For the moment, suppose that there are no external inputs so that h(x, t) = 0 in Equation (1). A traveling pulse of velocity c is then defined according to u(x, t) = U(ξ), with ξ = x − ct a traveling wave coordinate such that lim_ξ→±∞ U(ξ) = 0. Moreover, the wave profile is restricted to be super threshold in a connected interval of width d. Since the neural field is equivariant with respect to uniform translations (in the absence of external stimuli), we choose the two threshold crossing points to be (7)U(0)=κ, U(d)=κ. Thus, U(ξ) > κ for 0 < ξ < d, U(ξ) < κ for ξ < 0, and ξ > d. It turns out the wave travels in the same direction as the offset so we restrict ourselves to the case x₀ > 0 and c > 0. Substituting the traveling pulse solution into Equation (1) gives (8)−c∂U(ξ)∂ξ=−U(ξ)+∫0dw(ξ−ξ′)dξ′ Multiplying both sides by e^−ξ/c and integrating gives the following equation for the wave solution: (9)U(ξ)=eξ/cc∫ξ∞​W(ξ′)e−ξ′/cdξ′, where W(ξ)≡∫ξ−dξ​w(x)dx. It is convenient to express the weight function in piecewise form as follows: (10)w(x)={aee−σe(x−x0)−aie−σi(x−x0)≡w1(x),if x≥x0aeeσe(x−x0)−aieσi(x−x0)≡w2(x),if x≤x0 We then obtain a piecewise expression for W(ξ) of the form (11)W(ξ)={W3(ξ)≡∫ξ−dξ​w2(x)dx,if ξ≤x0W2(ξ)≡∫ξ−dx0​w2(x)dx             +∫x0ξ​w1(x)dx,if x0≤ξ≤x0+dW1(ξ)≡∫ξ−dξ​w1(x)dx,if ξ≥x0+d We then have U(ξ)={1ceξ/c(M3(ξ)+M1(x0+d)                    +M2(x0)),if ξ≤x01ceξ/c(M2(ξ)+M1(x0+d)),if x0≤ξ≤x0+d1ceξ/cM1(ξ),if ξ≥x0+d. where (12)Mn(ξ)=∫ξξnWn(ξ′)e−ξ′/cdξ′ with ξ₁ = ∞, ξ₂ = x₀ + d and ξ₃ = x₀.

Having obtained the piecewise wave profile U(ξ), the threshold conditions (Equation 7) can now be used to determine the pulse speed c and width d; the resulting transcendental equations have to be solved numerically. Figure 2 shows solutions for the pulse speed and width as functions of the threshold. It turns out that the solution with slower speed (and larger width) is stable (see below). This differs from traveling pulse solutions found in adaptive neural fields, where the faster wave (with larger width) tends to be stable (Pinto and Ermentrout, 2001; Kilpatrick and Bressloff, 2010a). Figure 3A shows a typical pulse waveform and Figure 3B shows a numerical simulation of the neural field Equation (1) using the wave solution as the initial condition. The pulse propagates at the predicted speed without changing shape significantly. This occurs because the parameters were chosen to make the pulse solution linearly stable.

In order to determine the linear stability of a traveling pulse solution U(ξ) in the moving frame, we linearize Equation (1) with h(x, t) = 0 by setting U(ξ, t)=U(ξ)+φ(ξ, t), and Taylor expanding to first order in ϕ. This gives (13)∂φ(ξ, t)∂t=c∂φ(ξ, t)∂ξ−φ(ξ, t)+∫−∞∞​w(ξ−y)×F′(U(y))φ(y, t)dy. In the case of the Heaviside rate function (Equation 3), we have (14)F′(U(ξ))=δ(ξ)|U′(0)|+δ(ξ−d)|U′(d)|. Moreover, differentiating Equation (9) with respect to ξ shows that U′(ξ)=1c(U(ξ)−W(ξ)). Substituting the previous two results into Equation (13) gives (15)∂φ(ξ, t)∂t=ℒφ(ξ, t)               ≡c∂φ(ξ, t)∂ξ−φ(ξ, t)+cφ(0, t)|κ−W(0)|w(ξ)+cφ(d, t)|κ−W(d)|w(ξ−d). where κ is the threshold. Looking for solutions of the form (16)φ(ξ, t)=eλtφ(ξ). then leads to the spectral problem (17)ℒφ(ξ, t)=λφ(ξ, t).

We take the linear operator ℒ to act on a Banach space ℬ of continuous, bounded functions ψ(ξ) that are defined for ξ ∈ ℝ, and that decay exponentially as ξ→±∞. Let σ(ℒ) denote the spectrum of the linear operator ℒ, and define the associated resolvent operator according to ℛ_λ ≡ (ℒ − λ I)⁻¹, where I is the identity operator. The spectrum can be defined as those values of λ for which Tλ≡ℒ−λI is not bijective. The spectrum is composed of three disjoint sets, the point or discrete spectrum, the residual spectrum, and the continuous spectrum. The point spectrum is defined as the values of λ (eigenvalues) for which the resolvent does not exist. The residual spectrum are the spectral values for which the resolvent exists but is not defined on a dense subset of ℬ. The continuous spectrum are the spectral values for which the resolvent exists and is densely defined but is unbounded (Kreyszig, 1978). Given these definitions, the traveling pulse is said to be linearly stable if (1) Re(λ) < 0 for all λ ın σ(ℒ), λ ≠ 0 and (2) the zero eigenvalue is simple. The existence of a zero eigenvalue with corresponding eigenfunction ϕ(ξ, t) = U′(ξ) reflects translation invariance, and immediately follows from differentiating (Equation 9) with respect to ξ.

We first consider the discrete spectrum by solving the eigenvalue equation dψ(ξ)dξ−λ+1cψ(ξ)+K0ψ(0)w(ξ)+Kdψ(d)w(ξ−d)=0, where we have introduced the constants K0=1|κ−W(0)| and Kd=1|κ−W(d)|. Multiplying both sides by e^{−(λ+1)ξ/c} and integrating gives ψ(ξ)=K0ψ(0)∫ξ∞​w(y)e(λ+1)(ξ−y)/cdy+Kdψ(d)∫ξ−d∞​w(y)e(λ+1)(ξ−d−y)/cdy, which can be rewritten in the more compact form (18)ψ(ξ)=K0ψ(0)(w×Pλ)(ξ)+Kdψ(d)(w×Pλ)(ξ−d) with (19)(w×Pλ)(ξ)=∫−∞∞​w(y)Pλ(ξ−y)dy, Pλ(ξ)=H(−ξ)e(λ+1)ξ/c The eigenvalues are now determined by imposing self-consistency at ξ = 0 and ξ = d. Setting ξ = 0 and ξ = d in Equation (18) leads to the vector equation (20)[K0(w×Pλ)(0)−1Kd(w×Pλ)(−d)K0(w×Pλ)(d)Kd(w×Pλ)(0)−1][ψ(0)ψ(d)]=0 This has a non-trivial solution if and only if the determinant of the matrix is zero. The determinant expressed as a function of λ, ℰ(λ), is a complex analytic function known as the Evans function: (21)ℰ(λ)=[K0(w×Pλ)(0)−1][Kd(w×Pλ)(0)−1]−K0Kd(w×Pλ)(d)(w×Pλ)(−d). Thus, the zeros of the Evans function determine the discrete spectrum of the linear operator formed by linearizing the neural field equation about the pulse solution. Evans functions were originally introduced within the context of the stability of solitary pulses in diffusive Hodgkin–Huxley type equations describing action potential propagation in nerve axons (Evans, 1975). Since then the Evans function construction has been extended to a wide range of PDEs, see the review (Sandstede, 2002). It has also recently been applied to neural field equations (Zhang, 2003; Coombes and Owen, 2004; Rubin, 2004; Folias and Bressloff, 2005; Pinto et al., 2005; Sandstede, 2007) and more general non-local problems (Kapitula et al., 2004). An example plot of the real and imaginary parts of ℰ(λ) = 0 on the complex plane is shown in Figure 4. It can be seen that there is a zero eigenvalue and one negative real eigenvalue, indicating that the corresponding traveling pulse is linearly stable.

To find the essential spectrum, which is the union of the residual and continuous spectra, we will derive an explicit expression for the resolvent ℛ_λ. We start by writing an inhomogeneous equation of the form Tλψ(ξ)=h(ξ), where h(ξ) represents a general function from the Banach space ℬ. This can be manipulated as before to give (22)ψ(ξ)=−(h×Pλ)(ξ)+K0ψ(0)(w×Pλ)(ξ)+Kdψ(d)(w×Pλ)(ξ−d) By evaluating at ξ = 0 and ξ = d as before, we arrive at the following vector equation, which differs from (Equation 20) only on the right-hand side: [K0(w×Pλ)(0)−1Kd(w×Pλ)(−d)K0(w×Pλ)(d)Kd(w×Pλ)(0)−1][ψ(0)ψ(d)]                     =[(h×Pλ)(0)(h×Pλ)(d)]. Since we are looking for spectral values outside the discrete spectrum, the determinant of the matrix satisfies ℰ(λ)≠ 0. Therefore, multiplying both sides by the inverse matrix yields expressions for ψ(0) and ψ(d) in terms of h: ψ(0)=S0hℰ(λ), ψ(d)=Sdhℰ(λ) where S0h=(Kd(w×Pλ)(0)−1)(h×Pλ)(0)−Kd(w×Pλ)(−d)(h×Pλ)(d) and Sdh=−K0(w×Pλ)(d)(h×Pλ)(0)+(K0(w×Pλ)(0)−1)(h×Pλ)(d). Substituting into Equation (22) gives the following expression for the resolvent operator, of the form ℛ_λh = ϕ: (23)−(h×Pλ)(ξ)+K0S0hℰ(λ)(w×Pλ)(ξ)+KdSdhℰ(λ)(w×Pλ)(ξ−d)=ψ(ξ) The resolvent is well-defined for all h in ℬ, so the residual spectrum of ℒ is empty. To find the continuous spectrum, we Fourier transform Equation (23): (24)−h^(k)P^λ(k)+K0S0hε(λ)w^(k)P^λ(k)+KdSdhε(λ)w^(k)P^λ(k)e−2πidk=ψ^(k) It follows that the resolvent operator is unbounded when P^λ is unbounded. Equation (19) implies that (25)P^λ(k)=∫−∞∞​Pλ(ξ)e2πikξ dξ=1λ+1c+2πik. Hence, P^λ is unbounded for λ = −1 − 2πikc so that the continuous spectrum of ℒ is a vertical line in the complex plane at Re(λ) = −1. Since Re(λ) < 0, the continuous spectrum will not make any pulse solution of our model unstable.

Now suppose that the neural field is driven by a moving external pulse stimulus of speed v so that Equation (1) becomes (26)∂u(x, t)∂t=−u(x, t)+∫−∞∞​w(x−y)H(u(y, t)−κ)dy+ I(x−vt). In order to study the existence of stimulus-locked pulses, we will define a “stimulus coordinate” ξ = x − vt and look for pulse solutions that move at the same speed as the stimulus, that is, u(x, t) = U(ξ) with (27)−v∂U(ξ)∂ξ=−U(ξ)+∫−∞∞​w(ξ−y)H(U(y)−κ)dy+I(ξ). For concreteness, the stimulus will be represented by a rectangular wave of amplitude I₀ and width d, defined formally as I(ξ)={I0,if 0≤ξ≤d0,if ξ<0 or ξ>d. Since translation invariance no longer holds, it is necessary to determine both threshold crossing points, which we denote by ξ = d₁ and ξ = d₂. Proceeding in a similar fashion to the case of freely propagating pulses, we find that U(ξ)=eξ/vv∫ξz0​e−y/vW(y) dy+eξ/vv∫ξz0​e−y/vI(y) dy, where z₀ = ∞ if v > 0, z₀ = −∞ if v < 0, and W(ξ)≡∫ξ−d2ξ−d1​w(x) dx. The latter can be expressed in the piecewise form (28)W(ξ)={W3(ξ)≡∫ξ−d2ξ−d1​w2(x) dx,if ξ≤x0+d1W2(ξ)≡∫ξ−d2x0​w2(x) dx+∫x0ξ−d1​w1(x) dx,if x0+d1≤ξ≤x0+d2W1(ξ)=∫ξ−d2ξ−d1​w1(x) dx,if ξ≥x0+d2 where w₁ and w₂ are defined as in Equation (10). After evaluating the integrals along similar lines to section “Neural Field Model of Direction Selectivity,” we obtain the following expressions for the pulse solution, defined independently for positive and negative stimulus directions:

v > 0:: U(ξ)={U3(ξ),if ξ≤x0+d1U2(ξ),if x0+d1≤ξ≤x0+d2,U1(ξ),if ξ≥x0+d2, with U3(ξ)=1veξ/v(M3(ξ)+M1(x0+d2)+M2(x0+d1))+Z(ξ)U2(ξ)=1veξ/v(M2(ξ)+M1(x0+d2))+Z(ξ)U1(ξ)=1veξ/vM1(ξ)+Z(ξ),Mn(ξ)=∫ξξnWn(ξ′)e−ξ′/vdξ′ for ξ₁ = ∞, ξ₂ = x₀ + d₂, ξ₃ = x₀ + d₁, and Z(ξ)={(eξ/v−e(ξ−d)/v)I0,if ξ<0(1−e(ξ−d)/v)I0,if 0≤ξ≤d.0,if ξ>d v < 0: U(ξ)={U3(ξ),if ξ≤x0+d1U2(ξ),if x0+d1≤ξ≤x0+d2,U1(ξ),if ξ≥x0+d2 with U3(ξ)=−1veξ/vN3(ξ)+Z(ξ)U2(ξ)=−1veξ/v(N2(ξ)+N3(x0+d1))+Z(ξ)U1(ξ)=−1veξ/v(N1(ξ)+N2(x0+d1)+N3(x0+d1))+Z(ξ)Nn(ξ)=∫ξnξWn(ξ′)e−ξ′/vdξ′ for ξ₃ = −∞, ξ₂ = x₀ + d₁, ξ₁ = x₀ + d₂, and Z(ξ)={0,if ξ<0(1−eξ/v)I0,if 0≤ξ≤d.(e(ξ−d)/v−eξ/v)I0,if ξ>d

The threshold crossing points (d₁ and d₂) are determined in the same way the pulse speed and width were determined in the no-stimulus case, which is by numerically solving a system of two transcendental equations. The first equation is given by U₃(d₁) = κ. The second equation is U₃(d₂) = κ if d₂ < x₀ + d₁, else it is given by U₂(d₂) = κ. Figure 5 shows a plot of d₁ (black curves) and d₂ (gray curves) vs. the threshold κ. It can be seen that for a certain range of thresholds there exists more than one stable/unstable pair of pulses. Figure 6 shows the linear stability and the number of solutions for different combinations of stimulus speed (v) and strength (I₀). The offset x₀ = 3 and the corresponding spontaneous wave speed is c = 4. (Note that for smaller offsets x₀ and thus smaller wave speeds c, one finds stimulus-locked waves for negative values of v). The stability of solutions in the presence of a stimulus is determined in much the same way as without a stimulus. We again define u(x, t) = U(ξ) + ϕ(ξ,t) and look at the behavior of the perturbations described by ϕ(ξ,t). Substituting into Equation (26), the stimulus term drops out when we perform the linearization, so that ∂φ(ξ, t)∂t=v∂φ(ξ, t)∂ξ−φ(ξ, t)+∫−∞∞​w(ξ−y)× F′(U(y))φ(y, t) dy.

Setting ϕ(ξ,t) = e^λtϕ ultimately yields the spectral problem (29)λφ(ξ)≡ℒφ(ξ)          =v∂φ(ξ, t)∂ξ−φ(ξ, t)          +|v|φ(d1)|κ−W(d1)−I(d1)|w(ξ−d1)          +|v|φ(d2)|κ−W(d2)−I(d2)|w(ξ−d2).

The corresponding Evans function is now (30)ℰ(λ)=[K1(w × Pλ)(0)−1][K2(w × Pλ)(0)−1]−K1K2(w × Pλ)(d2−d1)(w × Pλ)(d1−d2), where Kn=sgn(v)|κ−W(dn)−I(dn)|, n=1, 2, and Pλ(ξ)=H(−sgn(v)ξ)e(λ+1)ξ/v. It is easy to establish as before that the residual spectrum is empty and that the continuous spectrum consists of a vertical line in the complex plane at Re(λ) = −1. So the stability is again determined only by the discrete spectrum, which consists of the zeros of the Evans function. Figure 7A shows an example of a numerical simulation of a stable stimulus-locked pulse solution of Equation (26) with the analytical pulse solution U(x) as an initial condition. Figure 7B shows the same simulation except with the zero initial condition u(x,0) = 0. It can be seen from Figure 7C that both initial conditions converge to the same pulse profile.

The effects of additive or multiplicative extrinsic noise on traveling waves can be analyzed using multiple time-scale methods originally developed for reaction-diffusion equations (Schimansky-Geier et al., 1983; de Pasquale et al., 1992; Armero et al., 1998; Sagues et al., 2007), which were recently extended to neural field equations in Ref. Bressloff and Webber (2012b). The main idea is to assume that the fluctuating term generates two distinct phenomena that occur on different time-scales: a diffusive-like displacement of the traveling wave from its uniformly translating position at long time-scales, and fluctuations in the wave profile around its instantaneous position at short time-scales. It is important to point out that, in contrast to traveling front solutions of scalar neural field equations (Bressloff and Webber, 2012b), we are now considering traveling pulse solutions. Thus in addition to the center-of-mass of the traveling pulse wave, which moves with speed c in the absence of noise, there is an additional degree of freedom corresponding to the “width” of the pulse. (In the case of a Heaviside rate function, the width Δ is determined by the threshold crossing points). For simplicity, we assume that the width of the wave is only weakly affected by the noise; this is consistent with what is found numerically. We now express the solution U of Equation (33) as a combination of a fixed wave profile U₀ that is displaced by an amount Δ(t) from its uniformly translating position ξ = x − c_ϵt, where c_ϵ is a noise-dependent speed, and a time-dependent fluctuation Φ in the wave shape about its instantaneous position: (37)U(x, t)=U0(ξ−Δ(t))+ϵ1/2Φ(ξ−Δ(t), t). The wave profile U₀ and associated wave speed/width c_ϵ,Δ_ϵ are obtained by solving the modified deterministic equation (38)−cϵdU0(ξ)dξ−h(U0(ξ))=∫∞∞​w(ξ−ξ′)F(U0(ξ′))dξ′. The results depend on ϵ due to the ϵ-dependence of h. Equation (38) is chosen so that that to leading order, the stochastic variable Δ(t) undergoes unbiased Brownian motion with a diffusion coefficient D(∈)=O(∈) (see below). The next step is to substitute the decomposition Equation (37) into (33) and expand to first order in O(∈^1/2): −[cε+Δ˙]U0′(ξΔ)dt+ϵ1/2[dΦ(ξΔ, t)−[cϵ+Δ˙]Φ′(ξΔ, t)dt] =h(U0(ξΔ))dt+ϵ1/2h′(U0(ξΔ))Φ(ξΔ, t)dt  +∫−∞∞w(ξ−ξ′)F(U0(ξΔ′))dξ′dt  + ϵ1/2∫−∞∞w(ξ−ξ′)F′(U0(ξΔ′))Φ(ξΔ′, t)dξ′dt  + ϵ1/2dR(U0(ξΔ),ξ, t)+O(ϵ). where we have set ξ_Δ = ξ − Δ(t) and ξ_Δ' = ξ' − Δ(t). We now use Equation (38) for U₀, after shifting ξ→ ξ − Δ(t), to eliminate terms and then divide through by ϵ. This gives the inhomogeneous equation to O(ϵ^1/2) (39)dΦ(ξΔ, t)−L^Φ(ξΔ, t)dt=ϵ−12U′0(ξΔ)dΔ(t)+ dR(U0(ξΔ),ξ, t) where the non-self-adjoint linear operator (40)L^A(ξ)≡cϵdA(ξ)dξ+h′(U0(ξ))A(ξ)+ ∫−∞∞​w(ξ−ξ′)F′(U0(ξ′))A(ξ′)dξ′ is defined for all functions A(ξ) in ℒ₂(ℝ). Note that for all terms in Equation (40) to be of the same order we have taken Δ(t) = O(ε^1/2). It then follows that U₀(ξ − Δ(t)) = U₀(ξ) + O(ε^1/2) etc., and Equation (39) reduces to (41)dΦ(ξ, t)−L^Φ(ξ, t)dt=ϵ−12U′0(ξ)dΔ(t)+dRu(U0(ξ),ξ, t)

If U₀(ξ) were a traveling front solution of a neural field model with a symmetric, excitatory weight distribution w, then it could be proven that the operator L^ has a 1D null space spanned by U₀'(ξ) (Ermentrout and McLeod, 1993). We will assume that such a result carries over to traveling pulse solutions of a neural field with w given by an asymmetric Mexican hat function; the fact that U₀'(ξ) belongs to the null space follows immediately from differentiating Equation (38) with respect to ξ. We then have the solvability condition for the existence of a non-trivial bounded solution of Equation (41), namely, that the inhomogeneous part is orthogonal to all elements of the null space of the adjoint operator L^∗. The latter is defined with respect to the inner product ∫−∞∞​B(ξ)L^A(ξ) dξ=∫−∞∞​[L^∗B(ξ)]A(ξ) dξ. Integrating by parts and using (Equation 14) leads to (42)L^∗B(ξ)=−cϵdB(ξ)dξ+h′(U0(ξ))B(ξ)+ F′(U0(ξ))∫−∞∞w(ξ′−ξ)B(ξ′)dξ′. We will assume that the null space of the adjoint operator L^∗ is also one-dimensional and is spanned by some yet to be determined function V(ξ). (In the case of a Heaviside firing function, we will determine the null space explicitly). Hence, we can write the solvability condition as ∫−∞∞V(ξ)[U0′(ξ)dΔ(t)+ϵ1/2dR(U0(ξ),ξ, t)] dξ=0. which leads directly to the stochastic differential equation dΔ(t)=−ϵ1/2∫−∞∞​V(ξ)dR(U0, ξ, t) dξ∫−∞∞​V(ξ)U0′(ξ)dξ. Using the lowest order approximations dR(U₀, ξ, t) = g(U₀(ξ))dW(ξ, t), we deduce that [for Δ(0) = 0] (43)〈Δ(t)〉=0, 〈Δ(t)2〉=2D(ε)t, where D(ϵ) is the effective diffusivity (44)D(ϵ)=ϵ∫−∞∞​V2(ξ)g(U0(ξ))2 dξ[∫−∞∞​V(ξ)U0′(ξ) dξ]2.

In order to illustrate the above analysis, we consider a particular example where the mean speed c_ϵ and diffusion coefficient D(ϵ) can be calculated explicitly. That is, set g(U) = g₀U for the multiplicative noise term and take F(U) = H(u − κ). (The constant g₀ has units of length/time). Note that the choice for g(U) can be interpreted physiologically in terms of an effective modification in the membrane time constant of neurons due to stochastic background synaptic activity (Bernander et al., 1991; Rapp et al., 1992; Bressloff, 1994). The deterministic Equation (38) for U₀ then reduces to (45)−dU0(ξ)dξ+Γ(ϵ)U0(ξ)=1cϵ∫∞∞​w(ξ−ξ′)H(U0(ξ′)−κ)dξ′, where Γ(ϵ) = (1 − ϵC(0)g²₀)/c_ϵ. Hence, (46)U(ξ)=eΓξcϵ∫ξ∞W(ξ′)e−Γξ′ dξ′. The deterministic pulse profile can be evaluated along identical lines to section “Neural Field Model of Direction Selectivity.” In order to calculate the diffusion coefficient, it is first necessary to determine the null vector V(ξ) of the adjoint linear operator L^∗. Substituting F(U) = H(U − κ) and g(U) = g₀U into Equation (42) shows that (47)dV(ξ)dξ+Γ(ϵ)V(ξ)=δ(ξ)c|U0′(0)|∫−∞∞w(z)V(z)dz+ δ(ξ−Δ)c|U0′(Δ)|∫−∞∞w(z−Δ)V(z)dz. Proceeding along similar lines to Bressloff (2001) and Kilpatrick et al. (2008), we make the ansatz that (48)V(ξ)=AH(ξ)e−Γξ+BH(ξ−Δ)e−Γ(ξ−Δ). Substituting into Equation (47) shows that A = 1|U0′(0)|[Ab(0) + Bb(Δ)], B = 1|U0′(Δ)|[Ab(−Δ) + Bb(0)] where (49)b(z)≡1c∫z∞e−Γ(ξ′−z)w(ξ′) dξ′. Differentiating Equation (46) shows that U'(ξ) = b(ξ)−b(ξ − Δ), so that we obtain the vector equation [b(0)b(0)−b(−Δ)−1b(Δ)b(0)−b(−Δ)b(−Δ)b(0)−b(Δ)b(0)b(0)−b(Δ)−1][AB]=0 The matrix has rank 1, confirming that the linear operator L^∗x has a 1D null-space. The latter is spanned by the function (50)V(ξ)=b(Δ)H(ξ)e−Γξ−b(−Δ)H(ξ−Δ)e−Γ(ξ−Δ). In Figure 8 we show the temporal evolution of a freely propagating stochastic traveling pulse, which is obtained by numerically solving the Langevin Equation (31) for F(U) = H(U − κ), g(U) = U and the asymmetric difference-of-exponentials (Equation 4). Note that the location of the stochastic wave appears to coincide with the underlying mean solution. However, over longer time-scales the wandering of the pulse about its mean position would be seen. In Figure 9 we plot the mean position X¯(t) and variance σ²_X(t) of the leading and trailing edges of the pulse as a function of t. It can be seen that they all vary linearly with t, consistent with the assumption that there is a diffusive-like displacement of the center-of-mass of the pulse from its uniformly translating position at long time-scales. The slopes of these curves then determine the effective wave speed and diffusion coefficient according to X¯(t) ~ c_εt and σ²_X(t)~ 2D(ε)t. Both the leading and trailing edges exhibit the same speeds and diffusivities (after a transient phase). The transients are caused by fluctuations in the mean width of the pulse which can be neglected for large t, where the difference in the size of fluctuations of the leading and trailing edges can be neglected.

In order to find the mean location of the leading or trailing edge of the pulse as a function of time, we numerically carry out a large number of level set position measurements. That is, we determine the positions X_a(t) such that U(X_a(t),t) = a, for various level set values a and then define the mean location to be X¯(t)=E[Xa(t)], where the expectation is first taken with respect to the sampled values a and then averaged over N trials. The corresponding variance is given by σX2(t)=E[(Xa(t)−X¯(t))2]. In order to compare the numerical results with our theoretical analysis, we assume that X_a(t) = Δ(t) + c_ϵt + x_a(0) for each a on either the leading or trailing edge. It then follows that xX¯(t) = c_ϵt + x_a(0) and σ²_X(t) = 〈 Δ(t)²〉. In Figure 10 we plot the numerically estimated diffusion coefficient for various values of the threshold κ and compare these to the corresponding theoretical curves obtained using the above analysis. It can be seen that there is excellent agreement with our theoretical predictions. Finally, note that we can also use the level set data to estimate fluctuations in the width of the pulse. Suppose that X_d(t) and Y_d(t) denote the threshold crossing points of the leading and trailing edges of the pulse at time t. Then the stochastic width of the pulse can be defined according to D(t) = X_d(t) − Y_d(t). We find that after a transient phase, 〈D(t)²〉 − 〈D(t)〉² « σ²_X(t).

We now add a stimulus term I to the stochastic neural field Equation (33), that is (51)dU(x, t)=[h(U(x, t))+∫−∞∞​w(x−y)F(U(y, t)) dy]dt+ I(x−vt)dt+ϵ1/2dR(U, x, t), where the stimulus is again a rectangular wave of amplitude I₀ and width d, moving with speed v. Here h and dR are defined by Equations (34) and (35). The stochastic activity variable is now decomposed according to Equation (37), with ξ = x − vt, and the modified deterministic equation (52)−vdU0dξ−h(U0(ξ))−I(ξ)=∫∞∞​w(ξ−ξ′)F(U0(ξ′)) dξ′. Through a similar process as in the previous section, we expand to O(ϵ^1/2) to obtain the inhomogeneous equation (53)dΦ(ξ, t)−L^Φ(ξ, t)dt=−ϵ−1/2[U0′(ξ)−I′(ξ)]dΔ(t)−dR(U0(ξ), ξ, t)+O(ϵ1/2), where L^x is defined as in Equation (40) but with c_ϵ → v. The solvability condition is now (54)∫−∞∞V(ξ)[U0′(ξ)dΔ(t)+I′(ξ)dΔ(t)+ϵ1/2dR(U0(ξ), ξ, t)]dξ=0. This can be manipulated to give, to leading order, the Ornstein–Uhlenbeck equation (Gardiner, 2009): (55)dΔ(t)+AΔ(t)dt=dW^(t), where A=∫−∞∞​V(ξ)I′(ξ)dξ∫−∞∞​V(ξ)U0′(ξ)dξ, and dW^(t)=−ϵ1/2g0∫−∞∞​V(ξ)U0(ξ)dW(ξ, t) dξ∫−∞∞​V(ξ)U0′(ξ)dξ. Solving the stochastic differential equation in Equation (55) and taking averages shows that 〈Δ(t)〉 = Δ(0)e^−At and (56)〈Δ(t)2〉−〈Δ(t)〉2≈D(ϵ)A[1−e−2At], where D(ϵ) is given by Equation (44) except for a modified null vector V(ξ). Thus the variance of Δ(t) approaches D(ϵ)/A in the large t limit.

As in the case of freely propagating pulses, we can explicitly solve for V(ϵ) and thus calculate the diffusion coefficient D(ϵ) when F(U) = H(U − κ) and g(U) = U. Since the steps are similar to the previous case, we simply present our results here. In Figure 11 we show the temporal evolution of a single stimulus-locked front, which is obtained by numerically solving the Langevin Equation (51) for F(U) = H(U − κ), g(U) = U and the weight distribution (Equation 4). The external input is taken to be a square pulse of amplitude I₀ = 5, width d = 5, and speed v = 5. Next we determine the mean X¯(t) and variance σ²_X(t) of the position of the leading and trailing edges by averaging over level sets along identical lines to the freely-propagating case. The results are shown in Figure 12. It can be seen that, as predicted by the analysis, X¯(t) varies linearly with t with a slope equal to the stimulus speed v = 5. Moreover, the variance σ²_X(t) approaches a constant value as t → ∞ for both the trailing and leading edges. Thus, we find that stimulus-locked pulses are much more robust to noise than freely propagating pulses, since the variance of the mean position of the leading and trailing edges saturate as t→ ∞. Consequently, stimulus locking persists in the presence of noise over most of the parameter range for which stimulus locking is predicted to occur. However, the trailing edge has an asymptotic variance that is at least an order of magnitude larger than the leading edge, which implies that fluctuations in the width of the pulse can no longer be neglected.