# Approaches to Parameter Estimation from Model Neurons and Biological Neurons
## 4 Discussion  
Gradient descent algorithms have reliably recovered the 10-100 parameters of neuron-based conductance models. When the problem is well-posed, these parameters are systematically found to lie within 0.1% of the true parameter values. These are the parameters set in the model used to synthesize the time series data. An accuracy better than 0.1% supposes that the criteria of observability and identifiability are satisfied. Observability can be further improved by increasing the number of state variables which are observed. This means complementing electrophysiological recordings with calcium ([PERSON] et al., 2017) or optogenetic recordings ([PERSON] et al., 2017). More observed state variables also means that the parameter search becomes less likely to develop positive Lyapunov exponents ([PERSON] et al., 2017; [PERSON] et al., 2017; [PERSON] et al., 2017) and to converge ([PERSON] et al., 2017; [PERSON] et al., 2017). Identifiability is conditional to the nature of stimulation applied to the system. Although much theoretical work remains to be done to design stimulation protocols that transfer a maximum of information from the data to the model, numerical studies show that protocols eliciting both hyperpolarized and depolarized oscillations, while covering the full spectrum of ionic gate kinetics, are sufficient to recover all original parameters in twin experiments. Noise in experimental data is easily dealt with by smoothing membrane voltage time  
Figure 5: **Correlations between estimated parameters. ****(a)** Covariance matrix of the 71 parameters of a multichannel conductance models, synchronized to electrophysiological recordings of a songbird neuron. A total of 81 sets of parameters were estimated from different assimilation windows to generate the statistical sample used to compute the covariance. Parameters pertaining to each ion channel tend to be more correlated. The greater error/variance is in the recovery time constants of ionic gates (\(\tau_{nn}\), \(\tau_{h}\)). The principal values of the covariance matrix give the length of the principal semi-axes of the data misfit ellipsoid \(\delta c=kT\). (**b**) Spectrum of principal values of the covariance matrix. series to reduce the uncertainty on parameters. Additive Gaussian noise may be used in a constructive manner to regularize convergence towards the global minimum of the cost function. Currently the major roadblock in estimating the parameters of neurons is model error. Conventional approaches insert error in the cost function alongside data error. This requires balancing data and model error, either through ab-initio calculations of the model error covariance matrix, or by using a variational parameter such as \(R_{\lambda}\) as a majoration of model error. In either case, the data assimilation algorithm must relax the enforcement of constraints without sending the model error term of the cost function to zero. Reservoir computing [1] can, in principle, also help correct model error. The non-physical models, built by training a neural network on time series data, can be used to identify error in the conductance equations. The excellent predictions of neuron oscillations made by conductance models have suggested two empirical methods which revealed ion channel properties in good agreement with voltage clamp measurements and IC50 analysis. The first approach consists of taking the mean expectation value of a statistical sample of parameters computed from many different initial guesses. This method was shown to give consistent activation thresholds and ion channel conductances across the complement of ion channels of nominally equivalent songbird neurons. The second approach used the observation that parameter correlations tend to be greatest within each ion channel.