Solution techniques for inverse problems in neural field theory

In this thesis we present new solution techniques for inverse problems in neural field theory. Neural fields are a continuum limit of neural networks and describe the spatiotemporal evolution of neural activity in the brain. This evo- lution is described by an integro-differential equation called the Amari equation. One of the inverse problems in neural field theory is to describe the connections between neurons based on this spatiotemporal evolution. The inverse problem is ill-posed. In other work, this ill-posedness was dealt with using Tikhonov regularization. We present three methods that reduce the ill-posedness of the problem and improve the quality of the reconstruction. We compare these meth- ods to the use of Tikhonov regularization and also show what happens when we combine these methods with Tikhonov regularization. The first method we use is parameter optimization. We show that parameter optimization is necessary when dealing with data generated for fixed parameters. The second method we introduce is subsampling. We show that subsampling is a tool to reduce the error of the reconstruction and reduce the ill-posedness. At some point we reach a trade-off between accuracy and stability. We furthermore show that a combination of subsampling and Tikhonov regularization is sometimes the best method. The third method we present is combining data. Sometimes we are dealing with insufficiently informative data. To overcome this, we can combine data that is qualitatively different.