PaperlayerView exact revision on arXiv

Source-linked AI summary

Analysis of Nonlinear Noisy Integrate\&Fire Neuron Models: blow-up and steady states

María J. Cáceres, José A. Carrillo, Benoît Perthame

arXiv:1010.4683v1q-bio.NCmath.AP

TL;DR

The paper studies finite-time blow-up and steady states in the nonlinear noisy integrate-and-fire model. It analyzes weak solutions and stationary states, and proves that sufficiently concentrated initial data can produce blow-up for any positive connectivity parameter. The proof does not identify the nature of the resulting singularity.

  • Problem

    Finite-time blow-up had been observed numerically in the NNLIF model, but had not been stated as such or proved for a class of situations.

    Source: §6 · p. 21

  • Method

    The paper analyzes weak solutions and a priori estimates, characterizes stationary states, and applies the relative entropy principle to the linear case.

    Source: §1 · p. 4

  • Results

    For any b > 0, sufficiently concentrated initial data can produce finite-time blow-up; the number of steady states can be zero, one, or two depending on connectivity.

    Source: §1 · p. 4, §6 · pp. 21–22

  • Takeaways & Limitations

    When steady states exist, numerical simulations can exhibit convergence, while several questions about their multiplicity and stability remain open.

    Source: §6 · p. 22

    The blow-up proof establishes finite maximal existence time but does not characterize the first singularity.

    Source: §2 · p. 6

Abstract

from arXiv · show

Nonlinear Noisy Leaky Integrate and Fire (NNLIF) models for neurons networks can be written as Fokker-Planck-Kolmogorov equations on the probability density of neurons, the main parameters in the model being the connectivity of the network and the noise. We analyse several aspects of the NNLIF model: the number of steady states, a priori estimates, blow-up issues and convergence toward equilibrium in the linear case. In particular, for excitatory networks, blow-up always occurs for initial data concentrated close to the firing potential. These results show how critical is the balance between noise and excitatory/inhibitory interactions to the connectivity parameter.

1 Introduction

The introduction sets up a probability-density model of neuron voltages, with network activity coupled through the firing rate. It previews results on blow-up, steady states, and convergence in the linear case.

1 Introduction

  • The model describes a representative neuron's voltage stochastically and formulates network dynamics as an evolution equation for the voltage probability density.The firing rate is determined by the flux of neurons reaching the firing threshold.Source: §1 · pp. 1–2
  • Synaptic input is approximated by a continuous Ornstein–Uhlenbeck process, which preserves the spike-train current's mean and variance and becomes exact in the infinitely large network limit with appropriate scaling.The average input distinguishes excitatory from inhibitory networks through the sign of b.Source: §1 · p. 2
  • When voltage reaches the firing threshold, a neuron discharges and resets to the reset voltage; the network firing rate is computed self-consistently from threshold-crossing flux.The model specifies firing rate as external input plus the network's mean firing rate.Source: §1 · pp. 2–3
  • A weak formulation lets the analysis treat both the equation with a Dirac source at reset and its equivalent jump-condition form.The authors state that their analysis applies to both settings.Source: §1 · p. 3
  • For b > 0, suitable sufficiently concentrated initial data yield weak solutions that fail to exist for all times; stationary states are unique for b ≤ 0 with constant diffusion.For b > 0, there may be one, two, or no stationary states depending on b; for the linear problem b = 0 with a constant, the relative entropy principle implies exponential convergence toward equilibrium.Source: §1 · p. 4

2 Finite time blow-up and apriori estimates for weak solutions

This section establishes that sufficiently concentrated initial data can preclude global-in-time weak solutions in average-excitatory networks. It also develops a priori estimates for solutions that exist globally under stated decay and coefficient assumptions.

2 Finite time blow-up and apriori estimates for weak solutions

  • In an average-excitatory network, sufficiently concentrated initial data near the firing voltage can yield no global-in-time weak solution.The blow-up result also holds under less stringent coefficient hypotheses than the model's linear drift and diffusion specification.Source: §2 · pp. 4–5
  • The proof establishes finite maximal existence time but does not determine the nature of the first singularity.The authors identify loss of decay at negative infinity or a singular-measure firing rate as possible alternatives; numerical computations show finite-time growth of the firing rate.Source: §2 · p. 6
  • For global-in-time solutions that are fast-decaying at negative infinity, the section establishes a priori estimates under the specified drift and diffusion assumptions.A corollary adds a finite initial second moment and 0 < b < VF − VR as assumptions.Source: §2 · pp. 6–7

3 Steady states

The stationary-state problem reduces to finding positive firing rates satisfying a scalar normalization condition. The number of states depends on connectivity and, when diffusion varies with firing rate, on the diffusion parameters as well.

3.1 Generalities

  • For the stated regularity, stationary solutions have the derived form with firing rate N equal to any positive solution of the scalar condition (3.3).The firing rate is fixed by the unit-mass normalization condition.Source: §3.1 · pp. 8–9

3.2 Case of a(N) = a0.

  • With constant diffusion, b < 0 and sufficiently small b > 0 yield one steady state; intermediate conditions permit at least two, while high connectivity can eliminate steady states.The authors do not establish whether these are the only possible cases.Source: §3.2 · pp. 9, 10, 13
  • For b < 0, I(N) is increasing and strictly convex; for b > 0, I(N) is decreasing and convex, in the respective cases.The cited text also states that I(N) crosses 1/N at a single point.Source: §3.2 · pp. 11–12
  • The authors cannot prove or disprove that NI(N) has a unique maximum, so the apparent three-case pattern for constant diffusion is not established as exhaustive.The three observed cases are one, two, or no stationary states.Source: §3.2 · p. 13

3.3 Case of a(N) = a0 + a1N

  • With linearly increasing diffusion, parameter conditions give at least one, at least two, or no steady states; for b < 0, a state exists but its number is difficult to determine.For b < 0, I(N) may have a minimum, complicating the crossing analysis.Source: §3.3 · p. 15

4 Linear case and relaxation

For the linear problem, the paper uses relative entropy to study relaxation toward the unique steady state. Under the stated solution assumptions, it obtains exponential decay and identifies diffusion and firing as two relaxation processes.

4 Linear case and relaxation

  • The analysis assumes the needed solution regularity and a bound p(v,t) ≤ C_T p∞(v) on each finite time interval.The paper notes that such solutions might be obtained by a method developed elsewhere.Source: §4 · p. 16
  • Fast-decaying solutions of the linear problem converge toward the unique steady state, with diffusion and firing each contributing a relaxation process.The solutions are assumed to have the regularity required by the results and to remain bounded by a time-dependent multiple of the steady state.Source: §4 · p. 16
  • For fast-decaying solutions, the relative-entropy result yields exponential decay toward equilibrium.The proof combines a Poincaré inequality with the entropy inequality and Gronwall's lemma.Source: §4 · p. 16

5 Numerical results

The simulations illustrate steady-state behavior across connectivity regimes and show firing-rate blow-up for excitatory networks with sufficiently concentrated initial data. They also highlight slow dynamics near the higher-firing steady state and sensitivity to initial conditions.

5 Numerical results

  • The numerical scheme uses WENO finite differences for advection and second-order finite differences for diffusion; the discussed Chang–Cooper method is unsuitable when a(N) is constant, b > 0, and N is large.The simulations use a uniform voltage mesh, with Vmin adjusted so p(Vmin, t) ≈ 0 and VF fixed to 2, VR = 1.Source: §5 · pp. 18–19
  • At b = 0.5, the distribution reaches the steady state numerically after 3.5 time units, and the firing rate becomes constant.The distribution also shows Lipschitz behavior at VR.Source: Figure 3 · p. 18, Figure 4 · p. 19, §5 · p. 20
  • At b = 1.5, simulations suggest the lower-firing steady state is locally asymptotically stable, while the higher-firing state is unstable.Depending on initial firing rate, trajectories approach the lower steady state or the firing rate grows without bound; from N0 = 2.31901, evolution is slow before the distribution approaches the lower state.Source: §5 · p. 20, Figure 5 · p. 20
  • For negative b, simulations suggest local asymptotic stability of the unique steady state.The numerical example uses b = −1.5.Source: §5 · pp. 19–20
  • At b = 3, where the analysis predicts no steady state, the density concentrates around VR and VF as the firing rate appears to blow up in finite time.Figure 6 shows the evolving density; the corresponding firing-rate evolution is in Figure 4.Source: §5 · p. 21, Figure 6 · p. 21
  • For b > 0, sufficiently concentrated initial data near the firing voltage produces firing-rate growth without bound up to the computing time, even when p's derivative there does not diverge.The simulations show this for parameter values with either one or two steady states.Source: §5 · p. 21

6 Conclusion

The conclusion establishes blow-up for positive connectivity despite stabilizing noise, while leaving the singularity's precise nature unproved. It also summarizes connectivity-dependent steady-state counts and distinguishes this blow-up from a deterministic adaptive model.

6 Conclusion

  • For every b>0, the system can blow up despite stabilizing noise; Figure 7 numerically shows this for data concentrated near VF.Figure 7 uses a=1 and b=1.5, a parameter setting with two steady states.Source: §6 · p. 21, Figure 7 · p. 22
  • Uniform second-moment control suggests the singularity is neither vanishing near v≈−∞ nor loss of fast decay at infinity.These are estimates-based indications, not a mathematical characterization of the singularity.Source: §6 · p. 21
  • Numerics indicate that N(t) blows up in finite time whenever a singularity occurs, but the mathematical result does not identify its nature.The proof ensures finite maximal existence time but does not characterize the first singularity.Source: §6 · p. 22
  • Blow-up in the deterministic quadratic adaptive model differs: it is a single-neuron linear model here, with singularity on an adaptation variable absent from NNLIF.Source: §6 · p. 22
  • The steady-state set may be empty, contain one state, or contain two depending on connectivity; when states exist, numerics can exhibit convergence.The conclusion leaves open whether three or more steady states can occur and which steady states are stable.Source: §6 · p. 22
Loading 1010.4683v1…