From invasion to extinction in heterogeneous neural fields
© Bressloff; licensee Springer. 2012
Received: 15 December 2011
Accepted: 26 March 2012
Published: 26 March 2012
In this paper, we analyze the invasion and extinction of activity in heterogeneous neural fields. We first consider the effects of spatial heterogeneities on the propagation of an invasive activity front. In contrast to previous studies of front propagation in neural media, we assume that the front propagates into an unstable rather than a metastable zero-activity state. For sufficiently localized initial conditions, the asymptotic velocity of the resulting pulled front is given by the linear spreading velocity, which is determined by linearizing about the unstable state within the leading edge of the front. One of the characteristic features of these so-called pulled fronts is their sensitivity to perturbations inside the leading edge. This means that standard perturbation methods for studying the effects of spatial heterogeneities or external noise fluctuations break down. We show how to extend a partial differential equation method for analyzing pulled fronts in slowly modulated environments to the case of neural fields with slowly modulated synaptic weights. The basic idea is to rescale space and time so that the front becomes a sharp interface whose location can be determined by solving a corresponding local Hamilton-Jacobi equation. We use steepest descents to derive the Hamilton-Jacobi equation from the original nonlocal neural field equation. In the case of weak synaptic heterogenities, we then use perturbation theory to solve the corresponding Hamilton equations and thus determine the time-dependent wave speed. In the second part of the paper, we investigate how time-dependent heterogenities in the form of extrinsic multiplicative noise can induce rare noise-driven transitions to the zero-activity state, which now acts as an absorbing state signaling the extinction of all activity. In this case, the most probable path to extinction can be obtained by solving the classical equations of motion that dominate a path integral representation of the stochastic neural field in the weak noise limit. These equations take the form of nonlocal Hamilton equations in an infinite-dimensional phase space.
Keywordsneural fields invasive pulled fronts heterogeneous media Hamilton-Jacobi equation path integrals multiplicative noise large fluctuations population extinction.
Reaction-diffusion equations based on the Fisher-Kolmogorov-Petrovskii-Piskunov (F-KPP) model and its generalizations have been used extensively to describe the spatial spread of invading species including plants, insects, diseases, and genes in terms of propagating fronts [1–7]. One fundamental result in the theory of deterministic fronts is the difference between fronts propagating into a linearly unstable (zero) state and those propagating into a metastable state (a state that is linearly stable but nonlinearly unstable). In the latter case, the front has a unique velocity that is obtained by solving the associated partial differential equation (PDE) in traveling wave coordinates. The former, on the other hand, supports a continuum of possible velocities and associated traveling wave solutions; the particular velocity selected depends on the initial conditions. Fronts propagating into unstable states can be further partitioned into two broad categories: the so-called pulled and pushed fronts  emerging from sufficiently localized initial conditions. Pulled fronts propagate into an unstable state such that the asymptotic velocity is given by the linear spreading speed υ*, which is determined by linearizing about the unstable state within the leading edge of the front. That is, perturbations around the unstable state within the leading edge grow and spread with speed υ*, thus 'pulling along' the rest of the front. On the other hand, pushed fronts propagate into an unstable state with a speed greater than υ*, and it is the nonlinear growth within the region behind the leading edge that pushes the front speeds to higher values. One of the characteristic features of pulled fronts is their sensitivity to perturbations in the leading edge of the wave . This means that standard perturbation methods for studying the effects of spatial heterogeneities  or external noise fluctuations  break down.
Nevertheless, a number of analytical and numerical methods have been developed to study propagating invasive fronts in heterogeneous media. Heterogeneity is often incorporated by assuming that the diffusion coefficient and the growth rate of a population are periodically varying functions of space. One of the simplest examples of a single population model in a periodic environment was proposed by Shigesada et al. [5, 12], in which two different homogeneous patches are arranged alternately in one-dimensional space so that the diffusion coefficient and the growth rate are given by periodic step functions. The authors showed how an invading population starting from a localized perturbation evolves to a traveling periodic wave in the form of a pulsating front. By linearizing around the leading edge of the wave, they also showed how the minimal wave speed of the pulsating front could be estimated by finding solutions of a corresponding Hill equation . The theory of pulsating fronts has also been developed in a more general and rigorous setting [13–15]. An alternative method for analyzing fronts in heterogeneous media, which is applicable to slowly modulated environments, was originally developed by Freidlin [16–18] using large deviation theory and subsequently reformulated in terms of PDEs by Evans and Sougandis . More recently, it has been used to study waves in heterogeneous media (see for example [10, 13]). The basic idea is to rescale space and time so that the front becomes a sharp interface whose location can be determined by solving a corresponding Hamilton-Jacobi equation.
Another important topic in population biology is estimating the time to extinction of a population in the presence of weak intrinsic or extrinsic noise sources, after having successfully invaded a given spatial domain . The zero state (which is unstable in the deterministic limit) now acts as an absorbing state, which can be reached via noise-induced transitions from the nontrivial metastable steady state. The most probable path to extinction is determined in terms of classical solutions of an effective Hamiltonian dynamical system [21–24]. The latter can be obtained in the weak noise limit by considering a Wentzel-Kramers-Brillouin (WKB) approximation of solutions to a master equation (intrinsic noise) or Fokker-Planck equation (extrinsic noise) [25–27]; alternatively, the Hamilton equations can be obtained from a corresponding path integral representation of the stochastic population model .
In this paper, we extend the Hamiltonian-based approaches to invasion and extinction in reaction-diffusion models to the case of a scalar neural field model. Neural fields represent the large-scale dynamics of spatially structured networks of neurons in terms of nonlinear integrodifferential equations, whose associated kernels represent the spatial distribution of neuronal synaptic connections. Such models provide an important example of spatially extended dynamical systems with nonlocal interactions. As in the case of reaction diffusion systems, neural fields can exhibit a rich repertoire of wave phenomena, including solitary traveling fronts, pulses, and spiral waves [28–30]. They have been used to model wave propagation in cortical slices [31, 32] and in vivo . A common in vitro experimental method for studying wave propagation is to remove a slice of brain tissue and bathe it in a pharmacological medium that blocks the effects of inhibition. Synchronized discharges can then be evoked by a weak electrical stimulus to a local site on the slice, and each discharge propagates away from the stimulus at a characteristic speed of about 60 to 90 mm/s [32, 34]. These waves typically take the form of traveling pulses with the decay of activity at the trailing edge resulting from some form of local adaptation or refractory process. On the other hand, a number of phenomena in visual perception involve the propagation of a traveling front, in which a suppressed visual percept replaces a dominant percept within the visual field of an observer. A classical example is the wave-like propagation of perceptual dominance during binocular rivalry [35–38].
In the case of a scalar neural field equation with purely excitatory connections and a sigmoidal or Heaviside firing rate function, it can be proven that there exists a traveling front solution with a unique speed that depends on the firing threshold and the range/strength of synaptic weights [39, 40]. The wave, thus, has characteristics typical of a front propagating into a metastable state. Various generalizations of such front solutions have also been developed in order to take into account the effects of network in-homogeneities [41–43], external stimuli [44, 45], and network competition in a model of binocular rivalry waves . As far as we are aware, however, there has been very little work on neural fields supporting pulled fronts, except for an analysis of pulsating pulled fronts in . One possible motivation for considering such fronts is that they arise naturally in the deterministic limit of stochastic neural fields with a zero absorbing state. Indeed, Buice and Cowan  have previously used path integral methods and renormalization group theory to establish that a stochastic neural field with an absorbing state belongs to the universality class of directed percolation models and consequently exhibits power law behavior suggestive of several measurements of spontaneous cortical activity in vitro and in vivo [47, 48].
We begin by considering a neural field model that supports propagating pulled fronts, and determine the asymptotic wave speed by linearizing about the unstable state within the leading edge of the front (Section 2). We then introduce a spatial heterogeneity in the form of a slow modulation in the strength of synaptic connections (Section 3). Using a WKB approximation of the solution of a rescaled version of the neural field equation and carrying out steepest descents, we derive a local Hamilton-Jacobi equation for the dynamics of the sharp interface (in rescaled space-time coordinates). We then use perturbation methods to solve the associated Hamilton equations under the assumption that the amplitude of the spatial modulations is sufficiently small (Section 4). The resulting solution determines the location of the front as a function of time, from which the instantaneous speed of the front can be calculated. In the case of linearly varying modulations, the position of the front is approximately a quadratic function of time. In the second part of the paper, we investigate how time-dependent heterogeneities in the form of extrinsic multiplicative noise can induce rare noise-driven transitions to the zero-activity state, which now acts as an absorbing state signaling the extinction of all activity (on a shorter time scale, noise results in a subdiffusive wandering of the front ). We proceed by first constructing a path integral representation of the stochastic neural field (Section 5). The most probable path to extinction can then be obtained by solving the classical equations of motion that dominate the path integral representation in the weak noise limit; these equations take the form of nonlocal Hamilton equations in an infinite-dimensional phase space (Section 6).
2 Neural fields with propagating pulled fronts
One major difference from a mathematical perspective is that Equation 2.2 supports traveling fronts even when the range of the interaction kernel K goes to zero, that is, K(x) → δ(x), since we recover the standard local F-KPP equation [1, 2]. In particular, as the nonlocal interactions appear nonlinearly in Equation 2.2, they do not contribute to the linear spreading velocity in the leading edge of the front. On the other hand, nonlocal interactions play a necessary role in the generation of fronts in the neural field equation (Equation 2.1).
Assuming that the full nonlinear system supports a pulled front, then a sufficiently localized initial perturbation (one that decays faster than ) will asymptotically approach the traveling front solution with the minimum wave speed c0 = c(λ0). Note that c0 ~ σ and λ0 ~ σ-1. In Figure 2b, we show an asymptotic front profile obtained by numerically solving the neural field equation (Equation 2.1) when W0 = 1.2 (see Section 4.3). The corresponding displacement of the front is a linear function of time with a slope consistent with the minimal wave speed c0 ≈ 0.7 of the corresponding dispersion curve shown in Figure 2a. This wave speed is independent of κ.
In the remainder of the paper, we consider two distinct problems associated with the presence of an unstable zero-activity state in the neural field model (Equation 2.1). First, how do slowly varying spatial heterogeneities in the synaptic weights affect the speed of propagating pulled fronts (Sections 3 and 4)? Second, what is the estimated time for extinction of a stable steady state in the presence of multiplicative extrinsic noise (Sections 5 and 6)? Both problems are linked from a methodological perspective since the corresponding analysis reduces to solving an effective Hamiltonian dynamical system.
3 Hamilton-Jacobi dynamics and slow spatial heterogeneities
where 2πε is the period of the modulation with K(x) = K(x + 2π) for all x. In the case of a sigmoidal or Heaviside nonlinearity, two alternative methods for analyzing the effects of periodic wave modulation have been used: one is based on homogenization theory for small ε [41, 51], and the other is based on analyzing interfacial dynamics [42, 43]. Both approaches make use of the observation that for sufficiently small amplitude modulations, numerical simulations of the inhomogeneous network show a front-like wave separating high and low activity metastable states. However, the wave does not propagate with constant speed but oscillates periodically in an appropriately chosen moving frame. This pulsating front solution satisfies the periodicity condition a(x, t) = a(x + Δ, t + T), so we can define the mean speed of the wave to be c = Δ/T.
in which there is a slow (nonperiodic) spatial modulation J(εx) of the synaptic weight distribution with ε ≪ 1. The synaptic heterogeneity is assumed to occur on a longer spatial scale than the periodic-like microstructures associated with stimulus feature maps. Although we do not have a specific example of long wavelength modulations in mind, we conjecture that these might be associated with inter-area cortical connections. For example, it has been shown that heterogeneities arise as one approaches the V1/V2 border in the visual cortex, which has a number of effects including the generation of reflected waves . It is not yet clear how sharp is the transition across the V1/V2 border.
We have used the fact that for x > x(t) and ε ≪ 1, the solution is in the leading edge of the front, so we can take F to be linear.
which is independent of s due to conservation of 'energy,' that is, the Hamiltonian is not an explicit function of time.
4 Calculation of wave speed
4.1 Homogeneous case: J(x) ≡ 1
and c0 = c(λ0). This recovers the result of Section 2. Thus, in the case of a Gaussian weight distribution, λ0 is related to c0 according to Equation 2.15.
4.2 Synaptic heterogeneity: J(x) = 1 + β f(x), 0 β≪ 1
4.3 Numerical results
where we have used Equation 4.3. We assume that the initial position of the front is x(0) = l. Hence, our perturbation theory predicts that a linearly increasing modulation in synaptic weights results in the leading edge of the front tracing out a downward parabola in a space-time plot for times . This is consistent with our numerical simulations for ε2 = 0.005, as can be seen in the space-time plot of Figure 6b. However, our approximation for the time-dependent speed breaks down when , as illustrated in Figure 7c for ε2 = 0.01.
5 Path integral formulation of a stochastic neural field
Here, λ is the spatial correlation length of the noise such that C(x/λ) → δ(x) in the limit λ → 0, and ε determines the strength of the noise, which is assumed to be weak. We will assume that g(0) = 0, so the zero-activity state A = 0 is an absorbing state of the system; any noise-induced transition to this state would then result in the extinction of all activity. An example of multiplicative noise that vanishes at A = 0 is obtained by carrying out a diffusion approximation of the neural master equation previously introduced by Buice et al. [46, 54] (see Bressloff [55, 56]). Before considering the problem of extinction, a more immediate question is how multiplicative noise affects the propagation of the invasive pulled fronts analyzed in Section 2. Since this particular issue is addressed in some detail elsewhere , we only summarize the main findings here.
6 Hamiltonian-Jacobi dynamics and population extinction in the weak-noise limit
We now use the path integral representation of the conditional probability (Equation 5.13) in order to estimate the time to extinction of a metastable nontrivial state. We proceed along analogous lines to our previous study of a neural master equation with x-independent steady states [55, 56], where the effective Hamiltonian dynamical system obtained by extremizing the associated path integral action can be used to determine the most probable or optimal path to the zero absorbing state. Alternatively, one could consider a WKB approximation of solutions to the corresponding functional Fokker-Planck equation or master equation, as previously applied to reaction-diffusion systems [21, 24–27]. The connection between the two approaches is discussed in .
It can be shown that q(x, t) and p(x, t) satisfy the same boundary conditions as the physical neural field A(x, t) . Thus, in the case of periodic boundary conditions, q(x+L, t) = q(x, t) and p(x+L, t) = p(x, t). It also follows from the Hamiltonian structure of Equations 6.5 and 6.6 that there is an integral of motion given by the conserved 'energy' E = ℋ[q, p].
Note that since the extinction time is exponentially large in the weak noise limit, it is very sensitive to the precise form of the action S0 and thus the Hamiltonian ℋ. This implies that when approximatingthe neural master equation of Buice et al. [46, 54] by a Langevin equation in the form of Equation 5.1 with , where N is the system size, the resulting Hamiltonian differs from that obtained directly from the master equation and can thus generate a poor estimate of the extinction time. This can be shown either by comparing the path integral representations of the generating functional for both stochastic processes or by comparing the WKB approximation of the master equation and corresponding Fokker-Planck equation. This particular issue is discussed elsewhere for neural field equations [55, 56].
In this paper, we have explored some of the consequences of having an unstable zero-activity state in a scalar neural field model. First, we considered invasive activity fronts propagating into the unstable zero state. These waves exhibit a behavior analogous to pulled fronts in reaction-diffusion systems, in which the wave speed is determined by the spreading velocity within the leading edge of the front. Consequently, front propagation is sensitive to perturbations in the leading edge, which we investigated within the context of spatial heterogeneities in the synaptic weights. We showed how time-dependent corrections to the wave speed could be estimated using a Hamilton-Jacobi theory of sharp interface dynamics combined with perturbation theory. Second, we considered a stochastic version of the neural field model, in which the zero-activity fixed point acts as an absorbing state. By constructing an integral representation of the neural Langevin equation, we showed how to estimate the time to extinction from a nontrivial steady state using a Hamiltonian formulation of large fluctuation paths in the weak noise limit.
It is clear from the results of this paper that neural fields with an unstable zero-activity state exhibit considerably different behaviors compared to models in which the zero state (or down state) is stable. Of course, one important question is whether or not real cortical networks ever exist in a regime where the down state is unstable. From a mathematical viewpoint, one has to choose a very specific form of the firing rate function F for such a state to occur (see Figure 1). Nevertheless, Buice and Cowan  have demonstrated that a stochastic neural field operating in a regime with an absorbing state belongs to the universality class of directed percolation models and consequently exhibits power law behavior suggestive of several measurements of spontaneous cortical activity in vitro and in vivo [47, 48]. On the other hand, the existence of power law behavior is still controversial . Irrespective of these issues, exploring the connections between nonlocal neural field equations and reaction-diffusion PDES is likely to be of continued mathematical interest.
This work was supported by the National Science Foundation (DMS 1120327).
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