title: Dynamic behavior analysis of fractional-order Hindmarsh-Rose neuronal model.
abstract: Previous experimental work has shown that the firing rate of multiple time-scales of adaptation for single rat neocortical pyramidal neurons is consistent with fractional-order differentiation, and the fractional-order neuronal models depict the firing rate of neurons more verifiably than other models do. For this reason, the dynamic characteristics of the fractional-order Hindmarsh-Rose (HR) neuronal model were here investigated. The results showed several obvious differences in dynamic characteristic between the fractional-order HR neuronal model and an integer-ordered model. First, the fractional-order HR neuronal model displayed different firing modes (chaotic firing and periodic firing) as the fractional order changed when other parameters remained the same as in the integer-order model. However, only one firing mode is displayed in integer-order models with the same parameters. The fractional order is the key to determining the firing mode. Second, the Hopf bifurcation point of this fractional-order model, from the resting state to periodic firing, was found to be larger than that of the integer-order model. Third, for the state of periodically firing of fractional-order and integer-order HR neuron model, the firing frequency of the fractional-order neuronal model was greater than that of the integer-order model, and when the fractional order of the model decreased, the firing frequency increased.
authors.affiliation: College of Science, Air Force Engineering University, Xi'an, 710051 China ; The First Aeronautical Institute of Air Force, Xinyang, 464000 Henan China.
authors.forename: Dong
authors.initials: D
authors.lastname: Jun
authors.affiliation: College of Science, Air Force Engineering University, Xi'an, 710051 China ; School of Life Science and technology, Xi'an Jiao tong University, Xi'an, 710049 China.
authors.forename: Zhang
authors.initials: Z
authors.lastname: Guang-Jun
authors.affiliation: School of Aerospace, Xi'an Jiao tong University, Xi'an, 710049 China.
authors.forename: Xie
authors.initials: X
authors.lastname: Yong
authors.affiliation: College of Science, Air Force Engineering University, Xi'an, 710051 China.
authors.forename: Yao
authors.initials: Y
authors.lastname: Hong
authors.affiliation: School of Life Science and technology, Xi'an Jiao tong University, Xi'an, 710049 China.
authors.forename: Wang
authors.initials: W
authors.lastname: Jue
keywords: Fractional-order; HR model; Hopf bifurcation; Transition of firing mode