Correlated fluctuations in neural networks on the microscopic and mesoscopic scale

There is broad consent that understanding the brain's function relies on the investigation of the multi-scale dynamics of cortical networks. Rapidly advancing multi-electrode technology for recordings of extracellular brain activity allows monitoring of massively parallel spiking activity and local field potentials (LFPs). Both signals are related, in fact, they show brain dynamics on different spatial scales. While the spiking activity represents the output signal of individual neurons, the LFP is a spatially coarse-grained population measure with contributions from thousands up to millions of cells. The spike output as well as the LFP largely result from coordinated activity between neurons in complex networks. This activity is determined by the structure in network connections as well as intrinsic single-neuron dynamics. Proper understanding of the spiking activity and the LFP in recurrent networks requires mathematical modeling. In this thesis, we employ a connectionist modeling approach by regarding neurons as simplified computational units, and investigate which features of experimentally observed activity arise from the connectivity between such units. In particular, we concentrate on correlated fluctuations as arising in networks which fulfill the basic features of brain connectivity, that is, we consider networks of excitatory and inhibitory neurons, each receiving inputs from hundreds to thousands of other neurons in the network. In this setting, a balance between excitation and inhibition causes activity which is driven by fluctuations in the total input. Mean-field theories have employed this fact and approximated the input by effective Gaussians to explain activity and correlations on the level of neuronal populations. Here, we go beyond the population level and the approximation of Gaussian input by extending the applied mean-field theory to finite-size corrections that capture the experimentally observed first- and second-order statistics of correlations across neurons. Starting from randomly coupled Ising spins which constitute the classical model of collective phenomena in disordered systems, we derive equivalent networks of rate-based models, which can be analyzed using functional methods from statistical physics and field theory. In order to validate mean-field theoretical mappings between spiking and rate-based descriptions of neurons, and to foster the development of multi-scale models of brain activity combining both, we provide a simulation framework to efficiently integrate rate-based neuron dynamics in a spiking network simulator. Finally, we discuss a particular type of balanced network, the cortical microcircuit, and show how the microscopic dynamics for spontaneous and evoked activity are reflected in the LFP. Thereby we illustrate a generally applicable hybrid scheme for modeling local field potentials from spiking point-neuron networks which we derive and implement in the open-source software package hybridLFPy. In conclusion, this thesis combines mean-field theoretical and forward-modeling methods to build a bridge between microscopic and mesoscopic brain activity and derive model predictions for cortical dynamics that become increasingly accessible in experiments