Eigenspectrum bounds for semirandom matrices with modular and spatial structure for neural networks

Abstract

The eigenvalue spectrum of the matrix of directed weights defining a neural network model is informative of several stability and dynamical properties of network activity. Existing results for eigenspectra of sparse asymmetric random matrices neglect spatial or other constraints in determining entries in these matrices, and so are of partial applicability to cortical-like architectures. Here we examine a parameterized class of networks that are defined by sparse connectivity, with connection weighting modulated by physical proximity (i.e., asymmetric Euclidean random matrices), modular network partitioning, and functional specificity within the excitatory population. We present a set of analytical constraints that apply to the eigenvalue spectra of associated weight matrices, highlighting the relationship between connectivity rules and classes of network dynamics.

Functional selectivity and specific connectivity of inhibitory neurons in primary visual cortex

“…x N ] T is a vector of neuron activations, I(t) is an external input provided to the network, τ is a lumped neuron time constant and [·] + is the linear-threshold activation function [x] + = max(x, 0). Under this formulation, the eigenspectrum of the weight matrix W provides information about the stability of various network activity patterns, as a function of the network connectivity parameters 23 . We examined the stability of the networks in the state where all neurons were active (i.e.…”

“…In the presence of uniform, non-specific inhibition, specific excitatory connectivity within subnetworks quickly led to network instability 23 e. An unstable network with strong E Spec and no I Spec. f. In a stable network with strong E Spec balanced by moderate I Spec, recruitment and competition lead to strong anticorrelated network activity.…”

“…Partitioned network model Partitioned networks were generated and analysed as described previously 23 . Parameters used to build the network models are defined in Table 2.…”

Assessing the role of inhibition in stabilizing neocortical networks requires large-scale perturbation of the inhibitory population

“…However, in mammalian cortex, approximately 20 % of neurons are inhibitory (Gabott and Somogyi, 1986). We therefore redefined our network following Muir and Mrsic-Flogel (2015), and set the proportion of inhibitory neurons in the network to 20 % while maintaining all-to-all non-specific connectivity. We numerically computed the proportion of the inhibitory population that must be stimulated to observe the paradoxical effect in the stimulated neurons ( Fig.…”

“…To examine the effect of sparse connectivity we expanded upon the work in Muir and Mrsic-Flogel (2015) by introducing connection sparsity parameters that describe the number of synaptic connections made between nearby neurons, as a proportion of all possible partners. We estimated separate sparsity parameters for recurrent excitatory, exc.…”