PS another nice feature of the I_{inj} loss I described:
If the only parameters you are interested in are maximal conductances (or more generally, parameters that enter the equation for \dot{V} but not the time-derivatives of the ion channel states)…
Then you don’t even need to solve the differential equation more than once to calculate the loss at different parameter values! Since if you injected the I_{inj}(t) current, voltage (and hence ion channel) dynamics would be the same as for the nominal model.
So you can
- Solve the ion channel dynamics corresponding to your observed voltage trace (i.e. solve a differential equation).
- Trapezoidally integrate I_{inj}(t)^2 over time, using
I_{inj}(t) = \dot{V}^*(t) - \dot{V}(parameters, t)
and the fact that \dot{V}(parameters,t) + I_{inj}(t) follows the (solved) dynamics of V^*(t) over time.
Of course, this needs accurate knowledge of \dot{V}^*(t)…