# Selective Inhibition and Recruitment in Linear-Threshold Thalamocortical
Networks††thanks: This work was supported by NSF Award CMMI-1826065.

Michael McCreesh Jorge Cortés M. McCreesh and J. Cortés are with Department of
Mechanical and Aerospace Engineering, UC San Diego,
<EMAIL_ADDRESS>
###### Abstract

Neuroscientific evidence shows that for most brain networks all pathways
between cortical regions either pass through the thalamus or a transthalamic
parallel route exists for any direct corticocortical connection. This paper
seeks to formally study the dynamical behavior of the resulting
thalamocortical brain networks with a view to characterizing the inhibitory
role played by the thalamus and its benefits. We employ a linear-threshold
mesoscale model for individual brain subnetworks and study both hierarchical
and star-connected thalamocortical networks. Using tools from singular
perturbation theory and switched systems, we show that selective inhibition
and recruitment can be achieved in such networks through a combination of
feedback and feedforward control. Various simulations throughout the
exposition illustrate the benefits resulting from the presence of the thalamus
regarding failsafe mechanisms, required control magnitude, and network
performance.

##  I Introduction

The brain is a complex network composed of billions of individual neurons,
with their interconnections forming subnetworks that perform a myriad of
different functions. Communication of information between regions and its
subsequent processing is one such function. Brain regions, such as the
neocortex for example, have a hierarchical structure in which different
cognitive levels operate on distinct timescales. Within this hierarchy,
information travels from faster lower-level sensory brain regions to slower
higher-level cognitive brain regions (_bottom-up_ communication). Upon
processing in the higher-level regions, information regarding decisions made
by these regions is passed back down the hierarchy to perform some task (_top-
down_ communication). In this process, certain regions are selectively
recruited to perform the given task, while other areas are selectively
inhibited to ignore other inputs into the brain network.

Such hierarchies are not restricted to the neocortex, and neither top-down and
bottom-up communications occur entirely inside the neocortex. In fact, most,
if not all, direct corticocortical communications have a parallel
transthalamic pathway upon which the information is transmitted and modulated
[1]. Our goal here is to understand the role of transthalamic communication in
enabling selective attention, with a view to characterizing its benefits. We
seek to provide a dynamical explanation of this phenomena and validate the
hypothesis that selective inhibition and recruitment are feasible in
thalamocortical networks via feedback and feedforward mechanisms.

_Literature Review:_ The hierarchical organization of the brain has been known
for decades [2, 3] and has been extensively studied from different viewpoints
[4, 5, 6, 7, 8]. The role of the communication between the thalamus and
cortical regions in a thalamocortical hierarchy is a more recently studied
problem. Historically, the thalamus has been viewed as a relay of sensory
signals to the cortex. However, in recent literature, it has been shown to
also play a further role in cognitive processes [9]. In particular, [1, 10]
show that the thalamus transfers both sensory signals to the cortex using
first-order relays, but that for most direct corticocortical connection, there
exists a parallel transthalamic path made of higher-order thalamic relays.
Further evidence is given that these paths operate using feedforward
inhibitory control to communicate information from thalamic to cortical areas
[11, 12, 13, 14, 15]. The works [16, 11] show that depending on the purpose
(e.g., visual, auditory, somatomotor) of the hierarchical network, the
thalamus connects to the hierarchy at different levels. In general, little
theoretical understanding is available about the network properties of
thalamocortical structures and their role in the hierarchical nature of the
brain. To address this gap, here we employ linear-threshold dynamics [17, 18]
as a mesoscale model for the behavior of neuronal populations and build on the
hierarchical selective recruitment framework introduced in [19, 20] for
corticocortical networks. The latter provides us with a baseline to compare
against when analyzing the properties and performance of thalamocortical
networks. We also rely on results from switched piecewise and affine systems
[21, 22] and singular perturbation theory [23, 24, 25].

_Statement of Contributions:_ We deal with thalamocortical brain networks
where each brain region is governed by a linear-threshold rate dynamics. Given
our focus on selective attention, the neuronal populations in each region are
divided into task-relevant and task-irrelevent nodes. Inspired by the types of
pathways in thalamic circuitry, we consider two interconnections topologies,
multilayer hierarchical networks and star-connected networks. Our first
contribution provides an analysis of selective inhibition and recruitment in
hierarchical thalamocortical networks. We establish that the equilibrium maps
of individual layers can be described as piecewise-affine maps. Using singular
perturbation theory for non-smooth differential equations, we provide
conditions for the existence of feedback-feedforward control laws that achieve
selective inhibition and recruitment of the desired nodes. Our second
contribution deals with star-connected thalamocortical networks, both with and
without timescale separation between regions. For the latter class, we build
on a generalization of stability results on slowly varying nonlinear systems
to the case of exponential stability to provide conditions for the existence
of a feedback-feedforward controller providing selective inhibition and
recruitment. We achieve analogous results for the case of star-connected
networks with timescale separation using again singular perturbation theory.
Examples illustrate the beneficial role played by the thalamus in these
networks with respect to metrics such as failsafe mechanisms, control
magnitude, and network performance.

##  II Neuroscientific Background

Here we provide a summary of the neuroscientific background behind the
modeling assumptions adopted in the paper. We focus on observations about
brain organization, information transmission among brain regions, and the role
that the thalamus is believed to play.

_Task-Relevant and Task-Irrelevant Neuron Populations:_ in the nervous system,
stimuli are represented using series of electric spikes generated by neurons
that travel down nerve fibres [26]. A given stimulus is defined by a
characteristic pattern of spikes traveling between neurons, which can also be
represented as the firing rate of the neurons over time. In such a
representation, some neurons generate spikes (have a non-zero firing rate)
during the transmission of the stimuli, known as being excited, while other
neurons do not (have a firing rate of zero), which is referred to as being
inhibited111In network models, individual nodes are frequently considered to
be populations of neurons with similar firing rates. Our discussion is not
impacted by considering individual neurons or populations.. We refer to the
subset of neuron populations that are excited during the transmission of a
stimuli as the ‘task-relevant’ nodes and the remaining populations as the
‘task-irrelevant’ nodes.

_Information Pathways:_ in the brain, there exist information pathways between
different spatial regions allowing for the transmission of stimuli between
processing areas. The transmission of information can be seen as the activity
in one region (that is, the firing rates of the neuron populations defined by
the stimuli) driving the activity in the following region by exciting the
task-relevant nodes and inhibiting the task-irrelevant nodes to propagate the
stimuli (and any processing of it) through the pathway. This enables the brain
to generate appropriate responses to the stimuli by propagating a response
through the information pathway. This response, with its own set of task
relevant/irrelevant nodes, selectively _recruits_ (excites) the task-relevant
nodes, and selectively _inhibits_ the task-irrelevant nodes. Information
pathways between brain regions form both spatial and temporal hierarchies,
allowing for different levels of processing occurring in different regions
[27]. The temporal timescales are directly related to the complexity of
processing occurring in a given region. For a low-level sensory area, where
the inputs are brief and are processed quickly, the timescales are fast.
Meanwhile, further up the information pathway, in regions such as the
prefrontal cortex, higher-level cognitive processes use more complex inputs
from earlier in the pathway by integrating them over time, resulting in slower
timescales [17]. These pathways have been studied in strictly cortical
networks [17, 19, 20]. However, elements of such networks also have multiple
connections to the thalamus while maintaining a temporal hierarchy [1],
leading to the interest here in the role of the thalamus.

$\vdots$$\vdots$Subnetwork $i-1$Subnetwork $i$Subnetwork $i+1$ Thalamus (a)
hierarchical thalamocortical network

ThalamusSubnetwork $1$Subnetwork $4$Subnetwork $2$Subnetwork $3$ (b) star-
connected thalamocortical network

Figure 1: Topologies for thalamocortical networks considered in the paper. In
(a), each layer is connected directly to the thalamus, as well as the layers
directly above and below it. In (b), each layer is connected to the thalamus,
but no direct connections exist between cortical regions. In both plots, task-
relevant excitatory and inhibitory nodes are depicted in red and blue, resp.,
and task-irrelevant nodes are depicted in grey.

_Pathways between the Thalamus and Cortical Regions:_ in studying
thalamocortical networks, we consider two topologies of interest, cf. Fig. 1.
Both share the common trait that each cortical layer is connected to the
thalamus layer: however, connections between cortical layers differ in each
case. These topologies are inspired by the fact that pathways in thalamic
circuitry can be identified into two classes [28]. The first represents the
role of the thalamus as a modulator of information being passed between
cortical regions along a transthalamic route parallel to the existing
corticocortical information pathways (higher-order relay) [1]. Since these
cortical regions then form a temporal hierarchy, this leads to the
hierarchical thalamocortical network shown in Fig. 1(a). The second class
represents the thalamus as the main route for the transfer of information
between two (or more) brain areas. In this case, the thalamus is relaying an
input to the cortical regions (first-order relay) [1], which gives rise to the
star-connected network shown in Fig. 1(b). The cortical regions to which
information is being relayed can be parts of separate temporal hierarchies,
however, as the regions in the network do not form a temporal hierarchy
themselves, the timescales of the subnetworks are not directly related.

_Role of the Thalamus:_ The neuroscientific literature [13] discusses a number
of potential roles for the thalamus. The thalamus can significantly increase
the information contained in signals both being transmitted to and between
cortical regions [1]. In hierarchical networks within the cortex,
transthalamic pathways allow for layers near the top of the hierarchy to
directly receive the outputs from the lower levels, in addition to the more
processed inputs they receive from other cortical regions [13]. Thalamic
signals into the higher cortical regions can also allow for receiving further
details of motor signals, such as distinguishing between self-induced stimuli
(such as those generated by eye movements) and stimuli from the external
environment [1]. The thalamus also has a role in controlling recurrent
cortical dynamics, as the cortical networks are not able to self-sustain
activity [29, 12]. Other roles include contributions to learning, memory,
decision-making and inhibitory control [30, 31].

##  III Problem Setup

We start222We let $\mathbb{R}$, $\mathbb{R}_{\geq 0}$, denote the reals and
nonnegative reals, resp. Vectors and matrices are identified by bold-faced
letters. For vectors (matrices) $\mathbf{x},\mathbf{y}\in\mathbb{R}^{n}$
(resp. $\mathbb{R}^{n\times m}$), $\mathbf{x}\leq\mathbf{y}$ is the component-
wise comparison (analogously with $<,>,\geq)$. The identity matrix of
dimension $n$ is $\mathbf{I}_{n}$. $\mathbf{0}_{n}$ and $\mathbf{0}_{n\times
m}$ denote the $n$-vector of zeros and the $n\times m$ matrix of zeros, resp.
For a matrix $\mathbf{W}\in\mathbb{R}^{n\times n}$, we denote its element-wise
absolute value, spectral radius and induced $2$-norm by
$|\mathbf{W}|,\rho(\mathbf{W})$ and $\left\|\mathbf{W}\right\|$, resp.
Similarly, we let $\left\|\mathbf{x}\right\|$ denote the $2$-norm of a vector
$\mathbf{x}\in\mathbb{R}^{n}$. For $x\in\mathbb{R}$ and $m\in\mathbb{R}_{>0}$,
$[x]_{0}^{m}$ denotes $\min\\{\max\\{x,0\\},m\\}$. For
$\mathbf{x}\in\mathbb{R}^{n}$, $\mathbf{m}\in\mathbb{R}_{>0}^{n}$, this
operation is done component-wise as
$[\mathbf{x}]_{\mathbf{0}}^{\mathbf{m}}=[[x_{1}]_{0}^{m_{1}},\dots,[x_{n}]_{0}^{m_{n}}]$.
For a $2\times 2$-partitioned block matrix $\mathbf{W}=\begin{bmatrix}{\bf
W}^{00}&{\bf W}^{01}\\\ {\bf W}^{10}&{\bf W}^{11}\end{bmatrix}$, we use the
notation $\mathbf{W}^{\ell,\mathrm{all}}=\begin{bmatrix}{\bf W}^{\ell 0}&{\bf
W}^{\ell 1}\end{bmatrix}$ and
$\mathbf{W}^{\mathrm{all},\ell}=\begin{bmatrix}(\mathbf{W}^{0\ell})^{\top}&(\mathbf{W}^{1\ell})^{\top}\end{bmatrix}^{\top}$
for $\ell\in\\{0,1\\}$.  by providing details on the dynamic modeling of the
thalamocortical network layers, then describe the effect that the
interconnection topology has on the input to each layer, and finally formalize
the problem under consideration.

###  III-A Network Modeling

We consider a thalamocortical network $\mathcal{N}$ composed of $N$ cortical
layers and the thalamus. We use linear-threshold rate dynamics to model the
evolution of each region in the network. These dynamics provide a mesoscale
model of the evolution of the average firing rate of populations of neurons,
rather than individual spike trains, by looking at the electrical currents
flowing through synaptic connections, see [26]. The dynamics of cortical layer
$\mathcal{N}_{i}$ composed of $n_{i}$ nodes are

$\displaystyle\tau_{i}\dot{\mathbf{x}}_{i}=-\mathbf{x}_{i}+[\mathbf{W}_{i}\mathbf{x}_{i}+\mathbf{d}_{i}(t)]_{\mathbf{0}}^{\mathbf{m}_{i}}\qquad\mathbf{0}\leq\mathbf{x}_{i}(0)\leq\mathbf{m}_{i},$
(1)

where $\mathbf{x}_{i}\in\mathbb{R}_{\geq 0}^{n_{i}}$ represents the state of
the nodes within the layer, and each component of $\mathbf{x}_{i}$ represents
a population of neurons with similar firing rate. The matrix
$\mathbf{W}_{i}\in\mathbb{R}^{n_{i}\times n_{i}}$ is the synaptic connectivity
between neuron populations within the layer and
$\mathbf{d}_{i}(t)\in\mathbb{R}^{n_{i}}$ encapsulates the input into the
model,

$\displaystyle\mathbf{d}_{i}(t)=\mathbf{w}_{i}(t)+\mathbf{B}_{i}\mathbf{u}_{i}(t)+\mathbf{c}_{i}.$
(2)

Here $\mathbf{w}_{i}(t)$ models the interconnections between layers,
$\mathbf{u}_{i}(t)$ is the control used for the $n_{i}$ nodes in
$\mathcal{N}_{i}$, and $\mathbf{c}_{i}$ includes any unmodeled background
activity or external inputs. Finally $\tau_{i}\in\mathbb{R}_{\geq 0}^{n_{i}}$
is the timescale of the dynamics.

In our study, the control $\mathbf{u}_{i}$ inhibits $r_{i}\leq n_{i}$ task-
irrelevant nodes (drives their state to zero), while the remaining terms in
(2) recruit the remaining $n_{i}-r_{i}$ nodes and determine the desired
equilibrium or trajectory. To distinguish between task-relevant and task-
irrelevant nodes, we use the following partition of network variables in each
layer $\mathcal{N}_{i}$,

$\displaystyle\mathbf{x}_{i}=\begin{bmatrix}\mathbf{x}_{i}^{0}\\\
\mathbf{x}_{i}^{1}\end{bmatrix}\qquad\mathbf{W}_{i}$
$\displaystyle=\begin{bmatrix}\mathbf{W}_{i}^{00}&\mathbf{W}_{i}^{01}\\\
\mathbf{W}_{i}^{10}&\mathbf{W}_{i}^{11}\end{bmatrix}$ (3a)
$\displaystyle\mathbf{B}_{i}=\begin{bmatrix}\mathbf{B}_{i}^{0}\\\
\mathbf{0}\end{bmatrix}\qquad\mathbf{c}_{i}$
$\displaystyle=\begin{bmatrix}\mathbf{c}_{i}^{0}\\\
\mathbf{c}_{i}^{1}\end{bmatrix}\qquad\mathbf{m}_{i}=\begin{bmatrix}\mathbf{m}_{i}^{0}\\\
\mathbf{m}_{i}^{1}\end{bmatrix},$ (3b)

where $\mathbf{x}_{i}^{0}\in\mathbb{R}_{\geq 0}^{r_{i}}$ represent the task-
irrelevant nodes, $\mathbf{x}_{i}^{1}\in\mathbb{R}_{\geq 0}^{n_{i}-r_{i}}$ the
task-relevant nodes and $\mathbf{B}_{i}\in\mathbb{R}^{n_{i}\times p_{i}}$ is
such that the task-relevant nodes are not impacted by the control term
$\mathbf{u}_{i}\in\mathbb{R}^{p_{i}}$. Throughout we shall assume that
$p_{i}\geq r_{i}$, and that the matrices $\mathbf{B}_{i}^{0}$ have all full
rank.

The thalamus layer $\mathcal{N}_{T}$ is modeled in the same way as the
cortical layers with the difference appearing in the interconnection term
$\mathbf{w}_{T}(t)$. A final word about the brain mechanisms for the
inhibition of regions [32]. Feedforward inhibition between two layers,
$\mathcal{N}_{i}$ and $\mathcal{N}_{j}$, refers to when $\mathcal{N}_{i}$
sends an inhibitory signal to $\mathcal{N}_{j}$ to achieve its goal activity
pattern for $\mathcal{N}_{j}$ regardless of the current state of
$\mathcal{N}_{j}$. In contrast, feedback inhibition refers to when the
inhibition applied in a layer $\mathcal{N}_{i}$ is dependent upon the current
activity level of the nodes desired to be inhibited [32]. We employ a
combination of feedback and feedforward inhibition.

###  III-B Interconnection Topology Among Network Layers

We detail here the dynamical interconnection between the network layers for
each of the topologies depicted in Fig. 1.

#### Hierarchical Thalamocortical Networks

Consider the hierarchical thalamocortical network depicted in Fig. 1(a). The
hierarchical structure is encoded by the ordered timescales of the layers:
$\tau_{1}\gg\tau_{2}\gg\dots\gg\tau_{N}$, prescribing progressively faster
dynamics as one moves down the hierarchy. For generality, the timescale
$\tau_{T}$ of the thalamus might fit anywhere within the hierarchy. For each
cortical layer $\mathcal{N}_{i}$, the interconnection term $\mathbf{w}_{i}(t)$
in (2) takes the form

$\displaystyle\mathbf{w}_{i}(t)$
$\displaystyle=\mathbf{W}_{i,i-1}\mathbf{x}_{i-1}(t)+\mathbf{W}_{i,i+1}\mathbf{x}_{i+1}(t)+\mathbf{W}_{i,T}\mathbf{x}_{T}(t).$
(4)

Here, the terms $\mathbf{W}_{i,i-1},\mathbf{W}_{i,i+1}$, and
$\mathbf{W}_{i,T}$ represent the weights of the synaptic connections between
layers $\mathcal{N}_{i}$ and $\mathcal{N}_{i-1}$, $\mathcal{N}_{i+1}$ and
$\mathcal{N}_{T}$, resp. It is important to note that since the thalamus
impacts the cortical regions using feedforward inhibition, the interconnection
matrix between the thalamus and the cortical layer satisfies
$\mathbf{W}_{i,T}\leq 0$. Substituting (4) into the linear-threshold dynamics
(1), we get the dynamics for a cortical layer $\mathcal{N}_{i}$

$\displaystyle\tau_{i}\dot{\mathbf{x}}_{i}$
$\displaystyle=-\mathbf{x}_{i}+[\mathbf{W}_{i,i}\mathbf{x}_{i}+\mathbf{W}_{i,i-1}\mathbf{x}_{i-1}$
(5)
$\displaystyle\quad+\mathbf{W}_{i,i+1}\mathbf{x}_{i+1}+\mathbf{W}_{i,T}\mathbf{x}_{T}+\mathbf{B}_{i}\mathbf{u}_{i}(t)+\mathbf{c}_{i}]_{\mathbf{0}}^{\mathbf{m}_{i}},$

for $i\in\\{1,\dots,N\\}$. For consistency
$\mathbf{W}_{1,0}=\mathbf{0}=\mathbf{W}_{N,N+1}$, and we assume that
$r_{1}=0$, meaning no nodes are being inhibited in the top layer of the
network.

For the thalamus layer, to reflect the different connectivity it has in the
network, the interconnection term $\mathbf{w}_{T}(t)$ is

$\displaystyle\mathbf{w}_{T}(t)=\sum_{i=1}^{N}\mathbf{W}_{T,i}\mathbf{x}_{i}(t).$
(6)

Here $\mathbf{W}_{T,i}$ represents the weight of the synaptic connections
between layers $\mathcal{N}_{T}$ and $\mathcal{N}_{i}$ for
$i\in\\{1,\dots,N\\}$. Then, substituting (6) into (1), the dynamics for the
thalamus layer is

$\displaystyle\tau_{T}\dot{\mathbf{x}}_{T}$
$\displaystyle=-\mathbf{x}_{T}+[\mathbf{W}_{T}\mathbf{x}_{T}+\sum_{i=1}^{N}\mathbf{W}_{T,i}\mathbf{x}_{i}+\mathbf{B}_{T}\mathbf{u}_{T}(t)+\mathbf{c}_{T}]_{\mathbf{0}}^{\mathbf{m}_{T}}.$
(7)

We denote the timescale ratio between layers by
$\epsilon=(\epsilon_{1},\dots,\epsilon_{N},\epsilon_{T})$, where

$\displaystyle\epsilon_{i}=\frac{\tau_{i}}{\tau_{i-1}},\quad\epsilon_{T}=\frac{\tau_{T}}{\min_{\tau_{j}>\tau_{T}}\tau_{j}}.$

For a subnetwork $\mathcal{N}_{i}$ such that the thalamus timescale fits in
the hierarchy directly above it, i.e. $\tau_{T}>\tau_{i}$ but there does not
exist $j$ such that $\tau_{T}>\tau_{j}>\tau_{i}$, we let the timescale ratio
between $\mathcal{N}_{T}$ and $\mathcal{N}_{i}$ be given by
$\bar{\epsilon}_{i}={\tau_{i}}/{\tau_{T}}$.

#### Star-Connected Thalamocortical Networks

Consider the star-connected thalamocortical network depicted in Fig. 1(b). In
contrast to the hierarchical network, this topology does not form a
hierarchical timescale, and as such there is no direct relationship satisfied
by the timescales. The lack of an explicit relation encodes the thalamus’ role
as a sensory relay to multiple brain regions, each part of potentially
unrelated temporal hierarchies. Without loss of generality, we assume
subnetwork $\mathcal{N}_{1}$ represents a subcortical structure and provides
the input to the network for the thalamus to relay to the other brain regions.

As such, there are no nodes in $\mathcal{N}_{1}$ that are desired to be
inhibited, meaning $r_{1}=0$. We model the subcortical input subnetwork with a
linear-threshold dynamics, but without an independent control term, instead
modeling input changes by allowing $\mathbf{c}_{1}$ to be time-varying. That
is

$\displaystyle\tau_{1}\dot{\mathbf{x}}_{1}=-\mathbf{x}_{1}+[\mathbf{W}_{1,1}\mathbf{x}_{1}+\mathbf{W}_{1,T}\mathbf{x}_{T}+\mathbf{c}_{1}(t)]_{\mathbf{0}}^{\mathbf{m}_{1}}.$
(8)

For the cortical regions $\mathcal{N}_{i}$, $i\in\\{2,\dots,N\\}$, the
interconnection term $\mathbf{w}_{i}(t)$ is given by
$\mathbf{w}_{i}(t)=\mathbf{W}_{i,T}\mathbf{x}_{T}(t)$, and we recall that as
the thalamus utilizes feedforward inhibition, $\mathbf{W}_{i,T}\leq 0$.
Meanwhile, the interconnection of the thalamus with the cortical regions is
defined by
$\mathbf{w}_{T}(t)=\sum_{i=1}^{N}\mathbf{W}_{T,i}\mathbf{x}_{i}(t)$. Then,
from the linear-threshold model (1), for $i\in\\{2,\dots,N\\}$ the dynamics
takes the form

$\displaystyle\tau_{i}\dot{\mathbf{x}}_{i}$
$\displaystyle=-\mathbf{x}_{i}+[\mathbf{W}_{i,i}\mathbf{x}_{i}+\mathbf{W}_{i,T}\mathbf{x}_{T}+\mathbf{B}_{i}\mathbf{u}_{i}(t)+\mathbf{c}_{i}]_{\mathbf{0}}^{\mathbf{m}_{i}},$
(9) $\displaystyle\tau_{T}\dot{\mathbf{x}}_{T}$
$\displaystyle=-\mathbf{x}_{T}+[\mathbf{W}_{T}\mathbf{x}_{T}+\sum_{i=1}^{N}\mathbf{W}_{T,i}\mathbf{x}_{i}+\mathbf{B}_{T}\mathbf{u}_{T}(t)+\mathbf{c}_{T}]_{\mathbf{0}}^{\mathbf{m}_{T}}.$

###  III-C Problem Statement

For a purely cortical hierarchical brain network, selective inhibition and
recruitment can be achieved using a combination of feedback and feedforward
control, dependent on the subnetwork dynamics satisfying a set of stability
properties, cf. [19, 20]. However, with the presence of the thalamus and given
its impact on the dynamics of the individual subnetworks, such results are
inapplicable. In addition, for star-connected thalamocortical topology the
results do not apply, as the assumption of a hierarchical relationship between
subnetwork timescales, which plays a critical role, no longer holds. As such,
in this work we study the problem of selective inhibition and recruitment for
thalamocortical networks, both hierarchical and star-connected, which we
formalize next.

We seek to determine the existence of control mechanisms, resulting from the
combination of feedback and feedforward inhibition, such that for a
thalamocortical network, with either interconnection topology as described in
Section III-B, each of the individual subnetworks $\mathcal{N}_{i}$ and
$\mathcal{N}_{T}$ achieves selective inhibition and recruitment. In
particular, we look for a control law of the form

$\displaystyle\mathbf{u}_{i}(t)=\bar{\mathbf{u}}_{i}(t)+\mathbf{K}_{i}\mathbf{x}_{i}(t),$

for each $i\in\\{2,\dots,N,\\}\cup\\{T\\}$, that results in the state of each
subnetwork converging to a desired equilibrium trajectory. Solving this
problem requires determining the stability properties that the dynamics of
each of the individual subnetworks must satisfy to guarantee the existence of
such a control law.

In addition, we seek to evaluate how the addition of the thalamus results in
improved performance relative to a purely cortical network. In particular we
investigate how the addition of the thalamus impacts the control magnitude
needed to achieve selective inhibition and the convergence speed of the
network. Further, we study differences in the hierarchical and star-connected
topologies, particularly in how the latter can act as a failsafe for the
former.

##  IV Hierarchical Thalamocortical Networks

In this section, we consider a hierarchical multilayer thalamocortical
network, cf. Fig. 1(a), where the cortical layers are governed by (5) and the
thalamus layer is governed by (7).

###  IV-A Equilibrium Maps for Individual Layers

We note that, for a general linear-threshold dynamics (1) with
$\mathbf{W}\in\mathbb{R}^{n\times n}$, its equilibrium map
$h:\mathbb{R}^{n}\rightarrow\mathbb{R}_{\geq 0}^{n}$

$\displaystyle
h(\mathbf{c})=h_{\mathbf{W},\mathbf{m}}(\mathbf{c})=\\{\mathbf{x}\in\mathbb{R}_{\geq
0}^{n}~{}|~{}\mathbf{x}=[\mathbf{W}\mathbf{x}+\mathbf{c}]_{\mathbf{0}}^{\mathbf{m}}\\},$
(10)

maps a constant input $\mathbf{c}\in\mathbb{R}^{n}$ to the set of equilibria
of (1). For thalamocortical networks, the timescale of the thalamus impacts
the specific form of the equilibrium maps. For simplicity, we assume the
thalamus lies inside the hierarchy, i.e., there exist $a,b\in\\{1,\dots,N\\}$
such that $b=a+1$ and $\tau_{a}\gg\tau_{T}\gg\tau_{b}$. This choice results in
a layer $\mathcal{N}_{a}$ with an equilibrium map different than any that
appear in the cases when the thalamus timescale is on the boundary of the
hierarchy. Our results can also be stated for the latter case with appropriate
adjustments to the derived control law for selective inhibition and
recruitment.

The hierarchical nature of the topology plays an important role in defining
these equilibrium maps. At the theoretical limit of timescale separation
between two layers, the state of a given layer becomes constant at the
timescale of layers lower in the hierarchy as well as being a static function
of the above layer’s states. As such, when defining the equilibrium maps for a
given layer, the constant input into an equilibrium map can represent the
state of layers higher in the hierarchy, given that they are relative
constants at that level of the dynamics. The equilibrium maps for the layers
fall into three categories: below or above the thalamus, and the thalamus
itself. In all cases, the maps are defined recursively from the bottom to the
top of the network. At each layer, the equilibrium map takes a constant input,
representing the inputs from higher levels in the hierarchy along with any
external inputs to the system, and outputs the set of equilibrium values. We
next give explicit expressions for the equilibrium maps of the task-relevant
component of a layer in each of the categories, which we denote
$h_{i}^{1}:\mathbb{R}^{n_{i}-r_{i}}\rightarrow\mathbb{R}_{\geq
0}^{n_{i}-r_{i}}$, by combining the hierarchical model described in Section
III-B with (10).

#### Equilibrium Maps for Layers below Thalamus

We begin by considering layers below the thalamus, i.e., with $i\geq b$. Given
values of $\mathbf{x}_{T}\in\mathbb{R}^{n_{T}}$ and
$\mathbf{c}_{i+1}^{1}\in\mathbb{R}^{n_{i+1}-r_{i+1}}$,

$\displaystyle h_{i}^{1}(\mathbf{c})$
$\displaystyle=\\{\mathbf{x}_{i}^{1}~{}|~{}\mathbf{x}_{i}^{1}=[\mathbf{W}_{i,i}^{11}\mathbf{x}_{i}^{1}+\mathbf{W}_{i,i+1}^{11}h_{i+1}^{1}(\mathbf{W}_{i+1,i}^{11}\mathbf{x}_{i}^{1}$
$\displaystyle\qquad+\mathbf{W}_{i+1,T}^{11}\mathbf{x}_{T}^{1}+\mathbf{c}_{i+1}^{1})+\mathbf{c}]_{\mathbf{0}}^{\mathbf{m}_{i}^{1}}\\}.$
(11)

We note that since $\mathbf{W}_{N,N+1}=\mathbf{0}$ by convention, this
recursion is well-defined for $\mathcal{N}_{N}$, the bottom layer in the
network. In particular, for layer $N$, (IV-A) reduces to (10), the standard
equilibrium map for linear-threshold models.

#### Equilibrium Map for Thalamus

Since the thalamus is connected to all the cortical layers, the recursive
definition of its equilibrium map is dependent on the equilibrium maps of all
of the layers below it. Due to this, using the notation of (IV-A) for writing
the recursion becomes intractable for large networks. Instead, we introduce
the following simplifying notation. For a layer $\mathcal{N}_{i}$,
$i\in\\{1,\dots,N\\}$, in the thalamocortical hierarchy, let
$\mathbf{x}_{i_{e}}^{1}\in\mathbb{R}_{\geq 0}^{n_{i}-r_{i}}$ be such that

$\displaystyle\mathbf{x}_{i_{e}}^{1}\in
h_{i}^{1}(\mathbf{W}_{i,T}^{11}\mathbf{x}_{T}^{1}+\mathbf{W}_{i,i-1}^{11}\mathbf{x}_{i-1}^{1}+\mathbf{c}_{i}^{1}).$

Note that, despite $\mathbf{x}_{i_{e}}$ being dependent on a set of inputs, we
do not specify these. As such, when this notation is used, it is implicitly
assumed that the input values are given or determined in the recursion.
Depending on the point in the recursion being considered, the inputs
$\mathbf{x}_{T}$ and/or $\mathbf{x}_{i-1}$ will be replaced by their
equilibrium values. Now, given values
$\mathbf{c}_{b}^{1},\dots,\mathbf{c}_{N}^{1}$, where
$\mathbf{c}_{i}^{1}\in\mathbb{R}^{n_{i}-r_{i}}$, the thalamus equilibrium map
$h_{T}^{1}:\mathbb{R}^{n_{T}-r_{T}}\rightarrow\mathbb{R}_{\geq
0}^{n_{T}-r_{T}}$ is

$\displaystyle
h_{T}^{1}(\mathbf{c})=\\{\mathbf{x}_{T}^{1}~{}|~{}\mathbf{x}_{T}^{1}=[\mathbf{W}_{T}^{11}\mathbf{x}_{T}^{1}+\sum_{j=b}^{N}\mathbf{W}_{T,j}^{11}\mathbf{x}_{j_{e}}^{1}+\mathbf{c}]_{\mathbf{0}}^{\mathbf{m}_{T}^{1}}\\}.$
(12)

#### Equilibrium Maps for Layers above Thalamus

We note that, in the recursive definition for the equilibrium maps above the
thalamus, the inputs to the thalamus equilibrium map (12) will include
equilibrium values of layers above the thalamus in the hierarchy, in addition
to the equilibrium values from lower in the hierarchy. To distinguish which
maps are inputting equilibrium maps in a condensed manner, we introduce the
following notation. For $i\in\\{1,\dots,a\\}$, we let
$\mathbf{x}_{T_{e(i)}}^{1}\in\mathbb{R}^{n_{T}-r_{T}}$ denote a value inside
the thalamus equilibrium set satisfying

$\displaystyle\mathbf{x}_{T_{e(i)}}^{1}\in
h_{T}^{1}(\sum_{j=1}^{i}\mathbf{W}_{T,j}^{11}\mathbf{x}_{j}^{1}+\sum_{j=i+1}^{a}\mathbf{W}_{T,j}^{11}\mathbf{x}_{j_{e}}^{1}+\mathbf{c}_{T}^{1}).$

Using this notation, we define the remaining equilibrium maps. We first
provide the equilibrium maps for layers $\\{\mathcal{N}_{i}\\}_{i=1}^{a-1}$
and finish with the map for $\mathcal{N}_{a}$. For $i\in\\{1,\dots,a-1\\}$,
given $\mathbf{c}_{i+1}^{1}\in\mathbb{R}^{n_{i}-r_{i}}$, the equilibrium map
$h_{i}^{1}:\mathbb{R}^{n_{i}-r_{i}}\rightarrow\mathbb{R}_{\geq
0}^{n_{i}-r_{i}}$ is

$\displaystyle h_{i}^{1}(\mathbf{c})$
$\displaystyle=\\{\mathbf{x}_{i}^{1}~{}|~{}\mathbf{x}_{i}^{1}=[\mathbf{W}_{i,i}^{1,\mathrm{all}}\mathbf{x}_{i}+\mathbf{W}_{i,i+1}h_{i+1}^{1}(\mathbf{W}_{i+1,i}^{1,\mathrm{all}}\mathbf{x}_{i}+\mathbf{c}_{i+1}^{1})$
$\displaystyle\qquad+\mathbf{W}_{i,T}^{11}\mathbf{x}_{T_{e(i)}}+\mathbf{c}]_{\mathbf{0}}^{\mathbf{m}_{i}^{1}}\\}.$
(13)

The expression for the equilibrium map of the layer $\mathcal{N}_{a}$, which
is directly above the thalamus, differs from the other layers due to the fact
that it depends directly on a layer below the thalamus in addition to being
dependent on the thalamus. Given
$\mathbf{c}_{b}^{1}\in\mathbb{R}^{n_{b}-r_{b}}$, the equilibrium map
$h_{a}^{1}:\mathbb{R}^{n_{a}-r_{a}}\rightarrow\mathbb{R}_{\geq
0}^{n_{a}-r_{a}}$ is

$\displaystyle h_{a}^{1}(\mathbf{c})$
$\displaystyle=\\{\mathbf{x}_{a}^{1}~{}|~{}\mathbf{x}_{a}^{1}=[\mathbf{W}_{a,a}^{11}\mathbf{x}_{a}^{1}+\mathbf{W}_{a,b}^{11}h_{b}^{1}(\mathbf{W}_{b,a}^{11}\mathbf{x}_{a}^{1}$
(14)
$\displaystyle\qquad+\mathbf{W}_{b,T}^{11}\mathbf{x}_{T_{e(a)}}^{1})+\mathbf{c}_{b}^{1})+\mathbf{W}_{a,T}^{11}\mathbf{x}_{T_{e(a)}}^{1}+\mathbf{c}]_{\mathbf{0}}^{\mathbf{m}_{a}^{1}}\\}.$

We conclude this section by establishing that all the equilibrium maps in the
thalamocortical network (IV-A)-(14) are piecewise-affine and use this fact to
justify they are globally Lipschitz too. To begin, we note that since general
linear-threshold dynamics are switched affine, their equilibrium map (10) can
be written in a piecewise-affine form. In particular, this equilibrium map can
be written as follows

$\displaystyle h(\mathbf{c})$
$\displaystyle=\\{\mathbf{F}_{\sigma}\mathbf{c}+\mathbf{f}_{\sigma}~{}|~{}\mathbf{G}_{\sigma}\mathbf{c}+\mathbf{g}_{\sigma}\geq\mathbf{0},~{}\sigma\in\\{0,\ell,s\\}^{n}\\},$
(15)

for some matrices and vectors of the form

$\displaystyle\mathbf{F}_{\sigma}$
$\displaystyle=(\mathbf{I}-\Sigma^{\ell}\mathbf{W})^{-1},\qquad\mathbf{f}_{\sigma}=(\mathbf{I}-\Sigma^{\ell}\mathbf{W})^{-1}\Sigma^{\mathbf{s}}\mathbf{m},$
$\displaystyle\mathbf{G}_{\sigma}$
$\displaystyle=\begin{bmatrix}\Sigma^{\ell}+\Sigma^{s}-\mathbf{I}&\Sigma^{\ell}&-\Sigma^{\ell}&\Sigma^{s}\end{bmatrix}^{\top}\mathbf{F}_{\sigma},$
$\displaystyle\mathbf{g}_{\sigma}$
$\displaystyle=\begin{bmatrix}\mathbf{f}_{\sigma}^{\top}(\Sigma^{\ell}+\Sigma^{s}-\mathbf{I})&\mathbf{f}_{\sigma}^{\top}\Sigma^{\ell}&(\mathbf{m}-\mathbf{f}_{\sigma})^{\top}\Sigma^{\ell}&(\mathbf{f}_{\sigma}-\mathbf{m})^{\top}\Sigma^{s}\end{bmatrix}^{\top},$

where $\Sigma^{\ell}$ is a diagonal matrix with $\Sigma^{\ell}_{ii}=1$ if
$\sigma_{i}=\ell$ and zero otherwise, and $\Sigma^{s}$ is defined analogously.
Now, since $\mathbf{W}_{N,N+1}=\mathbf{0}$, the equilibrium map for the bottom
layer in the hierarchy, $h_{N}^{1}$, has the same form as (10), and hence can
be written in the form (15). The following result, which is a generalization
of [20, Lemma IV.1], illustrates that in fact all the equilibrium maps in the
hierarchical thalamocortical network are piecewise-affine.

######  Lemma IV.1.

(Piecewise-affinity of equilibrium maps in hierarchical thalamocortical
linear-threshold models): Let $h_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$,
$i\in\\{1,\dots,k\\}$, be piecewise-affine functions,

$\displaystyle
h_{i}(\mathbf{c})=\mathbf{F}^{i}_{\lambda_{i}}\mathbf{c}+\mathbf{f}^{i}_{\lambda_{i}},\qquad\forall\mathbf{c}\in\Psi_{\lambda_{i}}\triangleq\\{\mathbf{c}~{}|~{}\mathbf{G}_{\lambda_{i}}^{i}\mathbf{c}+\mathbf{g}_{\lambda_{i}}^{i}\geq\mathbf{0}\\},$

for $\lambda_{i}\in\Lambda_{i}$, where $\Lambda_{i}$ is a finite index set
such that
$\bigcup_{\lambda_{i}\in\Lambda_{i}}\Psi_{\lambda_{i}}=\mathbb{R}^{n}$. Define
$\Lambda=\Lambda_{1}\times\Lambda_{2}\times\dots\times\Lambda_{k}$,
$\lambda=(\lambda_{1},\dots,\lambda_{k})$ and
$\Psi_{\lambda}=(\Psi_{\lambda_{1}},\dots,\Psi_{\lambda_{k}})$. Given matrices
$\mathbf{W}_{1},\mathbf{W}_{2}^{i},\mathbf{W}_{3}^{i}$ and vectors
$\bar{\mathbf{c}}_{i},\mathbf{c}^{\prime}$ for all $i\in\\{1,\dots,k\\}$
assume

$\displaystyle\mathbf{x}=\left[\mathbf{W}_{1}\mathbf{x}+\sum_{i=1}^{k}\mathbf{W}_{2}^{i}h_{i}(\mathbf{W}_{3}^{i}\mathbf{x}+\bar{\mathbf{c}}_{i})+\mathbf{c}^{\prime}\right]_{\mathbf{0}}^{\mathbf{m}},$
(16)

is known to have a unique solution
$\mathbf{x}^{\prime}\in\mathbb{R}^{n^{\prime}}$ for each
$\mathbf{c}^{\prime}\in\mathbb{R}^{n^{\prime}}$, and let
$h^{\prime}(\mathbf{c}^{\prime})$ be this solution. Then, there exists a
finite index set $\Lambda^{\prime}$ and
$\\{(\mathbf{F}_{\lambda}^{\prime},\mathbf{f}_{\lambda}^{\prime},\mathbf{G}_{\lambda}^{\prime},\mathbf{g}_{\lambda}^{\prime})\\}_{\lambda^{\prime}\in\Lambda^{\prime}}$
such that

$\displaystyle
h^{\prime}(\mathbf{c}^{\prime})=\mathbf{F}_{\lambda^{\prime}}^{\prime}\mathbf{c}^{\prime}+\mathbf{f}_{\lambda^{\prime}}^{\prime},\quad\forall\mathbf{c}^{\prime}\in\Psi_{\lambda^{\prime}}^{\prime}\triangleq\\{\mathbf{c}^{\prime}~{}|~{}\mathbf{G}_{\lambda^{\prime}}^{\prime}\mathbf{c}^{\prime}+\mathbf{g}_{\lambda^{\prime}}^{\prime}\geq\mathbf{0}\\}$

for $\lambda^{\prime}\in\Lambda^{\prime}$ and
$\bigcup_{\lambda^{\prime}\in\Lambda^{\prime}}\Psi_{\lambda^{\prime}}=\mathbb{R}^{n^{\prime}}$.

The equilibrium maps for the hierarchical thalamocortical network satisfy
Lemma IV.1 with $r=1$, $r=N-b$ and $r=2$ for the layers below the thalamus,
the thalamus, and the layers above the thalamus, resp., and therefore the maps
are piecewise-affine. Finally, the fact that these maps are globally Lipschitz
follows from [20, Lemma IV.2]. This property is necessary to be able to apply
later the generalization of Tikhonov’s singular perturbation stability theorem
to non-smooth ODEs given in [25, Proposition 1]333To apply the result to a
non-smooth ODE such as (1) we need to justify the following: $1)$
Lipschitzness of the dynamics uniformly in $t$, $2)$ Existence, uniqueness and
Lipschitzness of the equilibrium map of the fast dynamics, $3)$ Lipschitzness
and boundedness of the reduced-order model, $4)$ Asymptotic stability of the
fast dynamics uniformly in $t$ and the slow variable, and $5)$ Global
attractivity of the fast dynamics for any fixed slow variable. to take
advantage of the timescale separation between the layers.

###  IV-B Stability Assumptions and Conditions

In the hierarchical thalamocortical model described in Section III, only the
task-irrelevant components of the dynamics are directly controlled over, cf.
(3b). This means that assumptions on the stability of the task-relevant
components of the dynamics are needed to guarantee their stability and
recruitment to an equilibrium trajectory. In particular, we are interested in
the reduced-order task-relevant dynamics, that is, the system dynamics in
which the inputs from layers lower in the hierarchy have been replaced by
their equilibrium values. Here, we provide details on these assumptions and
identify sufficient conditions for them to hold. We split our discussion on
the assumptions based upon the location of a layer in the hierarchy as
different locations maintain both different roles and equilibrium maps.

#### Top Layer of the Hierarchy

The top layer $\mathcal{N}_{1}$ in the hierarchy does not have any nodes that
are to be inhibited. In addition, its role is in driving the selective
recruitment in the lower levels, rather than being recruited itself. As such,
we only require that the trajectories of its dynamics are bounded. Formally,
for all sets of constants $\mathbf{c}_{1}\in\mathbb{R}^{n_{1}}$,
$\mathbf{c}_{i}^{1}\in\mathbb{R}^{n_{i}-r_{i}}$, $i\in\\{2,\dots,N\\}$, and
$\mathbf{c}_{T}\in\mathbb{R}^{n_{T}-r_{T}}$, we assume

$\displaystyle\tau_{1}\dot{\bar{\mathbf{x}}}_{1}$
$\displaystyle=-\mathbf{x}_{1}+[\mathbf{W}_{1,1}\bar{\mathbf{x}}_{1}+\mathbf{W}_{1,2}^{11}h_{2}^{1}(\mathbf{W}_{2,1}\bar{\mathbf{x}}_{1}+\mathbf{c}_{2}^{1})$
$\displaystyle\qquad+\mathbf{W}_{1,T}^{11}\mathbf{x}_{T_{e(1)}}+\mathbf{c}_{1}]_{\mathbf{0}}^{\mathbf{m}_{1}},$
(17)

has bounded solutions. We note that our earlier assumption that the thalamus
is in the middle of the hierarchy makes the top layer a cortical one. If,
instead, the thalamus was the top layer, one would instead assume here that
its dynamics has bounded solutions, replacing (17) accordingly.

#### Lower Layers in the Hierarchy

In each of the layers below the top one, we seek to accomplish selective
inhibition and recruitment by having the dynamics converge to a parameter-
dependent equilibrium trajectory. For the task-irrelevant components, we aim
to use a control law to stabilize them to zero. If successful, what remains is
the task-relevant dynamics, which we then need to be globally exponentially
stable to a parameter-dependent equilibrium trajectory. In such a case, an
additional challenge is then to identify conditions on the interconnected
network layers that ensure the coupled system displays the desired behavior.
Due to the different reduced-order dynamics throughout the network, we provide
the details for this assumption in four categories: layers above the thalamus,
layer directly above the thalamus, thalamus, and layers below the thalamus.

#### Layers above the Thalamus

For each layer $\mathcal{N}_{i}$, with $i\in\\{1,\dots,a-1\\}$, above the
thalamus, the dynamics are directly dependent on two layers below it, and
hence the reduced-order dynamics are dependent on two equilibrium maps. For
achieving selective inhibition and recruitment, we assume that, for all sets
of constants $\mathbf{c}_{i+1}^{1}\in\mathbb{R}^{n_{i+1}-r_{i+1}}$,
$\mathbf{c}_{i}^{1}\in\mathbb{R}^{n_{i}-r_{i}}$, and
$\mathbf{c}_{j}^{1}\in\mathbb{R}^{n_{j}-r_{j}}$,
$j\in\\{i+2,\dots,N\\}\cup\\{T\\}$, the reduced-order dynamics

$\displaystyle\tau_{i}\dot{\mathbf{x}}_{i}^{1}$
$\displaystyle=-\mathbf{x}_{i}^{1}+[\mathbf{W}_{i,i}^{11}\mathbf{x}_{i}^{1}+\mathbf{W}_{i,i+1}^{11}h_{i+1}^{1}(\mathbf{W}_{i+1,i}^{11}\mathbf{x}_{i}^{1}$
$\displaystyle\quad+\mathbf{c}_{i+1}^{1})+\mathbf{W}_{i,T}^{11}\mathbf{x}_{T_{e(i)}}+\mathbf{c}_{i}^{1}]_{\mathbf{0}}^{\mathbf{m}_{i}^{1}},$
(18)

are GES to an equilibrium trajectory defined by the chosen constants.

#### Layer Directly above the Thalamus

For the layer $\mathcal{N}_{a}$ directly above the thalamus, while still
dependent on two layers below it in the hierarchy, one of these layers is
below the thalamus. In the reduced-order model, this changes the equilibrium
maps. Thus, for all constants $\mathbf{c}_{a}^{1}\in\mathbb{R}^{n_{a}-r_{a}}$,
$\mathbf{c}_{b}^{1}\in\mathbb{R}^{n_{b}-r_{b}}$, and
$\mathbf{c}_{i}^{1}\in\mathbb{R}^{n_{i}-r_{i}}$,
$i\in\\{b+1,\dots,N\\}\cup\\{T\\}$, we assume its reduced-order dynamics

$\displaystyle\tau_{a}\dot{\mathbf{x}}_{a}^{1}$
$\displaystyle=-\mathbf{x}_{a}^{1}+[\mathbf{W}_{a,a}^{11}\mathbf{x}_{a}^{1}+\mathbf{W}_{a,b}^{11}h_{b}^{1}(\mathbf{W}_{b,a}^{11}\mathbf{x}_{a}^{1}$
$\displaystyle\quad+\mathbf{W}_{b,T}\mathbf{x}_{T_{e(a)}}+\mathbf{c}_{b}^{1})+\mathbf{W}_{a,T}^{11}\mathbf{x}_{T_{e(a)}}+\mathbf{c}_{a}^{1}]_{\mathbf{0}}^{\mathbf{m}_{a}^{1}},$
(19)

are GES to an equilibrium trajectory determined by the chosen constants.

#### Thalamus Layer

Since the thalamus layer depends on all the other layers in the network, its
reduced-order dynamics are dependent on the $N-b$ equilibrium maps of all the
layers (from $\mathcal{N}_{b}$ to $\mathcal{N}_{N}$) below the thalamus. For
all constants
$\mathbf{c}_{b}^{1}\in\mathbb{R}^{n_{b}-r_{b}},\dots,\mathbf{c}_{N}^{1}\in\mathbb{R}^{n_{N}-r_{N}}$
and $\mathbf{c}_{T}^{1}\in\mathbb{R}^{n_{T}-r_{T}}$, we assume the reduced-
order dynamics

$\displaystyle\tau_{T}\dot{\mathbf{x}}_{T}^{1}$
$\displaystyle=-\mathbf{x}_{T}^{1}+\big{[}\mathbf{W}_{T}^{11}\mathbf{x}_{T}^{1}+\sum_{j=1}^{a}\mathbf{W}_{T,j}^{1,\mathrm{all}}\mathbf{x}_{j}^{1}$
$\displaystyle\quad+\sum_{j=b}^{N}\mathbf{W}_{T,j}^{11}\mathbf{x}_{j_{e}}^{1}+\mathbf{c}_{T}^{1}\big{]}_{\mathbf{0}}^{\mathbf{m}_{T}^{1}},$
(20)

are GES to an equilibrium trajectory defined by the constants
$\mathbf{c}_{b}^{1},\dots,\mathbf{c}_{N}^{1}$ and $\mathbf{c}_{T}^{1}$.

#### Layers below the Thalamus

Finally, the layers below the thalamus are directly dependent on only one
layer below them in the hierarchy. For all constants
$\mathbf{c}_{i+1}^{1}\in\mathbb{R}^{n_{i+1}-r_{i+1}}$ and
$\mathbf{c}_{i}^{1}\in\mathbb{R}^{n_{i}-r_{i}}$, we assume its reduced-order
dynamics

$\displaystyle\tau_{i}\dot{\mathbf{x}}_{i}^{1}$
$\displaystyle=-\mathbf{x}_{i}^{1}+[\mathbf{W}_{i,i}^{11}\mathbf{x}_{i}^{1}+\mathbf{W}_{i,i+1}^{11}h_{i+1}^{1}(\mathbf{W}_{i+1,i}^{1,\mathrm{all}}\mathbf{x}_{i}$
$\displaystyle\quad+\mathbf{W}_{i+1,T}^{1,\mathrm{all}}\mathbf{x}_{T}+\mathbf{c}_{i+1}^{1})+\mathbf{W}_{i,T}^{1,\mathrm{all}}\mathbf{x}_{T}+\mathbf{c}_{i}^{1}]_{\mathbf{0}}^{\mathbf{m}_{i}^{1}},$
(21)

and we assume they are GES to an equilibrium trajectory dependent upon the
constants $\mathbf{c}_{i+1}^{1}$ and $\mathbf{c}_{i}^{1}$.

We next turn our attention to identifying conditions under which the above
assumptions on the individual subnetworks hold, specifically, ensuring that
the reduced-order dynamics for layers
$\mathcal{N}_{2},\dots,\mathcal{N}_{N},\mathcal{N}_{T}$ are GES. The following
result makes use of the fact that the equilibrium maps are piecewise-affine,
cf. Lemma IV.1.

######  Lemma IV.2.

(Sufficient condition for existence and uniqueness of equilibria and GES in
multilayer linear-threshold networks with parallel connections): Let
$h_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$, $i\in\\{1,\dots,K\\}$, be
piecewise-affine functions,

$\displaystyle
h_{i}(\mathbf{c})=\mathbf{F}^{i}_{\lambda_{i}}\mathbf{c}+\mathbf{f}^{i}_{\lambda_{i}},\qquad\forall\mathbf{c}\in\Psi_{\lambda_{i}}\triangleq\\{\mathbf{c}~{}|~{}\mathbf{G}_{\lambda_{i}}^{i}\mathbf{c}+\mathbf{g}_{\lambda_{i}}^{i}\\}$

for all $\lambda_{i}\in\Lambda_{i}$ where $\Lambda_{i}$ is a finite index set
such that
$\bigcup_{\lambda_{i}\in\Lambda_{i}}\Psi_{\lambda_{i}}=\mathbb{R}^{n}$. Define
$\bar{\mathbf{F}}_{i}\triangleq\max_{\lambda_{i}\in\Lambda_{i}}|\mathbf{F}^{i}_{\lambda_{i}}|$
as the matrix made of the entry-wise maximum of the elements in
$\\{|\mathbf{F}_{\lambda_{i}}^{i}|\\}_{\lambda_{i}\in\Lambda_{i}}$. For
$i\in\\{1,\dots,K\\}$ let the matrices
$\mathbf{W}_{1}^{i},\mathbf{W}_{2}^{i},$ be arbitrary and also consider
arbitrary matrix $\mathbf{W}$. Then, if

$\displaystyle\rho\left(|\mathbf{W}|+\sum_{i=1}^{K}|\mathbf{W}_{1}^{i}|\bar{\mathbf{F}}_{i}|\mathbf{W}_{2}^{i}|\right)<1,$

the dynamics

$\displaystyle\tau\dot{\mathbf{x}}$
$\displaystyle=-\mathbf{x}+\left[\mathbf{W}\mathbf{x}+\sum_{i=1}^{K}\mathbf{W}_{1}^{i}h_{i}(\mathbf{W}_{2}^{i}\mathbf{x}+\bar{\mathbf{c}}_{i})+\mathbf{c}\right]_{\mathbf{0}}^{\mathbf{m}}$

is GES to a unique for all constants $\bar{\mathbf{c}}_{i},\mathbf{c}$.

Lemma IV.2 generalizes [20, Theorem IV.4] to thalamocortical networks and its
proof follows a similar line of arguments. The application of this result to
our setting results in, if
$\rho(|\mathbf{W}_{i,i}^{11}|+|\mathbf{W}_{i,i+1}^{11}|\bar{\mathbf{F}}_{i+1}|\mathbf{W}_{i+1,i}^{11}|+|\mathbf{W}_{i,T}^{11}|\bar{\mathbf{F}}_{T}|\mathbf{W}_{T,i}^{11}|)<1$
for $i\in\\{2,\dots,a-1\\}$,
$\rho(|\mathbf{W}_{i,i}^{11}|+|\mathbf{W}_{i,i+1}^{11}|\bar{\mathbf{F}}_{i+1}|\mathbf{W}_{i+1,i}^{11}|)<1$,
for $i\in\\{b,\dots,N\\}$ and
$\rho(|\mathbf{W}_{T}^{11}|+\sum_{i=b}^{N}|\mathbf{W}_{T,i}^{11}|\bar{\mathbf{F}}_{i}|\mathbf{W}_{i,T}^{11}|)<1$,
then the dynamics (18)-(21) are GES to an equilibria.

######  Remark IV.3.

(Comparison with conditions for a strictly cortical hierarchical network):
Sufficient conditions for GES of the reduced-order dynamics of a strictly
cortical network can be obtained from the above by allowing
$\mathbf{W}_{i,T}^{11}=0$ for all $i$, and then these conditions reduce to
those found in [20], as expected. These sufficient conditions to guarantee
GES, for layers above the thalamus, are harder to satisfy in the
thalamocortical network than in a cortical one. However, as the conditions are
only sufficient, this does not mean that it is in fact more difficult to
achieve GES of the reduced-order dynamics in the thalamocortical case, as the
proof above does not explicitly invoke the inhibitory nature of the thalamus.
$\bullet$

###  IV-C Selective Inhibition and Recruitment

We are ready to illustrate how selective inhibition and recruitment can be
achieved in the hierarchical thalamocortical network model. Here, we first
formalize the concept mathematically and then provide a feedforward-feedback
control that achieves it. Recall that selective inhibition corresponds to the
task-irrelevant components of the network converging to zero, and recruitment
corresponds to having the task-relevant components converge to an equilibrium
trajectory. The timescale ratio between layers, $\epsilon$, must approach zero
to encode the separation of timescales observed in the brain. As such, we
require convergence of the task-relevant components of the network to an
equilibrium as this ratio approaches zero. Formally, selective inhibition and
recruitment is achieved if the following equations are satisfied for any
$0<t_{1}<t_{2}<\infty$: first, for all layers $\mathcal{N}_{i}$,
$i\in\\{2,\dots,N\\}\cup\\{T\\}$, it holds that

[inhibition]: $\displaystyle\quad\lim_{\epsilon\to
0}\sup_{t\in[t_{1},t_{2}]}\left\|\mathbf{x}_{i}^{0}(t)\right\|=0;$ (22a)
second, for the top layer $\mathcal{N}_{1}$ in the hierarchy, [driving layer]:
$\displaystyle\quad\lim_{\epsilon\to
0}\sup_{t\in[t_{1},t_{2}]}\left\|\mathbf{x}_{1}^{1}(t)-\bar{\mathbf{x}}_{1}^{1}(t)\right\|=0;$
(22b) for all layers $\\{\mathcal{N}_{i}\\}_{i=2}^{a}$ above the thalamus,
[recruitment]: $\displaystyle\quad\lim_{\epsilon\to
0}\sup_{t\in[t_{1},t_{2}]}\|\mathbf{x}_{i}^{1}(t)$ $\displaystyle\quad\quad-
h_{i}^{1}(\mathbf{W}_{i,i-1}^{11}\mathbf{x}_{i-1}^{1}(t)+\mathbf{c}_{i}^{1})\|=0;$
(22c) for the thalamus layer, [recruitment]:
$\displaystyle\quad\lim_{\epsilon\to
0}\sup_{t\in[t_{1},t_{2}]}\left\|\mathbf{x}_{T}^{1}(t)-x_{T_{e(a)}}\right\|=0;$
(22d) and finally, for the layers $\\{\mathcal{N}_{i}\\}_{i=b}^{N}$ below the
thalamus, [recruitment]: $\displaystyle\quad\lim_{\epsilon\to
0}\sup_{t\in[t_{1},t_{2}]}\|\mathbf{x}_{i}^{1}(t)-h_{i}^{1}(\mathbf{W}_{i,i-1}^{1,\mathrm{all}}\mathbf{x}_{i-1}(t)$
$\displaystyle\quad\qquad+\mathbf{W}_{i,T}^{1,\mathrm{all}}\mathbf{x}_{T}(t)+\mathbf{c}_{i}^{1})\|=0.$
(22e)

Intuitively, we note from (22) that achieving selective inhibition and
recruitment means that one can make the error between the network trajectories
and the equilibrium trajectories arbitrarily small if the timescale ratio is
small enough. The following result shows that selective inhibition and
recruitment can be achieved in the hierarchical thalamocortical network by
means of a combination of feedforward and feedback control.

######  Theorem IV.4.

(Selective inhibition and recruitment in hierarchical thalamocortical networks
with Feedforward-Feedback Control): Consider an $N$-layer thalamocortical
network as shown in Fig. 1(a), with layer dynamics given by (5) and (7).
Without loss of generality, let
$\tau_{1}\gg\tau_{2}\gg\dots\gg\tau_{T}\gg\dots\gg\tau_{N}$ and
$a,b\in\\{1,\dots,N\\}$ such that $\tau_{a}\gg\tau_{T}\gg\tau_{b}$, with
$b=a+1$. Assume the stability conditions (17)-(21) for the reduced-order
subnetworks are satisfied. Then, for $i\in\\{1,\dots,N\\}\cup\\{T\\}$ and
constants $\mathbf{c}_{i}\in\mathbb{R}^{n_{i}}$ and
$\mathbf{c}_{T}\in\mathbb{R}^{n_{T}}$, there exist control laws
$\mathbf{u}_{i}(t)=\mathbf{K}_{i}\mathbf{x}_{i}(t)+\bar{\mathbf{u}}_{i}(t)$,
with $\mathbf{K}_{i}\in\mathbb{R}^{p_{i}\times n_{i}}$ and
$\bar{\mathbf{u}}_{i}:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq
0}^{p_{i}}$, such that the closed-loop system achieves selective inhibition
and recruitment (22).

###### Proof.

We prove the result by constructing a control and iteratively applying the
generalization of Tikhonov’s theorem from [25]. Throughout the proof, we make
use of the following notation. For $i\leq a$, let
$\mathbf{x}_{1:i}=\begin{bmatrix}\mathbf{x}_{1}^{\top}&\dots&\mathbf{x}_{i}^{\top}\end{bmatrix}^{\top}$
and, for $i\geq b$, let
$\mathbf{x}_{1:T:i}=\begin{bmatrix}\mathbf{x}_{1}^{\top}\dots\mathbf{x}_{a}^{\top}&\mathbf{x}_{T}^{\top}&\mathbf{x}_{b}^{\top}&\dots&\mathbf{x}_{i}^{\top}\end{bmatrix}^{\top}$.
We first define the control for layer $\mathcal{N}_{N}$. Let $\mathbf{K}_{N}$
and $\bar{\mathbf{u}}_{N}(t)$ be such that

$\displaystyle\mathbf{B}_{N}^{0}\mathbf{K}_{N}$
$\displaystyle\leq-\mathbf{W}_{N,N}^{0,\mathrm{all}},$ (23a)
$\displaystyle\mathbf{B}_{N}^{0}\bar{\mathbf{u}}_{N}(t)$
$\displaystyle\leq-\mathbf{W}_{N,N-1}^{0,\mathrm{all}}\mathbf{x}_{N-1}(t)-\mathbf{W}_{N,T}^{0,\mathrm{all}}\mathbf{x}_{T}(t)-\mathbf{c}_{N}^{0}.$
(23b)

The inequalities (23) can be satisfied due to our assumption that the matrices
$\mathbf{B}_{i}^{0}$ have full rank and $p_{i}\geq r_{i}$ for all
$i\in\\{1,\dots,N\\}\cup\\{T\\}$. Substituting (23) into the dynamics,

$\displaystyle\tau_{1}\dot{\mathbf{x}}_{1}$
$\displaystyle=-\mathbf{x}_{1}+[\mathbf{W}_{1,1}\mathbf{x}_{1}+\mathbf{W}_{1,2}\mathbf{x}_{2}+\mathbf{W}_{1,T}\mathbf{x}_{T}+\mathbf{c}_{1}]_{\mathbf{0}}^{\mathbf{m}_{1}},$
$\displaystyle\vdots$ $\displaystyle\tau_{T}\dot{\mathbf{x}}_{T}$
$\displaystyle=-\mathbf{x}_{T}+\bigg{[}\mathbf{W}_{T}\mathbf{x}_{T}+\sum_{j=1}^{N}\mathbf{W}_{T,j}\mathbf{x}_{j}+\mathbf{B}_{T}\mathbf{u}_{T}+\mathbf{c}_{T}\bigg{]}_{\mathbf{0}}^{\mathbf{m}_{T}},$
$\displaystyle\vdots$
$\displaystyle\epsilon_{N}\tau_{N-1}\dot{\mathbf{x}}_{N}^{0}$
$\displaystyle=-\mathbf{x}_{N}^{0},$
$\displaystyle\epsilon_{N}\tau_{N-1}\dot{\mathbf{x}}_{N}^{1}$
$\displaystyle=-\mathbf{x}_{N}^{1}+[\mathbf{W}_{N,N}^{1,\mathrm{all}}\mathbf{x}_{N}+\mathbf{W}_{N,N-1}^{1,\mathrm{all}}\mathbf{x}_{N-1}$
$\displaystyle\qquad+\mathbf{W}_{N,T}^{1,\mathrm{all}}\mathbf{x}_{T}+\mathbf{c}_{N}^{1}]_{\mathbf{0}}^{\mathbf{m}_{1}^{1}}.$

Taking $\epsilon_{N}\to 0$ then provides a separation of timescales between
$\mathbf{x}_{N}$ and $\mathbf{x}_{1:T:N-1}$. Note that, by assumption, the
reduced-order dynamics (21) is GES. Using then the fact that the cascaded
interconnection of a GES system with an exponentially vanishing system is also
GES, cf. [19, Lemma A.1], we deduce that, for any constants $\mathbf{x}_{N-1}$
and $\mathbf{x}_{T}$, $\mathbf{x}_{N}$ is GES to
$(\mathbf{0}_{r_{N}},h_{N}^{1}(\mathbf{W}_{N,N-1}^{1,\mathrm{all}}\mathbf{x}_{N-1}+\mathbf{W}_{N,T}^{1,\mathrm{all}}\mathbf{x}_{T}+\mathbf{c}_{N}^{1}))$.
Recalling that, by Lemma IV.1 and [20, Lemma IV.2], the equilibrium maps
$h_{i}^{1}$ are globally Lipschitz for all $i\in\\{1,\dots,N\\}\cup\\{T\\}$,
and noting that the entire network is Lipschitz due to the Lipschitzness of
the linear-threshold function $[\cdot]_{\mathbf{0}}^{\mathbf{m}}$, we can
apply [25, Proposition 1], giving for $0<t_{1}<t_{2}<\infty$,

$\displaystyle\lim_{\epsilon_{1}\to
0}\sup_{[t_{1},t_{2}]}\left\|\mathbf{x}_{N}^{0}(t)\right\|=0,$
$\displaystyle\lim_{\epsilon_{1}\to
0}\sup_{[t_{1},t_{2}]}\big{\|}\mathbf{x}_{N}^{1}(t)-h_{N}^{1}(\mathbf{W}_{N,N-1}^{1,\mathrm{all}}\mathbf{x}_{N-1}(t)$
$\displaystyle\qquad+\mathbf{W}_{N,T}^{1,\mathrm{all}}\mathbf{x}_{T}(t)+\mathbf{c}_{N}^{1})\big{\|}=0,$
$\displaystyle\lim_{\epsilon_{1}\to
0}\sup_{[t_{1},t_{2}]}\left\|\mathbf{x}_{1:T:N-1}-\mathbf{x}_{1:T:N-1}^{(1)}\right\|=0,$
(24)

where $\mathbf{x}_{1:T:N-1}^{(1)}$ represents the first-step reduced-order
model coming from replacing $\mathbf{x}_{N}$ by its equilibrium value
$(\mathbf{0}_{r_{N}},h_{N}^{1}(\mathbf{W}_{N,N-1}^{1,\mathrm{all}}\mathbf{x}_{N-1}+\mathbf{W}_{N,T}^{1,\mathrm{all}}\mathbf{x}_{T}+\mathbf{c}_{N}^{1}))$.
We continue the proof by iterating the above process for each network layer,
utilizing constructed control laws and applying [25, Proposition 1] to the
corresponding reduced-order models built from substitution of the equilibrium
values.

We now construct control laws for layers $\mathcal{N}_{i}$ with
$i\in\\{b,\dots,N-1\\}$. We choose $\mathbf{K}_{i}$ and
$\bar{\mathbf{u}}_{i}(t)$ such that

$\displaystyle\mathbf{B}_{i}^{0}\mathbf{K}_{i}$
$\displaystyle\leq-|\mathbf{W}_{i,i}^{0,\mathrm{all}}|-|\mathbf{W}_{i,i+1}^{01}|\bar{\mathbf{F}}_{i+1}|\mathbf{W}_{i+1,i}^{1,\mathrm{all}}|,$
(25a) $\displaystyle\mathbf{B}_{i}^{0}\bar{\mathbf{u}}_{i}(t)$
$\displaystyle\leq-\mathbf{W}_{i,i-1}^{0,\mathrm{all}}\mathbf{x}_{i-1}(t)-\mathbf{W}_{i,T}^{0,\mathrm{all}}\mathbf{x}_{T}(t)-\mathbf{c}_{i}^{0}$
(25b)
$\displaystyle\qquad-|\mathbf{W}_{i,i+1}^{01}|\bar{\mathbf{F}}_{i+1}|\mathbf{W}_{i+1,T}^{1,\mathrm{all}}\mathbf{x}_{T}(t)+\mathbf{c}_{i+1}^{1}|,$
(25c)

in which
$\bar{\mathbf{F}}_{i}\in\mathbb{R}^{(n_{i}-r_{i})\times(n_{i}-r_{i})}$ is the
entry-wise maximal gain of the equilibrium map $h_{i}^{1}(\cdot)$ as defined
in Lemma IV.2. We use the control law (25) to construct the reduced-order
model, consider the timescale separation by letting $\epsilon_{i}\to 0$ for
$i\in\\{b,\dots,N-1\\}$, and finally apply [25, Proposition 1] to obtain

$\displaystyle\lim_{\epsilon\to
0}\sup_{t\in[t_{1},t_{2}]}\left\|\mathbf{x}_{i}^{(N-i)^{0}}(t)\right\|=0,$
$\displaystyle\lim_{\epsilon\to
0}\sup_{t\in[t_{1},t_{2}]}\|\mathbf{x}_{i}^{(N-i)^{1}}(t)-h_{i}^{1}(\mathbf{W}_{i,i-1}^{1,\mathrm{all}}\mathbf{x}_{i-1}^{(N-i)}(t)$
$\displaystyle\qquad+\mathbf{W}_{i,T}^{1,\mathrm{all}}\mathbf{x}_{T}^{(N-i)}(t)+\mathbf{c}_{i}^{1})\|=0,$
$\displaystyle\lim_{\epsilon\to
0}\sup_{t\in[t_{1},t_{2}]}\left\|\mathbf{x}_{1:T:i-1}^{(N-i)}(t)-\mathbf{x}_{1:T:i-1}^{(N-i+1)}(t)\right\|=0,$

for all $i\in\\{b,\dots,N-1\\}$. For the thalamus layer, we note that provided
the initial conditions lie in $[\mathbf{0},\mathbf{m}_{i}]$ for all
$i\in\\{1,\dots,N\\}\cup\\{T\\}$, by the properties of the linear-threshold
dynamics, we have that $\mathbf{x}_{i}(t)\leq\mathbf{m}_{i}$ for all $t\geq
0$. Utilizing these bounds, we define the control for the thalamus layer such
that it satisfies

$\displaystyle\mathbf{B}_{T}^{0}\mathbf{K}_{T}$
$\displaystyle\leq-|\mathbf{W}_{T}^{0,\mathrm{all}}|-\sum_{j=b}^{N}|\mathbf{W}_{T,j}^{01}|\bar{\mathbf{F}}_{j}|\mathbf{W}_{j,T}^{0,\mathrm{all}}|,$
(26a) $\displaystyle\mathbf{B}_{T}^{0}\bar{\mathbf{u}}_{T}(t)$
$\displaystyle\leq\sum_{j=1}^{1}\mathbf{W}_{T,j}^{0,\mathrm{all}}\mathbf{x}_{j}(t)-\mathbf{c}_{T}^{1}$
(26b)
$\displaystyle\quad-|\mathbf{W}_{T,b}^{0,\mathrm{all}}|\bar{\mathbf{F}}_{b}|\mathbf{W}_{b,a}^{1,\mathrm{all}}\mathbf{x}_{a}(t)+\mathbf{c}_{b}^{1}|$
(26c)
$\displaystyle\quad-\sum_{j=b+1}^{N}|\mathbf{W}_{T,j}^{01}|\bar{F}_{j}|\mathbf{W}_{j,j-1}^{1,\mathrm{all}}\mathbf{m}_{j-1}+\mathbf{c}_{j}^{1}|.$
(26d)

Then, after constructing the reduced-order model using the control laws (26a)
and (26d) and letting $\epsilon_{T}\to 0$ to create the timescale separation,
we again apply [25, Proposition 1] to get

$\displaystyle\lim_{\epsilon\to
0}\sup_{t\in[t_{1},t_{2}]}\left\|\mathbf{x}_{T}^{(T)^{0}}(t)\right\|=0,$
$\displaystyle\lim_{\epsilon\to
0}\sup_{t\in[t_{1},t_{2}]}\left\|\mathbf{x}_{T}^{(T)^{1}}(t)-h_{T}^{1}(\sum_{j=1}^{a}\mathbf{W}_{T,j}^{1,\mathrm{all}}\mathbf{x}_{j}^{(T)}(t)+\mathbf{c}_{T}^{1})\right\|=0,$
$\displaystyle\lim_{\epsilon\to
0}\sup_{t\in[t_{1},t_{2}]}\left\|\mathbf{x}_{1:a}^{(T)}(t)-\mathbf{x}_{1:a}^{(N-b+1)}(t)\right\|=0.$

What remains is to consider the layers above the thalamus. These layers
maintain the same form of control law except for the layer immediately above
the thalamus. For layer $\mathcal{N}_{a}$ we define terms $\mathbf{K}_{a}$ and
$\bar{\mathbf{u}}_{a}(t)$ such that

$\displaystyle\mathbf{B}_{a}^{0}\mathbf{K}_{a}$
$\displaystyle\leq-|\mathbf{W}_{a,a}^{0,\mathrm{all}}|-|\mathbf{W}_{a,b}^{01}|\bar{\mathbf{F}}_{b}|\mathbf{W}_{b,a}^{1,\mathrm{all}}|,$
(27a) $\displaystyle\mathbf{B}_{a}^{0}\bar{\mathbf{u}}_{a}(t)$
$\displaystyle\leq-\mathbf{W}_{a,a-1}^{0,\mathrm{all}}\mathbf{x}_{a-1}(t)-|\mathbf{W}_{a,b}^{01}|\bar{\mathbf{F}}_{b}|\mathbf{W}_{b,T}^{1,\mathrm{all}}\mathbf{x}_{T_{e(a)}}+\mathbf{c}_{b}^{1}|.$
(27b)

Now, for layers $\mathcal{N}_{i}$, $i\in\\{2,\dots,a-1\\}$, we let the
controls $\mathbf{K}_{i}$ and $\bar{\mathbf{u}}_{i}(t)$ be such that the
following hold

$\displaystyle\mathbf{B}_{i}^{0}\mathbf{K}_{i}$
$\displaystyle\leq-|\mathbf{W}_{i,i}^{0,\mathrm{all}}|-|\mathbf{W}_{i,i+1}^{01}|\bar{\mathbf{F}}_{i+1}|\mathbf{W}_{i+1,i}^{1,\mathrm{all}}|,$
(28a) $\displaystyle\mathbf{B}_{i}^{0}\bar{\mathbf{u}}_{i}(t)$
$\displaystyle\leq-\mathbf{W}_{i,i-1}^{0,\mathrm{all}}\mathbf{x}_{i-1}(t)-|\mathbf{W}_{i,i+1}^{01}|\bar{\mathbf{F}}_{i+1}|\mathbf{c}_{i+1}^{1}|,$
(28b)

Once the reduced-order models are constructed, letting $\epsilon_{i}\to 0$ for
$i\in\\{2,\dots,a\\}$, applying [25, Proposition 1] gives

$\displaystyle\lim_{\epsilon\to
0}\sup_{t\in[t_{1},t_{2}]}\left\|\mathbf{x}_{i}^{(N-i)^{0}}(t)\right\|=0,$
$\displaystyle\lim_{\epsilon\to
0}\sup_{t\in[t_{2},t_{2}]}\left\|\mathbf{x}_{i}^{(N-i)^{1}}(t)-h_{i}^{1}(W_{i,i-1}^{1,\mathrm{all}}\mathbf{x}_{i-1}^{(N-i)}(t)+\mathbf{c}_{i}^{1})\right\|=0,$
$\displaystyle\lim_{\epsilon\to
0}\sup_{t\in[t_{1},t_{2}]}\left\|\mathbf{x}_{1:i-1}^{(N-i)}(t)-\mathbf{x}_{1:i-1}^{(N-i+1)}(t)\right\|=0.$

for $i\in\\{2,\dots,a\\}$. The equations (22) for selective inhibition and
recruitment, are then obtained through repeated application of the triangle
inequality, completing the proof. ∎

Figure 2: Trajectories of a three-layer hierarchical thalamocortical network
under a periodic input. Red and blue lines correspond with excitatory and
inhibitory nodes, resp. Each layer consists of three nodes with the top layer
in the network containing only excitatory nodes, the thalamus being strictly
inhibitory and layers two and three having both excitatory and inhibitory
nodes. Using a control law designed through Theorem IV.4, in each layer other
than the top layer, nodes are able to be either selectively inhibited to zero
or recruited into a periodic equilibrium trajectory. Timescale ratios are
$\epsilon_{2}=0.54$, $\epsilon_{3}=0.15$ and $\epsilon_{T}=0.61$.

Fig. 2 illustrates Theorem IV.4. While the result is proven for the case in
which the thalamus has a timescale inside the hierarchy, it still holds when
the thalamus timescale is at the top or bottom of the hierarchy. In these
cases, the proof method remains the same, with the appropriate modifications
to the inequalities derived in (23)-(28).

######  Remark IV.5.

(Comparison with strictly cortical networks): Regarding selective inhibition
and recruitment, Theorem IV.4 is to multilayer thalamocortical networks what
[20, Theorem IV.3] is to multilayer cortical networks. Despite the analytical
similarities, the consideration of the thalamus provides a significant
generalization from a biological perspective, as transthalamic connections
exist in most brain networks [1]. From a technical viewpoint, the addition of
the thalamus, while only adding a layer, significantly complicates the
analysis due to its connection with all of the cortical layers. These
connections result in every layer having connections from timescales not
simply immediately above or below it in the hierarchy, which impacts the
determination of convergence to equilibrium values for layers above the
thalamus. Finally,the control laws (23)-(28) allow for smaller magnitude
sufficient controls than for the strictly cortical networks determined in [20]
due to the inhibitory properties of the thalamic connection matrices
$\mathbf{W}_{i,T}$, cf Section VI. $\bullet$

##  V Star-connected Thalamocortical Networks

In this section we consider star-connected thalamocortical networks, cf. Fig.
1(b), where the cortical regions are each connected only to the thalamus and
the dynamics are governed by (8)-(9). In this topology, there is no direct
relationship between the timescales of each layer and as such, no hierarchical
structure. Without timescale separation, the specifics of selective inhibition
and recruitment, both in terms of equilibria and stability criteria, differ
from the hierarchical case.

###  V-A Equilibria and Stability Conditions

With the lack of timescale separation, the decomposition described in Section
IV-A of the network equilibrium map as a collection of equilibrium maps for
each layer, with the state of layers higher in the hierarchy represented by a
constant input, no longer holds. As such, equilibrium values must be
determined concurrently for all the layers. Since the task-irrelevant
components get selectively inhibited to zero, the equilibrium for the task-
relevant components is given by the solution $\mathbf{x}_{i}^{1^{*}}$,
$i\in\\{1,\dots,N\\}\cup\\{T\\}$ to the system of equations:

$\displaystyle\mathbf{x}_{i}^{1}$
$\displaystyle=[\mathbf{W}_{i,i}^{1,\mathrm{all}}\mathbf{x}_{i}+\mathbf{W}_{i,T}^{1,\mathrm{all}}\mathbf{x}_{T}+\mathbf{c}_{i}^{1}]_{\mathbf{0}}^{\mathbf{m}_{i}^{1}},\qquad
i=1,\dots,N,$ $\displaystyle\mathbf{x}_{T}^{1}$
$\displaystyle=[\mathbf{W}_{T}\mathbf{x}_{T}+\sum_{i=1}^{N}\mathbf{W}_{T,i}\mathbf{x}_{i}+\mathbf{c}_{T}^{1}]_{0}^{\mathbf{m}_{T}^{1}}.$
(29)

The task-relevant equilibrium to which the system converges is dependent upon
$\mathbf{c}_{i}\in\mathbb{R}^{n_{i}-r_{i}}$, $i\in\\{1,\dots,N\\}$ and
$\mathbf{c}_{T}\in\mathbb{R}^{n_{T}-r_{T}}$, and so we represent it by
$\mathbf{x}_{i}^{1^{*}}(\mathbf{c}_{1},\dots,\mathbf{c}_{N},\mathbf{c}_{T})$.
In keeping with the role of layer $\mathcal{N}_{1}$ as driving the selective
recruitment in the other layers, rather than being recruited itself, we
consider in what follows an input signal $\mathbf{c}_{1}:\mathbb{R}_{\geq
0}\rightarrow\mathbb{R}^{n_{i}-r_{i}}$, rather than a constant, that gives
rise to an equilibrium trajectory
$\mathbf{x}_{i}^{1^{*}}(\mathbf{c}_{1}(t),\dots,\mathbf{c}_{N},\mathbf{c}_{T})$
for the dynamics.

As per the description of the star-connected thalamocortical network, cf
Section III, only the task-irrelevant component of the dynamics is directly
controlled, and as such the ability to achieve selective inhibition and
recruitment is dependent on the stability properties of the task-relevant
components. To ensure this, we employ below the fact [19, Theorem IV.8] that,
for a generic linear-threshold network mode
$\tau\dot{\mathbf{x}}=-\mathbf{x}+[\mathbf{W}\mathbf{x}+\mathbf{c}]_{\mathbf{0}}^{\mathbf{m}}$,
the condition $\rho(|\mathbf{W}|)<1$ is sufficient to ensure that, for all
$\mathbf{c}\in\mathbb{R}^{n}$, the dynamics is GES to an equilibrium.

###  V-B Selective Inhibition and Recruitment

We are ready to formalize selective inhibition and recruitment for star-
connected networks and provide conditions for its achievement. We recall that
the subnetwork $\mathcal{N}_{1}$ corresponds with a subcortical region
applying a sensory input signal to the thalamus to be relayed to the cortical
regions. As such, we do not inhibit any components in this subnetwork and
instead assume that it is stable to a trajectory
$\bar{\mathbf{x}}_{1}(\mathbf{c}_{1}(t))$ dictated by its own input signal.
For the remaining layers, we wish to inhibit the task-irrelevant components to
zero and recruit the task-relevant components to the equilibria trajectory
$\mathbf{x}_{i}^{1^{*}}(\mathbf{c}_{1}(t),\dots,\mathbf{c}_{N},\mathbf{c}_{T})$,
$i\in\\{2,\dots,N\\}\cup\\{T\\}$. This can be formalized to selective
inhibition and recruitment is achieved if for the input layer
$\mathcal{N}_{1}$,

[driving layer]:
$\displaystyle\quad\lim_{t\to\infty}\left\|\mathbf{x}_{1}(t)-\bar{\mathbf{x}}_{1}(\mathbf{c}_{1}(t))\right\|=0;$
(30a) and for all layers $\\{\mathcal{N}_{i}\\}_{i=2}^{N}$ and
$\mathcal{N}_{T}$, [inhibition]:
$\displaystyle\quad\lim_{t\to\infty}\left\|\mathbf{x}_{i}^{0}(t)\right\|=0;$
(30b) [recruitment]:
$\displaystyle\quad\lim_{t\to\infty}\left\|\mathbf{x}_{i}^{1}(t)-\mathbf{x}_{i}^{1^{*}}(\mathbf{c}_{1}(t),\dots,\mathbf{c}_{N},\mathbf{c}_{T})\right\|=0.$
(30c)

We also employ a weaker notion, referred to as $\epsilon$-selective inhibition
and recruitment, which is met if there exists $t^{*}$ such that the functions
in (30) are all upper bounded by $\epsilon>0$ for $t>t^{*}$. For convenience,
we also introduce the notation:

$\displaystyle\bar{\mathbf{W}}^{11}$
$\displaystyle=\begin{bmatrix}\mathbf{W}_{1,1}&\mathbf{0}&\dots&\mathbf{0}&\mathbf{W}_{1,T}^{11}\\\
\mathbf{0}&\mathbf{W}_{2,2}^{11}&\dots&\mathbf{0}&\mathbf{W}_{2,T}^{11}\\\
\vdots&\dots&\ddots&\vdots&\vdots\\\
\mathbf{0}&\mathbf{0}&\dots&\mathbf{W}_{N,N}^{11}&\mathbf{W}_{N,T}^{11}\\\
\mathbf{W}_{T,1}^{11}&\mathbf{W}_{T,2}^{11}&\dots&\mathbf{W}_{T,N}^{11}&\mathbf{W}_{T}^{11}\end{bmatrix}.$
(31)

Note the Schur decomposition
$\bar{\mathbf{W}}^{11}=\mathbf{Q}^{\top}(\mathbf{D}_{\bar{\mathbf{W}}^{11}}+\mathbf{N}_{\bar{\mathbf{W}}^{11}})\mathbf{Q}$,
where $\mathbf{Q}$ is unitary, $\mathbf{D}_{\bar{\mathbf{W}}^{11}}$ is
diagonal, and $\mathbf{N}_{\bar{\mathbf{W}}^{11}}$ is upper triangular with a
zero diagonal [33]. The next result establishes conditions to achieve
selective inhibition and recruitment in star-connected systems without a
hierarchy of timescales.

######  Theorem V.1.

(Selective inhibition and recruitment of star-connected networks): Consider an
$N$-layer star-connected thalamocortical network as shown in Fig. 1(b), with
layer dynamics given by (8) and (9). Suppose the following hold for all values
of $\mathbf{c}_{i}\in\mathbb{R}^{n_{i}}$, $i\in\\{1,\dots,N\\}$, and
$\mathbf{c}_{T}\in\mathbb{R}^{n_{T}}$:

  1. (i)

The input layer $\mathcal{N}_{1}$ has no nodes to be inhibited,
$\rho(|\mathbf{W}_{1,1}|)<1$, and the input $\mathbf{c}_{1}(t)$ lies in a
compact set and has a bounded rate derivative;

  2. (ii)

For each $i\in\\{2,\dots,N\\}\cup\\{T\\}$, the matrix $\mathbf{W}_{i,i}^{11}$
satisfies $\rho(|\mathbf{W}_{i,i}^{11}|)\leq\alpha_{i}$, with $\alpha_{i}<1$;

  3. (iii)

If
$\rho\big{(}\sum_{i=1}^{N}|\mathbf{W}_{i,T}^{11}\mathbf{W}_{T,i}^{11}|\big{)}\neq
0$, then $\alpha+\max(\delta,\delta^{1/p})<1$, where $p$ is the dimension of
$\mathbf{N}_{\bar{\mathbf{W}}^{11}}$ and

$\displaystyle\alpha$
$\displaystyle=\max_{i\in\\{2,\dots,N\\}\cup\\{T\\}}\\{\alpha_{i}\\},\qquad\delta=\gamma\sum_{j=1}^{p-1}\left\|\mathbf{N}_{\bar{\mathbf{W}}^{11}}\right\|^{j},$
$\displaystyle\gamma$
$\displaystyle=\max\\{\sum_{i=1}^{N-1}\mathbf{W}_{i,T}^{11}\mathbf{W}_{i,T}^{11^{\top}},\sum_{i=1}^{N-1}\mathbf{W}_{T,i}^{11}\mathbf{W}_{T,i}^{11^{\top}}\\}.$

Then, there exist control laws
$\mathbf{u}_{i}(t)=\mathbf{K}_{i}\mathbf{x}_{i}(t)+\bar{\mathbf{u}}_{i}(t)$,
with $\mathbf{K}_{i}\in\mathbb{R}^{p_{i}\times n_{i}}$ and
$\bar{\mathbf{u}}_{i}:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq
0}^{p_{i}}$, and $\epsilon>0$ such that the cortical and thalamic regions
within the closed-loop system achieve $\epsilon-$selective inhibition and
recruitment. Furthermore, if $\left\|\dot{\mathbf{c}}_{1}(t)\right\|\to 0$ as
$t\to\infty$, then the network achieves selective inhibition and recruitment
(30).

###### Proof.

First, for cortical region $\mathcal{N}_{i}$, $i\in\\{2,\dots,N\\}$, define
the control laws
$\mathbf{u}_{i}(t)=\mathbf{B}_{i}\mathbf{K}_{i}\mathbf{x}_{i}(t)+\mathbf{B}_{i}\bar{\mathbf{u}}(t)$
such that

$\displaystyle\mathbf{B}_{i}\mathbf{K}_{i}$
$\displaystyle\leq-\mathbf{W}_{i,i}^{0,\mathrm{all}},$ (32a)
$\displaystyle\mathbf{B}_{i}\bar{\mathbf{u}}_{i}(t)$
$\displaystyle\leq-\mathbf{W}_{i,T}^{0,\mathrm{all}}\mathbf{x}_{T}(t)-\mathbf{c}_{i}^{0}.$
(32b)

In a similar fashion, define the control law for the thalamus,
$\mathcal{N}_{T}$ by
$\mathbf{u}_{T}(t)=\mathbf{B}_{T}\mathbf{K}_{T}\mathbf{x}_{T}(t)+\mathbf{B}_{T}\bar{\mathbf{u}}(t)$
such that it satisfies

$\displaystyle\mathbf{B}_{T}\mathbf{K}_{T}$
$\displaystyle\leq-\mathbf{W}_{T}^{0,\mathrm{all}},$ (33a)
$\displaystyle\mathbf{B}_{T}\bar{\mathbf{u}}_{T}(t)$
$\displaystyle\leq-\mathbf{W}_{T,1}^{0,\mathrm{all}}\mathbf{x}_{1}(t)-\mathbf{c}_{T}^{0}.$
(33b)

Now, we permute the system variables and define corresponding timescale
matrices as follows

$\displaystyle\bar{\mathbf{x}}$
$\displaystyle=\begin{bmatrix}\mathbf{x}^{0}\\\
\mathbf{x}^{1}\end{bmatrix}\qquad\mathbf{x}^{0}=\begin{bmatrix}\mathbf{x}_{2}^{0}&\dots&\mathbf{x}_{N}^{0}&\mathbf{x}_{T}^{0}\end{bmatrix}^{\top}$
$\displaystyle\mathbf{x}^{1}$
$\displaystyle=\begin{bmatrix}\mathbf{x}_{1}&\mathbf{x}_{2}^{1}&\dots&\mathbf{x}_{N}^{1}&\mathbf{x}_{T}^{1}\end{bmatrix}^{\top}\qquad\tau^{0}=\text{diag}(\tau_{2},\dots,\tau_{N},\tau_{T})$
$\displaystyle\tau^{1}$
$\displaystyle=\text{diag}(\tau_{1},\tau_{2},\dots,\tau_{N},\tau_{T}).$

Substituting in the control laws (32) and (33), we have the following
controlled system dynamics

$\displaystyle\tau^{0}\dot{\mathbf{x}}^{0}$ $\displaystyle=-\mathbf{x}^{0}$
(34a) $\displaystyle\tau^{1}\dot{\mathbf{x}}^{1}$
$\displaystyle=-\mathbf{x}^{1}+[\bar{\mathbf{W}}\bar{\mathbf{x}}+\mathbf{c}(t)]_{\mathbf{0}}^{\bar{\mathbf{m}}},$
(34b)

where
$\bar{\mathbf{W}}=\begin{bmatrix}\bar{\mathbf{W}}^{10}&\bar{\mathbf{W}}^{11}\end{bmatrix},$
with $\bar{\mathbf{W}}^{11}$ as in (31), and $\mathbf{c}(t)$ the permutation
of the signal $\mathbf{c}_{1}(t)$ and the constants $\mathbf{c}_{i}$,
$i\in\\{2,\dots,N\\}\cup\\{T\\}$ corresponding to the permuted variables. Now,
we consider a ‘frozen’ version of the dynamics (34), in which we fix
$\mathbf{c}_{1}(t)$ to a constant $\bar{\mathbf{c}}_{1}$. By [19, Lemma A.1]
the frozen version of the dynamics (34) is GES to an equilibrium
$\mathbf{x}^{*}$, with $\mathbf{x}^{0}\to\mathbf{0}$, if $\dot{\bf y}=-{\bf
y}+[\bar{\mathbf{W}}^{11}{\bf y}+\mathbf{c}]_{\mathbf{0}}^{\bar{\mathbf{m}}}$
is GES to an equilibrium. By assumptions (ii) and (iii), along with [33,
Theorem 2], we have that $\rho(\bar{\mathbf{W}}^{11})<1$ and therefore, the
dynamics $\dot{\bf y}=-{\bf y}+[\bar{\mathbf{W}}^{11}{\bf
y}+\mathbf{c}]_{\mathbf{0}}^{\bar{\mathbf{m}}}$ is GES to an equilibrium, cf.
[19, Theorem IV.8]. Therefore the frozen version of (34) is GES to an
equilibrium using the control laws defined above. Given this, we can apply the
converse Lyapunov result given in Theorem .2 to get a Lyapunov function for
the dynamics (34) that satisfies the requirements of Theorem .1. The direct
application of this result then gives that selective inhibition and
recruitment (30) is achieved if $\left\|\dot{\mathbf{c}}_{1}(t)\right\|\to 0$
and is $\epsilon$-selectively inhibited and recruited if it is bounded but
does not tend to zero. ∎

Note that Theorem V.1 relies on the stability of the task-relevant dynamics of
each network layer when considered independently and the assumption that the
magnitude of the combination of thalamocortical and corticothalamic
interconnections does not exceed a certain stability margin. The latter
condition is consistent with neuroscientific observations: in fact, enhanced
corticothalamic feedback may result in pathological behavior [34]. In
particular, excessive corticothalamic feedback has been found to coincide with
epileptic loss of consciousness in absence seizures due to over-inhibition in
the cortical regions [35].

######  Remark V.2.

(Remote synchronization in star-connected brain networks): Remote
synchronization is a phenomenon observed in the brain in which distant brain
regions with similar structure synchronize their activity despite the lack of
a direct link [36]. This should then naturally arise in star-connected
networks if there is morphological symmetry between cortical regions, as this
topology directly shows regions without direct links. From (V-A), we note that
if any two cortical regions $\mathcal{N}_{i}$ and $\mathcal{N}_{j}$ have
identical task-relevant dynamics, i.e.,
$\mathbf{W}_{i,i}^{1,\mathrm{all}}=\mathbf{W}_{j,j}^{1,\mathrm{all}}$,
$\mathbf{W}_{i,T}^{1,\mathrm{all}}=\mathbf{W}_{j,T}^{1,\mathrm{all}}$ and
$\mathbf{c}_{i}^{1}=\mathbf{c}_{j}^{1}$, it follows that the equilibrium
points will satisfy $\mathbf{x}_{i}^{1^{*}}=\mathbf{x}_{j}^{1^{*}}$, meaning
that remote synchronization is achieved provided that the conditions of
Theorem V.1 are satisfied.

Remarkably, this conclusion seems to be independent of the particular dynamics
of the individual layers. In fact, the work [37] studies remote
synchronization in star-connected brain networks, cf. Fig. 1(b), with layer
dynamics given by Kuramoto oscillator dynamics,

$\displaystyle\dot{\theta}_{T}$
$\displaystyle=\omega_{0}+\sum_{i=1}^{N}K_{i}\sin(\theta_{i}-\theta_{0}-\xi)$
$\displaystyle\dot{\theta}_{i}$
$\displaystyle=\omega+A_{i}\sin(\theta_{0}-\theta_{i}-\xi),\qquad
i=1,\dots,N.$ (35)

According to [37], the outer cortical regions in star-connected brain networks
can remotely synchronize despite no direct links between the regions provided
the network dynamics satisfy conditions that parallel those required for the
star-connected linear-threshold networks studied above. In particular, to be
able to achieve remote synchronization, the network weights must satisfy
$A_{i}\geq(N-1)K_{i}$ for all $i\in\\{1,\dots,N\\}$. This condition guarantees
the existence of a locally stable equilibrium point, and is equivalent the
requirement of the matrices defining the task-relevant components of the
linear-threshold network being individually stable. $\bullet$

We conclude by noting that the thalamus, as a relay, can function as a
failsafe for the hierarchical thalamocortical network, allowing for selective
inhibition and recruitment even if corticocortical connections become damaged.
In fact, observe that the hierarchical topology where the matrices
$\mathbf{W}_{i,i-1},\mathbf{W}_{i,i+1}$ are equal to zero for all $i$ reduces
to the star-connected topology, while maintaining the timescale separation
between layers. Therefore, the star-connected topology with a hierarchical
timescale structure can be considered as a failsafe for the hierarchical
thalamocortical network. The next result provides conditions for selective
inhibition and recruitment for this topology.

######  Corollary V.3.

(Selective inhibition of a star-connected hierarchical thalamocortical
network): Consider a hierarchical star-connected thalamocortical network of
the form shown in Fig. 1(b) with timescales
$\tau_{1}\gg\tau_{2}\gg\dots\gg\tau_{N}$ and layer dynamics given by (5) and
(7). Without loss of generality let $\tau_{T}$ be such that
$\tau_{1}\gg\tau_{T}\gg\tau_{N}$ and let $a,b\in\\{1,\dots,N\\}$ such that
$\tau_{a}\gg\tau_{T}\gg\tau_{b}$ and $b=a+1$. Assume the stability assumptions
(17)-(21) for the reduced-order subnetworks are satisfied. Then, for
$i\in\\{1,\dots,N\\}\cup\\{T\\}$ and constants
$\mathbf{c}_{i}\in\mathbb{R}^{n_{i}}$ and
$\mathbf{c}_{T}\in\mathbb{R}^{n_{T}}$, there exist control laws
$\mathbf{u}_{i}(t)=\mathbf{K}_{i}\mathbf{x}_{i}(t)+\bar{\mathbf{u}_{i}}(t)$,
with $\mathbf{K}_{i}\in\mathbb{R}^{p_{i}\times n_{i}}$ and
$\bar{\mathbf{u}}_{i}:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq
0}^{p_{i}}$, such that the closed-loop system achieves selective inhibition
and recruitment (22).

The proof of the result is similar to that of Theorem IV.4, with differences
occurring in the constructed control laws on the basis that
$\mathbf{W}_{i,i+1}=\mathbf{W}_{i,i-1}=\mathbf{0}$ for all
$i\in\\{1,\dots,N\\}$. The loss of these connections plays a significant role
in the form of the control. In particular, the amount of feedforward control
coming from the thalamus to the cortical regions increases, due to the fact
that direct feedforward control between cortical regions is not possible. Fig.
3 illustrates Corollary V.3 on the star-connected network obtained by removing
the direct connections between cortical regions in the network of Fig. 2.

Figure 3: Trajectories of the network in Fig. 2 with the connections between
the cortical regions removed. The star-connected topology allows for
successful selective inhibition of the desired set of nodes despite the lack
of connections between cortical regions (cf. Corollary V.3), providing a
failsafe topology.

##  VI Quantitative Comparison of Cortical and Thalamocortical Networks

This section seeks to quantitatively illustrate ways in which the presence of
the thalamus might have a beneficial effect in the behavior and performance of
the dynamic models for brain networks adopted here. Fig. 3 has already
illustrated the failsafe role played by the thalamus in hierarchical
thalamocortical networks. Here we focus on two other beneficial impacts of the
thalamus that we have observed in simulation: the required control magnitude
to achieve selective inhibition and the convergence time in thalamocortical
networks versus cortical ones.

######  Example VI.1.

(Reduced average control magnitude in thalamocortical vs cortical networks):
We investigate the control magnitude required to achieve selective inhibition.
Control magnitude here refers to the aggregate of the inputs at all layers
integrated over time and averaged across trials. We consider hierarchical
pairs of thalamocortical and cortical networks, where the latter is obtained
by disconnecting the thalamus in the former. Fig. 4 shows that thalamocortical
networks require a lower control magnitude to achieve selective inhibition in
the cortical regions relative to the corresponding strictly cortical networks,
matching the intuition that they are easier to selectively inhibit due to the
thalamus impacting the cortical regions in an inhibitory fashion.

$\bullet$

Figure 4: Comparison of average control magnitude in cortical and
thalamocortical networks. The plot displays the control magnitude required to
selectively inhibit the bottom cortical layer averaged over $100$ random
thalamocortical networks and the corresponding cortical network in which the
thalamus is removed. Shaded regions correspond to the error bars. All networks
are composed of two three-node cortical regions. The top cortical layer has
two excitatory nodes and the bottom layer has only one. The thalamus is
composed of two inhibitory nodes. Each thalamocortical and cortical network
generated satisfies the assumptions of Theorem IV.4 and [20, Theorem IV.3],
resp., along with biological sign constraints. To make the required control
magnitude directly comparable, the thalamocortical and cortical networks are
inhibiting the same set of nodes within the bottom cortical layer.

######  Example VI.2.

(Convergence time of thalamocortical and cortical networks): Here we consider
the speed at which thalamocortical networks converge to an equilibrium as
another metric to evaluate the role of the thalamus. We compare the
convergence time for a cortical network with that of a thalamocortical network
maintaining the same cortical regions. While performing the comparison is
interesting as a function of multiple network parameters (e.g., network size,
layer size, ratio of excitatory-inhibitory nodes), we focus here specifically
on the thalamus and, in particular, on varying its timescale with respect to
the cortical regions in the model. Thalamocortical networks with varying
timescales are of particular interest due to their existence in the brain, as
even restricting only to the visual thalamus, the thalamus operates at both
slow and fast timescales [38]. Fig. 5 shows that thalamocortical networks have
faster average convergence time, with the margin between the two networks
decreasing as the timescale $\tau_{T}$ increases. This validates the the
beneficial role played by the thalamus, with faster thalamus dynamics (smaller
$\tau_{T}$) helping the cortical regions converge faster, leading to overall
decreased convergence time. $\bullet$

Figure 5: Convergence time of thalamocortical network with changing
timescales. The graph illustrates the average convergence time to the
equilibrium for $30$ randomly generated cortical and thalamocortical networks
with $20$ layers, each containing two excitatory and two inhibitory nodes. For
each simulation the initial condition is set to be a uniform distance away
from the equilibrium and the set of nodes to be inhibited is randomly
selected. The average convergence time for nodes within the thalamocortical
network is lower than for the cortical network. As the timescale $\tau_{T}$ of
the thalamus increases, the margin of improvement decreases.

##  VII Conclusions

We have studied the properties of both multilayer hierarchical and star-
connected thalamocortical brain networks modeled with linear-threshold
dynamics. Our primary motivation was understanding the role played by the
thalamus in achieving selective inhibition and recruitment of neural
populations. For both types of interconnection topologies, we have described
how the equilibria at each layer depends on the rest of the network and
identified suitable stability conditions. For hierarchical networks, these
take the form of GES requirements of the reduced-order dynamics of individual
layers. For star-connected thalamocortical network without a hierarchy of
timescales, these take the form of stability of the task-relevant dynamics of
each layer when considered independently and the magnitude of the combination
of thalamocortical and corticothalamic interconnections not exceeding a
certain stability margin. Future work will seek to analytically characterize
the robustness and performance of thalamocortical networks, study the role of
the thalamus in other cognitive tasks beyond selective attention (e.g., sleep
and consciousness, oscillations, and learning), and explore the impact of the
addition and deletion of neuronal populations (neurogenesis) in the
performance and expressivity of brain networks.

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|  Michael McCreesh received his B.A.Sc degree in Mathematics and Engineering
and his M.A.Sc in Mathematics and Engineering from Queen’s University,
Kingston, Canada in 2017 and 2019, resp. He is currently a Ph.D. student in
the Department of Mechanical and Aerospace Engineering at the University of
California San Diego. His current research interests include control theory
and its application to theoretical neuroscience, in particular the application
of dynamical systems to model brain networks.
---|---
|  Jorge Cortés (M’02, SM’06, F’14) received the Licenciatura degree in
mathematics from Universidad de Zaragoza, Zaragoza, Spain, in 1997, and the
Ph.D. degree in engineering mathematics from Universidad Carlos III de Madrid,
Madrid, Spain, in 2001. He held postdoctoral positions with the University of
Twente, Twente, The Netherlands, and the University of Illinois at Urbana-
Champaign, Urbana, IL, USA. He was an Assistant Professor with the Department
of Applied Mathematics and Statistics, University of California, Santa Cruz,
CA, USA, from 2004 to 2007. He is currently a Professor in the Department of
Mechanical and Aerospace Engineering, University of California, San Diego, CA,
USA. He is the author of Geometric, Control and Numerical Aspects of
Nonholonomic Systems (Springer-Verlag, 2002) and co-author (together with F.
Bullo and S. Martínez) of Distributed Control of Robotic Networks (Princeton
University Press, 2009). He is a Fellow of IEEE and SIAM. His current research
interests include distributed control and optimization, network science,
nonsmooth analysis, reasoning and decision making under uncertainty, network
neuroscience, and multi-agent coordination in robotic, power, and
transportation networks.
---|---

For completeness, here we include two results on the stability of slowly
varying nonlinear systems to a continuum equilibria, generalized from [39,
Theorem 3.1] to the case of exponential stability. Let

$\displaystyle\dot{\mathbf{x}}=f(\mathbf{x}(t),\mathbf{d}(t)),$ (36)

where $\mathbf{x}(t)\in\mathbb{R}^{n}$ and $\mathbf{d}(t)\in\mathcal{D}$, and
$\mathcal{D}$ is a compact subset of $\mathbb{R}^{m}$. We assume $f$ is
continuous on $\mathbb{R}^{n}\times\mathcal{D}$ and is locally Lipschitz in
both $\mathbf{x}$ and $\mathbf{d}$. We further consider the ‘frozen’ version
of the system (36) with a fixed parameter $\mathbf{d}$,

$\displaystyle\dot{\mathbf{x}}=f(\mathbf{x}(t),\mathbf{d}),\quad\mathbf{d}\in\mathcal{D}.$
(37)

We denote the solution to (37) for each initial condition
$\mathbf{x}(t_{0})=\mathbf{x}_{0}$ and $\mathbf{d}\in\mathcal{D}$ by
$\mathbf{x}(t,\mathbf{x}_{0},\mathbf{d})$. Let $\mathcal{A}$ be a forward
invariant set of the system (37).

######  Theorem .1.

(Exponential Stability of Slowly Varying Nonlinear Systems): Consider the
nonlinear system (36) and assume there exists a continuously differentiable
function $V:\mathbb{R}^{n}\times\mathcal{D}\rightarrow\mathbb{R}$ such that

$\displaystyle k_{1}|\mathbf{x}|_{\mathcal{A}}^{2}$ $\displaystyle\leq
V(\mathbf{x},\mathbf{d})\leq k_{2}|\mathbf{x}|_{\mathcal{A}}^{2}$ (38a)
$\displaystyle\frac{\partial
V(\mathbf{x},\mathbf{d})}{\partial\mathbf{x}}f(\mathbf{x},\mathbf{d})$
$\displaystyle\leq-k_{3}|\mathbf{x}|_{\mathcal{A}}^{2}$ (38b)
$\displaystyle\left|\frac{\partial
V(\mathbf{x},\mathbf{d})}{\partial\mathbf{d}}\right|$ $\displaystyle\leq k,$
(38c)

for all $\mathbf{x}\in\mathbb{R}^{n}$ and $\mathbf{d}\in\mathcal{D}$, with
$k_{1},k_{2},k_{3}$ and $k$ nonnegative constants. If
$\left\|\dot{d}(t)\right\|$ is uniformly bounded in time, then there exist
constants $\gamma,\lambda$ and $T$ such that

$\displaystyle|\mathbf{x}(t,\mathbf{x}_{0},\mathbf{d})|_{\mathcal{A}}$
$\displaystyle\leq\gamma\left\|\mathbf{x}_{0}\right\|e^{-\lambda(t-t_{0})}\quad\forall~{}t_{0}\leq
t\leq t_{0}+T$ (39a)
$\displaystyle|\mathbf{x}(t,\mathbf{x}_{0},\mathbf{d})|_{\mathcal{A}}$
$\displaystyle\leq\frac{k_{3}}{k_{2}}|\mathbf{x}_{0}|_{\mathcal{A}}\quad\forall~{}t\geq
t_{0},$ (39b) If $\lim_{t\to\infty}\left\|\dot{\mathbf{d}}(t)\right\|=0$, then
the system is exponentially stable.

The proof follows a similar line of reasoning as in [39, Theorem 3.1] with
slight modifications in the use of stability results from [40] to account for
exponential stability. We now provide a converse Lyapunov result, modified
from [39, Theorem 3.2], complementary to the above result.

######  Theorem .2.

(Converse Lyapunov Theorem for Slowly Varying Nonlinear Systems): If, for
every fixed $\mathbf{d}\in\mathcal{D}$, the nonlinear system (36) is GES to an
equilibrium, then there exists a Lyapunov function that satisfies (38a), (38b)
and (38c).

The proof of this result is identical to [39, Theorem 3.2], with the
assumption of GES giving the desired function form in the final step.