Symmetric coactivity weights do not distinguish reversed episodes

Suppose storing activities $a_1,\dots ,a_T$ changes a matrix by $\Delta W=\eta {\sum }_{t=1}^T{a}_t{a}_t^T$. Any permutation of the activities leaves this sum unchanged. Therefore the update alone cannot distinguish an episode from its reversal. This is a restriction on the stated order-insensitive update, not on all Hebbian learning: trace learning, an evolving context or directed transitions can introduce order information.