Seminar by David Nelson

Non-Hermitian Localization in Biological Networks

David Nelson, Lyman Laboratory, Harvard University

In the 70 years since Phillip Anderson proposed his ideas about localized states in disordered systems, it has become clear that virtually all electronic states are localized in one-dimension. We show that the situation is quite different when the hopping matrix becomes non-Hermitian.   Non-Hermitian matrices, with complex eigenvalue spectra, arise naturally in simple models of complex ecosystems, with many interacting predator and prey species.  Recent work has revealed particularly striking departure from the conventional wisdom in the one-dimensional non-Hermitian random matrices that describe sparse neural networks.   Approximately equal numbers of random excitatory and inhibitory connections lead to  an intricate fractal eigenvalue spectrum that controls the spontaneous activity and induced response.   When rings of neurons become directed, with a systematic  bias for the transfer of excitations in the clockwise direction, an hole centered on the origin opens up in the density of states in the complex plane.   All states are extended on the rim of this hole, while the states outside the hole are localized.