Statistical problems related to excitation threshold and reset value of membrane potentials [Elektronische Ressource] / von Patrick Jahn

Statistical Problems Related toExcitation Threshold and ResetValue of Membrane PotentialsDissertationzur Erlangung des Grades”Doktor der Naturwissenschaften”am Fachbereich Physik, Mathematik und Informatikder Johannes Gutenberg-Universit¨at MainzvonPatrick Jahngeboren in SimmernMainz, 03. Februar 2009Tag der mu¨ndlichen Pru¨fung: Montag, 30. M¨arz 2009(D77) Dissertation, Johannes Gutenberg-Universit¨at MainzPatrick JahnStatistical Problems Related to Excitation Thresholdand Reset Value of Membrane PotentialsAbstractThe present work is motivated by biological questions about the behaviorof membrane potentials in neurons. A commonly used model for spikingneurons is to assume that between spikes the membrane potential is givenby a diffusion process X = (X ) which is a solution of an SDEt t≥0dX =β(X )dt+σ(X )dBt t t twhere(B ) isastandardBrownianmotion. Thespikingbehaviorisusuallyt t≥0explained as follows. Whenever X reaches a certain excitation threshold S,a spike occurs. Thereafter the potential is set down to a certain reset valuex again.0In applications it is sometimes possible to observe the diffusion processX between spikes by estimating the drift coefficient β() and the diffusioncoefficientσ() from real data. Nevertheless, S andx have to be determined0in order to fix the model. However, in real data this is not obvious at all.One possible approach is to view x and S as parameters in a statistical0model and to estimate them.