Detecting the breakup of spiral waves in small-world networks of neurons due to channel block

The breakup of a spiral wave by blockade of sodium and potassium channels in a small-world network of Hodgkin-Huxley neurons is investigated in detail. The influence of ion channel block in poisoned excitable membrane patches of a certain size is measured, by varying channel noise and channel densities resulting from the change in conductance. For example, tetraethylammonium is known to cause a block (poisoning) of potassium channels, while tetrodotoxin blocks sodium channels. We observed the occurrence of spiral waves, which are ordered waves believed to play an important role in facilitating the propagation of electric signals across quiescent regions of the brain. In this paper, the effect of channel block was measured by the factors x (K) and x (Na), which represent the ratios of unblocked, or active, ion channels, to the overall number of potassium or sodium ion channels, respectively. To quantify these observations, we use a simple but robust synchronization measure, which succinctly captures the transition from spiral waves to other collective states, such as broken segments resulting from the breakup of the spiral wave. The critical thresholds of channel block can be inferred from the abrupt changes occurring in plots of the synchronization measure against different values of x (K) and x (Na). Notably, small synchronization factors can be tightly associated with states where the formation of spiral waves is robust to mild channel block.