Network dynamics of adaptive spiking neurons: model reduction and numerical analysis

The brain is a complex dynamical system. It can operate in many different states of neuronal network activity and switch between them, depending on cognitive demands. To better understand functionally relevant network states and pathological activity it is beneficial to employ computational models for predictions and mechanistic analyses. In this thesis, I consider a biophysical neuronal network model which is specified by parameters that can be related to experimental data. For characterizing the stochastic network dynamics I use and extend methods mainly from statistical physics and numerical mathematics. Particularly, I derive a mean-field description in terms of the Fokker-Planck equation for a large recurrent network of adaptive exponential integrate-and-fire neurons that receive fluctuating input. The derivation is based on the diffusion approximation of the synaptic input and the separation of fast (membrane voltage) and slow (adaptation current) neuronal time scale. The resulting mean-field system allows to obtain the dynamics of the spike rate through solving a partial differential equation. It is extended in a self-consistent way to yield in addition to the spike rate also the second and higher order spiking statistics. I further derive – based on a spectral decomposition of the Fokker-Planck operator and alternatively on a linear-nonlinear cascade ansatz – efficient spike rate models that describe the network activity with only a few ordinary differential equations. These models can be rapidly integrated and allow for convenient mathematical investigations of asynchronous, oscillatory or chaotic network states because linear stability analyses and powerful related methods are readily applicable. To efficiently solve the mean-field system and the derived simpler models I provide novel numerical methods implemented in the Python programming language: (1) a simulation scheme for the FokkerPlanck model using a finite volume discretization, (2) an algorithm for solving the eigenvalue problem associated with the Fokker-Planck operator, and (3) the time integration of the low-dimensional spike rate models. The source code is available as open source software under a free license on GitHub. Applying the Fokker-Planck model I analyze how neuronal properties, particularly adaptation currents, affect the spike rate dynamics of recurrent networks and the spiking statistics of single neurons. With this approach and by exploration of the relevant parameter space I find that the type of the adaptation current has a distinct role in regulating the activity state of a recurrent network and in determining gain, threshold and variability of neuronal spiking. A spike-driven adaptation current is capable of generating spike rate oscillations, and reducing gain as well as spiking irregularity. A subthreshold adaptation mechanism on the other hand destabilizes such rhythms leading to asynchronous states, and increases the spiking threshold and its variability. This thesis demonstrates the benefits of deriving population activity descriptions using bottom-up approaches and contributes concrete models and corresponding efficient numerics, advancing the methodology of the theoretical neuroscience community.