I'm studying a simplification of a biological neuron model with $n$ neurons. We are describing the evolution of the membrane potential of each neuron. Let $(X_t)_{t\geq 0}$ be a Markov Jump Process in $\mathbb{N}_0^n$ with the following generator $Q$:

\begin{equation} \begin{cases} (i)\ \ \ \ q(x,(x)_i^c) = b(x_i) \\ \\ (ii)\ \ \ q(x,x+e_i)=\lambda x_i\ \ \ \ i=1,...,n \\ \\ (iii)\ \ q(x,x-e_i) =\mu x_i \end{cases} \end{equation}

Where $(x)_i^c: = x+c(1,...,1)-(x_i+c)e_i\ \ \ (c\in \mathbb{N})$; $\ \ \mu > \lambda\ $; and $b:\mathbb{R}_{\geqslant 0}\to \mathbb{R}_{\geqslant 0}$ is a non-negative, non-decreasing function with $b(0)=0$.

Looking carefully at $(x)_i^c$, you can see that it represents the discharge of neuron $i$. It goes to zero, whereas the other neurons recieve a fixed potential $c>0$.

The goal is to prove that this process has almost certain extinction. That is, we need to prove that

$$ \mathbb{P}(T_0<+\infty) =1 $$

Where $T_0:=\inf\{t>0:X_t=(0,...,0)\}$

Here are some observations and an idea of what may be a path towards a proof:

For a start, see that if we ignore condition $(i)$, then $(ii)$ and $(iii)$ represent in each coordinate an independent Birth-and-Death process in $\mathbb{N}_0$ with almost certain extinction, so if we ignored $(i)$ then we would have almost certain extinction for $(X_t)$.

I expect that the perturbation introduced by $(i)$ does not affect so much: What $(i)$ says is that when a coordinate is too "high", then it increasingly tends to "discharge". The total potential $||X_t||_1$ is increased by $c(n-1)$ and decreased by $x_i(t)$, so it only increases if the potential $x_i(t)$ was below the threshold $c(n-1)$. A discharge of a neuron with such a potential occurs with rate at most $b(c(n-1))$.

The idea I have for a proof is to define some neighborhood around $(0,...,0)$ (say $\mathcal{C}:=\{ ||X||_1 \leqslant k\}$ for some $k>0$) where I can have positive probability of extinction $p$, independently of the starting point. And then I'd like to prove that if you are out of $\mathcal{C}$, then you come back with probability 1.

Finally I would define an associated random variable with Geometric distribution $p$ , thinking as the experiment " whether or not $(X_t)$ starting inside $\mathcal{C}$ reaches extinction".

Are my ideas correct?

What about the idea of a proof?

If correct, are there some details I need to be careful with?

EDIT 1: I've already proved that

$$ \mathbb{P}_x(T_{\mathcal{C}}<+\infty) = 1 \ \ \ \forall x\notin \mathcal{C}$$

Where $\mathcal{C}=\{x\in\mathbb{N}_0^n:||x||_1\leqslant K \}$ for some $K>0$ (which is a finite set).

Now the issue would be to write properly the idea that I've exposed above.