Interaction function of coupled bursting neurons

The interaction functions of electrically coupled Hindmarsh–Rose（HR） neurons for different firing patterns are investigated in this paper.By applying the phase reduction technique,the phase response curve（PRC） of the spiking neuron and burst phase response curve（BPRC） of the bursting neuron are derived.Then the interaction function of two coupled neurons can be calculated numerically according to the PRC（or BPRC） and the voltage time course of the neurons.Results show that the BPRC is more and more complicated with the increase of the spike number within a burst,and the curve of the interaction function oscillates more and more frequently with it.However,two certain things are unchanged：Φ = 0,which corresponds to the in-phase synchronization state,is always the stable equilibrium,while the anti-phase synchronization state with Φ = 0.5 is an unstable equilibrium.

Phase-response synchronization in neuronal population

In this study,we have formulated the phase description of the neuronal oscillator with non-instantaneous synaptic inputs and external periodic stimulus by using the phase sensitivity function.By numerical simulation,we have found that the phase of a neuronal oscillator undergoes periodic evolution or locked state,which is determined by the synaptic time constant.The synaptic time constant is also an important condition under which the global network is synchronized.When the synaptic time constant is relatively small,perfectly synchronized behavior quickly occurs in the neuronal population.As the synaptic time constant becomes slightly larger,periodic synchronization emerges in the neuronal population.However,synchronized activity in the neuronal population is lost for larger synaptic time constant.The external periodic stimulus can change the synchronization patterns in the neuronal population.With a weak low-frequency stimulus,the neuronal populations quick synchronized bursting;whereas a high-frequency stimulus can produce synchronized overlapping bursting.We have also found that neuronal oscillators with type-II phase response curves are more susceptible to synchronization than those with type-I phase response curves.

Entrainment mechanism of the cyanobacterial circadian clock induced by oxidized quinone

The circadian clock is a self-sustained biological oscillator which can be entrained by environmental signals.The cyanobacteria circadian clock is the simplest one,which is composed of the proteins KaiA,KaiB and KaiC.The phosphorylation/dephosphorylation state of KaiC exhibits a circadian oscillator.KaiA and KaiB activate KaiC phosphorylation and dephosphorylation respectively.CikA competing with KaiA for the same binding site on KaiB affects the phosphorylation state of KaiC.Quinone is a signaling molecule for entraining the cyanobacterial circadian clock which is oxidized at the onset of darkness and reduced at the onset of light,reflecting the environmental light-dark cycle.KaiA and CikA can sense external signals by detecting the oxidation state of quinone.However,the entrainment mechanism is far from clear.We develop an enhanced mathematical model including oxidized quinone sensed by KaiA and CikA,with which we present a detailed study on the entrainment of the cyanobacteria circadian clock induced by quinone signals.We find that KaiA and CikA sensing oxidized quinone pulse are related to phase advance and delay,respectively.The time of oxidized quinone pulse addition plays a key role in the phase shifts.The combination of KaiA and CikA is beneficial to the generation of entrainment,and the increase of signal intensity reduces the entrainment phase.This study provides a theoretical reference for biological research and helps us understand the dynamical mechanisms of cyanobacteria circadian clock.