The pre-Bötzinger complex located at the ventrolateral medulla in the brainstem is believed to have an important role in generating the respiratory rhythm in mammals, specially the inspiratory process [56]. Keeping this in mind, we will study a small network of such cells by means of a minimal model suggested and experimentally tested by Butera et al [6, 7]. A thorough analysis of the Butera model was done for two very small networks of pre-Bötzinger cells: a self coupled single cell and a network of two coupled cells [5]. In order to understand the role of coupling and heterogeneity in these two particular networks we reduce the self coupled single cell network to a one dimensional map using a similar approach as in [37]. Using this one dimensional map, some analytical conditions for switching from one regime to another are determined and numerical results are shown. Using the same idea as for the self coupled single cell case, two identical coupled cells are reduced to a two dimensional iterated map which is a composition of many one dimensional maps. Using the form of these maps, mechanisms for the transition between previously observed regimes [5] are determined and linear analysis is performed for a particular set of parameters.

Introducing heterogeneity on the network of two coupled identical cells, for a fixed level of synaptic input, shows that depending on the level of the synaptic input some different behaviors arise which were not previously observed in a network of homogenous cells [5]. These results suggest that introducing heterogeneity can increase the range in the parameter space for which cells are bursting. This is desirable, since from experiments it is observed that bursting is associated with the inspiratory rhythm of respiration.