**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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