Interface dynamics in planar neural field models
© Coombes et al.; licensee Springer 2012
Received: 21 November 2011
Accepted: 13 February 2012
Published: 2 May 2012
Neural field models describe the coarse-grained activity of populations of interacting neurons. Because of the laminar structure of real cortical tissue they are often studied in two spatial dimensions, where they are well known to generate rich patterns of spatiotemporal activity. Such patterns have been interpreted in a variety of contexts ranging from the understanding of visual hallucinations to the generation of electroencephalographic signals. Typical patterns include localized solutions in the form of traveling spots, as well as intricate labyrinthine structures. These patterns are naturally defined by the interface between low and high states of neural activity. Here we derive the equations of motion for such interfaces and show, for a Heaviside firing rate, that the normal velocity of an interface is given in terms of a non-local Biot-Savart type interaction over the boundaries of the high activity regions. This exact, but dimensionally reduced, system of equations is solved numerically and shown to be in excellent agreement with the full nonlinear integral equation defining the neural field. We develop a linear stability analysis for the interface dynamics that allows us to understand the mechanisms of pattern formation that arise from instabilities of spots, rings, stripes and fronts. We further show how to analyze neural field models with linear adaptation currents, and determine the conditions for the dynamic instability of spots that can give rise to breathers and traveling waves.
which can be useful in determining the stability of equilibrium solutions.
The main topic of this article is the development of an equivalent interface description for neural field models of the type exemplified by (1). We show that activity patterns can be described by dynamical equations of reduced dimension, and that these depend only on the shape of the interface (requiring no knowledge of activity away from the interface). Not only is this description amenable to fast numerical simulation strategies, it allows for the construction of localized states and an analysis of their linear stability. Given the computational overheads in simulating the full neural field model this enhances our ability to study pattern formation and suggests more generally that modeling the interfaces of patterns, rather than the patterns themselves, may lead to novel, efficient descriptions of brain activity. Indeed the use of interface dynamics to analyze patterns that arise in partial differential equation models of chemical and physical systems has a strong history , and it is natural to translate some of the ideas and technologies from these studies to non-local neural field models. The work by Goldstein [12, 13] and Muratov  on pattern formation in two-dimensional excitable reaction-diffusion systems is especially relevant in this context, as both authors have developed effective descriptions of interface dynamics in terms of non-local interactions. See also the book by Desai and Kapral  for a recent overview.
It is worth pointing out that whether computing interface dynamics can compete with other numerical schemes will depend on the problem at hand. In general, boundaries that remain relatively short and do not pinch guarantee a speed advantage. In practice, we expect this approach to be especially relevant for (semi-) analytical work aiming at qualitative understanding, as illustrated by some of the examples presented in this article.
In Section 2 we present some of the key ideas behind an interface dynamics in the setting of a one-dimensional neural field model. This is particularly useful for introducing the definition of normal velocity from a level-set condition, as well as establishing what it means for an interface to be linearly stable. The extension of these ideas to two-dimensional systems is presented in Section 3. By writing the synaptic connectivity in terms of a linear combination of Bessel functions, we show that dynamics for the interface can be constructed in terms of line-integrals along the interface, and that the normal velocity of the interface is driven by Biot-Savart-style interactions. Thus we obtain a reduced description for the evolution of a pattern boundary solely in terms of quantities on the boundary itself. Numerical simulations of the interface dynamics are shown to be in direct correspondence with those of the full neural field model. The notion of linear stability of stationary solutions in the interface framework is fleshed out in a series of examples (for spots, rings, stripes and fronts) in Sections 4 and 5, and allows us to understand some of the mechanisms for pattern formation. In Section 6 we add linear adaptation to (1) and extend our analysis to cover this important neural phenomenon. This can introduce dynamic instabilities of stationary structures, and we calculate where breathing and drift instabilities for localized spots occur. Moreover, we use a perturbation argument to determine the shape of traveling spots that emerge beyond a drift instability and show that spots contract in the direction of propagation and widen in the orthogonal direction. Finally, in Section 7 we discuss extensions of the work in this article.
2 A one-dimensional primer
A front is stable if .
The equation only has the solution . We also have that , showing that is a simple eigenvalue. Hence, the traveling wave front for this example is neutrally stable.
Given this preliminary exposition of interface dynamics we are now ready to describe the extension to two dimensions and to address the additional challenges that working in the plane gives rise to.
3 Interface dynamics in two dimensions
which is shown for and in Figure 2B.
From the form of (22), (23), and (24), we see that the evolution of the interface does not require any knowledge of the neural field away from the contour, and rather just depends on the shape of the sets where the field is above threshold. We now exploit the choice of as basis function for constructing the synaptic kernel to show how the double integrals in (23) and (24) can be reduced to line integrals. This yields an elegant description of the interface dynamics that emphasizes how the geometry of drives the evolution of spatiotemporal patterns. The key step in this reformulation is the use of Green’s identity. For a two-dimensional vector field F this identity is the two-dimensional version of the divergence theorem, which we write symbolically as . Using this first identity we may generate a second for a scalar field Ψ as .
Note that the choice of as a basis for w is merely a convenience to allow explicit calculations. As long as we can write the connectivity function w as the divergence of a vector field then we can exploit Green’s first identity to turn the right hand side of (23) into a line integral.
From the Biot-Savart form of (29) we see that for every part i of the synaptic kernel there is an effective repulsion between two arc length positions with anti-parallel tangent vectors, although the combined effect when including all N terms will depend on the choice of the amplitudes . Now with (22), (27), and (28) the normal velocity on the interface can be written solely in terms of certain line-integrals around the interface. From a computational perspective this leads to a substantial advantage in that one no longer needs to solve the full non-local neural field model (17) across the entire plane, and can instead simply evolve the interface in time by discretizing the boundary and translating the points with the normal velocity from (22) in the direction of n. One possible practical disadvantage of this is the need to monitor for possible self-intersections of the evolving boundary, splitting, where a connected region pinches off into two or more disconnected regions, or indeed the creation of new boundaries where none existed before. However, numerical schemes for coping with similar situations in fluid models are well developed in the literature and it is natural to turn to these for more refined numerical schemes and ones that can automate the process of contour surgery [22, 23]. In Figure 1B we illustrate the simple numerical implementation of the interface dynamics described in Section A.2 in the Appendix, showing the effectiveness of the dimensionally reduced system at capturing the spatiotemporal pattern formation of the full model shown in Figure 1A.
and the observation that .
which can also be evaluated as a line integral. In order to analyze the stability of stationary solutions in the original neural field formalism defined by (1) one would perturb the field variable u and linearize to derive an eigenvalue equation or Evans function . Here we determine stability using the interface dynamics, generalizing the approach described in Section 2.
The dynamics for is given by (23) with replaced by . The perturbation affects the normal vector as well as the displacement vector that occurs in (27). Thus to evaluate (35) it is necessary to linearize about the unperturbed contour. In the case of interfaces without curvature the linear contribution to is zero. In contrast for curved interfaces an addition theorem for Bessel functions shows that there is a non-zero contribution. To clarify this statement and show how the above machinery is used in practice, we now give some explicit examples of localized solutions and their stability.
4 Localized states: spots
The zeros of the first derivative of with respect to R give the stationary circular solutions, including the trivial case , as expected.
5 Rings, fronts and stripes
In this section we show how to treat other simple interface shapes, namely rings, fronts and stripes, and determine their stability. We recover previous results in  for rings (obtained with an Evans function method), whilst calculations for the other structures are shown to be straight-forward using the interface dynamics approach.
To determine stability we consider a front along and write the perturbed front as .
6 Neural field models with linear adaptation
which has zeros when and . Hence, the stationary front changes from stable to unstable as α is increased through .
Here R is determined by (71). A further weakly nonlinear analysis to understand the competition between drifting and breathing at is beyond the scope of this article.
In this article we have formulated an interface dynamics for planar neural fields with a Heaviside firing rate. This has allowed us to (i) develop an economical computational framework for the evolution of spatiotemporal patterns, and (ii) perform linear stability analyses of localized structures. For simplicity we have focused on single population models. However, the extension to population models that treat the dynamics of both excitatory and inhibitory populations is straightforward. Perhaps a more interesting extension is to consider neural field models that incorporate feature selectivity such as that observed in visual cortex for orientation , spatial frequency  and texture . Denoting this feature label by χ then all of these models are expressed in terms of some non-local integro-differential equation for . We note that the notion of an interface is still well defined and that the level set condition gives a constraint between local geometrical data and features. As an alternative to simulating the neural field models an interface approach (incorporating feature space) may be more useful for understanding how local data can be integrated into global geometrical structures, as advocated in the neurogeometry framework of Petitot  (say for understanding models of contour completion in models of primary visual cortex where the feature space is orientation). The extension of this work to treat sigmoidal firing rates remains an open challenge. However, recent techniques for dealing with a certain class of firing rate functions in one spatial dimension, which includes smooth firing rate functions connecting zero to one, are likely to be useful in this regard . We have included an adaptive current in the standard Amari model here, but it would be informative to develop interface treatments for other forms of modulation, e.g., arising from threshold accommodation  or synaptic depression , as well as the inclusion of axonal delays . These models can readily support spiral wave activity, and it would be interesting to see if an interface description, possibly adapting techniques by Hagberg and Meron , could shed light on their properties. Another possible extension of the work in this article, motivated by our numerical results for scattering spots, is to develop an interface theory of quasi-particle interactions along the lines for reaction-diffusion models described in [38, 39], using ideas developed by Bressloff  and Venkov  for weakly interacting systems in one spatial dimension. All of the above are topics of ongoing research and will be reported upon elsewhere.
Appendix: Numerical schemes
A.1 Fourier technique for neural field evolution
Because of its non-local character, the model described by (1), or its extension (61), is challenging to solve with conventional numerical methods. However, exploiting the convolution structure of (1) allows one to write the Fourier transform of as a product. Here and can be taken either as a Heaviside or a more general sigmoidal form. Introducing a spectral wave-vector k then this product is simply , where functions with arguments k denote two-dimensional spatial Fourier transforms. We may evaluate directly, at every time step, using fast Fourier transforms (FFTs). Note that can be pre-computed, by FFT or here even analytically, so that the procedure iterated over time amounts to computing by FFT, followed by a (complex) multiplication with , and finally an inverse FFT to obtain the result of the integral. We wish to employ a parallel compute cluster for rapid computation over large grids, and hence use the free software package FFTW 3.3 , which includes a parallel MPI-C version. Note that the use of Fourier methods implies that the discretization grid has periodic boundaries, or in other words, the solution is effectively computed on a torus. We use a grid spacing of about 0.03 or better in our computations here.
In order to compute the time evolution, we use DOPRI5 , a well-known implementation of an explicit Dormand-Prince (Runge-Kutta) method of order 5(4) with step size control and dense output of the order 4. A version in C due to J. Colinge is available on the web thanks to E. Hairer. However, in our case we perform parallel computations, so we have adapted this code accordingly using MPI-C. In particular, we now consider the maximum error across all compute nodes and all variables, rather than the mean error over local variables, and communicate the resulting time step adaptation over the cluster to achieve a unified evolution of the entire distributed grid. Numerical tolerances are set to where represents all variables, i.e., u and potentially a at all grid points.
This numerical method is robust against effects of the underlying grid. This is due to the employed Fourier method, which performs the spatial convolution as a multiplication in Fourier space. The discrete Fourier transform used to transfer this calculation to Fourier space calculates a trigonometric interpolation polynomial, and the influence of the grid is effectively smoothed by implicit interpolation.
Computing an evolution as shown in Additional File 1 takes several hours on the 32 to 64 Infiniband-connected compute nodes we have typically employed, and yields many gigabytes of data. We note that computation with a sigmoidal firing rate instead of the Heaviside one is over an order of magnitude faster, reflecting the numerical difficulty of dealing with sharp edges.
A.2 Interface dynamics
Equations (22) and (28) can be used to develop a numerical scheme. The contour is discretized into a set of points, and the normal vectors and the displacement vectors are found by computing the orientation and distance between points. Hence the computation of the contour integrals in (28) is straight-forward and yields the normal velocity, cf. (22), which is used to displace the points of the contour in the normal direction at every time step. We employed a simple Euler method to calculate the dynamics of the contour. As the contour grows/shrinks, additional points have to be created/eliminated along the contour.
This method does not provide any means to deal with the splitting or emergence of contours. It is faster than the Fourier technique (see Section A.1 in the Appendix) for small contours, yet the time to compute the normal velocity is proportional to (N being the number of points discretizing the contour), as opposed to for the Fourier technique (where M is the number of grid points). Hence it becomes slower for larger contours due to the absence of suitable spectral methods to compute the line integrals. The main advantage of this method is the fact that no underlying grid has to be deployed across the specified domain.
Electronic Supplementary Material
SC, HS and IB contributed equally. All authors read and approved the final manuscript.
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