# Approaches to Parameter Estimation from Model Neurons and Biological Neurons
## 2 Model Neurons: Lessons Learnt from Well-Posed Models  
### Choosing Neuron Models  
Neuron-based conductance models [42] describe the rate of change of the membrane voltage as a function of the ionic currents (Na, K, Ca, Cl, leak) crossing the neuron membrane. Each current flows through an ion-specific channel that is gated by the membrane voltage. Activation gates (\(m\)) and inactivation gates (\(h\)) open (resp. close) above a threshold voltage, albeit with a delay described by a first order kinetics. The voltage thresholds,recovery time constants, and maximum ionic conductances are the parameters that we seek to estimate here. The most common ionic currents are of the form:  
\[J_{i}(V,t)=\overline{g}_{i}\;m_{i}^{a}(V,t)\;h_{i}^{b}(V,t)\;(E_{i}-V) \tag{1}\]  
where \(\overline{g}_{i}\) is the maximal conductance of channel subtype \(i\), \(m_{i}\) and \(h_{i}\) are the probabilities of the activation and inactivation gates being open, \(a\geq 0\) and \(b\geq 0\) are the gate exponents, \(E_{i}\) is the reversal potential of the ionic species, and \(V\) is the membrane voltage. Choosing a neuron model entails selecting no two ionic currents that have the same mathematical dependence on \(V\) and \(t\) (Equation (1)). The maximal conductances might otherwise not be uniquely determined. To avoid parameter degeneracy, it is necessary that ion channel subtypes differ from each other through differences in gate exponents, \(a\) and \(b\), or through the voltage dependence of their inactivation time. This condition is sufficient if all other criteria in this section are met. The latter dependence can be bell-shaped (transient sodium current), rectifying (inactivating potassium current), or bi-exponential (low threshold calcium current, A-current) [8; 43]. Conversely, over-specified models, which include more ionic currents than are expressed in the data, are easily handled by data assimilation. The conductances of redundant ion channels are invariably set to zero when assimilating model data [10]. Reduced models have also been introduced to avoid over-specified parameters [44; 45; 46]. Biological considerations regarding the preferences of some ion channels to lie in greater density in the soma or the dendrites may also help select the ionic currents to include in the model. It is known, for example, that the density of calcium channels is greater in dendritic trees than in the soma [47; 48; 49; 50], and hence, calcium channels may be omitted when modelling the soma compartment. In addition, single compartment models are often sufficient to give excellent predictions of neuronal activity [7; 8; 13; 32; 51]. The minute contribution of dendritic compartments may therefore be treated as a perturbation alongside model error. Figure 1 shows the process of constructing a complete model from electrophysiological recordings of a songbird neuron [7; 8]. The conductance model used in this example is given in the supplementary information. The optimal fit of the model (green line) to the data (black line) is obtained by assimilating these data over a 590 ms time window. Once estimated, the parameters are inserted in the model equations. The completed model is then forward integrated from \(t=1090\) ms onwards (red trace), driven by the same current protocol applied to the interneuron (blue trace) [8]. Biological considerations also guide the choice of the parameter search intervals, which must be specified prior to assimilating the data. Figure 1: **Construction of a model neuron from electrophysiological recordings of interneurons from the songbird brain. ** The model (green trace) is synchronized to the membrane voltage time  
series (black trace) of an interneuron of the songbird brain. The model trace is directly overlayed over the data. The neuron membrane voltage was recorded under current stimulation by a protocol (blue trace) mixing chaotic oscillations (Lorenz system) and long current steps. Parameters are obtained from the model that best synchronizes its membrane voltage state variable to the data (green trace).