Mean-field Systems Research Articles

Analytical Approximations of Non-Linear SDEs of McKean-Vlasov Type

We provide analytical approximations for the law of the solutions to a certain class of scalar McKean- Vlasov stochastic differential equations (MKV-SDEs) with random initial datum. “Propagation of chaos” results (Sznitman 1991) connect this class of SDEs with the macroscopic limiting behavior of a particle, evolving within a mean-field interaction particle system, as the total number of particles tends to infinity. Here we assume the mean-field interaction only acting on the drift of each particle, this giving rise to a MKV-SDE where the drift coefficient depends on the law of the unknown solution. By perturbing the non-linear forward Kolmogorov equation associated to the MKV-SDE, we perform a two-steps approximating procedure that decouples the McKean-Vlasov interaction from the standard dependence on the state-variables. The first step yields an expansion for the marginal distribution at a given time, whereas the second yields an expansion for the transition density. Both the approximating series turn out to be asymptotically convergent in the limit of short times and small noise, the convergence order for the latter expansion being higher than for the former. The resulting approximation formulas are expressed in semi-closed form and can be then regarded as a viable alternative to the numerical simulation of the large-particle system, which can be computationally very expensive. Moreover, these results pave the way for further extensions of this approach to more general dynamics and to high-dimensional settings.

First-order phase transitions in outbreaks of co-infectious diseases and the extended general epidemic process

In co-infections, positive feedback between multiple diseases can accelerate outbreaks. In a recent letter Chen, Ghanbarnejad, Cai, and Grassberger (CGCG) introduced a spatially homogeneous mean-field model system for such co-infections, and studied this system numerically with focus on the possible existence of discontinuous phase transitions. We show that their model coincides in mean-field theory with the homogenous limit of the extended general epidemic process (EGEP). Studying the latter analytically, we argue that the discontinuous transition observed by CGCG is basically a spinodal phase transition and not a first-order transition with phase coexistence. We derive the conditions for this spinodal transition along with predictions for important quantities such as the magnitude of the discontinuity. We also shed light on a true first-order transition with phase coexistence by discussing the EGEP with spatial inhomogeneities.

A simple proof of a Kramers’ type law for self-stabilizing diffusions

We provide a new proof of a Kramers’ type law for self-stabilizing diffusion. These diffusions correspond to the hydrodynamical limit of a mean-field system of particles and may be seen as the probabilistic interpretation of the granular media equation. We use the same hypotheses as the ones used in the work “Large deviations and a Kramers’ type law for self-stabilizing diffusions” by Herrmann, Imkeller and Peithmann in which the authors obtain a first proof of the statement.

Fast and accurate representations of stochastic ion channel fluctuations

Random ion channel gating is an important source of noise at the single neuron level. For mesoscale ion channel population sizes, fast, accurate representation of channel noise fluctuations remains an important challenge. We highlight recent progress in three areas. In [1] we present an exact stochastic simulation algorithm, which takes into account the time dependence of the voltage sensitive transitions due to rapid voltage changes during action potentials. The exact algorithm is similar to a widely used approximate stochastic simulation algorithm, in which transition rates are held fixed during the intervals between channel state transitions. We compare the algorithms and show that they are inequivalent in a strong sense, meaning that sample paths diverge when driven with identical Poisson processes, leading to different precise firing times. But the stationary histograms produced are practically indistinguishable for modest channel numbers (circa N>100), indicating weak equivalence. For channel numbers in the hundreds or higher, numerical stochastic differential equations (SDE) algorithms based on the system size expansion can be significantly faster than simulations based on discrete Markov chain simulations, while retaining reasonable accuracy. For large networks, however, even SDE based simulations become costly, particularly as more complex channel gating schemes are introduced. Schmandt and Galan introduced a stochastic shielding approximation as a fast, accurate way of simulating stochastic ion channel kinetics [2]. In the SDE representation, each edge in the graph generates both a mean population flux between adjacent nodes, and a fluctuation about the mean. Only the fluctuations arising from edges connecting functionally distinct states directly affect fluctuations in the observed behavior of the cell. In [3] we analyze stochastic shielding both for the HH sodium and potassium gating models, and for an ensemble of random graphs. We derive a quantitative measure of edge importance related to the eigenvalue/eigenvector decomposition of the graph Laplacian matrix. Channel noise makes a regularly spiking neuron a stochastic oscillator. The classical definition for the asymptotic phase of an oscillator breaks down when stochasticity is taken into account. Alternative definitions of the ``phase'' have been based on the mean first passage time property of a system of isochronal surfaces [4] and in terms of the spectral decomposition of a the adjoint Kolmogorov operator [5,6]. In the vanishing noise limit, the spectrally derived isochrons approach those of the underlying mean field system, if the latter has a finite period limit cycle. Together, these results expand our analytic, numerical, and conceptual tools for understanding the effects of random ion channel gating in conductance based neural models.

Stochastic maximum principle for mean-field forward-backward stochastic control system with terminal state constraints

In this paper, we consider an optimal control problem with state constraints, where the control system is described by a mean-field forward-backward stochastic differential equation (MFFBSDE, for short) and the admissible control is mean-field type. Making full use of the backward stochastic differential equation theory, we transform the original control system into an equivalent backward form, i.e., the equations in the control system are all backward. In addition, Ekeland’s variational principle helps us deal with the state constraints so that we get a stochastic maximum principle which characterizes the necessary condition of the optimal control. We also study a stochastic linear quadratic control problem with state constraints.

Finite horizon mean-field stochastic [formula omitted] control for continuous-time systems with [formula omitted]-dependent noise

Stochastic H2/H∞ control of mean-field type for continuous-time systems with state- and disturbance-dependent noise is investigated in this paper. Based on a mean-field stochastic bounded real lemma (SBRL), a sufficient condition for the solvability of continuous-time mean-field H2/H∞ control is given via the solvability of coupled matrix-valued equations. In particular, for mean-field stochastic systems with state-dependent noise, a necessary and sufficient condition for the solvability of continuous-time mean-field H2/H∞ control is presented, which generalizes the previous results to the mean-field stochastic systems.

Landau-like theory for universality of critical exponents in quasistationary states of isolated mean-field systems.

An external force dynamically drives an isolated mean-field Hamiltonian system to a long-lasting quasistationary state, whose lifetime increases with population of the system. For second order phase transitions in quasistationary states, two nonclassical critical exponents have been reported individually by using a linear and a nonlinear response theories in a toy model. We provide a simple way to compute the critical exponents all at once, which is an analog of the Landau theory. The present theory extends the universality class of the nonclassical exponents to spatially periodic one-dimensional systems and shows that the exponents satisfy a classical scaling relation inevitably by using a key scaling of momentum.

Robust Mean Field Games

Recently there has been renewed interest in large-scale games in several research disciplines, with diverse application domains as in the smart grid, cloud computing, financial markets, biochemical reaction networks, transportation science, and molecular biology. Prior works have provided rich mathematical foundations and equilibrium concepts but relatively little in terms of robustness in the presence of uncertainties. In this paper, we study mean field games with uncertainty in both states and payoffs. We consider a population of players with individual states driven by a standard Brownian motion and a disturbance term. The contribution is threefold: First, we establish a mean field system for such robust games. Second, we apply the methodology to production of an exhaustible resource. Third, we show that the dimension of the mean field system can be significantly reduced by considering a functional of the first moment of the mean field process.

Nonequilibrium inhomogeneous steady state distribution in disordered, mean-field rotator systems

We present a novel method to compute the phase space distribution in the nonequilibrium stationary state of a wide class of mean-field systems involving rotators subject to quenched disordered external drive and dissipation. The method involves a series expansion of the stationary distribution in inverse of the damping coefficient; the expansion coefficients satisfy recursion relations whose solution requires computing a matrix where about three quarters of the elements vanish, making numerical evaluation simple and efficient. We illustrate our method for the paradigmatic Kuramoto model of spontaneous collective synchronization and for its two mode generalization, in the presence of noise and inertia, and demonstrate an excellent agreement between simulations and theory for the phase space distribution.

Order-disorder transitions in time-discrete mean field systems with memory: a novel approach via nonlinear autoregressive models

In a previous study strongly nonlinear autoregressive (SNAR) models have been introduced as a generalization of the widely-used time-discrete autoregressive models that are known to apply both to Markov and non-Markovian systems. In contrast to conventional autoregressive models, SNAR models depend on process mean values. So far, only linear dependences have been studied. We consider the case in which process mean values can have a nonlinear impact on the processes under consideration. It is shown that such models describe Markov and non-Markovian many-body systems with mean field forces that exhibit a nonlinear impact on single subsystems. We exemplify that such nonlinear dependences can describe order-disorder phase transitions of time-discrete Markovian and non-Markovian many-body systems. The relevant order parameter equations are derived and issues of stability and stationarity are studied.

Traveling wave in a three-dimensional array of conformist and contrarian oscillators.

We consider a system of conformist and contrarian oscillators coupled locally in a three-dimensional cubic lattice and explore collective behavior of the system. The conformist oscillators attractively interact with the neighbor oscillators and therefore tend to be aligned with the neighbors' phase. The contrarian oscillators interact repulsively with the neighbors and therefore tend to be out of phase with them. In this paper, we investigate whether many peculiar dynamics that have been observed in the mean-field system with global coupling can emerge even with local coupling. In particular, we pay attention to the possibility that a traveling wave may arise. We find that the traveling wave occurs due to coupling asymmetry and not by global coupling; this observation confirms that the global coupling is not essential to the occurrence of a traveling wave in the system. The traveling wave can be a mechanism for the coherent rhythm generation of the circadian clock or of hormone secretion in biological systems under local coupling.

One-Dimensional Population Density Approaches to Recurrently Coupled Networks of Neurons with Noise

Mean-field systems have been previously derived for networks of coupled, two-dimensional, integrate-and-fire neurons such as the Izhikevich, adapting exponential (AdEx) and quartic integrate and fire (QIF), among others. Unfortunately, the mean-field systems have a degree of frequency error and the networks analyzed often do not include noise when there is adaptation. Here, we derive a one-dimensional partial differential equation (PDE) approximation for the marginal voltage density under a first order moment closure for coupled networks of integrate-and-fire neurons with white noise inputs. The PDE has substantially less frequency error than the mean-field system, and provides a great deal more information, at the cost of analytical tractability. The convergence properties of the mean-field system in the low noise limit are elucidated. A novel method for the analysis of the stability of the asynchronous tonic firing solution is also presented and implemented. Unlike previous attempts at stability analysis with these network types, information about the marginal densities of the adaptation variables is used. This method can in principle be applied to other systems with nonlinear partial differential equations.

From Boltzmann to random matrices and beyond

These expository notes propose to follow, across fields, some aspects of the concept of entropy. Starting from the work of Boltzmann in the kinetic theory of gases, various universes are visited, including Markov processes and their Helmholtz free energy, the Shannon monotonicity problem in the central limit theorem, the Voiculescu free probability theory and the free central limit theorem, random walks on regular trees, the circular law for the complex Ginibre ensemble of random matrices, and finally the asymptotic analysis of mean-field particle systems in arbitrary dimension, confined by an external field and experiencing singular pair repulsion. The text is written in an informal style driven by energy and entropy. It aims to be recreative and to provide to the curious readers entry points in the literature, and connections across boundaries.

Super-Potts glass: A disordered model for glass-forming liquids

We introduce a new disordered system, the Super-Potts model, which is a more frustrated version of the Potts glass. Its elementary degrees of freedom are variables that can take M values and are coupled via pair-wise interactions. Its exact solution on a completely connected lattice demonstrates that for large enough M it belongs to the class of mean-field systems solved by a one step replica symmetry breaking Ansatz. Numerical simulations by the parallel tempering technique show that in three dimensions it displays a phenomenological behaviour similar to the one of glass-forming liquids. The Super-Potts glass is therefore the first long-sought disordered model allowing one to perform extensive and detailed studies of the Random First Order Transition in finite dimensions. We also discuss its behaviour for small values of M, which is similar to the one of spin-glasses in a field.

Optimal control of mean-field jump-diffusion systems with delay: A stochastic maximum principle approach

This paper is concerned with an optimal control problem under mean-field jump-diffusion systems with delay. Firstly, some existence and uniqueness results are proved for a jump-diffusion mean-field stochastic delay differential equation and a jump-diffusion mean-field advanced backward stochastic differential equation. Then necessary and sufficient maximum principles for control systems of mean-field type and with delay are established under certain conditions. A mean-field, delayed, linear-quadratic control problem is finally discussed using the obtained maximum principles.

Computing Earthquake Probabilities on Global Scales

Large devastating events in systems such as earthquakes, typhoons, market crashes, electricity grid blackouts, floods, droughts, wars and conflicts, and landslides can be unexpected and devastating. Events in many of these systems display frequency-size statistics that are power laws. Previously, we presented a new method for calculating probabilities for large events in systems such as these. This method counts the number of small events since the last large event and then converts this count into a probability by using a Weibull probability law. We applied this method to the calculation of large earthquake probabilities in California-Nevada, USA. In that study, we considered a fixed geographic region and assumed that all earthquakes within that region, large magnitudes as well as small, were perfectly correlated. In the present article, we extend this model to systems in which the events have a finite correlation length. We modify our previous results by employing the correlation function for near mean field systems having long-range interactions, an example of which is earthquakes and elastic interactions. We then construct an application of the method and show examples of computed earthquake probabilities.

Finite-size effects in a mean-field kinetically constrained model: dynamical glassiness and quantum criticality

On the example of a mean-field Fredrickson–Andersen kinetically constrained model, we focus on the known property that equilibrium dynamics take place at a first-order dynamical phase transition point in the space of time realizations. We investigate the finite-size properties of this first-order transition. By discussing and exploiting a mapping of the classical dynamical transition—an argued glassiness signature—to a first-order quantum transition, we show that the quantum analogy can be exploited to extract finite-size properties, which in many respects are similar to those in genuine mean-field quantum systems with a first-order transition. We fully characterize the finite-size properties of the order parameter across the first-order transition.

Cascades on a stochastic pulse-coupled network.

While much recent research has focused on understanding isolated cascades of networks, less attention has been given to dynamical processes on networks exhibiting repeated cascades of opposing influence. An example of this is the dynamic behaviour of financial markets where cascades of buying and selling can occur, even over short timescales. To model these phenomena, a stochastic pulse-coupled oscillator network with upper and lower thresholds is described and analysed. Numerical confirmation of asynchronous and synchronous regimes of the system is presented, along with analytical identification of the fixed point state vector of the asynchronous mean field system. A lower bound for the finite system mean field critical value of network coupling probability is found that separates the asynchronous and synchronous regimes. For the low-dimensional mean field system, a closed-form equation is found for cascade size, in terms of the network coupling probability. Finally, a description of how this model can be applied to interacting agents in a financial market is provided.

Energy density matrix formalism for interacting quantum systems: Quantum Monte Carlo study

We develop an energy density matrix that parallels the one-body reduced density matrix (1RDM) for many-body quantum systems. Just as the density matrix gives access to the number density and occupation numbers, the energy density matrix yields the energy density and orbital occupation energies. The eigenvectors of the matrix provide a natural orbital partitioning of the energy density while the eigenvalues comprise a single-particle energy spectrum obeying a total energy sum rule. For mean-field systems the energy density matrix recovers the exact spectrum. When correlation becomes important, the occupation energies resemble quasiparticle energies in some respects. We explore the occupation energy spectrum for the finite 3D homogeneous electron gas in the metallic regime and an isolated oxygen atom with ground-state quantum Monte Carlo techniques implemented in the qmcpack simulation code. The occupation energy spectrum for the homogeneous electron gas can be described by an effective mass below the Fermi level. Above the Fermi level evanescent behavior in the occupation energies is observed in similar fashion to the occupation numbers of the 1RDM. A direct comparison with total energy differences shows a quantitative connection between the occupation energies and electron addition and removal energies for the electron gas. For the oxygen atom, the association between the ground-state occupation energies and particle addition and removal energies becomes only qualitative. The energy density matrix provides an avenue for describing energetics with quantum Monte Carlo methods which have traditionally been limited to total energies.

Mean field analysis gives accurate predictions of the behaviour of large networks of sparsely coupled and heterogeneous neurons

Large networks of integrate-and-fire (IF) model neurons are often used to simulate and study the behaviour of biologically realistic networks. However, to fully study the large network behaviour requires an exploration of large regions of a multidimensional parameter space. Such exploration is generally not feasible with large network models, due to the computational time required to simulate a network with biologically significant size. To circumvent these difficulties we use a mean-field approach, based on the work of [1]. We consider a sparsely coupled, excitatory network of 10,000 Izhikevich model neurons [2], with Destexhe-type synapses [3]. The cellular models were fit to hippocampal CA1 pyramidal neurons and have heterogeneous applied currents with a normal distribution. We derived a mean-field system for the network which consists of differential equations for the mean of the adaptation current and the synaptic conductance. As CA1 is an area that displays prominent theta oscillations [4], we used the mean-field system to study how the frequency of bursting depends on various model parameters. Figure ​Figure1A1A shows an example study. These studies were successful in guiding numerical simulations of the large network. When parameter values determined from the mean-field analysis are used in a large network simulation, bursting of the predicted frequency occurs (Figure ​(Figure1B1B). Figure 1 A. Mean-field prediction of the bursting frequency as a function of the unitary synaptic conductance and the mean applied current. B. Simulation of a network of 10,000 neurons with unitary conductance 0.058 nS and mean current 250 pA, showing an oscillation ...