Mathematical models for neural networks and phase transitions in material
science include spatially long-range interaction between neurons or
particles, in the case of materials. Some of the interaction effects are
inhibitory (or anti-ferromagnetic) and some are excitatory
(ferromagnetic). This leads to evolution equations of the form
$u_t=d(J*u-u) + f(u)$, where $d>0$, $f $ is bistable, either $u \in
\el^\infty$ in the discrete case or $u \in L^\infty$ in the continuum, and
$*$ is convolution, discrete or continuous. The kernel $J$ may change sign
but has unit integral. We give conditions under which stable stationary
patterns exist and conditions under which traveling waves exist, even when
$J(x)$ changes sign with $x$. Thus, the presence of both excitatory and
inhibitory couplings can lead to both pattern formation or homogeneity
depending on finer details in the connections.
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