We propose a model of quantum-like (QL) processing of mental information. This model is based on quantum information theory. However, in contrast to models of "quantum physical brain" reducing mental activity (at least at the highest level) to quantum physical phenomena in the brain, our model matches well with the basic neuronal paradigm of the cognitive science. QL information processing is based (surprisingly) on classical electromagnetic signals induced by joint activity of neurons. This novel approach to quantum information is based on representation of quantum mechanics as a version of classical signal theory which was recently elaborated by the author. The brain uses the QL representation (QLR) for working with abstract concepts; concrete images are described by classical information theory. Two processes, classical and QL, are performed parallely. Moreover, information is actively transmitted from one representation to another. A QL concept given in our model by a density operator can generate a variety of concrete images given by temporal realizations of the corresponding (Gaussian) random signal. This signal has the covariance operator coinciding with the density operator encoding the abstract concept under consideration. The presence of various temporal scales in the brain plays the crucial role in creation of QLR in the brain. Moreover, in our model electromagnetic noise produced by neurons is a source of superstrong QL correlations between processes in different spatial domains in the brain; the binding problem is solved on the QL level, but with the aid of the classical background fluctuations.