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 10^{5} 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 cm^{2}, but its thickness amounts on average to only
2.5-3.0 mm^{2}. 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
*x* - *y*. 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.

- 9.1 Stationary patterns of neuronal activity
- 9.1.1 Homogeneous solutions
- 9.1.2 Stability of homogeneous states
- 9.1.3 `Blobs' of activity: inhomogeneous states

- 9.2 Dynamic patterns of neuronal activity

- 9.3 Patterns of spike activity

- 9.4 Robust transmission of temporal information
- 9.5 Summary

Cambridge University Press, 2002

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