h1

%0 Thesis
%A Tiberi, Lorenzo
%T The role of nonlinear interactions and connectivity in shaping critical and collective network dynamics
%I RWTH Aachen University
%V Dissertation
%C Aachen
%M RWTH-2023-09969
%P 1 Online-Ressource : Illustrationen
%D 2023
%Z Veröffentlicht auf dem Publikationsserver der RWTH Aachen University
%Z Dissertation, RWTH Aachen University, 2023
%X Neural computation is a collective phenomenon emerging from the complex interaction of a large number of neurons. Through evolution and learning, neural networks organize their structure, giving rise to rich collective dynamics that can support complex computational tasks. Some observables of these dynamics indeed appear to be optimally tuned for computation. Examples include the dimensionality and pricipal components' spectrum of neuronal activity, or the ubiquitous presence of observables following a power-law scaling, which suggests that the brain might tune itself into a critical state. Understanding which structural properties of neural networks allow the brain to optimally tune these dynamical observables is till an open question. Furthermore, it is not fully clear, in the very first place, what is the computational benefit of some of the brain's observed behaviors, such as criticality. In this thesis, we address these questions by characterizing the dynamics of two biologically inspired neural network models, the stochastic Wilson-Cowan model and the stochastic Sompolinsky-Crisanti-Sommers (SCS) model. To this end, we adapt tools from a field that has already seen great success in understanding collective phenomena - statistical physics. Seeking analytical understanding and interpretability, we focus on the two minimal and essential ingredients of neural computation: nonlinearities and connectivity. First, we focus on nonlinearities in the Wilson-Cowan model. We perform the first renormalization group (RG) analysis of a neural network model. By deriving the so-called flow of couplings, we are able to explore the computational implications of a fundamental property of critical systems: the presence of nonlinear interactions across all length-scales. This property has been so far inaccessible by previous studies, due to the use of mean-field approximations. We find nonlinearities to be in a Gell-Mann-Low regime: despite vanishing at very large length scales, they do so logarithmically slowly, thus remaining present on practically all intermediate scales. We argue this regime to be optimal for computation, striking a balance between linearity, optimal for memory, and nonlinearity, required for computation. Second, we focus on connectivity in the linearized SCS model. We ask which connectivity structures can directly control the network's dynamical observables, tuning them into the computationally optimal values observed experimentally. We develop a novel theory for random connectivity matrices, which shows that these structures are encoded in the shape of the connectivity's eigenvalue distribution. In particular, the density of nearly critical eigenvalues controls the power-law scaling of many dynamical observable, such as the autocorrelation, autoresponse, dimensionality, and principal components spectrum of neuronal activity. Differently than more traditional connectivity structures, such as motifs, these novel structures can account for phenomena such as a fine-tuned power-law scaling of the principal components spectrum, as observed in V1 of mice. Third, we focus on both nonlinearities and connectivity in the SCS model. We work on importing the RG analysis of nonlinear interactions to networks, especially in the general case of asymmetric heterogeneous networks, which is of most interest to neuroscience. This is - and still remains - an open problem, for which, however, we provide some novel steps forward. In the very first place, we make it formally possible to apply the RG methods to generic networks, by noticing an analogy with classical critical phenomena. Then, we identify new technical challenges specific to the asymmetric, heterogeneous case and propose formal methods to solve them, which rely on our newly developed random matrix theory. We also identify a novel mechanism, which causes a breakdown of the typical power-law scaling observed in classical critical phenomena.
%F PUB:(DE-HGF)11
%9 Dissertation / PhD Thesis
%R 10.18154/RWTH-2023-09969
%U https://publications.rwth-aachen.de/record/972001