The cerebral cortex is the central processing area of the brain. Nearly all higher cognitive functions depend on the activity patterns of cortical neuron populations. These activity patterns are shaped by the structure of connectivity between cortical neurons and, conversely, cortical connectivity is shaped by cortical activity through changes in connection strengths called "synaptic plasticity." This project will develop novel computational models and mathematical analyses to understand the interplay between cortical connectivity and cortical activity, and how it gives rise to cognitive functions like sensory processing and motor learning. The results will lead to a better understanding of how the cortex functions and how it dysfunctions in disease states. The results will also inform the development of biologically-inspired machine learning algorithms. Graduate and undergraduate students will be involved in all aspects of the research and benefit from coursework inspired by the research projects. Students at a public high school will be guided in a project to incorporate learning algorithms from the research into a robot, providing hands on experience with applied mathematics, computer programming, and robotics.
The project comprises three sub-projects that build on each other. The first sub-project utilizes the notion of excitatory-inhibitory balance in cortical networks to derive scaling laws for connection strengths at large network size. Previous theoretical work elucidates the computational utility of an inverse square root scaling law and recent experimental work provides biological evidence for it, but the mechanisms through which the law could emerge are unknown. Candidate biological mechanisms will be analyzed mathematically and evaluated in collaboration with an experimental neuroscientist. For the second project, the theory of neuronal networks with excitatory and inhibitory balance will be refined and extended to account for the imprecision of balance observed in experiments. Preliminary calculations suggest that this imprecision is mathematically necessary, produces biologically realistic activity, and invokes intricate response properties that have implications for artificial neural networks used for image analysis. The third project will develop and analyze a mathematical model of motor learning that combines Hebbian plasticity in a cortical pathway with reward modulated plasticity in a sub-cortical pathway, inspired by experimental observations. This two-pathway motor learning algorithm has implications for biological motor learning, motor disease, and robotics.
