Entering the information era, the formation of public opinion is largely associated with the complex system constructed by the Internet, thereby possessing new characteristics. The formation of public opinion is the result of the interaction of individual behavior with social environment. In reality, the environmental factor and the individual behavior are usually related to each other and co-evolve with time. Based on the Ising model, in this paper established is an opinion formation model that includes the process of the accumulation and digestion of the social tension. In the model, a parameter named effective dissolving factor c is designed to represent the extent of the interaction between the system and the social environment. A two-dimensional dynamical system is involved in the model to describe the dynamics of individual behavior and social tension. The co-evolution behavior of the system is studied. Based on the Landau mean field theory, the stationary states of the dynamical system under different parameter values, i.e. the value of effective dissolving factor c, their stability and bifurcation of the system, are analyzed. Finally, the computer simulation method is used to verify the results. The research shows that with the co-evolution mechanism of the system, our model exhibits certain self-organization characteristics. When the effective dissolving factor c is smaller than the threshold value, the system will reach final consensus opinion, resulting in a macroscopically ordered state. Otherwise, when the dissolving factor c exceeds a threshold value, the system is stable in the disordered state. It is interesting to find that there is such a critical value of the parameter that it leads the system to be self-organized into a critical state from any initial state. The future detailed investigation on the criticality of the co-evolving system is also suggested, such as testing whether the system has evolved into the critical state according to the finite-sized scaling theory and calculating the critical exponent of the system. In addition, in this paper provided is a new perspective to tackle practical problems in public opinion. Based on the mechanism of the formation of public opinion revealed by our model, researchers are encouraged to conduct studies on how to monitor the state of public opinion more precisely and to predict the tipping point of the system evolution.