# Shaping dynamical neural computations using spatiotemporal constraints

Jason Z. Kim<EMAIL_ADDRESS>Department of Physics, Cornell University,
Ithaca, NY 14853, USA Bart Larsen  Department of Pediatrics, Masonic Institute
for the Developing Brain, University of Minnesota Linden Parkes
<EMAIL_ADDRESS>Department of Psychiatry, Rutgers University,
Piscataway, NJ 08854, USA

(August 28, 2024)

###### Abstract

Dynamics play a critical role in computation. The principled evolution of
states over time enables both biological and artificial networks to represent
and integrate information to make decisions. In the past few decades,
significant multidisciplinary progress has been made in bridging the gap
between how we understand biological versus artificial computation, including
how insights gained from one can translate to the other. Research has revealed
that neurobiology is a key determinant of brain network architecture, which
gives rise to spatiotemporally constrained patterns of activity that underlie
computation. Here, we discuss how neural systems use dynamics for computation,
and claim that the biological constraints that shape brain networks may be
leveraged to improve the implementation of artificial neural networks. To
formalize this discussion, we consider a natural artificial analog of the
brain that has been used extensively to model neural computation: the
recurrent neural network (RNN). In both the brain and the RNN, we emphasize
the common computational substrate atop which dynamics occur—the connectivity
between neurons—and we explore the unique computational advantages offered by
biophysical constraints such as resource efficiency, spatial embedding, and
neurodevelopment.

recurrent neural networks, dynamics, computation, spatial constraints,
neurodevelopment

##  I Introduction

Dynamics have long underpinned computation. From the cycles of central pattern
generators that support locomotion [1] to the networks of large-scale brain
dynamics thought to regulate decision-making [2], it is clear that biological
systems make ample use of their time-evolution to respond to their
environment. Harnessing this dynamical computation, artificial recurrent
neural networks (RNNs) have been trained to successfully perform the same
computational tasks as humans [3, 4]. However, while inspired by the brain,
training of RNNs is typically carried out in an unconstrained manner, leading
to solutions that lack biophysical realism. Additionally, decades of
neuroscience research has demonstrated the importance of biological
constraints for achieving the brain’s unique structure and capabilities [5, 6,
7, 8]. Here, we draw on literature from dynamical systems and neuroscience to
discuss (i) how RNNs leverage dynamics to compute, and (ii) how biophysical
constraints may shape this computation by guiding the formation of network
structure. At the intersection of these goals exists an opportunity to study
how biologically-constrained RNNs may yield more powerful and more
interpretable computational models.

To understand how biologically realistic neurons compute, there has been a
long and rich history of modeling and interpreting neurobiological systems to
leverage their computational capabilities. These quantitative models fall
under the category of dynamical systems, whose evolution in time is determined
by mathematical functions. At the scale of a single neuron, detailed circuit
models of the ion channels that mediate membrane voltage have enabled
quantitative understanding of signal propagation and computation in dendrites
[9, 10]. At the scale of neural populations, mean-field models of excitatory
and inhibitory neurons have enabled the study of neural circuits for
biological sensing, imitation, and attention [11, 12, 13]. At the whole-brain
level, both linear [14, 15, 16, 17] and non-linear [18, 19] dynamical models
have been used to simulate large-scale activity patterns, and have examined
how those patterns spread across the brain’s white matter tracts. Across this
broad range of systems, scales, and models, there exists a diversity of ways
in which dynamics can be used for computation, as well as a crucial dependence
of these dynamics on biophysical parameters.

More recently, with technological advances in deep learning, the study of
neural computation has adopted a more functional direction that moves away
from biological realism. That is, rather than seeking a direct biophysical
model [9, 20, 11, 21], RNNs posit a general dynamical system that is Turing
complete [22], with parameters that are trained to solve computational tasks.
We focus on RNNs because they have been used extensively to understand how
general brain-like systems leverage dynamics to perform computation. Examples
of this use include time-series prediction [23], source-separation [24],
decision-making [3], odor classification [25], as well as concurrent
performance of multiple cognitive tasks [26]. However, these insights and
models often fail to translate into the real computational substrate of the
brain—the neural architecture—because RNNs are trained without regard for
biophysical constraints.

To merge biological realism with computational dynamics, we must first
understand the physical embedding and constraints of the brain. In contrast to
artificial RNNs, the brain is embedded within a circumscribed physical space
[27], and its inter-connectivity is subject to limited metabolic resources
[28]. This discrepancy makes it challenging to translate insights relating the
structure, dynamics, and computation of biological brains to artificial RNNs.
Neuroscience has studied these resource constraints for more than a century
[29], suggesting that the brain is pressured to make efficient use of space,
material, and time. That is, the brain must learn to communicate efficiently
(time) while leveraging limited physical (space) as well as metabolic and
cellular (material) resources. Critically, many of the brain’s topological
features of connectivity and communication are thought to emerge as a
consequence of navigating these pressures [28, 5, 6, 7, 8]. These findings
suggest that the brain’s resource constraints play a critical role in shaping
its dynamic repertoire and computational capacity.

Here, our goal is to lay out promising new directions for improving the
computational power and interpretability of RNN models of the brain. We posit
that this goal will be achieved by placing biological constraints on RNNs that
shape their structure and activity in systematic ways, which will in turn
produce computationally improved dynamics. We focus on two aspects of RNN
computational dynamics: the diversity of information that is represented by
the neurons (expressivity), and the manipulation of low-dimensional internal
representations (latent-spaces). In each section, we examine how biology
shapes brain networks—with particular emphasis on the spatially-patterned
macro-scale organizing principles of the cortex—and discuss how these
constraints may be ported to RNNs to improve performance with interpretable
structure and dynamics. Overall, we discuss how insights from biological and
artificial computation can enrich each other towards a new generation of
biophysically realistic RNNs.

##  II The RNN Model

To mathematically model the time-evolution of neural systems, we turn to
dynamical systems which posit that the next state of a neural system can be
written as a function of the current state and an input as

$\displaystyle\bm{r}_{t+1}=f(\bm{r}_{t},\bm{u}_{t}).$ (1)

Here, $\bm{r}_{t}\in\mathcal{R}^{n}$ is a vector of $n$ neural activity
states, $\bm{u}_{t}\in\mathcal{R}^{k}$ is a vector of $k$ inputs, and $f$ is a
function. As an example, let us consider a simple leaky integrator model with
a single neuron which evolves according to

$\displaystyle r_{t+1}=ar_{t}+bu_{t},$ (2)

where $0\leq a<1$ and $b$ are real numbers (Fig. 1A). As time evolves forward,
the neuron state integrates the input $bu_{t}$, and the accumulated history of
inputs decays at a rate set by $a$.

Figure 1: One-dimensional dynamics and RNN architecture. A, Schematic of a
single neuron model of a leaky integrator with an input connection in blue,
and a recurrent connection in gold (left), alongside examples of both a
continuous and discretized input into the neuron (center), and the state of
the input-driven neuron from the continuous (Eq. 5) and discretized (Eq. 2)
dynamics (right). B, Schematic of an RNN with $k$ inputs and $m$ outputs, with
input connections and matrix in blue, recurrent connections and matrix in
gold, and output connections and matrix in maroon.

While Eq. 1 is written with $t$ advancing in integer steps—thereby called a
discrete-time dynamical system—many physical neural models evolve forward
continuously in time as,

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\bm{r}(t)=\hat{f}(\bm{r}(t),\bm{u}(t)).$
(3)

We can approximate these continuous dynamics as discrete by evolving Eq. 3 in
time using steps of $\Delta t$ as,

$\displaystyle\bm{r}_{t+\Delta
t}=f(\bm{r}_{t},\bm{u}_{t})=\bm{r}_{t}+\int_{t}^{t+\Delta
t}\hat{f}(\bm{r}(\tau),\bm{u}(\tau))\mathrm{d}\tau.$ (4)

For example, the continuous-time version of the leaky integrator neuron is
given by

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}r(t)=\hat{a}r(t)+\hat{b}u(t),$ (5)

for $\hat{a}\leq 0$ (Fig. 1A). In this case, the parameters of these two
models can be interchanged through the transformation $a=e^{\hat{a}\Delta t}$,
$b=(a-1)\hat{b}/\hat{a}$, but fundamental differences exist between
continuous- and discrete-time systems [30]. Regardless of the system type,
neural models make tradeoffs between complexity in the level of detail and
tractability.

The RNN is a model that attempts to capture the biophysical quantity of the
interactions between neurons through the connectivity matrix $A$. In tandem,
the RNN simplifies the precise functional form of that interaction through the
activation function $f$. In its most basic form, an RNN is a subset of
dynamical systems (Eq. 3) that evolves in time as

$\displaystyle\begin{split}\bm{r}_{t+1}&=f({\color[rgb]{0.76171875,0.62109375,0.2421875}\definecolor[named]{pgfstrokecolor}{rgb}{0.76171875,0.62109375,0.2421875}A}\bm{r}_{t}+{\color[rgb]{0.1875,0.25390625,0.47265625}\definecolor[named]{pgfstrokecolor}{rgb}{0.1875,0.25390625,0.47265625}B}\bm{u}_{t}+\bm{d}),\\\
\bm{o}_{t}&=g(\bm{r}_{t}).\end{split}$ (6)

where $A\in\mathcal{R}^{n\times n}$ is the connectivity matrix between
neurons, $B\in\mathcal{R}^{n\times k}$ is a matrix that linearly maps the
inputs to the neurons, $\bm{d}\in\mathcal{R}^{n}$ is a vector of bias terms,
and $f$ is an activation function (Fig. 1B). Rather than having $f$ be a
complex and biophysically motivated function, it is often approximated as a
simple nonlinear function such as a sigmoid. The output of the RNN,
$\bm{o}_{t}$, is usually taken to be some function $g$ of the RNN state, and
is often a linear output
$\bm{o}_{t}={\color[rgb]{0.6484375,0.2890625,0.2109375}\definecolor[named]{pgfstrokecolor}{rgb}{0.6484375,0.2890625,0.2109375}W}\bm{r}_{t}$.
Typically, $A,B,\bm{d}$, and $W$ are treated as learnable parameters, some or
all of which can be trained using a wide variety of methods [4, 31, 32].

We focus primarily on the computational role of the connectivity matrix $A$,
as it dictates how the information in the RNN states is integrated as in the
leaky integrator example (Eq. 2, 5). Despite its crucial importance in
implementing computation, most uses of RNNs do not consider the biological
pressures experienced by the brain while training RNN connectivity. In the
following section, we describe how diversely the RNN states can express the
inputs by leveraging $A$, and how biological processes and constraints
reflect, mediate and accentuate this diversity.

##  III Tuning Expressivity through Regularilized Activity and Neuromodulation

When solving any computational problem, the expressivity of the language used
is of crucial importance. The more expressive a language is, the greater the
set of computations, formulae, and theorems comprise that language [33, 34].
For example, a programming language that supports conditional `if` statements
can represent many more programs than an equivalent language that is without
an `if` statement.

In the same way, neural networks can be viewed from the lens of expressivity.
Specifically, we can ask: given arbitrary weights, what is the space of
functions or dynamics that can be achieved? Previous work has demonstrated
that shallow multi-layer perceptrons (MLPs) are universal function
approximators [35, 36], and that RNNs are universal dynamics approximators
[37]. However, even if a particular function or dynamical pattern is
theoretically achievable, artificial neural networks—just like biological
neural networks—must be trained from an initial condition. Hence, the study of
expressivity extends beyond theoretical guarantees, and has been shown to rely
heavily upon architectural features such as depth [38], neuron activation
[39], and connectivity [40], which are crucial for explaining and engineering
the success of modern-day neural networks.

In this section, we study three consequences of biological processes for
expressivity. First, we introduce RNN expressivity as a richness of time
history information about the inputs, and tie this richness to spatially-
patterned temporal receptive fields in the brain that underpin information
integration. Second, we study constraints on expressivity induced by resource
constraints on neural activity as a putative learning mechanism. Finally, we
explore the potential for neural networks to modulate their expressivity at
short time-scales through neuromodulation. Together, the processes of the
brain offer enticing and novel paradigms for training and constructing more
expressive RNNs under biological constraints.

###  III.1 Expressivity as variable time-lagged integration of information

Figure 2: Expressivity and the representational basis formed by RNNs. A
Schematic of an RNN with a single input, the basis vectors formed by neurons
$i$ and $j$, and the reconstructable subspace spanned by those vectors. B
Schematic and reconstructable subspaces for a simple 2 neuron system with no
recurrent connections, C weak recurrent connections, and D strong recurrent
connections. E Example trajectories from a subset of 8 neurons of a 50 neuron
system after receiving an impulse (top), alongside the ideal and true output
of the RNN (bottom) after training a linear readout, $o_{t}=W\bm{r}_{t}$, to
reproduce a time-lagged version of the impulse with $|A|=0$, F $|A|=1.2$, and
G $|A|=1.6$.

Expressivity of RNNs is intricately tied to the concept of stability: how
quickly a perturbation to the RNN state decays [41]. If a perturbation decays
very quickly, then the information contained in the perturbation cannot be
used by the RNN for extended information processing. On the other hand, if the
perturbation grows uncontrollably, then the temporal information in the inputs
quickly becomes too complex to be represented by the finite number of neurons.
As a result, an optimal amount of controlled stability should maximally
preserve temporal information without saturating the RNN’s capacity.

We can quantify this intuition through a simple recursive substitution of Eq.
6,

$\displaystyle\begin{split}\bm{r}_{t+1}&=f(A\bm{r}_{t}+B\bm{u}_{t}+\bm{d})\\\
&=f(A(f(A\bm{r}_{t-1}+B\bm{u}_{t-1}+\bm{d})+B\bm{u}_{t}+\bm{d})\\\ &=\dotsm\\\
&=h(\bm{r}_{0},\bm{u}_{0},\bm{u}_{1},\dotsm,\bm{u}_{t}).\end{split}$ (7)

Hence, in a noiseless system, the RNN state $\bm{r}_{t+1}$ can be written as
an explicit function $h$ of the initial state, $\bm{r}_{0}$, and the full time
history of the inputs, $\bm{u}_{\tau}$, mediated by the recursive application
of $A,B,\bm{d}$, and the activation function $f$. The state of all neurons
$\bm{r}_{t+1}$ generates a basis for a subspace of the delay-embedded space of
inputs $\bm{u}_{\tau}$ (Fig. 2A), which means that the neuron states at time
$t+1$ hold information about the time history of the inputs mediated by $A$,
$B$, and $\bm{d}$ [42]. Thus, when computing an output using the neural
states, we are implicitly computing an output using a basis of time-lagged
input terms, where the connectivity defines the basis vectors, and therefore
the reconstructable subspace (Fig. 2A). The more expressive this time-lagged
basis, the greater the diversity of output functions which can be computed.

This expressivity is intimately tied to the connectivity matrix [43], and has
been studied through many different lenses such as computation at the edge of
chaos [40, 44, 45], criticality, and avalanches [46, 47]. The RNN’s stability
is set by the specific activation function $f$ and the connectivity matrix
$A$. To gain intuition for this dependence, let us consider a simple linear 2
neuron system driven by one input. When there is no connectivity between the
neurons, they store no time history of the inputs, and their state at time
$t=3$ is purely a function of the input at time $t=2$ (Fig. 2B). When we add
weak connections between the neurons, they begin to store some information
about the input at the previous time point $t=1$ (Fig. 2C). When we strengthen
these connections, the neurons begin to store information from further back in
time at $t=0$ (Fig. 2D).

To further develop this intuition for larger systems and a specific task, we
consider a 50 neuron system whose connectivity is randomly initialized, and
whose output is trained to recall an impulse from 30 time steps in the past.
At the trivial limit of an RNN with no connectivity where $|A|=0$, the
recursion of Eq. 7 yields $\bm{r}_{t+1}=f(B\bm{u}_{t}+\bm{d})$, and we see
that there is no time history of the input present in the RNN state. As a
result, the RNN is unable to recall the input at a later point in time (Fig.
2E) [48]. As we increase the strength of connectivity, the RNN state stores
more information about longer time lags of the input, $\bm{u}_{t-\tau}$, and
is thus able to more accurately recall the input later in time (Fig. 2F). As
the connectivity strength continues to further increase, the RNN state holds
increasingly more time lags of the input until it saturates such that the
number of neurons is smaller than the dimension of the space of time-lagged
input functions, thereby forming an incomplete basis for that space (Fig. 2G).

This link between storing long time histories and expressivity is displayed
prominantly and spatially in the brain. Specifically, there exists a tight
coupling between longer periods of temporal integration and higher-order
computation, and this relationship varies systematically across the cortex
[49]. At the macro-scale, cortical brain regions are thought to follow a
dominant axis of variation that encodes a global processing hierarchy [50].
This gradient of brain organization is broadly referred to as the
sensorimotor-association (S-A) axis [51, 52]. The S–A axis spans from primary
cortices supporting sensation and movement at the bottom, to multimodal
cortices supporting multisensory processing and integration in the middle, to
transmodal association cortices supporting higher-order cognition at the top.
The S-A axis is observed across multiple diverse features of brain structure
and function [51] and is conserved across species [52, 53], demonstrating its
evolutionary roots. Notably, as regions traverse up the S-A axis they undergo
a progressive lengthening of their temporal receptive windows [54, 49].
Specifically, regions’ intrinsic functional timescales vary over the S-A axis
[55, 56] with regions at the top showing slower fluctuations reflecting longer
temporal receptive windows. In turn, these longer windows are thought to
enable greater accumulation and integration of information over time,
facilitating higher-order cognition [49]. Conversely, regions at the bottom of
the S-A axis show relatively fast dynamics, which is thought to underpin rapid
integration of recent sensory information [49].

This spatial patterning of receptive windows suggests that the brain—unlike
naively constructed RNNs—distributes its computational expressivity
systematically across the cortex, and research suggests that this may be
critical for functional integration [49]. The above data suggest that RNNs too
may benefit from spatially varying periods of temporal integration.
Specifically, the stability of RNNs is typically only considered globally
across the entire system, as the connectivity of many RNNs are initialized
randomly. If the RNN is linear (i.e., if the activation function $f$ is the
identity matrix), then any ensuing dynamic instabilities are localized to
linear subspaces, or modes, of neural activity [57]. However, if the RNN is
nonlinear (e.g., $f=\tanh$), then instabilities bleed into other modes, making
it difficult for the RNN to form a clean segregation of time-scales. Hence,
varying periods of temporal integration could be achieved by specifying
spatially varying penalties on neuronal timescales into the RNN cost
functions. Such penalties may give rise to spatially segregated modules
responsible for processing inputs at different time-scales.

###  III.2 Learning by suppressing activity

While expressivity hinges on a careful balance of stable dynamics, biological
neural networks are constrained by energy; more active neurons require more
metabolic energy, which is a limited resource. As such, while artificial
networks can maximize their expressivity through unconstrained
backpropagation, the brain’s capacity to learn is restricted by resource
constraints; also, the extent to which the brain performs backpropagation
remains unclear [58], which has motivated the machine learning community to
consider more biologically-inspired optimization approaches. Hence, penalizing
activity in RNNs should intuitively penalize computational capability through
reduced expressivity. However, recent work has instead demonstrated important
computational benefits of minimizing energy usage during training [59]. For
example, Ali _et al._ [59] trained an RNN to predict sequences of handwritten
digits, and examined how different optimization functions impacted model
architecture and behavior. Specifically, Ali _et al._ [59] did not train their
RNN to minimize prediction error through backpropagation. Instead, they
trained their model to minimize absolute levels of neural activity prior to
passing that activity through neurons’ activation functions (here, ReLU). Such
_preactivation minimization_ is akin to selectively minimizing neurons’
presynaptic inputs in biological networks. Critically, executing this cost
function required no information about task performance, and instead simply
limited the RNN’s resources in a biologically plausible way.

Alongside good task performance, Ali _et al._ [59] observed dynamics in their
RNN indicative of predictive coding. Predictive coding describes the
hypothesis that the brain stores and updates expectations about its
environment which it compares with incoming sensory evidence for those
expectations [60]. Specifically, Ali _et al._ [59] observed activity patterns
in their RNN suggestive of (i) selective self-inhibition in neurons receiving
visual stimuli and (ii) prediction of future inputs in neurons not receiving
visual stimuli. These results accord with hierarchically-organized predictive
coding coupled to the S-A axis, wherein association cortices store predictions
and via their distributed connectivity modulate activity in sensorimotor
cortices [61, 62, 63]. Taken together, the findings of Ali _et al._ [59]
indicate that while limiting the neurons’ activation might intuitively reduce
their expressivity—thereby limiting their computational capability in RNNs—it
can also serve as a unique mechanism for distributed learning that is more
biophysically realistic than backpropagation.

###  III.3 Dynamically Tuning Expressivity via Neuromodulation

The preceding sections discussed expressivity as an emergent property of a
trained network that can be modified by placing certain constraints on RNN
training. This static expressivity has been shown to be effective at
performing a wide range of tasks, and is the basis of the success of reservoir
computing [64]. Unlike in an RNN, the internal connectivity of the reservoir
computer (RC) is not trained. Instead, only the output is trained, typically
as a weighted sum of RC states [31]. Hence, RCs rely completely on the
preexisting expressivity of their internal dynamics to generate a sufficiently
expressive basis representation of their inputs. Because RCs can be trained
without knowledge or modification of the internal system, a wide variety of
physical systems have been explored as efficient RCs [65], including photonics
[66], electrical circuits [67], hydrodynamic systems [24], and the brain [68].
However, expressivity in biological networks is not static, even in the
presence of fixed weights. Instead it can be modulated dynamically over short
time-scales and–––similar to regions’ temporal receptive windows–––this too is
spatially patterned.

Previously, we discussed how the S-A axis tracks functional specialization and
integration across the cortical mantle; cortical brain systems located at the
bottom of the S-A axis are responsible for processing sensory/motor
information while systems at the top of the S-A axis are involved in
processing higher-order cognition [50], and the brain’s connectivity allows
for the hierarchical flow of information across these systems [69, 70, 71].
However, the functional roles of these different brain systems are not static.
Instead the brain utilizes a complex array of neuromodulatory systems to
actively reconfigure the brain’s dynamic repertoire [72]. In turn, this
neuromodulation endows a relatively static network architecture (i.e.,
structural connectivity) with an increased capacity for functional
flexibility. Reviewing all of the brain’s neuromodulatory mechanisms, and
their effects on neural dynamics, is beyond the scope of this piece (see [73,
74, 75, 72] for reviews), as these mechanisms comprise myriad cortico-
cortical, cortico-subcortical, and subcortical-subcortical interactions. Here,
we focus on a specific example that we believe is well positioned to be
integrated into RNNs: the balance and modulation of cortical excitation and
inhibition.

One fundamental neuromodulatory effect is that of dynamic changes to cortical
excitation and inhibition. Neuron’s in the cortex receive a complex set of
excitatory and inhibitory inputs, and the ratio between these inputs (E/I
ratio) plays a critical role in coordinating an action potential. Following
the S-A axis [51], the E/I ratio varies systematically across the cortex [76,
77, 78], leading to baseline differences in regions’ dynamics and computation
[79, 80, 81, 56]. Moreover, incorporating regional variations to the E/I ratio
into biophysical models has been shown to improve their fit to empirical
functional data [80, 82], demonstrating that the E/I ratio shapes large-scale
brain dynamics. However, unlike features of brain stucture that track the S-A
axis [51], regions’ baseline E/I ratio can be dynamically shifted via up- or
downregulating the excitatory and inhibitory neurotransmitters of postsynaptic
cells [72]. This regulation is achieved via multiple neurochemical pathways
which can be driven exogenously—for example, via pharmaceutical agents [83] or
chemogenetics [84, 85]—or endogenously, for example via the ascending
noradrenergic arousal system (AAS) [72].

In dynamical systems, changes to neuronal excitation and inhibition are
thought to engender population-level changes in neural gain (Fig. 3) [72]; the
slope of a function that maps simulated neurons’ inputs to their outputs. By
tuning the neural gain between coupled oscillators, Shine _et al._ [86]
observed that increased gain lead to greater functional integration between
neural populations. Critically, functional integration is thought to be an
important computational property of the brain; in the human brain, functional
integration fluctuates over short time scales [87] and facilitates cross-talk
between the brain’s many functionally-specialized communities [88]. Thus, on-
the-fly changes to neurons’ E/I ratio facilitates a diverse range of dynamic
behaviors [11]. This diversity allows brain function to flexibly decouple from
its underlying structural architecture [89, 90, 91, 92], which in turn
supports a broader range of computations than would otherwise be possible.

Figure 3: Tuning neural gain in RNNs. A, Neural gain can be dynamically tuned
in a trained RNN by modifying the slope of neurons’ activation function. B,
Neuroscientific research demonstrates that neural gain varies systematically
across the cortex (left) and is tuned dynamically by neuromodulators (right).
C, This neuromodulation can be used to change the dynamics of RNNs.

The above data suggests that modulation of regions’ E/I ratio gives rise to
state-dependent dynamics that facilitate the brain’s computational
expressivity. Recently, researchers have begun examining how E/I modulation
might be instantiated in RNNs, with a particular focus on the aforementioned
AAS. The AAS stems from the locus coeruleus, a small brainstem structure that
provides diffuse noradrenergic projections spanning the cerebral cortex [93,
72]. These projections modulate neuronal excitability via the neurotransmitter
noradrenaline, granting the AAS the capacity to modulate the E/I ratio.
Drawing on this mechanism, Wainstein _et al._ [94] trained an RNN to perform a
perceptual switching task, wherein one visual stimuli (a plane) gradually
morphed into another (a shark) and the RNN was tasked with reporting which
stimuli it perceived at each time point. Once trained, Wainstein _et al._ [94]
modified the slope of the artificial neurons’ activation function (i.e., the
neural gain) and examined the corresponding change in perceptual switching.
The authors observed that higher gain caused perceptual switches to occur
earlier than expected, while lower gain caused the opposite. Additionally,
Wainstein _et al._ [94] modeled the energy landscape of the RNN state-space
and observed that increasing neurons’ gain flattened the landscape, allowing
for easier state transitions (perceptual switches). Finally, the authors
supported these modeling results with task-based fMRI data as well as
pupillometry data, which is thought to be an indirect measure of
noradrenaline-mediated arousal [95]. Together, the authors’ results
demonstrate that a system’s computational function can be dynamically
modulated in behaviorally meaningful ways, and that this reconfiguration may
be underpinned by an internal capacity to regulate neural excitability.
Critically, this dynamic reconfiguration unfolds on top of a static network
architecture, wherein only neurons’ activation functions are tweaked while
their trained weights are preserved.

The results of Wainstein _et al._ [94] demonstrate that the affects of
neuromodulation can be introduced to RNNs, modifying their functional outcomes
in behaviorally meaningful ways. However, as touched on above, the brain
comprises multiple neuromodulatory systems that are capable of influencing
regions’ excitation and inhibition, each of which subserve different
functional goals [72] and each exhibit unique spatial patterning of their
associated neurotransmitters and genes [77]. Future work examining how each of
these neurotransmitter maps affect RNN behavior, across a diverse range of
tasks, will be important to characterize how different neuromodulatory
mechanisms influence expressivity. Indeed, other fields of dynamical systems
(e.g. linear systems) have already begun pursuing these goals [96, 97, 98].

##  IV Computing with the latent spaces of RNNs via constrained connectivity

While expressivity tells us what information about the input is contained in a
specific state, it does not tell us about the computational meaning behind
that state. Specifically, although Eq. 7 provides us with a map of how any
input series $\bm{u}_{\tau}$ is expressed as a specific neural state
$\bm{x}_{t+1}$, the meaning of that state depends on the context of the
problem being solved. For example, while the dynamics of the transistors in a
microprocessor can be known and simulated, the computational meaning of the
transistor state depends on its internal, or latent representation [99].

As an illustration of latent representation, consider one of the fundamental
memory elements of computers, the set-reset latch (SR-latch) [100], which
simply remembers which of two inputs were pulsed most recently. A single
nonlinear neuron with two inputs can be designed to mimic this behavior (Fig.
4A), where its state remains high if input $u_{1}$ was last pulsed, and
remains low if input $u_{2}$ was last pulsed. Here, the state of the neuron is
directly the output of a latch. Alternatively, this latch functionality can be
defined in a distributed manner into a system of multiple neurons, where the
high state is represented as some pattern of activity $\bm{r}^{*}$, the low
state is represented as another pattern of activity $\bm{r}^{\dagger}$, and
the input pulses transition the RNN state between these two (Fig. 4B). Here,
no single neuron is responsible for the latch dynamics. Rather, these latent-
space latch dynamics depend on the connectivity between neurons, as well as
how that connectivity was formed by training. In this section, we discuss how
RNNs represent and manipulate information in their latent space, and the
consequence of biological constraints on these latent representations. Then,
we draw on recent advances from the field of neurodevelopment to put forth new
directions for studying biologically-constrained RNNs.

Figure 4: Distributed latent-space representations. A, A nonlinear single-
neuron model with two inputs (top), which has been designed to act as a memory
circuit such that an impulse from input $u_{1}$ or $u_{2}$ will cause the
state of the neuron to stay fixed at a high or low value, respectively
(bottom). B, A nonlinear multi-neuron RNN which has been designed to act as a
memory circuit (top), except that the RNN represents the states “high” and
“low” as two stable fixed-points across all of its states, and these fixed-
points are determined in a distributed manner via all of the connections
between the neurons.

###  IV.1 Sparsity and attractor stability

Neural networks harness the power of internal, or latent, representations for
computational tasks such as path integration [101], tracking [102], and
spatial working memory [103]. In RNNs and dynamical systems, these latent
representations are often referred to as attractors: sets of points,
$\mathcal{S}=\\{\bm{s}_{i}\\}$, to which the dynamics evolve after a
relatively long period of time. An early example of latent representations is
associative memory in Hopfield networks [104], wherein a specific set of
neural activity patterns, $\mathcal{S}=\\{\bm{s}_{i}\\}$, are stored as
memories that could represent information such as an image. Specifically,
$\mathcal{S}$ are stored as fixed-point attractors [105] such that after
stimulating neurons close to a specific pattern,
$\bm{x}_{0}=\bm{s}_{i}+\bm{\epsilon}$, the neural states will evolve toward a
stored memory, $\bm{x}_{t\rightarrow\infty}\approx\bm{s}_{i}$. In general, a
fixed-point is a state $\bm{x}^{*}$ such that

$\displaystyle\bm{x}^{*}=f(\bm{x}^{*}).$ (8)

The computational power of this Hopfield network is in using inputs to
retrieve pre-defined information (i.e., _memories_) stored in attractors, and
substantial work has gone into improving their computational capability [106]
and biophysical realism [107]. Hence, fixed-point attractors are latent
dynamical properties that can be harnessed for computation.

In addition to discrete memory states, RNNs can make use of the geometry of
their attractors to form representations and make decisions. For example,
continuous-attractor neural networks (CANNs) extend the concept of an
attracting point to higher-dimensional manifolds, thereby forming attracting
curves and surfaces such that the geometric position along these manifolds
holds latent computational meaning. For example, the geometric trajectory of
the neural network state along these manifolds can reflect a path traversed in
real physical space by an agent, [101], the continuous tracking of a moving
stimulus [102], and the recall of spatial location in the prefrontal cortex
[103]. While the connectivity and dynamics of CANNs are precisely engineered
to preserve translation-invariance along their structure, this continuum of
attractors also emerges in trained RNN models [108], and even in models of the
prefrontal cortex trained to integrate information given different contexts
[3]. Hence, rather than the activity of one or a collection of specific
neurons [109], it is the geometry of the attractor manifold that forms the
internal representation of information in the RNN, and the RNN integrates
external information by moving its representation along that manifold [110].

Formation of attractors occurs as a consequence of a loss of energy in the
system, which in turn results in the stabilization of dynamics [111]. This
stabilization can be characterized using methods such as Lyapunov
functions—energy-like quantities that monotonically decrease or dissipate
throughout the dynamics [112, 113]—and Lyapunov exponents—the rate of
convergence towards an attracting manifold [114]—to study existing systems.
However, what is less clear is how connectivity can develop to improve the
stabilization of these attractors. Intuitively, this energy dissipation
usually takes the form of a loss in the neural activity (e.g. the “leaky”
component of the leaky integrator in Eq. 2), whereas the parameters that can
be learned in the RNN are the connectivity. Thus, the question becomes: how do
we modify the RNN connectivity to achieve greater attractor stability?

A biologically-inspired optimization process that has proven useful for
stabilizing RNN dynamics is _sparseness_. In order to minimize energy
expenditure [111], the brain substantially prunes its connectivity [115]
retaining only a sparse set of weights that are finely tuned to achieve its
functional goals. In RNNs, inducing sparseness via weight pruning has been
shown to provide several computational benefits. For example, Averbeck [116]
trained RNNs with and without weight pruning to complete a working memory
task. Compared to their unpruned counterparts, moderate amounts of pruning
yielded RNNs that (i) exhibited better task performance; (ii) required fewer
training epochs; (iii) had stronger connectivity weights; and (iv) were more
resistant to task distractors [116]. Notably, regarding distractor resistance,
pruned RNNs showed a smaller departure from their dynamic trajectories when
they were perturbed by a distracting probe within the task. This result
demonstrates that the sparse connectivity in the pruned RNNs strengthened
their attractor basins (Fig. 5), making them more stable and resistant to
undesired inputs.

Figure 5: Sparse connectivity leads to stable attractors. Pruning connectivity
weights in RNNs leads to sparse weight matrices that deepen the attractor
basins of RNNs dynamics, making dynamics more stable.

###  IV.2 Deduction: Learning Problem Structure through Iterative Algorithms

While the geometry of dynamical attractors can be used for computational
purposes, they also arise as the solutions to complex problems. For example,
iterative methods are commonly used in optimization problems such as iterative
refinement, [117], root-finding methods [118], and feasibility problems [119].
Critically, in these and many other examples, the solution is not learned in
the typical deep learning sense (i.e., training), but rather emerges as an
attractor to satisfy the conditions of an iterative algorithm. In the same
manner, RNNs need not only learn the attractor structure of specific input-
output relations (i.e., through training), but have the potential to encode a
specific algorithm in the iteration of the neural states, such that solutions
to problems (i.e., tasks) emerge as attractors.

Biological neural networks demonstrate the ability to run iterations within
their latent representations [120]. A prominant example is hippocampal replay,
whereby hippocampal place cells will reactivate along the same sequence as in
a prior navigation experience [121], even when the subject is not actively
performing a navigation task. Another prominant example is dynamical
inference, whereby neural activity in the dlPFC can reliably predict the
future nonlinear trajectory of a ball, and RNN models which best replicate
prediction behavior are trained on the sequence of the ball’s trajectory [122,
123]. Hence, RNNs are not only capable of learning attractor geometries, but
also of learning and simulating the sequence of the problem structure, which
may enable more generalizable solutions.

RNNs can be engineered to run iterative deductions through several means. One
approach involves assigning to each neuron the state variable of an algorithm,
and defining complex interaction dynamics such that the RNN state will settle
on the solution as a stable attractor. For example, an RNN can be designed to
solve $k$-satisfiability problems, which seek an assignment of $n$ Boolean
variables that satisfy a set of $c$ constraints, where each constraint places
a condition on subsets of $k$ Boolean variables (Fig. 6A) [124]. These RNNs
evolve until the neural states find a solution [124]. Surprisingly, a wide
variety of different dynamics and architectures can lead to different
algorithms for solving the same satisfiability problem [125, 126], and other
difficult optimization problems such as integer linear programming feasibility
[127] or the $n$-queens problem [128]. Hence, algorithm variables can be
directly represented by individual neurons, and the algorithm rules can be
directly encoded in the connectivity and update rules $f$ of RNNs.

Figure 6: Programming Algorithms into RNNs A, Schematic of an RNN whose
neurons $s_{i}$ and $a_{j}$ represent states in a $k$-SAT problem, and whose
connectivity is defined such that the stable attracting manifold is guaranteed
to be a solution (left). The dynamics of the RNN evolve in a complex manner as
it searches and finds the assignment of states that satisfies the $k$-SAT
problem (center, right). B, Schematic of the process for designing algorithms
into RNNs, where the recurrent connectivity of the RNN (gold) can be
constructed from a low-rank product of matrices ($B$,$W$). Through this
construction, the evolution of the RNN states concurrently evolves the state
variables of the algorithm within a low-dimensional projection. This designed
connectivity was used to program and simulate the SR-latch in Fig. 4B).

In addition to a direct, one-to-one encoding of problem structure as an RNN,
we can also design algorithms into the latent spaces of RNNs [129]. Rather
than ascribing each neuron a specific variable in an algorithm, we can embed
an algorithm into the distributed connectivity of an RNN (Fig. 6B). One
technique for this embedding is the neural engineering framework (NEF) which
enables the design of iterations of the latent-space variables through the
engineering of low-rank connectivity matrices [130, 131]. Extensions enable
the programming of iterations in pre-existing, higher-rank connectivity, and
the ability to reverse-engineer representations from conventionally trained
RNNs [42]. Other engineered architectures such as the differentiable neural
computer [132] and the neural turing machine [133] emulate the structure of
conventional computers using differentiable neural elements. Hence, RNNs have
the capability to explicitly run complex algorithms in their latent spaces,
which is crucial for generalizable computation; rather than learning
individual solutions, we posit that training RNNs on solution sequences will
enable them to learn and generalize problem-solving strategies [134], even
into nonlinear, out-of-sample regimes [135].

While the above approaches may provide more generalizable RNN solutions than
task-specific training, the added computational capabilities of engineered
RNNs is accompanied by a further deviation of the corresponding connectivity
from biology. Whether through the enforcing of low-rank connectivity [130,
136] or the segregation of memory and processing units in a neural von Neumann
architecture [132], engineered neural connectivities lack many of the costs
and constraints experienced by biological networks. A critical question then
is how the brain formulates connectivities that permit sophisticated latent
space computations while adhering to biological constraints. Prior work has
demonstrated the capability of largely disordered RNNs to produce sequences
that rely on recurrent connectivity [137], and the importance of the sequence
of learning over many learning iterations—a curriculum—for RNN performance
[138]. Hence, rather than forming low-rank structures ab initio, the brain
defines its structure and dynamics through learning and plasticity on long
time scales. We explore insights into the governing principles and
computational advantages of this progression in neurodevelopment.

###  IV.3 Latent-space computational capability throughout neurodevelopment

Just as the topology of an RNN is sculpted over training epochs (Fig. 7A), the
topology of the human connectome is sculpted throughout development (Fig. 7B).
However, unlike the RNN, which may be trained in an unconstrained manner,
neurodevelopment follows a carefully orchestrated and stereotyped program that
unfolds dynamically across space and time. Specifically, cortical
neurodevelopment is thought to spatially track the aforementioned S-A axis in
a temporally staged manner [51, 139], and this staging is thought to underpin
the emergence of cortical regions’ functional specialization and inter-
connectivity. Crucially, the asynchronous nature of this developmental program
is thought to underpin the sequential emergence of increasingly complex
cognitive functions [140, 141], suggesting that neurodevelopment stages the
brain’s acquisition of lower- and higher-order computational processes.
Mechanistically, this program may be underpinned by windows of heightened
neural plasticity that cascade up the S-A axis [142, 139] priming specific
neural circuits at specific points in time for experience-dependent neural
change.

Figure 7: Computational capabilities of RNNs and biological brains throughout
development. RNNs and brains are subject to different environments. A, RNNs
comprise abstract nodes and edges, and their training is unconstrained. This
allows RNNs to evolve toward any network architecture that optimally performs
a given task. B, By contrast, brains comprise physically embedded nodes and
edges, and the formations of their network architecture is tightly constrained
by space and time. C, The extent to which pre-existing connectivities can
follow latent representations can be captured through generalized
synchronization, which measures how well an RNN that is driven by an
attracting manifold (in this case, the chaotic Lorenz attractor) can
reconstruct the driving signal from its states. D, The ability of pre-existing
connectivities to internalize and generate latent representations is captured
through invertible generalized synchronization, whereby an RNN that has
learned output weights $W$ that copy the attractor manifold can then drive
itself by feeding those outputs back as inputs to autonomously generate the
attractor without any input.

Regions in the cortex are defined in part according their laminar structure
[143, 144], with different regions exhibiting variations in the number and
size of their distinct layers, as well as different distributions of cells
throughout those layers. Critically, cortical variations in cytoarchitecture
conform to the S-A axis [145], and animal research demonstrates that this
spatial patterning predicts regions’ extrinsic connectivity [146], including
their strength [147], distance [147], and layer-wise projections [147, 148].
In humans, structural connectivity between regions at the bottom of the S-A
axis refines relatively early in development, while connectivity at the top of
the S-A axis does so later in development [149, 150, 151, 152, 71].
Furthermore, recent work has shown that cytoarchitecture plays an important
role in shaping how dynamics spread across the connectome throughout
development [69]. Thus, the spatial patterning embedded in the S-A axis plays
a key role in shaping connectome topology throughout development. But what
about RNNs? Recent work by Achterberg _et al._ [153] regularized RNNs by using
the Euclidean distance between regions to constrain training. The authors
found that modularity [154] and small-worldness [155]—two complex topological
features that are hallmarks of the human connectome [28]—emerged to a greater
extent in these spatially-embedded RNNs compared to standard RNNs (see also
recent work by Tanner _et al._ [156] for evidence of modularity in RNNs
trained without spatial constraints). Additionally, this effect coincided with
achieving higher out-of-sample task performance earlier in training compared
to standard RNNs (though performance eventually converged; see their Figure
2A).

The results of Achterberg _et al._ [153] demonstrate that incorporating space-
based inductive biases into RNN training causes them to converge on
topological features observed in the human connectome. However, it remains
unclear whether neurodevelopmentally-informed spatial constraints, like those
embedded in the S-A axis [51], show similar effects. We posit that
constraining RNNs using the S-A axis may outperform Euclidean distance-based
spatial embedding, as the former is rooted in evolutionary programs of
connectivity formation and functional specialization [53]. Additionally, the
spatial constraints deployed by Achterberg _et al._ [153] were static
throughout training. As mentioned above, the S-A axis scaffolds connectome
development in a temporally varying way [139], and incorporating this dynamic
information will be critical to achieving realistic brain-like topology in
RNNs. One approach would be to code spatially varying periods of heightened
learning into RNNs, simulating traveling waves of heightened neural plasticity
[142, 139]. Such an inductive bias could be achieved by including temporal
cascades of weight training that flow bottom-up across the S-A axis.

In addition to injecting spatially constrained inductive biases into RNNs, we
can also directly assess the ability of the developing connectome to support
latent-space computation. This analysis can be achieved through studying the
synchronization between a given RNN and a particular latent attractor.
Historically, synchronization has been shown to be crucial for computation in
both biological cortical networks [157] and artifical RNNs [158], and is
deeply related to consensus dynamics [159]. Intuitively, synchronization
between two systems implies that both systems are evolving identically. This
concept can be extended to generalized synchronization, which stipulates
conditions under which a response system, $\bm{r}_{t}$, has synchronized in a
general sense to a driving system, $\bm{d}_{t}$ [160], such that rather than
evolving identically such that $\bm{r}_{t}=\bm{d}_{t}$ [161], the joint system
has collapsed onto a function $\phi$ of only the driver system such that
$\bm{r}_{t}=\phi(\bm{d}_{t})$. Under these conditions, the response system has
followed the attractor structure of the driving system. If we choose the
response system to be an RNN with an empirically-derived connectivity taken at
a specific point in neurodevelopment, and the driver system to be a specific
latent attractor dynamical system, then we can assess whether the RNN can
follow the attractor structure of the driving system (Fig. 7C).

Of course, just because the RNN can synchronize to a particular latent-space
attractor does not guarantee that it can maintain that attractor once the
driving system is gone. In order for the RNN to internalize the attractor
dynamics, theories of invertible generalized synchronization (IGS) stipulate
conditions for which the attractor of the driver system can be invertibly
reproduced and stabilized by the response system, and thus can be learned
autonomously [162]. Hence, rather than modifying RNN connectivity to learn a
latent space conditioned on spatial and resource constraints, IGS tests
whether a given RNN connectivity that already obeys those spatial and resource
constraints can stably generate a latent attractor (Fig. 7D). The IGS theory
also indicates that the ability to internalize latent attractors from driving
systems depends not only on the RNN’s structure, but also the latent
attractor. Thus, given RNNs of different structures throughout development, we
may examine the IGS on their most-likely encountered driving signals
corresponding to their position along the S-A axis.

##  V Conclusions

Neural systems compute using dynamics, and the dynamics of biological brains
evolve atop the computational substrate of a spatially and resource
constrained network. Here, we sought to jointly discuss advances in
neuroscience and dynamical systems with a view to improving the computational
power and interpretability of RNNs. Within the context of two computational
capabilities—expressivity and latent-space computing—we highlighted several
avenues for future research that we believe will advance our understanding of
dynamical computation. Through these avenues, we envision biologically
interpretable, computationally improved RNN models of how the brain computes.

Central to this research program is the application of biophysical constraints
on the computational dynamics of RNNs. While not exhaustive, the constraints
discussed herein represent the diverse influences of neurobiology on network
structure, and they have been shown to influence the emergence of complex
behavior in humans. Furthermore, the influence of these constraints on RNN
connectivity can studied in combination. For example, the spatial-patterning
of regions’ baseline E/I ratio emerges throughout development [139],
indicating it’s connection to the S-A axis. Another example is to explore how
to extend low-rank RNN design approaches to higher-rank connectivities that
more closely match the spatial gradients of the S-A axis. Thus, these
biophysical constraints provide fertile ground for future experimental,
computational, and theoretical work into biologically-informed RNNs.

The methodological strategies for incorporating these constraints into
computational RNN modeling are vast. Here, we have discussed several
approaches: using known cortical structure and function as target
connectivities of additional training constraints [51, 153], using resource
constraints as an alternative learning mechanism [32], dynamically altering
connectivity with neuromodulation for increased expressivity [86], stabilizing
latent attractors through pruning [116], engineering and modifying low-rank
latent representations and algorithms [130, 42], probing the teachability of
RNNs via synchronization [162], among many others. Hence, there is no one
prescription that serves as a panacea for the rich problems that lie at the
intersection of computation, dynamics, and neurobiology. Instead, we must
continue developing diverse and creative approaches for maximizing the
computational capabilities of neurodynamical models through the use of
biological and developmental constraints.

## Acknowledgements

We gratefully acknowledge comments and conversations with Dr. Zhixin Lu, Dr.
Harang Ju, Dr. Dale Zhou, Dr. Melody Lim, and Dr. Stephen Hanson. JZK was
supported by the Bethe/KIC/Wilkins postdoctoral fellowship. BL was supported
by the National Institute Of Mental Health of the National Institutes of
Health under Award Number R00MH127293. LP was supported by the National
Institute Of Mental Health of the National Institutes of Health under Award
Number R00MH127296.

## References

## References

  * [1] Ijspeert, A. J.  Central pattern generators for locomotion control in animals and robots: a review.  _Neural networks_ 21, 642–653 (2008).
  * [2] Seeley, W. W. _et al._ Dissociable intrinsic connectivity networks for salience processing and executive control.  _Journal of Neuroscience_ 27, 2349–2356 (2007).
  * [3] Mante, V., Sussillo, D., Shenoy, K. V. & Newsome, W. T.  Context-dependent computation by recurrent dynamics in prefrontal cortex.  _nature_ 503, 78–84 (2013).
  * [4] Werbos, P. J.  Backpropagation through time: what it does and how to do it.  _Proceedings of the IEEE_ 78, 1550–1560 (1990).
  * [5] Betzel, R. F. _et al._ Generative models of the human connectome.  _NeuroImage_ 124, 1054–1064 (2016).
  * [6] Oldham, S. & Fornito, A.  The development of brain network hubs.  _Developmental Cognitive Neuroscience_ 36, 100607 (2019).
  * [7] Oldham, S. _et al._ Modeling spatial, developmental, physiological, and topological constraints on human brain connectivity.  _Science Advances_ 8, eabm6127 (2022).
  * [8] Akarca, D. _et al._ A generative network model of neurodevelopmental diversity in structural brain organization.  _Nature Communications_ 12, 4216 (2021).
  * [9] Hodgkin, A. L. & Huxley, A. F.  A quantitative description of membrane current and its application to conduction and excitation in nerve.  _The Journal of physiology_ 117, 500 (1952).
  * [10] Petousakis, K.-E., Apostolopoulou, A. A. & Poirazi, P.  The impact of hodgkin–huxley models on dendritic research.  _The Journal of Physiology_ (2022).
  * [11] Wilson, H. R. & Cowan, J. D.  Excitatory and inhibitory interactions in localized populations of model neurons.  _Biophysical journal_ 12, 1–24 (1972).
  * [12] Pinto, D. J., Brumberg, J. C., Simons, D. J., Ermentrout, G. B. & Traub, R.  A quantitative population model of whisker barrels: re-examining the wilson-cowan equations.  _Journal of computational neuroscience_ 3, 247–264 (1996).
  * [13] Sadeghi, S., Mier, D., Gerchen, M. F., Schmidt, S. N. & Hass, J.  Dynamic causal modeling for fmri with wilson-cowan-based neuronal equations.  _Frontiers in Neuroscience_ 14, 593867 (2020).
  * [14] Parkes, L. _et al._ Using network control theory to study the dynamics of the structural connectome.  preprint, Neuroscience (2023).
  * [15] Lynn, C. W. & Bassett, D. S.  The physics of brain network structure, function and control.  _Nature Reviews Physics_ 1, 318–332 (2019).
  * [16] Seguin, C., Sporns, O. & Zalesky, A.  Brain network communication: concepts, models and applications.  _Nature Reviews Neuroscience_ (2023).
  * [17] Srivastava, P. _et al._ Models of communication and control for brain networks: distinctions, convergence, and future outlook.  _Network Neuroscience_ 4, 1122–1159 (2020).
  * [18] Breakspear, M.  Dynamic models of large-scale brain activity.  _Nature Neuroscience_ 20, 340–352 (2017).
  * [19] Roberts, J. A. _et al._ Metastable brain waves.  _Nature Communications_ 10, 1056 (2019).
  * [20] FitzHugh, R.  Impulses and physiological states in theoretical models of nerve membrane.  _Biophysical journal_ 1, 445–466 (1961).
  * [21] McCulloch, W. S. & Pitts, W.  A logical calculus of the ideas immanent in nervous activity.  _The bulletin of mathematical biophysics_ 5, 115–133 (1943).
  * [22] Chung, S. & Siegelmann, H.  Turing completeness of bounded-precision recurrent neural networks.  _Advances in Neural Information Processing Systems_ 34, 28431–28441 (2021).
  * [23] Liu, Y., Gong, C., Yang, L. & Chen, Y.  Dstp-rnn: A dual-stage two-phase attention-based recurrent neural network for long-term and multivariate time series prediction.  _Expert Systems with Applications_ 143, 113082 (2020).
  * [24] Lu, Z., Kim, J. Z. & Bassett, D. S.  Supervised chaotic source separation by a tank of water.  _Chaos: An Interdisciplinary Journal of Nonlinear Science_ 30 (2020).
  * [25] Wang, P. Y., Sun, Y., Axel, R., Abbott, L. & Yang, G. R.  Evolving the olfactory system with machine learning.  _Neuron_ 109, 3879–3892.e5 (2021).
  * [26] Yang, G. R., Joglekar, M. R., Song, H. F., Newsome, W. T. & Wang, X.-J.  Task representations in neural networks trained to perform many cognitive tasks.  _Nature Neuroscience_ 22, 297–306 (2019).
  * [27] Roberts, J. A. _et al._ The contribution of geometry to the human connectome.  _NeuroImage_ 124, 379–393 (2016).
  * [28] Bullmore, E. & Sporns, O.  The economy of brain network organization.  _Nature Reviews Neuroscience_ 13, 336–349 (2012).
  * [29] Ramón y Cajal, S.  _Histology of the nervous system of man and vertebrates_.  No. no. 6 in History of neuroscience (Oxford University Press, New York, 1995).
  * [30] Dupont, E., Doucet, A. & Teh, Y. W.  Augmented neural odes.  _Advances in neural information processing systems_ 32 (2019).
  * [31] Lukoševičius, M. & Jaeger, H.  Reservoir computing approaches to recurrent neural network training.  _Computer science review_ 3, 127–149 (2009).
  * [32] Ali, A., Ahmad, N., de Groot, E., van Gerven, M. A. J. & Kietzmann, T. C.  Predictive coding is a consequence of energy efficiency in recurrent neural networks.  _Patterns_ 3 (2022).
  * [33] Felleisen, M.  On the expressive power of programming languages.  _Science of computer programming_ 17, 35–75 (1991).
  * [34] Sipser, M.  Introduction to the theory of computation.  _ACM Sigact News_ 27, 27–29 (1996).
  * [35] Hornik, K., Stinchcombe, M. & White, H.  Multilayer feedforward networks are universal approximators.  _Neural networks_ 2, 359–366 (1989).
  * [36] Cybenko, G.  Approximation by superpositions of a sigmoidal function.  _Mathematics of control, signals and systems_ 2, 303–314 (1989).
  * [37] Schäfer, A. M. & Zimmermann, H. G.  Recurrent neural networks are universal approximators.  In _Artificial Neural Networks–ICANN 2006: 16th International Conference, Athens, Greece, September 10-14, 2006. Proceedings, Part I 16_ , 632–640 (Springer, 2006).
  * [38] Poole, B., Lahiri, S., Raghu, M., Sohl-Dickstein, J. & Ganguli, S.  Exponential expressivity in deep neural networks through transient chaos.  _Advances in neural information processing systems_ 29 (2016).
  * [39] Raghu, M., Poole, B., Kleinberg, J., Ganguli, S. & Sohl-Dickstein, J.  On the expressive power of deep neural networks.  In _international conference on machine learning_ , 2847–2854 (PMLR, 2017).
  * [40] Bertschinger, N. & Natschläger, T.  Real-time computation at the edge of chaos in recurrent neural networks.  _Neural computation_ 16, 1413–1436 (2004).
  * [41] Sompolinsky, H., Crisanti, A. & Sommers, H.-J.  Chaos in random neural networks.  _Physical review letters_ 61, 259 (1988).
  * [42] Kim, J. Z. & Bassett, D. S.  A neural machine code and programming framework for the reservoir computer.  _Nature Machine Intelligence_ 1–9 (2023).
  * [43] Rajan, K. & Abbott, L. F.  Eigenvalue spectra of random matrices for neural networks.  _Physical review letters_ 97, 188104 (2006).
  * [44] Langton, C. G.  Computation at the edge of chaos: Phase transitions and emergent computation.  _Physica D: nonlinear phenomena_ 42, 12–37 (1990).
  * [45] Sussillo, D. & Abbott, L. F.  Generating coherent patterns of activity from chaotic neural networks.  _Neuron_ 63, 544–557 (2009).
  * [46] Munoz, M. A.  Colloquium: Criticality and dynamical scaling in living systems.  _Reviews of Modern Physics_ 90, 031001 (2018).
  * [47] Hochstetter, J. _et al._ Avalanches and edge-of-chaos learning in neuromorphic nanowire networks.  _Nature Communications_ 12, 4008 (2021).
  * [48] Ju, H., Kim, J. Z., Beggs, J. M. & Bassett, D. S.  Network structure of cascading neural systems predicts stimulus propagation and recovery.  _Journal of Neural Engineering_ 17, 056045 (2020).
  * [49] Wolff, A. _et al._ Intrinsic neural timescales: temporal integration and segregation.  _Trends in Cognitive Sciences_ S1364661321002928 (2022).
  * [50] Mesulam, M.  Representation, inference, and transcendent encoding in neurocognitive networks of the human brain.  _Annals of Neurology_ 64, 367–378 (2008).
  * [51] Sydnor, V. J. _et al._ Neurodevelopment of the association cortices: Patterns, mechanisms, and implications for psychopathology.  _Neuron_ 109, 2820–2846 (2021).
  * [52] Margulies, D. S. _et al._ Situating the default-mode network along a principal gradient of macroscale cortical organization.  _Proceedings of the National Academy of Sciences_ 113, 12574–12579 (2016).
  * [53] Xu, T. _et al._ Cross-species functional alignment reveals evolutionary hierarchy within the connectome.  _NeuroImage_ 223, 117346 (2020).
  * [54] Hasson, U., Yang, E., Vallines, I., Heeger, D. J. & Rubin, N.  A Hierarchy of Temporal Receptive Windows in Human Cortex.  _Journal of Neuroscience_ 28, 2539–2550 (2008).
  * [55] Gao, R., van den Brink, R. L., Pfeffer, T. & Voytek, B.  Neuronal timescales are functionally dynamic and shaped by cortical microarchitecture.  _eLife_ 9, e61277 (2020).
  * [56] Sydnor, V. J. _et al._ Intrinsic activity development unfolds along a sensorimotor–association cortical axis in youth.  _Nature Neuroscience_ 26, 638–649 (2023).
  * [57] Hespanha, J. P.  _Linear systems theory_ (Princeton university press, 2018).
  * [58] Marblestone, A. H., Wayne, G. & Kording, K. P.  Toward an Integration of Deep Learning and Neuroscience.  _Frontiers in Computational Neuroscience_ 10 (2016).
  * [59] Ali, A., Ahmad, N., De Groot, E., Johannes Van Gerven, M. A. & Kietzmann, T. C.  Predictive coding is a consequence of energy efficiency in recurrent neural networks.  _Patterns_ 3, 100639 (2022).
  * [60] Millidge, B., Seth, A. & Buckley, C. L.  Predictive Coding: a Theoretical and Experimental Review (2021).  Publisher: arXiv Version Number: 4.
  * [61] Bastos, A. M. _et al._ Canonical Microcircuits for Predictive Coding.  _Neuron_ 76, 695–711 (2012).
  * [62] Singer, W.  Recurrent dynamics in the cerebral cortex: Integration of sensory evidence with stored knowledge.  _Proceedings of the National Academy of Sciences_ 118, e2101043118 (2021).
  * [63] Friston, K.  Computational psychiatry: from synapses to sentience.  _Molecular Psychiatry_ (2022).
  * [64] Jaeger, H.  Tutorial on training recurrent neural networks, covering bppt, rtrl, ekf and the" echo state network" approach (2002).
  * [65] Tanaka, G. _et al._ Recent advances in physical reservoir computing: A review.  _Neural Networks_ 115, 100–123 (2019).
  * [66] Van der Sande, G., Brunner, D. & Soriano, M. C.  Advances in photonic reservoir computing.  _Nanophotonics_ 6, 561–576 (2017).
  * [67] Soriano, M. C. _et al._ Delay-based reservoir computing: noise effects in a combined analog and digital implementation.  _IEEE transactions on neural networks and learning systems_ 26, 388–393 (2014).
  * [68] Suárez, L. E., Richards, B. A., Lajoie, G. & Misic, B.  Learning function from structure in neuromorphic networks.  _Nature Machine Intelligence_ (2021).
  * [69] Parkes, L. _et al._ Asymmetric signaling across the hierarchy of cytoarchitecture within the human connectome.  _Science Advances_ 8, eadd2185 (2022).
  * [70] Pines, A. _et al._ Development of top-down cortical propagations in youth.  _Neuron_ S0896627323000387 (2023).
  * [71] Baum, G. L. _et al._ Modular Segregation of Structural Brain Networks Supports the Development of Executive Function in Youth.  _Current Biology_ 27, 1561–1572.e8 (2017).
  * [72] Shine, J. M. _et al._ Computational models link cellular mechanisms of neuromodulation to large-scale neural dynamics.  _Nature Neuroscience_ 24, 765–776 (2021).
  * [73] Marder, E. & Thirumalai, V.  Cellular, synaptic and network effects of neuromodulation.  _Neural Networks_ 15, 479–493 (2002).
  * [74] Bucher, D. & Goaillard, J.-M.  Beyond faithful conduction: Short-term dynamics, neuromodulation, and long-term regulation of spike propagation in the axon.  _Progress in Neurobiology_ 94, 307–346 (2011).
  * [75] McCormick, D. A., Nestvogel, D. B. & He, B. J.  Neuromodulation of Brain State and Behavior.  _Annual Review of Neuroscience_ 43, 391–415 (2020).
  * [76] Kim, Y. _et al._ Brain-wide Maps Reveal Stereotyped Cell-Type-Based Cortical Architecture and Subcortical Sexual Dimorphism.  _Cell_ 171, 456–469.e22 (2017).
  * [77] Burt, J. B. _et al._ Hierarchy of transcriptomic specialization across human cortex captured by structural neuroimaging topography.  _Nature Neuroscience_ 21, 1251–1259 (2018).
  * [78] Anderson, K. M. _et al._ Transcriptional and imaging-genetic association of cortical interneurons, brain function, and schizophrenia risk.  _Nature Communications_ 11, 2889 (2020).
  * [79] Deco, G. _et al._ How Local Excitation-Inhibition Ratio Impacts the Whole Brain Dynamics.  _Journal of Neuroscience_ 34, 7886–7898 (2014).
  * [80] Deco, G. _et al._ Dynamical consequences of regional heterogeneity in the brain’s transcriptional landscape.  _Science Advances_ 7, eabf4752 (2021).
  * [81] Gao, R., Peterson, E. J. & Voytek, B.  Inferring synaptic excitation/inhibition balance from field potentials.  _NeuroImage_ 158, 70–78 (2017).
  * [82] Zhang, S. _et al._ In-vivo whole-cortex estimation of excitation-inhibition ratio indexes cortical maturation and cognitive ability in youth.  preprint, bioRxiv (2023).
  * [83] Larsen, B. _et al._ A developmental reduction of the excitation:inhibition ratio in association cortex during adolescence.  _Science Advances_ 8, eabj8750 (2022).
  * [84] Rocchi, F. _et al._ Increased fMRI connectivity upon chemogenetic inhibition of the mouse prefrontal cortex.  _Nature Communications_ 13, 1056 (2022).
  * [85] Markicevic, M. _et al._ Cortical Excitation:Inhibition Imbalance Causes Abnormal Brain Network Dynamics as Observed in Neurodevelopmental Disorders.  _Cerebral Cortex_ 30, 4922–4937 (2020).
  * [86] Shine, J. M., Aburn, M. J., Breakspear, M. & Poldrack, R. A.  The modulation of neural gain facilitates a transition between functional segregation and integration in the brain.  _eLife_ 7, e31130 (2018).
  * [87] Shine, J. M. _et al._ Human cognition involves the dynamic integration of neural activity and neuromodulatory systems.  _Nature Neuroscience_ 22, 289–296 (2019).
  * [88] Bertolero, M. A., Yeo, B. T. T. & D’Esposito, M.  The modular and integrative functional architecture of the human brain.  _Proceedings of the National Academy of Sciences_ 112 (2015).
  * [89] Vázquez-Rodríguez, B. _et al._ Gradients of structure–function tethering across neocortex.  _Proceedings of the National Academy of Sciences_ 116, 21219–21227 (2019).
  * [90] Preti, M. G. & Van De Ville, D.  Decoupling of brain function from structure reveals regional behavioral specialization in humans.  _Nature Communications_ 10, 4747 (2019).
  * [91] Mišić, B. _et al._ Network-Level Structure-Function Relationships in Human Neocortex.  _Cerebral Cortex_ 26, 3285–3296 (2016).
  * [92] Zamani Esfahlani, F., Faskowitz, J., Slack, J., Mišić, B. & Betzel, R. F.  Local structure-function relationships in human brain networks across the lifespan.  _Nature Communications_ 13, 2053 (2022).
  * [93] Samuels, E. & Szabadi, E.  Functional Neuroanatomy of the Noradrenergic Locus Coeruleus: Its Roles in the Regulation of Arousal and Autonomic Function Part II: Physiological and Pharmacological Manipulations and Pathological Alterations of Locus Coeruleus Activity in Humans.  _Current Neuropharmacology_ 6, 254–285 (2008).
  * [94] Wainstein, G. _et al._ Gain neuromodulation mediates perceptual switches: evidence from pupillometry, fMRI, and RNN Modelling.  preprint, Research Square (2023).
  * [95] Joshi, S. & Gold, J. I.  Pupil Size as a Window on Neural Substrates of Cognition.  _Trends in Cognitive Sciences_ 24, 466–480 (2020).
  * [96] Singleton, S. P. _et al._ Receptor-informed network control theory links LSD and psilocybin to a flattening of the brain’s control energy landscape.  _Nature Communications_ 13, 5812 (2022).
  * [97] Luppi, A. I. _et al._ Transitions between cognitive topographies: contributions of network structure, neuromodulation, and disease.  preprint, bioRxiv (2023).
  * [98] Aitken, K. & Mihalas, S.  Neural population dynamics of computing with synaptic modulations.  _Elife_ 12, e83035 (2023).
  * [99] Jonas, E. & Kording, K. P.  Could a neuroscientist understand a microprocessor?  _PLoS computational biology_ 13, e1005268 (2017).
  * [100] Torii, N. _et al._ Asic implementation of random number generators using sr latches and its evaluation.  _EURASIP Journal on Information Security_ 2016, 1–12 (2016).
  * [101] Samsonovich, A. & McNaughton, B. L.  Path integration and cognitive mapping in a continuous attractor neural network model.  _Journal of Neuroscience_ 17, 5900–5920 (1997).
  * [102] Fung, C. A., Wong, K. M. & Wu, S.  A moving bump in a continuous manifold: a comprehensive study of the tracking dynamics of continuous attractor neural networks.  _Neural Computation_ 22, 752–792 (2010).
  * [103] Wimmer, K., Nykamp, D. Q., Constantinidis, C. & Compte, A.  Bump attractor dynamics in prefrontal cortex explains behavioral precision in spatial working memory.  _Nature neuroscience_ 17, 431–439 (2014).
  * [104] Hopfield, J. J.  Neural networks and physical systems with emergent collective computational abilities.  _Proceedings of the national academy of sciences_ 79, 2554–2558 (1982).
  * [105] Strogatz, S. H.  _Nonlinear dynamics and chaos with student solutions manual: With applications to physics, biology, chemistry, and engineering_ (CRC press, 2018).
  * [106] Ramsauer, H. _et al._ Hopfield networks is all you need.  _arXiv preprint arXiv:2008.02217_ (2020).
  * [107] Storkey, A.  Increasing the capacity of a hopfield network without sacrificing functionality.  In _Artificial Neural Networks—ICANN’97: 7th International Conference Lausanne, Switzerland, October 8–10, 1997 Proceeedings 7_ , 451–456 (Springer, 1997).
  * [108] Smith, L. M., Kim, J. Z., Lu, Z. & Bassett, D. S.  Learning continuous chaotic attractors with a reservoir computer.  _Chaos: An Interdisciplinary Journal of Nonlinear Science_ 32 (2022).
  * [109] Nichols, M. J. & Newsome, W. T.  Middle temporal visual area microstimulation influences veridical judgments of motion direction.  _Journal of Neuroscience_ 22, 9530–9540 (2002).
  * [110] Jazayeri, M. & Ostojic, S.  Interpreting neural computations by examining intrinsic and embedding dimensionality of neural activity.  _Current opinion in neurobiology_ 70, 113–120 (2021).
  * [111] Howarth, C., Gleeson, P. & Attwell, D.  Updated Energy Budgets for Neural Computation in the Neocortex and Cerebellum.  _Journal of Cerebral Blood Flow & Metabolism_ 32, 1222–1232 (2012).
  * [112] Giesl, P. & Hafstein, S.  Review on computational methods for lyapunov functions.  _Discrete and Continuous Dynamical Systems-B_ 20, 2291–2331 (2015).
  * [113] Bellman, R.  Vector lyapunov functions.  _Journal of the Society for Industrial and Applied Mathematics, Series A: Control_ 1, 32–34 (1962).
  * [114] Wolf, A., Swift, J. B., Swinney, H. L. & Vastano, J. A.  Determining lyapunov exponents from a time series.  _Physica D: nonlinear phenomena_ 16, 285–317 (1985).
  * [115] Petanjek, Z. _et al._ Extraordinary neoteny of synaptic spines in the human prefrontal cortex.  _Proceedings of the National Academy of Sciences_ 108, 13281–13286 (2011).
  * [116] Averbeck, B. B.  Pruning recurrent neural networks replicates adolescent changes in working memory and reinforcement learning.  _Proceedings of the National Academy of Sciences_ 119, e2121331119 (2022).
  * [117] Moler, C. B.  Iterative refinement in floating point.  _Journal of the ACM (JACM)_ 14, 316–321 (1967).
  * [118] Ypma, T. J.  Historical development of the newton–raphson method.  _SIAM review_ 37, 531–551 (1995).
  * [119] Aragón Artacho, F. J., Campoy, R. & Tam, M. K.  The douglas–rachford algorithm for convex and nonconvex feasibility problems.  _Mathematical Methods of Operations Research_ 91, 201–240 (2020).
  * [120] Flesch, T., Juechems, K., Dumbalska, T., Saxe, A. & Summerfield, C.  Orthogonal representations for robust context-dependent task performance in brains and neural networks.  _Neuron_ 110, 1258–1270 (2022).
  * [121] Gillespie, A. K. _et al._ Hippocampal replay reflects specific past experiences rather than a plan for subsequent choice.  _Neuron_ 109, 3149–3163 (2021).
  * [122] Rajalingham, R., Sohn, H. & Jazayeri, M.  Dynamic tracking of objects in the macaque dorsomedial frontal cortex.  _bioRxiv_ 2022–06 (2022).
  * [123] Rajalingham, R., Piccato, A. & Jazayeri, M.  Recurrent neural networks with explicit representation of dynamic latent variables can mimic behavioral patterns in a physical inference task.  _Nature Communications_ 13, 5865 (2022).
  * [124] Ercsey-Ravasz, M. & Toroczkai, Z.  Optimization hardness as transient chaos in an analog approach to constraint satisfaction.  _Nature Physics_ 7, 966–970 (2011).
  * [125] Molnar, B., Toroczkai, Z. & Ercsey-Ravasz, M.  Continuous-time neural networks without local traps for solving boolean satisfiability.  In _2012 13th International Workshop on Cellular Nanoscale Networks and their Applications_ , 1–6 (IEEE, 2012).
  * [126] Yamashita, H., Suzuki, H., Toroczkai, Z. & Aihara, K.  Bounded continuous-time satisfiability solver.  In _International Symposium on Nonlinear Theory and its Applications (NOLTA2019)_ (2019).
  * [127] Li, C. & MacLennan, B. J.  Continuous-time systems for solving 0-1 integer linear programming feasibility problems.  _arXiv preprint arXiv:1905.04612_ (2019).
  * [128] Ding, Y., Li, Y., Xiao, M., Wang, Q. & Li, D.  A high order neural network to solve n-queens problem.  In _The 2010 International Joint Conference on Neural Networks (IJCNN)_ , 1–6 (IEEE, 2010).
  * [129] Mastrogiuseppe, F. & Ostojic, S.  Linking connectivity, dynamics, and computations in low-rank recurrent neural networks.  _Neuron_ 99, 609–623 (2018).
  * [130] Eliasmith, C. & Anderson, C. H.  _Neural engineering: Computation, representation, and dynamics in neurobiological systems_ (MIT press, 2003).
  * [131] DeWolf, T., Jaworski, P. & Eliasmith, C.  Nengo and low-power ai hardware for robust, embedded neurorobotics.  _Frontiers in Neurorobotics_ 14, 568359 (2020).
  * [132] Graves, A. _et al._ Hybrid computing using a neural network with dynamic external memory.  _Nature_ 538, 471–476 (2016).
  * [133] Tkačík, J. & Kordík, P.  Neural turing machine for sequential learning of human mobility patterns.  In _2016 International joint conference on neural networks (IJCNN)_ , 2790–2797 (IEEE, 2016).
  * [134] Fawzi, A. _et al._ Discovering faster matrix multiplication algorithms with reinforcement learning.  _Nature_ 610, 47–53 (2022).
  * [135] Kim, J. Z., Lu, Z., Nozari, E., Pappas, G. J. & Bassett, D. S.  Teaching recurrent neural networks to infer global temporal structure from local examples.  _Nature Machine Intelligence_ 3, 316–323 (2021).
  * [136] Valente, A., Ostojic, S. & Pillow, J. W.  Probing the relationship between latent linear dynamical systems and low-rank recurrent neural network models.  _Neural computation_ 34, 1871–1892 (2022).
  * [137] Rajan, K., Harvey, C. D. & Tank, D. W.  Recurrent network models of sequence generation and memory.  _Neuron_ 90, 128–142 (2016).
  * [138] Kepple, D., Engelken, R. & Rajan, K.  Curriculum learning as a tool to uncover learning principles in the brain.  In _International Conference on Learning Representations_ (2022).
  * [139] Larsen, B., Sydnor, V. J., Keller, A. S., Yeo, B. T. & Satterthwaite, T. D.  A critical period plasticity framework for the sensorimotor–association axis of cortical neurodevelopment.  _Trends in Neurosciences_ S0166223623001674 (2023).
  * [140] Kim, N. Y. & Kastner, S.  A biased competition theory for the developmental cognitive neuroscience of visuo-spatial attention.  _Current Opinion in Psychology_ 29, 219–228 (2019).
  * [141] Tervo-Clemmens, B. _et al._ A Canonical Trajectory of Executive Function Maturation During the Transition from Adolescence to Adulthood.  preprint, PsyArXiv (2022).
  * [142] Larsen, B. & Luna, B.  Adolescence as a neurobiological critical period for the development of higher-order cognition.  _Neuroscience & Biobehavioral Reviews_ 94, 179–195 (2018).
  * [143] García-Cabezas, M. Á., Zikopoulos, B. & Barbas, H.  The structural model: a theory linking connections, plasticity, pathology, development and evolution of the cerebral cortex.  _Brain Structure and Function_ 224, 985–1008 (2019).
  * [144] García-Cabezas, M. Á., Hacker, J. L. & Zikopoulos, B.  A protocol for cortical type analysis of the human neocortex applied on histological samples, the atlas of von economo and koskinas, and magnetic resonance imaging.  _Frontiers in Neuroanatomy_ 14, 576015 (2020).
  * [145] Paquola, C. _et al._ Microstructural and functional gradients are increasingly dissociated in transmodal cortices.  _PLOS Biology_ 17, e3000284 (2019).
  * [146] Barbas, H.  General Cortical and Special Prefrontal Connections: Principles from Structure to Function.  _Annual Review of Neuroscience_ 38, 269–289 (2015).
  * [147] Markov, N. T. _et al._ Anatomy of hierarchy: Feedforward and feedback pathways in macaque visual cortex.  _Journal of Comparative Neurology_ 522, 225–259 (2014).
  * [148] Beul, S. F.  Towards a "canonical" agranular cortical microcircuit.  _Frontiers in Neuroanatomy_ 8 (2015).
  * [149] Huttenlocher, P. R., de Courten, C., Garey, L. J. & Van der Loos, H.  Synaptogenesis in human visual cortex — evidence for synapse elimination during normal development.  _Neuroscience Letters_ 33, 247–252 (1982).
  * [150] Peter R., H.  Synaptic density in human frontal cortex — Developmental changes and effects of aging.  _Brain Research_ 163, 195–205 (1979).
  * [151] Semple, B. D., Blomgren, K., Gimlin, K., Ferriero, D. M. & Noble-Haeusslein, L. J.  Brain development in rodents and humans: Identifying benchmarks of maturation and vulnerability to injury across species.  _Progress in Neurobiology_ 106-107, 1–16 (2013).
  * [152] Buckner, R. L. & Krienen, F. M.  The evolution of distributed association networks in the human brain.  _Trends in Cognitive Sciences_ 17, 648–665 (2013).
  * [153] Achterberg, J., Akarca, D., Strouse, D. J., Duncan, J. & Astle, D. E.  Spatially embedded recurrent neural networks reveal widespread links between structural and functional neuroscience findings.  _Nature Machine Intelligence_ (2023).  URL https://www.nature.com/articles/s42256-023-00748-9.
  * [154] Sporns, O. & Betzel, R. F.  Modular Brain Networks.  _Annual Review of Psychology_ 67, 613–640 (2016).
  * [155] Bassett, D. S. & Bullmore, E. T.  Small-World Brain Networks Revisited.  _The Neuroscientist_ 23, 499–516 (2017).
  * [156] Tanner, J., L., S. M., Coletta, L., Gozzi, A. & Betzel, R. F.  Functional connectivity modules in recurrent neural networks: function, origin and dynamics (2023).  URL https://arxiv.org/abs/2310.20601.  Publisher: arXiv Version Number: 1.
  * [157] Fries, P.  Neuronal gamma-band synchronization as a fundamental process in cortical computation.  _Annual review of neuroscience_ 32, 209–224 (2009).
  * [158] Loebel, A. & Tsodyks, M.  Computation by ensemble synchronization in recurrent networks with synaptic depression.  _Journal of computational neuroscience_ 13, 111–124 (2002).
  * [159] Li, Z., Duan, Z., Chen, G. & Huang, L.  Consensus of multiagent systems and synchronization of complex networks: A unified viewpoint.  _IEEE Transactions on Circuits and Systems I: Regular Papers_ 57, 213–224 (2009).
  * [160] Rulkov, N. F., Sushchik, M. M., Tsimring, L. S. & Abarbanel, H. D.  Generalized synchronization of chaos in directionally coupled chaotic systems.  _Physical Review E_ 51, 980 (1995).
  * [161] Pecora, L. M. & Carroll, T. L.  Synchronization in chaotic systems.  _Physical review letters_ 64, 821 (1990).
  * [162] Lu, Z. & Bassett, D. S.  Invertible generalized synchronization: A putative mechanism for implicit learning in neural systems.  _Chaos: An Interdisciplinary Journal of Nonlinear Science_ 30 (2020).