9. Spatially Structured Networks

So far the discussion of network behavior in Chapters 6 - 8 was restricted to homogeneous populations of neurons. In this chapter we turn to networks that have a spatial structure. In doing so we emphasize two characteristic features of the cerebral cortex, namely the high density of neurons and its virtually two-dimensional architecture.

Each cubic millimeter of cortical tissue contains about 10⁵ neurons. This impressive number suggests that a description of neuronal dynamics in terms of an averaged population activity is more appropriate than a description on the single-neuron level. Furthermore, the cerebral cortex is huge. More precisely, the unfolded cerebral cortex of humans covers a surface of 2200-2400 cm², but its thickness amounts on average to only 2.5-3.0 mm². If we do not look too closely, the cerebral cortex can hence be treated as a continuous two-dimensional sheet of neurons. Neurons will no longer be labeled by discrete indices but by continuous variables that give their spatial position on the sheet. The coupling of two neurons i and j is replaced by the average coupling strength between neurons at position x and those at position y, or, even more radically simplified, by the average coupling strength of two neurons being separated by the distance $\left\vert\vphantom{ x-y }\right.$x - y$\left.\vphantom{ x-y }\right\vert$. Similarly to the notion of an average coupling strength we will also introduce the average activity of neurons located at position x and describe the dynamics of the network in terms of these averaged quantities only. The details of how these average quantities are defined, are fairly involved and often disputable. In Sect. 9.1 we will - without a formal justification - introduce field equations for the spatial activity A(x, t) in a spatially extended, but otherwise homogeneous population of neurons. These field equations are particularly interesting because they have solutions in the form of complex stationary patterns of activity, traveling waves, and rotating spirals - a phenomenology that is closely related to pattern formation in certain nonlinear systems that are collectively termed excitable media. Some examples of these solutions are discussed in Sect. 9.1. In Sect. 9.2 we generalize the formalism so as to account for several distinct neuronal populations, such as those formed by excitatory and inhibitory neurons. The rest of this chapter is dedicated to models that describe neuronal activity in terms of individual action potentials. The propagation of spikes through a locally connected network of SRM neurons is considered in Section 9.3. The last section, finally, deals with the transmission of a sharp pulse packet of action potentials in a layered feed-forward structure. It turns out that there is a stable wave form of the packet so that temporal information can be faithfully transmitted through several brain areas despite the presence of noise.