The inverse problem of neuron identification

Depending on the state of neuron membrane, the inverse problem of neuron identification is divided into two categories: the passive neuron identification and the active neuron identification. In the first category, we provided a more efficient way to recover neuron parameters than the traditional approach. By exploring the impedance function meticulously, our method reveals a clean and analytical relation between the electrical properties of neurons and their response to sub-threshold current stimulation.;Mathematical equations like the Hodgkin-Huxley equations and the Fitzhugh-Nagumo equations that model active neurons have been established for many years. However, the inverse problem in this category has barely started. Our research in this direction attempts to establish a proper formulation of the inverse problem and to investigate possible mathematical techniques that are needed to solve it. For the relatively simple Fitzhugh-Nagumo equations, we successfully reconstructed the nonlinear membrane conductance function and the coefficients of the recovering variable. The method is then extended to a more realistic neuron model, the Morris-Lecar model. We provide a computational strategy for systematically recovering the nonlinearity of both the calcium and the potassium channels.