The simulation in Figure 1 indicates that a stable travelling anti-pulse solution exists. We will consider the existence and stability for a Heaviside firing rate function, making use of an Evans function approach. This is a powerful tool for the stability analysis of nonlinear waves on unbounded domains. It was originally formulated by Evans [4] in the context of a stability theorem about excitable nerve axon equations of Hodgkin-Huxley type. The extension to integral models is far more recent, see [21] for a discussion. For neural field models with axonal delays it has previously been noted these do not typically induce any change of wave stability [22, 23] (though it will affect the shape and speed of a wave). Since our simulations indicate similar effects for the anti-pulse, we will initially consider the case 1/*v*=0 for simplicity. The discussion below is adapted from that presented in [22, 23], to which we refer the reader for further details.

### Existence of the anti-pulse

To show existence and stability it is convenient to first rewrite the integro-differential equation (2) in integral form. We define *η*(*t*)=e^{−t} and *η*_{
a
}=e^{−t/τ}/*τ* with *η*(*t*)=*η*_{
a
}(*t*)=0 if *t*<0. The model with adaptation can then be written in the integral form

u=\eta \ast \psi -\mathrm{\kappa \eta}\ast \left({\eta}_{a}\ast u\right),

(6)

where ∗ denotes temporal convolution (e.g. [\phantom{\rule{0.3em}{0ex}}\eta \ast \psi ](x,t)={\int}_{0}^{\infty}\eta \left(s\right)\psi (x,t-s)\mathrm{d}s). The construction of travelling waves, e.g. fronts and pulses, proceeds by introducing the co-moving frame *ξ*=*x*−*c* *t* with *c*>0. Travelling waves are then stationary solutions *u*(*x*,*t*)=*u*(*ξ*) where

u\left(\xi \right)={\int}_{0}^{\infty}\mathrm{d}\mathrm{s\eta}\left(s\right)\left(\psi (\xi +\mathit{\text{cs}})-\kappa {\int}_{0}^{\infty}\mathrm{d}{s}^{\prime}{\eta}_{a}({s}^{\prime}-s)u\left({s}^{\prime}\right)\right),

(7)

with

\psi \left(\xi \right)={\int}_{-\infty}^{\infty}\mathrm{d}\mathit{\text{yw}}\left(y\right)f\left(u\right(\xi -y\left)\right).

(8)

Introducing the Fourier transform,

\hat{g}\left(k\right)={\int}_{-\infty}^{\infty}g\left(\xi \right){e}^{-\mathrm{ik\xi}}\mathrm{d}\xi ,

(9)

equation (6) can be arranged to the following form

\hat{u}\left(k\right)={\hat{\eta}}_{c}\left(k\right)\hat{\psi}\left(k\right),\phantom{\rule{2em}{0ex}}{\hat{\eta}}_{c}\left(k\right)=\frac{\hat{\eta}\left(k\right)}{1+\kappa \hat{\eta}\left(k\right){\hat{\eta}}_{a}\left(k\right)}.

(10)

Since \hat{\eta}\left(k\right)={(1+\mathit{\text{ik}})}^{-1} and \phantom{\rule{2.77626pt}{0ex}}{\hat{\eta}}_{a}\left(k\right)={(1+\mathrm{ik\tau})}^{-1} we find

{\hat{\eta}}_{c}\left(k\right)=\frac{-(1+\mathrm{ik\tau})}{\tau (k-i{\lambda}_{+})(k-i{\lambda}_{-})},\phantom{\rule{1em}{0ex}}{\lambda}_{\pm}=\frac{1}{2\tau}\left(1+\tau \pm \sqrt{{(1-\tau )}^{2}-4\mathrm{\tau \kappa}}\right).

(11)

Exploiting the pole structure of (11), an inverse Fourier transform yields

{\eta}_{c}\left(t\right)=\frac{1}{\tau ({\lambda}_{-}-{\lambda}_{+})}\left((1-\tau {\lambda}_{+}){\mathrm{e}}^{-{\lambda}_{+}t}-(1-\tau {\lambda}_{-}){\mathrm{e}}^{-{\lambda}_{-}t}\right),\phantom{\rule{1em}{0ex}}t\ge 0.

(12)

A travelling wave solution can then be written succinctly as

u\left(\xi \right)={\int}_{0}^{\infty}\mathrm{d}s{\eta}_{c}\left(s\right)\psi (\xi +\mathit{\text{cs}}).

(13)

For *f*(*u*)=*H*(*u*−*θ*), a Heaviside step function, the pulse solution can be found by specifying where the solution is above the threshold *θ*. An anti-pulse has a profile *u*(*ξ*) above threshold everywhere except for *ξ*=*x*−*c* *t*∈ [ −Δ,0]. So the profile is formally given by

u\left(\xi \right)={\int}_{0}^{\infty}\mathrm{d}s{\eta}_{c}\left(s\right)\left(1-{\int}_{\xi +\mathit{\text{cs}}}^{\xi +\mathit{\text{cs}}+\mathrm{\Delta}}w\left(y\right)\mathrm{d}y\right),

(14)

where we note that the width Δ and speed *c* are yet unknown. These can be found by imposing the crossing conditions *u*(−Δ)=*u*(0)=*θ*. This yields the following two equations

\begin{array}{cc}2\theta & =\frac{2\tau {c}^{2}+(2+\tau -\mathrm{\tau \kappa})c+(1+\kappa )+(1+\kappa )(\mathrm{c\tau}+1){\mathrm{e}}^{-\mathrm{\Delta}}}{(\tau {c}^{2}+(1+\tau )c+1+\kappa )(1+\kappa )},\end{array}

(15)

\begin{array}{ll}\tau \left({\lambda}_{-}-{\lambda}_{+}\right)2\theta =& \frac{1-{\lambda}_{+}\tau}{{\lambda}_{+}\left({c}^{2}-{\lambda}_{+}^{2}\right)}\left(2{c}^{2}{\mathrm{e}}^{-{\lambda}_{+}\mathrm{\Delta}/c}+{\lambda}_{+}c\left(1-{\mathrm{e}}^{-\mathrm{\Delta}}\right)+{\lambda}_{+}^{2}\left(1+{\mathrm{e}}^{-\mathrm{\Delta}}\right)\right)\\ -\frac{1-{\lambda}_{-}\tau}{{\lambda}_{-}\left({c}^{2}-{\lambda}_{-}^{2}\right)}\left(2{c}^{2}{\mathrm{e}}^{-{\lambda}_{-}\mathrm{\Delta}/c}+{\lambda}_{-}c\left(1-{\mathrm{e}}^{-\mathrm{\Delta}}\right)+{\lambda}_{-}^{2}\left(1+{\mathrm{e}}^{-\mathrm{\Delta}}\right)\right).\end{array}

(16)

When we take the limit Δ→*∞*, we see that (15) and (16) with *ε*=*τ*^{−1} reduce to

2\theta -\frac{2{c}^{2}+c(1-\kappa +2\epsilon )+\epsilon (1+\kappa )}{({c}^{2}+c(1+\epsilon )+\epsilon (1+\kappa \left)\right)(1+\kappa )}=0,

(17)

which recovers a result in [24] for right-moving inactivating fronts (left-moving waves in their case). For completeness we note that the travelling pulse satisfies

\begin{array}{cc}2\theta & =\frac{(1+\mathrm{c\tau})\left(1-{\mathrm{e}}^{-\mathrm{\Delta}}\right)}{(1+c)(1+\mathrm{c\tau})+\kappa},\end{array}

(18)

\begin{array}{ll}2({\lambda}_{-}-{\lambda}_{+})\mathrm{\tau \theta}=& \frac{(1-{\lambda}_{-}\tau )}{\left({c}^{2}-{\lambda}_{-}^{2}\right){\lambda}_{-}}\left(2{c}^{2}\left({\mathrm{e}}^{-{\lambda}_{-}\mathrm{\Delta}/c}-1\right)+{\lambda}_{-}\left({\lambda}_{-}+c\right)\left(1-{\mathrm{e}}^{-\mathrm{\Delta}}\right)\right)\\ -\frac{\left(1-{\lambda}_{+}\tau \right)}{\left({c}^{2}-{\lambda}_{+}^{2}\right){\lambda}_{+}}\left(2{c}^{2}\left({\mathrm{e}}^{-{\lambda}_{+}\mathrm{\Delta}/c}-1\right)+{\lambda}_{+}\left({\lambda}_{+}+c\right)\left(1-{\mathrm{e}}^{-\mathrm{\Delta}}\right)\right).\end{array}

(19)

These are determined analogously as for the anti-pulse but with the requirement that the profile is above threshold only for *ξ*∈(−Δ,0) [24]. The speed of activating fronts (moving to the right) can be obtained from (18) in the limit Δ→*∞*. The above equations apply to right-moving waves, i.e. *c*>0.

### Stability of the anti-pulse

We will now construct the Evans function. First, we linearise (6) around the travelling solution \u016b\left(\xi \right) by writing u(\xi ,t)=\u016b\left(\xi \right)+z(\xi ,t). In particular, we will look for exponentially decaying/growing separable solutions of the form *z*(*ξ*,*t*)=*z*(*ξ*)*e*^{λt}. Collecting *O*(*z*) terms yields the eigenvalue equation

z\left(\xi \right)=\frac{1}{c}{\int}_{-\infty}^{\infty}\mathrm{d}\mathit{\text{yw}}\left(y\right){\int}_{\xi -y}^{\infty}\mathrm{d}r{\eta}_{c}\left(\right(r+y-\xi )/c){\mathrm{e}}^{\lambda (\xi -r-y)/c}{f}^{\prime}\left(\u016b\right(r\left)\right)z\left(r\right).

(20)

For the Heaviside firing rate function we have *f*^{′}(*u*(*r*))=*δ*(*u*(*r*)−*θ*)/|*u*^{′}(*r*)|, which appears inside the integral only and hence poses no difficulties. Since we have crossing points at *ξ*=0 and *ξ*=−Δ, we get *z*(*ξ*)=*A*(*λ*,*ξ*)*z*(0)+*B*(*λ*,*ξ*)*z*(−Δ), where

\begin{array}{ll}A(\lambda ,\xi )=& \frac{1}{c\left|{u}^{\prime}\right(0\left)\right|}{\int}_{\xi}^{\infty}\mathrm{d}y\phantom{\rule{0.3em}{0ex}}w\left(y\right){\eta}_{c}\left(\frac{y-\xi}{c}\right){\mathrm{e}}^{\lambda (\xi -y)/c},\phantom{\rule{2em}{0ex}}\end{array}

(21)

\begin{array}{ll}B(\lambda ,\xi )=& \frac{1}{c\left|{u}^{\prime}\right(-\mathrm{\Delta}\left)\right|}{\int}_{\xi +\mathrm{\Delta}}^{\infty}\mathrm{d}y\phantom{\rule{0.3em}{0ex}}w\left(y\right){\eta}_{c}\left(\frac{y-\xi -\mathrm{\Delta}}{c}\right){\mathrm{e}}^{\lambda (\xi +\mathrm{\Delta}-y)/c}.\phantom{\rule{2em}{0ex}}\end{array}

(22)

The eigenvalue problem should be self-consistent at *ξ*=0 and *ξ*=−Δ giving the system of equations

\left[\begin{array}{c}z\left(0\right)\\ z(-\mathrm{\Delta})\end{array}\right]=\mathcal{A}\left(\lambda \right)\left[\begin{array}{c}z\left(0\right)\\ z(-\mathrm{\Delta})\end{array}\right],\phantom{\rule{2em}{0ex}}\mathcal{A}\left(\lambda \right)=\left[\begin{array}{cc}A(\lambda ,0)& B(\lambda ,0)\\ A(\lambda ,-\mathrm{\Delta})& B(\lambda ,-\mathrm{\Delta})\end{array}\right].

(23)

A nontrivial solution for *z* exists if \mathcal{E}\left(\lambda \right):=det\left(\mathcal{A}\right(\lambda )-I)=0. The function \mathcal{E}\left(\lambda \right) is the Evans function for the travelling anti-pulse. Note that the same construction applies to the travelling pulse. Solving (15) and (16) we find that *c*(*κ*) has a turning point, so that we have a fast and a slow branch, denoted by *c*_{+} and *c*_{−} respectively. Figure 2 shows the roots of the Evans function along the pulse and anti-pulse branches. There is a trivial eigenvalue *λ*=0 due to translation invariance. The nontrivial eigenvalue is positive for the slow (anti-) pulse *c*_{−}, and negative for the fast branch *c*_{+}. Hence, the latter is stable.

### A combined view on fronts and pulses

From equations (15) and (16), it is straightforward to determine the speed as a function of the adaptation strength *κ*, see Figure 3. For completeness, we also plot the speeds of the (right-moving) travelling anti-front, front and pulse. What is apparent is that all curves meet at the point *C* given by (*κ*_{
c
},*c*)=(1/(2*θ*)−1,1/(2*θ*)−1−1/*τ*). Here we have a codimension 2 heteroclinic cycle bifurcation. We have checked using simulations that varying *κ* for Gaussian connectivity induces a similar transition between pulse and anti-pulse behaviour. The unfolding of this bifurcation involves two heteroclinic and two homoclinic bifurcation curves. It implies that the anti-pulse exists generically.

We have also plotted the profiles for *κ*=0.65 and *κ*=0.75 with *τ*=7 and *θ*=0.3, see Figure 4. This shows the difference between the slower and faster solutions. The after-overshoot in activity is due to diminished adaptation during the anti-pulse. When *τ* is decreased, two things happen. First, the profile develops a slow and strongly damped oscillation as *λ*_{±} become complex. Second, the fast branch of stable pulse solutions disappears, see Figure 3 (right) as the adaptation is fast and pulls down all activity quickly. This happens when the pulse and anti-pulse branch have become tangent to the inactivating front and front branches for Δ→*∞*, respectively, i.e. they have interchanged their position in the (*c*,*κ*) plane.