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We find that the Hindmarsh-Rose model neuron demonstrates different types of firing behavior with the change in the fractional order by using the fractional-order Hindmarsh-Rose model in this study.There exists a clear difference in the bifurcation values between the fractional-order model and the corresponding integer-order model, where the neuron starts a period-doubling bifurcation cascade route to chaos when the externally applied current is considered to be the bifurcation parameter.And then we show that the complexity of firing behavior of the fractional-order model is greater than that of the integer-order model.Finally, we also find that the firing frequency of the fractional-order model neuron is larger than that of the integer-order counterpart irrespective of whether the neuron exhibits periodic firing or chaos.