In previous papers (Smythies et al., 2012, 2014) we presented the general hypothesis that the claustrum may be concerned with information processing operations on synchronized gamma oscillations in the brain at three levels. At the first level (subhypothesis 1) it just magnifies the oscillations in cortico-claustral circuits. At the second level (subhypothesis 2) it may integrate these oscillations. At the third level (subhypothesis 3) it might process the contained spike codes. These previous papers were concentrated upon the first level. In this paper we will review arguments that both subhypotheses 2 and 3 are, at present, unsatisfactory. Subhypothesis 1, however, remains satisfactory. The claustrum consists of a body of densely interconnected P (pyramidal) cells and GABAergic interneurons (INs) arranged in a sheet of gray matter underlying the insula in the basal telencephalon of the mammalian brain. These cells are arranged in functional units, each of which is connected with a particular cortical or subcortical area with which the claustrum maintains reciprocal relationships, which encompass practically every such area. There is evidence that it is particularly concerned with salient activities requiring integration between two or more of these areas [see Smythies et al. (2012, 2014), for details]. One structurally facilitative feature of this anatomy is that the claustrum has a number of afferents/efferents along a topographical spectrum. There are well-defined arrays of visual-, auditory-, and somatosensory-associated zones (or “maps”), plus extensive limbic connections with the ventral claustrum. To the extent studied, individual P cells have been shown to receive a (mostly) modality-specific input. For example, visual zone neurons respond (almost) exclusively to visual stimuli, auditory neurons respond (almost) exclusively to auditory stimuli, etc. (Remedios et al., 2010). The afferent glutamatergic axons of cortical layer VI P cells densely synapse on claustral P cells. In turn, these claustral P cells reciprocate by sending efferents back to the very same cortical area from which their afferent input originated. It has been demonstrated that some claustral P cells have bifurcated axonal projections to different cortical areas. Of particular significance is that claustral P cells also maintain direct contact with claustral GABAergic INs via local collaterals forming a dense axonal array. The INs quite likely form an interactive gap-junction syncytium. This structure entails that the afferent inflow to the claustrum will carry multiple impulses with different power spectra and synchronized at different frequencies that will have ample opportunity to interact in complex ways within the densely packed amorphous syncytium that constitutes the interior of the claustrum. The results of these interactions, in the form of processed and integrated information, might then be transported to selected areas of the brain via the efferent network. The question that we will address now is what is the validity of subhypotheses 2 and 3.