Abstract.
We discuss a method by which the dynamics of a network of neurons, coupled by mutual inhibition, can be reduced to a one-dimensional map. This network consists of a pair of neurons, one of which is an endogenous burster, and the other excitable but not bursting in the absence of phasic input. The latter cell has more than one slow process. The reduction uses the standard separation of slow/fast processes; it also uses information about how the dynamics on the slow manifold evolve after a finite amount of slow time. From this reduction we obtain a one-dimensional map dependent on the parameters of the original biophysical equations. In some parameter regimes, one can deduce that the original equations have solutions in which the active phase of the originally excitable cell is constant from burst to burst, while in other parameter regimes it is not. The existence or absence of this kind of regulation corresponds to qualitatively different dynamics in the one-dimensional map. The computations associated with the reduction and the analysis of the dynamics includes the use of coordinates that parameterize by time along trajectories, and “singular Poincaré maps” that combine information about flows along a slow manifold with information about jumps between branches of the slow manifold.
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Received: 19 May 1997 / Revised version: 6 April 1998
Rights and permissions
About this article
Cite this article
LoFaro, T., Kopell, N. Timing regulation in a network reduced from voltage-gated equations to a one-dimensional map. J Math Biol 38, 479–533 (1999). https://doi.org/10.1007/s002850050157
Issue Date:
DOI: https://doi.org/10.1007/s002850050157