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 $\operatorname H(t) \exp(-t/\tau)$, where $\operatorname H(\cdot)$ is the Heaviside step function:
$$\begin{aligned}x(t) &= h(t) * u(t)\\h(t)&=\operatorname H(t) \exp(-t/\tau).\end{aligned}$$
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):
