  Consider a large number $n$ of neurons, each being connected to approximately
$N$ other ones, chosen at random. When a neuron spikes, which occurs randomly
at some rate depending on its electric potential, its potential is set to a
minimum value $v_{min}$, and this initiates, after a small delay, two fronts on
the (linear) dendrites of all the neurons to which it is connected. Fronts move
at constant speed. When two fronts (on the dendrite of the same neuron)
collide, they annihilate. When a front hits the soma of a neuron, its potential
is increased by a small value $w_n$. Between jumps, the potentials of the
neurons are assumed to drift in $[v_{min},\infty)$, according to some
well-posed ODE. We prove the existence and uniqueness of a heuristically
derived mean-field limit of the system when $n,N \to \infty$ with $w_n \simeq
N^{-1/2}$. We make use of some recent versions of the results of Deuschel and
Zeitouni \cite{dz} concerning the size of the longest increasing subsequence of
an i.i.d. collection of points in the plan. We also study, in a very particular
case, a slightly different model where the neurons spike when their potential
reach some maximum value $v_{max}$, and find an explicit formula for the
(heuristic) mean-field limit.
