Class 1 neural excitability, conventional synapses, weakly connected networks, and mathematical foundations of pulse-coupled models

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Abstract
Many scientists believe that all pulse-coupled neural networks are toy models that are far away from the biological reality. We show, however, that a huge class of biophysically detailed and biologically plausible neural-network models can be transformed into a canonical pulse-coupled form by a piece-wise continuous, possibly noninvertible, change of variables. Such transformations exist when a network satisfies a number of conditions; e,g., it is weakly connected; the neurons are Class 1 excitable (i.e., they can generate action potentials with an arbitrary small frequency); and the synapses between neurons are conventional (i.e., axo-dendritic and axe-somatic). Thus, the difference between studying the pulse-coupled model and Hodgkin-Huxley-type neural networks is just a matter of a coordinate change. Therefore, any piece of information about the pulse-coupled model is valuable since it tells something about all weakly connected networks of Class 1 neurons. For example, we show that the pulse-coupled network of identical neurons does not synchronize in-phase. This confirms Ermentrout's (1996) result that weakly connected Class 1 neurons are difficult to synchronize, regardless of the equations that describe dynamics of each cell.