A model of the phase-sensitivity of the pacemaking cell in the bullfrog heart

A mathematical criterion based on phase response curves for stability in a ring of coupled oscillators

“…They established two criteria: one for determining when entrainment is possible based on PRCs, and the second for determining whether an entrained mode is stable. These criteria have been supported experimentally by a variety of studies (Moore et al 1963;Perkel et al 1964;Guevara et al 1981;Shumaker et al 1991;Wessel 1995).…”

“…This approximation ignores the possibilities that: (a) a stimulus to a given neuron will have some eect on the waveform with which that neuron ®res and (b) a stimulus may have a longer-lasting eect, such as lengthening or shortening periods other than the one during which it is applied. These eects are typically minor, however, compared with the modi®cation of the length of the current cycle of the postsynaptic neuron (Perkel et al 1964;Shumaker et al 1991). For example, in a study employing a computational model of the endogenously bursting neuron R15 of the abdominal ganglion of Aplysia, Demir et al (1997) showed that a neuron stimulated during the ®rst 3/4 of its period will return to its limit cycle by the beginning of the next burst; a neuron stimulated in the last quarter of its period will experience only a slight modi®cation in the waveform of the following burst and in the duration of the following period.…”

“…Recently, Canavier et al (1997) and DeFranceschi (1995) applied phase-sensitivity analysis to unidirectional ring networks. These two studies employed phase response curves (PRCs), which have been used extensively to study phase-locked entrainment of biological oscillators, including neurons, cardiac cells, and circadian oscillators Demir et al 1997;Glass and Mackey 1988;Guevara et al 1981;Moore et al 1963;Pavlidis 1973;Perkel et al 1964;Shrier et al 1990;Shumaker et al 1991;Wessel 1995;Winfree 1980). A unidirectional ring network operating in a steady-state mode may be viewed as an example of entrainment, wherein each neuron entrains the following one.…”