Abstract
We are interested in the optimal control problem associated with certain quadratic cost functionals depending on the solution X=X^alpha of the stochastic mean-field type evolution equation in {mathbb {R}}^ddXt=b(t,Xt,L(Xt),αt)dt+σ(t,Xt,L(Xt),αt)dWt,X0∼μ(μgiven),(1)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} dX_t=b(t,X_t,{\\mathcal {L}}(X_t),\\alpha _t)dt+\\sigma (t,X_t,{\\mathcal {L}}(X_t),\\alpha _t)dW_t\\,, \\quad X_0\\sim \\mu (\\mu \\text { given),}\\qquad (1) \\end{aligned}$$\\end{document}under assumptions that enclose a system of FitzHugh–Nagumo neuron networks, and where for practical purposes the control alpha _t is deterministic. To do so, we assume that we are given a drift coefficient that satisfies a one-sided Lipschitz condition, and that the dynamics (2) satisfies an almost sure boundedness property of the form pi (X_t)le 0. The mathematical treatment we propose follows the lines of the recent monograph of Carmona and Delarue for similar control problems with Lipschitz coefficients. After addressing the existence of minimizers via a martingale approach, we show a maximum principle for (2), and numerically investigate a gradient algorithm for the approximation of the optimal control.
Highlights
MotivationsBased on a modification of a model by van der Pol, FitzHugh [17] proposed in 1961 the following system of equations in order to describe the dynamics of a single neuron subject to an external current I :v = v − 1 v3 − w + I (2)w = c(v + a − bw) for some constants a, b, c > 0, where the unknowns v, w correspond respectively to the so-called voltage and recovery variables
Under assumptions that enclose a system of FitzHugh–Nagumo neuron networks, and where for practical purposes the control αt is deterministic
After addressing the existence of minimizers via a martingale approach, we show a maximum principle for (2), and numerically investigate a gradient algorithm for the approximation of the optimal control
Summary
Based on a modification of a model by van der Pol, FitzHugh [17] proposed in 1961 the following system of equations in order to describe the dynamics of a single neuron subject to an external current I :. One has to enlarge the previous pair by an additional unknown y that counts a fraction of open channels (synaptic channels), and which is sometimes referred to as gating variable When it comes to an interacting network of neurons, it is customary to assume that the corresponding graph is fully connected, which is arguably a good approximation at small scales [24]. When interaction is present (J = 0), the model describes the situation where each neuron of the network affects its adjacent neurons by releasing chemical transmitters, causing particular ion channels of adjacent neurons to open This induces a current to the adjacent neuron, affecting its membrane potential.
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