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.

We study a family of non-linear McKean-Vlasov SDEs driven by a Poisson measure, modelling the mean-field asymptotic of a network of generalized Integrate-and-Fire neurons. We give sufficient conditions to have periodic solutions through a Hopf bifurcation. Our spectral conditions involve the location of the roots of an explicit holomorphic function. The proof relies on two main ingredients. First, we introduce a discrete time Markov Chain modeling the phases of the successive spikes of a neuron. The invariant measure of this Markov Chain is related to the shape of the periodic solutions. Secondly, we use the Lyapunov-Schmidt method to obtain self-consistent oscillations. We illustrate the result with a toy model for which all the spectral conditions can be analytically checked.

We study the long time behavior of a family of McKean-Vlasov stochastic differential equations. We give conditions ensuring the local stability of an invariant probability measure. Our criterion involves the location of the roots of an explicit holomorphic function associated to the dynamics. When all the roots lie on the left-half plane, local stability holds and we prove convergence in Wasserstein norms. We also provide the optimal rate of convergence. Our probabilistic proof makes use of Lions derivatives and relies on a new "integrated sensibility" formula.