Figure 1

Comparison of Markov probabilities and colour map of correlations. In (a), the open circles are the prior probabilities $P_i$ (the probability of observing an active variable i at a given phase), the upwards pointing triangles are sustaining probabilities $P(x_i(t)=1|x_i(t-1)=1)$ (the probability of observing an active i given that i is already active at a preceding phase), and the downwards pointing triangles are the first activation probabilities $P(x_i(t)=1|x_i(t-1)=0)$ (the probability of observing an active i given that i inactive at a preceding phase). We ranked the variables in increasing order of prior probabilities. We see clearly that a vast majority of the sustaining probabilities are at least an order of magnitude larger than their corresponding prior probabilities. These variables have the tendency to self-sustain which suggests that we should treat multiple-phase activation of these variables as a single event. In (b) we compare the lag-1 sustaining probabilities against the probabilities $P(x_i(t)=1|x_i(t-1)=1,x_i(t-2)=0)$ and $P(x_i(t)=1|x_i(t-1)=1,x_i(t-2)=1)$ (i.e. treating variables as a lag-2 dynamical process). Unlike in (a), there is no clear increase in probability when the second preceding phase is accounted for. In (c) we show, on a colour map, the standard scores of observed conditional probabilities $P_{\mathrm{observed}}(x_j(t)=1|x_i(t-1)=1)$ against their expected values if i and j were independent. If i and j were independent and characterized solely by their respective prior probabilities $P_i$ and $P_j$, the expected conditional probabilities should be equal to the prior, i.e. $P_{\mathrm{expected}}(x_j(t)=1|x_i(t-1)=1)=P_j$ and the distribution of conditional probabilities should have a standard deviation of $\sigma =\sqrt{P_j(1-P_j)/N_i}$. The standard score is obtained from $Z=(P_{\mathrm{observed}}(x_j(t)=1|x_i(t-1)=1)-P_j)/\sigma$. The colours on the map range from blue (lower than expected) to teal (equals expectation), and finally maroon (higher than expected). The dominant feature of this map is a diagonal of higher-than-expected conditional probabilities for self-transitions.