Topics in Mathematical Neuroscience

Jonathan Rubin
Department of Mathematics, University of Pittsburgh

(September 11, 2012 10:20 AM - 11:15 AM)

Topics in Mathematical Neuroscience

Abstract

Professor Jon Rubin (University of Pittsburgh)

Topics in Mathematical Neuroscience

The communication among neurons that underlies mental function is inherently a multiple timescale process. Perhaps as a result of this property, the study of neural systems has attracted theorists interested in multi-timescale dynamics and has driven the development of new mathematics for understanding multi-timescale phenomena. In these lectures, I will discuss the basic ideas of fast-slow decomposition and their application, with varying degrees of sophistication, in the study both of small networks of model neurons and of complicated dynamics in single neuron models. In the single neuron case, the dynamics to be considered include activity patterns known as bursting oscillations and mixed-mode oscillations. While the applications to be discussed are neural, these forms of activity are known to arise in cells in other physiological systems and likely have general relevance across settings where evolution on multiple timescales occurs. I will also discuss how simple ideas of fast-slow decomposition have led to a theory about pathological activity in Parkinson's disease and how deep brain stimulation therapy can help alleviate parkinsonian pathology, along with some other /recent ideas on these topics.

 

Professor Steven Schiff (Penn State)

Topics in Mathematical Neuroscience

Observer models in engineering and control theory have long been used in robotics and numerical weather prediction. But they have had far less applicability to biology. As our control frameworks have improved to be capable of handling the nonlinearities inherent in biological systems, our knowledge of biology embedded within our evolving mathematical models of biology have similarly improved. I will detail an emerging intersection between computational biology and control theory which is now emerging in a series of four lectures:

  1. The principles of nonlinear ensemble Kalman filters and their application to models of disease including epilepsy, Parkinson's disease, and migraines.
  2. The role of symmetry in observability, controllability, and data assimilation in neuroscience.
  3. The application of particle filters to the unsolved problem of edge finding, with application to the growth of the brain, hydrocephalus, and the symmetries of epilepsy.
  4. The unsolved problem of observability of the neonatal sepsitisome, and the influence of climate dynamics in driving infant infections of the brain in East Africa.