The previous section investigated situations where joining multiple channels does not lead to a higher score than that of individual channels. The sufficient condition for the corresponding results is that channels have non-overlapping activity. Next, in order to understand which conditions must be met such that a join can have a higher score than its individual channels, the join op- eration is investigated closely. Therefore, different re-formulations of the SSS are discussed to highlight aspects that facilitate high score values of joins. For this consider two nodes i and j, which are potential parents of a child node. The score of configurations where nodes iand j are both exclusive parents of the child will be compared to that of their joint parenthood. Therefore let ak = (ak,t)t=1,...,T (k=i, j) denote the corresponding activity level series and
lets= (st)t=1,...,T be the child’s spike train.
First, the definition of the SSS is repeated by plugging in the join operation [equation (3.2)] directly into the score’s formula [equation (3.3)]:
SSS a(i,j), s; ∆t = P tmax{ai,t, aj,t} ·st+∆t P tmax{ai,t, aj,t} . (4.19)
If the activity of channelsiandjis non-overlapping proposition 1 can be applied to replace the join operation max by the sum of its arguments, which yields:
SSS a(i,j), s; ∆t i,jnon-overlapping = P tai,t·st+∆t + Ptaj,t·st+∆t P tai,t + Ptaj,t . (4.20)
Here, the score of the join a(i,j) is the sum of the snapshots of the individual channels and accumulated activity, respectively. However, equation (4.20) does not hold when channels have overlapping activity (Fig. 4.1) because the condi- tion of proposition 1 is violated. In order to yield generally valid re-formulations of equation (4.19), the join operation is replaced by different identities, which hold whether channels have non-overlapping activity or not. The first identity to consider is given in
Proposition 3 Let a, b∈R. Then
max{a, b}=a+b−min{a, b} (4.21)
holds.
activity max{ai,t, aj,t} at any timet. This yields:
SSS a(i,j), s; ∆t
=
P
tai,t·st+∆t + Ptaj,t·st+∆t − Ptmin{ai,t, aj,t} ·st+∆t
P tai,t + P taj,t − P tmin{ai,t, aj,t} . (4.22)
In this formula we find the left and the middle sums corresponding to the terms that make up the score of configurations where nodeiand nodejare exclusive parents. The two sums on the right gather snapshots and activity of both channelsiandjwhile their activity overlaps. The score is calculated by adding the snapshots and accumulated activity of both channels and correcting it by the overlap of channels. Indeed, if no such overlap exists, i.e. in the case where channels i and j have non-overlapping activity, the right sums both evaluate as 0, such that equation (4.22) reduces to (4.20). Finding equation (4.20) to be a special case of (4.22) verifies its more general validity; however, it does not give any explicit insights into what maximises the score of a join. This is because it is not sufficient to maximise the ratio of the left and middle sums while minimising the right sums ratio; in order to optimise the score of the join the overall ratio must of course be maximised. Due to the sheer number of possible combinations of values for the six sums, it is not possible to derive a simple, general condition, which is sufficient to guarantee a high score value of a join. It is, however, possible to identify special cases of overlapping activity in which joining channels cannot have higher score values than their separate channels. In order to spot these cases we use the following basic
Proposition 4 Let a, b∈R≥0. Then
max{a, b}=a+ max{b−a,0} (4.23)
holds.
Proposition 4 is applicable to activity level series as these, by definition, are non-negative at all times. We thus find equation (4.19) equivalent to
SSS a(i,j), s; ∆t = P tai,t·st+∆t + P tmax{aj,t−ai,t,0} ·st+∆t P tai,t + P tmax{aj,t−ai,t,0} . (4.24) This formulation of the SSS can be interpreted when recognising that the sums on the left resemble the score of channel i alone (without joining with j); the right sums constitute the change to the score of channeliwhen joiningiandj. If the activity aj,t of channel j is lower or equal than that of channel i at all
timest, the right sums both evaluate as 0, such that the score of the join is equal to that of solely using channeli. (This has already been observed in examples 5 and 6 in section 3.2.) However, if channel idoes not out-value channelj at all times the score of the join will be different; it can either be higher or lower than that of i’s score. Whether the score actually increases or decreases by joining channels iandj depends on the ratio of the right sums compared to the score of node ias single parent: Proposition 2 shows that the score of the joina(i,j) will only be higher, if the ratio of the right sums is higher than that of channeli alone.
The preceding investigations in this chapter aimed at specifying the SSS’s gen- eral characteristics. Under specific assumptions about the (non-)simultaneous activity on different channels strong results could be obtained. Considerations of more general cases gave at least an impression about mechanisms at work. In order to be more specific, the next section addresses particular questions, which might have arisen from previous results.