Regimes and mechanisms of transient amplification in abstract and biological neural networks

Abstract

Neuronal networks encode information through patterns of activity that define the networks’ function. The neurons’ activity relies on specific connectivity structures, yet the link between structure and function is not fully understood. Here, we tackle this structure-function problem with a new conceptual approach. Instead of manipulating the connectivity directly, we focus on upper triangular matrices, which represent the network dynamics in a given orthonormal basis obtained by the Schur decomposition. This abstraction allows us to independently manipulate the eigenspectrum and feedforward structures of a connectivity matrix. Using this method, we describe a diverse repertoire of non-normal transient amplification, and to complement the analysis of the dynamical regimes, we quantify the geometry of output trajectories through the effective rank of both the eigenvector and the dynamics matrices. Counter-intuitively, we find that shrinking the eigenspectrum’s imaginary distribution leads to highly amplifying regimes in linear and long-lasting dynamics in nonlinear networks. We also find a trade-off between amplification and dimensionality of neuronal dynamics, i.e., trajectories in neuronal state-space. Networks that can amplify a large number of orthogonal initial conditions produce neuronal trajectories that lie in the same subspace of the neuronal state-space. Finally, we examine networks of excitatory and inhibitory neurons. We find that the strength of global inhibition is directly linked with the amplitude of amplification, such that weakening inhibitory weights also decreases amplification, and that the eigenspectrum’s imaginary distribution grows with an increase in the ratio between excitatory-to-inhibitory and excitatory-to-excitatory connectivity strengths. Consequently, the strength of global inhibition reveals itself as a strong signature for amplification and a potential control mechanism to switch dynamical regimes. Our results shed a light on how biological networks, i.e., networks constrained by Dale’s law, may be optimised for specific dynamical regimes. Author summary The architecture of neuronal networks lies at the heart of its dynamic behaviour, or in other words, the function of the system. However, the relationship between changes in the architecture and their effect on the dynamics, a structure-function problem, is still poorly understood. Here, we approach this problem by studying a rotated connectivity matrix that is easier to manipulate and interpret. We focus our analysis on a dynamical regime that arises from the biological property that neurons are usually not connected symmetrically, which may result in a non-normal connectivity matrix. Our techniques unveil distinct expressions of the dynamical regime of non-normal amplification. Moreover, we devise a way to analyse the geometry of the dynamics: we assign a single number to a network that quantifies how dissimilar its repertoire of behaviours can be. Finally, using our approach, we can close the loop back to the original neuronal architecture and find that biologically plausible networks use the strength of inhibition and excitatory-to-inhibitory connectivity strength to navigate the different dynamical regimes of non-normal amplification.