Simplified Models of Individual Neurons

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Abstract
In the previous thirteen chapters, we met and described, sometimes in excruciating detail, the constitutive elements making up the neuronal hardware: dendrites, synapses, voltagedependent conductances, axons, spines and calcium. We saw how, different from electronic circuits in which only very few levels of organization exist, the nervous systems has many tightly interlocking levels of organization that codepend on each other in crucial ways. It is now time to put some of these elements together into a functioning whole, a single nerve cell. With such a single nerve cell model in hand, we can ask functional questions, such as: at what time scale does it operate, what sort of operations can it carry out, and how good is it at encoding information. We begin this Herculean task by (1) completely neglecting the dendritic tree and (2) replacing the conductance-based description of the spiking process (e.g., the Hodgkin- Huxley equations) by one of two canonical descriptions. These two steps dramatically reduce the complexity of the problem of characterizing the electrical behavior of neurons. Instead of having to solve coupled, nonlinear partial differential equations, we are left with a single ordinary differential equation. Such simplifications allow us to formally treat networks of large numbers of interconnected neurons, as exemplified in the neural network literature, and to simulate their dynamics. Understanding any complex system always entails choosing a level of description that retains key properties of the system while removing those nonessential for the purpose at hand. The study of brains is no exception to this. Numerous simplified single-cell models have been proposed over the years, yet most of them can be reduced to just one of two forms. These can be distinguished by the form of their output: spike or pulse models generate discrete, all-or-none impulses.

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