a certain physical data set. Topological changes in the graph’s
connectivity can be induced, starting from a maximal graph, by solely considering the sum over the irreps.
This is due to the fact that assigning a vanishing irrep to a link of a graph corresponds to removing that link
from the graph. Then all the interpolating topologies can be obtained summing over all the compatible irreps
and intertwiner quantum numbers assigned respectively to the graph nodes and links.
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Neuromorph. Comput. Eng. 2(2022) 042002 C Fields et al
The minimal number of spatial (i.e., ξ) dimensions required to embed 1-complex (graph and/or loop)
degrees of freedom in a boundary manifold Bis 2, but a 2d spatial embedding does not allow linking and
knotting (i.e., entangling) of these states to take place, which instead requires at least 3 spatial dimensions. A
larger number of Hausdorff dimensions is also achievable, and this would induce an encoding of
higher-dimensional topological features depending of the number of dimensions of the ambient space.
As noted in [76], the construction of TQNNs presents several formal analogies with quantum gravity
(QG) models. Inspecting the dynamics within the framework of the BF formalism, just taking into account
BF Lagrangians, we can easily convince ourselves that topological BF theory can be accomplished: in
(1+1)-dimensions resorting to the Jackiw–Teitelboim gravity [197,198], either for a SO(2,1)or for a SO(3)
gauge symmetry, in (2+1)-dimensions, considering either the topological Einstein–Hilbert action, without
accounting for the cosmological constant, or a double Chern–Simons theory, the quantization of which has
been proved in [199] to be equivalent to the Turaev–Viro [200,201] quantization of the Einstein–Hilbert
action with cosmological constant, and in (3+1)-dimensions, producing the topological Ooguri [202] and
Crane–Yetter [203,204] models. That different realizations that can be recovered shows that dynamics is
affected by the number of dimensions, even when considering simplified topological theories described by
kinetic BF actions.
Having considered a mathematical framework in which functorial evolutions of graphs are supported on
2-complexes, of which graphs are slices of co-dimension 1, and having embedded both the structures
respectively in boundary space sectors Band bulk spaces, of which Bare slices, it is natural to be convinced
that ‘spatially’ organized data, localized on the Bsectors, are analyzed hierarchically, in the space or
parameter (respectively, in the Euclidean and Lorentzian signature case) flow that induces the slicing of the
bulk. This simple consideration has profound consequences since it ensures that TQNNs are ‘neuromorphic’
in a relevant sense. In general, boundary states can be thought as holographic states embedded in lower
dimensional projections of the bulk. Boundary states may then encode information about the local curvature
when quantum group irreps are considered. For instance, if the irreps that are considered participate in the
evaluation of the partition function of the double Chern–Simons theory in 3-dimensions, i.e., the
Turaev–Viro model endowed with SUq(2)irreps, the cosmological constant provides the curvature of the
faces that belong to the polyhedra dual to the lattice structure.
We can cast the previous framework in terms of CCCDs, as models of QRFs. Since data are organized
spatially, as quantum boundary states/TQNNs are embedded in auxiliary spaces, the boundary sectors B,
CCCDs automatically turn out to be hierarchical in their representations. This implies an orientation for the
convolution of CCCD, the inputs of which, thus, are not all processed by only one combinatoric criterion
extended to the whole system. The auxiliary spatial degree of freedom participates in the coarse-graining.
Because of the proven hierarchical structure, local correlations turn out to be more informative than distant
ones, as correlations are suppressed spatially, in inverse powers of the distances involved in the auxiliary
spaces. This is a relevant feature for this framework, and involves a confrontation among possible alternative
scenarios for coarse-graining: the renormalization group flow `
a la Wilson versus the Kadanov group—see
e.g. [205]. Within the TQNN framework accounted for here, an invariance under coarse-graining of the
simplicial tessellation of the space slices of the manifolds emerges, unless one involves a more refined and
theoretically challenging extra geometrical renormalization group flow construction—see e.g. the Ricci flow
renormalization group approach of [206]. A paradigmatic example is once again provided by the
Turaev–Viro model, which is invariant under refinement of the triangulations. This is not true, for instance,
for the Ponzano–Regge model [207], which instantiates the spin-foam/BF like quantization of the Einstein
Hilbert action.
It is crucial to notice, in order to make our considerations in this paper sharper, that the dynamics of
TQNNs is instantiated by quantum curvature constraints proper of TQFTs in a way that is equivalent to
imposing the FEP on the system. The imposition at the quantum level of the curvature constraints amounts
indeed to an extremization of the classical action of either the TQFTs or the effective QFTs describing the
specific systems at hand. Indeed, within the semiclassical limit, a constraint that imposes the limitation of the
free energy in computational tasks is automatically recovered, imposing tight requirements to the efficiency
of TQNN algorithms. Efficiency, which can be modeled as a cost per link in either the TQNN or the CCCD
picture, then corresponds to a path minimization for the two-complexes structures intertwining among the
boundary states. A detailed analysis of the link between the FEP and the semiclassical limit of TQNNs, which
is relevant to unveil generalization in deep-learning systems (DNNs) [143], will be addressed elsewhere.
Finally, in concluding this section we emphasize that we have established TQNNs as a general framework
for neuromorphic computation. Notoriously, this is not the case for standard classical DNNs, which rather
constitute a less general framework, corresponding to a specific (semi-classical) limit of TQNNs.
16

Neuromorph. Comput. Eng. 2(2022) 042002 C Fields et al
Table 1. Correspondence between features of biological or neuromorphic architectures and the formal or physical constructs employed
here.
Bio/neuromorphic architecture Formal or physical constructs
System-environment boundary MB, holographic screen B
System identity preserved through time Separability, conditional independence
Homeostasis, allostasis VFE minimization, asymptotic approach to NESS
Metabolism Free energy transfer across B
Information processing TQNN(s) implemented by internal dynamics
Memory for previous observational outcomes Write/read QRFs acting on B
Object identification, categorization, object-directed behavior Implementation of object-specific QRFs
8. Conclusion
The results reviewed here show how any system with morphological degrees of freedom and locally limited
free energy will, under the constraints imposed by the FEP, evolve toward a neuromorphic morphology that
supports hierarchical computations in which each ‘level’ of the hierarchy enacts a coarse-graining of its
inputs, and dually a fine-graining of its outputs. Such hierarchies occur throughout biology, from the
architectures of intracellular signal-transduction pathways to the large-scale organization of perception and
action processing in the mammalian brain. The close formal connections between CCCDs as models of
QRFs on the one hand, and between CCCDs and TQFTs on the other, allow the representation of such
computations in the fully-general quantum-computational framework of TQNNs. The mapping between
biological or neuromorphic architectures and the formal or physical constructs employed here is
summarized in table 1. As noted in the Introduction, we employ these quantum-theoretic constructs because
they are scale-free and completely general, but defer discussion of any particular ‘quantum brain’ type
models to future work.
One practical implication of the above analysis—that inherits from the distinction between states and
parameters of a generative model—is a fundamental distinction between biomimetic computation on Turing
machines and neuromorphic computing. From a classical perspective, optimizing the states of a neural
network can be read as inference, while optimizing the model parameters (i.e. connection weights in an
ANN) corresponds to learning at a slow timescale. In biomimetic schemes, the connection weights or model
parameters are generally stored in working memory in the form of tensors to compute the messages that are
passed along nodes of a factor graph to instantiate inference at a fast timescale. However, in practice, the vast
majority of compute time (and, thermodynamic expenditure) is taken by reading and writing the
connectivity tensors from memory. This means that the arguments based upon minimizing the complexity
of generative models only provide a lower bound on the thermodynamics of belief updating [42,208–211].
This lower bound that can only be realized if the connection weights are physically realized as in
neuromorphic architectures. This may be an important motivation that goes beyond biomimetic aspirations
[142], especially in applications such as edge computing (e.g. surveillance drones).
A further pragmatic perspective on recent trends in machine learning is afforded by the notion of
hierarchical computation. In virtue of the fact that these entail a local minimization of VFE (with locally
limited thermodynamic free energy), efficient computing on deep networks should conform to these local
constraints. Indeed, this is apparent in the move away from backpropagation schemes to local energy-based
schemes [212]. This is nicely illustrated by the comparative analyses of backpropagation with predictive
coding implementations of deep learning [213–215]. In the current setting, hierarchical predictive coding
can be regarded as an implementation of VFE minimization, under hierarchical generative models [216].
More generally, the above results offer some directions for future research. The first is understanding how
the pressures that result in neuromorphic architectures impact evolutionary developmental biology, which
seeks to determine the origin of specific nervous system patterns [217–219]. More than looking backwards,
however, this kind of work can drive advances in both bio-hybrid (biorobotics, chimeric) and software-based
AI. A variety of hybrots, organoids, and biobots are being created [220–222] as a way to escape the fact that
all of Earth’s biological forms are basically an N=1 example of evolution (barring advances in exobiology).
The inclusion of neural (and non-neural bio-electrical) components in these synthetic beings, often made in
the absence of any genetic change [124,125,223–225], will help test predictions of generic laws driving the
structure and function of the body-wide communication system. Similarly, these principles could be of use
in designing unconventional and traditional connectionist computational systems, as well as help drive the
discovery of interventions guiding cell behavior in regenerative medicine settings [108].
17

Neuromorph. Comput. Eng. 2(2022) 042002 C Fields et al
Data availability statement
No new data were created or analysed in this study.
Acknowledgments
K J F is supported by funding for the Wellcome Centre for Human Neuroimaging (Reference:
205103/Z/16/Z), a Canada-UK Artificial Intelligence Initiative (Reference: ES/T01279X/1) and the European
Union’s Horizon 2020 Framework Programme for Research and Innovation under the Specific Grant
Agreement No. 945539 (Human Brain Project SGA3). M L gratefully acknowledges funding from the Guy
Foundation and the Finding Genius Foundation. A M wishes to acknowledge support by the Shanghai
Municipality, through the Grant No. KBH1512299, by Fudan University, through the Grant No. JJH1512105,
the Natural Science Foundation of China, through the Grant No. 11875113, and by the Department of
Physics at Fudan University, through the Grant No. IDH1512092/001.
Conflict of interest
The authors declare no competing, financial, or commercial interests in this research.
ORCID iDs
Chris Fields https://orcid.org/0000-0002-4812-0744
Karl Friston https://orcid.org/0000-0001-7984-8909
James F Glazebrook https://orcid.org/0000-0001-8335-221X
Michael Levin https://orcid.org/0000-0001-7292-8084
Antonino Marcian`
ohttps://orcid.org/0000-0003-4719-110X
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