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Dynamics from a Time Series: Can We Extract the Phase Resetting Curve from a Time Series?

S. A. Oprisan ^*, V. Thirumalai  and C. C. Canavier ^*

^* Department of Psychology, University of New Orleans, New Orleans, Louisiana; and Volen Center for Complex Systems, Brandeis University, Waltham, Massachusetts

Correspondence: Address reprint requests to Carmen C. Canavier, University of New Orleans, Dept. of Psychology, 2000 Lakeshore Dr., GP 2001, New Orleans, LA 70148. Tel.: 504-280-6775; Fax: 504-280-6049; E-mail: ccanavie{at}uno.edu.

Recordings of the membrane potential from a bursting neuron were used to reconstruct the phase curve for that neuron for a limited set of perturbations. These perturbations were inhibitory synaptic conductance pulses able to shift the membrane potential below the most hyperpolarized level attained in the free running mode. The extraction of the phase resetting curve from such a one-dimensional time series requires reconstruction of the periodic activity in the form of a limit cycle attractor. Resetting was found to have two components. In the first component, if the pulse was applied during a burst, the burst was truncated, and the time until the next burst was shortened in a manner predicted by movement normal to the limit cycle. By movement normal to the limit cycle, we mean a switch between two well-defined solution branches of a relaxation-like oscillator in a hysteretic manner enabled by the existence of a singular dominant slow process (variable). In the second component, the onset of the burst was delayed until the end of the hyperpolarizing pulse. Thus, for the pulse amplitudes we studied, resetting was independent of amplitude but increased linearly with pulse duration. The predicted and the experimental phase resetting curves for a pyloric dilator neuron show satisfactory agreement. The method was applied to only one pulse per cycle, but our results suggest it could easily be generalized to accommodate multiple inputs.

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