Fig. 12. Qualitative model for spontaneous activity in hyperexcitable systems. We assume that such a system can be described by only two variables, the average activity (a) and the relative network excitability (s), and therefore any state of the system corresponds to a point in thea–s plane [for simplicity, we have not included a faster modulation of the positive feedback that is responsible for oscillations seen during an episode (Tabak et al., 2000b)]. The black S-shaped curve represents the possible activity states for each value of the positive feedback gain. There is a range of s values for which the system has several possible states, one being unstable (dashed) and representing a threshold value between the two stable states. For a given value ofs, any system with activity that is above threshold will reach the high activity state (equivalent to an episode), whereas a system with activity that is below threshold will fall to the low activity state (equivalent to the inter-episode interval). During an episode, because activity is high, network excitability declines; that is, the amount of positive feedback in the network decreases, moving the state of the system to the left as indicated by thearrowhead. For a critical value of s, only the low activity state persists, and the network falls back to low activity: this is the end of the episode. Network excitability can then recover, so the system state moves toward the right (arrowhead on low activity state). In the actual spinal network there are transient depolarizations in motoneurons and interneurons that may arise from the random coincidence of interneuronal spiking (Wenner and O'Donovan, 2001). Such events are represented in the diagram by the small vertical lineslabeled activity transients on the low activity state. When the maximum amplitude of such events becomes above threshold after sufficient recovery of the network, some of these events can trigger an episode. Because of the random nature of these events, there is no unique value of s for which episodes will occur, but rather a range of values. In the diagram, this range spans the point at which the maximum transient amplitude line crosses the threshold curve to the point at which the low activity state and the threshold curves coincide (gray segment ofabscissa). When the low activity state reaches threshold, even the slightest amount of noise will trigger an episode. Because episodes stop at a unique value of s, episodes that start after a longer inter-episode interval will have a longer duration. That is, the length of the episode is determined by the value of s at which an episode starts, and it does not affect the terminating value of s. As a result, episode length is correlated with the previous inter-episode duration and not the following one.