Energy spectrum and thermodynamics of the proton subsystem of the crystal with chains of hydrogen bonds are investigated. The consideration is based on the orientational-tunneling model, which includes two types of proton transfer. The strong short-range interactions and the influence of the internal ordering field arising due to the inter-chain correlations are taken into account. It is shown that under certain conditions the uniform proton distribution along the chain becomes unstable. A possible explanation of the precursor effect observed in the CsDSO 4 crystal at temperatures below the superionic transition point is proposed.