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Introduction

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.

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 [1]–[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).

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.

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.

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 Neural Modes and Masses, 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 Neural Field Models section. In Numerical Simulations: Ensemble Activity from Neuronal to Whole Brain Scale, 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 Cognitive and Clinical Applications, we illustrate applications of neural ensemble modeling in health and disease; namely, decision-making, auditory scene analysis, and absence seizures.

A summary of the notation for all the main dynamical variables and physiological parameters is given in Table 1.

Table 1. List of notation and symbols.

doi:10.1371/journal.pcbi.1000092.t001

Mean-Field Models

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., [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.

Ensemble density models.

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, V, capacitive current, I, or the time since the last action potential, T. 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 ν = {V,I,T} Єℜ³ or particle in phase space. Imagine a very large number of neurons that populate phase space with a density p(ν,t). As the state of each neuron evolves, the points will flow through phase space, and the ensemble density p(ν,t) will evolve until it reaches some steady state or equilibrium. p(ν,t) 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 equation
(1)
This equation comprises a flow and a dispersion term; these terms embed the assumptions about the dynamics (phase flow, f(ν,t)) and random fluctuations (dispersion, D(ν,t)) 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 with
(2)
where, D = ½σ² and ω is a standard Wiener process; i.e., w(t)−w(t+Δt)~N(0, Δt). 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 Q
(3)

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.

From spiking neurons to mean-field models.

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 [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 [7] used below. Other models relevant for systems neuroscience can be found in [4],[8],[9]. In the LIF model, each neuron i can be fully described in terms of a single internal variable, namely the depolarization V_i(t) of the neural membrane. The basic circuit of a LIF model consists of a capacitor, C, in parallel with a resistor, R, driven by a synaptic current (excitatory or inhibitory postsynaptic potential, EPSP or IPSP, respectively). When the voltage across the capacitor reaches a threshold θ, the circuit is shunted (reset) and a δ pulse (spike) is generated and transmitted to other neurons. The subthreshold membrane potential of each neuron evolves according to a simple RC circuit, with a time constant τ = RC given by the following equation:
(4)
where I_i(t) is the total synaptic current flow into the cell i and V_L 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 [10] for extensions considering detailed synaptic dynamics such as AMPA, NMDA, and GABA). The total synaptic current coming into the cell i is therefore given by the sum of the contributions of δ-spikes produced at presynaptic neurons. Let us assume that N neurons synapse onto cell i and that J_ij is the efficacy of synapse j, then the total synaptic afferent current is given by
(5)
where is the emission time of the k^th spike from the j^th presynaptic neuron. The subthreshold dynamical Equation 4, given the input current (from Equation 5), can be integrated, and yields
(6,7)
if the neuron i is initially (t = 0) at the resting potential (V_i(0) = V_L). In Equation 7, H(t) is the Heaviside function (H(t) = 1 if t>0, and H(t) = 0 if t<0). Thus, the incoming presynaptic δ-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.

The population density approach.

Realistic neuronal networks comprise a large number of neurons (e.g., a cortical column has O(10⁴)−O(10⁸) neurons) which are massively interconnected (on average, a neuron makes contact with O(10⁴) 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., [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., [12]) and neurobiology (e.g., [13],[14]).

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 [15] (see also [16],[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 V(t) at a given time t have a different history because of random fluctuations in the input current I(t). 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 V(t) in a population are indistinguishable. Consequently, the dynamics are described by the evolution of the probability density function:
(8)
which expresses the population density, which is the fraction of neurons at time t that have a membrane potential V(t) in the interval [ν,ν+dν]. The evolution of the population density is given by the Chapman-Kolmogorov equation
(9)
where ρ(ε|ν) = Prob{V(t+dt) = ν+ε|V(t) = ν} is the conditional probability that generates an infinitesimal change ε = V(t+dt)−V(t) in the infinitesimal interval dt. The Chapman-Kolmogorov equation can be written in a differential form by performing a Taylor expansion in p(ν′,t) ρ(ε|ν′) around ν′ = ν; i.e.,
(10)
In the derivation of the last equation, we have assumed that p(ν′,t) and ρ(ε| ν′) are infinitely many times differentiable in ν. Inserting this expansion in Equation 9, and replacing the time derivative in ν′ by the equivalent time derivative in ν, we obtain
(11,12)
where 〈…〉_ν denotes the average with respect to ρ(ε| ν) at a given ν. Finally, taking the limit for dt → 0, we obtain:
(13)

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.

The diffusion approximation.

The temporal evolution of the population density as given by Equation 13 requires the moments 〈ε^k〉_υ due to the afferent current during the interval dt. 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, dV(t), in the membrane potential of all neurons as:
(14)
where N is the number of neurons, and Q(t) is the mean population firing rate. This is determined by the proportion of active neurons by counting the number of spikes n_spikes(t,t+dt) in a small time interval dt and dividing by N and by dt [18]; i.e.,
(15)

In Equation 14, 〈J〉_J denotes the average of the synaptic weights in the population. The moments of the infinitesimal depolarization, ε = dV(t), 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:
(16)

(17)
In general, keeping only the leading term linear in dt, it is easy to prove that for k>1,
(18)
and hence,
(19)

The diffusion approximation arises when we neglect high-order (k>2) terms. The diffusion approximation is exact in the limit of infinitely large networks, i.e., N → ∞, if the synaptic efficacies scale appropriately with network size, such that J → 0 but NJ² → const. In other words, the diffusion approximation is appropriate, if the minimal kick step, J, 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 (μ) and diffusion (σ²) coefficients.

The diffusion approximation allows us to omit all higher orders k>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
(20)

In the particular case that the drift is linear and the diffusion coefficient, σ²(t), is given by a constant, the Fokker-Planck equation describes a well-known stochastic process called the Ornstein-Uhlenbeck process [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 V(t) when the input afferent currents are given by
(21)
where ω(t) 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 μ(t) and variance σ².

The mean-field model.

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.

The Fokker-Planck equation describing the Ornstein-Uhlenbeck process, with μ = 〈J〉_J NQ(t) and σ² = 〈J²〉_J NQ(t), can be rewritten as a continuity equation:
(22)
where F is the flux of probability defined as follows:
(23)
The stationary solution should satisfy the following boundary condition:
(24)
and
(25)
which expresses the fact that the probability current at threshold gives, by a self-consistent arguments, the average firing rate, Q, of the population. Furthermore, at ν→−4 the probability density vanishes fast enough to be integrable; i.e.,
(26)
and
(27)
In addition, the probability mass leaving the threshold at time t has to be re-injected at the reset potential at time t+t_ref (where t_ref is the refractory period of the neurons), which can be accommodated by rewriting Equation 22 as follows:
(28)
where H(.) is the Heaviside function. The solution of Equation 28 satisfying the boundary conditions (Equations 24–27) is:
(29)
Taking into account the fraction of neurons, Qt_ref, in the refractory period and the normalization of the mass probability,
(30)
Finally, substituting Equation 29 into Equation 30, and solving for Q, we obtain the population transfer function, φ, of Ricciardi [13]:
(31)
where .

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, Q_i, for different populations, i, in the network can be found by solving a set of coupled self-consistency equations, given by:
(32)

To solve the equations defined by Equation 32 for all i, we integrate the differential equation below, describing the approximate dynamics of the system, which has fixed-point solutions corresponding to Equation 32:
(33)

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 [10],[20]. In our case, the derived transfer function, φ, 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 [10]. Before turning to neural mass models, we consider some applications of mean-field modeling that will be reprised in the last section.

Competition and cooperation.

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 [21], and neurophysiological evidence [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 [21],[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.

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 [6], [24]–[33]. In the section entitled Cognitive and Clinical Applications, we present one of these examples, in the context of decision-making.

Neural Modes and Masses

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 [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.

An alternative approach to dimension reduction is to approximate the ensemble densities with a linear superposition of probabilistic modes or basis functions η(ν) 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.

Moments and modes of density dynamics.

Instead of characterising the density dynamics explicitly, one can summarize it in terms of coefficients parameterising the expression of modes:
(34)
where μ = η⁻p, η⁻ being the generalized inverse of the matrix encoding the basis set of modes.

A useful choice for the basis functions are the eigenfunctions (i.e., eigen vectors) of the Fokker-Planck operator, Q [17], where Qη = ηλ⇒η⁻Qη = λ and λ 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
(35)

The last expression means that, following perturbation, each mode decays exponentially, to disclose the equilibrium mode, η₀, 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
(36)

See [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.

Neural mass models.

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 [5],[36],[37]. The term mass action model was coined by [38] as an alternative to density dynamics. These models can be motivated as a description in terms of the expected values of neuronal states, μ, 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, ς(μ_ν), relating expected depolarisation to expected firing rate [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, μ = {μ_ν,μ_i} where
(37)
Here μ_a can be regarded as capacitive current. The constant γ controls the rise time of voltage, in response to inputs (see also the Neural Field Models section). These differential equations can be expressed as a convolution of inputs, ς(μ_ν), to give the expected depolarization, μ_ν; i.e., the convolution of the input signal with an impulse response kernel W(t)
(38)

The input is commonly construed to be a firing rate (or pulse density) and is a sigmoid function, ς, of mean voltage of the same or another ensemble. The coupling constant, κ, scales the amplitude of this mean-field effect. This form of neural mass model has been used extensively to model electrophysiological recordings (e.g., [39]–[41]) and has been used recently as the basis of a generative model for event-related potentials that can be inverted using real data [42].

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.

Neural Field Models

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, x, and time, μ(t)→μ(x,t). 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 [43],[44]. Key forms for neural field equations were proposed and analysed by [45]–[47]. These models were generalized by [48],[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.

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.,
(39)
where |x−x′| is the distance between the spatial locations x and x′, c is the characteristic speed of spike propagation, and γ reflects the spatial decay of lateral interactions. The corresponding second order equations of motion are a neural wave equation (see [48],[49] and below)
(40)
where γ = c/r and ▽² 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., [50]–[53]). The generic form of neural field dynamics can be written as (see also [53]):
(41)
where μ = μ(x,t) is the neural field, capturing the neural mass activity at time t and position x. f(μ) captures the local dynamics of the neural field, and T_c = t−|x−x′|/c is the time delay due to signal propagation. h is a constant threshold value and Γ is the spatial domain of the neural field, where x Є Γ = [0,L]. The kernel W(|x−x′|) 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 f(μ), , 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 c is the propagation velocity of action potentials traveling down an axon. If f(μ) = −ζ μ, then the constant ζ>0Єℜ represents the growth rate of the neural mass. α>0 is a scaling constant. Under instantaneous interactions, c→∞, 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) [47]. This class of models are also sometimes referred to as continuous attractor neural networks (CANN). When the firing rate, ς, is a Heaviside step function, [45] was able to construct an explicit one-bump solution of the form
(42)
where the value a 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 [4],[54],[55]. Spatially localized bump solutions are equivalent to persistent activity and have been linked to working memory in prefrontal cortex [56],[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 [58],[59], have been put forward. The latter will be captured by specific choices of the local dynamics, f(μ), in Equation 41; for instance, [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 [61]–[64], place cells [65]–[68], movement initiation [69], and feature selectivity in visual cortex, where bump formation is related to the tuning of a particular neuron's response [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 [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 [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 [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 [74].

Much experimental evidence, supporting the existence of neural fields, has been accumulated (see [53] for a summary). Most of these results are furnished by slice studies of pharmacologically treated tissue, taken from the cortex [75]–[77], hippocampus [78], and thalamus [79]. In brain slices, these waves can take the form of synchronous discharges, as seen during epileptic seizures [80], and spreading excitation associated with sensory processing [81]. For traveling waves, the propagation speed depends on the threshold, h, which has been established indirectly in real neural tissue (rat cortical slices bathed in the GABA-A blocker picrotoxin) by [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, h, in Equation 42.

Recent developments in neural field models.

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 [38], [43], [44], [48], [50], [83]–[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 [88], or hemodynamic responses that determine the blood oxygen level–dependent (BOLD) signals [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.

Recent work in this area has resulted in numerous quantitatively verified predictions about brain electrical activity, including EEG time series [86],[87],[90], spectra [50],[86],[87],[90],[91], coherence and correlations, evoked response potentials (ERPs) [87], and seizure dynamics [86],[90],[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 [86],[93],[94].

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.