The unique electronic and optical properties of recently discovered two-dimensional (2D) crystals, such as graphene, graphene oxide, molybdenum disulphide etc., demonstrate their enormous potential in creating ultrahigh density electronics for image recognition systems and information storage. Synapse-like memristive heterostructures are considered as a new type of electronic switches with extremely low power consumption and footprint that can be used to overcome the limit of current CMOS technology. Memristors with a floating photogate, called photomemristors, based on graphene and MoS2, are considered. Photocatalytic oxidation of graphene is considered as an effective method for creating memristive heterostructures with photoresistive switching for non-volatile electronic memory of ultrahigh density for the formation of self-assembled nanoscale memristive elements interfacing with neural networks. 2D photomemristors with a floating photogate exhibit multiple states that can be monitored over a wide range of electromagnetic radiation and can be used in neurohybrid systems for image processing and pattern recognition, as well as for selective manipulation of neurons by light. Copyright © VBRI Press.