Chapter 6
Discussion and conclusions
Our study pursued two main goals: (1) to investigate the possible factors that could underlie a rich Ca2+ dynamics in astrocytes both in terms of inherent cell properties and
in terms of feedback mechanisms, and (2) to characterize the impact of astrocytic Ca2+
signals in terms of information processing performed by a tripartite synapse. Results and predictions are following summarized and discussed.
6.1
Astrocyte calcium dynamics revisited
6.1.1
Multi-facets of astrocyte excitability
One of the original motivations of our work was the experimental observation that Ca2+
oscillations in astrocytes can be extremely complex by being greatly variable both in their amplitude and in their frequency (Cornell-Bell et al., 1990; Lee and Parpura, 2007; Pasti et al., 1997). This aspect is crucial for the understanding of the physiological meaning of Ca2+ signalling of astrocytes because it could hint the way that these cells
use to encode synaptic activity. For this reason, following the approach of recent studies (Nadkarni and Jung, 2007; Volman et al., 2007), we considered a Li-Rinzel (LR) formal-ism (Li and Rinzel, 1994) for the astrocyte Ca2+ dynamics and we investigated on how
such dynamics could be affected by heterogenities of the ER stores both in terms of their filling Ca2+ capacity and their expression of IP
3Rs, SERCAs or Ca2+ leakage channels
(Toescu, 1995). We modelled such heterogeneities by varying d5, c0, c1, rC, rL, vER, kER
with respect to their original values1 and we were able to generate completely different
Ca2+ responses characterized by modulations of Ca2+ oscillations either in amplitude
(AM), or in frequency (FM) or in both (AFM). As Ca2+ dynamics in a LR astrocyte
depends on the intracellular IP3 concentration which is directly controlled by synaptic
stimulation (Porter and McCarthy, 1996; Verkhratsky et al., 1998), we looked at these
1rC, r
L, vER and kER correspond to v1, v2, v3 and k3 in the original Li-Rinzel notation (Li and
three dynamics as a hint for the existence of three possible different ways adopted by astrocytes to encode synaptic activity.
For the original set of parameter values, a LR astrocyte displays an AM-encoding Ca2+ dynamics consistent with oscillations that can greatly vary with IP
3 levels but are
confined in a narrow frequency band. By varying one or two of the above mentioned parameters we found that such dynamics could turn into FM-encoding which is con-sistent with the opposite situation of Ca2+ oscillations almost constant in amplitude
but with frequency that is highly dependent on the intracellular IP3 concentration. In
particular a reduction in the IP3R affinity for Ca2+ activation (i.e. d5 > 0.16 µM), like
the one observed in presence of increased IP3 concentrations (Kaftan et al., 1997), could
account for FM-encoding Ca2+ oscillations characterized by a prolonged latency and a
sharp rising phase. Similar effects were also observed with small Ca2+ leakage rates
(rL < 3 · 10−3 s−1) as well as with high SERCA Ca2+-affinities (kER < 0.06 µM), two
conditions that can be found experimentally in concurrence of low luminal (ER) Ca2+
levels (Mogami et al., 1999, 1998).
An inspection of Ca2+ dynamics on the fast time scale (by setting the slow gating
variable h to h∞), revealed that in the case of FM-encoding, the slope of the Ca2+ efflux
from the ER was less steep at basal Ca2+ levels than the one of the SERCA Ca2+
re-uptake. This fact was read as an equivalent onset by SERCAs of an energy barrier that prevented CICR to occur unless a threshold IP3 concentration was reached. In these
conditions in fact, from low to threshold IP3 concentrations, we observed that the onset
of CICR was promptly contrasted by a fast predominant SERCA Ca2+ sequestration.
When the IP3 threshold was reached instead, the slope of the Ca2+ efflux curve became
steeper than that of the SERCA influx and a sudden Ca2+ discharge could be observed
in the form of a relaxation-like oscillation (De Young and Keizer, 1992; Mishchenko and Rozov, 1980).
A phase-plane analysis of the Li-Rinzel system for both AM and FM dynamics, al-lowed us to associate these two Ca2+ responses with two well distinct bifurcation
struc-tures. Interestingly, we found that such bifurcation structures were strikingly similar to those underlying the two classes of neuronal excitability (Hodgkin, 1948; Izhikevich, 2000). As for type-1 neurons, in FM-encoding astrocytes the basal resting state dis-appeared through a saddle-node bifurcation on an invariant circle (SNIC) to which the onset of Ca2+ oscillations at arbitrarily small frequency followed. Hence we termed
these cells as type-1/FM-encoding astrocytes. On the contrary, we defined type-2/AM-encoding astrocytes as those LR astrocytes in which, similarly to type-2 neurons, os-cillations rose via a supercritical Hopf bifurcation at finite frequency but (differently from neurons) with arbitrarily small amplitudes. Interestingly, recent studies confirmed that the astrocyte population of the brain could be effectively heterogeneous (Matthias et al., 2003) and distinguished between two groups of cells with fundamentally different functional properties (Grass et al., 2004). However no characterization of the properties of Ca2+ excitability was performed on these two groups, therefore future experiments
are required to test whether such a diversity in terms of functional properties, could also account for our predicted existence of different classes of astrocyte Ca2+ excitability.
In spite of our distinction between AM-encoding and FM-encoding, the emerging view is that Ca2+ signals use both these modes to encode information (Berridge et al., 1998;
Carmignoto, 2000). The existence of such “AFM-encoding” in astrocytes remains cur-rently elusive due to inherent limits of the available experimental techniques (Berridge et al., 1998). Indeed astrocytes are generally recognized to perform FM-encoding of synaptic activity (Pasti et al., 1997) but there is not hitherto any reliable evidence that these cells could perform a concurrent AM-encoding of the same stimulus. This possi-bility however, is substantiated by the consideration that differences in the amplitude of oscillations such as those experimentally reported, could be required to trigger the exocytosis of different gliotransmitters (G. Carmignoto, personal communication, July 2007). Notwithstanding it remains unclear whether the existence of AFM-encoding could result from inherent properties of astrocytes or rather it is related to the existence of complex intracellular feedback mechanisms. As this latter possibility is substantiated by numerous studies (Houart et al., 1999; Kummer et al., 2000; Marhl et al., 1998), but the former is not, we considered our choice of parameters to identify under what conditions a LR astrocyte could exhibit AFM-encoding. Our analysis culminated with the formulation of the so-called “CPB rule”, which represents one of the main results of this study.
By tools of bifurcation theory we focused on the region of the parameter space in which changes in d5, rL or kER could make the system to switch from AM- to
FM-encoding and back. We found that in such a transition the LR system underwent a characteristic sequence of codim-2 bifurcations consisting first in a Bautin bifurcation through which the supercritical Hopf point at lower IP3 values turned into subcritical,
and then in a cusp bifurcation which accounted for the appearance/disappearance of the SNIC at the basis of FM-encoding2. We demonstrated that such sequence was not
casual but rather the occurrence of the Bautin could be regarded as a hint toward the loss/gain of AM-encoding and the cusp as a clue for the onset/loss of FM-encoding. This fact suggested us to consider the interval comprised between these two bifurcations as the possible region for the occurrence of AFM-encoding. For this reason we defined the Cusp-Bautin (CPB) factor as the distance of the cusp from the Bautin relative to the cusp in units of bifurcation parameter3, i.e. δCPB = |(CP − B)/CP|. Then,
we considered all the possible transitions, from AM to FM and vice versa, triggered by progressively changing all possible bifurcation parameters (among those previously listed). From AM→FM (FM→AM), we defined the appearance (disappearance) of FM-encoding when the period of oscillations doubled (halved). AM-FM-encoding was recognized instead only if the range of the amplitudes of oscillations was no less than the maximum amplitude of oscillations exhibited by the system at the Bautin point. We found that AFM-encoding could be observed in all cases in which 0.2 < δCPB < 0.8. These re-sults were eventually summarized in the above mentioned “CPB rule” which states that: “Given a Li-Rinzel system, it is a necessary condition for the coexistence of amplitude
and frequency encoding, that the tuning of a second parameter in addition to the IP3
concentration changes the bifurcation diagram so that only two Hopf bifurcation points exist and the one at lower IP3 values is subcritical. Moreover, this condition is also
2This sequence was reversed in the case of a transition from FM to AM.
3The CPB factor was computed only in FM→AM transitions. For AM→FM transitions instead, the
sufficient if it is possible to estimate the CPB factor such that it is: 0.2 < δCPB < 0.8.” On the basis of the CPB rule, we were able to identify several ranges of parameter values for which AFM-encoding occurred. In particular we found that AFM-encoding is likely to be exhibited mainly by type-1 rather than by type-2 astrocytes. Furthermore in all the considered cases of type-1 excitability, both an increase of the cell-averaged free Ca2+ concentration (c
0 ↑) or a diminished SERCA rate (vER ↓), this latter though at
values too low to be comprised in their estimated physiological range, could account for the coexistence of amplitude and frequency encoding of synaptic activity. Further cases, specific to the nature of type-1 excitability, were also recognized. For example type-1 astrocytes with a low IP3R affinity for Ca2+ activation (d5 ↑) exhibited AFM Ca2+
dy-namics in concurrence with a reduced SERCA Ca2+ affinity (k
ER ↑) or a fast leakage
rate (rL↑). A similar condition in terms of leakage was also found for type-1 cells with
an increased SERCA Ca2+ affinity, for which AFM encoding could also be observed in
concurrence with a slightly higher IP3R affinity for Ca2+ activation (d5 < 0.127 µM).
Interestingly, neither the ratio of ER volume over the cytoplasm volume (c1) nor the
maximal (total) CICR rate (rC) seemed to have any effect on the encoding mode of
astrocytes of both classes of excitability.
In spite of the fact that all these cases call for an experimental validation, our analysis remains instrumental for the understanding of the possible AFM-encoding capabilities of astrocyte Ca2+ signalling. In this regard, we provided theoretical support for the
possi-bility that inherent cellular properties before complex intracellular feedbacks or crosstalk pathways, could account for a great variety of astrocyte Ca2+ responses (Toescu, 1995).
By contrast, we can predict that the inclusion of such feedbacks/crosstalk pathways in the description of Ca2+ dynamics, will account for a repertoire of complex Ca2+
re-sponses (Borghans et al., 1997; Schuster et al., 2002) in which AFM is inherent or even other encoding modes, different from AM, FM or AFM, are present.
It is noteworthy to point out that we focused on the ideal Ca2+ dynamics, that is
we performed a bifurcation study in which all bifurcation parameters were (necessar-ily) treated as constants, but this is not in general what happens in real cells and even more so in astrocytes. This is the case of the cell-averaged free Ca2+ concentration
(c0), which is dynamically related to fluxes through the plasma membrane that are not
included in the original Li-Rinzel description. Another example is constituted by the SERCA uptake rate (vER) which may be modulated by a specific CaM kinase on the
ER membrane (Girard and Clapham, 1993; Golovina et al., 1996; Toyofuku et al., 1994). Once such dynamical changes are taken into account, it is likely that we will observe a complex concurrence of encoding mechanisms as a result of a much richer variety of Ca2+ responses (Li et al., 1994; Sneyd et al., 2004).
In spite of these considerations however, the CPB rule remains valid and represents a general heuristic criterion for the study of dynamical systems of Li-Rinzel type. Further theoretical efforts will be devoted towards a possible rigorous formalization of such a rule, with the aim to understand whether its applicability could be extended to other types of systems or under what conditions we could do so.
6.1.2
Revisiting astrocytic encoding capacity by coupling of IP
3metabolism with Ca
2+dynamics
The second part of our study was devoted to the characterization of the coupling be-tween Ca2+ dynamics and IP
3 metabolism. Previous theoretical investigations showed
how the existence of Ca2+ feedbacks on IP
3 production could account for extremely rich
Ca2+ responses, notwithstanding the physiological role of such feedbacks remains to be
elucidated. (Borghans et al., 1997; Dupont and Goldbeter, 1993; Kummer et al., 2000). In particular we addressed the question of how different feedbacks could affect astrocyte encoding of synaptic activity and for this purpose, we considered prototypical positive and negative feedbacks of Ca2+ ions on IP
3 production such as Ca2+ activation of PLC
and Ca2+ activation of IP
3 3-kinase (IP3K). Accordingly, the Li-Rinzel description of
astrocytic Ca2+ dynamics was modified to include a third equation for IP
3 metabolism
that could account for such feedbacks.
We found that positive and negative feedbacks have opposite effects whose entity depends on the astrocyte class of Ca2+ excitability. Positive feedback broadens the
fre-quency band of oscillations if Ca2+ sensitivity of PLC is only somewhat above basal
Ca2+ concentrations, namely it is k
P LC > 0.05 µM. In such conditions, the onset of
Ca2+ oscillations is delayed because both IP
3 and Ca2+ must rise to a certain level for
triggering explosive opening of IP3 receptors. Therefore, long oscillation periods arise
for low levels of stimulation, whereas for strong stimuli, the high IP3 level obviates the
need for additional Ca2+ activation of PLC. In type-2/AM-encoding astrocytes, this
mechanism may also account for the appearance of FM- or mixed AFM-encoding. On the contrary, negative feedback generally decreases both frequency and ampli-tude of Ca2+ oscillations by lowering the mean level of intracellular IP
3. Moreover, if
the action of IP3 5-phosphatase (IP5P) is fast, namely the IP3 turnover time is short,
oscillations may even be abolished. These results agree with previous theoretical studies by Dupont and Erneux (1997) and are also supported by several experimental observa-tions. For example, overexpression of IP3K in fibroblasts of rat brain have been reported to decrease Ca2+ signalling (Bhalla et al., 1991). Similarly, overexpression of IP5P
ac-tivity in Chinese hamster ovary cells is known to abolish Ca2+ oscillations in response
to stimulation (De Smedt et al., 1997).
An inspection of the parameter space allowed to explain the diversity of effects due to opposite feedbacks in terms of different bifurcations sequences. Moreover, we found that positive feedback could account for many more different mechanisms for the onset of oscillations with respect to negative feedback. This possibility is in general agreement with previous studies which demonstrated a predominant role in the modulation of Ca2+
signals for positive feedbacks in general (Brandman et al., 2005), and for Ca2+-dependent
activation of PLC in particular (Venance et al., 1997).
The variety of dynamical behaviors that is in association with positive feedback could be put in relation with the possible existence of multistationarity (Kwon and Cho, 2007). High Ca2+ affinities of PLC in fact, could exchange the reciprocal positions between Hopf
points and saddle-node points through fold-Hopf bifurcations making the system bistable or even tristable. In these conditions, depending on the time scales of the stimulus with
respect to Ca2+ signalling, various types of Ca2+ bursts could be observed. Accordingly,
because Ca2+ bursts consist in intraburst oscillations in concurrence with interburst
oscillations (Hoppensteadt and Izhikevich, 1997), we may speculate that each of these oscillations could account for a distinct encoding of the stimulus. From this perspective, astrocytic Ca2+ signals could carry information of synaptic activity in much more
com-plex fashions than only AM-, FM- or AFM-encoding.
Different modes of action of positive and negative feedbacks are also reflected by opposite requirements on the lifetime of IP3. Interestingly, these requirements are also
opposite between the two classes of astrocytic Ca2+ excitability. In the case of positive
feedback, IP3 turnover must be fast in type-2 astrocytes and slow in type-1 cells in order
to broaden the frequency band of oscillations. By contrast, in presence of negative feed-back, the period of oscillations increases and accordingly, the frequency band widens, only if IP3 degradation is fast in type-1 astrocytes and slow in type-2 cells. These results
put in evidence that the rate of IP3 degradation is a critical factor in determining the
properties of astrocytic Ca2+ signals (Wang et al., 1995). As IP
3 signalling constitutes
the intracellular pathway of signal transduction of synaptic stimuli into Ca2+ responses
(Li et al., 1994), it follows that the lifetime of IP3 could set the time frame over which
IP3 accumulations following different synaptic stimuli, are integrated as input signals.
Furthermore, as astrocytes are spatially extended structures, the rate of degradation of IP3 could also dictate the spatial (and temporal) limits for specific modes of astrocytic
encoding of synaptic activity.
IP3 turnover could also influence integrative properties of astrocytic Ca2+ signalling
by affecting the threshold and the stimulus range for which oscillations occur. In anal-ogy with a previous study by Politi et al. (2006), we found that following an increase in the rate of IP3 degradation, the threshold for the onset of oscillations increases whereas
the extent of the oscillatory range decreases. Notwithstanding, the steady-state IP3
concentration in the absence of any feedback remains essentially constant. This quan-tity was defined as ¯vP LC = τdvP LC where τd is the IP3 turnover time and vP LC is the
maximal rate of PLC activity. As this latter can be put in direct relation with receptor occupancy and hence with synaptic activity, ¯vP LC itself can be taken as a measure of
the stimulus. Accordingly, the fact that ¯vP LC must be approximately constant dictates
a condition in terms of τd and stimulus features, i.e. the stimulus intensity and the
associated occupancy of astrocytic receptors, that must be satisfied in order to generate a Ca2+ response. From this perspective, such a condition provides theoretical support
to the experimental observation that astrocytes integrate only those inputs that satisfy specific cellular intrinsic properties (Perea and Araque, 2005b). Furthermore it allows for some predictions on what cellular properties could be critical for the integration of the stimulus. These are the rate of IP3 degradation and clearly, the density of receptors
on the astrocyte plasma membrane.
We considered typical parameter values of glutamatergic synapses and on the ba-sis of our computations of ¯vP LC we estimated the density of astrocytic mGlu
recep-tors at perisynaptic processes: a parameter that is currently not known experimentally. Although our estimated density values are in the same range of those for AMPA re-ceptors at Bergmann glia somas (Matsui et al., 2005), we found a clear distinction between mGluR densities of astrocytes of different classes of excitability. This
re-sult could be regarded as a morphological hypothesis for the existence of astrocytes which exhibit different Ca2+dynamics and consequently adopt different modes to encode
synaptic activity. In particular, our estimated values for mGluR densities are between 1–50 receptors/µm2 in type 2/AM-encoding cells and about 10-fold higher, namely
be-tween 100-500 receptors/µm2, in type-1/FM-encoding cells. Interestingly, we found that
mGluR density scales approximately as ∼ 1/τd, namely it is inversely proportional to
the IP3 turnover time. This result, which follows directly from our definition of ¯vP LC, is
however of general validity and leads to intriguingly speculations. IP3 turnover may be
a critical factor in Ca2+ dynamics not only because it characterizes both the extent of
Ca2+ feedbacks on IP
3 metabolism and the integrative properties of Ca2+ signalling, but
also because it could define the expression of mGlu receptors on astrocytic processes en-folding synapses. This possibility hints that, in a similar way to dendritic spines (Ethell and Pasquale, 2005), IP3 turnover, and in particular the rate of IP3K activity (which is
inherent in our definition of τd), could set developmental constraints on the morphology
of astrocytic processes. Furthermore, although our estimation refers to mGluRs, the above results are of general validity for any other Gq-protein coupled receptor, namely
for any other metabotropic receptor coupled with the phosphotidylinositol reaction cas-cade (Hille, 2001). Nonetheless further investigations are needed to ascertain that our results do not depend on the adopted description of IP3 dynamics and could be found
therefore in presence of other feedback mechanisms.
6.2
Neuron-glial processors
Following the study of the properties of astrocyte Ca2+ dynamics, we focused on the
characterization of the modulatory action of this latter on synaptic transmission. For this purpose, we developed a mathematical description of neuron-astrocyte interactions that could realistically reproduce the essential physiology of a glutamatergic tripartite synapse (Araque et al., 1999a). The description of astrocytic IP3/Ca2+ dynamics
dis-cussed in the previous section was modified to include the coupling of IP3 production
with synaptic activity. Synaptic transmission was described by Tsodyks-Uziel-Markram formalism (Tsodyks et al., 1998) whereas postsynaptic currents were distinguished be-tween AMPAR- and NMDAR-mediated currents. Finally, astrocyte feedback on synap-tic transmission was modelled according to Volman et al. (2007) with the addition of an equation for astrocyte-mediated postsynaptic SICs.
Much effort was devoted to the estimation of physiological ranges for the fifty-four pa-rameters of the model with particular attention to those relative to astrocyte exocytosis. As the mechanism of glutamate exocytosis from astrocytes is not known, we considered an equivalent description of it in terms of a mixed Hill/Michaelis-Menten kinetic scheme borrowed from a model of Ca2+-dependent synaptic exocytosis (Destexhe et al., 1994).
Interestingly, this choice was shown to lead to general consistency with experimental data thus supporting the notion that the mechanism of exocytosis in astrocytes could be analogous to the synaptic one (Ni et al., 2007; Zhang et al., 2004b). In addition, we were able to estimate the apparent Ca2+ affinity (EC
astrocyte exocytosis, namely Syt IV (Zhang et al., 2004a), obtaining ECSyt IV50 < 1 µM.
Although a comparison of this value with experimental data is not possible for the lack of these latter, the fact that glutamate exocytosis occurs at Ca2+ concentrations
close to basal levels (Parpura and Haydon, 2000) supports our estimation. Moreover, recent measures of Ca2+-binding capabilities of Syt IV and Syt I by means of AFM4
single-molecule force spectroscopy have shown that Syt IV binds calcium at lower levels than Syt I, hinting that Syt IV Ca2+ affinity is higher than Syt I Ca2+ affinity, that is
ECSyt IV50 < 10 µM (V. Parpura, unpublished observations).
General consistency of our model with experimental data was assessed on the basis of simulations of key experiments that allowed to characterize astrocyte feedback on synaptic terminals. Astrocyte-induced presynaptic depression was described according to experiments by Araque et al. (1998a). Increase in frequency of spontaneous mPSCs was modelled to seek consistency with data of Araque et al. (1998b). Finally, synchro-nized postsynaptic SICs were described in order to qualitatively reproduce experimental recordings by Angulo et al. (2004) and Fellin et al. (2004). All simulations were robust with respect to parameter changes and qualitatively analogous modulatory actions on synaptic terminals were observed for both classes of astrocyte excitability. In addition, thanks to our model we have been able to make several predictions on the possible com-putational properties of neuron-astrocyte interactions.
In our simulations, astrocytes of different classes of excitability responded differently to stimuli of equal intensities and identical ISI statistics. By contrast, stimuli at the same frequency but with different ISI statistics triggered distinct Ca2+ responses in cells
of the same type. Moreover, Ca2+ dynamics was different for identical stimuli that
im-pinged astrocytes through distinct axonal pathways. Hence, not only inherent cellular properties (i.e. the nature of Ca2+ excitability), but also the nature of the stimulatory
pathway, which is part of the “ultrastructure” of synapse-astrocyte interactions (Haber et al., 2006), could critically define the properties of Ca2+ signalling. This possibility,
which has been experimentally demonstrated by Perea and Araque (Perea and Araque, 2005b), strongly supports the notion that astrocyte Ca2+ dynamics does not simply
mirror synaptic activity but rather it represents a form of integration and processing of this latter (Araque et al., 2001).
A further implication of our results is that Ca2+ responses of astrocytes are likely
to depend on both the timing of the stimulus and its spatial characteristics (Zonta and Carmignoto, 2002). It follows that Ca2+-dependent exocytosis of glutamate, and
accordingly astrocytic feedback on synaptic activity, could be themselves specific to the spatiotemporal features of stimulation so that distinct stimuli could be modulated differently by the same cell. Astrocytes could ultimately exhibit multiple integrative properties which might underlie both their adaptability and their inherent plasticity with respect to different stimuli (Pasti et al., 1997). Moreover, integration of stimuli over their duration hints that astrocyte Ca2+ signalling could encode information on the
past history of synaptic activity. This information could eventually be transferred back to neurons through Ca2+ dependent astrocytic feedback on synaptic terminals, thus
pro-viding a possible mechanism for learning and memory in neural networks of the brain
(Fields and Stevens-Graham, 2002).
The difference between the time scales of astrocyte-induced presynaptic depression and postsynaptic SICs is such that SICs occur when synaptic depression is maximal. Therefore SICs could constitute the main depolarizing agent of postsynaptic neurons during astrocyte Ca2+ activation (Fellin et al., 2004; Perea and Araque, 2002). As
ob-served from our simulations, an intriguing consequence of this mechanism could be the modulation of neuronal activity from pseudo-random/periodic firing to bursting, that is the clustering of action potentials followed by relative quiescent states (Rinzel, 1987). In such case, SICs would account for fast/intraburst oscillations whereas astrocyte-regulated presynaptic depression would determine the period of slow/interburst oscilla-tions. In addition, given that the time course of SICs and presynaptic depression both depend on the temporal evolution of astrocytic Ca2+ dynamics, we may predict that
intraburst and interburst oscillations could exhibit long-range correlations which may ultimately account for an additional coding dimension of neuronal activity (Sejnowski and Paulsen, 2006).
We have been able to substantiate this prediction by calculating Fano factors (FFs) (Gabbiani and Koch, 1998; Teich et al., 1996) of simulated sequences of action potentials following stimulation of a regular spiking neuron (Izhikevich, 2007; Tateno et al., 2004) through five identical synapses. We considered first the case of synaptic stimulation without any astrocytic feedback: a condition that could be experimentally reproduced by inhibition of Ca2+ activity through astrocyte intracellular injections of the calcium
chelator BAPTA (Araque et al., 1998a). Then we performed identical synaptic stimula-tion in presence of astrocytic feedback. Because we chose neuronal parameters such that the neuron could not affect the ISI statistics of the stimulus, any additional variability in the ISI sequence that emerged following the inclusion of astrocytic feedback, was taken as a measure of the astrocyte action on the timing of neuronal firing.
We found that synapse-astrocyte interactions deeply affected the ISI statistics of neuronal spikes in a way that was essentially independent from the stimulus. Fano factor curves, plotted as function of the counting time (Teich et al., 1996) exhibited a characteristic straight-line increase on bilogarithmic scale above the lower limit of the counting time set by neuronal refractoriness. For sufficiently low frequencies, this be-havior was consistent with a spike-train power spectral density of the type 1/f , hinting that astrocyte feedback could modulate neural spiking in a fractal fashion (Lowen and Teich, 1995). Fractal-like spike sequences are indeed those observed in the case of neu-ronal bursting as in our simulations, in which spike clusters are nested over multiple time scales. Interestingly, bursting and clustering of spikes seem to be a general feature of the activity of neuron-glial networks both in vitro (Segev et al., 2004; Segev and Ben-Jacob, 2001; Segev et al., 2003; Volman et al., 2004) and in vivo (Bassett et al., 2006; Lee et al., 2005), hinting that neuron-glia interactions could be substantially involved in the fractal character of brain activity (Lutzenberger et al., 1995). If this was true then neuron-astrocyte interactions could substantially influence the final complexity of the neural code by increasing the coding dimension of information in the brain through the addition of long-range correlations (Volman et al., 2007). This possibility strongly favors the hypothesis that neuron-astrocyte interactions could be functionally inherent in the processing of information by the brain, so that not neurons but rather “neuron-glial
6.3
Future directions
Experimental evidences supporting a predominant role of astrocytes in brain functions are continuously growing, but theoretical efforts devoted to the understanding of the possible implications of such a renovated role for these cells are only at their beginning. Although we hope that our work could represent a step towards the filling of this gap, many questions still remain to be answered concerning astrocytic regulation of synaptic transmission.
As seen in our analysis, changes of cellular properties could substantially modify Ca2+
signalling of astrocytes and their encoding capabilities accordingly. Many of these prop-erties could be subjected to dynamical regulation by means of complex Ca2+ feedbacks
(Mishra and Bhalla, 2002). Hence future efforts should be devoted to the development of a description of astrocytic Ca2+ dynamics that could reliably include these mechanisms.
In parallel the inclusion in such a description of ionic fluxes through the astrocyte plasma membrane could provide additional clues on the origin of the rich dynamical repertoire of astrocytic Ca2+ responses (Sneyd et al., 2004).
The realistic possibility that Ca2+ signalling could encode information both in
am-plitude and frequency of its oscillations, was assessed in our work by considering only the Li-Rinzel description of Ca2+ dynamics. Hence, future theoretical investigations
should consider what analytical conditions underlie AFM-encoding in order to develop if possible, a general mathematical description of such mechanism. We might expect that unravelling the origin of AFM-encoding in astrocytes could be instrumental in un-derstanding the spatiotemporally multi-scaled functions of these cells (De Pitt`a et al., 2007).
No geometrical details relative to the ultrastructure of neuron-glial interactions were taken into account in our study. Future versions of our model therefore should consider this aspect too, given that the geometry of the extracellular space between synapses and astrocytes could define the modalities of interactions for these cells by directly influ-encing the diffusion of neuroactive substances (Perea and Araque, 2005a; Ventura and Harris, 1999).
Theoretical efforts should also address the study of neuron-glial interactions at the level of neuronal networks. In this case the inherent differences between the time scales of neural activity and astrocyte Ca2+ signalling could lead to new perspectives in
net-work studies of the brain (Boccaletti et al., 2006; Izhikevich, 2007). Neurons could be regarded as digital elements interspersed in glial syncytia which could be treated as analog-like excitable media. Nonetheless, the dynamical consequences of such a concep-tualization on network activity in general and even more so in the brain have never been investigated so far.
What role do astrocytes, and glia in general, play in information processing, learning and memory? And most importantly, what are the fundamental elements of neuron-glial interactions that account for such a role? In our study we provided theoretical support to the notion that astrocytes could substantially affect the final complexity of neural code by adding long-range correlations that could account for additional coding dimen-sions of information. As this possibility currently remains at the level of theoretical
prediction, further investigations both experimental and theoretical, are needed to sub-stantiate it. Moreover, theoretical efforts should also be devoted to the understanding of how to characterize the complexity of neuron-astrocyte interactions and their effect on neural activity. Once we will have answered all these questions we will probably know the essential alphabet of neuron-glial signalling. Then we will be ready to move a further step towards the decryption of the language of our brain.
