Classical Solutions for a Nonlinear Fokker-Planck Equation Arising in Computational Neuroscience

Carrillo, José A.; González, María del Mar; Gualdani, Maria Pia; Schonbek, María E.

doi:10.1080/03605302.2012.747536

Cited by 60 publications

(121 citation statements)

References 28 publications

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“…With the choice φ(r) = φ 1 (r) = sign(r) one recovers the first proposed equation (6). As we said, the numerical simulations with φ 1 (r) work well, except that for large values of R solutions with the wrong sign in some parts appear.…”

Section: Introduction and Summary Of Resultssupporting

confidence: 74%

“…As in the case of the thermostat, the changes made at the level of the free boundary are plugged back into the whole function as a form of feed-back for the market agents; this transfer of information produces instabilities if the feed-back has an excessive strength. Note that a reaction term of the form Rp (t)δ p(t)±a also recalls the nonlinear part of the equation studied in [6] after a suitable transformation.…”

Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

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Instability and Bifurcation in a Trend Depending Price Formation Model

2016

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A well-known model due to J.-M. Lasry and P.L. Lions that presents the evolution of prices in a market as the evolution of a free boundary in a diffusion equation is modified in order to show instabilities for some values of the parameters. This loss of stability is associated to the appearance of new types of solutions, namely periodic solutions, due to a Hopf bifurcation and representing price oscillations; and traveling waves, that represent either inflationary or deflationary behavior.

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Section: Introduction and Summary Of Resultssupporting

confidence: 74%

Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

Instability and Bifurcation in a Trend Depending Price Formation Model

2016

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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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“…The simulations therein suggest that the blow-up phenomenon is reflected in a divergence in finite time of the firing rate. Following these observations, in [9] a criterion for the maximal time of existence for classical solutions was derived. Essentially, it ensures that the solutions exist while the firing rate is finite.…”

Section: Introductionmentioning

confidence: 99%

“…There we prove the global existence and uniqueness of classical solutions for both the average-inhibitory and the average-excitatory case when D > 0. First, we extend to the case at hand the characterization of the maximal time of existence of the solutions in terms of the size of the firing rate, provided in [9] for the case without delay (D = 0). Mainly, this ensures that local solutions exist and are unique as long as the firing rate is finite.…”

Section: Introductionmentioning

confidence: 99%

Global-in-time solutions and qualitative properties for the NNLIF neuron model with synaptic delay

Cáceres

Roux

Salort

et al. 2019

Communications in Partial Differential Equations

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The Nonlinear Noisy Leaky Integrate and Fire (NNLIF) model is widely used to describe the dynamics of neural networks after a diffusive approximation of the mean-field limit of a stochastic differential equation system. When the total activity of the network has an instantaneous effect on the network, in the averageexcitatory case, a blow-up phenomenon occurs. This article is devoted to the theoretical study of the NNLIF model in the case where a delay in the effect of the total activity on the neurons is added. We first prove global-in-time existence and uniqueness of classical solutions, independently of the sign of the connectivity parameter, that is, for both cases: excitatory and inhibitory. Secondly, we prove some qualitative properties of solutions: asymptotic convergence to the stationary state for weak interconnections and a non-existence result for periodic solutions if the connectivity parameter is large enough. The proofs are mainly based on an appropriate change of variables to rewrite the NNLIF equation as a Stefan-like free boundary problem, constructions of universal super-solutions, the entropy dissipation method and Poincaré's inequality.Key-words: Leaky integrate and fire models, noise, blow-up, relaxation to steady state, neural networks, delay, global existence, Stefan problem. AMS Class. No: 35K60, 82C31, 92B20where the function ρ(·, t) is the probability density of the electric potential of a randomly chosen neuron at time t, D ≥ 0 is the delay between emission and reception of the spikes (synaptic delay), V R is the reset potential

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Classical Solutions for a Nonlinear Fokker-Planck Equation Arising in Computational Neuroscience

Cited by 60 publications

References 28 publications

Instability and Bifurcation in a Trend Depending Price Formation Model

Instability and Bifurcation in a Trend Depending Price Formation Model

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

Global-in-time solutions and qualitative properties for the NNLIF neuron model with synaptic delay

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