Identification problems in distributed parameter neuron models

-One-dimensional cable theory provides the theoretical framework to evaluate the role of linear electrical membrane properties and dendritic geometry in neuronal signal reception and processing. This article presents the basic physical principles as well as mathematical problems arising in neuronal modelling. The main types of passive dendritic neuron models are reviewed and their use in exploring important aspects of a neuron's signal processing capabilities is demonstrated. Employing the continuum approach, a new model of current flow in neurons with branching dendrites is introduced. Applying results on parameter identifiability in linear one-dimensional partial differential equations of parabolic type, each of the models is shown to be not parameter identifiable, i.e. system unidentifiable. Identifiable combinations of parameters corresponding to finite regions in the parameter space, however, can be determined which are limited by physiological restrictions as derived from experimental data. The relevance of the parameter identification problem in distributed parameter neuron models to explain neuronal function is discussed.