Channel noise-induced phase transition of spiral wave in networks of Hodgkin-Huxley neurons

Jun Ma,Ying Wu,Heping Ying,Ya Jia

doi:10.1007/s11434-010-4281-2

Abstract

The phase transition of spiral waves in networks of Hodgkin-Huxley neurons induced by channel noise is investigated in detail. All neurons in the networks are coupled with small-world connections, and the results are compared with the case for regular networks, in which all neurons are completely coupled with nearest-neighbor connections. A statistical variable is defined to study the collective behavior and phase transition of the spiral wave due to the channel noise and topology of the network. The effect of small-world connection networks is described by local regular networks and long-range connection with certain probability p. The numerical results confirm that (1) a stable rotating spiral wave can be developed and maintain robust with low p, where the breakup of the spiral wave and turbulence result from increasing the probability p to a certain threshold; (2) appropriate intensity of the optimized channel noise can develop a spiral wave among turbulent states in small-world connection networks of H-H neurons; and (3) regular connection networks are more robust to channel noise than small-world connection networks. A spiral wave in a small-world network encounters instability more easily as the membrane temperature is increased to a certain high threshold.

Highlights

The phase transition of spiral waves in networks of Hodgkin-Huxley neurons induced by channel noise is investigated in detail
The importance of studying spiral waves is that it gives important clues as to how to remove spiral waves in cardiac tissue and prevent ventricular fibrillation [19] and allows a better understanding of the nonlinear dynamics from a spiral wave to turbulence
The dynamics of spiral waves and control pattern selection in reaction-diffusion systems have been studied extensively while few works have been reported on the development and phase transition of the spiral wave in the networks of neurons, and its role in signal communica

Summary

Mathematical model and discussion

The H-H neuron mode is more realistic than other presented neuron models. Small-world networks of H-H neurons are described as follows: Cm dVij dt. The variable Vi,j describes the membrane potential of the neuron in site (i, j) and the subscripts (i, j) indicate the site of the neuron. Dm, Dn and Dh describe the intensity of noise, function δ(t–t′) = 1 at t = t′ and δ(t–t′) = 0 at t ≠ t′, and NNa and NK are the total numbers of sodium and potassium channels present in a given patch of the membrane, respectively. Further numerical results have confirmed that a spiral wave can develop in networks (regular or small-world type) of neurons even if there is no external forcing current. The following section presents a numerical investigation of the robustness and phase of spiral waves in the small-world networks of H-H neurons in the presence of channel noise where there are no external forcing currents acting on neurons

Numerical results and discussion

Findings

Conclusions