Numerical solution of inverse problems in computational neuroscience models - Mandujano Valle, Jemy Alex (2019)
In this thesis, we studied three mathematical models in computational neuroscience. A mathematical model for the initiation and propagation of an action potential in a neuron was named after its creators in 1952. Since then, the Hodgkin-Huxley (H-H) model, or conductance-based model, has been used vastly in the world of physiology. A system of four coupled nonlinear ordinary differential equations, such as the H-H model is usually difficult to analyze. In this context, some new models developed appear to satisfactorily reduce it from four to three or two differential equations. One of the reduced models, with a system of two coupled nonlinear ordinary differential equations, is the FitzHugh–Nagumo (F-N) model. On the other hand, in mathematics, it is easier to work with linear differential equations than with nonlinear equations (H-H and F-N models). For this reason, we begin our research with the cable equation, one linear partial differential equation that describes the voltage in a straight cylindrical cable. This model has been applied to model the electrical potential in dendrites and axons. However, sometimes this equation might result in incorrect predictions for some realistic geometries, in particular when the radius of the cable changes significantly. The main goal of this work was then to estimate parameters in the previously mentioned models, given the membrane potential measurement (inverse problems). To solve the inverse problems we consider iterative regularization methods, as the Landweber and the minimal error methods. We compute the adjoint of the Gateaux derivative using different approaches for each one of our problems. Also, we numerically implement the methods in order to show their efficiency, using the forward Euler and backward Euler numerical methods. Next, we describe our inverse problems. In the cable equation, we determine approximate conductances with non-uniform distribution, both in a single branch and in a tree. To obtain the unknown parameters in the cable equation we used the Landweber iteration. We apply the minimal error method to find an approximate unknown function in the FitzHugh- Nagumo model. In the Hodgkin-Huxley model, we estimate the maximal conductances (three constants), the number of activation and inactivation particles in the ion channels (three constants), and also parameters with non-uniform distribution. We use the minimal error method again, in the H-H equation, to approximate the unknown parameters.