In biochemical and gene networks, there is an important distinction between intrinsic and extrinsic noise; extrinsic noise refers to external sources of randomness associated with environmental factors, whereas intrinsic noise refers to random fluctuations arising from the discrete and probabilistic nature of chemical reactions at the molecular level, which are particularly significant when the number of reacting molecules is small. The main goal of this research project is to develop a mathematical theory of intrinsic and extrinsic noise in neuronal population dynamics, adapting analytical methods from the study of chemical master equations such as Langevin approximations, stochastic hybrid systems, and large deviation theory. It is assumed that intrinsic noise at the network level arises from fluctuations about an asynchronous state due to finite size effects, whereas extrinsic noise arises from fluctuating external inputs. The theory is applied to a variety of neurobiological phenomena where noise is thought to play a crucial role, including the stimulus-induced synchronization of neural oscillators during sensory processing, and the generation of oscillations and waves during binocular rivalry. The latter forms the basis for non-invasive studies of human vision.
Noise has recently emerged as a key component of many biological systems including the brain. Randomness arises at multiple levels of brain function, ranging from molecular processes such as gene expression and the opening of ion channel proteins to complex networks of brain cells (neurons) that generate behavior. Indeed, the presence of noise has direct behavioral consequences, from setting perceptual and decision thresholds to influencing movement precision. Noise also contributes to the generation of spontaneous activity patterns during resting brain states, which are thought to play an important role in cognition. From one perspective, neuroscientists are interested in how, in spite of significant levels of noise, the brain appears to function reliably, consistent with the idea that it has evolved under the constraints that are imposed by noise. From another perspective, neuroscientists are interested in situations where the presence of noise can either be harmful to or, in certain cases, actually enhance brain function. The main goal of this project is to use mathematical and computational modeling to develop our understanding of how noise that is present at the molecular and cellular levels affects dynamics and information processing at the network level, both in healthy and diseased brains.
Numerous behaviors ranging from locomotion to cognitive tasks rely on oscillatory activity generated by networks of neurons in the brain. Despite the predominance and indispensability of brain oscillations, few theoretical tools are available for understanding how such oscillations are generated or controlled. A novel approach is used that combines biological experiments and mathematical analysis to break apart the complex interactions present in network components into simple building blocks. This allows core elements that are important in the generation of oscillations to be extracted and will clarify the role of other existing components in sculpting behavior using mathematical models. The models are tested through experiments that connect real time computer-simulated neurons to small oscillatory network in the crab central nervous systems. This project provides a framework for developing neural-based control systems with potential applications in robotics and bio-inspired computing.