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

Jun Ma,Heping Ying,Zhongsheng Pu,Long Huang

doi:10.1007/s11434-012-5114-2

Abstract

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.

Highlights

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
We focus on ion channel block-induced breakup of spiral waves in a two-dimensional square array, in which Hodgkin-Huxley neurons are connected in a small-world network
Some interesting results are reported and can be summarized as follows: (1) Breakup of a spiral wave can be induced by block of either sodium or potassium channels, and the critical threshold of xK and xNa can be detected from the abrupt decay points in plots of factors of synchronization vs. xK and xNa. (2) The threshold of factor xNa is higher the factor xK, which may be related to the fact that sodium and potassium channels affect the membrane potential in different ways, and potassium channels are more important in controlling the neuronal membrane potential

Summary

Mathematical model and discussion

Huxley neurons in a small-world network, and the stochastic properties of this model [25], are described by the following equations: Cm dVij dt g K ni4j (VK Vij ) g Na mi3j hij (VNa Vij ). The total channel number is decided by NNa=ρNas and NK=ρKs, where s describes the size of the membrane patch. The statistical properties [32] of the channel noise [25] are defined by m t m t According to these equations, the intensity of channel noise will change when different values for xK and xNa are selected. The statistical properties of the measurable variables (e.g. the neuronal membrane potential) are described by. The factors of synchronization are plotted as a function of the parameters xK and xNa, to detect the critical condition for spiral wave breakup induced by block of ion channels. A spiral wave can develop and occupy most of the area in a network with 300×300 neurons, within a transient period of about 3000 time units.

Numerical results and discussion

Findings

Conclusions