Source-linked AI summary
Particle systems with a singular mean-field self-excitation. Application to neuronal networks
F. Delarue, J. Inglis, S. Rubenthaler, E. Tanré
TL;DR
Singular self-excitation makes the neuronal mean-field equation too irregular for standard McKean–Vlasov results, and continuation after blow-up has been unclear. The paper constructs physical solutions and approximates the equation with finite particle systems or delayed interactions using M1 topology. These results include solutions with synchronized spikes, while uniqueness after synchronization remains open.
Problem
The singular firing-rate dependence falls outside standard McKean–Vlasov theory, and prior work left continuation after blow-up unclear.
Source: §1 · pp. 1–2Method
The paper uses physical solutions and M1-topology tightness to approximate the equation by finite particle systems and delayed self-interactions.
Source: §1 · p. 2Results
Weak limits of the particle system solve the equation, convergence holds when uniqueness holds, and physical solutions can include synchronization events.
Source: §1 · p. 2, §4.2 · p. 17Takeaways & Limitations
Discontinuous firing-rate jumps encode synchronized proportions and provide a framework for continuing solutions after synchronization.
Source: §1 · p. 2Uniqueness of physical solutions that experience synchronization remains open.
Source: §1 · p. 3
Abstract
from arXiv · showhide
We discuss the construction and approximation of solutions to a nonlinear McKean-Vlasov equation driven by a singular self-excitatory interaction of the mean-field type. Such an equation is intended to describe an infinite population of neurons which interact with one another. Each time a proportion of neurons 'spike', the whole network instantaneously receives an excitatory kick. The instantaneous nature of the excitation makes the system singular and prevents the application of standard results from the literature. Making use of the Skorohod M1 topology, we prove that, for the right notion of a 'physical' solution, the nonlinear equation can be approximated either by a finite particle system or by a delayed equation. As a by-product, we obtain the existence of 'synchronized' solutions, for which a macroscopic proportion of neurons may spike at the same time.
1. Introduction
The paper addresses a singular mean-field equation for interacting spiking neurons, whose irregular self-excitation falls outside standard McKean–Vlasov theory. It develops particle and delayed approximations using physical solutions and the M1 topology, including solutions that continue through synchronization events.
1. Introduction
- The mean-field nonlinearity is singular because its firing-rate derivative depends irregularly on the law of the neuron’s potential, beyond standard McKean–Vlasov theory.The rate is e′(t) = −(1/2)∂yp(t, 1), tying it to the density at the firing threshold.Source: §1 · p. 1
- For some parameters, the firing rate blows up in finite time as a macroscopic group spikes simultaneously, while sufficiently small α can yield a unique global solution [8].Before this work, continuation after blow-up was unclear.Source: §1 · p. 2
- Particle-system limits yield solutions of the nonlinear equation, and converge to its solution when uniqueness holds, giving propagation of chaos; delayed self-interaction provides a second approximation.The delayed equation does not exhibit blow-up, even when α is close to 1.Source: §1 · p. 2
- M1-topology tightness and a sequential notion of physical spiking support both approximation routes, including nondecreasing, discontinuous spike counts.The approach uses M1 on càdlàg paths because it is convenient for nondecreasing counting processes.Source: §1 · p. 2
- Physical solutions allow discontinuous e(t) with specified jump sizes, representing the proportion of neurons synchronizing and permitting continuation after blow-up [8].Their uniqueness after synchronization is left open.Source: §1 · pp. 2–3
- A related portfolio-default model also uses threshold events, but the neuronal model’s more singular interactions produce blow-up absent from that setting.Source: §1 · p. 3
2. The nonlinear equation: background
This section formulates the nonlinear spike-count equation and explains why defining its solutions requires more than the usual expectation-based coupling. It specifies physical solutions, standing assumptions, prior existence results, and a criterion for recognizing physicality.
2. The nonlinear equation: background
- The spike count M is coupled to the potential through e(t) = E(Mt), so the definition of a solution must handle a self-referential interaction.The model assumes α ∈ (0, 1), Brownian noise, and a Lipschitz drift.Source: §2 · p. 3
2.1. The right notion of a solution.
- Physical solutions have strictly increasing spike times that cannot accumulate in finite time, consistent with càdlàg potential paths.The paper assumes a globally Lipschitz drift and an initial potential bounded away from 1 with finite moments of every order; the support restriction is a technical simplification.Source: §2.1 · p. 4, §2.2 · p. 5
- The jump relation alone can admit multiple sizes, so physicality selects the largest kick magnitude that can absorb particles approaching the threshold.For a uniform pre-spike potential on [1 − α, 1], the cited jump equation is satisfied for every η ∈ [0, 1].Source: §2.1 · p. 4
- A physical solution restricts each instantaneous count jump to at most one and constrains jumps of e to select a physically admissible kick size.A spike occurs when the potential reaches the threshold after accounting for the kick; at a jump in e, the potential can spike below threshold and retain a positive value after reset.Source: §2.1 · pp. 3–5
2.2. Standing assumptions and related literature.
- Earlier results establish blow-up for some initial conditions at every α ∈ (0, 1), and unique global solutions for sufficiently small α, within the continuous-e class [8].Source: §2.2 · p. 5
- Existence and uniqueness for the broader physical-solution class remain open; the paper reformulates the equation through an auxiliary process Z and spike count M.In the reformulation, Z’s discontinuities are dictated by the deterministic map e.Source: §2.2 · pp. 5–6
2.3. A criterion for a physical solution.
- A sufficient criterion for physicality is given, alongside a bound on moments of the reformulated process Z over finite time intervals.The cited lemma states E[sup_{t∈[0,T]} |Zt|^p] < +∞ for p ≥ 1.Source: §2.3 · p. 6
- The criterion’s proof uses stochastic comparison to contradict right-continuity if e increases too slowly immediately after the relevant time.The argument derives a positive lower bound on the short-time increment of e, conflicting with right-continuity.Source: §2.3 · pp. 6–8
3. Two candidates for approximate solutions
This section introduces two candidate approximations of the nonlinear equation: a fully connected spiking-neuron particle system and a delayed equation that prevents instantaneous cascades. The delayed system has a unique strong solution up to any finite time, with the expected process continuously differentiable in time.
3.1. The particle system approximation.
- The finite particle system models a fully connected network of noisy integrate-and-fire neurons, and the paper aims to prove its convergence to the nonlinear equation as the population grows.The authors note that the singular interaction makes this convergence different from previously studied mean-field settings.Source: §3.1 · pp. 8–9
- The particle system can have multiple solutions, but ordering each spike cascade from the neuron that first reaches threshold yields a unique solution called physical.This natural ordering determines which additional neurons spike after each successive kick.Source: §3.1 · p. 9, Non-physical solutions · p. 10, Physical solutions · pp. 11–12
- Physical particle solutions satisfy a discrete version of the physicality condition for the nonlinear equation, motivating that solution concept.The section states this relationship and gives a proposition for the physical particle solution.Source: Physical solutions · pp. 13–14
3.2. The system with delays.
- A positive delay separates induced spikes in time, preventing a macroscopic synchronous cascade; the delayed system has a unique global solution with finite mean-firing rate for any α and initial condition.Its firing-rate map is continuously differentiable.Source: §3.2 · pp. 14–15
- The interval-by-interval construction can be iterated through step ⌈T/δ⌉ to obtain a solution up to any T > 0.Source: §3.2 · p. 15
- The delayed system given by (3.11) and (3.12) has a unique strong solution up to any T > 0.Source: §3.2 · p. 15
- For this solution, t ↦ E(Mδ_t) is continuously differentiable on [0, T].Source: §3.2 · p. 15
4. Results
The results use weak limits of particle and delayed systems to construct physical solutions, including global-in-time solutions that may blow up. Under uniqueness, particle and delayed systems converge in the M1 topology, and particle convergence yields propagation of chaos.
4.1. The M1 topology.
- M1 convergence is characterized through ordered parametrizations of completed graphs, while the oscillation modulus vanishes for monotone paths.The section also states that M1 is Polish, supporting weak-convergence arguments.Source: §4.1 · pp. 16–17
4.2. Existence of weak solutions with simultaneous spikes.
- Weak limits of particle and delayed systems yield physical solutions up to any fixed time, and diagonal construction extends existence to infinite time, including solutions that blow up.The existence results use tightness in M1; the particle-system result gives physical solutions under weak limits, while the delayed-system result does likewise as the delay vanishes.Source: §4 · p. 15, §4.2 · pp. 17–18
4.3. Convergence of the particle system and propagation of chaos.
- For X-particles, M1 convergence of full paths does not follow because addition may not be continuous in that topology.The passage limits what can be inferred for X-particles from convergence of the reformulated particles.Source: §4.3 · p. 19
- Assuming a unique physical solution, the particle system converges in M1, and finite-dimensional marginals of tagged particles converge to those of independent copies.The stated convergence applies at times outside the discontinuities of the mean firing function.Source: §4.3 · pp. 18–19
4.4. Convergence of the delayed system.
- Assuming a unique physical solution, the delayed system converges in M1 at times outside discontinuities of the mean firing function.Source: §4.4 · p. 19
5. Proofs
The proofs develop particle-system estimates and M1 tightness and continuity arguments to analyze limits of particle and empirical-measure laws. They identify limits with physical solutions, establish convergence under uniqueness, and prove Theorems 4.4, 4.6, and 4.9 by tightness and limit passage.
5.1. Preliminary estimates for the particle system.
- The estimates show that a macroscopic proportion of particles spiking twice or more in a short interval has very small probability, extending Proposition 3.4 to intervals of non-zero length.The proposition considers intervals [t, t + h] with h in (0, 1).Source: §5.1 · p. 22
5.2. Tightness properties and convergent subsequences.
- For any T > 0, the laws of a representative particle and the empirical-measure laws are tight under the weak topology induced by the M1 topology.Source: §5.2 · pp. 26–27
5.3. Proof of Theorem 4.4.
- Using tightness, the proof of Theorem 4.4 extracts a convergent subsequence of the empirical-measure laws as N tends to infinity.Source: §5.3 · p. 27
- For Π∞-almost every µ, the canonical process solves (2.6) with Brownian law and initial law X0, and satisfies physical conditions (2) and (4).The proof establishes the Brownian law using the continuous mapping theorem, identifies the initial distribution, and verifies both physical conditions.Source: §5.3 · pp. 29–30
5.4. Proof of Theorem 4.7.
- The empirical mean converges in probability in the M1 topology on ˆD([0, S], R) to s↦E(Ms), and for Π∞-almost every µ, γS♯µ equals the law of Z on [0, S].Source: §5.4 · pp. 30–31
5.5. Continuity of related mappings.
- Under Lemma 5.6’s crossing property, M1 convergence yields convergence of integer-valued running maxima away from at most countably many times.The lemma shows that convergence can fail at only finitely many continuity points where a positive integer level is first crossed.Source: §5.5 · p. 33
- With uniform second-moment bounds and the stated crossing and centering assumptions, expectations of mt converge at continuity points of t↦⟨µ, mt⟩.For almost every µ under Π∞, the crossing property also implies this convergence for M1-weakly convergent measures.Source: §5.5 · pp. 33–34
- The continuity lemma is proved by representing an M1-convergent sequence on a common space and using continuity at selected times to pass mapped quantities to the limit.The selected times are almost-sure continuity points for the canonical process and the expectation of η under the limiting measure.Source: §5.5 · pp. 36–37
5.6. Proof of Theorems 4.6 and 4.9.
- Theorem 4.6 is proved by obtaining tightness of the approximating laws, checking the limit’s crossing property, and passing to the limit to establish condition (2).The proof uses moment bounds and eδ(t) ≤ Ct^1/2; the limit argument considers continuity points outside a countable exceptional set.Source: §5.6 · p. 37
- The hardest step is verifying condition (3), using Lemma 5.11 as the particle-system counterpart of Proposition 5.3 to recover (5.19).The lemma gives an h^1/8 error bound for the relevant inequality; the proof of Theorem 4.9 follows the model of Theorem 4.7.Source: §5.6 · p. 38