Mounting evidence in recent years suggests that astrocytes, a sub-type of glial cells, not only serve metabolic and structural support for neurons and synapses but also play critical roles in regulation of proper functioning of the nervous system. In this work, we investigate the effect of astrocyte on the spontaneous firing activity of a neuron through a combined model which includes a neuron-astrocyte pair. First, we show that an astrocyte may provide a kind of multistability in neuron dynamics by inducing different firing modes such as random and bursty spiking. Then, we identify the underlying mechanism of this behavior and search for the astrocytic factors that may have regulatory roles in different firing regimes. More specifically, we explore how an astrocyte can participate in occurrence and control of spontaneous irregular spiking activity of a neuron in random spiking mode. Additionally, we systematically investigate the bursty firing regime dynamics of the neuron under the variation of biophysical facts related to the intracellular environment of the astrocyte. It is found that an astrocyte coupled to a neuron can provide a control mechanism for both spontaneous firing irregularity and burst firing statistics, i.e., burst regularity and size.Astrocytes are the most numerous glial cells in the mature brain, and often surround neuronal somata and dendrites and provide fine enmeshment of synapses. Unlike neurons, these star-shaped cells do not elicit action potentials, yet they contribute to information processing via feedback to the cells by listening to the synaptic activity. We here investigate the their role on shaping the spontaneous firing behavior of a neuron. By using a neuron-astrocyte pair model, we identify distinct astrocyte-controlled neural activity patterns and explore their individual statistical characteristics under the variation of different biologicallyplausible intrinsic astrocytic factors.