Mean-Field of 2D Spiking Neurons

We consider the following McKean-Vlasov equation: $$ \begin{aligned} \mathrm{d} V^\mu_t &= b ( V^\mu_t) \mathrm{d} t + J \mathbb{E} [X^\mu_t f(V^\mu_t)] \mathrm{d} t - V^\mu_{t-} \int_{ \mathbb{R}_+} \mathbf{1}_{\{z \leq f(V^\mu_{t-})\}} N(\mathrm{d} t, \mathrm{d} z) , \\ \mathrm{d} X^\mu_t &= \frac{1 - X^\mu_t}{\tau} \mathrm{d} t - U X^\mu_{t-} \int_{ \mathbb{R}_+} \mathbf{1}_{\{z \leq f(V^\mu_{t-})\}} N(\mathrm{d} t, \mathrm{d} z) , \end{aligned} $$ where $b(v) = m - \lambda v$, and $f(v) = v^2$ or $f(v) = v^{1/\beta}$ or $f(v) = \frac{1}{\beta} 1_{v \geq 1}$.