Professor Guckenheimer studies dynamical models of a small neural system, the stomatogastric ganglion of crustaceans, in collaboration with the laboratory of Ronald Harris-Warrick. The STG is a central pattern generator, a group of neurons that control movement. Through this research, there is a hope to learn more about neuromodulation, the ways in which the rhythmic output of the STG is modified by chemical and electrical inputs. The models that are investigated are systems of differential equations that describe the currents contributing to the membrane potential of each neuron in the network. The interaction of these currents is complex , making it difficult to predict what the effects of varying the conductances or other properties of the currents will be on the oscillations of an individual cell or the entire network.
Guckenheimer, J. 1979. Sensitive dependence to initial conditions for one dimensional maps. Communications in Mathematical Physics 70:133-60.
Guckenheimer, J., and P. Holmes. 1983. Nonlinear oscillations, dynamical systems and bifurcations of vector fields. New York: Springer-Verlag.
Armbruster, D., J. Guckenheimer, and P. Holmes. 1988. Heteroclinic cycles and modulated travelling waves in systems with 0(2) symmetry. Physica D 24:257-82.
Guckenheimer, J., and S. Johnson. 1990. Distortion of S-unimodal maps. Annals of Mathematics 132:71-130.
Guckenheimer, J., S. Gueron and R. Harris-Warrick. 1993. The dynamics of a conditionally bursting neuron. Philosophical Transactions of the Royal Society of London B 341:345-59.
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