Neuronal networks quantified as vector fields

The function of the brain is defined by the interactions between its neurons. But these neurons exist in tremendous numbers, are continuously active, and densely interconnected. Thereby, they form one of the most complex dynamical systems known and there is a lack of approaches to characterize the functional properties of such biological neuronal networks without resorting to dimensionality reduction methods. Here, we introduce an approach to describe these functional properties by using its activity-defining constituents, the weights of the synaptic connections and the current activity of its neurons. We show how a high-dimensional vector field, which describes how the activity distribution across the neuron population is impacted at each instant of time, naturally emerges from these constituents. We show why a mixture of excitatory and inhibitory neurons and a diversity of synaptic weights are critical to obtain a network vector field with a structural richness. We argue that this structural richness can be the foundation of achieving the diverse, dynamic activity patterns across the neuron population observed in recordings in vivo and thereby an underpinning of the behavioral flexibility and adaptability that characterizes biological creatures.