Mathematical Analysis of Neural Dynamics with Multiple Frequencies

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The nervous system produces electrical activity in all cognitive states, and this activity generally displays significant power at any given time in multiple frequency bands. This project focuses on the mathematical issues that have been raised by previous large-scale simulations, and seeks to get a deeper understanding of how the qualitative properties of intrinsic and ionic currents in single cells shape the complex behaviour that has been seen in a variety of simulations of large and small networks. The work focuses on two specific situations which present many of the general issues. The first is the interaction of the gamma (40-90 Hz) and theta (4-12 Hz) rhythms in the hippocampus. As shown in previous numerical and experimental work, the gamma and theta rhythms appear to be produced in vitro by different sub-networks of hippocampal neurons, with some components in common; simulations have shown that changes in parameters can switch control of the common elements and change the power in the different frequency bands. The project considers the global bifurcations involved in switches of control. The second situation concerns changes of brain rhythms in the presence of the anesthetic propofol which, at the biophysical level, acts mainly by increasing the decay time and amplitude of GABA_A mediated inhibition. A central question of the previous modeling work is the origin of the so-call "beta buzz", in which a low dose of propofol excites, rather than sedates, the patient, with an increase in the power in the beta frequency bands (13-30 Hz) and a decrease in lower and higher frequency bands. Simulations have shown this un-intuitive behaviour in model networks having multiple components, notably by the creation of "clustering" of inhibitory cells into subgroups firing in antiphase, transforming low frequencies into higher ones. Kopell mentors many graduate students and postdoctoral fellows; this project ties mathematical analysis to other work focusing on function, and thus allows trainees to see how mathematics can be used to bridge from biophysics to function.The nervous system produces electrical activity in all cognitive states, and this activity generally displays significant power at any given time in multiple frequency bands. This project focuses on the mathematical issues that have been raised by previous large-scale simulations, and seeks to get a deeper understanding of how the qualitative properties of intrinsic and ionic currents in single cells shape the complex behaviour that has been seen in a variety of simulations of large and small networks. The work addresses the general issues of how the brain produces its multiple frequencies, and how changes in biophysics can change the mixture of dynamic components. In the first sub-project, this is linked with the central question of how networks react to inputs with spatial and temporal structure reflecting the coding of information. In the second, the work helps to bridge the knowledge of biophysical effects of an anesthetic to its functional properties (inducing loss of consciousness) by tying the biophysical changes to changes in dynamics known to be related to cognitive state. The analytical tools proposed have been used in much simpler contexts. The work will require the development of extensions to allow application to larger and more complex networks. The extensions should be applicable to a wide range of neural applications. Kopell is co-Director of the Center for BioDynamics and the Program in Mathematical and Computational Neuroscience at Boston University. In this context, she works with and mentors a large number of graduate students and postdocs, including many women. This project ties mathematical analysis to other work focusing on function, and thus allows these and other trainees to see how mathematics can be used to bridge from biophysics to function.