A given neural network in the brain is involved in many different tasks. This implies that, when considering a specific task, the network's connectivity contains a component which is related to the task and another component which can be considered random. Understanding the interplay between the structured and random components, and their effect on network dynamics and functionality is an important open question. Recent studies addressed the co-existence of random and structured connectivity, but considered the two parts to be uncorrelated. This constraint limits the dynamics and leaves the random connectivity non-functional. Algorithms that train networks to perform specific tasks typically generate correlations between structure and random connectivity. Here we study nonlinear networks with correlated structured and random components, assuming the structure to have a low rank. We develop an analytic framework to establish the precise effect of the correlations on the eigenvalue spectrum of the joint connectivity. We find that the spectrum consists of a bulk and multiple outliers, whose location is predicted by our theory. Using mean-field theory, we show that these outliers directly determine both the fixed points of the system and their stability. Taken together, our analysis elucidates how correlations allow structured and random connectivity to synergistically extend the range of computations available to networks.

Recently, a theoretical framework aimed at separating the roles of dynamics and topology in multidimensional systems has been developed [Gao et al., Nature (London) 530, 307 (2016)10.1038/nature16948]. The validity of their method is assumed to hold depending on two main hypotheses: (i) The network determined by the the interaction between pairs of nodes has negligible degree correlations; (ii) the node activities are uniform across nodes on both the drift and the pairwise interaction functions. Moreover, the authors consider only positive (mutualistic) interactions. Here we show the conditions proposed by Gao and collaborators [Nature (London) 530, 307 (2016)10.1038/nature16948] are neither sufficient nor necessary to guarantee that their method works in general and validity of their results are not independent of the model chosen within the class of dynamics they considered. Indeed we find that a new condition poses effective limitations to their framework and we provide quantitative predictions of the quality of the one-dimensional collapse as a function of the properties of interaction networks and stable dynamics using results from random matrix theory. We also find that multidimensional reduction may work also for an interaction matrix with a mixture of positive and negative signs, opening up an application of the framework to food webs, neuronal networks, and social and economic interactions.

1Neurons in different layers of sensory cortex generally have different functional 2properties. But what determines firing rates and tuning properties of neurons in 3 different layers? Orientation selectivity in primary visual cortex (V1) is an inter-4 esting case to study these questions. Thalamic projections essentially determine 5 the preferred orientation of neurons that receive direct input. But how is this 6 tuning propagated though layers, and how can selective responses emerge in lay-7 ers that do not have direct access to the thalamus? Here we combine numerical 8 simulations with mathematical analyses to address this problem. We find that a 9 large-scale network, which just accounts for experimentally measured layer and cell-10 type specific connection probabilities, yields firing rates and orientation selectivities 11 matching electrophysiological recordings in rodent V1 surprisingly well. Further 12 analysis, however, is complicated by the fact that neuronal responses emerge in 13 a dynamic fashion and cannot be directly inferred from static neuroanatomy, as 14 some connections tend to have unintuitive effects due to recurrent interactions and 15 strong feedback loops. These emergent phenomena can be understood by lineariz- 16 ing and coarse-graining. In fact, we were able to derive a low-dimensional linear 17 dynamical system effectively describing stimulus-driven activity layer by layer. This 18 low-dimensional system explains layer-specific firing rates and orientation tuning by 19 accounting for the different gain factors of the aggregate system. Our theory can 20 also be used to design novel optogenetic stimulation experiments, thus facilitating 21 further exploration of the interplay between connectivity and function. 22 Author summary 23 Understanding the precise roles of neuronal sub-populations in shaping the ac-24 tivity of networks is a fundamental objective of neuroscience research. In complex 25 neuronal network structures like the neocortex, the relation between the connec-26 tome and the algorithm implemented in it is often not self-explaining. To this end, 27 our work makes three important contributions. First, we show that the connectivity 28 extracted by anatomical and physiological experiments in visual cortex suffices to 29 explain important properties of the various sub-populations, including their selec-30 tivity to visual stimulation. Second, we introduce a novel system-level approach 31 for the analysis of input-output relations of recurrent networks, which leads to the 32 observed activity patterns. Third, we present a method for the design of future 33 optogenetic experiments that can be used to devise specific stimuli resulting in a 34 predictable change of neuronal activity. In summary, we introduce a novel frame- 35 work to determine the relevant features of neuronal microcircuit function that can 36 be applied to a wide range of neuronal systems. 37 Understanding the complex computations performed by neural networks in the nervous 39 system is ...