D_2 \xi_2(t)
    \end{aligned}$$ where $x_1, y_1$ and $x_2, y_2$ correspond to single neurons (or neuron populations), which are linearly coupled with coupling strength $C$. The variables $x_1$, $x_2$ are related to the transmembrane voltage and $y_1$, $y_2$ refer to various quantities connected to the electrical conductance of the relevant ion currents. Here $a$ is an excitability parameter whose value defines whether the system is excitable ($a>1$) or exhibits self-sustained periodic firing ($a<1$), $\epsilon_{1}$ and $\epsilon_2$ are the timescale parameters that are usually chosen to be much smaller than unity, corresponding to fast activator variables $x_1$, $x_2$, and slow inhibitor variables $y_1$, $y_2$.

The synaptic coupling between two neurons is modelled as a diffusive coupling considered for simplicity to be symmetric [@LIL94; @PIN00; @DEM01]. More general delayed couplings are considered in [@BUR03]. The coupling strength $C$ summarizes how information is distributed between neurons. The mutual delay $\tau$ in the coupling is motivated by the propagation delay of action potentials between the two neurons $x_1$ and $x_2$.

Each neuron is driven by Gaussian white noise $\xi_i(t)$ $(i=1,2)$ with zero mean and unity variance. The noise intensities are denoted by parameters $D_1$ and $D_2$, respectively.

Besides the delayed coupling we will also consider 