Category Archives: Publications

P. Mazzarisi, S. Zaoli, C. Campajola, F. Lillo (2020). Tail Granger causalities and where to find them: Extreme risk spillovers vs spurious linkages, Journal of Economic Dynamics and Control 121, 104022

Identifying risk spillovers in financial markets is of great importance for assessing systemic risk and portfolio management. Granger causality in tail (or in risk) tests whether past extreme events of a time series help predicting future extreme events of another time series. The topology and connectedness of networks built with Granger causality in tail can be used to measure systemic risk and to identify risk transmitters. Here we introduce a novel test of Granger causality in tail which adopts the likelihood ratio statistic and is based on the multivariate generalization of a discrete autoregressive process for binary time series describing the sequence of extreme events of the underlying price dynamics. The proposed test has very good size and power in finite samples, especially for large sample size, allows inferring the correct time scale at which the causal interaction takes place, and it is flexible enough for multivariate extension when more than two time series are considered in order to decrease false detections as spurious effect of neglected variables. An extensive simulation study shows the performances of the proposed method with a large variety of data generating processes and it introduces also the comparison with the test of Granger causality in tail by Hong et al. (2009). We report both advantages and drawbacks of the different approaches, pointing out some crucial aspects related to the false detections of Granger causality for tail events. An empirical application to high frequency data of a portfolio of US stocks highlights the merits of our novel approach.

P. Mazzarisi, F.Lillo, S. Marmi (2019). When panic makes you blind: A chaotic route to systemic risk, Journal of Economic Dynamics and Control , 100, 176-199

We present an analytical model to study the role of expectation feedbacks and overlapping portfolios on systemic stability of financial systems. Building on Corsi et al. (2016), we model a set of financial institutions having Value-at-Risk capital requirements and investing in a portfolio of risky assets, whose prices evolve stochastically in time and are endogenously driven by the trading decisions of financial institutions. Assuming that they use adaptive expectations of risk, we show that the evolution of the system is described by a slow-fast random dynamical system, which can be studied analytically in some regimes. The model shows how the risk expectations play a central role in determining the systemic stability of the financial system and how wrong risk expectations may create panic-induced reduction or over-optimistic expansion of balance sheets. Specifically, when investors are myopic in estimating the risk, the fixed point equilibrium of the system breaks into leverage cycles and financial variables display a bifurcation cascade eventually leading to chaos. We discuss the role of financial policy and the effects of some market frictions, as the cost of diversification and financial transaction taxes, in determining the stability of the system in the presence of adaptive expectations of risk.

P. Mazzarisi, P. Barucca, F. Lillo, D. Tantari (2020), A dynamic network model with persistent links and node-specific latent variables, with an application to the interbank market , European Journal of Operational Research, 281, 1, 50-65

We propose a dynamic network model where two mechanisms control the probability of a link between two nodes: (i) the existence or absence of this link in the past, and (ii) node-specific latent variables (dynamic fitnesses) describing the propensity of each node to create links. Assuming a Markov dynamics for both mechanisms, we propose an Expectation-Maximization algorithm for model estimation and inference of the latent variables. The estimated parameters and fitnesses can be used to forecast the presence of a link in the future. We apply our methodology to the e-MID interbank network for which the two linkage mechanisms are associated with two different trading behaviors in the process of network formation, namely preferential trading and trading driven by node-specific characteristics. The empirical results allow to recognize preferential lending in the interbank market and indicate how a method that does not account for time-varying network topologies tends to overestimate preferential linkage.

D. Di Gangi, F. Lillo, D. Pirino (2018) , Assessing systemic risk due to fire sales spillover through maximum entropy network reconstruction , Journal of Economic Dynamics and Control, 94, 117-141

Monitoring and assessing systemic risk in financial markets is of great importance but it often requires data that are unavailable or available at a very low frequency. For this reason, systemic risk assessment with partial information is potentially very useful for regulators and other stakeholders. In this paper we consider systemic risk due to fire sales spillovers and portfolio rebalancing by using the risk metrics defined by Greenwood et al. (2015). By using a method based on the constrained minimization of the Cross Entropy, we show that it is possible to assess aggregated and single bank’s systemicness and vulnerability, using only the information on the size of each bank and the capitalization of each investment asset. We also compare our approach with an alternative widespread application of the Maximum Entropy principle allowing to derive graph probability distributions and generating scenarios and we use it to propose a statistical test for a change in banks’ vulnerability to systemic events.

A. Barra, G. Genovese, P. Sollich, D. Tantari, Phase diagram of restricted Boltzmann machines and generalized Hopfield networks with arbitrary priors , Physical Review E 97 (2), 022310, 2018

Restricted Boltzmann machines are described by the Gibbs measure of a bipartite spin glass, which in turn can be seen as a generalized Hopfield network. This equivalence allows us to characterize the state of these systems in terms of their retrieval capabilities, both at low and high load, of pure states. We study the paramagnetic-spin glass and the spin glass-retrieval phase transitions, as the pattern (i.e., weight) distribution and spin (i.e., unit) priors vary smoothly from Gaussian real variables to Boolean discrete variables. Our analysis shows that the presence of a retrieval phase is robust and not peculiar to the standard Hopfield model with Boolean patterns. The retrieval region becomes larger when the pattern entries and retrieval units get more peaked and, conversely, when the hidden units acquire a broader prior and therefore have a stronger response to high fields. Moreover, at low load retrieval always exists below some critical temperature, for every pattern distribution ranging from the Boolean to the Gaussian case.

L.M. Calcagnile, G. Bormetti, M. Treccani, S. Marmi, F. Lillo, Collective synchronization and high frequency systemic instabilities in financial markets, Quantitative Finance 18 (2), 237-247

We present some empirical evidence on the dynamics of price instabilities in financial markets and propose a new Hawkes modelling approach. Specifically, analysing the recent high frequency dynamics of a set of US stocks, we find that since 2001 the level of synchronization of large price movements across assets has significantly increased. We find that only a minor fraction of these systemic events can be connected with the release of pre-announced macroeconomic news. Finally, the larger is the multiplicity of the event—i.e. how many assets have swung together—the larger is the probability of a new event occurring in the near future, as well as its multiplicity. To reproduce these facts, due to the self- and cross-exciting nature of the event dynamics, we propose an approach based on Hawkes processes. For each event, we directly model the multiplicity as a multivariate point process, neglecting the identity of the specific assets. This allows us to introduce a parsimonious parametrization of the kernel of the process and to achieve a reliable description of the dynamics of large price movements for a high-dimensional portfolio.

Letizia E., Barucca P., Lillo F. (2018). Resolution of ranking hierarchies in directed networks

Identifying hierarchies and rankings of nodes in directed graphs is fundamental in many applications such as social network analysis, biology, economics, and finance. A recently proposed method identifies the hierarchy by finding the ordered partition of nodes which minimises a score function, termed agony. This function penalises the links violating the hierarchy in a way depending on the strength of the violation.
To investigate the resolution of ranking hierarchies we introduce an ensemble of random graphs, the Ranked Stochastic Block Model. We find that agony may fail to identify hierarchies when the structure is not strong enough and the size of the classes is small with respect to the whole network. We analytically characterise the resolution threshold and we show that an iterated version of agony can partly overcome this resolution limit.

A.Barra, G.Genovese, P.Sollich, D.Tantari (2017), Phase transitions in Restricted Boltzmann Machines with generic priors

We study generalized restricted Boltzmann machines with generic priors for units and weights, interpolating between Boolean and Gaussian variables. We present a complete analysis of the replica symmetric phase diagram of these systems, which can be regarded as generalized Hopfield models. We underline the role of the retrieval phase for both inference and learning processes and we show that retrieval is robust for a large class of weight and unit priors, beyond the standard Hopfield scenario. Furthermore, we show how the paramagnetic phase boundary is directly related to the optimal size of the training set necessary for good generalization in a teacher-student scenario of unsupervised learning.

P.Barucca, D.Tantari, F.Lillo (2016), Centrality metrics and localization in core-periphery networks

Two concepts of centrality have been defined in complex networks. The first considers the centrality of a node and many different metrics for it have been defined (e.g. eigenvector centrality, PageRank, non-backtracking centrality, etc). The second is related to large scale organization of the network, the core-periphery structure, composed by a dense core plus an outlying and loosely-connected periphery. In this paper we investigate the relation between these two concepts. We consider networks generated via the stochastic block model, or its degree corrected version, with a core-periphery structure and we investigate the centrality properties of the core nodes and the ability of several centrality metrics to identify them. We find that the three measures with the best performance are marginals obtained with belief propagation, PageRank, and degree centrality, while non-backtracking and eigenvector centrality (or MINRES [10], showed to be equivalent to the latter in the large network limit) perform worse in the investigated networks.

Agliari, Elena, et al. “Two-particle problem in comblike structures.” Physical Review E 93.5 (2016): 052111.

Encounters between walkers performing a random motion on an appropriate structure can describe a wide variety of natural phenomena ranging from pharmacokinetics to foraging. On homogeneous structures the asymptotic encounter probability between two walkers is (qualitatively) independent of whether both walkers are moving or one is kept fixed. On infinite comblike structures this is no longer the case and here we deepen the mechanisms underlying the emergence of a finite probability that two random walkers will never meet, while one single random walker is certain to visit any site. In particular, we introduce an analytical approach to address this problem and even more general problems such as the case of two walkers with different diffusivity, particles walking on a finite comb and on arbitrary bundled structures, possibly in the presence of loops. Our investigations are both analytical and numerical and highlight that, in general, the outcome of a reaction involving two reactants on a comblike architecture can strongly differ according to whether both reactants are moving (no matter their relative diffusivities) or only one is moving and according to the density of shortcuts among the branches.

N. Angelini, G. Bormetti, S. Marmi, F. Nardini, A Stylized Model for Long-Run Index Return Dynamics , Essays in Economic Dynamics, 111-122

We introduce a discrete-time model of stock index return dynamics grounded on the ability of Shiller’s Cyclically Adjusted Price-to-Earning ratio to predict long-horizon market performances. Specifically, we discuss a model in which returns are driven by a fundamental term and an autoregressive component perturbed by external random disturances. The autoregressive component arises from the agents’ belief that expected returns are higher in bullish markets than in bearish markets. The fundamental term, driven by the value towards which fundamentalists expect the current price should revert, varies in time and depends on the initial averaged price-to-earnings ratio. The actual stock price may deviate from the perceived reference level as a combined effect of an idyosyncratic noise component and local trends due to trading strategies. We demonstrate both analytically and by means of numerical experiments that the long-run behavior of our stylized dynamics agrees with empirical evidences reported in literature.

F. Corsi, S. Marmi, F. Lillo, When micro prudence increases macro risk: The destabilizing effects of financial innovation, leverage, and diversification, Operations Research 64 (5), 1073-1088

By exploiting basic common practice accounting and risk-management rules, we propose a simple analytical dynamical model to investigate the effects of microprudential changes on macroprudential outcomes. Specifically, we study the consequence of the introduction of a financial innovation that allows reducing the cost of portfolio diversification in a financial system populated by financial institutions having capital requirements in the form of Value at Risk (VaR) constraint and following standard mark-to-market and risk-management rules. We provide a full analytical quantification of the multivariate feedback effects between investment prices and bank behavior induced by portfolio rebalancing in presence of asset illiquidity and show how changes in the constraints of the bank portfolio optimization endogenously drive the dynamics of the balance sheet aggregate of financial institutions and, thereby, the availability of bank liquidity to the economic system and systemic risk. The model shows that when financial innovation reduces the cost of diversification below a given threshold, the strength (because of higher leverage) and coordination (because of similarity of bank portfolios) of feedback effects increase, triggering a transition from a stationary dynamics of price returns to a nonstationary one characterized by steep growths (bubbles) and plunges (bursts) of market prices.

Clemente De Rosa, Elisa Luciano, Luca Regis (2016). Basis risk in static versus dynamic longevity-risk hedging. Scandinavian Actuarial Journal

This paper provides a tractable, parsimonious model for assessing basis risk in longevity and its effect on the hedging strategies of Pension Funds and annuity providers. Basis risk is captured by a single parameter, that measures the co-movement between the portfolio and the reference population’s longevity. The paper sets out the static, full and customized swap-hedge for an annuity, and compares it with a dynamic, partial, and index-based hedge. We calibrate our model to the UK and Scottish populations. The effectiveness of static versus dynamic strategies depends on the rebalancing frequency of the second, on the relative costs, and on basis risk, which does not affect fully-customized, static hedges. We show that appropriately calibrated dynamic hedging strategies can still be reasonably effective, even at low rebalancing frequencies.

Luca Cattivelli, Elena Agliari, Fabio Sartori, and Davide Cassi. 2015 Lévy flights with power-law absorption. Phys. Rev. E 92, 042156

We consider a particle performing a stochastic motion on a one-dimensional lattice with jump lengths distributed according to a power law with exponent μ+1. Assuming that the walker moves in the presence of a distribution a(x) of targets (traps) depending on the spatial coordinate x, we study the probability that the walker will eventually find any target (will eventually be trapped). We focus on the case of power-law distributions a(x)xα and we find that, as long as μ<α, there is a finite probability that the walker will never be trapped, no matter how long the process is. This result is shown via analytical arguments and numerical simulations which also evidence the emergence of slow searching (trapping) times in finite-size system. The extension of this finding to higher-dimensional structures is also discussed.

Agliari, E., Sartori, F., Cattivelli, L. and Cassi, D., 2015. Hitting and trapping times on branched structures. Physical Review E, 91(5), p.052132.

In this work we consider a simple random walk embedded in a generic branched structure and we find a close-form formula to calculate the hitting time H(i,f) between two arbitrary nodes i and j. We then use this formula to obtain the set of hitting times {H(i,f)} for combs and their expectation values, namely, the mean first-passage time, where the average is performed over the initial node while the final node f is given, and the global mean first-passage time, where the average is performed over both the initial and the final node. Finally, we discuss applications in the context of reaction-diffusion problems.

G. Bormetti, L. M. Calcagnile, M. Treccani, F. Corsi, S. Marmi, F. Lillo, Modelling systemic price cojumps with Hawkes factor models , Quantitative Finance 15 (7), 1137-1156

Instabilities in the price dynamics of a large number of financial assets are a clear sign of
systemic events. By investigating portfolios of highly liquid stocks, we find that there are a
large number of high-frequency cojumps. We show that the dynamics of these jumps is
described neither by a multivariate Poisson nor by a multivariate Hawkes model. We
introduce a Hawkes one-factor model which is able to capture simultaneously the time
clustering of jumps and the high synchronization of jumps across assets.

D.H. Kim, S. Marmi, Distribution of asset price movement and market potential, Journal of Statistical Mechanics: Theory and Experiment 2015 (7), P07001

In this article we discuss the distribution of asset price movements by introducing a market potential function. From the principle of free energy minimization we analyze two different kinds of market potentials. We obtain a U-shaped potential when market reversion (i.e. contrarian investors) is dominant. On the other hand, if there are more trend followers, flat and logarithmic potentials appear. By using the cyclically adjusted price-to-earning ratio, which is a common valuation tool, we empirically investigate the market data. By studying long term data we observe the historical change of the market potential of the US stock market. Recent US data show that the market potential looks more like a trend-following potential. Next, we compare the market potentials for 12 different countries. Though some countries have similar market potentials, there are specific examples like Japan which exhibits a very flat potential.

Amendola, G., Marengo, L., Pirino, D., Settepanella, S. and Takemura, A. (2015). Decidability in complex social choices. Evolutionary and Institutional Economics Review, 12(1), 141-168.

In this paper, we develop on a geometric model of social choice among bundles of interdependent elements (objects). Social choice can be seen as a process of search for optima in a complex multidimensional space and objects determine a decomposition of such a space into subspaces. We present a series of numerical and probabilistic results which show that such decompositions in objects can greatly increase decidability, as new kind of optima (called local and u-local) are very likely to appear also in cases in which no generalized Condorcet winner exists in the original search space.

Bottazzi, G., Tamagni F., and Pirino, D. (2015). Zipf law and the firm size distribution: a critical discussion of popular estimators. Journal of Evolutionary Economics, 25(3), 585-610.

The upper tail of the firm size distribution is often assumed to follow a Power Law. Several recent papers, using different estimators and different data sets, conclude that the Zipf Law, in particular, provides a good fit, implying that the fraction of firms with size above a given value is inversely proportional to the value itself. In this article we compare the asymptotic and small sample properties of different methods through which this conclusion has been reached. We find that the family of estimators most widely adopted, based on an OLS regression, is in fact unreliable and basically useless for appropriate inference. This finding raises doubts about previously identified Zipf behavior. Based on extensive numerical analysis, we recommend the adoption of the Hill estimator over any other method when individual observations are available.

Lillo, F., & Pirino, D. (2015). The impact of systemic and illiquidity risk on financing with risky collateral. Journal of Economic Dynamics and Control, 50, 180–202.

Repurchase agreements (repos) are one of the most important sources of funding liquidity for many financial investors and intermediaries. In a repo, some assets are given by a borrower as collateral in exchange of funding. The capital given to the borrower is the market value of the collateral, reduced by an amount termed as haircut (or margin). The haircut protects the capital lender from loss of value of the collateral contingent on the borrower׳s default. For this reason, the haircut is typically calculated with a simple Value at Risk estimation of the collateral for the purpose of preventing the risk associated to volatility. However, other risk factors should be included in the haircut and a severe undervaluation of them could result in a significant loss of value of the collateral if the borrower defaults. In this paper we present a stylized model of the financial system, which allows us to compute the haircut incorporating the liquidity risk of the collateral and, most important, possible systemic effects. These are mainly due to the similarity of bank portfolios, excessive leverage of financial institutions, and illiquidity of assets. The model is analytically solvable under some simplifying assumptions and robust to the relaxation of these assumptions, as shown through Monte Carlo simulations. We also show which are the most critical model parameters for the determination of haircuts.

Taranto, D. E., Bormetti, G., and Lillo, F. (2014) The adaptive nature of liquidity taking in limit order books. Journal of Statistical Mechanics: Theory and Experiment 2014.6: P06002


In financial markets, the order flow, defined as the process assuming value one for buy market orders and minus one for sell market orders, displays a very slowly decaying autocorrelation function. Since orders impact prices, reconciling the persistence of the order flow with market efficiency is a subtle issue. A possible solution is provided by asymmetric liquidity, which states that the impact of a buy or sell order is inversely related to the probability of its occurrence. We empirically find that when the order flow predictability increases in one direction, the liquidity in the opposite side decreases, but the probability that a trade moves the price decreases significantly. While the last mechanism is able to counterbalance the persistence of order flow and restore efficiency and diffusivity, the first acts in the opposite direction. We introduce a statistical order book model where the persistence of the order flow is mitigated by adjusting the market order volume to the predictability of the order flow. The model reproduces the diffusive behaviour of prices at all time scales without fine-tuning the values of parameters, as well as the behaviour of most order book quantities as a function of the local predictability of the order flow.

S. Marmi, C. Pacati, R. Renò, W.A. Risso, A quantitative approach to Faber’s tactical asset allocation, International Journal of Computational Economics and Econometrics 3 (1-2), 91-101

Routinely, practitioners and academics alike propose the use of trading strategies with an
alleged improvement on the risk–return relation, typically entailing a considerably higher
return for the given level of risk. A very popular example is” A quantitative approach to
tactical asset allocation” by the fund manager M. Faber, a real hit in the SSRN online library.
Is this paper a counterexample to market efficiency? We reject this conclusion, showing that
a lot of caution should be used in this field, and we indicate a series of bootstrapping
experiments which can be easily implemented to evaluate the performance of trading

G. Buccheri, S. Marmi, R.N. Mantegna, Evolution of correlation structure of industrial indices of US equity markets, Physical Review E 88 (1), 012806

We investigate the dynamics of correlations present between pairs of industry indices of U.S. stocks traded in U.S. markets by studying correlation-based networks and spectral properties of the correlation matrix. The study is performed by using 49 industry index time series computed by K. French and E. Fama during the time period from July 1969 to December 2011, which spans more than 40 years. We show that the correlation between industry indices presents both a fast and a slow dynamics. The slow dynamics has a time scale longer than 5 years, showing that a different degree of diversification of the investment is possible in different periods of time. Moreover, we also detect a fast dynamics associated with exogenous or endogenous events. The fast time scale we use is a monthly time scale and the evaluation time period is a 3-month time period. By investigating the correlation dynamics monthly, we are able to detect two examples of fast variations in the first and second eigenvalue of the correlation matrix. The first occurs during the dot-com bubble (from March 1999 to April 2001) and the second occurs during the period of highest impact of the subprime crisis (from August 2008 to August 2009).

Pirino D., Rigosa J., Ledda A. and Ferretti, L. (2012). Detecting correlations among functional-sequence motifs. Physical Review E, 85, 066124.

Sequence motifs are words of nucleotides in DNA with biological functions, e.g., gene regulation. Identification of such words proceeds through rejection of Markov models on the expected motif frequency along the genome. Additional biological information can be extracted from the correlation structure among patterns of motif occurrences. In this paper a log-linear multivariate intensity Poisson model is estimated via expectation maximization on a set of motifs along the genome of E. coli K12. The proposed approach allows for excitatory as well as inhibitory interactions among motifs and between motifs and other genomic features like gene occurrences. Our findings confirm previous stylized facts about such types of interactions and shed new light on genome-maintenance functions of some particular motifs. We expect these methods to be applicable to a wider set of genomic features.

Pirino D. and Renò R. (2010). Electricity prices: a non-parametric approach. International Journal of Theoretical and Applied Finance, 13(2), 285-299.

We propose a simple univariate model for the dynamics of spot electricity prices. The model is nonparametric in the sense that it is free from parametric model assumptions and flexible in capturing the dynamics of the data. The estimation is performed in two steps. Preliminarily, spikes are identified by means of an iterative filtering technique. The series of spikes is used to estimate a seasonal spike intensity function and fitted with an exponential law. We then implement Nadaraya-Watson estimators for the drift and the diffusion coefficients on the filtered series. Monte Carlo simulations are used to evaluate estimation errors.
We fit the model on European and American time series of spot day-ahead electricity prices; in spite of the simplicity of the proposed model, our specification tests indicate successful goodness-of-fit. We provide evidence for mean-reversion, nonlinear volatility and seasonal spike intensity; moreover we find that American markets show a very low level of mean reversion and a lower volatility with respect to their European counterparts.

Corsi, F., Pirino, D., and Renò, R. (2010). Threshold bipower variation and the impact of jumps on volatility forecasting. Journal of Econometrics, 159(2), 276-288

This study reconsiders the role of jumps for volatility forecasting by showing that jumps have a positive and mostly significant impact on future volatility. This result becomes apparent once volatility is separated into its continuous and discontinuous components using estimators which are not only consistent, but also scarcely plagued by small sample bias. With the aim of achieving this, we introduce the concept of threshold bipower variation, which is based on the joint use of bipower variation and threshold estimation. We show that its generalization (threshold multipower variation) admits a feasible central limit theorem in the presence of jumps and provides less biased estimates, with respect to the standard multipower variation, of the continuous quadratic variation in finite samples. We further provide a new test for jump detection which has substantially more power than tests based on multipower variation. Empirical analysis (on the S&P500 index, individual stocks and US bond yields) shows that the proposed techniques improve significantly the accuracy of volatility forecasts especially in periods following the occurrence of a jump.

Roma A. and Pirino D. (2009). The extraction of natural resources: the role of thermodynamic efficiency. Ecological Economics, 68(10), 2594–2606.

The modelling of production in microeconomics has been the subject of heated debate. The controversial issues include the substitutability between production inputs, the role of time and the economic consequences of irreversibility in the production process. A case in point is the use of Cobb–Douglas type production functions, which completely ignore the physical process underlying the production of a good. We examine these issues in the context of the production of a basic commodity (such as copper or aluminium). We model the extraction and the refinement of a valuable substance which is mixed with waste material, in a way which is fully consistent with the physical constraints of the process. The resulting analytical description of production unambiguously reveals that perfect substitutability between production inputs fails if a corrected thermodynamic approach is used. We analyze the equilibrium pricing of a commodity extracted in an irreversible way. We force consumers to purchase goods using energy as the means of payment and force the firm to account in terms of energy. The resulting market provides the firm with a form of reversibility of its use of energy. Under an energy numeraire, energy resources will naturally be used in a more parsimonious way.


Pirino D. (2009). Jump detection and long range dependence. Physica A, 388(7), 1150-1156.

Memory properties of financial assets are investigated. Using Detrended Fluctuation Analysis we show that the long memory detection in volatility is affected by the presence of jumps, realized volatility being a biased volatility proxy. We propose threshold bipower variation as an alternative volatility estimator unaffected by discontinuous variations. We also show that, with typical sample sizes, DFA is unable to disentangle long memory from short range dependence with characteristic time comparable to the whole sample length.

Allegrini P., Fronzoni L. and Pirino, D. (2009). The influence of the astrocyte field on neuronal dynamics and synchronization. Journal of Biological Physics, 35 (4), 413-423.

Astrocytes can sense local synaptic release of glutamate by metabotropic glutamate receptors. Receptor activation in turn can mediate transient increases of astrocytic intracellular calcium concentration through inositol 1,4,5-trisphosphate production. Notably, the perturbation of calcium concentration can propagate to other adjacent astrocytes. Astrocytic calcium signaling can therefore be linked to synaptic information transfer between neurons. On the other hand, astrocytes can also modulate neuronal activity by feeding back onto synaptic terminals in a fashion that depends on their intracellular calcium concentration. Thus, astrocytes can also be active partners in neuronal network activity. The aim of our study is to provide a computationally simple network model of mutual neuron-astrocyte interactions, in order to investigate the possible roles of astrocytes in neuronal network dynamics. In particular, we focus on the information entropy of neuronal firing of the whole network, considering how it could be affected by neuron-glial interactions.