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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.

##### MSC:
 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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