• mGluR density
• synaptotagmin IV
• IP3 lifetime
• Fano factor
• modelling
• mixed modulation
• cusp
• bautin
• information processing
• calcium dynamics
• neuron-glia interactions
• tripartite synapse
• astrocytes

A vast amount of experimental evidence hints that astrocytes could be active players in information processing of the brain. It remains unclear nevertheless how these cells could encode synaptic stimuli through variations of their intracellular Ca2+ levels as well as how they could influence the timing of neuronal activity. In this study, we adopt a dynamical system approach and use tools of bifurcation theory and statistics in order to address both these issues. We consider a Li-Rinzel description of astrocyte Ca2+ signalling and we show that thanks to specific choices of biophysical parameters, synaptic activity could be encoded by modulations of Ca2+ oscillations in amplitude (AM), in frequency (FM), or in both (AFM). Interestingly, AM- and FM-encoding pertain to different classes of Ca2+ excitability that are reminiscent of the analogous neuronal ones. In addition, any transition from AM to FM and viceversa is accomplished through a characteristic “Bautin-cusp" bifurcation sequence which could hint the conditions for the coexistence of both these encoding modes. Such a possibility is throughout investigated and eventually formalized in the “CPB rule", a heuristic criterion valid for any system of the Li-Rinzel type that allow us to determine several biophysical conditions under which AFM Ca2+ dynamics could occur in astrocytes. Successively, we demonstrate that different encoding modes could be accomplished not only on the basis of inherent heterogeneities of cellular properties but also thanks to the existence of different (opposite) Ca2+ feedbacks on IP3 production. In this regard we modify the Li-Rinzel system in order to include a third equation for IP3 metabolism which also considers Ca2+ activation of PLC (positive feedback) and Ca2+ activation of IP3 3-kinase (negative feedback). In agreement with experimental data and recent theoretical studies, our analysis hints that Ca2+-dependent activation of PLC could account for a much richer variety of oscillatory regimes and encoding modes with respect to the case of negative feedback. An inspection of the parameter space reveals that this is possible because positive Ca2+ feedback on IP3 production modifies the structure of the system towards the appearance of multistationarity which could also account for Ca2+ dynamics of bursting type. Moreover, we show that the lifetime of IP3 could be a critical limiting factor for the effects of both feedbacks. Meaningfully, IP3 turnover could influence the integrative properties of astrocytic Ca2+ signalling by affecting both the frequency band of Ca2+ oscillations and the threshold stimulus for their onset. In addition, IP3 turnover could regulate the expression of mGlu receptors on the astrocyte plasma membrane. In this regard, we show that the density of mGlu receptors could be proportional to the rate of IP3 turnover and accordingly we provide an estimation of this parameter which otherwise would remain experimentally unknown. In particular, on the basis of our bifurcation data, we estimate that AM-encoding astrocytes could express an average mGluR density between 1-50 receptors/um^2 that is of the same order of AMPA receptor density measured on Bergmann glia somas. We show however that FM-encoding astrocytes could be consistent with an overexpression of mGluRs by a factor of 8 or 10 with respect to AM-encoding cells. In the last part of our study, we finally consider the characterization of the possible integrative properties of astrocyte Ca2+ signalling and the effects of astrocytic Ca2+-dependent glutamate exocytosis on neuronal activity. For this purpose we develop a mathematical description of neuron-glial interactions at the level of a single astrocytic microdomain in which our modified Li-Rinzel model of astrocyte Ca2+/IP3 dynamics is coupled with an ensemble of Tsodyks-Uziel-Markram synapses on the soma a regular spiking Izhikevich neuron. We show that astrocytes of different classes of excitability could respond differently to stimuli of equal intensities and identical interspike-interval (ISI) statistics. On the other hand, stimuli at the same frequency but with different ISI statistics could trigger distinct Ca2+ responses in cells of the same type. All these possibilities are also dependent on the nature of the stimulatory pathway. Astrocyte Ca2+ signalling could therefore result from a complex integration of spatiotemporal features of synaptic stimuli which could represent a form of processing of neuronal activity. Perhaps even more intriguingly, Ca2+ signals could encode information on the past history of synaptic activity which in turn would be transferred back to neuron through Ca2+-dependent glutamate exocytosis with deep consequences on the informational content of postsynaptic neuronal activity. Computation of Fano Factors on simulated time series of postsynaptic action potentials reveal in fact that neuron-astrocyte interactions could substantially affect the rate of neuronal firing by adding long-range correlations to the timing of neuronal spikes. These results are consistent with the possibility that neuron-astrocyte bidirectional signalling could influence information processing of the brain by increasing the information-coding dimension of the neural code.