Population activity models derived from neuronal networks
For characterizing the dynamics of biophysically modeled, recurrent neuronal networks we derive in the first part of the talk efficient models that describe the network activity in terms of a few ordinary differential equations. These systems are simple to solve and allow for convenient investigations of asynchronous, oscillatory or chaotic network states because linear stability analyses and powerful related methods are readily applicable. We build upon two research lines on which substantial efforts have been exerted in the last two decades: (i) the development of single neuron models of reduced complexity, particularly adaptive nonlinear integrate-and-fire neurons, that can accurately reproduce a large repertoire of observed neuronal behavior, and (ii) different approaches to approximate the Fokker-Planck equation that represents the collective dynamics of large neuronal networks. We combine these advances and extend recent approximation methods of the latter kind to obtain spike rate models that surprisingly well reproduce the macroscopic dynamics of the underlying neuronal network. At the same time the microscopic properties are retained through the single neuron model parameters.
In the second part of the talk we outline how non-invasive electrical stimulation could be used to control the population dynamics of spatially extended neurons. Based on a large system of cable equations in terms of multiple coupled partial differential equationswe derive a reduced description that is computationally feasable. This allows to analyze the resonance behaviour of recurrent neuronal networks in the presence of an externally applied current. Using that approach we show that an electric field amplifies distinct frequency bands that are not pronounced without stimulation which could be used as a mechanism to entrain oscillatory network dynamics.