MATHEMATICAL MODELS IN THE BEHAVIOR OF THE CENTRAL NERVOUS SYSTEM BY H. D. LANDAHL It has been suggested that it would be appropriate to discuss some of the applications of mathematics to problems of the central nervous system. To do this we shall select a number of different problems some of which have been approached in several ways. In order to cover a number of topics, it will be necessary to limit the background information to that required to show how a particular model is formulated mathematically. It will only be possible to indicate the solution required and point out what questions can be answered by the model, what success is achieved in accounting for known facts, and what insight might be gained by the use of the model. The type of mathematics applied will, of course, depend on the type of pheno- mena which is being considered. There are a number of phenomena which suggest "field" properties. Some aspects of these might well be considered in terms of special formulations in which these properties determine the mathematical approach. On the other hand, many phenomena are more naturally considered in terms of networks of neural elements which have properties approximating those of neurons which have been studied in relative isolation. Even with relatively simple networks, fairly complex behavior results. Hence this latter approach has usually been followed. Furthermore, certainfieldproperties arise rather naturally. When an electrical impulse is started in a neuron, A, it travels along the axon of the neuron until it reaches a synaptic region where the axon terminates on a second neuron, B. The impulse dies out here but initiates changes in neuron B. If these changes are fast enough, they may leave no residual excitation when a second impulse comes down the axon of A. In this case, it may be impossible to have temporal summation so that if thefirstimpulse could not start an impulse in neuron B, two or more in succession would be inadequate also. But in this case, if a second impulse arives from another neuron A' at the same time as does the impulse from A, then neuron B may "fire," that is, initiate an impulse. But it will only initiate a single impulse, and that, only if the impulses from A and A' arrive within a sufficiently short time. On the other hand, if the processes in neuron B have a more prolonged effect, then the summation time may be longer and more than one impulse can be generated by a single input. It is clear that these two cases can be treated differently (Landahl, 1960). Consider the first of these situations and add the assumption that time can be quantized in units of the characteristic time of the excitation process. For convenience also let the impulses from A, A' have equal effect on B. Let the threshold (0) of B be the number of simultaneously impinging impulses required l
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