Connexins are specialized ionic channels that control the action potential propagation between cardiac myocytes. In this paper, we study the connexin dynamics in a one-dimensional model of cardiac tissue. We show that the connexin dynamics may lead to a spatial organization of the gap junction conductance. In the numerical simulations presented in this paper we have found two different regimes for the spatial organization of the conductances: (a) a spatially uniform conductance; (b) a spatially complex pattern of local values of high and low conductances. in addition, we have observed that, locally, the two final states are limit cycles with a period equal to the period associated with the external excitation of the tissue strand. The conductance dispersion usually takes place on a very large time scale, i.e. thousands of heart beats, and on a very short spatial scale. Due to its simplicity, the one-dimensional setting allows a detailed study of the emerging structure and in particular very long simulations. We have studied the transition between the two aforementioned states as a function of the gap junction conductance characteristics. Furthermore, we have studied the effect of initially added noises on the outcome of the system. Finally, using spatial autocorrelation functions we have characterized the spatial dispersion in conductance values.