 
Summary: ELSEVIER Physica D 121 (1998) 367395
On the behavior of a neural oscillator electrically coupled
to a bistable element
NancyKopella,,, L.F. Abbott b, CristinaSotoTrevifioa
a Department of Mathematics, Boston University, Boston, MA 02215, USA
b Volen Centerfor Complex Systems, Brandeis University, Waltham, MA 02254, USA
Received 11 July 1997; received in revised form 28 January 1998; accepted 4 February 1998
Communicated by C.K.R.T. Jones
Abstract
We study the periodic solutions of a twocell network consisting of a relaxation oscillator and a bistable element. The aim
is to understand how the frequency and wave form of the network depend on the intrinsic properties of the cells and on the
strength of the coupling between them. The network equations constitute a fastslow system; we show that there are four
curves of saddlenode points of the fast system whose geometry in parameter space encodes information about the wave form
and frequency. These curves give information about the value of the variables at which transitions are made between high and
low voltage states for either of the elements, and how those transition points in phase space depend on the coupling strength.
Furthermore, we develop a new geometric method to construct the curves of saddlenodes from families of curves associated
with the equations for each of the two cells. The construction allows one to see how changes in either of the elements affects
the wave form of the network output. The analysis also shows that the network can produce unintuitive behavior. For example,
though electric coupling may keep the network pinned longer at a higher or lower voltage level than the uncoupled oscillator,
larger values of the coupling strength may be less effective at this pinning. © 1998 Elsevier Science B.V. All rights reserved.
