Pulsating fronts in periodically modulated neural field models

We consider a coarse-grained neural field model for synaptic activity in spatially extended cortical tissue that possesses an underlying periodicity in its microstructure. The model is written as an integrodifferential equation with periodic modulation of a translationally invariant spatial kernel. This modulation can have a strong effect on wave propagation through the tissue, including the creation of pulsating fronts with widely varying speeds and wave-propagation failure. Here we develop a new analysis for the study of such phenomena, using two complementary techniques. The first uses linearized information from the leading edge of a traveling periodic wave to obtain wave speed estimates for pulsating fronts, and the second develops an interface description for waves in the full nonlinear model. For weak modulation and a Heaviside firing rate function the interface dynamics can be analyzed exactly and gives predictions that are in excellent agreement with direct numerical simulations. Importantly, the interface dynamics description improves on the standard homogenization calculation, which is restricted to modulation that is both fast and weak.

Figure 1

An example of a pulsating front solution of Eqs. (1) and (2) for $W(x)={\mathrm{e}}^{-\left|x\right|}/2$, $J(x)=J_0+\epsilon \sin (2\pi x/\sigma )$, with $f(u)=H(u-h),$ using parameter values $J_0=1$, $\epsilon =0.4$, $h=0.3$, and $\sigma =2\pi$.

Figure 2

A plot of the wave speed estimate $c$ as a function of $\lambda$ defined by $detA(c,\lambda )=0$ with $N=20$ for three values of $\epsilon$. $\epsilon =1.5$ [dash-dotted (red) curve], $\epsilon =0.5$ [dashed (green) curve], and $\epsilon =0$ [solid (blue) curve]. Other parameters are $\gamma =2$, $J_0=1$, and $\sigma =2\pi$.

Figure 3

Minimum wave speed $c^{*}$ as a function of $\epsilon$ for three values of $\gamma$. Other parameter values are $J_0=1$ and $\sigma =2\pi$. Also shown are values measured from numerical simulations for $\gamma =1.1$ (circles), $\gamma =2$ (triangles), and $\gamma =3$ (squares).

Figure 4

Minimum wave speed $c^{*}$ as a function of $\sigma$ (log scale) for three values of $\epsilon$. Other parameter values are $J_0=1$ and $\gamma =2$. Also shown are values measured from numerical simulations for $\epsilon =2$ (squares), $\epsilon =1$ (triangles), and $\epsilon =0.3$ (circles).

Figure 5

An example of a pair of pinned fronts in a model with $W(x)={\mathrm{e}}^{-\left|x\right|}/2$, $J(x)=J_0+\epsilon \sin (2\pi x/\sigma )$, and $f(u)=H(u-h)$. The solid (dashed) line shows a stable (unstable) solution. Here $J_0=1$, $\epsilon =0.3$, $\sigma =2\pi$, and $h=0.5$.

Figure 6

A plot of the existence condition on the firing threshold $h$ for solutions shown in Fig. 5. Linear stability analysis shows that solutions, corresponding to the points at which $q(\eta )=h$, with $\mathrm{d}q(\eta )/\mathrm{d}\eta <0$, are stable and those with $\mathrm{d}q(\eta )/\mathrm{d}\eta >0$ are unstable.

Figure 7

Solid (red) line: result from an interface dynamics analysis given by Eq. (42). Circles show results measured from simulations. Dashed (black) line: result from homogenization theory given by Eq. (43). Parameters are $\epsilon =0.3$ and $h=0.3$.

Figure 8

Solid (black) line: result from an interface dynamics analysis given by Eq. (47). Symbols show the results of numerical simulations for different values of $\sigma$. Parameters are $h=0.3$.

Figure 9

Solid (black) line: result from an interface dynamics analysis given by Eq. (50). Symbols show the results of numerical simulations for different values of $\sigma$. Parameters are $h=0.3$.

Figure 10

Solid (black) line: result from an interface dynamics analysis given by Eq. (53). Symbols show the results of numerical simulations for different values of $\sigma$. Parameters are $h=0.3$.

Figure 11

A stationary solution of Eqs. (1) and (2) for $f=f_1$. Parameters are $\gamma =0.95$, $J_0=1$, $\sigma =2\pi$, and $\epsilon =0.3$.