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Noise-induced behaviors in neural mean field dynamics

Jonathan Touboul, Geoffroy Hermann, Olivier Faugeras

arXiv:1104.5425v2math.DSq-bio.NC

TL;DR

Large neural assemblies have collective behaviors shaped by neuronal noise, but analyzing those effects is difficult. The paper derives a probabilistic mean-field model whose Gaussian solutions yield closed ODEs for their moments, then compares its regimes with finite-network simulations. Those simulations agree well even for networks of a few hundred to a few thousand neurons, with transitions matching mean-field bifurcations.

  • Problem

    Analyzing noise effects in large neural assemblies is difficult because rigorous mean-field equations are generally implicit stochastic-process or probability-distribution equations.

    Source: Introduction · pp. 1, 3

  • Method

    A probabilistic infinite-population limit yields Gaussian mean-field solutions whose means and variances satisfy closed nonlinear ODEs.

    Source: Introduction · p. 2, §4 · p. 23

  • Results

    Finite-network simulations agree very well with mean-field solutions for networks of a few hundred to a few thousand neurons, and qualitative transitions match mean-field bifurcations.

    Source: §4 · p. 24

  • Takeaways & Limitations

    Regular oscillations under increasing noise are a collective effect absent from the described two-neuron networks.

    Source: §3.3 · p. 21

    The rate model describes population activity rather than precise individual-neuron activity, and realistic individual-neuron models will not generally yield Gaussian mean-field solutions.

    Source: §4.3 · p. 27

Abstract

from arXiv · show

The collective behavior of cortical neurons is strongly affected by the presence of noise at the level of individual cells. In order to study these phenomena in large-scale assemblies of neurons, we consider networks of firing-rate neurons with linear intrinsic dynamics and nonlinear coupling, belonging to a few types of cell populations and receiving noisy currents. Asymptotic equations as the number of neurons tends to infinity (mean field equations) are rigorously derived based on a probabilistic approach. These equations are implicit on the probability distribution of the solutions which generally makes their direct analysis difficult. However, in our case, the solutions are Gaussian, and their moments satisfy a closed system of nonlinear ordinary differential equations (ODEs), which are much easier to study than the original stochastic network equations, and the statistics of the empirical process uniformly converge towards the solutions of these ODEs. Based on this description, we analytically and numerically study the influence of noise on the collective behaviors, and compare these asymptotic regimes to simulations of the network. We observe that the mean field equations provide an accurate description of the solutions of the network equations for network sizes as small as a few hundreds of neurons. In particular, we observe that the level of noise in the system qualitatively modifies its collective behavior, producing for instance synchronized oscillations of the whole network, desynchronization of oscillating regimes, and stabilization or destabilization of stationary solutions. These results shed a new light on the role of noise in shaping collective dynamics of neurons, and gives us clues for understanding similar phenomena observed in biological networks.

Introduction.

Large neural assemblies can produce coherent population-level responses, but how noise shapes their collective behavior is difficult to analyze. This paper derives a probabilistic mean-field description whose Gaussian solutions reduce the analysis to closed ODEs for their moments.

  • Large neural assemblies matter because collective dynamics can produce mesoscopic signals associated with cortical columns and areas.Cortical columns span about 50µm to 1mm and contain hundreds to hundreds of thousands of neurons.Source: Introduction · p. 1
  • Traditional continuum models represent network activity by a deterministic population-averaged firing rate, implicitly assuming noise vanishes in large populations.These models have been used to study phenomena including spatio-temporal pattern formation.Source: Introduction · p. 2
  • Other approaches incorporate stochastic activity, but population-density methods may require moment-hierarchy truncation, while many Markov-chain models use one-neuron updates and analyze only finitely many moments.The cited passage notes technical concerns about closure, biological plausibility, and adequacy of finite-moment equations.Source: Introduction · p. 2
  • The paper derives infinite-population limit equations probabilistically from microscopic firing-rate neurons, without assuming sparse connectivity, and accounts for distinct neuronal populations.Because the resulting mean-field solutions are Gaussian, their first two moments satisfy ODEs, enabling analysis of parameter and noise effects.Source: Introduction · pp. 2–3
  • The mean-field equations for nonlinear networks with multiple neuronal populations are difficult to analyze because they are implicit stochastic-process equations or non-local equations on probability distributions.Even basic properties and the qualitative effects of noise are challenging to establish.Source: Introduction · pp. 2–3

1. Model and mean field equations.

The paper models noisy, nonlinearly coupled firing-rate neurons in several populations and derives their large-network limit. Under the stated assumptions, the limit has Gaussian solutions whose moments obey a closed ODE system, with uniform-in-time convergence properties.

1. Model and mean field equations.

  • The model uses multiple populations of firing-rate neurons with linear intrinsic dynamics, nonlinear sigmoidal coupling, and additive noisy currents.Synaptic weights are scaled by the size of the presynaptic population so total input does not depend on network size; random synaptic weights are not covered.Source: §1 · pp. 3–4
  • As network size grows, finite sets of neurons converge to independent mean-field processes, with uniform-in-time convergence established for this system.The result applies simultaneously across populations with nontrivial limiting proportions.Source: §1 · pp. 4, 5, 7, 8
  • Under Gaussian initial conditions, the mean-field solutions remain Gaussian and their means and variances satisfy a closed system of coupled ODEs.This replaces analysis of the implicit stochastic equations with a finite-dimensional dynamical system.Source: §1 · pp. 8, 10
  • Uniform propagation of chaos implies that the empirical mean converges uniformly in time to the deterministic mean-field mean.For non-Gaussian initial conditions, the mean-field solution converges exponentially toward a Gaussian solution.Source: §1 · p. 9
  • For an erf sigmoid, the Gaussian expectation term in the moment equations has a closed-form expression.The paper assumes smooth, increasing sigmoid transforms that approach 0 and 1 at the two extremes.Source: §1 · pp. 4, 10

2. Noise-induced phenomena.

The section uses mean-field ODEs to analyze how noise alters neuronal population dynamics. Depending on the network and noise level, it can stabilize stationary states, suppress oscillations, or induce cycles.

2. Noise-induced phenomena.

  • Noise smooths the sigmoid by reducing its slope, shifting the bifurcations that determine the system’s collective behavior.The section analyzes these effects through mean-field ODEs, justified by uniform convergence of the empirical mean and propagation of chaos.Source: §2 · pp. 10–11

2.1. The external noise can destroy a pitchfork bifurcation.

  • For J > 0 and λ < J/√π, the mean-field null state changes stability at g = g*; for larger λ, it remains stable for any g.This noise-dependent threshold replaces the deterministic pitchfork threshold.Source: §2.1 · p. 12

2.2. The external noise can destroy oscillations.

  • For J > 0 and λ < J/√π, the mean-field system undergoes a supercritical Hopf bifurcation at g = g*, beyond which a stable periodic orbit exists.For λ > J/√π, the null state is stable for any g, so noise can eliminate the oscillatory regime.Source: §2.2 · p. 13, Figure 2.1 · p. 14

2.3. The external noise can induce oscillations.

  • As noise increases, large-amplitude, slow cycles can emerge, then shrink and speed up before disappearing; the two-population system’s bifurcation diagram also contains cusps and a Bogdanov-Takens point.For fixed I1 = 0, these transitions proceed through bistability and loss of the stable fixed point before a supercritical Hopf bifurcation restores a single fixed point.Source: §2.3 · p. 15
  • Across six noise intervals, cycles exist for some inputs at low noise but none exist for any input at sufficiently high noise.The authors suggest using these partitions to evaluate a noise range by varying input to the excitatory population.Source: §2.3 · p. 15

3. Back to the network dynamics.

The section tests whether finite stochastic networks reproduce the regimes predicted by the mean-field analysis. Across the simulations, network statistics and noise-dependent transitions closely track the mean-field solutions.

3. Back to the network dynamics.

  • The noise-dependent diagram distinguishes fixed-point, periodic, and bistable regimes, with a small λ interval where cycles coexist with a fixed point near the saddle-homoclinic orbit.The diagram varies I1 and λ; at I1 = 0, it shows saddle-node, Hopf, and saddle-homoclinic bifurcations.Source: Figure 2.2 · p. 16

3.1. Numerical Simulations.

  • Simulations use Euler–Maruyama with a fixed time step below 0.01; computation time grows approximately linearly with network size after cache memory saturates.The authors identify trajectory-storage memory as the main constraint on simulating very large networks.Source: §3.1 · pp. 16–17

3.2. A one population case.

  • For the pitchfork, loss of zero-state stability occurs at the predicted λ for networks as small as 50 neurons, while simulated bifurcation values are close to mean-field values above 250.Finite simulation time smooths the pitchfork in both network and mean-field simulations.Source: §3.2 · pp. 17–18, Figure 3.2 · p. 18

3.3. Two populations case and oscillations.

  • For λ ∈ [1.12, 1.33], stable periodic orbits coexist with stable fixed points; in network simulations, initial conditions determine which regime is reached.The mean-field system has a unique stable limit cycle for λ ∈ [1.33, 1.97] and a unique attractive fixed point for λ > 1.97.Source: §3.3 · p. 18
  • During the oscillatory regime, neurons oscillate in phase, producing coherent network activity, while individual trajectories form a tube around the mean-field cycle.The phase does not depend on the realization; the tube grows in size with λ.Source: §3.3 · p. 19
  • At 10,000 neurons, network simulations reproduce the mean-field oscillation onset and disappearance at matching λ values, with a similar route to oscillations.The authors conclude that the mean-field equations accurately reproduce network dynamics at this size; near the homoclinic point, precise spectral characterization would require very long simulations.Source: §3.3 · pp. 19, 21

4. Discussion.

The paper derives Gaussian mean-field dynamics with closed moment equations and compares them with finite-network simulations. It argues that noise can structure collective neural activity, while noting that the model’s simplicity limits its representation of individual neurons.

4. Discussion.

  • Finite networks of a few hundred to a few thousand neurons agree well with the mean-field solutions, and their qualitative transitions match the predicted bifurcations.Figure 4.1 compares empirical voltage distributions with the theoretical mean-field distribution.Source: §4 · p. 24, Figure 4.1 · p. 24

4.1. Noise-induced phenomena.

  • At particular noise levels, regular mean-rate oscillations and network-wide synchronization can emerge even when the noiseless system has a stable fixed point.The authors distinguish these transitions from stochastic and coherence resonance, which they say their system does not exhibit.Source: §4.1 · pp. 24–25
  • The noise-induced transitions arise from the particular form of these mean-field equations and generally do not occur in standard stochastic differential equations.The authors also propose that bifurcation classes could help infer noise levels from responses to different stimuli.Source: §4.1 · pp. 25–26

4.2. Understanding the functional role of noise.

  • Noise-triggered synchronized oscillations resemble the sudden, large-amplitude, low-frequency oscillations associated with epileptic seizures, suggesting a possible cortical-mass-model application.The model’s saddle-homoclinic bifurcation produces this oscillatory regime as noise increases.Source: §4.2 · p. 26

4.3. Perspectives.

  • The linear intrinsic dynamics make quantitative analysis possible, but the rate model may describe population activity better than individual neurons.The authors suggest trial-averaged firing rates or interpreting each diffusion as a whole cortical column as alternative viewpoints.Source: §4.3 · pp. 26–27
  • For more realistic individual-neuron models, mean-field solutions will generally not be Gaussian, and analyzing them requires harder process-level or Fokker–Planck approaches.The authors identify nonlinear intrinsic dynamics and multiple ionic populations as possible extensions.Source: §4.3 · pp. 26–27
  • With random synaptic weights frozen during network evolution, propagation of chaos still applies in the simple model, but the mean and covariance no longer close as ordinary differential equations.The authors also identify stochastic synaptic weights as a route to multiplicative noise and possible new transitions.Source: §4.3 · p. 27

Appendix A. Proof of lemma 1.3.

The appendix proves a simplification of the mean-field functions for erf-shaped sigmoidal transforms by evaluating an integral after a geometric change of variables.

Appendix A. Proof of lemma 1.3.

  • For sigmoidal transforms Sα(x) = erf(gαx + γα), the appendix proves that the mean-field functions fα take the simple form given in equation (1.9).The claim is stated for Gaussian initial conditions.Source: Appendix A · p. 27
  • The proof recasts the integral over an affine domain, illustrated by Figure A.1.The figure is identified as a change of variable for the erf function.Source: Appendix A · p. 27, Figure A.1 · p. 28
  • A rotation aligns the affine boundary with the new coordinates, with tan(θ) = a.The rotated domain is then used to compute the integral.Source: Appendix A · p. 28

Appendix B. Bifurcations Diagram as a function of

The appendix organizes the bifurcation diagrams into six λ zones as input I1 varies, with transitions between fixed points, limit cycles, and bistability.

Appendix B. Bifurcations Diagram as a function of

  • Zone A has four saddle-node bifurcations and a supercritical Hopf bifurcation; its stable cycles disappear through a saddle-homoclinic bifurcation, with bistability confined to a narrow parameter range.The six λ zones and their typical diagrams are summarized in Table B.1 and Figure B.1.Source: Appendix B · pp. 28–29, Table B.1 · p. 28
  • In zone B, a cusp removes the two inferior saddle-node bifurcations, while the other behavior remains broadly like zone A.The narrow bistable behavior noted for zone A is absent.Source: Appendix B · p. 29
  • In zone C, a Bogdanov–Takens bifurcation gives rise to supercritical Hopf and saddle-homoclinic curves, with two coexisting families of limit cycles.Both cycle families disappear through saddle-homoclinic bifurcations.Source: Appendix B · p. 29
  • Zone D lies between the turning points of the saddle-homoclinic and Hopf curves; it has two supercritical Hopf bifurcations but no homoclinic bifurcation.The cycle families from the two Hopf bifurcations are identical.Source: Appendix B · pp. 29–30
  • In zone E, after the Hopf-curve turning point, two Hopf bifurcations remain and the system has pure bistability without cycles..Source: Appendix B · p. 30
  • In zone F, the two saddle-node bifurcations merge into a cusp, after which there is one attractive equilibrium for every I1.This is the appendix’s stated trivial regime.Source: Appendix B · p. 30
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