Inverse Problems in Neural Field Theory

Inverse Problems in Neural Field Theory

We study inverse problems in neural field theory, i.e., the construction of synaptic weight kernels yielding a prescribed neural field dynamics. We address the issues of existence, uniqueness, and stability of solutions to the inverse problem for the Amari neural field equation as a special case, and prove that these problems are generally ill-posed. In order to construct solutions to the inver...

Dimensional Reduction for the Inverse Problem of Neural Field Theory

Inverse problems in computational neuroscience comprise the determination of synaptic weight matrices or kernels for neural networks or neural fields respectively. Here, we reduce multi-dimensional inverse problems to inverse problems in lower dimensions which can be solved in an easier way or even explicitly through kernel construction. In particular, we discuss a range of embedding techniques...

Inverse Scale Space Theory for Inverse Problems

Abstract. In this paper we derive scale space methods for inverse problems which satisfy the fundamental axioms of fidelity and causality and we provide numerical illustrations of the use of such methods in deblurring. These scale space methods are asymptotic formulations of the Tikhonov-Morozov regularization method. The analysis and illustrations relate diffusion filtering methods in image pr...

Inverse Problems in Darboux’ Theory of Integrability

The Darboux theory of integrability for planar polynomial differential equations is a classical field, with connections to Lie symmetries, differential algebra and other areas of mathematics. In the present paper we introduce the concepts, problems and inverse problems, and we outline some recent results on inverse problems. We also prove a new result, viz. a general finiteness theorem for the ...

Direct and Inverse Bioelec- Tric Field Problems