Temporal Behaviour

Cyclic Networks

The relation between the input and output can be described by a set of equations, where the input states I are known, and all other states are variable. If the network graph is cyclic, there exists a least one series of transitions

$$\langle (A,A_1),(A_1,A_2), \ldots (A_n,A) \rangle$$

which begins and end at the same node $A$. Thus, the equation for $a$, the state of $A$, will contain $a$ itself and may therefore have no solution.

Networks as Dynamic Systems

Cyclic networks are dynamic systems which must be described by their temporal behaviour, thus all states $s_k$ become a function of time $s_k(t)$ (or $s_{k,t}$ for discrete time). In the case ${\bf S}$ is a real interval, the propagation function $f_i(I_i)$ must be replaced by a corresponding temporal operator $F_i(I_i)$. If the evaluation takes place in continuous time, $F_i(I_i)$ would be a differential operator, if discrete timesteps are assumed, $F_i(I_i)$ would be a difference operator, using the $\Delta$-operator ( $\Delta x_t=x_t-x_{t-1}$) to refer to previous values of states. The dynamic behaviour of the network could then be described by a set of differential or difference equations with the following boundary conditions, of which the last one is optional, depending whether the input is hold fixed during the simulation.

$$s_k=s_k(t), \qquad (\forall k) \; s_k(0) = s_k^{(0)}, \qquad (\forall X_i \in {\bf I}) \; s_i(t) = s_i$$

Another possibility of dealing with discrete time is the use of recursion, where the new states is calculated using the old states. The temporal operator $F_i$ of a neurone can be defined as:

$$F_i(I_i)\:O_{i,t} = f_i(I_{i,t-1}), \quad \mbox{with} \quad I_{i,t} = (x_{1,t},x_{2,t}, \ldots x_{k,t})$$

The description of the network would then result in a system in a recursive formula.