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A constructive mean field analysis of multi population neural networks with random synaptic weights and stochastic inputs

Olivier Faugeras, Jonathan Touboul, Bruno Cessac

arXiv:0808.1113v2physics.bio-phcond-mat.dis-nnmath-phmath.MPq-bio.NC

TL;DR

Deriving mesoscopic neural equations from individual-neuron dynamics is challenging, while mesoscopic descriptions matter for understanding measurements that reflect many neurons. The paper formulates mean-field behavior as functional equations on stochastic processes and proves existence and uniqueness with a constructive solution method, subject to limits on the stationary and spiking-neuron results.

  • Problem

    Deriving mean-field evolution equations from microscopic dynamics is difficult, and stationary solutions are not known to exist in general.

    Source: §1 · pp. 2–3

  • Method

    The authors treat mean-field equations as functional equations on stochastic processes and construct solutions by iterating a map from an initial process.

    Source: Conclusion · p. 33, §6 · p. 51

  • Results

    A unique mean-field solution exists for every nondegenerate Gaussian initial condition, for any initial and final times.

    Source: Conclusion · p. 33

  • Takeaways & Limitations

    The model allows smooth bounded interaction functions, making it more general than the instantaneous rate model underlying Ermentrout’s voltage-based model.

    Source: §6 · p. 51

    The spiking-neuron extension remains obstructed by the need to characterize parameter regions where neurons can take arbitrarily long to fire for the first time.

    Source: §7 · p. 53

Abstract

from arXiv · show

We deal with the problem of bridging the gap between two scales in neuronal modeling. At the first (microscopic) scale, neurons are considered individually and their behavior described by stochastic differential equations that govern the time variations of their membrane potentials. They are coupled by synaptic connections acting on their resulting activity, a nonlinear function of their membrane potential. At the second (mesoscopic) scale, interacting populations of neurons are described individually by similar equations. The equations describing the dynamical and the stationary mean field behaviors are considered as functional equations on a set of stochastic processes. Using this new point of view allows us to prove that these equations are well-posed on any finite time interval and to provide a constructive method for effectively computing their unique solution. This method is proved to converge to the unique solution and we characterize its complexity and convergence rate. We also provide partial results for the stationary problem on infinite time intervals. These results shed some new light on such neural mass models as the one of Jansen and Rit \cite{jansen-rit:95}: their dynamics appears as a coarse approximation of the much richer dynamics that emerges from our analysis. Our numerical experiments confirm that the framework we propose and the numerical methods we derive from it provide a new and powerful tool for the exploration of neural behaviors at different scales.

1 Introduction

The paper develops a rigorous, constructive population-level mean-field framework for multi-population neural networks with stochastic inputs, addressing finite-time dynamics and stationary solutions. It examines how random synaptic variability and nonlinear averaging shape neural-mass descriptions, including in the Jansen–Rit model.

1 Introduction

  • For large neuronal populations, averaging can yield a mesoscopic mean-field description with a few effective parameters rather than a separate parameter set for every neuron.Such models are motivated by measurements and structures, including cortical columns, that reflect collective activity.Source: §1 · p. 2
  • Dynamic mean-field methods address networks with random independent synaptic weights and capture fluctuations and correlations that distinguish stable and chaotic regimes even when both have zero mean.The introduction notes that rigorous results using this method remain limited.Source: §1 · p. 3
  • Prior dynamic mean-field approaches are computationally demanding, difficult to extend to multiple populations, and leave stationary-solution existence and characterization unclear.The stationary formulation also requires an initial condition that can only be characterized through the nonstationary case; the introduction identifies Crisanti and coworkers as the only prior investigation of the stationary question it knew of.Source: §1 · p. 3

mean-field equations.

  • The paper develops a stochastic-analysis approach for neural-mass evolution and studies models including a simple network and a neuronal assembly exemplified by Jansen and Rit’s cortical column model.It derives equations from neuron dynamics but focuses on proving their well-posedness and providing an algorithm to compute solutions.Source: §1 · p. 4, mean-field equations · p. 4
  • Under mild assumptions, the dynamic mean-field equations have a unique solution on a finite time interval given an initial condition; a constructive algorithm also computes stationary solutions for the simple model.Both proofs are constructive and provide algorithms for numerical computation.Source: mean-field equations · pp. 4–5

The dynamics

  • The dynamics add Brownian noise to the first k −1 equations, a technical assumption needed for well-posedness proofs that also generates a rich class of stochastic inputs.When the added noise vanishes, the system recovers the derivative interpretation of those components and the original equations.Source: The dynamics · pp. 8–9

2.2 The Mean-Field equations

2.2.1 General derivation of the mean-field equation

  • The authors derive the mean-field equations, prove they are well-posed, and provide an algorithm for computing their solution.Their population-by-population description relies on neurons within each population being interchangeable under the joint law of connectivity weights and Brownian motion.Source: §2.2.1 · p. 12
  • Random synaptic weights yield time-dependent Gaussian interaction fields that depend on the full past, so neural-mass trajectories can depart strongly from naive mean-field predictions.The naive projection is valid when synaptic weights do not fluctuate, but generally fails for random weights.Source: §2.2.1 · p. 13

2.2.2 Example I: The Simple Model

  • With zero synaptic-weight variability and zero external-noise amplitude, the simple model’s mean-field equations reduce to the naive equations; generally, solving for the mean requires the process’s whole past.The dependence on past values illustrates the non-Markovian nature of the mean-field solution.Source: §2.2.2 · p. 18

2.2.3 Example II: The model of Jansen and Rit

  • In the Jansen–Rit model, connectivity weights are originally treated as constant, but their variability affects the solution at the neural-mass level.The model represents pyramidal, excitatory-feedback, and inhibitory-interneuron populations.Source: §2.2.3 · p. 18
  • First-order synaptic integration yields exponentially decaying postsynaptic potentials and a first-order Jansen–Rit equation, with distinct parameters for inhibitory versus pyramidal and excitatory feedback populations.The synaptic time constant and efficiency define the g-shape; the inhibitory population uses different values from the pyramidal and excitatory feedback populations.Source: §2.2.3 · p. 20
  • The original Jansen–Rit equations [23, 27] follow by taking the mean of the mean-field process and commuting expectation with the sigmoid, a strong assumption when solutions fluctuate.Because the sigmoid is generally nonlinear, the expectation need not commute with it.Source: §2.2.3 · p. 21
  • The higher-order model of van Rotterdam and colleagues uses second-order synaptic integration to better represent synaptic integration and real postsynaptic-potential characteristics.Its g-shapes satisfy a second-order differential equation.Source: §2.2.3 · p. 21
  • Higher-order synaptic integration can yield richer behavior: bifurcations producing epileptic-like oscillations and alpha activity appear in the second-order model but not the first-order model.Grimbert and Faugeras reported these behaviors for the second-order Jansen–Rit model.Source: §2.2.3 · p. 21

3 Existence and uniqueness of solutions in finite time

This section establishes existence and uniqueness of finite-time mean-field solutions under mild boundedness and nondegeneracy assumptions, using Gaussian-process iterates. It proves convergence of their means and covariances and gives a constructive iteration method applicable on any interval [t0, T].

3.1 Convergence of Gaussian processes

  • For Gaussian-process convergence, the proof uses uniform tightness together with uniform convergence of the mean functions and covariance operators.The section recalls this convergence criterion before defining the sequence of Gaussian processes.Source: §3.1 · p. 22

3 Existence and uniqueness of solutions in finite time

  • The mean-field equation has a unique solution under the stated assumptions, and the Gaussian-process iteration converges in distribution to its unique fixed point.The assumptions include bounded continuous L(t), singular values of F(t) bounded above and away from zero, and bounded external input I(t).Source: §3 · pp. 21–22, §3.1 · p. 23

3.2 Existence and uniqueness of a solution for the mean-field equations

  • The proof proceeds in two steps: establish uniform tightness of the iterates, then show their mean functions and covariance operators are Cauchy in the uniform norm.Together these steps establish the conditions needed for convergence of the Gaussian-process sequence.Source: §3.2 · p. 24

3.2.1 Uniform tightness

  • The Gaussian-process iterates are shown to satisfy Kolmogorov’s tightness criterion, yielding uniform tightness on the finite time interval.The argument verifies the criterion with β = 4, using bounded input current among its conditions.Source: §3.2.1 · pp. 24, 27

3.2.2 The mean and covariance sequences are Cauchy sequences

  • The iterates’ covariance and mean functions are Cauchy sequences in the uniform norm; an integral bound of kn/(n −1)! supports the convergence argument.The proof bounds successive differences recursively, using the stated lemmas and a convergent series.Source: §3.2.2 · p. 29
  • The mean and covariance sequences are Cauchy because successive covariance differences are bounded by a constant to the nth power divided by (n−1)!.The proof obtains the corresponding Cauchy property for the mean sequence as well.Source: §3.2.2 · pp. 32–33

3.2.3 Existence and uniqueness of a solution of the mean-field equations

  • For any nondegenerate kP-dimensional Gaussian Z0 independent of Brownian motion, Fk has a unique fixed point, and Gaussian iterates initialized at X(t0)=Z0 converge to it in law.The fixed point is the solution of the mean-field equations.Source: §3.2.3 · p. 33

Conclusion

  • The constructive proof computes the solution by iterating Fk from any initial process X satisfying X(t0)=Z0, and existence and uniqueness hold for any t0 and T.An Ornstein–Uhlenbeck process is given as one possible starting process.Source: Conclusion · p. 33

4 Existence and uniqueness of stationary solutions

The stationary mean-field problem is studied through a constructive fixed-point framework. Under contraction conditions, it has a unique stationary solution and iteration converges to that solution in law.

4 Existence and uniqueness of stationary solutions

  • A stationary solution has a probability distribution unchanged by the equation's flow; under a contraction condition, the framework gives a constructive way to find and simulate it.The paper calls this equation the long-term mean-field equation.Source: §4 · pp. 34–35
  • The stationary analysis formally takes the neuron-number limit before sending the initial time to minus infinity, and these limits need not commute.The paper notes known noncommuting examples in spin glasses.Source: §4 · p. 34
  • The stochastic convolution is well-defined, Gaussian, and stationary as the initial time tends to minus infinity under the stated assumptions.Its limiting covariance is a function of the time difference.Source: §4 · p. 35
  • Under contraction conditions, the long-term mean-field function has a unique stationary fixed point, and its iterates converge in law to that solution.The convergence holds for iteration from any process.Source: §4 · p. 40
  • Stationary existence and uniqueness are not guaranteed without the contraction condition; too-small leaks can lead to divergence or multiple fixed points.The paper associates too-small leaks with long membrane-potential decay time constants.Source: §4 · pp. 39, 41

5 Numerical experiments

The section develops an iterative method for computing mean-field solutions and analyzes its convergence, then tests it on one- and two-population examples. The experiments show that fluctuations can materially change the dynamics of the mean membrane potential.

5.1 Simulation algorithm

  • Iterations of Fk from any initial condition converge to the mean-field solution under the stated noise-covariance assumptions, providing the basis for the numerical algorithm.For Gaussian processes, computing the iterate’s law amounts to computing its mean and covariance functions.Source: §5.1 · p. 41

5.1.1 Computing Fk.

  • The algorithm computes the mean and covariance of F1(X) from the law of Gaussian process X, with the interaction covariance requiring a two-time function and the mean generally retaining whole-past dependence.The covariance equation cannot be reduced to an ordinary differential equation unless stationarity is assumed.Source: §5.1.1 · pp. 42–43

5.1.2 Analysis of the algorithm

  • The computed uniform error in mean and covariance is bounded by O((N + T)dt + R_N(kmax)), combining discretization error with an iteration remainder.The discretization scheme converges in O(dt), and the remainder decreases rapidly with N.Source: §5.1.2 · p. 44

5.2.1 Stationary solutions

  • In the stationary one-population analysis, the potential-based equation is self-consistent because q = C(0) depends on the solution; the cited analysis distinguishes a zero-covariance regime from a positive-covariance chaotic regime.The reported boundary condition is g²J²⟨S′⟩²q ≥ 1 for the homoclinic trajectory.Source: §5.2.1 · pp. 45–46

5.2.2 Numerical experiments

  • Without external noise, covariance degeneracy coincides with numerical instabilities; adding small noise removes them, while the resulting solution no longer matches Sompolinsky et al.’s theory.The no-noise iterations still produce a curve well fitted by the theoretical results.Source: §5.2.2 · p. 47, Figure 3 · p. 47

5.3.1 Theoretical framework.

  • For two populations, Gaussian fluctuations make fixed-point stability depend on v(t), which can affect oscillation periods and allow complex dynamics beyond the naive mean-field picture.Without fluctuations, the examined oscillatory regime has a Hopf bifurcation; with J ≠ 0, the simulations show periodic oscillations in both µ1(t) and v1(t).Source: §5.3.1 · p. 49, §5.3.2 · p. 50

5.3.2 Numerical experiments

  • With J ≠ 0, small Gaussian fluctuations accompany periodic oscillations of both µ1(t) and v1(t), unlike the J = 0 case with regular µ1(t) oscillations and stable v1(t) = 0.Figure 4 magnifies v1(t) by a factor of 100.Source: §5.3.2 · p. 50, Figure 4 · p. 50

6 Discussion

The discussion presents a constructive mean-field framework for neuronal populations and argues that its solutions retain non-Markovian mean–fluctuation dynamics. In the sigmoidal case, existing neural mass models appear as approximations that neglect this coupling.

6 Discussion

  • The framework derives mean-field equations from a microscopic model with stochastic and deterministic inputs and random connectivity, and proves existence and uniqueness under mild assumptions.A sufficient convergence condition is a full-rank noise matrix with bounded eigenvalues.Source: §6 · pp. 50–51
  • The constructed mean-field solutions are fundamentally non-Markovian, avoiding an ad hoc parameter such as q to summarize the process’s history.Source: §6 · p. 51
  • For sigmoidal nonlinearities, neural mass models approximate the mean-field equations by neglecting the coupling between mean membrane-potential changes and fluctuations represented by covariance functions.Source: §6 · p. 51
  • The discussion identifies integrate-and-fire models as a possible comparison: their history-cutoff representation removes the need for an ad hoc parameter summarizing the non-Markovian process.The authors suggest comparing the approaches to provide a rigorous basis for that analysis and examine field fluctuations.Source: §6 · p. 52

7 Conclusion and further work

The paper outlines future work on analyzing mean-field dynamics, investigating non-stationary regimes, and extending the theory to spiking-neuron models.

7 Conclusion and further work

  • The authors identify several directions for extending the present work.These include bifurcation analysis, non-stationary regimes, and a larger class of models.Source: §7 · pp. 52–53
  • The authors propose bifurcation analysis of the trajectory-space map and ask whether generalized mean-field equations reveal regimes absent from Jansen–Rit equations.They also identify experimental investigation of any such regimes as future work.Source: §7 · p. 52
  • Stationary-solution existence and uniqueness are proved only for low values of gain g, while the approach applies on any finite time interval.The authors identify transient regimes, including networks with non-stationary inputs, as a direction for investigation.Source: §7 · p. 53
  • Applying the theory to spiking models, including integrate-and-fire neurons with conductance-based synapses, remains challenging.The main obstruction is characterizing parameter regions where neurons can take arbitrarily long to fire for the first time.Source: §7 · p. 53

Aknowledgement

The research received support from European Union, foundation, and research-network funding sources.

Aknowledgement

A Identification of the mean-field equations

This appendix relates a large-network spin-glass approach to the paper’s mean-field equation and establishes convergence to a unique solution in the one-population case. It also notes that extending the proof to multiple populations is outside the paper’s scope.

A Identification of the mean-field equations

  • For one population, the authors adapt the spin-glass proof approach despite the model’s nonlinear intrinsic dynamics and linear interaction.The cited spin-glass work obtained a weak limit for a spin’s law and proved independence.Source: A Identification of the mean-field… · p. 54
  • The empirical measure satisfies a large deviation principle, and annealed and quenched tightness lead to convergence toward the unique process minimizing the rate function.The limiting process has a density with respect to the no-interaction Ornstein–Uhlenbeck law, satisfying an implicit equation.Source: A Identification of the mean-field… · p. 54
  • The implicit equation has a unique solution, which satisfies a nonlinear non-Markovian stochastic differential equation.Source: A Identification of the mean-field… · p. 54
  • The resulting equation can also be written as the mean-field equation averaged over connectivities.Source: A Identification of the mean-field… · p. 55
  • Extending the proof to multiple populations using multidimensional Girsanov’s theorem is described as possible but left outside the paper’s scope.Source: A Identification of the mean-field… · p. 55

B The resolvent

The appendix defines the resolvent for a homogeneous differential equation and gives its properties. It then uses the resolvent to express solutions of an inhomogeneous linear stochastic differential equation.

B The resolvent

  • The resolvent is defined as the unique solution of a linear equation, with the identity matrix specified as the P × P identity.Source: B The resolvent · p. 55
  • The resolvent’s listed properties include a composition rule and bounds under a bound on the coefficient matrix.The proof attributes the flow properties to reversible ODEs and the bound to Gronwall’s lemma.Source: B The resolvent · pp. 55–56
  • The inhomogeneous linear SDE solution can be written using the resolvent, and the proof establishes pathwise uniqueness.The proof verifies the solution using Itô’s formula.Source: B The resolvent · p. 56

C Matrix norms

This section recalls vector and matrix norms and specifies the norms used in the article.

C Matrix norms

  • Given a vector norm on R^n, the corresponding induced matrix norm is defined on M_n×n.The matrix space is finite-dimensional, so all norms are equivalent.Source: C Matrix norms · p. 57
  • The spectral norm of L is the square root of the largest singular value of L, equivalently the largest eigenvalue of L^T L.For positive definite L, the passage identifies this with its largest eigenvalue, also called its spectral radius.Source: C Matrix norms · p. 57

D Important Constants

Table 1 collects important notation introduced in the article and used in multiple places.

D Important Constants

E Proof of lemma 3.5

Lemma E.1 gives upper bounds valid for every n ≥ 1 and s,t in [t0,T], with its proof deriving the inequalities from earlier assumptions and results.

E Proof of lemma 3.5

  • Lemma E.1 states upper bounds valid for all n ≥ 1 and s,t ∈ [t0,T].The displayed bounds are not included in the supplied passage.Source: E Proof of lemma 3.5 · p. 58
  • The proof obtains its first and second inequalities by taking infinite norms of equations (41) and (42), respectively, and applying assumptions and earlier results.The first uses assumption 1 and assumption 3 in section 3.1 with equation (64) and Lemma 3.2; the second uses assumptions 1 and 2 with equations (64)–(65) and Lemma 3.2.Source: E Proof of lemma 3.5 · p. 58

F Proof of lemma 3.6

Lemma F.1 gives bounds over the finite interval, and its proof establishes positivity using the nondegenerate initial Gaussian and invertibility of the transition matrix.

F Proof of lemma 3.6

  • Lemma F.1 applies for every t ∈ [t0,T], α = 1,…,kP, and n ≥ 1.The supplied passages do not include the displayed bound itself.Source: F Proof of lemma 3.6 · p. 58
  • The proof lower-bounds a diagonal covariance term using the smallest eigenvalue of the propagated covariance matrix.The initial covariance is positive definite because Z0 is assumed to be a nondegenerate Gaussian, and ΦL(t,t0) is invertible on the interval, making the relevant smallest eigenvalues strictly positive.Source: F Proof of lemma 3.6 · p. 58

G Proof of lemma 3.7

The proof of Lemma G.1 rewrites equation (42) using the resolvent’s group property, then represents the relevant covariance terms through transformed basis vectors. It derives nonnegativity from covariance structure and uses a positive singular-value lower bound in its estimates.

G Proof of lemma 3.7

  • The estimates invoke the strictly positive lower bound on the singular values of F(u) over [t0,T].The proof writes Γ(u) = F(u)F(u)T under assumption 3.1.Source: G Proof of lemma 3.7 · pp. 59–60
  • The proof rewrites equation (42) using the resolvent’s group property before assuming s < t and introducing notation for the argument.It defines Eα(t) from the transposed resolvent applied to the canonical basis vector eα.Source: G Proof of lemma 3.7 · p. 59
  • The covariance terms Cαα(t,s), Cαα(s,s), and Cαα(t,t) are expressed using Eα and covariance matrices.The last expression is independent of s because A(s) + B(s,t) = A(t).Source: G Proof of lemma 3.7 · p. 59
  • The proof establishes nonnegativity of intermediate expressions using the covariance-matrix properties of a, A, and B.It uses E1(s,t) ≥ 0 and E2(s,t) ≥ 0 to infer E3(s,t) ≥ 0, then lower-bounds a covariance determinant expression.Source: G Proof of lemma 3.7 · pp. 59–60

H Proof of lemma 3.8

The proof of Lemma H.1 bounds a 2n-dimensional integral by a factorial-decaying expression. It reduces the estimate using bounds on the integrand, symmetry, and a change of variables.

H Proof of lemma 3.8

  • The integral is upper-bounded by kn/(n −1)! for a positive constant k when each ρi is either 1 or 1/θ(ui,vi).The proof notes that the integral is well-defined by Lemma 3.7.Source: H Proof of lemma 3.8 · p. 61
  • The proof reduces the integral bound to the case where every ρi(ui,vi) equals 1/|ui −vi|, up to a constant factor.It sets t0 = 0 without loss of generality and uses the cases n = 1, 2, 3 to illustrate the process.Source: H Proof of lemma 3.8 · p. 61
  • Symmetry in s and t and a change of variables rewrite the integral into a form illustrated by Figure 5.The figure is identified as showing the change of coordinates.Source: H Proof of lemma 3.8 · p. 61, Figure 5 · p. 62
  • The proof bounds parts of the transformed integral using the relation u ∨ v = v = V − U in the integration region.It also reports that symbolic computation verifies 0 < αi < 1 for all i ≥ 1.Source: H Proof of lemma 3.8 · pp. 61–62
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