Conceptualizing networks of neurons as vector fields to analyze their high-dimensional behavior

Observations that have received little attention indicate that complexity is important for flexibility and adaptability of natural behavior and the remarkably efficient learning of the brain. The term complexity implies meaningful structural richness, which exists in systems that offer a plurality of possible solutions. The existence of multiple solutions in the neuronal network enables the brain to interpret a diverse set of sensory inputs and can allow it to operate and respond flexibly across a wide array of contexts. Hence it is important to study factors that can enable network complexity. However, there is currently a lack of tools to quantify network complexity, where many popular dimensionality reduction methods are instead doing the opposite, i.e. eliminating data that could reflect network complexity. Therefore, here we develop a new tool to quantify the infrastructure of network complexity, using a vector field description as a starting point, and illustrate its application to cortical neuron population activity in the rat in vivo. We show that considering network properties on a vector field basis can be used to analyze the number of possible neuron population activity evolutions, or solutions, as they fall into one of the multiple attractor states. We developed a new tool of high-dimensional plane extraction to quantify the structure of the vector field across the entire neuron population, thus eliminating the previous problem of dimensionality reduction. This method hence allows an analysis of what impacts learnt synaptic weight changes have on high-dimensional network behavior, and therefore the complexity of general behavior.