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The multiplication of distributions and the Tsodyks model of synapses dynamics. (English) Zbl 1276.46031
By using three formulas of the \(\alpha\)-product of distributions (where \(\alpha\in \mathcal D(\mathbb{R})\) is a real function with \(\int^\infty_{-\infty}\alpha= 1\)), the author defines the concept of an \(\alpha\)-generalized solution of the ordinary differential equation \(DX= UX+V\) where \(U\), \(V\) belong to certain spaces of distributions. Then this framework is applied to the equation \(X'= -X+ k(1- X)(\tau_a\delta)\) (Tsodyks model of synapses dynamics). The explicit \(\alpha\)-generalized solution of this equation is: \[ X(t)= \begin{cases} e^{-t}[c_1+ {2k\over k+2} (e^a- c_1) H(t- a)]\quad & \text{if }k\neq -2,\\ e^{-t}[e^a+ (c_2- e^a) H(t-a)]\quad & \text{if }k=-2,\end{cases} \] which is independent of \(\alpha\). It is shown that these solutions may differ from solutions obtained by other approaches where the multiplication of distributions is defined by approximation algorithms.

46F10 Operations with distributions and generalized functions
34A37 Ordinary differential equations with impulses
46F30 Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)
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