I'm working on a neuroscience problem which looks at the time a neuron spikes. A neuron spikes when its voltage increases to a level, $\theta$. The simplest such model is one of the form:
$$\tau_m\frac{dV}{dt} = -V(t) + RI(t)$$
where $I$ is a constant current. Assuming $V(0) = 0$, one can solve the above equation exactly to give:
$$V(t) = RI[1-\exp(t/\tau_m)]$$
If the time of the first spike is $t^{(1)}$, then one can solve the equation $$\theta = RI[1-\exp(t^{(1)}/\tau_m)]$$ to find the spike time. However, this method requires being able to solve the equation explicitly. A substantially more accurate model is the following equation:
$$\frac{dV}{dt} = -g_L(V-V_{rest}) + g_L\Delta_T\exp(\frac{V-V_T}{\Delta_T}) + I$$
Parameter values for the above equation are: $g_L = 0.1, V_T = -30, \Delta_T = 3.48$ (and I is a constant).
This equation is no longer solveable (it can be separated, but not solved). Is there anyway to analytically estimate the timing of the first spike when the voltage reaches $\theta$? It can easily be done computationally but I wanted to see if there's a way to do it analytically.