Dynamics of spatially extended dendrites - Svensson, Carl-Magnus (2009)
Dendrites are the most visually striking parts of neurons. Even so many neuron models are of point type and have no representation of space. In this thesis we will look at a range of neuronal models with the common property that we always include spatially extended dendrites. First we generalise Abbott’s “sum-over-trips” framework to include resonant currents. We also look at piece-wise linear (PWL) models and extend them to incorporate spatial structure in the form of dendrites. We look at the analytical construction of orbits for PWL models. By using both analytical and numerical Lyapunov exponent methods we explore phase space and in particular we look at mode-locked solutions. We will then construct the phase response curve (PRC) for a PWL system with compartmentally modelled dendrites. This sets us up so we can look at the effect of multiple PWL systems that are weakly coupled through gap junctions. We also attach a continuous dendrite to a PWL soma and investigate how the position of the gap junction influences network properties. After this we will present a short overview of neuronal plasticity with a special focus on the spatial effects. We also discuss attenuation of distal synaptic input and how this can be countered by dendritic democracy as this will become an integral part of our learning mechanisms. We will examine a number of different learning approaches including the tempotron and spike-time dependent plasticity. Here we will consider Poisson’s equation around a neural membrane. The membrane we focus on has Hodgkin-Huxley dynamics so we can study action potential propagation on the membrane. We present the Green’s function for the case of a one-dimensional membrane in a two-dimensional space. This will allow us to examine the action potential initiation and propagation in a multi-dimensional axon.