<title>Introduction</title> <p>The brain appears to adhere to two fundamental principles of functional organization,
functional integration and functional specialization, where the integration within
and among specialized areas is mediated by connections among them. The distinction
relates to that between localisationism and connectionism that dominated thinking
about cortical function in the nineteenth century. Since the early anatomic theories
of Gall, the identification of a particular brain region with a specific function
has become a central theme in neuroscience. In this paper, we address how
distributed and specialized neuronal responses are realized in terms of microscopic
brain dynamics; we do this by showing how neuronal systems, with many degrees of
freedom, can be reduced to lower dimensional systems that exhibit adaptive
behaviors.</p> <p>It is commonly accepted that the information processing underlying brain functions,
like sensory, motor, and cognitive functions, is carried out by large groups of
interconnected neurons ##REF##11156197##[1]##–##REF##15319516##[4]##. Neurons are the cells
responsible for encoding, transmitting, and integrating signals originating inside
or outside the nervous system. The transmission of information within and between
neurons involves changes in the so-called resting membrane potential, the electrical
potential of the neurons at rest, when compared to the extracellular space. The
inputs one neuron receives at the synapses from other neurons cause transient
changes in its resting membrane potential, called postsynaptic potentials. These
changes in potential are mediated by the flux of ions between the intracellular and
extracellular space. The flux of ions is made possible through ion channels present
in the membrane. The ion channels open or close depending on the membrane potential
and on substances released by the neurons, namely neurotransmitters, which bind to
receptors on the cell's membrane and hyperpolarize or depolarize the cell.
When the postsynaptic potential reaches a threshold, the neuron produces an impulse.
The impulses or spikes, called action potentials, are characterized by a certain
amplitude and duration and are the units of information transmission at the
interneuronal level. Information is thought to be encoded in terms of the frequency
of the action potentials, called spiking or firing rate (i.e., rate coding), as well
as in the timing of action potentials (i.e., temporal coding).</p> <p>One way to investigate the biological basis of information processing in the brain is
to study the response of neurons to stimulation. This can be done in experimental
animals using implanted electrodes to record the rates and timing of action
potentials. However, this invasive approach is generally not possible in humans. To
study brain function in humans, techniques allowing the indirect study of neuronal
activity have been developed. An example is functional magnetic resonance imaging
(fMRI), measuring regional changes in metabolism and blood flow associated with
changes in brain activity. This approach to measuring regional differences in brain
activity is possible because at a macroscopic level the cortex is organized into
spatially segregated regions known to have functionally specialized roles. A
technique such as fMRI allows the mapping of brain regions associated with a
particular task or task component.</p> <p>Understanding the fundamental principles underlying higher brain functions requires
the integration of different levels of experimental investigation in cognitive
neuroscience (from single neurons, neuroanatomy, neurophysiology, and neuroimaging,
to neuropsychology and behavior) via a unifying theoretical framework that captures
the neural dynamics inherent in the elaboration of cognitive processes. In this
paper, we review and integrate a variety of computational approaches that have been
used to characterize the dynamics of the cortex, as evidenced at different levels of
measurement.</p> <p>The paper is structured as follows. The central theme of this review is that the
activity in populations of neurons can be understood by reducing the degrees of
freedom from many to few, hence resolving an otherwise intractable computational
problem. The most striking achievement in this regard is the reduction of a large
population of spiking neurons to a distribution function describing their
probabilistic evolution—that is, a function that captures the likely
distribution of neuronal states at a given time. In turn, this can be further
reduced to a single variable describing the mean firing rate. This reduction is
covered first, in the next section. In the section entitled <xref ref-type="sec" rid="s3">Neural Modes and Masses</xref>, we return to the full probability
distribution function and show how it can be represented by a set of scalars that
parameterize it parsimoniously. These parameters are equivalent to the moments of
the distribution. In many instances, a few—possibly even one (equivalent
to the center of mass)—are sufficient to summarize activity. These are
known as Neural Mass Models. These models capture the dynamics of a neuronal
population. Naturally, it is useful to understand how neuronal activity unfolds on
the spatially continuous cortical sheet. This can be addressed with neural field
models; involving differential operators with both temporal and spatial terms. That
is, neuronal activity depends on its current state as well as spatial gradients,
which allow its spread horizontally across the cortical surface. These models are
covered in the <xref ref-type="sec" rid="s4">Neural Field Models</xref> section. In
<xref ref-type="sec" rid="s5">Numerical Simulations: Ensemble Activity from
Neuronal to Whole Brain Scale</xref>, we provide numerical simulations of
neuronal ensemble dynamics across a hierarchy of spatial and temporal scales. At the
microscopic scale, we simulate an entire array of spiking neurons in response to a
sensory-evoked synaptic current. By comparing the response to that of a mesoscopic
neural mass model, we show what is gained and what is lost by abstracting to a more
tractable set of evolution equations. The spread of activity across the cortical
surface, in a neural field model, is also illustrated. Finally, in the section
entitled <xref ref-type="sec" rid="s6">Cognitive and Clinical Applications</xref>,
we illustrate applications of neural ensemble modeling in health and disease;
namely, decision-making, auditory scene analysis, and absence seizures.</p> <p>A summary of the notation for all the main dynamical variables and physiological
parameters is given in ##TAB##0##Table
1##.</p>
<title>Discussion</title> <p>In conclusion, we have seen that statistical descriptions of neuronal ensembles can
be formulated in terms of a Fokker-Planck equation, a functional differential
equation prescribing the evolution of a probability density on some phase space. The
high dimensionality and complexity of these Fokker-Planck formalisms can be finessed
with a mean-field approximation to give nonlinear Fokker-Planck equations,
describing the evolution of separable ensembles that are coupled by mean-field
effects. By parameterizing the densities in terms of basis functions or probability
modes, these partial differential equations can be reduced to coupled differential
equations describing their evolution. In the simplest case, we can use a single mode
that can be regarded as encoding the location of a probability mass, hence neural
mass models. Neural mass models can be generalized to neural field models by making
the expectations a function of space, thereby furnishing wave equations that
describe the spatiotemporal evolution of expected neuronal states over the cortical
surface. We have tried to show the conceptual and mathematical links among the
ensuing levels of description and how these models can be used to characterize key
dynamical mechanisms in the brain.</p>
<p>The cortex is a complex system, characterized by its dynamics and architecture,
which underlie many functions such as action, perception, learning, language,
and cognition. Its structural architecture has been studied for more than a
hundred years; however, its dynamics have been addressed much less thoroughly.
In this paper, we review and integrate, in a unifying framework, a variety of
computational approaches that have been used to characterize the dynamics of the
cortex, as evidenced at different levels of measurement. Computational models at
different space–time scales help us understand the fundamental
mechanisms that underpin neural processes and relate these processes to
neuroscience data. Modeling at the single neuron level is necessary because this
is the level at which information is exchanged between the computing elements of
the brain; the neurons. Mesoscopic models tell us how neural elements interact
to yield emergent behavior at the level of microcolumns and cortical columns.
Macroscopic models can inform us about whole brain dynamics and interactions
between large-scale neural systems such as cortical regions, the thalamus, and
brain stem. Each level of description relates uniquely to neuroscience data,
from single-unit recordings, through local field potentials to functional
magnetic resonance imaging (fMRI), electroencephalogram (EEG), and
magnetoencephalogram (MEG). Models of the cortex can establish which types of
large-scale neuronal networks can perform computations and characterize their
emergent properties. Mean-field and related formulations of dynamics also play
an essential and complementary role as forward models that can be inverted given
empirical data. This makes dynamic models critical in integrating theory and
experiments. We argue that elaborating principled and informed models is a
prerequisite for grounding empirical neuroscience in a cogent theoretical
framework, commensurate with the achievements in the physical sciences.</p>
<title>Mean-Field Models</title> <p>This section provides an overview of mean-field models of neuronal dynamics and their
derivation from models of spiking neurons. These models have a long history spanning
a half-century (e.g., ##UREF##2##[5]##) and are formulated using concepts from
statistical physics. In this section, we try to clarify some key concepts and show
how they relate to each other. Models are essential for neuroscience, in the sense
that the most interesting questions pertain to neuronal mechanisms and processes
that are not directly observable. This means that questions about neuronal function
are generally addressed by inference on models or their parameters, where the model
links neuronal processes that are hidden from our direct observation. Broadly
speaking, models are used to generate data, to study emergent behaviors, or they can
be used as forward or observation models, which are inverted given empirical data.
This inversion allows one to select the best model (given some data) and make
probabilistic comments about the parameters of that model. Mean-field models are
suited to data which reflect the behavior of a population of neurons, such as the
electroencephalogram (EEG), magnetoencephalogram (MEG), and fMRI. The most prevalent
models of neuronal populations or ensembles are based upon something called the
mean-field approximation. The mean-field approximation is used extensively in
statistical physics and is essentially a technique that finesses an otherwise
computationally or analytically intractable problem. An exemplary approach, owing to
Boltzmann and Maxwell, is the approximation of the motion of molecules in a gas by
mean-field terms such as temperature and pressure.</p> <title/> <title>Ensemble density models</title> <p>Ensemble models attempt to model the dynamics of large (theoretically
infinite) populations of neurons. Any single neuron could have a number of
attributes; for example, post-synaptic membrane depolarization,
<italic>V</italic>, capacitive current, <italic>I</italic>, or the time
since the last action potential, <italic>T</italic>. Each attribute induces
a dimension in the phase space of a neuron; in our example the phase space
would be three dimensional and the state of each neuron would correspond to
a point
<italic>ν</italic> = {<italic>V</italic>,<italic>I</italic>,<italic>T</italic>}
∈ℜ<sup>3</sup> or particle in phase space. Imagine a
very large number of neurons that populate phase space with a density
<italic>p</italic>(<italic>ν</italic>,<italic>t</italic>). As
the state of each neuron evolves, the points will flow through phase space,
and the ensemble density
<italic>p</italic>(<italic>ν</italic>,<italic>t</italic>) will
evolve until it reaches some steady state or equilibrium.
<italic>p</italic>(<italic>ν</italic>,<italic>t</italic>) is a
scalar function returning the probability density at each point in phase
space. It is the evolution of the density per se that is characterized in
ensemble density methods. These models are particularly attractive because
the density dynamics conform to a simple equation: the Fokker-Planck equationThis equation comprises a flow and a dispersion term; these
terms embed the assumptions about the dynamics (phase flow,
<italic>f</italic>(<italic>ν</italic>,<italic>t</italic>)) and
random fluctuations (dispersion,
<italic>D</italic>(<italic>ν</italic>,<italic>t</italic>)) that
constitute our model at the neuronal level. This level of description is
usually framed as a (stochastic) differential equation (Langevin equation)
that describes how the states evolve as functions of each other and some
random fluctuations withwhere,
<italic>D</italic> = ½<italic>σ</italic>
<sup>2</sup>
and <italic>ω</italic> is a standard Wiener process; i.e.,
<italic>w</italic>(<italic>t</italic>)−<italic>w</italic>(<italic>t</italic>+Δ<italic>t</italic>)∼<italic>N</italic>(0,
Δ<italic>t</italic>). Even if the dynamics of each neuron are
complicated, or indeed chaotic, the density dynamics remain simple, linear,
and deterministic. In fact, we can write the density dynamics in terms of a
linear operator or Jacobian <italic>Q</italic>

</p> <p>In summary, for any model of neuronal dynamics, specified as a stochastic
differential equation, there is a deterministic linear equation that can be
integrated to generate ensemble dynamics. In what follows, we will explain
in detail the arguments that take us from the spiking behavior of individual
neurons to the mean-field dynamics described by the Fokker-Planck equation.
We will consider the relationship between density dynamics and neural mass
models and how these can be extended to cover spatiotemporal dynamics in the
brain.</p> <title>From spiking neurons to mean-field models</title> <p>The functional specialization of the brain emerges from the collective
network dynamics of cortical circuits. The computational units of these
circuits are spiking neurons, which transform a large set of inputs,
received from different neurons, into an output spike train that constitutes
the output signal of the neuron. This means that the spatiotemporal spike
patterns produced by neural circuits convey information among neurons; this
is the microscopic level on which the brain's representations and
computations rest ##UREF##3##[6]##. We assume that the nonstationary temporal
evolution of the spiking dynamics can be captured by one-compartment,
point-like models of neurons, such as the leaky integrate-and-fire (LIF)
model ##UREF##4##[7]## used below. Other models relevant for
systems neuroscience can be found in ##REF##15319516##[4]##,##UREF##5##[8]##,##UREF##6##[9]##. In the LIF
model, each neuron <italic>i</italic> can be fully described in terms of a
single internal variable, namely the depolarization
<italic>V<sub>i</sub></italic>(<italic>t</italic>) of the neural
membrane. The basic circuit of a LIF model consists of a capacitor,
<italic>C</italic>, in parallel with a resistor, <italic>R</italic>,
driven by a synaptic current (excitatory or inhibitory postsynaptic
potential, EPSP or IPSP, respectively). When the voltage across the
capacitor reaches a threshold <italic>θ</italic>, the circuit is
shunted (reset) and a <italic>δ</italic> pulse (spike) is generated
and transmitted to other neurons. The subthreshold membrane potential of
each neuron evolves according to a simple <italic>RC</italic> circuit, with
a time constant
<italic>τ</italic> = <italic>RC</italic>
given by the following equation:where <italic>I<sub>i</sub></italic>(<italic>t</italic>) is
the total synaptic current flow into the cell <italic>i</italic> and
<italic>V<sub>L</sub></italic> is the leak or resting potential of
the cell in the absence of external afferent inputs. In order to simplify
the analysis, we neglect the dynamics of the afferent neurons (see ##REF##11524578##[10]##
for extensions considering detailed synaptic dynamics such as AMPA, NMDA,
and GABA). The total synaptic current coming into the cell
<italic>i</italic> is therefore given by the sum of the contributions of
<italic>δ</italic>-spikes produced at presynaptic neurons. Let
us assume that <italic>N</italic> neurons synapse onto cell
<italic>i</italic> and that <italic>J<sub>ij</sub></italic> is the efficacy
of synapse <italic>j</italic>, then the total synaptic afferent current is
given bywhere  is the emission time of the
<italic>k<sup>th</sup></italic> spike from the
<italic>j<sup>th</sup></italic> presynaptic neuron. The subthreshold
dynamical Equation 4, given the input current (from Equation 5), can be
integrated, and yieldsif the neuron <italic>i</italic> is initially
(<italic>t</italic> = 0) at the resting
potential
(<italic>V<sub>i</sub></italic>(0) = <italic>V<sub>L</sub></italic>).
In Equation 7, <italic>H</italic>(<italic>t</italic>) is the Heaviside
function
(<italic>H</italic>(<italic>t</italic>) = 1
if <italic>t</italic>&gt;0, and
<italic>H</italic>(<italic>t</italic>) = 0
if <italic>t</italic>&lt;0). Thus, the incoming presynaptic
<italic>δ</italic>-pulse from other neurons is basically
low-pass filtered to produce an EPSP or IPSP in the post-synaptic cell.
Nevertheless, the integrate-and-fire (IF) model is not only defined by the
subthreshold dynamics but includes a reset after each spike generation,
which makes the whole dynamics highly nonlinear. In what follows, we present
a theoretical framework which is capable of dealing with this.</p> <title>The population density approach</title> <p>Realistic neuronal networks comprise a large number of neurons (e.g., a
cortical column has O(10<sup>4</sup>)−O(10<sup>8</sup>) neurons)
which are massively interconnected (on average, a neuron makes contact with
O(10<sup>4</sup>) other neurons). The underlying dynamics of such
networks can be described explicitly by the set of coupled differential
equations (Equation 4) above. Direct simulations of these equations yield a
complex spatiotemporal pattern, covering the individual trajectory of the
internal state of each neuron in the network. This type of direct simulation
is computationally expensive, making it very difficult to analyze how the
underlying connectivity relates to various dynamics. In fact, most key
features of brain operation seem to emerge from the interplay of the
components; rather than being generated by each component individually. One
way to overcome these difficulties is by adopting the population density
approach, using the Fokker-Planck formalism (e.g., ##REF##8373886##[11]##). As noted
above, the Fokker-Planck equation summarizes the flow and dispersion of
states over phase space in a way that is a natural summary of population
dynamics in genetics (e.g., ##UREF##7##[12]##) and neurobiology (e.g., ##REF##508846##[13]##,##REF##7578480##[14]##).</p> <p>In what follows, we derive the Fokker-Planck equation for neuronal dynamics
that are specified in terms of spiking neurons. This derivation is a little
dense but illustrates the approximating assumptions and level of detail that
can be captured by density dynamics. The approach we focus on was introduced
by ##UREF##8##[15]## (see also ##REF##10798499##[16]##,##REF##10769319##[17]##). In this approach, individual IF neurons are
grouped together into populations of statistically similar neurons. A
statistical description of each population is given by a probability density
function that expresses the distribution of neuronal states (i.e., membrane
potential) over the population. In general, neurons with the same state
<italic>V</italic>(<italic>t</italic>) at a given time
<italic>t</italic> have a different history because of random fluctuations
in the input current <italic>I</italic>(<italic>t</italic>). The main source
of randomness is from fluctuations in recurrent currents (resulting from
“quenched” randomness in the connectivity and
transmission delays) and fluctuations in the external currents. The key
assumption in the population density approach is that the afferent input
currents impinging on neurons in one population are uncorrelated. Thus,
neurons sharing the same state <italic>V</italic>(<italic>t</italic>) in a
population are indistinguishable. Consequently, the dynamics are described
by the evolution of the probability density function:which expresses the population density, which is the fraction
of neurons at time <italic>t</italic> that have a membrane potential
<italic>V</italic>(<italic>t</italic>) in the interval
[<italic>ν</italic>,<italic>ν</italic>+<italic>dν</italic>].
The evolution of the population density is given by the Chapman-Kolmogorov equationwhere
<italic>ρ</italic>(<italic>ε</italic>|<italic>ν</italic>) = <italic>Prob</italic>{<italic>V</italic>(<italic>t</italic>+<italic>dt</italic>) = <italic>ν</italic>+<italic>ε</italic>|<italic>V</italic>(<italic>t</italic>) = <italic>ν</italic>}
is the conditional probability that generates an infinitesimal change
<italic>ε</italic> = <italic>V</italic>(<italic>t</italic>+<italic>dt</italic>)−<italic>V</italic>(<italic>t</italic>)
in the infinitesimal interval <italic>dt</italic>. The Chapman-Kolmogorov
equation can be written in a differential form by performing a Taylor
expansion in
<italic>p</italic>(<italic>ν</italic>′,<italic>t</italic>)
<italic>ρ</italic>(<italic>ε</italic>|<italic>ν</italic>′)
around
<italic>ν</italic>′ = <italic>ν</italic>; i.e.,In the derivation of the last equation, we have assumed that
<italic>p</italic>(<italic>ν</italic>′,<italic>t</italic>)
and <italic>ρ</italic>(<italic>ε</italic>|
<italic>ν</italic>′) are infinitely many times
differentiable in <italic>ν</italic>. Inserting this expansion in
Equation 9, and replacing the time derivative in
<italic>ν</italic>′ by the equivalent time derivative in
<italic>ν</italic>, we obtainwhere 〈…〉<italic><sub>ν</sub></italic> denotes the average with respect to
<italic>ρ</italic>(<italic>ε</italic>|
<italic>ν</italic>) at a given <italic>ν</italic>. Finally,
taking the limit for <italic>dt</italic> → 0, we obtain:
</p> <p>Equation 13 is known as the Kramers-Moyal expansion of the original integral
Chapman-Kolmogorov equation (Equation 9). It expresses the time evolution of
the population density in differential form.</p> <title>The diffusion approximation</title> <p>The temporal evolution of the population density as given by Equation 13
requires the moments 〈<italic>ε<sup>k</sup></italic>〉<italic><sub>υ</sub></italic> due to the afferent current during the interval
<italic>dt</italic>. These moments can be calculated by the mean-field
approximation. In this approximation, the currents impinging on each neuron
in a population have the same statistics, because as we mentioned above, the
history of these currents is uncorrelated. The mean-field approximation
entails replacing the time-averaged discharge rate of individual cells with
a common time-dependent population activity (ensemble average). This assumes
ergodicity for all neurons in the population. The mean-field technique
allows us to discard the index denoting the identity of any single neuron
and express the infinitesimal change,
<italic>dV</italic>(<italic>t</italic>), in the membrane potential of all
neurons as:where <italic>N</italic> is the number of neurons, and
<italic>Q</italic>(<italic>t</italic>) is the mean population firing
rate. This is determined by the proportion of active neurons by counting the
number of spikes
<italic>n<sub>spikes</sub></italic>(<italic>t</italic>,<italic>t</italic>+<italic>dt</italic>)
in a small time interval <italic>dt</italic> and dividing by
<italic>N</italic> and by <italic>dt</italic>
##REF##10636933##[18]##; i.e.,
</p> <p>In Equation 14, 〈<italic>J</italic>〉<italic><sub>J</sub></italic> denotes the average of the synaptic weights in the population. The
moments of the infinitesimal depolarization,
<italic>ε</italic> = <italic>dV</italic>(<italic>t</italic>),
can now be calculated easily from Equation 14. The first two moments in the
Kramers-Moyal expansion are called drift and diffusion coefficients,
respectively, and they are given by:
In general, keeping only the leading term linear in
<italic>dt</italic>, it is easy to prove that for <italic>k</italic>&gt;1,and hence,
</p> <p>The diffusion approximation arises when we neglect high-order
(<italic>k</italic>&gt;2) terms. The diffusion approximation is exact in
the limit of infinitely large networks, i.e., <italic>N</italic> →
∞, if the synaptic efficacies scale appropriately with network
size, such that <italic>J</italic> → 0 but
<italic>NJ<sup>2</sup></italic> → <italic>const</italic>. In other
words, the diffusion approximation is appropriate, if the minimal kick step,
<italic>J</italic>, is very small but the overall firing rate is very
large. In this case, all moments higher than two become negligible, in
relation to the drift (<italic>μ</italic>) and diffusion
(<italic>σ</italic>
<sup>2</sup>) coefficients.</p> <p>The diffusion approximation allows us to omit all higher orders
<italic>k</italic>&gt;2 in the Kramers-Moyal expansion. The resulting
differential equation describing the temporal evolution of the population
density is called the Fokker-Planck equation, and reads
</p> <p>In the particular case that the drift is linear and the diffusion
coefficient, <italic>σ</italic>
<sup>2</sup>(<italic>t</italic>), is
given by a constant, the Fokker-Planck equation describes a well-known
stochastic process called the Ornstein-Uhlenbeck process ##UREF##9##[19]##. Thus, under the diffusion approximation, the
Fokker-Planck equation (Equation 20) expresses an Ornstein-Uhlenbeck
process. The Ornstein-Uhlenbeck process describes the temporal evolution of
the membrane potential <italic>V</italic>(<italic>t</italic>) when the input
afferent currents are given bywhere <italic>ω</italic>(<italic>t</italic>) is a
white noise process. Under the diffusion approximation, Equation 21 can also
be interpreted (by means of the Central Limit Theorem), as the case in which
the sum of many Poisson processes (Equation 5) becomes a normal random
variable with mean <italic>μ</italic>(<italic>t</italic>) and
variance <italic>σ</italic>
<sup>2</sup>.</p> <title>The mean-field model</title> <p>The simulation of a network of IF neurons allows one to study the dynamical
behavior of the neuronal spiking rates. Alternatively, the integration of
the non-stationary solutions of the Fokker-Planck equation (Equation 20)
also describes the dynamical behavior of the network, and this would allow
the explicit simulation of neuronal and cortical activity (single cells,
EEG, fMRI) and behavior (e.g., performance and reaction time). However,
these simulations are computationally expensive and their results
probabilistic, which makes them unsuitable for systematic explorations of
parameter space. However, the stationary solutions of the Fokker-Planck
equation (Equation 20) represent the stationary solutions of the original IF
neuronal system. This allows one to construct bifurcation diagrams to
understand the nonlinear mechanisms underlying equilibrium dynamics. This is
an essential role of the mean-field approximation: to simplify analyses
through the stationary solutions of the Fokker-Planck equation for a
population density under the diffusion approximation (Ornstein-Uhlenbeck
process) in a self-consistent form. In what follows, we consider stationary
solutions for ensemble dynamics.</p> <p>The Fokker-Planck equation describing the Ornstein-Uhlenbeck process, with
<italic>μ</italic> = 〈<italic>J</italic>〉<italic><sub>J</sub>
NQ</italic>(<italic>t</italic>) and
<italic>σ</italic>
<sup>2</sup> = 〈<italic>J<sup>2</sup></italic>〉<italic><sub>J</sub>
NQ</italic>(<italic>t</italic>), can be rewritten as a continuity equation:where <italic>F</italic> is the flux of probability defined
as follows:The stationary solution should satisfy the following boundary condition:andwhich expresses the fact that the probability current at
threshold gives, by a self-consistent arguments, the average firing rate,
<italic>Q</italic>, of the population. Furthermore, at
<italic>ν</italic>→−4 the probability density
vanishes fast enough to be integrable; i.e.,andIn addition, the probability mass leaving the threshold at
time <italic>t</italic> has to be re-injected at the reset potential at time
<italic>t</italic>+<italic>t<sub>ref</sub></italic> (where
<italic>t<sub>ref</sub></italic> is the refractory period of the
neurons), which can be accommodated by rewriting Equation 22 as follows:where <italic>H</italic>(.) is the Heaviside function. The
solution of Equation 28 satisfying the boundary conditions (Equations
24–27) is:Taking into account the fraction of neurons,
<italic>Qt<sub>ref</sub></italic>, in the refractory period and the
normalization of the mass probability,Finally, substituting Equation 29 into Equation 30, and
solving for <italic>Q</italic>, we obtain the population transfer function,
<italic>φ</italic>, of Ricciardi ##REF##508846##[13]##:where .</p> <p>The stationary dynamics of each population can be described by the population
transfer function, which provides the average population rate as a function
of the average input current. This can be generalized easily for more than
one population. The network is partitioned into populations of neurons whose
input currents share the same statistical properties and fire spikes
independently at the same rate. The set of stationary, self-reproducing
rates, <italic>Q<sub>i</sub></italic>, for different populations,
<italic>i</italic>, in the network can be found by solving a set of coupled
self-consistency equations, given by:
</p> <p>To solve the equations defined by Equation 32 for all <italic>i</italic>, we
integrate the differential equation below, describing the approximate
dynamics of the system, which has fixed-point solutions corresponding to
Equation 32:
</p> <p>This enables a posteriori selection of parameters, which induce the emergent
behavior that we are looking for. One can then perform full nonstationary
simulations using these parameters in the full IF scheme to generate true
dynamics. The mean-field approach ensures that these dynamics will converge
to a stationary attractor that is consistent with the steady-state dynamics
we require ##REF##11524578##[10]##,##REF##15242673##[20]##. In our
case, the derived transfer function, <italic>φ</italic>,
corresponds consistently to the assumptions of the simple LIF model
described in the From Spiking-Neurons to Mean-Field Models section. Further
extension for more complex and realistic models are possible. For example,
an extended mean-field framework, which is consistent with the IF and
realistic synaptic equations that considers both the fast and slow
glutamatergic excitatory synaptic dynamics (AMPA and NMDA) and the dynamics
of GABA inhibitory synapses, can be found in ##REF##11524578##[10]##. Before turning
to neural mass models, we consider some applications of mean-field modeling
that will be reprised in the last section.</p> <title>Competition and cooperation</title> <p>How are different cortical representations integrated to form a coherent
stream of perception, cognition, and action? The brain is characterized by a
massive recurrent connectivity between cortical areas, which suggests that
integration of partial representations might be mediated by cross talk via
interareal connections. Based on this view ##UREF##10##[21]##, and
neurophysiological evidence ##REF##7605061##[22]##, it has been
hypothesized that each cortical area represents a set of alternative
hypotheses, encoded in the activities of cell assemblies. Representations of
conflicting hypotheses compete with each other; however, each area
represents only a part of the environment or internal state. In order to
arrive at a coherent global representation, different cortical areas bias
each others' internal representations by communicating their
current states to other areas, thereby favoring certain sets of local
hypotheses over others. By recurrently biasing each others'
competitive internal dynamics, the neocortical system arrives at a global
representation in which each area's state is maximally consistent
with those of the other areas. This view has been referred to as the
biased-competition hypothesis. In addition to this competition-centered
view, a cooperation-centered picture of brain dynamics, where global
representations find their neural correlate in assemblies of coactivated
neurons, has been formulated ##UREF##10##[21]##,##REF##16257103##[23]##. Coactivation is
achieved by increased connectivity among the members of each assembly.
Reverberatory communication between the members of the assembly then leads
to persistent activation to engender temporally extended representations.</p> <p>The mean-field approach has been applied to biased-competition and
cooperation networks and has been used to model single neuronal responses,
fMRI activation patterns, psychophysical measurements, effects of
pharmacological agents, and effects of local cortical lesions ##UREF##3##[6]##,
##UREF##11##[24]##–##REF##15703227##[33]##. In the section
entitled <xref ref-type="sec" rid="s6">Cognitive and Clinical
Applications</xref>, we present one of these examples, in the context of
decision-making.</p> <title>Neural Modes and Masses</title> <p>The Fokker-Planck equation, (Equation 1), is a rather beautiful and simple expression
that prescribes the evolution of ensemble dynamics, given any initial conditions and
equations of motion that embed our neuronal model. However, it does not specify how
to encode or parameterize the density itself. There are several approaches to this.
These include binning the phase space and using a discrete approximation to a
continuous density. However, this can lead to a vast number of differential
equations, especially if there are multiple states for each population. One solution
to this is to reduce the number of states (i.e., dimension of the phase space) to
render the integration of the Fokker-Planck more tractable. One elegant example of
this reduction can be found in ##UREF##12##[34]##. Here, population dynamics are described by a set
of one-dimensional partial differential equations in terms of the distributions of
the refractory density (where the refractory state is defined by the time elapsed
since the last action potential). This furnishes realistic simulations of the
population activity of hippocampal pyramidal neurons, based on something known as
the refractory density equation and a single-neuron threshold model. The threshold
model is a conductance-based model with adaptation-providing currents.</p> <p>An alternative approach to dimension reduction is to approximate the ensemble
densities with a linear superposition of probabilistic modes or basis functions
<italic>η</italic>(<italic>ν</italic>) that cover phase space.
In this section, we overview this modal approach to ensemble dynamics, initially in
the general setting and then in the specific case, where the dynamics can be
captured by the activity of a single node.</p> <title/> <title>Moments and modes of density dynamics</title> <p>Instead of characterising the density dynamics explicitly, one can summarize
it in terms of coefficients parameterising the expression of modes:where
<italic>μ</italic> = <italic>η</italic>
<sup>−</sup>
<italic>p</italic>,
<italic>η</italic>
<sup>−</sup> being the generalized
inverse of the matrix encoding the basis set of modes.</p> <p>A useful choice for the basis functions are the eigenfunctions (i.e., eigen
vectors) of the Fokker-Planck operator, <italic>Q</italic>
##REF##10769319##[17]##, where
<italic>Qη</italic> = <italic>ηλ</italic>⇒<italic>η</italic>
<sup>−</sup>
<italic>Qη</italic> = <italic>λ</italic>
and <italic>λ</italic> is a leading-diagonal matrix of eigenvalues.
Because the Fokker-Planck operator conserves probability mass, all its real
eigenvalues are zero or negative. In the absence of mean-field effects, the
biorthogonality of the eigenfunctions effectively uncouples the dynamics of
the modes they represent
</p> <p>The last expression means that, following perturbation, each mode decays
exponentially, to disclose the equilibrium mode,
<italic>η</italic>
<sub>0</sub>, that has a zero eigenvalue.
Because the eigenvalues are complex (due to the fact that the Jacobian is
not symmetric), the decay is oscillatory in nature, with a frequency that is
proportional to the imaginary part of the eigenvalue and a rate constant
proportional to the real part. The key thing about this parameterisation is
that most modes will decay or dissipate very quickly. This means we only
have to consider a small number of modes, whose temporal evaluation can be
evaluated simply with
</p> <p>See ##REF##16087449##[35]## for an example of this approach, in which the
ensuing nonlinear differential equations were used in a forward model of
observed data. In summary, we can formulate the ensemble dynamics of any
neuronal system, given its equations of motion, using the equation above.
This specifies how the coefficients of probability modes would evolve from
any initial state or following a perturbation to the neuronal states. It
furnishes a set of coupled differential equations that can be integrated to
form predictions of real data or to generate emergent behaviors. We have
introduced parameterisation in terms of probability modes because it
provides a graceful link to neural mass models.</p> <title>Neural mass models</title> <p>Neural mass models can be regarded as a special case of ensemble density
models, where we summarize our description of the ensemble density with a
single number. Early examples can be found in the work of ##UREF##2##[5]##,##REF##13950415##[36]##,##REF##5832697##[37]##. The term mass action model was coined by
##UREF##13##[38]## as an alternative to density dynamics. These
models can be motivated as a description in terms of the expected values of
neuronal states, <italic>μ</italic>, under the assumption that the
equilibrium density has a point mass (i.e., a delta function). This is one
perspective on why these simple mean-field models are called neural mass
models. In short, we replace the full ensemble density with a mass at a
particular point and then summarize the density dynamics by the location of
that mass. What we are left with is a set of nonlinear differential
equations describing the evolution of this mode. But what have we thrown
away? In the full nonlinear Fokker-Planck formulation, different phase
functions or probability density moments could couple to each other; both
within and between populations or ensembles. For example, this means that
the average depolarisation in one ensemble could be affected by the
dispersion or variance of depolarisation in another. In neural mass models,
we ignore this possibility because we can only couple the expectations or
first moments. There are several devices that are used to compensate for
this simplification. Perhaps the most ubiquitous is the use of a sigmoid
function,
<italic>ς</italic>(<italic>μ<sub>ν</sub></italic>),
relating expected depolarisation to expected firing rate ##UREF##13##[38]##. This implicitly encodes variability in the
postsynaptic depolarisation, relative to the potential at which the neuron
would fire. A common form for neural mass equations of motion posits a
second order differential equation for expected voltage, or, equivalently,
two coupled first order equations,
<italic>μ</italic> = {<italic>μ<sub>ν</sub></italic>,<italic>μ<sub>i</sub></italic>} whereHere <italic>μ<sub>a</sub></italic> can be regarded
as capacitive current. The constant <italic>γ</italic> controls the
rise time of voltage, in response to inputs (see also the <xref ref-type="sec" rid="s4">Neural Field Models</xref> section). These
differential equations can be expressed as a convolution of inputs,
<italic>ς</italic>(<italic>μ<sub>ν</sub></italic>),
to give the expected depolarization,
<italic>μ<sub>ν</sub></italic>; i.e., the convolution of the
input signal with an impulse response kernel <italic>W</italic>(<italic>t</italic>)
</p> <p>The input is commonly construed to be a firing rate (or pulse density) and is
a sigmoid function, <italic>ς</italic>, of mean voltage of the same
or another ensemble. The coupling constant, <italic>κ</italic>,
scales the amplitude of this mean-field effect. This form of neural mass
model has been used extensively to model electrophysiological recordings
(e.g., ##REF##7578475##[39]##–##REF##15808977##[41]##) and has been
used recently as the basis of a generative model for event-related
potentials that can be inverted using real data ##REF##16473023##[42]##.</p> <p>In summary, neural mass models are special cases of ensemble density models
that are furnished by ignoring all but the expectation or mean of the
ensemble density. This affords a considerable simplification of the dynamics
and allows one to focus on the behavior of a large number of ensembles,
without having to worry about an explosion in the number of dimensions or
differential equations one has to integrate. The final sort of model we will
consider is the generalisation of neural mass models that allow for states
that are functionals of position on the cortical sheet. These are referred
to as neural field models and are discussed in the following sections.</p> <title>Neural Field Models</title> <p>The density dynamics and neural mass models above covered state the attributes of
point processes, such as EEG sources, neurons, or neuronal compartments. An
important extension of these models speaks to the fact that neuronal dynamics play
out on a spatially extended cortical sheet. In other words, states like the
depolarisation of an excitatory ensemble in the granular layer of cortex can be
regarded as a continuum or field, which is a function of space, <italic>x</italic>,
and time,
<italic>μ</italic>(<italic>t</italic>)→<italic>μ</italic>(<italic>x</italic>,<italic>t</italic>).
This allows one to formulate the dynamics of the expected field in terms of partial
differential equations in space and time. These are essentially wave equations that
accommodate lateral interactions. Although we consider neural field models last,
they were among the first mean-field models of neuronal dynamics ##REF##4332108##[43]##,##REF##4767470##[44]##. Key
forms for neural field equations were proposed and analysed by ##UREF##14##[45]##–##REF##911931##[47]##. These
models were generalized by ##REF##10062950##[48]##,##UREF##15##[49]## who, critically, considered delays in the
propagation of spikes over space. The introduction of propagation delays leads to
dynamics that are very reminiscent of those observed empirically.</p> <p>Typically, neural field models can be construed as a spatiotemporal convolution
(c.f., Equation 38) that can be written in terms of a Green's function; e.g.,where |<italic>x</italic>−<italic>x′</italic>| is
the distance between the spatial locations <italic>x</italic> and
<italic>x′</italic>, <italic>c</italic> is the characteristic speed of
spike propagation, and <italic>γ</italic> reflects the spatial decay of
lateral interactions. The corresponding second order equations of motion are a
neural wave equation (see ##REF##10062950##[48]##,##UREF##15##[49]## and below)where
<italic>γ</italic> = <italic>c</italic>/<italic>r</italic>
and ▽<sup>2</sup> is the Laplacian. The formal similarity with the neural
mass model in (37) is self-evident. These sorts of models have been extremely useful
in modeling spatiotemporally extended dynamics (e.g., ##UREF##16##[50]##–##UREF##18##[53]##). The
generic form of neural field dynamics can be written as (see also ##UREF##18##[53]##):where
<italic>μ</italic> = <italic>μ</italic>(<italic>x</italic>,<italic>t</italic>)
is the neural field, capturing the neural mass activity at time <italic>t</italic>
and position <italic>x</italic>. <italic>f</italic>(<italic>μ</italic>)
captures the local dynamics of the neural field, and
<italic>T<sub>c</sub></italic> = <italic>t</italic>−|<italic>x</italic>−<italic>x′</italic>|/<italic>c</italic>
is the time delay due to signal propagation. <italic>h</italic> is a constant
threshold value and Γ is the spatial domain of the neural field, where
<italic>x</italic> ∈
Γ = [0,<italic>L</italic>].
The kernel
<italic>W</italic>(|<italic>x</italic>−<italic>x′</italic>|)
denotes the connectivity function, which is translationally invariant in space,
i.e., the probability that two neural masses are connected depends only on the
distance between them. If we neglect the local dynamics
<italic>f</italic>(<italic>μ</italic>), , and use an exponential kernel as in Equation 39, we recover
Equations 39 and 40. This approximation is valid when the axonal delays contribute
mostly to the dynamics, for instance in large-scale networks, when the local
dynamics are much faster than the network dynamics. It is easy to show that most
realistic connectivity kernels provide a neural wave equation like Equation 40; this
is due to the fact that the connectivity must remain integrable. As above, the
parameter <italic>c</italic> is the propagation velocity of action potentials
traveling down an axon. If
<italic>f</italic>(<italic>μ</italic>) = −<italic>ζ
μ</italic>, then the constant
<italic>ζ</italic>&gt;0∈ℜ represents the growth rate
of the neural mass. <italic>α</italic>&gt;0 is a scaling constant. Under
instantaneous interactions, <italic>c</italic>→∞, single
population models with locally excitatory and laterally inhibitory connectivity can
support global periodic stationary patterns in one dimension as well as single or
multiple localized solutions (bumps and multi-bumps) ##REF##911931##[47]##. This class of models
are also sometimes referred to as continuous attractor neural networks (CANN). When
the firing rate, <italic>ς</italic>, is a Heaviside step function, ##UREF##14##[45]## was able
to construct an explicit one-bump solution of the formwhere the value <italic>a</italic> corresponds to the width of the
bump. Amari also identified criteria to determine if only one bump, multiple bumps,
or periodic solutions exist and if they are stable. This simple mathematical model
can be extended naturally to accommodate multiple populations and cortical sheets,
spike frequency adaptation, neuromodulation, slow ionic currents, and more
sophisticated forms of synaptic and dendritic processing as described in the review
articles ##REF##15319516##[4]##,##UREF##19##[54]##,##REF##16059785##[55]##.
Spatially localized bump solutions are equivalent to persistent activity and have
been linked to working memory in prefrontal cortex ##REF##8547793##[56]##,##REF##7695894##[57]##. During
behavioral tasks, this persistent elevated neuronal firing can last for tens of
seconds after the stimulus is no longer present. Such persistent activity appears to
maintain a representation of the stimulus until the response task is completed.
Local recurrent circuitry has received the most attention, but other theoretical
mechanisms for the maintenance of persistent activity, including local recurrent
synaptic feedback and intrinsic cellular bistability ##REF##11127836##[58]##,##REF##11476885##[59]##, have
been put forward. The latter will be captured by specific choices of the local
dynamics, <italic>f</italic>(<italic>μ</italic>), in Equation 41; for
instance, ##REF##15806392##[60]## choose a cubic-shaped function of the firing rate,
which, under appropriate parameters, allows for intrinsic bistability. Single bump
solutions have been used for neural modeling of the head-direction system ##REF##12895528##[61]##–##REF##11539168##[64]##, place cells ##UREF##21##[65]##–##REF##10495029##[68]##, movement initiation
##UREF##22##[69]##, and feature selectivity in visual cortex, where bump
formation is related to the tuning of a particular neuron's response ##REF##7731993##[70]##.
Here the neural fields maintain the firing of its neurons to represent any location
along a continuous physical dimension such as head direction, spatial location, or
spatial view. The mathematical analysis of the neural field models is typically
performed with linear stability theory, weakly nonlinear perturbation analysis, and
numerical simulations. With more than one population, nonstationary (traveling)
patterns are also possible. In two dimensions, many other interesting patterns can
occur, such as spiral waves ##UREF##23##[71]##, target waves, and doubly periodic patterns.
These latter patterns take the form of stripes and checkerboard-like patterns, and
have been linked to drug-induced visual hallucinations ##REF##486593##[72]##. For smooth
sigmoidal firing rates, no closed-form spatially localized solutions are known,
though much insight into the form of multibump solutions has been obtained using
techniques first developed for the study of fourth-order pattern forming systems
##UREF##24##[73]##.
Moreover, in systems with mixed (excitatory and inhibitory) connectivity or
excitatory systems with adaptive currents, solitary traveling pulses are also
possible. The bifurcation structure of traveling waves in neural fields can be
analysed using a so-called Evans function and has recently been explored in great
detail ##UREF##25##[74]##.</p> <p>Much experimental evidence, supporting the existence of neural fields, has been
accumulated (see ##UREF##18##[53]## for a summary). Most of these results are
furnished by slice studies of pharmacologically treated tissue, taken from the
cortex ##REF##3143812##[75]##–##REF##10366633##[77]##, hippocampus ##UREF##26##[78]##, and
thalamus ##REF##7500152##[79]##. In brain slices, these waves can take the form of
synchronous discharges, as seen during epileptic seizures ##UREF##27##[80]##, and spreading
excitation associated with sensory processing ##REF##11182079##[81]##. For traveling
waves, the propagation speed depends on the threshold, <italic>h</italic>, which has
been established indirectly in real neural tissue (rat cortical slices bathed in the
GABA-A blocker picrotoxin) by ##REF##15698234##[82]##. These experiments exploit the fact that (i)
cortical neurons have long apical dendrites and are easily polarized by an electric
field, and (ii) that epileptiform bursts can be initiated by stimulation. A positive
(negative) electric field applied across the slice increased (decreased) the speed
of wave propagation, consistent with the theoretical predictions of neural field
theory, assuming that a positive (negative) electric field reduces (increases) the
threshold, <italic>h</italic>, in Equation 42.</p> <title/> <title>Recent developments in neural field models</title> <p>More and more physiological constraints have been incorporated into neural
field models of the type discussed here (see Equations 39 and 40). These
include features such as separate excitatory and inhibitory neural
populations (pyramidal cells and interneurons), nonlinear neural responses,
synaptic, dendritic, cell-body, and axonal dynamics, and corticothalamic
feedback ##UREF##13##[38]##, ##REF##4332108##[43]##, ##REF##4767470##[44]##, ##REF##10062950##[48]##, ##UREF##16##[50]##, ##UREF##28##[83]##–##REF##12111274##[87]##. A key feature of recent models is that they
use parameters that are of functional significance for EEG generation and
other aspects of brain function; for example, synaptic time constants,
amount of neurotransmitter release or reuptake, and the speed of signal
propagation along dendrites. Inferences can also be made about the
parameters of the nonlinear IF response at the cell body, and about speeds,
ranges, and time delays of subsequent axonal propagation, both within the
cortex and on extracortical paths (e.g., via the thalamus). It is also
possible to estimate quantities that parametrize volume conduction in
tissues overlying the cortex, which affect EEG measurements ##UREF##32##[88]##, or hemodynamic responses that determine the
blood oxygen level–dependent (BOLD) signals ##UREF##33##[89]##. Each of
these parameters is constrained by physiological and anatomical
measurements, or, in a few cases, by other types of modeling. A key aim in
modeling is to strike a balance between having too few parameters to be
realistic, and too many for the data to be able to constrain them
effectively.</p> <p>Recent work in this area has resulted in numerous quantitatively verified
predictions about brain electrical activity, including EEG time series ##UREF##31##[86]##,##REF##12111274##[87]##,##UREF##34##[90]##, spectra ##UREF##16##[50]##,##UREF##31##[86]##,##REF##12111274##[87]##,##UREF##34##[90]##,##UREF##35##[91]##, coherence
and correlations, evoked response potentials (ERPs) ##REF##12111274##[87]##, and seizure
dynamics ##UREF##31##[86]##,##UREF##34##[90]##,##REF##16280462##[92]##. Inversion of these models has also furnished
estimates of underlying physiological parameters and their variations across
the brain, in different states of arousal and pathophysiology ##UREF##31##[86]##,##REF##15281141##[93]##,##REF##15501472##[94]##.</p> <p>There are several interesting aspects to these modeling initiatives, which
generalize the variants discussed in earlier sections: (i) synaptic and
dendritic dynamics and summation of synaptic inputs to determine potentials
at the cell body (soma), (ii) generation of pulses at the axonal hillock,
and (iii) propagation of pulses within and between neural populations. We
now look more closely at these key issues.</p> <title>
<italic>Synaptodendritic dynamics and the soma potential</italic>
</title> <p>Assume that the brain contains multiple populations of neurons, indexed by
the subscript <italic>a</italic>, which labels simultaneously the structure
in which a given population lies (e.g., a particular nucleus) and the type
of neuron (e.g., interneuron, pyramidal cell). Then the spatially continuous
soma potential, <italic>V<sub>a</sub></italic>, is the sum of contributions,
<italic>V<sub>ab</sub></italic>, arriving as a result of activity at
each type of (mainly) dendritic synapse <italic>b</italic>, where
<italic>b</italic> indexes both the incoming neural population and the
neurotransmitter type of the receptor. (Note that
<italic>V<sub>a</sub></italic> is linearly related to the current reaching
the soma, and to <italic>μ</italic> in earlier sections.) Thus we writewhere
<bold>r</bold> = (<italic>x</italic>,<italic>y</italic>)
denotes the spatial coordinates, <italic>t</italic> the time. The summation
is assumed to be linear, and all potentials are measured relative to the
resting potential ##UREF##36##[95]##. For moderate perturbations relative to a
steady state, the value of the resting potential can be subsumed into the
values of other parameters ##UREF##36##[95]##. As above, the cortex is approximated as
a 2-D sheet and <bold>r</bold> is assumed to be the actual position in the
case of the cortex; other structures, such as the thalamus, are linked to
the cortex via a primary topographic map. This map links points in a
one-to-one manner between structures; i.e., we assign the same value of r to
such points. Hence, in structures other than the cortex, this dimensional
map coordinate, <bold>r</bold>, denotes a rescaled physical dimension (i.e.
the physical coordinate multiplied by the ratio of the cortical scale to the
structure's scale), a point that should be remembered when
interpreting values of spatial parameters in these structures.</p> <p>The subpotentials, <italic>V<sub>ab</sub></italic>, respond in different ways
to incoming spikes, depending on their synaptic dynamics (ion-channel
kinetics, diffusion in the synaptic cleft, etc.), and on subsequent signal
dispersion in the dendrites. The resulting soma response to a delta-function
input at the synapse can be approximated via the differential equation ##UREF##16##[50]##.where <italic>ν<sub>ab</sub></italic> is a coupling
strength, <italic>N<sub>ab</sub></italic> is the mean number of synapses on
neuron <italic>a</italic> from neurons <italic>b</italic>,
<italic>s<sub>ab</sub></italic> is the mean time-integrated strength of
the response of <italic>V</italic> per incoming spike, and
<italic>θ<sub>ab</sub></italic> is the average rate of
incoming spikes (allowing for the possibility of a discrete time delay,
<italic>τ<sub>ab</sub></italic>, between populations
<italic>b</italic> and <italic>a</italic> in addition to any delays due
to spreading within populations). The parameter
<italic>α<sub>ab</sub></italic> is the mean decay rate of the
soma response to a delta-function synaptic input, while
<italic>β<sub>ab</sub></italic> is the mean rise rate: this
biexponential form has been found to be a good approximation ##UREF##5##[8]##,##UREF##16##[50]##,##UREF##37##[96]##,##UREF##38##[97]##. If the
<italic>α<sub>ab</sub></italic> and
<italic>β<sub>ab</sub></italic> are independent of
<italic>b</italic> (which is not generally the case), then the subscript
<italic>b</italic> on <italic>D<sub>ab</sub></italic> can be omitted and
<italic>V<sub>a</sub></italic> itself satisfies Equation 44 with the
right side of Equation 44 replaced by the sum of
<italic>P<sub>ab</sub></italic> over <italic>b</italic>. This approximation
is also valid if <italic>α</italic> and <italic>β</italic>
are interpreted as effective values, averaged over subpopulations.</p> <title>
<italic>Pulse generation</italic>
</title> <p>In cells with voltage-gated ion channels, action potentials are produced at
the axonal hillock when the soma potential exceeds some threshold
<italic>θ<sub>a</sub></italic>. When averaged over a
population of neurons, with normal response characteristics, a reasonable
approximation for the firing rate, <italic>Q</italic>, iswhere <italic>Q<sub>a</sub></italic>
<sub>max</sub> is the
maximum firing rate and <italic>S<sub>a</sub></italic> is a monotonic
increasing sigmoidal function that approaches zero as
<italic>V<sub>a</sub></italic>→−∞ and unity as
<italic>V<sub>a</sub></italic>→∞. A commonly used
approximation iswhere <italic>θ<sub>a</sub></italic> is the firing
threshold for channels of type <italic>a</italic> and  is the standard deviation of the threshold over the
population.</p> <title>
<italic>Axonal propagation</italic>
</title> <p>Spatiotemporal propagation of pulses within and between populations
determines the values of <italic>φ<sub>ab</sub></italic>. If we
indicate the firing rate <italic>Q<sub>a</sub></italic> for the cell type
<italic>a</italic> by a subscript, then
<italic>φ<sub>ab</sub></italic> can be expressed in terms of the
firing rate at other locations and earlier times. If we assume linear
propagation, signals propagate as described by the neural field equation
(Equation 40).where, as per Equation 40, <italic>r<sub>ab</sub></italic> is
the characteristic range of axons, including dendritic arborization,
<italic>c<sub>ab</sub></italic> is the characteristic velocity of
signals in these axons, and
<italic>γ<sub>ab</sub></italic> = <italic>c<sub>ab</sub></italic>
/ <italic>r<sub>ab</sub></italic> is the resulting temporal damping
coefficient in the absence of pulse regeneration. Note that, in comparision
to the use of the terms <italic>μ<sub>ν</sub></italic> and
<italic>ς</italic>(<italic>μ<sub>ν</sub></italic>)
in Equation 40, the present wave-equation is formalized in relationship to
population-specific pulse densities,
<italic>φ<sub>ab</sub></italic>, and firing rates,
<italic>Q<sub>a,b</sub></italic>. By employing population-specific
fields and parameters, it allows each population to generate a family of
outgoing fields that propagate to different populations in different ways.
Equation 49 is also satisfied if <italic>φ<sub>ab</sub></italic> is
replaced by the free propagator  and the right side is replaced by a source of the form
<italic>δ</italic>(<bold>r</bold>−<bold>r′</bold>)<italic>δ</italic>(<italic>t</italic>−<italic>t′</italic>).
In Fourier space, this giveswhere <bold><italic>k</italic></bold> = (k<sub>x</sub>,k<sub>y</sub>) is
the wave vector and <italic>ω</italic> is the angular frequency.
Critically, the neural field Equation 49 enables very diffuse (i.e., not
topographically specific) connections between populations to be handled
straightforwardly, simply by increasing <italic>r<sub>ab</sub></italic>
while reducing <italic>γ<sub>ab</sub></italic>, thereby allowing
influences to propagate long distances with little damping.</p> <title>
<italic>Parameters and modulations</italic>
</title> <p>The above equations contain a number of parameters encoding physiology and
anatomy (e.g., coupling strengths, firing thresholds, time delays,
velocities, etc.). In general, these can vary in space, due to differences
among brain regions, and in time, due to effects like habituation,
facilitation, and adaptation. In brief, time-dependent effects can be
included in neural field models by adding dynamical equations for the
evolution of the parameters. Typically, these take a form in which parameter
changes are driven by firing rates or voltages, with appropriate time
constants. The simplest such formulation is ##UREF##36##[95]##
where <italic>x</italic> is the evolving parameter,
<italic>y</italic> is the quantity that drives the evolution,
<italic>x</italic>
<sup>(0)</sup> and <italic>y</italic>
<sup>(0)</sup>
are steady state values, and <italic>x</italic>
<sup>(1)</sup> is a constant
that describes the strength of feedback. The symbol ⊗ indicates a
convolution of the driver with the temporal response function
<italic>H</italic>(<italic>t</italic>), which incorporates the time
constants of the dynamics. If we use the normalized formthen we find the differential equivalent of Equation 53:
</p> <title>
<italic>Steady states and dynamics</italic>
</title> <p>Here, we first discuss how to find the steady states of neural field models.
Important phenomena have been studied by linearizing these models around
their steady state solutions. Hence, we discuss linear properties of such
models, including how to make predictions of observable quantities from
them; including transfer functions, spectra, and correlation and coherence
functions. In doing this, we assume for simplicity that all the model
parameters are constant in time and space, although it is possible to relax
this assumption at some cost in complexity. Linear predictions from neural
field models have accounted successfully for a range of experimental
phenomena, as mentioned above. Nonlinear dynamics of such models have also
been discussed in the literature, resulting in successful predictions of
epileptic dynamics, for example ##UREF##31##[86]##,##REF##16280462##[92]##, but are not considered here (but see the
<xref ref-type="sec" rid="s6">Cognitive and Clinical Applications</xref>
section).</p> <p>
<italic>Steady states and global dynamics</italic>. Previous work has shown
that many properties of neuronal dynamics can be obtained by regarding
activity changes as perturbations of a steady state ##UREF##31##[86]##. Spatially
uniform steady states can be obtained by solving the preceding equations
with all time and space derivatives set to zero, assuming that the
parameters are spatially constant. The spatially uniform steady states are
thus the solutions of the set of equationswhich are generally transcendental in form.</p> <p>
<italic>Linear equations for activity</italic>. Of the relevant equations
above, all but Equation 48 are linear in <italic>Q</italic>. Equation 48 can
be linearized by replacing the sigmoid, <italic>S<sub>a</sub></italic>, by
its slope, <italic>ρ<sub>a</sub></italic>, at the steady state value
of <italic>V<sub>a</sub></italic>; we also approximate this quantity as
constant. If we Fourier transform the resulting set of linear equations, we
find for the fluctuating partswhere  is given by Equation 50 and we have assumed that all the
parameters of the equations (but not the fields of activity) are constant on
the timescales of interest. Note that we have assumed the system to be
unbounded in order to employ a continuous Fourier transform here. The case
of bounded systems with discrete spatial eigenmodes can be treated
analogously.</p> <p>For any given spatial wavenumber, <bold>k</bold>, and temporal frequency,
<italic>ω</italic>, Equations 56–60 can be rearranged
to obtainwhere the gains are defined by
<italic>G<sub>ab</sub></italic> = <italic>ρ<sub>a</sub>ν<sub>ab</sub></italic>.</p> <p>If there are <italic>N</italic>′ neural populations and
<italic>J</italic>′ stimulus sources, and we assume that there is
no feedback of stimuli on themselves, or of the brain on stimuli, then we
can write Equation 61 aswhere the sum on the left of Equation 63 extends only over
populations in the brain, while the sum on the right covers only stimulus
sources, denoted by <italic>j</italic>. <italic>Q<sub>j</sub></italic> is
written as <italic>N<sub>j</sub></italic> to make the distinction between
population firing rates and incoming stimulus rates absolutely clear. We can
now write Equation 63 in matrix form aswhere <bold>A</bold> is an
<italic>N</italic>′×<italic>N</italic>′
matrix, <bold><italic>Q</italic></bold> is an <italic>N</italic>′-element column vector,
<bold>B</bold> is an
<italic>N</italic>′×<italic>J</italic>′
matrix, and <bold>N</bold> is a <italic>J</italic>′-element column
vector. We can simply invert Equation 66 to find <bold><italic>Q</italic></bold> in terms of the stimuli <bold>N</bold>:where
<bold>T</bold> = <bold>A</bold>
<sup>−1</sup>
<bold>B</bold>
is the <italic>N</italic>′×<italic>J</italic>′
transfer matrix of the system. The element <italic>T<sub>aj</sub></italic>
is the response of <italic>Q<sub>a</sub></italic> to a change in
<italic>N<sub>j</sub></italic> at the same frequency and wave
vector.</p> <p>
<italic>Observables</italic>. A measurable scalar quantity
<italic>ψ</italic>, such as an EEG scalp voltage or voltage
difference, can generally be approximated by a linear function of the firing
rates, <italic>Q<sub>a</sub></italic>. For example, a scalp potential may
involve contributions from several populations, with various weights (that
may include filtering by volume conduction effects). In this case, at given
<italic>ω</italic>,where <bold>M</bold> is an
<italic>N</italic>′-element row vector of complex-valued
measurement coefficients that encode spatiotemporal filtering
characteristics, phase shifts, etc. For example, the coefficients of the
matrix <bold>M</bold> can be chosen such that
<italic>ψ</italic>(<bold>k</bold>,<italic>ω</italic>) = <italic>φ<sub>ab</sub></italic>(<bold>k</bold>,<italic>ω</italic>).
Further classes of measurement functions are those relating the neural
activity to, for example, local field potentials, multiunit activity, the
blood oxygen level–dependent (BOLD) response that forms the basis
of functional magnetic resonance imaging (fMRI), the metabolic responses
underlying positron emission tomography (PET), or single-photon emission
computed tomography (SPECT). In what follows, we will implicitly absorb
<bold>M</bold> into <bold>T</bold> for simplicity.</p> <p>
<italic>Dispersion and stability</italic>. The dispersion relation of linear
waves in the system is given byand the system is stable at a particular real <bold>k</bold>
if all the frequency roots of this equation have negative imaginary parts.
If the steady state is stable for all <bold>k</bold>, spectra and other
properties of the linear perturbations can be self-consistently defined;
otherwise a fully nonlinear analysis is needed.</p> <p>
<italic>Spectra</italic>. The power spectral density of
<italic>ψ</italic> at <bold>k</bold> and
<italic>ω</italic> isThe frequency and wavenumber spectra are thenA position-dependent frequency cross-spectrum can be
calculated from Equation 70:where the angle brackets denote an average over multiple
trials and/or over the phase of the exogenous stimuli that drive the system.
The spectrum at a particular point, <bold>r</bold>, is
<italic>P</italic>(<bold>r</bold>,<bold>r</bold>′,<italic>ω</italic>).</p> <p>
<italic>Correlation and coherence functions</italic>. In steady state, the
two-point correlation function can be obtained from Equation 73 via the
Wiener-Khinchtine theorem, givingIn the case where the system is statistically uniform,
Equation 74 depends only on the separation
<bold>R</bold> = <bold>r</bold>′−<bold>r</bold>, givingwherehas been used and the arguments of <bold>T</bold> and
<bold>N</bold> have been shown for emphasis. At
<bold>R</bold> = 0, Equation 74 becomes the
Fourier transform of the local power spectrum. In terms of the above
expressions, the normalized correlation function and the coherence function,
which are both used widely in the literature, arerespectively.</p> <p>
<italic>Time series and evoked potentials</italic>. The time series of
<italic>ψ</italic> at a given point can be obtained via the
transfer function, by first calculating the Fourier form of the stimuli that
generate it; these can be background noise sources, discrete impulses, or
sinusoidal drives, for example. In the case of an impulsive stimulus, the
resulting ERP is obtained by setting
</p> <p>Similarly, for a spatially uniform sinusoidal drive, the resulting steady
state evoked potential (SSEP) is obtained by usingwhere <italic>ω</italic>
<sub>0</sub> is the drive
frequency and <italic>φ</italic> is its phase.</p> <p>
<italic>Case of one long-range population</italic>. An important case, in
many applications, is the situation where spatial spreading of activity is
dominated by the axons of one population, typically because they have the
longest range, are most numerous, or have the highest axonal velocity. In
this case, one can ignore the <bold>k</bold> dependence in the other
propagators, and it becomes possible to express the transfer function with
elements of the formwhere  is typically a complicated expression depending on the
various <italic>J<sub>ab</sub></italic>(<italic>ω</italic>).</p> <title>
<italic>Heterogeneous connectivity in neural fields</italic>
</title> <p>The brain's network dynamics depend on the connectivity within
individual areas, as well as generic and specific patterns of connectivity
among cortical and subcortical areas ##REF##15319516##[4]##,##UREF##6##[9]##,##UREF##39##[98]##. Intrinsic or
intracortical fibers are confined to cortical gray matter in which the
cortical neurons reside; these intrinsic connections define the local
connectivity within an area. Intracortical fibers are mostly unmyelinated
and extend laterally up to 1 cm (in the human brain) with excitatory and
inhibitory connections. Their distribution is mostly invariant under spatial
translations (homogeneous) ##UREF##29##[84]##,##UREF##40##[99]##, which
fits the assumptions on the connectivity function in neural fields so far.
On the other hand, the corticocortical (extrinsic) fiber system contains
fibers which leave the gray matter and connect distant areas (up to 20 cm
##UREF##29##[84]##). This fiber system is myelinated, which
increases the transmission speed by an order of magnitude, and is not
invariant under spatial translations (heterogeneous); in fact it is patchy
##UREF##40##[99]##. Due to finite transmission speeds, time
delays of interareal communication can reach 50–100 ms ##UREF##29##[84]##,
which is not negligible. Several studies have focused on spatially
continuous neural fields, which describe the temporal change of neural
activity on local scales, typically within a brain area (see ##REF##15319516##[4]##,##UREF##19##[54]##,##REF##16059785##[55]## for reviews), assuming homogeneous
connectivity and time delays. As discussed in the previous section, early
attempts include neural field theories which approximate the large-scale
components of the connectivity matrix as translationally invariant and
decaying over space ##UREF##14##[45]##,##REF##10062950##[48]##,##UREF##16##[50]##. These approaches have been successful in
capturing key phenomena of large-scale brain dynamics, including
characteristic EEG power spectra ##UREF##14##[45]##,##UREF##16##[50]##, epilepsy ##REF##16280462##[92]##, and MEG
activity during sensorimotor coordination ##UREF##15##[49]##. Here we review
extensions of these efforts and address network stability under variation of
(i) intracortical (intrinsic) connectivity, (ii) transmission speed, and
(iii) length of corticocortical (extrinsic) fibers. All three anatomical
attributes undergo characteristic changes during the development of the
human brain and its function, as well changing in the aged and diseased
brain (see ##UREF##6##[9]## for an overview). As a first step, we can
split the connectivity function, <italic>W</italic>, into two parts, the
homogeneous connectivity,
<italic>W<sub>hom</sub></italic>(|<italic>x</italic>−<italic>y</italic>|),
which depends only on the distance, and the heterogeneous connectivity,
<italic>W<sub>het</sub></italic>(<italic>x</italic>,<italic>y</italic>),
which captures the effects of the extrinsic fiber system (for an alternative
approach with applications to visual gamma phenomena, see ##UREF##41##[100]##–##REF##17899164##[102]##). We can
then rewrite the neural field equation as follows:where
<italic>T<sub>c</sub></italic> = <italic>t</italic>−|<italic>x</italic>−<italic>y</italic>|
/<italic>c</italic> and <italic>c</italic> is the propagation speed
through the heterogeneous corticocortical or extrinsic connections. These
fibers are myelinated and hence to be distinguished from the typically
unmyelinated (hence slower) intracortical fibers. The latter intrinsic
fibers have a transmission speed of <italic>c<sup>hom</sup></italic> and a
transmission delay . If <italic>W<sub>het</sub></italic> describes the
connectivity of <italic>n</italic> areas, then it can always be written as a
sum of two-point connections viawhere <italic>ν<sub>ij</sub></italic> ∈
ℜ again represents the coupling strength between areas at
<italic>x<sub>i</sub></italic> and <italic>x<sub>j</sub></italic>.
The fixed point solution is given by
<italic>μ</italic>
<sub>0</sub>(<italic>x</italic>) with . To gain insight into the linear stability of this
equilibrium solution
<italic>μ</italic>
<sub>0</sub>(<italic>x</italic>), we perform a
mode expansion of
<italic>μ</italic>(<italic>x</italic>,<italic>t</italic>) into a set
of spatial basis functions
{<italic>ϕ<sub>k</sub></italic>(<italic>x</italic>)} such thatwhere
<italic>ξ<sub>k</sub></italic>(<italic>t</italic>) is the
time-dependent amplitude related to the spatial basis function
<italic>ϕ<sub>k</sub></italic>(<italic>x</italic>). The
adjoint set of spatial biorthogonal basis functions is denoted by . It will be generally true (except in degenerate cases)
that only one spatial pattern will become unstable first. For simplicity, we
consider the stationary solution
<italic>μ</italic>
<sub>0</sub>(<italic>x</italic>) = 0
to be the rest state and consider its deviations
<italic>μ</italic>(<italic>x</italic>,<italic>t</italic>) = <italic>ξ<sub>k′</sub></italic>(<italic>t</italic>)<italic>ϕ<sub>k′</sub></italic>
(<italic>x</italic>)+<italic>c</italic>.<italic>c</italic>.,
where <italic>c</italic>.<italic>c</italic>. denotes the complex conjugate.
Then the linear stability of each temporal mode is given bywhere
<italic>d</italic> = |<italic>x<sub>i</sub></italic>−<italic>x<sub>j</sub></italic>|&gt;0.
Also,  and  (but for simplicity, we drop the tilde in  from now on).</p> <p>Let us pause for a moment and reflect upon the significance of Equation 85.
Equation 85 describes the rate of (temporal) change,
∂<italic><sub>t</sub>ξ<sub>k</sub></italic>(<italic>t</italic>),
of its corresponding spatial neural activation pattern,
<italic>ϕ<sub>k</sub></italic>(<italic>x</italic>). This
pattern will change as a function of its own spatial configuration,
<italic>ϕ<sub>k</sub></italic>(<italic>x</italic>), the
connections (<italic>W<sub>hom</sub></italic> and
<italic>ν<sub>ij</sub></italic>), and, last but not least, the
transmission times of information exchange,  and <italic>d</italic>/<italic>c</italic>. If the rate of
change,
∂<italic><sub>t</sub>ξ<sub>k</sub></italic>(<italic>t</italic>),
is positive, then the particular pattern
<italic>ϕ<sub>k</sub></italic>(<italic>x</italic>) is unstable,
otherwise it is stable. In other words, Equation 85 identifies
quantitatively how a particular neural activation is impacted by its local
and global connectivity in a biologically realistic environment, including
signal exchange with finite and varying (intracortical versus
corticocortical) transmission speeds. Every treatment of the interplay of
anatomical connectivity (local and global connections) and functional
connectivity (network dynamics) will have to be represented in the form of
Equation 85 or a variation thereof. In this sense, we have here achieved our
goal stated in the introduction of this section.</p> <p>To illustrate the effects of interplay between anatomical and functional
connectivity, we discuss a simple example following ##UREF##43##[103]##,##UREF##44##[104]##. We assume that there exists only a single
corticocortical fiber with terminals at locations
<italic>x</italic>
<sub>1</sub> and <italic>x</italic>
<sub>2</sub>, that is
<italic>n</italic> = 2. Then we have an
architecture as shown in ##FIG##0##Figure
1##. Our objective is to identify the stability boundaries of the
rest state activity, here the equilibrium solution
<italic>μ</italic>
<sub>0</sub>(<italic>x</italic>) = 0.</p> <p>We will consider eigenfunctions of the form
<italic>ϕ<sub>k</sub></italic>(<italic>x</italic>) = <italic>e<sup>ikx</sup></italic>.
Changing the variables such that
<italic>z</italic> = <italic>y</italic>−<italic>x</italic>
and assuming a solution of the form
<italic>ξ<sub>k</sub></italic>(<italic>t</italic>) = <italic>e<sup>λt</sup></italic>,
<italic>λ</italic> ∈ <italic>C</italic>, the stability
condition can then be determined by the following characteristic equation:
</p> <p>Linear stability of Equation 86 is obtained if
<italic>Re</italic>[<italic>λ</italic>]&lt;0
and is lost, according to ##UREF##45##[105]##, at
<italic>Re</italic>[<italic>λ</italic>] = 0,
that is
<italic>λ</italic> = <italic>iω</italic>.
##FIG##1##Figure 2## shows
various connectivity kernels, <italic>W<sub>hom</sub></italic>, that are
often found in the literature.</p> <p>Qubbaj and Jirsa ##UREF##44##[104]## discussed the properties of the
characteristic Equation 86 in detail, considering separately the special
cases of symmetric and asymmetric connectivity, <italic>W</italic>. The
characteristic equation defines the critical boundary in the parameter space
of <italic>ν</italic>
<sub>12</sub>,
<italic>ν</italic>
<sub>21</sub>, <italic>c</italic>,
<italic>c</italic>
<sup>hom</sup>, at which the resting activity,
<italic>μ</italic>
<sub>0</sub>(<italic>x</italic>) = 0,
becomes unstable. Recall that <italic>c</italic> and
<italic>c</italic>
<sup>hom</sup> are the conduction velocities along
extrinsic and intrinsic axons, respectively. The general result of ##UREF##44##[104]## can be represented as a critical surface
separating stable from unstable regimes as shown in ##FIG##2##Figure 3##. Here the critical transmission
delay,
<italic>τ</italic> = <italic>d</italic>/<italic>c</italic>,
through the heterogeneous fiber is plotted as a function of the real and
imaginary part of the eigenvalue of the connectivity, <italic>W</italic>.
Essentially a heterogeneous fiber with symmetric weights,
<italic>ν</italic>
<sub>21</sub> = <italic>ν</italic>
<sub>12</sub> = <italic>ν</italic>,
has only real eigenvalues, whereas asymmetries result in imaginary
components. We find that for positive values of <italic>ν</italic>
greater than a critical value, the system becomes unstable through a
non-oscillatory instability for all values of <italic>c</italic>,
<italic>c</italic>
<sup>hom</sup>, (bold line in ##FIG##2##Figure 3##). Within the cylindrical
component of the surface, the equilibrium of the system remains always
stable for all values of <italic>c</italic>,
<italic>c</italic>
<sup>hom</sup>, and hence a time delay shows no effect. In
the other regimes of the critical surface, the system typically destabilizes
via an oscillatory instability, <italic>ω</italic>≠0, and is
sensitive to time delays. The surface shown in ##FIG##2##Figure 3## represents the lower bound of
stability with
<italic>c</italic>/<italic>c<sup>hom</sup></italic> = 1.
Increases of the ratio, <italic>c</italic>/<italic>c<sup>hom</sup></italic>,
equivalent to increases of degree of myelination, result in a larger
enclosed volume by this surface, i.e., increase of stability. The largest
stability region is that for a purely inhibitory kernel followed by that of
a local inhibitory and lateral excitatory kernel. The next largest involves
a local excitatory and lateral inhibitory kernel. The smallest stability
region is obtained for a purely excitatory kernel.</p> <p>A surprising result is that all changes of the extrinsic pathways have the
same qualitative effect on the stability of the network, independent of the
local intrinsic architecture. This is not trivial, since despite the fact
that extrinsic pathways are always excitatory the net effect on the network
dynamics could have been inhibitory, if the local architecture is dominated
by inhibition. Hence qualitatively different results on the total stability
could have been expected. Such is not the case, as we have shown here.
Obviously the local architecture has quantitative effects on the overall
network stability, but not qualitatively differentiated effects. Purely
inhibitory local architectures are most stable, purely excitatory
architectures are the least stable. The biologically realistic and
interesting architectures, with mixed excitatory and inhibitory
contributions, play an intermediate role. When the stability of the
network's fixed point solution is lost, this loss may occur through
an oscillatory instability or a nonoscillatory solution. The loss of
stability for the nonoscillatory solution is never affected by the
transmission speeds, a direct physical consequence of its zero frequency
allowing time for all parts of the system to evolve in unison. The only
route to a non-oscillatory instability is through the increase of the
heterogeneous connection strength. For oscillatory instabilities, the
situation is completely different. An increase of heterogeneous transmission
speeds always causes a stabilization of the global network state. These
results are summarized in ##FIG##3##Figure 4##.</p> <title>Numerical Simulations: Ensemble Activity from Neuronal to Whole Brain Scales</title> <p>This section illustrates neuronal ensemble activity at microscopic, mesoscopic, and
macroscopic spatial scales through numeric simulations. Our objective is to
highlight some of the key notions of ensemble dynamics and to illustrate
relationships between dynamics at different spatial scales.</p> <title/> <title>Ensemble dynamics at the microscopic scale</title> <p>To illustrate ensemble dynamics from first principles, we directly simulate a
network of coupled neurons which obey deterministic evolution rules and
receive both stochastic and deterministic inputs. The system is constructed
to embody, at a microscopic level, the response of the olfactory bulb to
sensory inputs, as originally formulated by Freeman ##UREF##13##[38]##, ##REF##497266##[106]##–##REF##3593783##[108]##.
Specifically, in the absence of a sensory input, neurons fire sporadically
due to background stochastic inputs. The presence of additional synaptic
currents due to a sensory input (e.g., inhaled odor) evokes a bifurcation
onto a limit cycle or chaotic attractor. Note that in this section we
simulate dynamics at the scale of coupled individual neurons. We can derive
directly the predicted ensemble mean response by simply summing over all
neurons. We compare this with an explicit model of neural mass dynamics at
the mesoscopic scale in the subsequent section.</p> <title>
<italic>Microscopic evolution equations</italic>
</title> <p>Each neuron is modeled as a planar reduction of the Hodgkin-Huxley model
##UREF##46##[109]##,##UREF##47##[110]##, namely,where <italic>f<sub>ion</sub></italic> introduces
conductance-determined transmembrane currents through voltage-dependent
channels,
<italic>ion</italic> = {<italic>Na<sup>+</sup></italic>,<italic>K<sup>+</sup></italic>}
and <italic>I</italic> are synaptic currents. The planar reduction has slow
potassium channel kinetics but fast sodium channels, whose states vary
directly with transmembrane potential ##REF##7260316##[111]##. Synaptic
currents are modeled, for the present purposes, to arise from three sources,
</p> <p>The first term represents recurrent feedback from neurons within the ensemble
due to their own firing. The coupling term, <italic>H<sub>c</sub></italic>,
incorporates both the nature of the (all-to-all) within-ensemble coupling
and the EPSP with parametric strength <italic>c</italic>. For the present
purposes, the EPSP consists of a brief steady current whenever the
presynaptic neuron is depolarized. The external currents,
<italic>I<sub>noise</sub></italic>, introduce stochastic inputs
(e.g., from brain stem inputs) and are modeled as a constant flow with a
superimposed Poisson train of discrete pulses. The final term,
<italic>I<sub>sensory</sub></italic>, models sensory input,
consisting of a constant synaptic current to a subset of neurons, whenever
the sensory stimulus is present. Hence this system permits an exploration of
the relative impact of the flow (deterministic) and diffusive (stochastic)
effects as embodied at the ensemble level by the Fokker-Planck equation
(Equation 20) at the neuronal network level. The Nernst potentials,
conductances, and background current are set so that, in the absence of
noise and sensory inputs, each neuron rests just below a saddle-node
bifurcation to a limit cycle ##UREF##46##[109]##. This
implies that neurons are spontaneously at rest (quiescent) but depolarize
with a small perturbation. If the perturbation is due to a stochastic train,
then the neuron fires randomly at an average rate proportional to the
stochastic inputs. However, following a small increase in the constant flow
term, due to a sensory input, <italic>I<sub>sensory</sub></italic>, the
quiescent state becomes unstable and the neuron evolves on a
(noise-modulated) limit cycle. ##FIG##4##Figure 5## shows a stochastically driven neuron (A) compared to a
noise-modulated periodic neuron (B). In the former case, the activity is
dominated by the stochastic terms. In the latter case, the limit cycle
dynamics dominate, although the stochastic inputs modulate the
depolarization amplitude.</p> <title>
<italic>Microscopic dynamics</italic>
</title> <p>
##FIG##5##Figure 6## shows the
results of simulating an ensemble of 250 neurons with a sensory input to all
neurons between <italic>t</italic> = 1,000
ms to <italic>t</italic> = 3,000 ms. ##FIG##5##Figure 6A## shows a raster
plot of the neural spike timing whilst ##FIG##5##Figure 6B## shows the simulated local field
potential from the ensemble ( = total
current flow across all neurons). As constructed, the effect of the input is
to effect a bifurcation in each neuron from stochastic to limit cycle
dynamics. The secondary effect of the appearance of limit cycle dynamics is
to suppress the impact of the spatially uncorrelated stochastic inputs.
Hence the neurons show an evolution towards phase locking, which was not
present prior to the stimulus. As evident in ##FIG##5##Figure 6B##, the increased firing synchrony
leads in turn to a marked increase in the simulated local field potentials
as individual neurons begin to contribute concurrent ion currents. Once the
stimulus ends, there is a brief quiescent phase because all of the neurons
have just fired and require a short train of stochastic inputs before they
commence firing again. Interestingly, there is evidence of damped mean-field
oscillations in the ensemble following stimulus termination, abating after
some further 800 ms. To underscore the observation that the mean synaptic
currents evidence an emergent phenomenon, and not merely the super-position
of a bursting neuron, the time series of a single neuron is provided in
##FIG##5##Figure 6C##. Clearly
no burst is evident at this scale.</p> <p>The impact of the stimulus input on the density of the ensemble is shown in
##FIG##6##Figure 7##, which
shows the spike-timing difference of all neurons in the ensemble with
respect to a randomly chosen seed-neuron. The mean spike-timing difference
is 0 ms throughout the simulation. This is because the system has complete
symmetry, so that all neurons fire, on average, symmetrically before or
after any other neuron. However, as evident in ##FIG##6##Figure 7A##, the variance in relative
spike-timing decreases dramatically during the stimulus interval. Of note is
that the ensemble variance does not simply step down with the onset of the
stimulus, but rather dynamically diminishes throughout the presence of the
stimulus. When this occurs, the mean-field term continues to increase in
amplitude. ##FIG##6##Figure 7B##
shows the evolution of the kurtosis (normalized so that a Gaussian
distribution has a kurtosis of zero). Prior to the stimulus, and reflecting
the weak network coupling, the ensemble has a mesokurtotic (broad)
distribution. It increases markedly following the stimulus onset, implying a
dynamical evolution towards a leptokurtotic (peaked) distribution. That is,
although the parameter values are static, the ensemble mean, variance, and
kurtosis evolve dynamically in an inter-related fashion. Hence this system
exhibits time-dependent interdependence between its first, second, and
fourth moments. This is the sort of coupling (between moments of the
ensemble density) that neural mass models do not capture.</p> <p>It is important to note that the spatiotemporal structure of the noise
remains constant throughout the simulation, as does the intra-ensemble
coupling. Hence the appearance of phase locking is an emergent feature of
the dynamics and has not been imposed. A dynamic contraction of the ensemble
cloud occurs whether the pre-existing noise continues unchanged during the
stimulus input—hence increasing the firing rate of each
neuron—or diminishes so that, in the absence of coupling, firing
rates are kept the same on average. In the latter case (as in ##FIG##4##Figure 5##), there is simply
a change from stochastic to periodic firing. The ensemble cloud is
visualized directly in ##FIG##7##Figure
8##. The upper row shows the first return map for the ensemble over
five consecutive time steps. For each neuron, this is defined as the
inter-spike delay at time
<italic>t</italic> = <italic>T</italic>
plotted against the inter-spike delay for the subsequent spike at
<italic>t</italic> = <italic>T</italic>+1.
Six such first return state-space values are plotted for all neurons. To
control for changes in spike rate, these plots are normalized to the average
firing rate. Values for the seed neuron used in ##FIG##6##Figure 7## are plotted in red. The left
column shows the ensemble state, prior to the stimulus current. The right
column shows the intra-stimulus activity. The contraction of the ensemble is
seen clearly. In addition, the first return map confirms that individual
neurons have stochastic dynamics prior to the stimulus, which change to
periodic (i.e., a fixed point in the first return map) during the stimulus.
The lower row of ##FIG##6##Figure
7## shows corresponding probability distributions of the inter-neuron
spike-timing differences. This reiterates that not only does the
distribution contract, but as the mean-field dynamics become strongly
nonlinear, the ensemble kurtosis increases markedly from sub- to
super-Gaussian.</p> <title>Neural mass dynamics at the mesoscopic scale</title> <p>Whilst such simulations are illustrative, they are computationally intensive;
even when limited to just 250 neurons at &lt;5 s of integration time. As
discussed in The <xref ref-type="sec" rid="s2">Mean-Field Model</xref>
section, it is possible to study a reduced model representing only the mean
ensemble dynamics. This is essentially achieved by generalizing parameter
values (such as ion channel thresholds) from individual point values to
population likelihood values. Freeman ##UREF##13##[38]## additionally
introduced synaptic effects through convolving the inputs with a suitable
response kernel as presented in Equation 37. For the simple illustration
here, we do not introduce synaptic filtering.</p> <title>
<italic>Mesoscopic evolution equations</italic>
</title> <p>For the present purpose, we simulate a single mass with both excitatory and
inhibitory neurons ##UREF##17##[52]##,##REF##12779875##[112]##. Expected mean
states of the ensemble excitatory neurons
<italic>μ<sub>e</sub></italic> = 〈<italic>V<sub>e</sub></italic>〉
are, as above, based upon Morris Lecar planar dynamics, with slow potassium
channel dynamics. Inhibitory neurons,
<italic>μ<sub>i</sub></italic>, respond passively to input from
excitatory neurons and feedback to induce additional outward (rectifying)
currents in excitatory cells. In the microscopic system considered above,
interneuron coupling was via a direct pulse during presynaptic
depolarization. At the mesoscopic scale, neuronal coupling is via neural
firing pulse densities, <italic>ς<sub>a</sub></italic>, which
capture the expected neuronal firing rate, given the mean neuronal
transmembrane potential
<italic>ς</italic>(<italic>μ<sub>a</sub></italic>) for
<italic>a</italic> = <italic>e</italic>,<italic>i</italic>.
Assuming a Gaussian distribution of individual neuronal firing thresholds,
one obtains a symmetric sigmoid-shaped function for
<italic>ς<sub>a</sub></italic> as per The <xref ref-type="sec" rid="s2">Mean-Field Model</xref> section. The dynamics are thus of the form
where the function <italic>G</italic> represents the coupling
between mean firing rates and induced synaptic currents. By targeting either
Na<sup>+</sup> or Ca<sup>++</sup> currents
and including (postsynaptic) voltage-dependent effects, this function can
incorporate, to a first-order approximation, a variable proportion of AMPA
or NMDA-like kinetics ##UREF##17##[52]##. The coefficients
<italic>ν<sub>αβ</sub></italic> represent
the synaptic density between excitatory (<italic>e</italic>) and inhibitory
(<italic>i</italic>) populations or from the stochastic/noise
(<italic>n</italic>) or sensory (<italic>s</italic>) inputs. Note that both
populations receive stochastic inputs but only the excitatory population
receives the sensory input <italic>I<sub>sensory</sub></italic>. The
functions <italic>f<sub>ion</sub></italic> are the same as for the
microscopic system including the slow potassium channel—although
they are now parameterized by population-wide estimates.</p> <title>
<italic>Mesoscopic dynamics</italic>
</title> <p>
##FIG##8##Figure 9## shows the
response of a single neural mass to sensory evoked synaptic currents with
the same temporal timing as for the microscopic system. Prior to the
stimulus, the system is in a stable fixed point regimen. The stochastic
inputs act as perturbations around this point, giving the time series a
noisy appearance, consistent with the prestimulus microscopic ensemble
activity. However, the mechanisms are quite distinct: Individual neurons
within the microscopic ensemble fired stochastically, but at uncorrelated
times. Hence, at the level of the ensemble, such individual events
contribute in a piecemeal fashion. That is, although individual neurons
exhibit nonlinear dynamics, the ensemble mean dynamics are (linearly) stable
to the stochastic inputs until the background current is increased. In the
mesoscopic case, the system as a whole is stable to small perturbations
prior to the stimulus current. The temporally uncorrelated stochastic inputs
are effectively filtered by the response properties of the system around
this fixed point to yield the simulated activity.</p> <p>In the mesoscopic neural mass, the fixed point state is rendered unstable by
the stimulus current and large amplitude oscillations occur. These cease
following stimulus termination. This accords with the appearance of
stimulus-evoked nonlinear oscillations in the ensemble-averaged response of
the microscopic system. In both models, such oscillations abate following
stimulus termination. Hence, at a first pass, this neural mass model
captures the mean-field response of the microscopic ensemble to a simulated
sensory stimulus.</p> <p>What is lost in the neural mass model? In this model, activity transits
quickly from a noise-perturbed fixed point to large amplitude nonlinear
oscillations. A brief, rapid periodic transient is evident at the stimulus
onset (1,000 ms). The system subsequently remains in the same dynamic state
until the stimulus termination. This hence fails to capture some of the
cardinal properties of the microscopic ensemble, namely the coupling between
the first and second moments (mean and variance). As discussed above, this
process underscores the dynamical growth in the mean-field oscillations and
the interdependent contraction of the interneuron spike timing variance
shown in ##FIG##5##Figures 6## and
##FIG##6##7##. Because of this
process the system is far more synchronized than prior to the stimulus. This
synchronization leads to the damped mean-field oscillations evident in the
ensemble system after the stimulus termination (3,200 ms→4,500 ms),
because there is a more coherent ensemble-wide response. What is gained in
the neural mass model? The addition of a third dimension (i.e., the
inhibitory mean activity) to the dynamics enables the expression of chaotic
dynamics ##UREF##17##[52]##,##REF##12779875##[112]##. Hence the
flow terms in the neural mass model contribute to the expression of
aperiodic dynamics in addition to the stochastic inputs. This is not
possible in the (planar) single neural dynamics of the microscopic system
because chaotic dynamics require at least three degrees of freedom. Thus the
dimension reduction afforded by the neural mass approximation allows the
introduction of more complex intrinsic dynamics, permitting dynamical chaos.
Whilst additional dimensions could be added to the microscopic neurons, this
would add to an already significant computational burden. The massive
reduction in the computational load of the neural mass approximation also
allows extension of the spatial scale of the model by an array of neural
masses, coupled to form a small patch of cortical tissue. Such a mesoscopic
system can be endowed with additional structure, such as hierarchical ##REF##3593783##[108]##, scale-free ##UREF##48##[113]##, multiscale
##REF##16087448##[114]##, or small world ##REF##17548818##[115]## properties. For
the present purposes, we couple a single input neural mass, as modeled
above, hierarchically to a sheet with internal hyperbolic (i.e., scale-free)
coupling. Intersystem coupling is purely excitatory-to-excitatory. Within
the sheet, the coupling drops in proportion to spatial separation and is
hence scale-free:where <italic>F</italic> incorporates all intrasystem
dynamics as per Equation 89 and the indices numerate either the sensory node
{<italic>sens</italic>} or the nodes within the sheet
{<italic>sheet</italic>}. As above, synaptic currents are induced by the
pulse density of the presynaptic neurons, rather than directly via
individual presynaptic depolarization. The sensory node receives the only
direct stimulus-induced currents,
</p> <p>The hierarchical nature of the system is embodied by the targeted nature of
the sensory inputs and the separate parameterization of parameters that
couple masses to or within the sheet, <italic>C<sub>sens</sub></italic> and
<italic>C<sub>sheet</sub></italic>, respectively. It would also be
possible to increase the degree of forward and backward asymmetry by
incorporating purely AMPA-like kinetics for the former and NMDA-like
kinetics for the latter, as has been proposed as a mechanism for perceptual
inference ##REF##14622888##[116]##,##REF##15937014##[117]##. ##FIG##9##Figure 10## shows the
response of the system when
<italic>C<sub>sens</sub></italic>&gt;<italic>C<sub>sheet</sub></italic>.
##FIG##9##Figure 10A## shows the
individual mean synaptic currents of all nonsensory nodes. ##FIG##9##Figure 10B## shows the total
synaptic currents averaged across the nonsensory sheet. Several features can
be noted. For a start, despite use of the same parameters, the coupling of
neural masses into an array or sheet leads to the appearance of spontaneous
prestimulus activity. Stimulus-evoked activity from the sensory node
reorganizes this activity from spatially incoherent to synchronized. Thus
the array-averaged synaptic currents increase during the stimulus period.
Hence the dynamics at this scale mirror those within the microscopic
ensemble, which each node in this simulation is constructed to represent.
However, at least in this simulation, the array does not exhibit a dynamic
growth in the system-wide currents during the stimulus period. Presumably,
if this did occur within the individual mesoscopic nodes, then the
array-wide current may also grow dynamically. That is, one would anticipate
coupling between moments across ensembles, as discussed in ##UREF##13##[38]##.</p> <p>
##FIG##10##Figure 11## shows the
simulated activity following an increase in the intrasheet coupling such
that
<italic>C<sub>sens</sub></italic>≈<italic>C<sub>sheet</sub></italic>.
All other parameters are unchanged. Spontaneous prestimulus activity is
clearly more coherent; consistent with a stronger internally determined
dynamical state. The injection of the externally evoked sensory currents
into this prior activity actually has a slightly desynchronizing effect, as
evident as a decrease in the array-wide average response (##FIG##10##Figure 11B##). That is, the
temporal mismatch between the within-sheet dynamics and the externally
induced activity leads to more spatially complex dynamics (an increase in
the spatial entropy and hence the information content of the system).</p> <p>If the stochastic inputs, <italic>I<sub>noise</sub></italic>, are decreased
below a threshold, then the spontaneous activity in the nonsensory array
diminishes. The feedback effect of this quiescent activity is to suppress
the stimulus-evoked activity in the sensory node. Hence there is a top-down
mechanism for the complete suppression of sensory-evoked activity.
Presumably, more subtle feedback effects may be possible if more forward
versus backward receptor detail was modeled. In summary, these mesoscopic
simulations impress a view of sensory-evoked effects as a reorganization of
ongoing activity. Depending upon the ratio of internal to sensory-related
coupling, this reorganization may lead to an increase or a decrease in the
information content of the system dynamics.</p> <title>Neural field dynamics at the whole-brain scale</title> <p>We now provide brief illustrations of sensory evoked and nonlinear activity
as modeled by macroscopic field equations. As discussed in the section
entitled Recent Developments in Neural Field Models, these incorporate
synaptic filtering and axonal conduction delays, in addition to the
population-wide conversion of membrane potentials into firing densities
##UREF##16##[50]##. Significantly, they also permit the
incorporation of subcortical systems, such as the thalamus ##UREF##35##[91]##. Recent developments (see the Heterogeneous
Connectivity in Neural Fields section) now allow elucidation of the impact
of biologically relevant connection heterogeneities on the stability and
conduction of cortical activity. The equations, their derivation, and
relevant references are provided in the Recent Developments in Neural Field
Models section.</p> <title>Macroscopic dynamics</title> <p>Two crucial differences occur when moving to the macroscopic scale of the
corticothalamic field model. Firstly, sensory inputs are modeled as entering
the specific nuclei of the thalamus rather than directly into a cortical
sensory node. The ensuing evoked corticothalamic activity can then be
studied in a biologically informed framework. Second, while prestimulus
activity is modeled as a noise-perturbed steady state, the system is not
destabilized by sensory inputs. Instead, inputs evoke damped oscillations in
the corticothalamic loop ##REF##12111274##[87]##. ##FIG##11##Figure 12## illustrates an example of
sensory-evoked activity (Aquino et. al., unpublished data). Evoked afferent
pulse densities are shown because they reflect more accurately the expected
synaptic currents, through their action on postsynaptic neurons. The smooth
spatiotemporal dispersion of the evoked cortical response and its time
delayed corticothalamic volley are evident.</p> <title>Summary of numerical simulations</title> <p>These simulations give insight into the rich neural ensemble dynamics at
different spatial scales that arise spontaneously, are evoked by sensory
inputs, or follow changes in state parameters. The intention is to
demonstrate concrete examples of ensemble dynamics under varying influences
of flow and dispersion. The resulting dynamics can be seen to emerge from
the interplay of stochastic dispersion and flow-determined ensemble
contraction. The view of stimulus-evoked synaptic currents as evoking a
bifurcation in neural ensemble activity derives largely from the formative
work of Freeman, following detailed physiological studies of the olfactory
bulb. One of the key outstanding problems is to reconcile the apparent
discrepancy between proposals involving a key role of nonlinear dynamics
(see also ##REF##9345546##[118]##) and the apparent success of mean-field
models to predict measured evoked responses, without recourse to nonlinear
dynamics. One approach is to construct a multiscale hierarchy, with
self-consistent evolution equations at each scale and to couple the emergent
dynamics from fine scales into the activity at coarser scales ##REF##16087448##[114]##. Although this permits small scale nonlinear
activity to coincide with and influence stochastic macroscopic activity, it
requires a somewhat elaborate framework. An alternative approach is to
recursively enslave micro- and mesoscopic activity to predicted macroscopic
field oscillations by driving them with the predicted mean-field synaptic
currents. A problem here concerns the resulting emergence of sustained
oscillations within mesoscopic activity and the possible causal
inconsistency that this may entail.</p> <p>The nature and strength of neuronal connectivity varies markedly when
considered across the heirarchy of spatial scales. At the microscopic scale,
connectivity is dense, concentrated equally in vertical and horizontal
directions and, more or less isotropic when considered across different
cortical regions. At mesoscopic scales, connectivity has a patchy,
colmunar-dominated structure. At macroscopic scales, connectivity is
sparser, can be considered exclusively horizontal, and is predominantly
excitatory in nature. It is also characterized by heterogenous connections
(large fiber tracts) which fulfill functionally defined roles. These rules
are reflected in the abstractions and refinements of the models which
address the different scales.</p> <title>Cognitive and Clinical Applications</title> <p>In this section, we present three distinct applications of neural ensemble modeling.
We first illustrate a computational example, namely decision-making, as implemented
in a mean-field model. We then illustrate healthy and pathological activity in
neural field models. The healthy example is of the well-known psychophysical
phenomenon of auditory streaming—the balance of segmentation versus
integration in auditory perception. We then illustrate examples of spatiotemporal
dynamics occurring in corticothalamic loops during Absence seizures.</p> <title/> <title>Spiking dynamics underlying decision-making</title> <p>Decision-making is a key brain function of intelligent behavior. A number of
neurophysiological experiments on decision-making reveal the neural
mechanisms underlying perceptual comparison, by characterising the neuronal
correlates of behavior ##REF##12368806##[119]##–##REF##14715143##[121]##. In particular,
##REF##12368806##[119]##–##REF##12612633##[124]## have studied the
neural mechanisms underlying perceptual comparison by measuring
single-neuron responses in monkeys trained to compare two mechanical
vibrations applied sequentially to the tip of a finger; the subjects have to
report which of the two stimuli has the higher frequency. They found neurons
in the ventral premotor cortex (VPC) whose firing rate depended only on the
difference between the two applied frequencies, the sign of that difference
being the determining factor for correct task performance ##REF##14715143##[121]##.
These neurons reflect the implementation of the perceptual comparison
process and may underlie the process of decision-making.</p> <p>
##FIG##12##Figure 13## shows a
biophysically realistic computational model for a probabilistic
decision-making network that compares two mechanical vibrations applied
sequentially (f1 and f2). The model implements a dynamical competition
between neurons: The model enables a formal description of the transients
(nonstationary) and probabilistic character of behavior (performance) by the
explicit use, at the microscopic level, of spiking and synaptic dynamics of
one-compartment IF neuron models. The network contains excitatory pyramidal
cells and inhibitory interneurons. The excitatory recurrent postsynaptic
currents (EPSCs) are mediated by AMPA (fast) and NMDA-glutamate (slow)
receptors, whereas external EPSCs imposed on the network are driven by AMPA
receptors only. Inhibitory postsynaptic currents (IPSCs) to both excitatory
and inhibitory neurons are mediated by GABA receptors. Neurons are clustered
into populations. There are two subtypes of excitatory population: namely,
specific and nonselective. Specific populations encode the result of the
comparison process in the two-interval vibrotactile discrimination task,
i.e., if f1&gt;f2 or f1&lt;f2. The neurons in the two specific
populations additionally receive external inputs encoding stimulus specific
information. They are assumed to originate from the somatosensory area S2
and from the PFC, encoding the frequency of both stimuli f1 (stored) and f2
(present) to be compared during the comparison period, i.e., when the second
stimuli is applied (see ##REF##16930418##[125]## for details).</p> <p>The attractors of the network of IF neurons can be studied exhaustively by
using the associated reduced mean-field equations. The set of stationary,
self-reproducing rates, <italic>ν<sub>i</sub></italic>, for the
different populations, <italic>i</italic>, can be found by solving a set of
coupled self-consistency equations. This enables a posteriori selection of
parameter regions that contain desired behaviors. In the present case, the
essential requirement is that, for the stationary conditions, different
attractors are stable. The attractors of interest for our task correspond to
the activation (high spiking rates) or inactivation (low spiking rates) of
the neurons in the specific populations f1&gt;f2 and f1&lt;f2. The
activation of the specific population f1&gt;f2 (f1&lt;f2) and the
simultaneous lack of activation of the complementary population f1&lt;f2
(f1&gt;f2), corresponds to an encoding single state associated with a
motor response reporting the categorical decision f1&gt;f2
(f1&lt;f2). The lack of activation of both specific populations
(spontaneous state) would correspond to an encoding state that cannot lead
to a behavioral decision; i.e., there is no answer, or a motor response is
generated randomly. The same happens if both specific populations are
activated to the same degree (pair state). Because responses in animals are
probabilistic in nature, the operating point of the network should be such
that both categorical decisions, i.e., both states, are bistable. In
addition, we have also shown that the model predicts a behavior consistent
with Weber's law if, and only if, the spontaneous state is also a
stable state, i.e., when the dynamical operating point of the network is in
a regime of multistability. In this way, Weber's law informs the
operating point of the network.</p> <p>
##FIG##13##Figure 14## shows numerical
simulations corresponding to the response of VPC neurons during the
comparison period (to be contrasted with the experimental results shown in
Figure 2 of ##REF##14715143##[121]##). This figure shows the average firing
rate as a function of f1 and f2, obtained with the spiking simulations
(diamond points correspond to the average values over 200 trials, and the
error bars to the standard deviation). The lines correspond to the
mean-field calculations. Black indicates f1&lt;f2
(f2 = f1+8 Hz) and gray indicates
f1&gt;f2 (f2 = f1−8 Hz). The
average firing rate of the population f1&lt;f2 depends only on the sign
of f2−f1 and magnitude of the difference, |f2−f1|;
confirming again that Weber's law cannot be encoded in the firing
rate, but only in the probability with which that firing rate can be reached
(that depends on the sign and magnitude of the difference between f1 and
f2).</p> <title>Auditory streaming</title> <p>One of the applications of neural fields in cognitive processing is found in
auditory scene analysis ##UREF##49##[126]##, particularly auditory streaming.
Intuitively, auditory streaming or stream segregation is like listening to
bass and soprano vocalists singing simultaneously. Although the two voices
overlap in time, they clearly form two distinct percepts. In the laboratory,
a similar effect can be created using sequences of tones. In a typical
streaming experiment, two sequences are created using sets of high and low
tones. Sequences vary in presentation rate and the frequency difference
between the tones. The basic finding (see e.g., ##UREF##50##[127]##,##UREF##51##[128]##) is: (i) when the frequency separation is
relatively small and/or the rate is relatively slow, listeners perceive a
single integrated melody (or stream) and can accurately report the ordering
of the tones, and (ii) when the frequency separation is relatively large
and/or the rate relatively fast, people clearly perceive two segregated
auditory streams, one with a higher pitch than the other. Essentially, there
is a frequency–time boundary (known as the Fission Boundary, FB)
beneath which all sequences are heard as integrated, regardless of
instructions. There is a frequency–time boundary (known as the
Temporal Coherence Boundary, TCB) above which all sequences are heard as
segregated, regardless of instructions. In between these two boundaries
exists a bistable region in which a sequence can be heard as either
integrated or segregated depending upon instructions. Hysteresis phenomena
are observed when traversing the bistable regime from either the FB or TCB.
Many other auditory phenomena of a related nature are discussed in ##UREF##51##[128]##.</p> <p>To capture the perceptual integration and segregation processes in the human
brain, while accommodating contemporary brain theories ##REF##11156197##[1]##,##UREF##0##[2]##,##REF##15319516##[4]##, the authors of
##UREF##49##[126]## proposed a tonotopically organized
neural field for peripheral processing with projections to the higher areas
that are responsible for cognitive integration. The neural field is
tonotopically organized such that the frequency of the acoustic stimulus
maps onto a location in neural space. The second nontonotopically organized
system may either be represented by a neural field or a subnetwork. Its
function is the classification of the peripheral spatiotemporal neural field
dynamics. This classification process is not just a measurement (in which
case the application of a simple measure to the neural field would suffice)
but is itself a dynamic process. In fact, bistability and hysteresis turn
out to be properties of the classification process rather than properties of
the neural field dynamics. The dynamics of the neural field
<italic>μ</italic>(<italic>x</italic>,<italic>t</italic>) are
given by the wave Equation 39, which has been extended to accommodate
auditory inputs <italic>s</italic>(<italic>x</italic>,<italic>t</italic>) as follows:where, as a reminder,
<italic>γ</italic> = <italic>c</italic>/<italic>r</italic>,
<italic>c</italic> is the speed of spike propagation, and
<italic>r</italic> parameterizes the spatial decay of lateral interactions.
The external input or stimulus to the neural sheet is
<italic>s</italic>(<italic>x</italic>,<italic>t</italic>):
<bold>ℜ</bold>
<sup>2</sup>→<bold>ℜ</bold>,
which contains all the spatiotemporal characteristics of the auditory input
stream. Periodic boundary conditions,
<italic>μ</italic>(0,<italic>t</italic>) = <italic>μ</italic>(<italic>L</italic>,<italic>t</italic>),
<italic>t</italic>≥0, are used.</p> <p>The second network is not tonotopically organized, hence its spatial
dimension is of no relevance, when we consider only the competition of two
streams. In fact, the ability to show multistable pattern formation is the
only relevant property of the network and can be realized in multiple
network architectures as discussed in previous sections. A simple
multistable subsystem with its scalar state variable
<italic>y</italic>(<italic>t</italic>) is given by the equationwhere <italic>ε</italic> is a constant that captures
all linear contributions. <italic>I</italic>
<sub>0</sub> contains all
constant contributions given rise to the rest state activity. The functional
<italic>I</italic>(<italic>t</italic>) is specified aswhere Ω is a neural activity threshold. Equations
93, 94, and 95 define the dynamics of a stream classification model in one
of its simplest forms. ##FIG##14##Figure
15## illustrates the architecture of the model.</p> <p>To understand van Noorden's results, we parametrize a sequence of
consecutive tones by their frequency difference,
Δ<italic>f</italic>, and their interonset interval,
<italic>IOI</italic>. As the neural field evolves, it is integrated across
space and time yielding the time-dependent, but scalar, activity,
<italic>I</italic>(<italic>t</italic>), driving the second system.
<italic>I</italic>(<italic>t</italic>) represents the relevant
information from the neural field, <italic>μ</italic>, as a
spatiotemporally integrated activity measure, which depends on the amount of
dispersion over space and time. The greater the dispersion, the greater will
be the value of <italic>I</italic>(<italic>t</italic>) at a given time
point. ##FIG##15##Figure 16## shows
the contour lines of neural field activity over space <italic>x</italic> and
time <italic>t</italic> for the bistable situation.</p> <p>The final state reached by the second system defined in Equation 94 with
activity <italic>y</italic> will depend on
<italic>I</italic>(<italic>t</italic>) and its own intrinsic dynamics. The
curve of the flow is shifted up or down depending on
<italic>I</italic>(<italic>t</italic>), creating either one positive or
one negative fixed point. For an intermediate value of
<italic>I</italic>(<italic>t</italic>), there is a bistable regime in
which <italic>y</italic> can assume either one of the fixed points. The
negative fixed point is identified with perceiving one stream and the
positive fixed point with perceiving two streams. The time series for
<italic>y</italic> are shown in ##FIG##16##Figure 17## for several different initial
conditions of the activity <italic>y</italic>. After a transient the
activity becomes stationary, displaying three possible scenarios (see ##FIG##16##Figure 17## from top to
bottom): one stream only, or the bistable situation, in which either one
integrated stream or two separate streams may be perceived, or finally two
streams only. For each choice of Δ<italic>f</italic> and
<italic>IOI</italic>, the model Equations 93 and 94 are solved numerically
and their stationary states determined. The results are plotted in the 2-D
parameter space in ##FIG##17##Figure
18##. TCB and the FB are reproduced in a manner that corresponds nicely
to van Noorden's (1975) results including a bistable region ##UREF##50##[127]##. Note that the exact experimental numerical
values at which the boundaries occur vary from subject to subject and depend
on the experimental methods employed ##UREF##51##[128]##.</p> <p>We will briefly illustrate another phenomenon. When two interleaved rising
and falling tone sequences, as shown in ##FIG##18##Figure 19##, are presented, human subjects
report them to be either crossing or bouncing perceptually ##UREF##51##[128]##,##UREF##52##[129]##. This
phenomenon is known as the crossing scales phenomenon. The implementation
within the neural field model of ##UREF##49##[126]## is
straightforward and illustrated in ##FIG##18##Figure 19##.</p> <title>Modeling seizures</title> <p>Experimental and theoretical arguments propose that the onset of a seizure
reflects a bifurcation in cortical activity from damped stochastic
activity—where peaks in the power spectrum reflect damped linear
resonances—to high amplitude nonlinear oscillations arising from
activity on a limit cycle or chaotic attractor ##UREF##31##[86]##, ##REF##12028360##[130]##–##REF##16508406##[134]##. ##FIG##19##Figure 20## presents an
example of a bifurcation arising from a 3 Hz oscillatory instability in the
corticothalamic neural field model of the Recent Developments in Neural
Field Models section. Stochastic activity either side of the seizure can be
seen, reflecting the response properties of the stable steady state mode.
The large amplitude oscillations arise from a transient change in a
corticothalamic state parameter from
<italic>t</italic> = 5 s to
<italic>t</italic> = 20 s. A more
systematic analysis of the bifurcations in this neural field model was
undertaken in ##REF##16280462##[92]##. It was argued that the study of these
bifurcations provides a parsimonious explanation of the unique time course,
symmetry, onset, and offset of both Absence and tonic clonic seizures,
capturing their similarities and the differences. Further analysis of the 3
Hz (Absence) bifurcation in a reduced model argues that interactions between
the reticular and specific nuclei of the thalamus contribute importantly to
the Absence seizure waveform ##UREF##54##[135]##. In the
present simulation, the 3 Hz seizure has inherently aperiodic dynamics, as
shown in the right panel of ##FIG##19##Figure 20##.</p> <p>Neural field formulations, through their implicit treatment of horizontal
synaptic coupling, also lend themselves naturally to studying the spatial
propagation of seizure activity, a clinically important phenomenon. An
analysis of the frequency and amplitude properties of spatially extended 3
Hz seizures (Knock et. al., unpublished data) is presented in ##FIG##20##Figure 21##, comparing data
recorded from a young male with Absence epilepsy (left panels) to a
simulated seizure in the corticothalamic model (right panels). The top row
shows the temporal dynamics of the dominant frequency across a spatially
extended array of (real and simulated) electrodes. The observed data (left)
shows that the seizure onsets (almost) simultaneously across the scalp,
although frontal electrodes lead fractionally. However, onset frequencies
range from 2.7 Hz at frontal electrodes to 4 Hz over temporal regions. There
follows a pattern of frequency convergence so that within 2 s of seizure
onset, all cortical regions express a common frequency of 3 Hz, slowing
progressively to 2.5 Hz. The seizure simulated in the corticothalamic model
(right) shows a similar pattern. Peak onset frequencies in this model
predominantly reflect corticothalamic conduction time lags, which have been
parameterized to reflect the varying separation of cortex and thalamus.
Subsequent frequency convergence in this model arises from corticortical
coupling (there is no intrathalamic coupling in this simulation).</p> <p>The lower panels show the temporal evolution of the amplitude envelope of
activity within the dominant mode. The principal feature of interest in the
observed data (left) is the increasing modulation of the amplitude envelope
as one moves from frontal electrodes, which have the strongest power, to
parietal electrodes, where the onset power is weaker. These differing
degrees of amplitude modulation are also present in the simulated seizure
(right). Importantly, all parameters of the model are constant during the
seizure. Hence the amplitude modulation is due to coupling between nonlinear
modes at different spatial locations.</p> <p>Whereas frequency locking is not surprising in a model with spatial coupling,
the amplitude modulation is a novel, emergent property of the nonlinear
dynamics.</p>
<p>VKJ thanks Felix Almonte for help with the generation of various figures. We thank
the attendees of Session IV at BCW2006* for the interesting discussions,
which represent the source of inspiration for the present review.</p>
<fig id="pcbi-1000092-g001" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1000092.g001</object-id><label>Figure 1</label><caption><title>Anatomical connectivity,
<italic>W</italic> = <italic>W<sub>hom</sub></italic>+<italic>W<sub>het</sub></italic>,
comprising homogeneous and heterogeneous connections.</title><p>The intracortical connections are illustrated as densely connected
fibers in the upper sheet and define the homogeneous connectivity
<italic>W<sub>hom</sub></italic>. A single fiber connects
the two distant regimes (A) and (B) and contributes to the
heterogeneous connectivity, <italic>W<sub>het</sub></italic>,
whereas regime (C) has only homogeneous connections.</p></caption></fig> <fig id="pcbi-1000092-g002" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1000092.g002</object-id><label>Figure 2</label><caption><title>Typical homogeneous connectivity kernels,
<italic>W<sub>hom</sub></italic>(<italic>z</italic>), used for
local architectures plotted as a function of spatial distance
<italic>z</italic>.</title><p>Purely excitatory connectivity is plotted in (A); purely inhibitory
in (B); center-on, surround-off in (C); and center-off, surround-on
in (D). The connectivity kernel in (C) is the most widely used in
computational neuroscience.</p></caption></fig> <fig id="pcbi-1000092-g003" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1000092.g003</object-id><label>Figure 3</label><caption><title>Minimal stable regions for the equilibrium state of a neural
field as a function of its connectivity and time delay
<italic>τ</italic> = <italic>d</italic>/<italic>c</italic>.</title><p>The critical surface, at which the equilibrium state undergoes an
instability, is plotted as a function of the real and imaginary part
of the eigenvalue of its connectivity, <italic>W</italic>. Regimes
below the surface indicate stability, above instability. The
vertical axis shows the time delay via transmission along the
heterogeneous fiber.</p></caption></fig> <fig id="pcbi-1000092-g004" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1000092.g004</object-id><label>Figure 4</label><caption><title>Summary of the stability changes of a neural field with mixed
(local/global) connectivity.</title><p>(Top) The relative size of stability area for different connectivity
kernels. (Bottom) Illustration of change of stability as a function
of various factors. Gradient within the arrows indicates the
increase of the parameter indicated by each arrow. The direction of
the arrow refers to the effect of the related factor on the
stability change. The bold line separating stable and unstable
regions indicates the course of the critical surface as the time
delay changes.</p></caption></fig> <fig id="pcbi-1000092-g005" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1000092.g005</object-id><label>Figure 5</label><caption><title>Planar spiking neuron.</title><p>(A) Stochastically perturbed fixed point. (B) Limit cycle
attractor.</p></caption></fig> <fig id="pcbi-1000092-g006" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1000092.g006</object-id><label>Figure 6</label><caption><title>Results of simulating an ensemble of 250 neurons with sensory
evoked synaptic currents to all neurons between
<italic>t</italic> = 1,000 ms and
t = 3,000 ms.</title><p>(A) Raster plot. (B) Mean synaptic currents. (C) Time series of a
single neuron. The effect of the input is to effect a bifurcation in
each neuron from stochastic to limit cycle dynamics (phase locking),
suppressing the impact of the spatially uncorrelated stochastic
inputs. As evident in (A), the increased firing synchrony leads in
turn to a marked increase in the simulated local field potentials.
The mean synaptic currents evidence an emergent phenomenon, and not
merely the superposition of a bursting neuron, as can be seen in
(C): clearly no burst is evident at this scale.</p></caption></fig> <fig id="pcbi-1000092-g007" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1000092.g007</object-id><label>Figure 7</label><caption><title>Contraction of spike-timing differences due to synaptic inputs.</title><p>A seed neuron is chosen at random and the interneuron spike
difference for all other neurons is plotted each time it spikes. (A)
Solid and dashed lines show ±1 and ±1.5
standard deviations of the ensemble spike timing. (B) The normalized
fourth moment (excess kurtosis) derived from a moving frame.</p></caption></fig> <fig id="pcbi-1000092-g008" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1000092.g008</object-id><label>Figure 8</label><caption><title>Contraction of spike-timing differences due to synaptic inputs.</title><p>The left column shows the ensemble state, prior to the stimulus
current. The right column shows the intrastimulus activity. Top row:
First return map for the cloud interspike delay over five
consecutive time steps, before (A) and following (B) synaptic input.
The plots are normalized to the average firing rate to control for
changes in spike rate. Values for the seed neuron used in ##FIG##6##Figure 7## are
plotted in red. Lower row (C,D): the corresponding spike timing
histograms. The ensemble kurtosis increases markedly from sub- to
super-Gaussian.</p></caption></fig> <fig id="pcbi-1000092-g009" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1000092.g009</object-id><label>Figure 9</label><caption><title>Mesoscopic neural mass model with sensory evoked synaptic
currents from
<italic>t</italic> = 1,000 ms to
t = 3,000 ms.</title><p>Prior to the stimulus, the system is in a stable fixed point regimen.
The stochastic inputs act as perturbations around this point.
Although individual neurons exhibit nonlinear dynamics, the ensemble
mean dynamics are (linearly) stable to the stochastic inputs until
the background current is increased. Then the fixed point state is
rendered unstable by the stimulus current and large amplitude
oscillations occur. These cease following stimulus termination.</p></caption></fig> <fig id="pcbi-1000092-g010" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1000092.g010</object-id><label>Figure 10</label><caption><title>Coupled mesoscopic neural masses with sensory evoked synaptic
currents into single sensory node from
<italic>t</italic> = 1,000 ms to
t = 3,000 ms
<italic>C<sub>sens</sub></italic>&gt;<italic>C<sub>sheet</sub></italic>.</title><p>(A) Individual mean synaptic currents of all nonsensory nodes. (B)
Total synaptic currents averaged across the nonsensory sheet.
Stimulus-evoked activity from the sensory node reorganizes this
activity from spatially incoherent to synchronized.</p></caption></fig> <fig id="pcbi-1000092-g011" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1000092.g011</object-id><label>Figure 11</label><caption><title>Coupled mesoscopic neural masses with sensory evoked synaptic
currents into single sensory node from
<italic>t</italic> = 1,000 ms to
<italic>t</italic> = 3,000 ms
<italic>C<sub>sens</sub></italic>≈<italic>C<sub>sheet</sub></italic>.</title><p>All other parameters as in ##FIG##9##Figure 10##. (A) Individual mean
synaptic currents of all nonsensory nodes. (B) Total synaptic
currents averaged across the nonsensory sheet. The injection of the
externally evoked sensory currents into the prior activity actually
has a slightly desynchronizing effect.</p></caption></fig> <fig id="pcbi-1000092-g012" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1000092.g012</object-id><label>Figure 12</label><caption><title>The spatiotemporal evolution of the evoked response of excitatory
pulse densities
<italic>μ</italic>(<italic>r</italic>,<italic>t</italic>)
in an example of sensory-evoked activity (Aquino et. al.,
unpublished data).</title><p>The smooth spatiotemporal dispersion of the evoked cortical response
and its time delayed corticothalamic volley are evident.</p></caption></fig> <fig id="pcbi-1000092-g013" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1000092.g013</object-id><label>Figure 13</label><caption><title>Decision-making neuronal network.</title><p>Minimal neurodynamical model for a probabilistic decision-making
network that performs the comparison of two mechanical vibrations
applied sequentially (f1 and f2). The model implements a dynamical
competition between different neurons. The network contains
excitatory pyramidal cells and inhibitory interneurons. The neurons
are fully connected (with synaptic strengths as specified in the
text). Neurons are clustered into populations. There are two
different types of population: excitatory and inhibitory. There are
two subtypes of excitatory population, namely: specific and
nonselective. Specific populations encode the result of the
comparison process in the two-interval vibrotactile discrimination
task, i.e., if f1&gt;f2 or f1&lt;f2. The recurrent arrows
indicate recurrent connections between the different neurons in a
population.</p></caption></fig> <fig id="pcbi-1000092-g014" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1000092.g014</object-id><label>Figure 14</label><caption><title>Average firing rate of a neuron as a function of f1 and f2,
obtained with the spiking simulations of the response of VPC neurons
during the comparison period (to be contrasted with the experimental
results shown in Figure 2 of ##REF##14715143##[121]##).</title><p>Diamond points correspond to the average values over 200 trials, and
the error bars to the standard deviation. The lines correspond to
the mean-field calculations: the black line indicates f1&lt;f2
(f2 = f1+8 Hz) and the red
dashed line f1&gt;f2
(f2 = f1−8 Hz). The
average firing rate of the population f1&lt;f2 depends only on
the sign of f2−f1 and magnitude of the difference,
|f2−f1|.</p></caption></fig> <fig id="pcbi-1000092-g015" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1000092.g015</object-id><label>Figure 15</label><caption><title>Cortical architecture of the model.</title><p>The neural field is illustrated by the rectangular box showing the
neural activity
<italic>μ</italic>(<italic>x</italic>,<italic>t</italic>)
composed of inhibitory and excitatory neurons. The input
<italic>s</italic>(<italic>x</italic>,<italic>t</italic>) is
provided at locations <italic>x<sub>i</sub></italic> via the
Gaussian localization function  with width . The explicit model parameters used in the
simulations are given in ##UREF##49##[126]##.</p></caption></fig> <fig id="pcbi-1000092-g016" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1000092.g016</object-id><label>Figure 16</label><caption><title>Bistable regime of auditory streaming.</title><p>The stimulus sequences (top) and its resulting neural field dynamics
(bottom).</p></caption></fig> <fig id="pcbi-1000092-g017" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1000092.g017</object-id><label>Figure 17</label><caption><title>Percept formation.</title><p>For multiple initial conditions, the time series of
<italic>y</italic>(<italic>t</italic>) are plotted for the three
regimes, one stream only (top), bistable (middle), and two streams
only (bottom).</p></caption></fig> <fig id="pcbi-1000092-g018" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1000092.g018</object-id><label>Figure 18</label><caption><title>van Noorden's bifurcation diagram.</title><p>Computational simulations yield the van Noorden's
bifurcation diagram as a function of the frequency difference
Δ<italic>f</italic> and the <italic>IOI</italic>. The
parameter space is partitioned into three regimes, one region with
the percept one stream, another region with the percept two streams
and a region in between which permits both.</p></caption></fig> <fig id="pcbi-1000092-g019" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1000092.g019</object-id><label>Figure 19</label><caption><title>Crossing scales phenomenon.</title><p>The input tone sequences used to form a percept of crossover are
shown in (A). The resulting contours of neural field activity are
plotted in (B), together with the final time series of
<italic>y</italic>(<italic>t</italic>) shown in (C). In this
particular case, the classification system
<italic>y</italic>(<italic>t</italic>) does traverse from the
positive (two streams) to the negative (one stream) fixed point and
back. This trajectory is identified with the percept of crossover.
In the case of the bouncing percept, the time series of
<italic>y</italic>(<italic>t</italic>) will not cross the x-axis
as shown in (D).</p></caption></fig> <fig id="pcbi-1000092-g020" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1000092.g020</object-id><label>Figure 20</label><caption><title>Simulated nonlinear oscillations arising from 3 Hz modal
instability in a corticothalamic neural field model.</title><p>(A) Stochastic activity either side of the seizure can be seen,
reflecting the response properties of the stable steady state mode.
The large amplitude oscillations arise from a transient change in a
corticothalamic state parameter from
<italic>t</italic> = 5 s to
<italic>t</italic> = 20 s. (B)
Shows more detail of the aperiodic oscillations.</p></caption></fig> <fig id="pcbi-1000092-g021" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1000092.g021</object-id><label>Figure 21</label><caption><title>Spatiotemporal activity during 3 Hz spike and wave seizures.</title><p>(Left) Shows analysis of an Absence seizure in an adolescent male
subject. (Right) Shows the results of a simulated seizure in the
corticothalamic field model described in the Recent Developments in
Neural Field Models section. Top row shows the evolution of the
center frequency of the dominant nonlinear mode. Lower row show the
amplitude envelope of these modes. Frequencies and amplitudes were
derived using a complex Morlet wavelet decomposition of the real and
simulated time series.</p></caption></fig>
<table-wrap id="pcbi-1000092-t001" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1000092.t001</object-id><label>Table 1</label><caption><title>List of notation and symbols.</title></caption><alternatives><table frame="hsides" rules="groups"><colgroup span="1"><col align="left" span="1"/><col align="center" span="1"/><col align="center" span="1"/></colgroup><thead><tr><td align="left" rowspan="1" colspan="1">Quantity</td><td align="left" rowspan="1" colspan="1">Symbol</td><td align="left" rowspan="1" colspan="1">SI Unit</td></tr></thead><tbody><tr><td align="left" rowspan="1" colspan="1">Neural membrane potential</td><td align="left" rowspan="1" colspan="1">
<italic>V</italic>
</td><td align="left" rowspan="1" colspan="1">V</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural membrane capacitive current</td><td align="left" rowspan="1" colspan="1">
<italic>I</italic>
</td><td align="left" rowspan="1" colspan="1">A</td></tr><tr><td align="left" rowspan="1" colspan="1">Time elapsed since most recent action potential</td><td align="left" rowspan="1" colspan="1">
<italic>T</italic>
</td><td align="left" rowspan="1" colspan="1">s</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural membrane capacitance</td><td align="left" rowspan="1" colspan="1">
<italic>C</italic>
</td><td align="left" rowspan="1" colspan="1">F</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural membrane resistance</td><td align="left" rowspan="1" colspan="1">
<italic>R</italic>
</td><td align="left" rowspan="1" colspan="1">
<italic>ω</italic>
</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural membrane time constant</td><td align="left" rowspan="1" colspan="1">
<italic>τ</italic> = <italic>RC</italic>
</td><td align="left" rowspan="1" colspan="1">s</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural leak or resting potential</td><td align="left" rowspan="1" colspan="1">
<italic>V<sub>L</sub></italic>
</td><td align="left" rowspan="1" colspan="1">V</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural firing threshold</td><td align="left" rowspan="1" colspan="1">
<italic>θ</italic>
</td><td align="left" rowspan="1" colspan="1">V</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural action potential spike</td><td align="left" rowspan="1" colspan="1">
<italic>δ</italic>
</td><td align="left" rowspan="1" colspan="1"/></tr><tr><td align="left" rowspan="1" colspan="1">Neural refractory period</td><td align="left" rowspan="1" colspan="1">
<italic>T<sub>ref</sub></italic>
</td><td align="left" rowspan="1" colspan="1">s</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural reset (post-firing) membrane potential</td><td align="left" rowspan="1" colspan="1">
<italic>V<sub>reset</sub></italic>
</td><td align="left" rowspan="1" colspan="1">V</td></tr><tr><td align="left" rowspan="1" colspan="1">Synaptic efficacy of cell <italic>j</italic> onto cell
<italic>i</italic>
</td><td align="left" rowspan="1" colspan="1">
<italic>J<sub>ij</sub></italic>
</td><td align="left" rowspan="1" colspan="1">A s<sup>−1</sup>
</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural membrane phase space state vector</td><td align="left" rowspan="1" colspan="1">
<italic>ν</italic>
</td><td align="left" rowspan="1" colspan="1">varies</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural ensemble probability density</td><td align="left" rowspan="1" colspan="1">
<italic>p</italic>(<italic>ν</italic>,<italic>t</italic>)</td><td align="left" rowspan="1" colspan="1">varies</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural ensemble dynamics (flow)</td><td align="left" rowspan="1" colspan="1">
<italic>f</italic>(<italic>ν</italic>,<italic>t</italic>)</td><td align="left" rowspan="1" colspan="1">varies</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural ensemble random fluctuations (dispersion)</td><td align="left" rowspan="1" colspan="1">
<italic>D</italic>(<italic>ν</italic>,<italic>t</italic>)</td><td align="left" rowspan="1" colspan="1">varies</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural ensemble jacobian</td><td align="left" rowspan="1" colspan="1">
<italic>Q</italic>
</td><td align="left" rowspan="1" colspan="1">varies</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural ensemble average synaptic efficacy</td><td align="left" rowspan="1" colspan="1">〈<italic>J</italic>〉<italic><sub>J</sub></italic>
</td><td align="left" rowspan="1" colspan="1">A s<sup>−1</sup>
</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural ensemble mean firing rate</td><td align="left" rowspan="1" colspan="1">
<italic>Q</italic>(<italic>t</italic>)</td><td align="left" rowspan="1" colspan="1">s<sup>−1</sup>
</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural ensemble infinitesimal depolarization</td><td align="left" rowspan="1" colspan="1">
<italic>ε</italic> = <italic>dV</italic>(<italic>t</italic>)</td><td align="left" rowspan="1" colspan="1">V</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural ensemble mean membrane potential (drift)</td><td align="left" rowspan="1" colspan="1">
<italic>μ</italic> = <italic>
˙μ<sub>ν</sub></italic>
</td><td align="left" rowspan="1" colspan="1">V</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural ensemble mean capacitive current</td><td align="left" rowspan="1" colspan="1">

</td><td align="left" rowspan="1" colspan="1">A</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural ensemble membrane potential variance (diffusion)</td><td align="left" rowspan="1" colspan="1">
<italic>σ<sup>2</sup></italic>
</td><td align="left" rowspan="1" colspan="1">(V)<sup>2</sup>
</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural ensemble probability density flux</td><td align="left" rowspan="1" colspan="1">
<italic>F</italic>
</td><td align="left" rowspan="1" colspan="1">varies</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural population transfer function</td><td align="left" rowspan="1" colspan="1">
<italic>φ</italic>(<italic>ν</italic>,<italic>σ</italic>)</td><td align="left" rowspan="1" colspan="1">s<sup>−1</sup>
</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural population probability basis functions (modes)</td><td align="left" rowspan="1" colspan="1">
<italic>η</italic>(<italic>ν</italic>)</td><td align="left" rowspan="1" colspan="1">varies</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural mass synaptic gain time constant</td><td align="left" rowspan="1" colspan="1">
<italic>κ</italic>
</td><td align="left" rowspan="1" colspan="1">s<sup>−1</sup>
</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural mass synaptic response coefficient</td><td align="left" rowspan="1" colspan="1">
<italic>γ</italic>
</td><td align="left" rowspan="1" colspan="1">dimensionless</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural field local membrane potential in population
<italic>a</italic>
</td><td align="left" rowspan="1" colspan="1">
<italic>V<sub>a</sub></italic>(<italic>r</italic>,<italic>t</italic>)</td><td align="left" rowspan="1" colspan="1">V</td></tr><tr><td align="left" rowspan="1" colspan="1">Neural field local firing rate in population
<italic>a</italic>
</td><td align="left" rowspan="1" colspan="1">
<italic>Q<sub>a</sub></italic>(<italic>r</italic>,<italic>t</italic>)</td><td align="left" rowspan="1" colspan="1">s<sup>−1</sup>
</td></tr><tr><td align="left" rowspan="1" colspan="1">Mean number of synapses on neuron <italic>a</italic>
from neurons <italic>b</italic>
</td><td align="left" rowspan="1" colspan="1">
<italic>N<sub>ab</sub></italic>
</td><td align="left" rowspan="1" colspan="1"/></tr><tr><td align="left" rowspan="1" colspan="1">Mean time-integrated strength of the response of
<italic>V<sub>a</sub></italic> per incoming spike from
neurons <italic>b</italic>
</td><td align="left" rowspan="1" colspan="1">
<italic>s<sub>ab</sub></italic>
</td><td align="left" rowspan="1" colspan="1">
<italic>μ</italic> V s</td></tr><tr><td align="left" rowspan="1" colspan="1">Average rate of incoming spikes (pulse density) between
populations <italic>a</italic> and <italic>b</italic>
</td><td align="left" rowspan="1" colspan="1">
<italic>φ<sub>ab</sub></italic>
</td><td align="left" rowspan="1" colspan="1">s<sup>−1</sup>
</td></tr><tr><td align="left" rowspan="1" colspan="1">Discrete time delay between populations
<italic>a</italic> and <italic>b</italic>
</td><td align="left" rowspan="1" colspan="1">
<italic>τ<sub>ab</sub></italic>
</td><td align="left" rowspan="1" colspan="1">s</td></tr><tr><td align="left" rowspan="1" colspan="1">Coupling strength between neural populations
<italic>a</italic> and <italic>b</italic>
</td><td align="left" rowspan="1" colspan="1">
<italic>ν<sub>ab</sub></italic> = <italic>N<sub>ab</sub>s<sub>ab</sub></italic>
</td><td align="left" rowspan="1" colspan="1">V s</td></tr><tr><td align="left" rowspan="1" colspan="1">Mean decay rate of the soma response to a
delta-function synaptic input</td><td align="left" rowspan="1" colspan="1">
<italic>α<sub>ab</sub></italic>
</td><td align="left" rowspan="1" colspan="1">s<sup>−1</sup>
</td></tr><tr><td align="left" rowspan="1" colspan="1">Mean rise rate of the soma response to a delta-function
synaptic input</td><td align="left" rowspan="1" colspan="1">
<italic>β<sub>ab</sub></italic>
</td><td align="left" rowspan="1" colspan="1">s<sup>−1</sup>
</td></tr><tr><td align="left" rowspan="1" colspan="1">Firing threshold for channels of type
<italic>a</italic>
</td><td align="left" rowspan="1" colspan="1">
<italic>θ<sub>a</sub></italic>
</td><td align="left" rowspan="1" colspan="1">V</td></tr><tr><td align="left" rowspan="1" colspan="1">Characteristic range of axons, including dendritic
arborization</td><td align="left" rowspan="1" colspan="1">
<italic>r</italic>, <italic>r<sub>ab</sub></italic>
</td><td align="left" rowspan="1" colspan="1">m</td></tr><tr><td align="left" rowspan="1" colspan="1">Characteristic action potential propagation velocity</td><td align="left" rowspan="1" colspan="1">
<italic>c</italic>, <italic>c<sub>ab</sub></italic>
</td><td align="left" rowspan="1" colspan="1">m s<sup>−1</sup>
</td></tr><tr><td align="left" rowspan="1" colspan="1">Temporal damping coefficient in the absence of pulse
regeneration</td><td align="left" rowspan="1" colspan="1">
<italic>γ<sub>a</sub></italic> = <italic>c<sub>a</sub>/r<sub>a</sub></italic>
</td><td align="left" rowspan="1" colspan="1">s<sup>−1</sup>
</td></tr><tr><td align="left" rowspan="1" colspan="1">Steady state sigmoid slope in population
<italic>a</italic>
</td><td align="left" rowspan="1" colspan="1">
<italic>ρ<sub>a</sub></italic>
</td><td align="left" rowspan="1" colspan="1">(V s)<sup>−1</sup>
</td></tr><tr><td align="left" rowspan="1" colspan="1">Macroscopic observable</td><td align="left" rowspan="1" colspan="1">
<italic>ψ</italic>(<italic>k</italic>,<italic>ω</italic>) = <bold>M</bold>
<bold><italic>Q</italic></bold>
</td><td align="left" rowspan="1" colspan="1"/></tr><tr><td align="left" rowspan="1" colspan="1">Spatiotemporal measurement matrix</td><td align="left" rowspan="1" colspan="1">
<bold>M</bold>
</td><td align="left" rowspan="1" colspan="1"/></tr><tr><td align="left" rowspan="1" colspan="1">Autonomous (uncoupled) growth/decay rate of neural mass</td><td align="left" rowspan="1" colspan="1">
<italic>ζ</italic>
</td><td align="left" rowspan="1" colspan="1">s<sup>−1</sup>
</td></tr></tbody></table></alternatives></table-wrap>
<inline-formula></inline-formula> <disp-formula><label>(1)</label></disp-formula> <disp-formula><label>(2)</label></disp-formula> <disp-formula><label>(3)</label></disp-formula> <disp-formula><label>(4)</label></disp-formula> <disp-formula><label>(5)</label></disp-formula> <inline-formula></inline-formula> <disp-formula><label>(6,7)</label></disp-formula> <disp-formula><label>(8)</label></disp-formula> <disp-formula><label>(9)</label></disp-formula> <disp-formula><label>(10)</label></disp-formula> <disp-formula><label>(11,12)</label></disp-formula> <disp-formula><label>(13)</label></disp-formula> <disp-formula><label>(14)</label></disp-formula> <disp-formula><label>(15)</label></disp-formula> <disp-formula><label>(16)</label></disp-formula> <disp-formula><label>(17)</label></disp-formula> <disp-formula><label>(18)</label></disp-formula> <disp-formula><label>(19)</label></disp-formula> <disp-formula><label>(20)</label></disp-formula> <disp-formula><label>(21)</label></disp-formula> <disp-formula><label>(22)</label></disp-formula> <disp-formula><label>(23)</label></disp-formula> <disp-formula><label>(24)</label></disp-formula> <disp-formula><label>(25)</label></disp-formula> <disp-formula><label>(26)</label></disp-formula> <disp-formula><label>(27)</label></disp-formula> <disp-formula><label>(28)</label></disp-formula> <disp-formula><label>(29)</label></disp-formula> <disp-formula><label>(30)</label></disp-formula> <disp-formula><label>(31)</label></disp-formula> <inline-formula></inline-formula> <disp-formula><label>(32)</label></disp-formula> <disp-formula><label>(33)</label></disp-formula> <disp-formula><label>(34)</label></disp-formula> <disp-formula><label>(35)</label></disp-formula> <disp-formula><label>(36)</label></disp-formula> <disp-formula><label>(37)</label></disp-formula> <disp-formula><label>(38)</label></disp-formula> <disp-formula><label>(39)</label></disp-formula> <disp-formula><label>(40)</label></disp-formula> <disp-formula><label>(41)</label></disp-formula> <inline-formula></inline-formula> <disp-formula><label>(42)</label></disp-formula> <disp-formula><label>(43)</label></disp-formula> <disp-formula><label>(44,45,46)</label></disp-formula> <disp-formula><label>(47)</label></disp-formula> <disp-formula><label>(48)</label></disp-formula> <inline-formula></inline-formula> <disp-formula><label>(49)</label></disp-formula> <inline-formula></inline-formula> <disp-formula><label>(50,51)</label></disp-formula> <disp-formula><label>(52)</label></disp-formula> <disp-formula><label>(53)</label></disp-formula> <disp-formula><label>(54)</label></disp-formula> <disp-formula><label>(55)</label></disp-formula> <disp-formula><label>(56,57,58,59,60)</label></disp-formula> <inline-formula></inline-formula> <disp-formula><label>(61,62)</label></disp-formula> <disp-formula><label>(63,64,65)</label></disp-formula> <disp-formula><label>(66)</label></disp-formula> <disp-formula><label>(67)</label></disp-formula> <disp-formula><label>(68)</label></disp-formula> <disp-formula><label>(69)</label></disp-formula> <disp-formula><label>(70)</label></disp-formula> <disp-formula><label>(71,72)</label></disp-formula> <disp-formula><label>(73)</label></disp-formula> <disp-formula><label>(74)</label></disp-formula> <disp-formula><label>(75)</label></disp-formula> <disp-formula><label>(76)</label></disp-formula> <disp-formula><label>(77,78)</label></disp-formula> <disp-formula><label>(79)</label></disp-formula> <disp-formula><label>(80)</label></disp-formula> <disp-formula><label>(81)</label></disp-formula> <inline-formula></inline-formula> <disp-formula><label>(82)</label></disp-formula> <inline-formula></inline-formula> <disp-formula><label>(83)</label></disp-formula> <inline-formula></inline-formula> <disp-formula><label>(84)</label></disp-formula> <inline-formula></inline-formula> <disp-formula><label>(85)</label></disp-formula> <inline-formula></inline-formula> <inline-formula></inline-formula> <inline-formula></inline-formula> <inline-formula></inline-formula> <disp-formula><label>(86)</label></disp-formula> <disp-formula><label>(87)</label></disp-formula> <disp-formula><label>(88)</label></disp-formula> <disp-formula><label>(89)</label></disp-formula> <disp-formula><label>(90)</label></disp-formula> <disp-formula><label>(91)</label></disp-formula> <disp-formula><label>(92)</label></disp-formula> <disp-formula><label>(93)</label></disp-formula> <disp-formula><label>(94)</label></disp-formula> <disp-formula><label>(95)</label></disp-formula> <inline-formula></inline-formula> <inline-formula></inline-formula>
<table-wrap-foot><fn id="nt101"><label/><p>The first column gives a brief description of the parameter, with its
symbol listed in the second. The unit of each quantity is given in the
third column.</p></fn></table-wrap-foot> <fn-group><fn fn-type="COI-statement"><p>The authors have declared that no competing interests exist.</p></fn><fn fn-type="financial-disclosure"><p>GD was supported by the European Union, grant EC005-024 (STREP Decisions in
Motion), by the Spanish Research Project BFU2007-61710 and CONSOLIDER
(Bilingualism and Cognitive Neuroscience). PR was supported by The Australian
Research Council. VKJ acknowledges funding by ATIP (CNRS) and Brain NRG
JSMF22002082. KJF is supported by the Wellcome Trust and is a member of the
Brain NRG JSMF22002082. MB acknowledges funding by the Australian Research
Council, the NHMRC, and Brain NRG JSMF22002082.</p></fn></fn-group>
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PLoS Comput Biol. 2008 Aug 29; 4(8):e1000092