Class 1 Neural Excitability, Conventional Synapses, Weakly Connected Networks, and Mathematical Foundations of Pulse-Coupled Models

IEEE Transactions On Neural Networks (1999), 10:499-507

Eugene M. Izhikevich

Systems Science Center, Box 7606,
Arizona State University,
Tempe, AZ 85287-7606.

Abstract. Many scientists believe all pulse-coupled neural networks are toy models that are far away from the biological reality. We show here, however, that a huge class of biophysically detailed and biologically plausible neural network models can be transformed into a pulse-coupled form by a piece-wise continuous, possibly non-invertible, 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 axo-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.

Keywords: Class 1 neural excitability, saddle-node on limit cycle bifurcation, weakly connected neural networks, conventional synapses, canonical model, integrate-and-fire, desynchronization

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