Dirac Delta Function

The Dirac Delta Function is a distribution over a curve to represent a unit impulse. Essentially, it is a generalized function on the real numbers, which has a value of zero at all places except at zero itself.

In reality, it is just a “spike” or a near direct upward line on an otherwise flat curve. The reason this is interesting is this “impossible” to model as a curve with a function, and thus we have to use calculus limits to estimate as we approach $\infty$.

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Use

This impossible function is quite useful for modeling instantaneous impulses. It is called the delta function because it is a continuous analogue of the Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1.

For my case, the function is quite usefull for modeling neuronal spike trains and firing rates.

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Representation

We can represent it heuristically as:

$$\delta (x)=\{\begin{array}{ll}0,& x\ne 0\\ \infty ,& x=0\end{array}$$

As an integral:

$${\int }_{-\infty }^{\infty }\delta (x)dx=1$$

As a interactive desmos graph:

Desmos