  We consider a stochastic model describing the spiking activity of a countable
set of neurons spatially organized into a homogeneous tree of degree $d$, $d
\geq 2$; the degree of a neuron is just the number of connections it has.
Roughly, the model is as follows. Each neuron is represented by its membrane
potential, which takes non-negative integer values. Neurons spike at Poisson
rate 1, provided they have strictly positive membrane potential. When a spike
occurs, the potential of the spiking neuron 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 number $d$ of
connections 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 almost surely goes extinct; For small
values of $\gamma$, a fixed neuron spikes infinitely many times with a positive
probability, and for "intermediate" values of $\gamma$, the system has a
positive probability of always presenting spiking activity, but, individually,
each neuron eventually stops spiking and remains at rest forever.
