energy sources are ATP, GTP and SAM [210, 211]. Furthermore, since the
landscape topography and flux in determining the functions and stability of
the functional states can be globally quantified, one can explore which
underlying neural network interactions or specific types of neurons these
global dynamic and thermodynamic measures are sensitive to. Through such
global sensitivity analysis, the key types of neurons and neural interactions
which are critical to the states’ stability and switching dynamics could be
identified, and hopefully this global system perspective could be used to
design strategies for perturbations of brain function that could mitigate
pathologies caused by neurological disorders.

##  VI Metastability, learning and neural network function

###  VI.1 Anatomical and functional underpinning of metastable dynamics

As reviewed in Sec. V.2, recent models point to the presence of neural
clusters as originators of metastable activity. In this section, we will
review evidence from experimental neuroscience suggesting that clustered
architectures predicted by the model can indeed be found in cortical networks.

####  VI.1.1 Preferential patterns of connectivity in cortex

In cortical circuits, evidence for the presence of preferentially connected
clusters of neurons has been reported using either paired recording
electrophysiology, which allows for the direct measurement of the properties
of a connection between two neurons [212], or using a variety of approaches
for circuit mapping including uncaging of neurotransmitters [213] or
optogenetic-assisted circuit mapping [214]. At the anatomical level, studies
in sensory cortex have found that excitatory neurons in the superficial layer
have a higher probability of being recurrently connected when they share a
common input [213], and that the probability of connection is higher when the
magnitudes of the incoming inputs are comparable [215]. While these results
suggest the presence of preferential connectivity, the data were obtained _ex
vivo_ , therefore there is no evidence that such groups of connected neurons
respond to similar stimuli. At the functional level, Ref. [216] reported that
excitatory neurons responding to a specific feature of a visual stimulus were
more likely to be connected, further supporting the presence of clusters of
connected neurons. However, whether such connectivity represents clusters of
neurons that participate in the functional aspects of neural processing
remains controversial. Indeed, the connection probability reported by [216] is
comparable to that reported by previous studies that simply measured the
probability of finding any connected pairs of neurons in _ex vivo_
preparations [217, 218]. Thus, the higher connection probability may rely on
proximity, not so much on responsiveness to a common functional feature.

####  VI.1.2 Metastability as a mechanism for multiplexing

An additional level of complexity arises when considering that cortical
neurons can respond to multiple features of a stimulus or encode multiple
cognitive variables, a property sometimes called multiplexing. In the visual
cortex, as well as auditory and somatosensory cortices, analysis of circuit
organization and functional responsiveness has focused primarily on neurons
responding to specific features; very little work investigated multiplexing.
Coding for complex variables is often ascribed to high order cortical areas
and areas of the cerebral cortex thought to be involved in bringing together
sensory and cognitive information including attention, expectation, and
reward. This hierarchical view of cortical circuits considers neurons as
structures specialized to respond to specific stimulus features, process these
features, and then transmit this information to neurons in the next
hierarchical order, which in turn will bring together information they
received from other circuits and encode a percept.

Recent work in the rodent GC provides an alternative view of multiplexing: via
metastability. First, neurons in GC are indeed involved in multiplexing: the
same neuron can respond to multiple taste stimuli and to cues promoting
expectations, and can participate in driving decisions [219, 220, 221, 222,
223, 224, 60]. Second, taste-related metastable activity in GC is compatible
with ongoing metastable dynamics (Sec. IV), a fact that the spiking network
model of Sec. V.2.1 predicts to rely on the clustered architecture of the
network, which provides a common mechanism for both taste-related and ongoing
metastability. Can the latter also result in multiplexing capabilities?
Metastable dynamics in GC may indeed facilitate the encoding of multiple
variables by leveraging the temporal dynamics of the neurons’ own activity:
this could dynamically rearrange their participation in coding for one or the
other variable by visiting different metastable states – a form of dynamic
population coding [225].

Further theoretical work is required to evaluate the plausibility of this
proposal, but it is also crucial to find experimental evidence for its basic
ingredient: recurrent neural clusters interacting so as to produce metastable
activity. Although some indirect evidence of cortical clusters has been
reported [226], ideally one would like to directly measure neural connections
and their susceptibility to being modified by experience to form functional
clusters.

####  VI.1.3 Methods for mapping cortical circuits

Paired recording approaches to local circuit mapping are highly effective in
identifying specific connections [227, 212], but suffer from a number of
limitations. With paired recordings, a small number of neurons can be
simultaneously recorded. The advantage of this approach is in allowing
detailed identification of recorded neurons and full control over their
membrane properties by using patch clamp electrophysiology. The resolution is
on suprathreshold as well as subthreshold events underlying neuron-to-neuron
communication, making it highly suitable for assessing connectivity. The
primary limitation is that the analysis of connectivity works well for near-
neighbor coupling, but is less effective in assessing less spatially
restricted connectivity. Other approaches like channelrhodopsin-assisted
circuit mapping [214] facilitate the identification of connectivity maps over
a larger spatial scale, and can resolve subthreshold and suprathreshold
events, but lose precise control over cell-to-cell connectivity as they can
recruit both direct and indirect inputs to the recorded neurons. Both paired
recordings and channelrhodopsin-assisted circuit mapping require the use of
_ex vivo_ preparations, thus they are not ideal for determining whether or not
connected neurons share common functional properties _in vivo_.

Multielectrode recordings in behaving animals allow for the simultaneous
recording of spiking activity from multiple neurons and have been instrumental
for the analyses that identified metastable dynamics in cortical circuits [11,
12, 7, 15, 18, 54, 55]. This experimental approach provides spontaneous and
stimulus-driven activity in animals that are engaged in a variety of tasks.
Correlations of activity across groups of neurons facilitates the
identification of functional clusters, although it cannot offer sufficient
resolution to determine whether neurons that share common functional
properties are recurrently connected or share a common input. If large scale
recordings are needed to add a spatial component to the analysis of metastable
dynamics, it is possible to increase the number of electrodes or recording
sites, an approach that facilitates analysis of spiking activity along the
vertical axis (depth) of the cortical mantle [100], or to use calcium imaging
for assessing neural activity along the horizontal axis [228]. While analysis
techniques are well-established for identifying metastable dynamics in spiking
activity, there are technical caveats that need to be resolved to extract
these dynamics from the much slower calcium signals. More generally, there is
a need to develop approaches to reliably link spiking activity to the activity
detected with fluorescent signals emitted by calcium indicators [229, 230,
231, 232, 233]. Solving this set of technical problems will facilitate
extraction of metastable states at the slower time scales typical of calcium
signals and provide experimental evidence for the theorized link between
metastability and cluster activation.

###  VI.2 Synaptic plasticity, learning and metastable dynamics

Cortical circuits subserving metastable dynamics may be genetically codified
and partially formed at birth, but given their putative role in coding for
sensory and cognitive processes (Sec. IV.1), it is assumed that they would
fully develop through experience via a process of learning and plasticity. As
indirect evidence, changes in metastable dynamics due to learning has been
reported in the rodent gustatory cortex [72]. Theoretical models suggest that
metastability is generated and maintained by a clustered network architecture
[16, 17, 18, 19, 20, 8, 21], and that neural plasticity may be involved in
establishing such architecture and modulating it [185, 234]. No study to date
has directly assessed the biological correlates of such predictions. In the
next section, we discuss experimental evidence from studies of experience-
dependent circuit refinement and learning that may provide evidence for the
involvement of plasticity in establishing, maintaining, or modulating
clustered connectivity. While none of the experimental findings we present
below is directly related to metastability, they help make inferences
regarding neural circuit organization and its modulation by experience and
learning that seems required to produce metastable dynamics in models. The
hypotheses we discuss are speculative at this stage, but provide clues on
whether or not a model of functional clustered connectivity is biologically
plausible, on how clustered connectivity may originate during postnatal
development, and plasticity and learning may affect it.

Learning, memory formation and storage and other adaptations induced by
experience are known to induce neural plasticity, which can manifest in
changes in the efficacy of transmission of signals between neurons,
alterations of neural connectivity or other modulation that affect a neuron’s
input/output function [235, 236, 237]. Changes in the efficacy of neuron-to-
neuron communication and circuit configurations that arise as a consequence of
neural plasticity have the potential of destabilizing circuit dynamics.
Indeed, many models of network function fail to reproduce network dynamics
unless normalizing functions are built into the model, as we will discuss in
Sec. VI.3. However, healthy brain circuits clearly have the capability to
preserve their activity throughout life, indicating that mechanisms are in
place to maintain the system within a functional working range. How such
dynamic patterns of activity may be maintained in the face of changes in
neural activity driven by cognitive processes is an open experimental and
theoretical question.

####  VI.2.1 Circuit refinement by synaptic plasticity

During postnatal development, neural plasticity is thought to help refine
circuits that allow for the establishment of metastable dynamics. For example,
visual experience is necessary for establishing preferential connectivity of
neurons responding to a common property of an incoming visual input [238].
These results, viewed in the context of the clustered network architecture
that supports metastability (Sec. V.2.1), suggest that an initial set of
activity-dependent patterns may be needed to establish the circuit
connectivity that facilitates the generation of metastable dynamics. Several
studies have also reported evidence for the modulation of the efficacy of
synaptic transmission in visual cortical circuits in response to the onset of
visual experience [217, 218, 239]. Plastic changes like those reported in
visual cortex have been observed in other sensory areas including
somatosensory [240] and auditory cortex [241, 242].

####  VI.2.2 Memory formation and Hebbian learning

In addition to circuit refinement, synaptic plasticity is thought to be the
neural underpinning of learning and memory. Changes in the efficacy of
synaptic transmission between neurons can contribute to forming and storing
new memories. A well established theory proposed by Donald O. Hebb in 1949
states that the formation of a memory requires the strengthening of synaptic
transmission between neurons [243]. Extrapolated to the scale of entire
circuits, this theory provided the basis for the idea that learning leads to
the strengthening of synaptic transmission between groups of neurons that have
all been activated by specific patterns of incoming activity [244]. Such
groups of neurons could be easily reactivated if the learned pattern or
stimulus is presented again after learning, possibly speeding up the neural
coding of stimulus properties. These co-activated groups of neurons are
thought to participate in the formation of what is sometimes defined as an
‘engram’, a signature circuit for a memory [245] (see also Sec. VI.4.1).
Storing of multiple memories, according to this idea, would occur by formation
of multiple, distinct or partially overlapping engrams over time [246].

This idea is the natural evolution of earlier proposals based on the concept
of ‘Hebb assembly’ [39, 40, 247, 159, 158] which can be generalized in several
ways. One such generalization is the inclusion of inhibitory neurons.
Theoretical work has proposed the possibility that inhibitory engrams may be
recruited following the formation of a memory. The idea behind the inhibitory
engram relies on the possibility that plasticity at a selected group of
neurons that release GABA, the inhibitory neurotransmitter, may offer a
counterbalance to the increased excitation and readjust circuit excitability
within a sensitive and stable range [248]. No experimental evidence for the
existence of inhibitory engrams is currently available, although there is
extensive evidence that GABAergic neurons can change their efficacy in an
activity-dependent fashion in response to patterned activity [249, 250, 251].
The study of the functional significance of inhibitory synaptic plasticity has
just began. Recent work demonstrated that a possible role for this form of
plasticity is to determine the sign of the change in synaptic efficacy at
converging excitatory inputs [252], possibly providing constraints on how
activity can modulate circuit excitability and connectivity. When considering
these results in the context of the clustered theoretical model for metastable
dynamics, one may speculate that functional connectivity of neurons recruited
for the formation of a memory suggests the presence of a clustered
architecture. Inhibitory circuits may contribute to separating the functional
clusters, facilitating the formation and stabilization of memories. Studies on
memory engrams may therefore support a role for plasticity in establishing and
maintaining the cluster architecture that underlies metastable dynamics as
predicted by the theoretical models (see Sec. VI.2).

####  VI.2.3 The potential link with cluster formation

The evidence discussed above provides grounds to speculate that experience-
dependent potentiation of synapses among neurons belonging to the same engram
may be part of the mechanism by which neural clusters are formed. Although it
is natural to postulate such a relationship, much experimental work is
required to clarify the link between cluster-based metastability and the
formation of engrams via synaptic plasticity. Formation of engrams via Hebb-
like synaptic plasticity is expected to partition cortical networks in
subpopulations of neurons, which in turn could give rise to metastability via
the mechanism reviewed in Sec. V.2.1. However, the temporal and spatial
constraints for co-activation of groups of neurons during the formation of
memories are unknown. The identification of engrams so far is limited to the
co-expression of genes known as ‘immediate early genes’ that are used as a
proxy for neural activity [253], therefore it does not provide any information
for a role of these groups of neurons in circuit dynamics, nor on the activity
patterns that may have led to their co-activation.

While experiments can tell us a great deal about the learning rules and the
induction of plasticity, the investigation of how changes in the efficacy of
synaptic transmission can be integrated in a complex neural circuit without
rapidly leading to instability can only be explored with theoretical models
that can provide predictive hypotheses. In the case of metastable dynamics, if
its maintenance is functional to the operation of a neural network, it is
fundamental to determine how the implementation of learning rules observed
experimentally may affect neural activity and circuit interactions, to
determine how these learning rules modulate state transitions, and what are
the neural and circuit requirements that preserve metastable dynamics in the
face of experience and learning. In the next section, we review some of the
theoretical efforts that have been made in this direction.

###  VI.3 Modeling cluster formation by experience-dependent plasticity

Clustered network models have been able to reproduce metastable dynamics
observed experimentally and make useful predictions about possible circuit
configurations that generate these dynamics. This is usually accomplished by
tuning the synaptic weights and other parameters and requires some degree of
ingenuity; mean field theory can aid this process, as reviewed in Sec. V.2.1.
A goal of ongoing research is to understand how the clustered structure
underlying metastable activity might develop and then be maintained in the
brain.

Synaptic plasticity is the most likely mechanism for generating metastable
clustered neural circuits. Recent theoretical efforts have focused on
variations of spike-timing dependent plasticity (STDP) combined with
inhibitory plasticity and homeostatic mechanisms for the generation of neural
clusters. The self-organization of neural circuits via STDP has been studied
in a number of works (see e.g. the Discussion of [254] and references
therein); here we focus on a couple of recent works that are highly relevant
to the development of metastable dynamics in neural circuits [185, 234].

The basic STDP rule produces a synaptic change based on the relative timing of
pre- and postsynaptic action potentials [255, 256], though a dependence on the
postsynaptic membrane potential can also be included [218]. However, it has
long been recognized that this type of plasticity would quickly saturate the
synaptic weights of co-activating neurons (e.g., [257, 258, 259]), resulting
in instabilities or, for finite synaptic weights, in the potential formation
of a single, giant cluster of neurons. This problem can be countered by
concurrent inhibitory plasticity [260], which acts to reduce the firing rates
of excitatory neurons, reducing the interference among clusters and slowing
down their formation.

Let us illustrate these ideas by describing the model of [185]. The network
comprises adaptive exponential integrate-and-fire neurons, a variation of the
LIF model neuron described in Sec. V.2.1 (whose details are not relevant for
the following). The synaptic weights between two excitatory neurons, $J_{ij}$,
where $i$ is the postsynaptic neuron and $j$ is the presynaptic neuron,
changes according to

$\frac{d}{dt}J_{ij}(t)=-A_{d}s_{j}(t)[u_{i}(t)-\theta_{d}]_{+}+A_{p}x_{j}(t)[V_{i}(t)-\theta_{p}]_{+}[v_{i}(t)-\theta_{d}]_{+}.$
(33)

Here, $V_{i}$ is the membrane potential and $u_{i},v_{i}$ are low-pass filters
of $V_{i}$ with different time constants ($10$ and $7$ ms, respectively);
$s_{j}(t)=\sum_{k}\delta(t-t_{k}^{j})$ is the presynaptic spike train and
$x_{j}(t)$ is a low-pass filter of $s_{j}(t)$ with a time constant of $15$ ms;
$\theta_{d,p}$ is a threshold for inducing long term depression (LTD) or long
term potentiation (LTP), respectively, with $\theta_{d}<\theta_{p}$; $[z]_{+}$
is the threshold linear function, equal to $z$ if $z\geq 0$ and zero
otherwise; and $A_{d,p}$ are the strength of LTD and LTP, respectively, with
$A_{d}<A_{p}$.

The first term in the right hand side of Eq. 33 causes LTD when the
presynaptic neuron fires an action potential and the (filtered) postsynaptic
neuron activity $u_{i}$ is small but above $\theta_{d}=-70$ mV (equal to the
neuron’s resting potential). For large values of the membrane potential
($V_{i}>\theta_{p}=-49$ mV), the second term will dominate, leading to LTP.
Synapses between neurons that co-activate in response to the same stimulus
will undergo LTP, while synapses onto neurons that remain inactive will
undergo LTD, in fairly Hebbian fashion (no change occurs if the presynaptic
neuron is inactive).

This learning rule leads to synchronized oscillations [185]. To avoid this,
inhibitory plasticity is implemented on the synapses from inhibitory to
excitatory neurons:

$\displaystyle J_{ij}$ $\displaystyle\leftarrow$ $\displaystyle
J_{ij}+\eta(y_{i}(t)-2r_{0}\tau_{y})\quad\mbox{if the presynaptic neuron
fired,}$ $\displaystyle J_{ij}$ $\displaystyle\leftarrow$ $\displaystyle
J_{ij}+\eta y_{j}(t)\hskip 48.36958pt\mbox{if the postsynaptic neuron fired}.$

Here, $J_{ij}$ is the absolute value of the synaptic weight, $y_{i,j}(t)$ is
the low-pass filter of the postsynaptic or presynaptic spike train,
respectively (with time constant $\tau_{y}=20$ ms), $\eta$ is the learning
rate, and $r_{0}=3$ spikes/s is a target firing rate to which the inhibitory
plasticity attempts to balance, homeostatically, the postsynaptic neuron. The
inhibitory-to-excitatory weights are further kept within a bounded range.

However, even this combination of excitatory and inhibitory plasticity seems
insufficient to generate stable clusters, unless a process of ‘synaptic
normalization’ is also added [257, 261, 262]. This is a procedure in which, at
regular intervals, the excitatory synaptic weights onto the same postsynaptic
neuron are reduced by a fixed amount (this will maintain a constant row sum in
the synaptic matrix). This procedure depends on the current values of all the
synaptic weights and it is not clear how it could occur in real synapses: as
synapses do not have a way to communicate their strengths to one another,
learning rules should be local, i.e., they should make use of information
coming only from the pre- and postsynaptic neuron [218] or from global
neuromodulatory signals [263, 264] (see e.g. Ch. 10 of [265] for theoretical
implications of locality). The timing of synaptic normalization is also
troublesome, since a mechanism that would inform all synapses in a circuit to
normalize themselves based on an internal clock is not known to exist
(although randomizing the times at which renormalization occurs would probably
achieve the same goal [266]). However, there is evidence that neurons can
sense their level of activity within the network and readjust their activity
state to remain within a functional range [267], or sense network activity and
engage distinct forms of plasticity to readjust it within a functional range
[268].

As experimental work investigating the functional implications of homeostatic
regulation of circuit excitability is still ongoing, identifying biophysically
plausible means of normalizing all synapses in a circuit to preserve circuit
dynamics can at the moment be explored only in theoretical models. If well
conceptualized, these models have predictive power and can provide strong
working hypotheses for future experiments. The recent models reviewed above
have a good degree of biological plausibility and have taught us many lessons,
including: (i) once clusters are formed, metastable activity generates
patterns of activity similar to those experienced by the neurons during
training. By mimicking the conditions occurring during training, metastable
activity can coexist with ongoing plasticity [185]. (ii) After training, the
clustered structure of the network can be further modified by training with
new stimuli while preserving metastability. (iii) Inhibitory plasticity is not
strictly necessary for the formation of stable clusters. If the correct level
of inhibitory activity is set prior to training, excitatory plasticity may be
sufficient. However, inhibitory plasticity provides an adaptive mechanism to
produce the necessary amount of inhibition to render learning stable [234].

###  VI.4 Landscape modifications induced by learning

In Sec. V.2.3 we have introduced the non-equilibrium landscape and flux
framework for general neural networks. As discussed there, local minima in
neural network systems can significantly influence their dynamical behaviors
by inducing sequences of metastable states. In fact, metastable dynamics
induced by external stimuli, changes in synaptic properties and ubiquitous
noise play a crucial role in different neural circuit functions. Below, we
discuss the relationship between network dynamics and their corresponding
neurobiological functions including associative memory, working memory,
decision making, and fear response. The ability to produce the relevant
dynamics can be predicted by the model landscape, which in turn can be
modified by learning. We shall focus on two critical functional properties of
neural systems that at first glance appear incompatible: robustness and
sensitivity. As we shall see, these properties are closely associated with
metastable dynamics.

####  VI.4.1 Associative memory

Associative memory is the ability to form memories that associate two stimuli,
for example, a sound or light cue and a footshock in the case of fear
conditioning. Once the association is learned, it can be retrieved with a
partial cue close to that information. As discussed in Sec. VI.2.2, it is
widely believed that associative memories are formed through the induction of
synaptic plasticity and stored in synapses that have been modified during
learning. Theoretically, the memories are represented by attractors of the
network dynamics. Early modeling studies of associative memory have often
focused on systems with symmetric couplings [40, 189], i.e., the connections
between pairs of neurons satisfy $J_{ij}=J_{ji}$. In this case, memory states
are local minima of a global energy function $E$. Once a partial cue about the
desired memory is presented, the system is put in the initial state that is
near the basin of attraction of the valley corresponding to that desired
memory. Thus, memory retrieval can be performed by following the gradient of
the computational energy $E$ (see Fig. 10A).

In real cortical networks synapses are not symmetric [269], a property that
can be related to learning and recalling sequences rather than single items.
Symmetric networks, whose dynamical behaviors are determined by purely
potential energy, lack the ability to retrieve a temporally ordered sequence
of memories using a single recalling input. Such a computation for temporal
association can be achieved in asymmetric circuits through appropriate
learning rules, which result in rapid transitions between quasi-stable states
that represent individual memories [41, 42, 270]. For general asymmetric
circuits, the energy function $E$ defined in the symmetric cases is no longer
a Lyapunov function. It cannot describe collective computational properties
such as temporal association. Fortunately, the intrinsic potential
$\phi_{0}(\bm{x})$ we introduced in Sec. V.2.3 is always a Lyapunov function
regardless of the symmetry of the synaptic weights. The dynamics of general
networks are dominated by both the nonequilibrium potential landscape related
to the steady-state probability distribution and the steady-state rotational
curl probability flux [23, 24]. The flux breaking the detailed balance can
provide the main driving force for transitions between states and
oscillations.

Fig. 10(B) and (C) shows landscapes of Lyapunov function $\phi_{0}(\bm{x})$
for symmetric and asymmetric circuits, respectively. Similar to the Hopfield
symmetric network model, we can see multiple attractors in Fig. 10(B). For
asymmetric networks, sequential ‘oscillatory’ motion can occur and the
underlying potential landscape has a Mexican-hat topography (Fig. 10(C)).
After being attracted down to the oscillation ring (white arrows represent the
driving force from the negative gradient of the potential landscape), the
system is mainly dominated by the curl flux shown by the green arrows.
Different memories can be associated by the flux, which may induce ordered
sequences of memories, due to the asymmetry resulting from specific learning
procedures. This produces associations among memories through the continuous
attractor (oscillation ring) which is driven by the flux due to the asymmetry
of the synaptic connections.

Figure 10: (A) Schematic diagram of the energy function landscape of the
Hopfield network. (B) The potential landscape $\phi_{0}(\bm{x})$ of a
symmetric neural network. (C) The potential landscape $\phi_{0}(\bm{x})$ as
well as the corresponding force for an asymmetric neural circuit: the green
arrows represent the flux, and the white arrows represent the force from the
negative gradient of the potential landscape. Reproduced from H. Yan, L. Zhao,
L. Hu, X. Wang, E. Wang, and J. Wang, Proc Natl Acad Sci U S A 110, E4185–94
(2013). Copyright 2013 the authors. [23]

Oscillatory patterns of neural activity are widely distributed in our brain
[271, 272, 273, 274, 275]. Oscillations may play a role in various aspects of
memory including spatial representation and memory maintenance [276, 277]. In
addition to the context of storage and associative recall of information, the
present approach can be useful in understanding mechanisms for generating
rhythmic motor patterns (such as swimming and locomotion) by central pattern
generators (CPG), the synchronization among different groups with coherent
oscillations in cognitive functions such as and physiological rhythm
regulations [270, 274, 278, 279, 280, 281, 282]. An important example of this
class of problems is the cycling of sleep phases. The rhythmic REM/non-REM
cycle in human sleep is regulated by the activation-repression of two neural
populations [281, 282]. A detailed description of the REM/non-REM sleep rhythm
with the landscape and flux approach is shown in [23]. A global sensitivity
analysis based on the global topography of the landscape shows the effects of
key factors such as the release of the neurotransmitters acetylcholine and
norepinephrine on the stability and function of the system.

####  VI.4.2 Working memory and decision making

Actively holding information online for a brief period of time (seconds) is an
important ability of the brain. This capability is a part of working memory
(WM), which is used for tasks such as planning, organizing, movement
preparation and decision-making [283, 284, 285, 286]. In contrast to long-term
memory, which requires structural changes in neural circuits and in the
connections between neurons, the mechanisms underlying working memory are
believed to depend on persistent neuronal activity [287, 288, 289]. In
general, positive reverberation driven by recurrent synaptic excitation in
interconnected neural clusters can work as the basic principle for generating
persistent activity. Triggered by incoming signals, working memory circuits
can sustain an elevated firing even after the inputs are withdrawn. As
discussed in Sec. V.2.1 (see e.g. Fig. 6A), dynamical models with local
feedback excitation between principal neurons that are controlled by global
feedback inhibition can exhibit multiple attractor states (each coding a
particular memory item) that coexist with a background (resting) state. This
is illustrated in Fig. 11 below for a circuit model of WM with two excitatory
populations.

Figure 11: (A-B) The schematic diagram of the circuit model for working memory
(WM). The model comprises two selective, excitatory populations, labeled 1 and
2. Each excitatory population is recurrently connected and inhibits each other
through a common pool of inhibitory interneurons. (C-D) The schematic diagrams
for the WM during different phases in a WM task. (E-H) The corresponding
potential landscapes in the (S1; S2) state space during different phases. The
dimensionless quantities S1 and S2 are average synaptic gating variables of
the two selective populations, which can represent the mean population
activities. The label ‘r’ indicates the attractor representing the resting
state. The attractors representing the target-related and distractor-related
memory state are labeled with ‘m1’ and ‘m2’, respectively. Reproduced from H.
Yan and and J. Wang, PLoS. Comput. Biol.16(10): e1008209 (2020); licensed
under a Creative Commons Attribution (CC BY) license. [202]

Although robustness in the face of small perturbations or noise is an
important requisite for a working memory system, the transient properties of
the system are also important [287, 290, 64]. Sometimes, it is needed to tilt
the balance in favor of increased flexibility rather than robustness depending
on environmental conditions and/or behavioral task demands, e.g., a foraging
task in which an animal uses visual input to catch a prey [291]. In addition
to robust maintenance of memory states, activity should be reset quickly when
there is a novel sensory cue that needs to be stored. In other words, a
working memory system should have the properties of robustness against
fluctuations while being very sensitive to incoming stimuli. In recent
theoretical works, this fundamental contradiction can be achieved by global
inhibitory connections, where the system can exhibit structurally stable
dynamics with fixed stimulus and qualitatively change its dynamics if the
stimulus is changed[202]. The non-equilibrium potential landscape and flux
approach has been applied to a biophysically based working memory model
composed of integrate-and-fire neurons through a mean-field approach, which
can reproduce most of the psychophysical and physiological results in delayed
response tasks [202, 286, 174, 127]. This model comprises two selective,
excitatory populations, labeled 1 and 2, whose dynamics are described by the
following equations:

$\frac{ds_{i}}{dt}=-\frac{s_{i}}{\tau_{s}}+\left(1-s_{i}\right)\gamma
f\left(I_{i,tot}\right),~{}i\in\\{1,2\\},$ (34)

where $s_{i}$ is the average gating variable of neural population $i$, which
can represent the mean population activities, and $r_{i}=f(I_{i,tot})$ is the
corresponding firing rate, a function of total input current $I_{i,tot}$,
$\tau_{S}$ is the gating timescale, and $\gamma$ is the kinetic parameter that
controls the rate of saturation of $s_{i}$. The robustness of working memory
was quantified by the underlying landscape topography (barrier heights) and
the corresponding mean transition time for varying recurrent excitations and
mutual inhibitions, shown in Fig. 12 [202]. A combination of both increased
self-excitation and mutual inhibition can enhance flexibility to external
signals without significantly reducing robustness to random fluctuations. The
key element of the underlying mechanism for achieving good performance in
working memory is the emergence of a new intermediate state with larger energy
consumption.

Cortical areas that are engaged in working memory, such as the prefrontal
cortex, are also involved in other cognitive functions such as decision
making, selective attention, and behavioral control [284, 286, 127, 292, 293].
In the two-choice visual motion discrimination task, trained monkeys were
presented for a few seconds with a pattern of randomly moving dots and asked
to make a decision regarding the direction of motion by saccadic eye movement
[293, 294, 295, 296]. Models originally developed for working memory can
account for decision-making processes. The only difference between a working
memory simulation and a decision simulation is that in a delayed-response task
only one stimulus is presented while for a perceptual discrimination task
conflicting sensory inputs are fed into competing neural subpopulations in the
circuit (Fig. 13A) [154, 286]. The external sensory input to selective neural
population in random dots motion tasks is determined by the motion coherence,
which is used to indicate the degree of direction bias of moving dots (Fig.
13B-C).

Figure 12: (A-F) The robustness against random fluctuations during the
maintenance phase. (A-C) The potential landscapes for different self-
excitations $J_{+}$. (D) The schematic diagram of the barrier heights on the
corresponding potential landscapes for increasing $J_{+}$. (E-F) Robustness of
WM against random fluctuations as a function of self-excitations $J_{+}$ and
mutual inhibition $J_{-}$ through quantifying the corresponding barrier height
and the mean first passage time across the barrier. (G-L) The robustness
against distractors during the maintenance phase. Reproduced from H. Yan and
J. Wang, PLoS. Comput. Biol.16(10): e1008209 (2020); licensed under a Creative
Commons Attribution (CC BY) license. [202]

With the landscape approach, the decision-making processes can be quantified
with the optimal paths from the undecided attractor states to the decided
attractor states, which are identified as basins of attraction on the
underlying landscape (Fig. 13C) [24, 297]. In addition to the speed-accuracy
tradeoff discussed in previous decision-making studies, actually there is a
speed-accuracy-dissipation tradeoff. When additional signals are presented,
reasonable accuracy performance can be reached with minimum dissipation cost
and fast decision time [297]. Moreover, making decisions is often accompanied
by situations in which we change our minds. The mechanism of mind changes may
be closely associated with new, intermediate states that emerge when large
inputs are presented (Fig. 13(D-G)). The initial incorrect choice is more
likely to be changed due to the attraction by the new intermediate state,
while the initial correct choice is still likely to be maintained. The speed-
accuracy tradeoff always works. Although time pressure may lead to more
initial errors, changes made to correct these errors are also more likely to
happen due to the emergence of a new intermediate state. The mechanism for
changes of mind guarantees a reasonable performance of the decision making
process with emphasis on speed.

Figure 13: (A) The schematic diagram of the reduced two population decision-
making model. This reduced model consists of two competing neural populations
that are selective for leftward or rightward directions, respectively. (B)The
schematic representation of the random dots motion. For higher motion
coherence, most dots move in one direction, whereas the dots move with no
directional bias at a low motion coherence level. (C) The potential landscape
of the decision-making network with varying inputs and pathways. The pink
lines indicate the paths of decision making from undecided state to decided
states. The red dotted lines represent the paths from the two decided states
back to the undecided state. (D-G) The mechanism of changes of mind based on
the emergence of the new intermediate state in the center. (D-E) The two-
dimensional potential landscapes for different large inputs at zero coherence
level. (F-G) The two-dimensional potential landscapes for large input when the
motion coherence $c^{\prime}=0.02$ and $0.06$, respectively. Reproduced with
permission from Yan H, Zhang K and Wang J., Chin. Phys. B. 25, 078702 (2016).
Copyright 2016 Chinese Physical Society and IOP Publishing. [297].

####  VI.4.3 Fear learning and expression

The non-equilibrium landscape and flux approach can also be used to analyze
the attractor landscape of cortical circuits involved in the phenomenon of
fear conditioning. These circuits support the rapid selection of the
appropriate response to a threat, a behavior that is crucial for animals’
survival. Such ability involves associative learning from experience to
predict, and then make appropriate responses, to danger. Pavlovian fear
conditioning experiments have often been used to understand the neural
circuitry underlying fear learning and expression [298, 299, 300, 301, 302].
During conditioning, subjects are presented with a conditioned stimulus (CS;
typically a sound or a light stimulus) paired with an aversive unconditioned
stimulus (US; typically a mild electric shock) to elicit conditioned defensive
responses, such as freezing behavior (a passive defensive response) [302, 303,
304] or active responses such as startle, escape, flight and avoidance [305,
306].

Figure 14: (A-C) Potential landscapes for different inputs. The increase of
the inputs induces symmetry breaking from the symmetric but featureless state
to the biased state with biological functions. The fluxes are indicated by
purple arrows. $\lambda_{1}$ and $\lambda_{2}$ represent the strength of the
inputs to the SOM+ and SOM- neurons, respectively. (D) A diagram of how a one-
dimensional potential landscape changes with stimulus inputs. (E) Average flux
landscape in the space of different inputs. The average flux is significantly
positively correlated with the external inputs, when the neural circuit is
away from its equilibrium state. (F) The entropy production rate landscape in
the space of different inputs. The neural circuit dissipates more energy with
larger inputs. It costs more energy to maintain the dominant freezing
responses than dominant no-freezing behaviors. $\lambda_{1}$ and
$\lambda_{2}$: same as is panels A-C. Reproduced with permission from Yan H,
Li B and Wang J., J. R. Soc. Interface. 16,20180756 (2019). Copyright 2019
Royal Society.[307].

The central amygdala plays a crucial role in both acquisition and expression
of conditioned fear [298, 299, 300, 301, 302]. Following fear conditioning,
the excitatory synapses from neurons in the lateral amygdala onto neurons in
central amygdala undergo different changes: those onto a category of
inhibitory neurons known to express somatostatin (SOM+) are strengthened,
while those onto non-somatostatin expressing neurons (SOM-) are weakened [302,
303, 304]. In turn, this elicits freezing behavior.

The mechanisms of active defensive response, as well as the rapid selection
between passive and active responses, are less understood. It is possible that
rapid selection is gated by the central amygdala. To understand the underlying
dynamic mechanism of how the central amygdala gates passive and active
defensive responses, [307] used landscape and flux theory to study a model of
the central amygdala dominated by local inhibitory connections between SOM+
and SOM- neurons. With this approach, the underlying attractor landscape of
the circuit model with varying inputs can be quantified (Fig. 14A-D).

In the model, freezing states observed in experiment (due to activated SOM+
neurons and inhibited SOM- neurons) emerge in the presence of a CS with biased
excitatory inputs to SOM+ neurons (due to fear conditioning-induced synaptic
modifications) [302, 306]. However, the model also shows that, if excitatory
synapses to SOM+ and SOM- neurons are both strengthened during fear
conditioning, for a range of inputs there exists a bi-stable phase with both a
freezing and a non-freezing state. In this bi-stable phase, the non-freezing
(active defense) state is induced by the same inputs that can elicit freezing
responses, but resulting instead in non-freezing active responses. Learning to
selecting the type of response (passive vs. active) also requires the
strengthened synaptic projections to both SOM+ and SOM-neurons. The underlying
topology of non-equilibrium landscape shaped by such a set of inputs supports
two distinct attractors with a clear barrier in between (Fig. 14A-D). The
switches between different defensive responses under threats can be physically
characterized by the transitions between the two attractors. Furthermore, the
maintenance of such bi-stable phase needs additional energy, which can be
measured through the entropy production rate closely related to the non-
equilibrium flux and discussed in Sec. V.2.5 (Fig. 14E-F).

Based on this model, we predict that in situations where active responses are
reinforced, such as in active avoidance learning, excitatory synaptic
transmission onto the SOM- neurons would be more robustly potentiated than in
situations where only passive responses are allowed, such as classical fear
conditioning. If only the excitatory projections to SOM+ neurons were
strengthened during fear conditioning, only the freezing state (passive fear
response) would be possible in response to threats.

####  VI.4.4 Lessons from the case studies

The above case studies show how the non-equilibrium landscape and flux
approach provides a general way to study neural circuit dynamics. As explained
in Sec. V.2.3, this approach allows to analyze modern models of neural
networks that have non-symmetric connections. This allows in particular to
analyze models with synaptic connections that respect Dale’s law (stating that
neurons are either excitatory or inhibitory), and therefore obey important
biological constraints. The relevant models reviewed in Sec. V.2 belong to
this class and therefore require the landscape and flux theory approach for a
proper analysis.

To summarize, the sequential functional states during metastable activity can
be identified through the underlying potential landscapes. In particular, the
robustness of these functional states can be quantified not only by the depth
and the breadth of the corresponding basins of attraction, but also according
to the distance between the basins and the transition times between states
corresponding to different basins. Metastable neural dynamics can be induced
by noise (whether from external sources or endogenously generated as in the
deterministic spiking model of Sec. V.2.1), or it can be induced by changes in
the landscape topography. The latter may result from varying key ingredients
such as relevant synaptic connections, inputs, or neurons. Regardless of its
origin, metastable dynamics in cortical circuits can be described in terms of
the optimal paths under the action of both the landscape and flux components
of the driving forces, and has uncovered important potential mechanisms for
various brain functions, including sleep cycle regulation [23], stability-
flexibility tradeoff in working memory [202], changes of mind in decision-
making [297], the selection of passive and active fear responses [307], and
network mechanisms of Parkinson disease [308]. The theory also allows the
quantification of the thermodynamic cost for maintaining neural network
function and can help to facilitate the design of strategies achieving an
optimal balance between performance and cost. These are not easily achieved
through more conventional theoretical methods.

##  VII Summary and conclusions

The field of neuroscience is moving towards an appreciation of the role of
neural dynamics in coding and computation. In this article we have reviewed
recent progress on the front of characterizing and modeling neural dynamics,
with particular emphasis on a type of metastable dynamics that unfolds as a
sequence of discrete states. This kind of metastable dynamics has been
quantified in several cortical areas of rodents, monkeys and humans, and seems
related to coding sensory stimuli as well as internal deliberations. Focusing
on metastable dynamics signals a departure from earlier and more traditional
views, which were centered on the notion of single neurons’ input-output
function and its modulation as predictors of stimulus features or behavioral
outcomes. One of the most relevant implications of the fact that cortical
activity evolves as a sequence of discrete, metastable states is that
transitions in neural activity are not just triggered by external events, such
as a stimulus or a reward, but are instead spontaneously generated and may
occur at anytime, including when the subject is idling and not engaged in a
task. This goes against the notion that neural activity is just a ‘reaction’
to external events or a static representation of incoming stimuli, and is
compatible with the presence of incessant ‘ongoing activity’ observed in
cortex.

Among the most salient characteristics of the type of cortical metastable
dynamics reviewed here are: (i) the hidden states are states of collective
behavior in populations of neurons that are putative fixed points of the
neural dynamics (Sec. V); (ii) state transitions are typically one order of
magnitude faster than the state durations, and are close to their
theoretically observable lower bound (Sec. V.1.8); (iii) the ongoing dynamics
of some cortical circuits is also highly structured and characterized by
repeatable metastable transitions, and does not resemble metastability en
route to a ground state configuration (Sec. IV.3); (iv) neural dynamics evoked
by a stimulus is often metastable, with some hidden states coding for specific
stimulus features, internal decisions, or upcoming actions (Sec. IV); (v)
metastable dynamics during a task can be modulated by task variables (such as
trial difficulty or internal expectation) or behavioral outcome (correct vs.
error) (Sec. IV.2).

We have also reviewed classical and contemporary approaches to infer hidden
states from the neural dynamics (Sec. V.1), as well as mechanistic models of
metastable dynamics based on clustered networks of spiking neurons (Sec.
V.2.1). A topological organization in potentiated clusters seem necessary to
explain the observed metastable dynamics and is able to predict its most
salient features. Network models are hard to study and are often analyzed with
mean field techniques; we have reviewed basic mean field theory for networks
of spiking neurons as well as the more general landscape and flux theory of
network dynamics (Sec. V.2). While the former allows to predict the fixed
points of the dynamics in a situation of equilibrium, with the fixed points
possibly becoming metastable in finite networks with random connectivity,
landscape and flux theory also allows to study the metastable dynamics out of
equilibrium.

Neural clusters have only been indirectly observed so far, and are presumably
learned through experience. We have reviewed the possible links between
plasticity and metastable dynamics in Sec. VI. This included recent efforts to
obtain metastable dynamics in spiking network models via experience-dependent
synaptic plasticity (Sec. VI.3), as well as theoretical investigations of the
consequences of learning on the dynamics of cortical networks (Sec. VI.4).

With the improvement of recording techniques and the ability to perform a
larger variety of behavioral tasks in the laboratory, we predict that evidence
for neural clusters and metastability will continue to accrue, together with
the refinement of theoretical tools for their analysis and modeling. In turn,
these endeavors will help solidify a dynamics-centric view of cortical
activity supporting sensory and cognitive processes.

###### Acknowledgements.

The authors wish to thank Tianshu Li for preparing Figure 1. This work was
partially supported by the National Institute Of Neurological Disorders And
Stroke of the National Institutes of Health under award number UF1NS115779
(BAWB, AM, IMP, AF, JW and GLC) and by the National Natural Science Foundation
of China under award number NSFC 21721003 (HY). The content of this article is
solely the responsibility of the authors and does not necessarily represent
the official views of the National Institutes of Health or the National
Natural Science Foundation of China.

## Author contributions

All authors contributed to the design of the manuscript, literature research
and manuscript writing.

## Data availability statement

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

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