# Approaches to Parameter Estimation from Model Neurons and Biological Neurons
## 1 Introduction  
Big data are driving the development of algorithms that predict the future evolution of complex systems and learn optimal strategies. _Data-based learning_ has demonstrated remarkable success in predicting chaotic dynamics through reservoir computing [1] and in mastering board games without human input [2]. _Model-based learning_ has instead relied on data assimilation to configure model equations with optimal parameters and initial conditions [3] that synchronize internal state variables to observations. The model is then completed by incorporating the estimated parameters and initial conditions in the model equations. Predictions are then made by forward integrating the completed model. This approach is routinely used by meteorologists, who forward integrate completed Navier-Stokes models to forecast weather conditions [3; 4] or map ocean flow [5; 6]. It is also increasingly being used by computational neuroscientists to predict the state of a neuron [7; 8; 9; 10; 11; 12; 13] and biocircuits [14; 15]. The estimation of parameters from neuron-based conductance models is also important as it has the potential to extract microscopic information on the complement of ion channels and intracellular dynamics, which are currently inaccessible to measurement. Parameters such as ionic conductances, gate thresholds, and gate kinetics are signatures of ion channel subtypes [7; 8; 12; 13; 16]. Inferring these parameters can potentially reveal the ion channels expressed in the electrical activity of a cell. Furthermore, changes in parameter values can potentially detect ion channel mutations which are difficult to classify using transcriptomics [17]. For example, channelopathies are involved in the early stages of brain cancer [18; 19] and Alzheimer disease [20]. Model-based approaches are also important because they allow reconstructing the dynamics of ionic gates which are inaccessible to experiment [7; 8]. Inferring biological information is met with three challenges. Firstly neuron-based conductance models respond nonlinearly to stimulation. Model optimization calls forconvex algorithms that minimize a cost function subject to equality and inequality constraints. Equality constraints are given by the equations of motion of the state vector linearized at each point of the assimilation window. Inequality constraints are given by the boundaries of the parameter search intervals. Early attempts at optimizing neuron models used linear solvers to extract ionic conductances, while setting the gate thresholds and kinetics (nonlinear parameters) at empirically known values [21; 22]. These solvers included evolutionary [23], genetic [24; 25; 26; 27; 28; 29], and particle swarm optimization algorithms [30]. More recently, interior point methods [7; 8; 9; 10; 30; 31], 4 DVar [4], and self-organizing state space methods [32] have successfully estimated both linear and nonlinear parameters (see links in supplementary information). The second challenge is to transfer from the data all the information needed to determine a unique set of model parameters and initial conditions. The added difficulty here is that many state variables cannot be observed, leaving only a few, such as the membrane voltage, as the entry points of experimental data into the model. It is important that the criteria of observability, identifiability, and data sampling be fulfilled to fully constrain the model parameters. Given the availability of powerful convex optimization solvers [30; 31], this paper focuses on setting the conditions for transferring information from the data to the model. The third challenge relates to the awareness that biological circuits may intrinsically incorporate redundant degrees of freedom. Experiments on crustacean central pattern generators have shown that the same electrical activity may arise from different circuit configurations, pointing to possible functional overlap between parameters [33; 34; 35] and compensation [36].