Neural Fields

Post-publication activity

Dr. Stephen Coombes, University of Nottingham, U.K.

Neural field equations are tissue level models that describe the spatiotemporal evolution of coarse grained variables such as synaptic or firing rate activity in populations of neurons.

Introduction

The number of neurons and synapses in even a small piece of cortex is immense. Because of this a popular modelling approach has been to take a continuum limit and study neural networks in which space is continuous and macroscopic state variables are mean firing rates. Perhaps the first attempt at developing a continuum approximation of neural activity can be attributed to Beurle (1956) in the 1950s and later by Griffith (1963, 1965) in the 1960s. By focusing on the proportion of neurons becoming activated per unit time in a given volume of model brain tissue, Beurle was able to analyse the triggering and propagation of large scale brain activity. However, this work only dealt with networks of excitatory neurons with no refractory or recovery variable. It was Wilson and Cowan (1972,1973) in the 1970s who extended Beurle’s work to include both inhibitory and excitatory neurons as well as refractoriness. Further work, particularly on pattern formation, in continuum models of neural activity was pursued by Amari (1975, 1977) under natural assumptions on the connectivity and firing rate function. Amari considered local excitation and distal inhibition which is an effective model for a mixed population of interacting inhibitory and excitatory neurons with typical cortical connections (commonly referred to as Mexican hat connectivity).

Since these seminal contributions to dynamic neural field theory similar models have been used to investigate EEG rhythms, visual hallucinations, mechanisms for short term memory and motion perception.

Physiological motivation

Neural fields consider populations of neurons embedded in a coarse-grained spatial area (Wilson & Cowan 1973). The grains in such an area reflect micro- or macro-columns as observed in primary sensory areas, such as the barrel cortex in rodents (Petersen 2007) or in the mammalian visual cortex (Hubel & Wiesel 1962, Saez et al. 1998). Moreover neural fields consider instantaneous population firing rates (population coding), i.e. the number of firing neurons in a certain short time interval of few milliseconds. The time variable in neural field models is a multiple of this time interval. Consequently neural fields are coarse-grained in time and space and represent a mean-field model. It is important to mention the similarity to neural mass models, which neglect spatial extensions but involve the same underlying assumptions on population coding and time-graining.

Beurle (1956) and Wilson & Cowan (1972, 1973) were among the first who have derived mathematically the neural field model equations. Recently Faugeras et al. (2009) and Bressloff (2009) have given different derivations considering statistical properties and stochastic dynamics of single neurons. Interestingly, they present an extended model taking into account the dynamics of both the mean activity and its variance, whereas previous models considered the mean activity only. The following paragraphs give a more heuristic derivation of neural field models taking into account three major elements of neural action: dendritic currents evoked at chemical synapses by incoming pulses, the firing of neurons subject to dendritic currents and the transmission of action potentials along axonal branches. The evoked dendritic current I(t) obeys

\( I(t) = \int_{-\infty}^t {\rm d}\tau^\prime h(t-t^\prime) P(t^\prime)\tag{1} \)

with the synaptic response function h(t) of a chemical receptor to a single incoming pulse and the incoming instantaneous pulse train P(t). Experimental investigations of single synapses suggest certain functions h(t) which approximate well the synaptic response behavior. In neural fields, one considers an effective average synaptic response function, that describes the mean response of many synapses to many incoming pulse trains in the neural population network. Of course the mean time response of a population of synapses may be different than the response function of a single synaptic receptor, however the shape may be the same. Hence Eq.(1) is assumed to be also valid for the population dendritic current I(t) involving effective parameters and the population firing rate P(t) related by the synaptic population response function h(t).

Mathematically, the conversion from P(t) to I(t) is a Volterra integral equation with kernel h(t). It is often more convenient to consider differential equations than integral equations and thus one may try to calculate the inverse of the integral operator. This is possible for certain integral kernel functions h(t). A simple and widely-applied model is \(h(t)=\exp(-t/\tau)/\tau\) with the decay time constant \(\tau\). Then

\( \begin{array}{lcl} \frac{dI(t)}{dt}&=&-\frac{1}{\tau}\int_{-\infty}^t {\rm d}t^\prime \exp\left(-(t-t^\prime)/\tau\right)/\tau P(t^\prime) + \int_{-\infty}^t \left({\rm d}t^\prime/dt\right)|_{\infty}^t \exp\left(-(t-t^\prime)/\tau\right)/\tau P(t^\prime)\\ &=&-\frac{1}{\tau}I(t) + \frac{1}{\tau} P(t) \end{array} \)

applying the chain rule and Eq. ((1)) can be written as a differential equation

\( \hat{L} I(t)=P(t)\quad,\quad \hat{L}=\tau\frac{d}{dt}+1 . \)

A more detailed response function and its corresponding differental operator is

\( h(t)=\frac{1}{\tau_1-\tau_2}\left(e^{-t/\tau_1}-e^{t/\tau_2}\right) \quad , \quad \hat{L}=\tau_1\tau_2\frac{d^2}{dt^2}+\left(\tau_1+\tau_2\right)\frac{d}{dt}+1 \)

with the short rise time \(\tau_2\) and the long decay time \(\tau_1\). For \(\tau_1=\tau_2=\tau\), the response function is the so-called alpha-function

\( h(t)=te^{-t/\tau}/\tau\quad , \quad \hat{L}=\tau^2\frac{d^2}{dt^2}+2\tau\frac{d}{dt}+1~. \)

Mathematical Framework

In many continuum models for the propagation of electrical activity in neural tissue it is assumed that the synaptic input current is a function of the pre-synaptic firing rate function (Wilson & Cowan 1973). These infinite dimensional dynamical systems are typically variations on the form

\( \frac{1}{\alpha} \frac{\partial u(x,t)}{\partial t} = -u + \int_{-\infty}^\infty {\rm d} y w(y) f(u(x-y,t - |y|/v)) . \)

Figure 1: A caricature of a neural field. Such models can include not only the effects of axonal delays, but the filtering of firing rate signals by both synapses and dendrites.

Here, u(x,t) is interpreted as a neural field representing the local activity of a population of neurons at position x and time t. The second term on the right represents the synaptic input, with f interpreted as the firing rate function of a single neuron. The strength of connections between neurons separated by a distance y is denoted w(y), and the function w is often referred to as the synaptic footprint. This formulation assumes that the system is spatially homogeneous and isotropic. Typically w reflects global excitation (w>0), global inhibition (w<0), local excitation - lateral inhibition (Mexican hat) describing, e.g. orientation tuning in the visual cortex (Somers et al. 1995, Ben-Yishai et al. 1995), or local inhibition - lateral excitation (inverse Mexican hat) reflecting short-range interactions of inhibitory interneurons and long-range interactions of excitatory pyramidal cells. Periodic w have also attracted some attention (Ben-Yishai et al. 1995, Laing and Troy 2003). The parameter \(\alpha\) is the temporal decay rate of the synapse. The delayed argument to u under the spatial integral represents the axonal conduction delay arising from the finite speed of signals travelling over a distance y (Wilson & Cowan 1972; Nunez 1974; Jirsa & Haken 1997); namely |y|/v where v is the velocity of an action potential along axonal fibres. Recent extensions involve distributions of axonal transmission speeds v (Atay and Hutt 2006).

There are several natural choices for the firing rate function, the simplest being a Heaviside step function. In this case a neuron fires maximally (at a rate set by its absolute refractory period) or not at all, depending on whether or not synaptic activity is above or below some threshold. In a statistical mechanics approach to formulating mean-field neural equations this all or nothing response is replaced by a smooth sigmoidal form (Wilson & Cowan 1972; Amari 1972).

The simple mathematical model above can be naturally extended to describe multiple populations, cortical sheets, spike frequency adaptation, neuromodulation, slow ionic currents and more sophisticated forms of synaptic and dendritic processing as described in the review articles below.

Dynamic behaviour

The sorts of dynamic behaviour that are typically observed in neural field models include spatially and temporally periodic patterns beyond a Turing instability (Ermentrout 1979; Tass 1995), localised regions of activity such as bumps (Kishimoto 1979) and travelling waves (Ermentrout 1993; Pinto & Ermentrout 2001). In the latter case corresponding phenomena may be observed experimentally using multi-electrode recordings and imaging methods. In particular it is possible to electrically stimulate slices of pharmacologically treated tissue taken from the cortex (Chervin et al. 1988; Golomb & Amitai 1997,Wu et al 1999}, hippocampus (Miles et al. 1995) and thalamus (Kim et al. 1995). In brain slices these waves can take the form of synchronous discharge seen during an epileptic seizure (Connors & Amitai 1993) and spreading excitation associated with sensory processing (Ermentrout & Kleinfeld 2001). Interestingly, spatially localised bumps of activity have been linked to working memory (the temporary storage of information within the brain) in prefrontal cortex (Colby et al. 1995, Goldman-Rakic 1995), representations in the head-direction system (Zhang 1996), and feature selectivity in the visual cortex, where bump formation is related to the tuning of a particular neuron's response (Ben-Yishai et al. 1995).

Pattern formation

Figure 2: An activity profile of a hexagonal pattern emerging beyond a Turing instability in a two-dimensional neural field model with short-range excitation and long-range inhibition.

Neural field models are nonlinear spatially extended systems and thus have all the necessary ingredients to support pattern formation. The analysis of such behaviour is typically performed with a mixture of linear Turing instability theory, weakly nonlinear perturbative analysis and numerical simulations. In one dimension single population models with Mexican-hat connectivity can support global periodic stationary patterns. With more than one population non-stationary (travelling) patterns are also possible. In two dimensions many other interesting patterns can occur such as spiral waves (Laing 2005), target waves and doubly periodic patterns. These latter patterns take the form of stripes and checkerboard like patterns, and have been linked by Ermentrout & Cowan (1979) and Bressloff et al (2001) to drug-induced visual hallucinations.

Figure 3: A spatially localised 3-bump solution in a two-dimensional neural field model.

Neural field models with short-range excitation and long-range inhibition are also able to support spatially localised solutions, commonly referred to as bumps or multi-bumps. For the case that the firing rate function is a Heaviside step function with threshold h Amari (1977) was able to construct an explicit one-bump solution of the form

\( u(x) = \int_0^\Delta w(x-y) {\rm d y}, \qquad u(0)=h=u(\Delta) , \)

such that below some critical threshold there co-exists both a wide and a narrow solution. Of the two, it is the wider solution that is stable. For smooth sigmoidal firing rates no closed-form spatially localised solutions are known, though much insight into the form of multi-bump solutions has been obtained using techniques first developed for the study of fourth order pattern forming systems (Laing & Troy 2003). A stationary activity bump can exhibit a variety of dynamical instabilities including a Hopf bifurcation to a spatially localized oscillatory solution or breather (Folias and Bressloff 2004, Coombes and Owen 2005).

One possible computational role for an activity bump is to encode a set of stimulus features in terms of the peak location of the bump within a spatially-structured network. In the case of a homogeneous network, the set of allowed features will form a continuous manifold (attractor) that reflects the underlying topology of the network. Thus, a periodic stimulus feature can be encoded by an activity bump in a homogeneous ring network (Ben-Yishai et al 1995, Zhang 1995). However, one of the consequences of a homogeneous network is that the bump will be marginally stable with respect to spatial translations tangential to the continuous manifold. This means that the activity bump will slowly drift over time in the presence of arbitrarily small levels of noise (Laing and Chow 2001). One way to construct neural field models that are robust to noise is to introduce some form of cellular bistability (Camperi and Wang 1998, Fall et al 2004).

Travelling Waves

Figure 4: A travelling pulse in a one dimensional neural field model with spike frequency adaptation.

For one-dimensional models with sigmoidal firing rate functions and excitatory coupling it is possible to find wave fronts joining an excited state to a resting state (Ermentrout & McLeod 1993). Moreover, in systems with mixed (excitatory and inhibitory) coupling or excitatory systems with adaptive currents, solitary travelling pulses are also possible. For a Heaviside firing rate function with threshold h many exact results about travelling waves have been obtained. For example the speed of a stable travelling front in a purely excitatory network with w(x)=exp(-|x|)/2 takes the explicit form

\( c = \frac{v(2h-1)}{2h-1 - 2 h v/\alpha} \)

Figure 5: A propagating wave of electrical activity in a slice experiment from the Pinto Lab (with permission). Left: local field potential recordings from a linear array of 16 electrodes. Right: a smooth colour coded version of the field.

The strong dependence of the wave speed on the threshold h has now been indirectly established in real neural tissue (rat cortical slices bathed in the GABA_A blocker picrotoxin) by Richardson et al. 2005. These experiments exploit the fact that cortical neurons have long apical dendrites and are easily polarisable by an electric field and that epileptiform bursts can be initiated by a stimulation electrode. An applied positive (negative) electric field 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.

The bifurcation structure of travelling waves can be analysed using a so-called Evans function. This was originally formulated by Evans (1975) in the context of a stability theorem about excitable nerve axon equations of Hodgkin–Huxley type. The zeros of this complex analytic function determine the normal spectrum of the operator obtained by linearising a system about its travelling wave solution. The extension to neural field models is more recent and, for the special case of a Heaviside firing rate function, several models have now been studied (Coombes & Owen 2004, 2005).

One of the common assumptions in most neural field models is that the network is homogeneous and isotropic, that is, the weight distribution depends on the distance between interacting populations within the network. The real cortex, however, is more realistically modeled as an anisotropic and inhomogeneous two-dimensional medium due to the patchy nature of long-range horizontal connections found in superficial layers of cortex (Bosking et al 1997). Anisotropies in the weight distribution could lead to variations in wave speed in different directions, whereas inhomogeneities could lead to time-varying wave profiles and possibly wave propagation failure (Bressloff 2001).

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Internal references

Yuri A. Kuznetsov (2006) Andronov-Hopf bifurcation. Scholarpedia, 1(10):1858.
John W. Milnor (2006) Attractor. Scholarpedia, 1(11):1815.
Eugene M. Izhikevich (2006) Bursting. Scholarpedia, 1(3):1300.
Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
S. Murray Sherman (2006) Thalamus. Scholarpedia, 1(9):1583.