# Approaches to Parameter Estimation from Model Neurons and Biological Neurons
## 2 Model Neurons: Lessons Learnt from Well-Posed Models
### Identifiability  
A model is identifiable if any two different sets of parameters produce different state trajectories starting from the same initial state [66; 67]. Over-specified models may still be identifiable if unexpressed ion channels are filtered out (i.e., \(\overline{g_{i}}=0\)), with all other ion channel parameters being fully constrained. This aside, over-specified models are not identifiable.  
Identifiability is critically dependent on the stimulation protocol applied to the neuron. Unfortunately, parameter estimation has mainly focused on autonomous oscillators. Examples of parameter estimation from systems driven by an external force are comparatively few. Consequently, there is a need for further research on stimulation protocols that maximize identifiability. Central pattern generators are one example of autonomous oscillators where disparate parameters may yield the same oscillation patterns under no or tonic stimulation [34]. Similarly, no or tonic stimulation would fail to reveal adaptation in the electrical activity of a cell [68] from which calcium properties could not be identified. On the other hand, perturbing a central pattern generator, even with a modest variation in temperature or pH, is sufficient to switch between oscillatory modes [69; 70; 71] and in the process lift parameter degeneracy [40; 41]. Maximum perturbation has been applied to neurons by designing current protocols that reveal adaptation mechanisms [7; 8; 14; 32]. The design of such protocols has so far been guided by intuitive considerations based on the need to excite the hyperpolarized, sub-threshold, and depolarized regimes of the neuron, and to elicit the different response times of the neuron/circuit [7; 8; 14].  
Identifiability may be achieved through careful design of current protocols. The aim is to retrieve the maximum of information from the data. The information contained in the current protocol may be quantified by the conditional Shannon entropy [72] relative to the conductance model. An alternative approach to testing identifiability has been suggested by [PERSON] et al. [73] and [PERSON] et al. [74]. Their method computes the Lie matrix linking the time derivatives of the data at different orders to the parameter vector at time \(t\). Identifiability is validated by the matrix having full rank. If the matrix is singular, identifiability may be recovered by removing the matrix column(s) responsible for the singularity. This approach implies reparametrizing the model to remove functional overlap between parameters. Because models are meant to describe physical reality rather than the other way round, such simplifications rarely help characterize the underlying biology. Instead, Lie matrices are useful to validate the stimulation protocols that warrant the identifiability of neuron-based conductance models.  
Numerical experiments have shown that identifiability criteria are fulfilled when current protocols elicit the full range of membrane voltage amplitudes and dynamics. Protocols that meet these conditions were found to constrain all model parameters to within 0.1% of their true value, independently of the waveform chosen for the current protocol.