In this chapter we have investigated dynamic properties of spatially structured networks with local interactions by means of two complementatory approaches. The first approach relies on a mean-field approximation that describes neuronal activity in terms of an averaged membrane potential and the corresponding mean firing rate; cf. Eq. (6.129). The second approach is directly related to the microscopic dynamics of the neurons. Equations for the firing time of single action potentials can be solved exactly in the framework of the Spike Response Model. In both cases, most of the observed phenomena - traveling waves, periodic wave trains, and rotating spirals - are very similar to those of chemical reaction-diffusion systems, though the underlying mathematical equations are rather different.

The last section deals with the reliable transmission of temporal information in a hierarchically organized network. We have seen that sharp packets of spike activity can propagate from one layer to the next without changing their shape. This effect, which is closely related to Abeles' synfire chain, has been analyzed in the framework of the population equation derived in Section 6.3.

There are several books that provide an in-depth introduction to wave propagation in excitable media, see, e.g., the standard reference for mathematical biology by Murray (1993) or the more recent book by Keener and Sneyd (1998). A text that is apart from the mathematics also very interesting from an asthetic point of view is the book by (Meinhardt, 1995) on pattern formation in sea shells. An interesting relation between pattern formation in neuronal networks and visual haluzination patterns can be found in a paper by Ermentrout and Cowan (1979). The standard reference for information transmission with packets of precisely timed spikes is the book of Abeles (1991).

Cambridge University Press, 2002

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