Physics of the extended neuron

We review recent work concerning the effects of dendritic structure on single neuron response and the dynamics of neural populations. We highlight a number of concepts and techniques from physics useful in studying the behaviour of the spatially extended neuron. First we show how the single neuron Green's function, which incorporates details concerning the geometry of the dendritic tree, can be determined using the theory of random walks. We then exploit the formal analogy between a neuron with dendritic structure and the tight-binding model of excitations on a disordered lattice to analyse various Dyson-like equations arising from the modelling of synaptic inputs and random synaptic background activity. Finally, we formulate the dynamics of interacting populations of spatially extended neurons in terms of a set of Volterra integro-differential equations whose kernels are the single neuron Green's functions. Linear stability analysis and bifurcation theory are then used to investigate two particular aspects of population dynamics (i) pattern formation in a strongly coupled network of analog neurons and (ii) phase-synchronization in a weakly coupled network of integrate-and-fire neurons.