Particle systems with a singular mean-field self-excitation. Application to neuronal networks

Delarue, François; Inglis, James; Rubenthaler, Sylvain; Tanré, Étienne

doi:10.1016/j.spa.2015.01.007

Cited by 88 publications

(164 citation statements)

References 16 publications

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“…Lastly, it would be interesting to investigate the extension to the models in Nadtochiy and Shkolnikov (2017) and Delarue et al (2015b)…”

Section: Discussionmentioning

confidence: 99%

Simulation of a simple particle system interacting through hitting times

Kaushansky¹,

Reisinger²

2017

Discrete &Amp; Continuous Dynamical Systems - B

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We develop an Euler-type particle method for the simulation of a McKean-Vlasov equation arising from a mean-field model with positive feedback from hitting a boundary. Under assumptions on the parameters which ensure differentiable solutions, we establish convergence of order 1/2 in the time step. Moreover, we give a modification of the scheme using Brownian bridges and local mesh refinement, which improves the order to 1. We confirm our theoretical results with numerical tests and empirically investigate cases with blow-up.

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“…Lastly, it would be interesting to investigate the extension to the models in Nadtochiy and Shkolnikov (2017) and Delarue et al (2015b)…”

Section: Discussionmentioning

confidence: 99%

Simulation of a simple particle system interacting through hitting times

Kaushansky¹,

Reisinger²

2017

Discrete &Amp; Continuous Dynamical Systems - B

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“…Finally, we conclude this short bibliography of mathematical mean-field models in neuroscience by some papers closer to our setting: models of networks of spiking neurons with soft (see De Masi et al [14] and Fournier and Löcherbach [17]) or hard (see Cáceres, Carrillo and Perthame [5], Carrillo, Perthame, Salort and Smets [7], Delarue, Inglis, Rubenthaler and Tanré [12,13] and Inglis and Talay [21]) bounds on the membrane potential have also been studied. In particular, in [21] the authors introduced a model of propagation of membrane potentials along the dendrites but it is very different from ours.…”

Section: 5mentioning

confidence: 99%

On a toy network of neurons interacting through their dendrites

Fournier¹,

Tanré²,

Veltz³

2020

Ann. Inst. H. Poincaré Probab. Statist.

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Consider a large number n of neurons, each being connected to approximately N other ones, chosen at random. When a neuron spikes, which occurs randomly at some rate depending on its electric potential, its potential is set to a minimum value v min , and this initiates, after a small delay, two fronts on the (linear) dendrites of all the neurons to which it is connected. Fronts move at constant speed. When two fronts (on the dendrite of the same neuron) collide, they annihilate. When a front hits the soma of a neuron, its potential is increased by a small value wn. Between jumps, the potentials of the neurons are assumed to drift in [v min , ∞), according to some well-posed ODE. We prove the existence and uniqueness of a heuristically derived mean-field limit of the system when n, N → ∞ with wn N −1/2 . We make use of some recent versions of the results of Deuschel and Zeitouni [15] concerning the size of the longest increasing subsequence of an i.i.d. collection of points in the plan. We also study, in a very particular case, a slightly different model where the neurons spike when their potential reach some maximum value vmax, and find an explicit formula for the (heuristic) mean-field limit.2010 Mathematics Subject Classification. 60K35, 60J75, 92C20. Key words and phrases. Mean-field limit, Propagation of chaos, nonlinear stochastic differential equations, Ulam's problem, Longest increasing subsequence, Biological neural networks. arXiv:1802.04118v2 [math.PR] 26 Apr 2019 1.2. Biological background. Although the above particle systems are toy models, they are strongly inspired by biology.General organization. A neuron is a specialized cell type of the central nervous system. It is composed of sub-cellular domains which serve different functions, see Kandel [23]. More precisely, the neuron is comprised of a dendrite, a soma (otherwise known as the cell body) and an axon. See Figure 1 for a schematic description. The neurons are connected with synapses which are the interface between the axons and the dendrites. On Figure 1, the axon of the neuron i is connected, through synapses, to the dendrites of the neurons j and k.

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“…Note that the global existence results obtained for this model differ from those obtained for the "standard" Integrate-and-Fire model with a fixed deterministic threshold. This situation, studied for instance in [4,6,12,13], corresponds (informally) to the choice f (x) = +∞½ {x≥ϑ} , ϑ > 0 being the fixed threshold. In these papers, a diffusion part is included in the modeling.…”

Section: Introductionmentioning

confidence: 99%

Long time behavior of a mean-field model of interacting neurons

Cormier¹,

Tanré²,

Veltz

2020

Stochastic Processes and their Applications

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We study the long time behavior of the solution to some McKean-Vlasov stochastic differential equation (SDE) driven by a Poisson process. In neuroscience, this SDE models the asymptotic dynamic of the membrane potential of a spiking neuron in a large network. We prove that for a small enough interaction parameter, any solution converges to the unique (in this case) invariant probability measure. To this aim, we first obtain global bounds on the jump rate and derive a Volterra type integral equation satisfied by this rate. We then replace temporary the interaction part of the equation by a deterministic external quantity (we call it the external current). For constant current, we obtain the convergence to the invariant probability measure. Using a perturbation method, we extend this result to more general external currents. Finally, we prove the result for the non-linear McKean-Vlasov equation.Keywords McKean-Vlasov SDE · Long time behavior · Mean-field interaction · Volterra integral equation · Piecewise deterministic Markov process Mathematics Subject Classification Primary: 60B10. Secondary 60G55 · 60K35 · 45D05 · 35Q92.Between two spikes, the potentials evolve according to the one dimensional equationThe functions b and f are assumed to be smooth. This process is indeed a PDMP, in particular Markov (see [10]). Equivalently, the model can be described using a system of SDEs driven by Poisson measures. Let (N i (du, dz)) i=1,··· ,N be a family of N independent Poisson measures on R + × R + with intensity measure dudz. Let (X i,N 0 ) i=1,··· ,N be a family of N random variables on R + , i.i.d. of law ν and independent of the Poisson measures. Then (X i,N ) is a càdlàg process solution of coupled SDEs:When the number of neurons N goes to infinity, it has been proved (see [11,18]) for specific linear functions b and under few assumptions on f that X 1,N t -i.e. the first coordinate of the solution to (1) -converges in law to the solution of the McKean-Vlasov SDE:The Picard iteration studied in Part 4.4 shows that ∀t ≥ 0, lim n→∞ |J E f (X t ) − a n (t)| = 0.We have proved that• Step 6 We now prove that there exists s ≥ 0 such that E f (X s ) ≤ min(ā (J) J ,r(J m ) + 1). ByStep 1, we have lim sup E f (X t ) ≤ r(J). Since r(J) < a(J)/J and since r(J) ≤ r (J m ), the conclusion follows. Consequently, Step 5 can be applied to the process (X t ) t≥s starting with ν = L(X s ). This proves the convergence of the jump rate.The convergence of the law of X t to the invariant measure then follows from Proposition 27. This ends the proof of Theorem 7.Remark 51. There is some freedom in the above construction of the constants λ and J * . We can choose any λ in [0, λ * ) and the value of J * depends both on λ and on a parameter α ∈ (0, 1), here chosen to be equals to 1/2 (see Step 4). We may optimize this construction to get either J * or λ as large as possible.

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Particle systems with a singular mean-field self-excitation. Application to neuronal networks

Cited by 88 publications

References 16 publications

Simulation of a simple particle system interacting through hitting times

Simulation of a simple particle system interacting through hitting times

On a toy network of neurons interacting through their dendrites

Long time behavior of a mean-field model of interacting neurons

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