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Disorganized electrical activity in the heart leads to sudden cardiac death.To what extent can this electrical turbulence be viewed as classical fluidturbulence,which is an important central problem in modern physics? Weinvestigate,for the first time,via extensive DNSs,the statistical properties ofspiral-and scroll-wave turbulence in two- and three-dimensional excitable mediaby using approaches employed in studies of classical turbulence. We use thePanfilov and the Aliev-Panfilov mathematical models for cardiac tissue. We showthat once electrical-wave turbulence has been initiated,there is a forwardcascade,in which spirals or scrolls form,interact,and break to yield aturbulent state that is statistically steady and,far away from boundaries,isstatistically homogeneous and isotropic. For the transmembrane potential $V$and the slow recovery variable $g$,which define our models,we define $E_V(k)$and $E_g(k)$,the electrical-wave analogs of the fluid energy spectrum $E(k)$ influid turbulence. We show that $E_V(k)$ and $E_g(k)$ are spread out overseveral decades in $k$. Thus spiral- and scroll-wave turbulence involves a widerange of spatial scales. $E_V(k)$ and $E_g(k)$ show approximate power laws,insome range of $k$, however,their exponents cannot be determined as accuratelyas their fluid-turbulence counterparts. The dimensionless ratio $L/\lambda$ isa convenient control parameter like the Reynolds number for fluidturbulence,where $L$ is the linear size of the domain and $\lambda$ thewavelength of a plane wave in the medium. By comparing several otherstatistical properties for spiral- and scroll-wave turbulence with theirfluid-turbulence counterparts,we show that,although spiral- and scroll-waveturbulence have some statistical properties like those of fluidturbulence,overall these types of turbulence are special and differ inimportant ways from fluid turbulence.

The statistical properties of spiral- and scroll-wave turbulence in cardiac tissue