Linear Cable Theory

Abstract

In the previous chapter, we briefly met some of the key actors of this book. In particular, we introduced the RC model of a patch of neuronal membrane and showed an instance where such a “trivial” model accounts reasonably well for the input-output properties of a neuron, as measured at its cell body. However, almost none of the excitatory synapses are made onto the cell body, contacting instead the very extensive dendritic arbor. As we will discuss in detail in Chap. 3, dendritic trees can be quite large, containing up to 98% of the entire neuronal surface area. We therefore need to understand the behavior of these extended systems having a cablelike structure. The basic equation governing the dynamics of the membrane potential in thin and elongated neuronal processes, such as axons or dendrites, is the cable equation. It originated in the middle of the last century in the context of calculations carried out by Lord Kelvin, who described the spread of potential along the submarine telegraph cable linking Great Britain and America. Around the turn of the century, Herman and others formulated the concept of Kemleitermodel, or core conductor model, to understand the flow of current in nerve axons. Such a core conductor can be visualized as a thin membrane or sheath surrounding a cylindrical and electrically conducting core of constant cross section placed in a solution of electrolytes. The study of the partial differential equations describing the evolution of the electrical potential in these structures gave rise to a body of theoretical knowledge termed cable theory. In the 1930s and 1940s concepts from cable theory were being applied to axonal fibers, in particular to the giant axon of the squid (Hodgkin and Rushton, 1946; Davis and Lorente de No, 1947). The application of cable theory to passive, spatially extended dendrites started in the late 1950s and blossomed in the 1960s and 1970s, primarily due to the work of Rail (1989). In an appropriate gesture acknowledging his role in the genesis of quantitative modeling of single neurons, Segev, Rinzel, and Shepherd (1995) edited an annotated collection of his papers, to which we refer the interested reader. It also contains personal recollections from many of Rail's colleagues as well as historical accounts of the early history of this field.