Dynamical Mean Field Theory and Applications to Ecological and Neural Systems

In this Master Thesis, we explore the intricate applications of Dynamical Mean Field Theory (DMFT) in unraveling the complexities of ecological and neural systems. The study commences with an in-depth analysis of the DMFT framework, emphasizing its utility in simplifying complex interacting systems through dimensionality reduction and the Gaussian noise approximation. This theoretical groundwork is pivotal for understanding the subsequent applications of the theory. The thesis then progresses to applying DMFT to ecological systems, with a specific focus on the Generalized Lotka-Volterra (GLV) systems characterized by quenched disorder. A significant aspect of this exploration is the examination of how functional response theory influences the dynamics of these systems. This inquiry delves into the various response mechanisms within the GLV systems and their effect on system behavior and stability. Extending the scope of DMFT, the thesis also investigates its application in the field of neuroscience, particularly within recurrent neural networks. This segment of the study is centered on identifying the conditions that lead to the emergence of a reservoir of multiple timescales, akin to the behaviors observed in real cortical neural networks. This exploration provides valuable insights into the mechanisms that underlie neural processing and information storage. Lastly, the thesis incorporates an empirical analysis of datasets to correlate theoretical findings with real-world data. This crucial part of the study not only grounds the theoretical aspects in practical scenarios but also serves as a critical evaluation of the DMFT framework. By juxtaposing theoretical predictions with actual data, the thesis aims to validate the effectiveness of DMFT in practical applications and highlights areas that might require further development or refinement. Overall, this Master Thesis offers a significant contribution to the fields of theoretical physics, ecology, and neuroscience, presenting new perspectives and methodologies for understanding the dynamics of complex systems in an interconnected world.