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1 Life Sciences Department and Zlotowski Center for Neurosciences, Ben-Gurion University of the Negev, Beer Sheva, Israel
2 Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, USA
3 Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ, USA
4 Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, USA; Department of Biological Sciences, Rutgers University, Newark, NJ, USA
* To whom correspondence should be addressed. E-mail: yairman{at}bgumail.bgu.ac.il.
In many rhythmic neuronal networks that operate in a wide range of frequencies, the time of neuronal firing relative to the cycle period (the phase) is invariant. This invariance suggests that, when frequency changes, firing time is precisely adjusted, either by intrinsic or synaptic mechanisms. We study the maintenance of phase in a computational model in which an oscillator neuron (O) inhibits a follower neuron (F) by comparing the dependency of phase on cycle period in two cases: when the inhibitory synapse is depressing and when it is non-depressing. Of the numerous ways of changing the cycle period, we focus on three cases where either the duration of the active state, the inactive state, or the duty cycle of neuron O remains constant. In each case, we measure the phase at which neuron F fires with respect to the onset of firing in neuron O. With a non-depressing synapse this phase is generally a monotonic function of cycle period, except in a small parameter range in the case of the constant inactive duration. In contrast, with a depressing synapse, there is always a parameter regime in which phase is a cubic function of cycle period: it decreases at short cycle periods, increases in an intermediate range and decreases at long cycle periods. This complex shape for the phase-period relationship arises because of the interaction between synaptic dynamics and intrinsic properties of the postsynaptic neuron. By choosing appropriate parameters, the cubic shape of the phase-period curve results in a small variation in phase for a large interval of periods. Consequently, we find that although a depressing synapse does not produce perfect phase maintenance, in most cases it is superior to a non-depressing synapse in promoting a constant phase difference.
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