Mean-field theory for complex neuronal networks

Understanding the working principles of the brain constitutes the major challenge in computational neuroscience. Reaching this goal is particularly difficult since the brain displays complex dynamics on various spatial and temporal scales, emerging from the interaction between a tremendous number of individual components, the neurons. Therefore a crucial step is to understand how collective dynamics in the brain arises from single neurons dynamics and the connectivity between the neurons. To better understand the collective dynamics in network of neurons, theoretical neuroscience has developed analytical tools, which reduce the complexity of the dynamics by averaging over spatial or temporal scales. In particular, this so-called mean-field theory relates single neuron properties and connectivity to network activity, which allows for a systematic understanding of the emerging activity. In this thesis we advance mean-field theory both on the level of single neuron properties and on the level of network structure. One building block of mean-field theory is the transfer function characterizing how a modulation of the neuron’s input is transmitted to a modulation of its output. A Neuron in the brain receives signals from a large number of presynaptic neurons, effectively causing a noisy input. Realistic activation kinetics of synaptic transmission amounts to a low-pass filtering of this input. Formally, the mathematical analysis of the transfer function therefore leads to a difficult colored-noise problem. We develop a general method to reduce this colored-noise to a white-noise system by capturing the color of the noise by effective boundary conditions. We apply this formalism to one of the most used neuron models, revealing a novel analytical expression for its transfer function. We further show how these results contribute to the understanding of fluctuations in a network of model neurons. On the level of structural connectivity we investigate the impact of connections between different parts of the brain on the stability of the network activity. To this end, we extend mean-field theory to devise a method allowing us to shape the phase space of a large-scale model by systematically refining its experimentally obtained connectivity map. Fundamental constraints on the activity, i.e., prohibiting quiescence and requiring global stability, prove sufficient to obtain realistic activity. Thus, the method contributes to the data integration process by constraining the empirical connectivity map to a realization that is compatible with physiological experiments. Moreover, dynamical mean-field theory relates connectivity to network generated fluctuations in the activity. We extend dynamical mean-field theory to stochastic systems where the noise mimics the spike emission process of neurons. This allows us to derive the self-consistent statistics of the activity as well as the point where the neural network displays a transition to chaos. We find that noise suppresses chaotic fluctuations by a dynamic mechanism effectively reducing the sensitivity of the network to small perturbations. The quantitative validation of the various mean-field approaches requires simulations of two model classes, i.e., rate-based and spike-based model neurons. We develop a unified simulation framework supporting both models, which facilitates this validation and increases reliability