At schools teachers may teach us that an electric current is a flow of electrically charged particles. Students understand that if such particles move, they form electric current.
Maxwell's equations, however, expand the concept of electric current, and one of its key features is that changing magnetic field causes changing electric field. Electric field causes electric current to flow: electrically charged particles would just oscillate without electric field. But in the presence of electric field they start moving along the electric field lines.
The Heliospheric Current Sheet is produced by the magnetic field of the Sun. The star rotates, thus, rotates its magnetic field, it means that the magnetic field in the space around the sun is changing. You are right, there are many ions and other charged particles forming plasma around the sun and within the interplanetary medium. The changing magnetic field creates electric fields, which makes the charged particles move. This is the mechanism behind the electric current in the Heliospheric Current Sheet
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