Applications of free probability to the diversity of response and variability in neuronal networks

Abstract: Cognition is supported by neuronal activity that spans several brain regions, and understanding how this activity is coordinated is essential for a coherent theory of neuronal function. We will study how the structure of a visual categorization task in non-human primates influences the coordinated activity of three brain areas known to be essential for proper task performance. While neuronal connectivity does influence how brains perform tasks on average, a key signature of connectivity is how it also determines the fluctuations (or variability) of brain activity during tasks. We will model the trial-to-trial variability of distributed population responses with simple, yet nonlinear, recurrent circuit models. A key advance of our proposal is to consider both the input to a brain region and the wiring within and between brain regions to be randomly structured. This choice will require the use of novel analysis techniques based in free probability theory as applied to random matrices, which can untangle how two sources of randomness contribute to the variability of circuit response. Our theory will make testable predictions which can be explored in our experimental framework. In total, our work will provide a deep understanding of how diverse population brain circuitry supports the underlying mechanics of neuronal codes.