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 $γ\geq 0$. 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 $γ$: For large values of $γ$, the neural spiking activity almost surely goes extinct; For small values of $γ$, a fixed neuron spikes infinitely many times with a positive probability, and for "intermediate" values of $γ$, the system has a positive probability of always presenting spiking activity, but, individually, each neuron eventually stops spiking and remains at rest forever.