  We consider a stochastic model describing the spiking activity of a countable
set of neurons, which are spatially organized into a homogeneous tree of degree
$d$, $d \geq 2$. Roughly, the model is as follows. Each neuron is represented
by its membrane potential, which assumes non-negative integer values. Neurons
spike at Poisson rate one, provided they have strictly positive membrane
potential. When a spike occurs, the spiking neuron's potential changes to 0,
and all neurons connected to it receive a positive amount of potential.
Moreover, between successive spikes and without receiving any spiking inputs
from other neurons, each neuron's potential behaves independently as a pure
death process with death rate $\gamma \geq 0$. In this article, we show that if
the degree $d$ is large enough, then the process exhibits at least two phase
transitions depending on the choice of rate $\gamma$: For large values of
$\gamma$, the neural spiking activity goes extinct almost surely; For small
values of $\gamma$, a fixed neuron spikes infinitely many times with positive
probability, and for "intermediate" values of $\gamma$, the system has a
positive probability of always presenting spiking activity, but this activity
drifts off to infinity, and each neuron eventually stops spiking and remains at
rest forever.
