Signal Integration over Time

Introducing time dependence requires the activation to be defined by its rate of change $\frac {da}{dt}$, the derivative. The input weighted sum is then

$$u_j = \sum_j w_{ij} \cdot x_i$$

and the activation is

$$y = \frac {da}{dt} = - \eta \space a + \beta \space u$$

where $\eta$ and $\beta$ are positive constants. $-\eta$ a gives rise to activation decay. $\beta u$ represents the input from other neurons.

Decay effect is summarised as follows:

1. $a > 0 \longrightarrow \frac {da}{dt} < 0$: $a$ decreases
2. $a < 0 \longrightarrow \frac {da}{dt} > 0$: $a$ increases

The neuron will reach equilibrium when $\frac {da}{dt} = 0$. That is, when $u = \frac {\beta u}{\eta}$.