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
where ∗ denotes temporal convolution (e.g. ). 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
(7)
with
(8)
Introducing the Fourier transform,
(9)
equation (6) can be arranged to the following form
(10)
Since and we find
(11)
Exploiting the pole structure of (11), an inverse Fourier transform yields
(12)
A travelling wave solution can then be written succinctly as
(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
(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
(15)
(16)
When we take the limit Δ→∞, we see that (15) and (16) with ε=τ−1 reduce to
(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
(18)
(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 by writing . In particular, we will look for exponentially decaying/growing separable solutions of the form z(ξ,t)=z(ξ)eλt. Collecting O(z) terms yields the eigenvalue equation
(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
(21)
(22)
The eigenvalue problem should be self-consistent at ξ=0 and ξ=−Δ giving the system of equations
(23)
A nontrivial solution for z exists if . The function 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/(2θ)−1,1/(2θ)−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.