Limit of an infinite chain of first-order exponential filters

First-order exponential filter

The simplest model how the voltage $x$ at a synapse responds to input $u$ is a first-order filter:

$$\tau \dot{x}=-x+u.$$

This corresponds to convolving signal $u(t)$ with exponential filter $\mathrm{H}(t)\exp (-t/\tau )$, where $\mathrm{H}(\cdot )$ is the Heaviside step function:

$$\begin{array}{rl}x(t)& =h(t)\ast u(t)\\ h(t)& =\mathrm{H}(t)\exp (-t/\tau ).\end{array}$$

The alpha function

A first-order filter has a discontinuous jump in response to an abrupt inputs (like spikes). A more realistic response is the "alpha function"  $t\cdot \exp (-t)$. The alpha function can be obtained by convolving two first decay functions (i.e. chaining together two first-order filters):