Theory of Physiological Properties of Dendrites

In this paper, I propose to focus attention upon the branching dendritic trees that are characteristic of many neurons, and to consider the contribution such dendritic trees can be expected to make to the physiological properties of a whole neuron. More specifically, I shall present> a mathematical theory relevant to the question: How does a neuron integrate various distributions of synaptic excitation and inhibition delivered to its soma-dendritic surface. A mathematical theory of such integration is needed to help fill a gap that exists between the mathematical theory of nerve membrane properties, on the one hand, and the mathematical theory of nerve nets and of populations of interacting neurons, on the other hand. Before plunging into the mathematics, I assume that the nonneurophysiologists present would like a few introductory remarks about the relation between neuronal structure and function. Three characteristic types of mammalian neurons are illustrated in FIGURE 1. It is a remarkable fact, established by the research of neuroanatomists and neurophysiologists, that, in spite of such complicated geometry the neuron is a single cell; we assume it to be bounded by one continuous membrane. Nevertheless, it is often convenient to make a distinction between the somadendritic portion and the axonal portion of the cell. The function of the soma-dendritic portion is to receive and to integrate the excitation and inhibition that is delivered to it by other cells. When the resultant excitation of this soma-dendritic system exceeds the threshold condition for the initiation of a nerve impulse, it is the function of the axon to provide reliable propagation of this nerve impulse to the distant endings of this axon. In the ease of the axon, no one seriously questions that its extended length, its all-or-nothing impulse, and such specializations as a nodally interrupted myelin sheath are all well suited to the function of information transmission. In the case of the soma-dendritic system, however, the relation between structure and function is less well established; (see however, Bishop, 1958; Bullock, 1959). In spite of the fact that such extensively branched dendritic trees provide a very large and well-distributed receptive surface area, many neurophysiologists and mathematical biologists have tended to neglect the importance of the dendrites to the receptive and integrative functions of the neuron. One reason for this was that the density of synaptic connections was said to be much less over the dendritic membrane surface than over the soma membrane surface. Furthermore, it was said that any excitatory disturbance delivered to the peripheral portion of the dendritic membrane surface would be dissipated in the process of spread-