The smallest number of coordinates needed to specify uniquely a point in space

'Dimension' stems from dnnetTri, Latin for 'to measure out'. Definitions 1 and 2 are the ones most relevant to this discussion, which does not (yet?) involve treatment of spaces with dimensions higher than the familiar four of space-time (definitions 3 and 4).1

Brains are considered to have evolved in evolution under pressures that were supposed to lead to the selection of faculties beneficial for survival in a limited niche of the universe.2 It is therefore likely, in spite of occasional sparks of unjustified hubris that claim otherwise, that the individual human brain perceives only limited dimensions of nature, in terms of both properties and magnitude. That segment of our physical ambience that is directly accessible to our senses and has shaped our intuition3 can be dubbed, based on its dimensions, as the 'mesoworld'. It refers to properties such as size, number, and location in a four-dimensional space; to scales of millimetres to kilometres, milligrams to kilograms, seconds to years; and to complexities that are subserved by processing a few chunks of information at a time. Whatever transcends the aforementioned properties, scales, or complexities, requires "culturally derived technical and conceptual tools for detection, qualification, quantification, and analysis (e.g. Nicolis and Prigogine 1989; Mainzer 1994; Wilson 1995). Science literates do learn to accommodate in their mind notions of entities such as electrons and atoms, the speed of light, galaxies, and black holes; but whether such dimensions are always assimilated in intuition is open to debate. Similarly, in the analysis of the brain itself, we should expect to encounter dimensions that are difficult to grasp intuitively. These may involve quantities (e.g. the number of synapses in a human brain), complexities (e.g. the spatiotemporal activity patterns of large "intracellular signalling networks or "cell assemblies), and qualities (e.g. "consciousness).

Contemplating the dimensions of the research object at the outset of the investigation is always useful. It yields a rough estimate of what lies ahead, assists in focusing on the appropriate "levels of analysis and on the right "methodologies, and even provides a safeguard against certain types of "artefacts. The measures (definition 1) that are within the realm of the 'mesoworld' are a straightforward business: we naturally tend to characterize an object in terms of size, location, or time. Other measures are invisible to the naive eye. These are 'latent dimensions', which could be real, or merely useful hypothetical constructs. They may pop out even in the absence of rigorous statistical analysis, although their verification should involve statistics (Martin and Bateson 1993; Kerlinger and Lee 2000). Alternatively, they become apparent only by factor analysis (Spearman 1904; Thurstone 1947; Kerlinger and Lee 2000). In factor analysis the correlations among variables are used to determine which variables vary together and hence could share an underlying factor. Such factors are candidate dimensions. For example, analysis of the results of a battery of common intelligence tests unveils the presence of two major factors, verbal and mathematical ability. We could then pursue the in-depth analysis of the verbal and mathematical dimensions of human cognition. In this case, but not necessarily in others, the unveiled dimensions agree with intuition.

Let's now illustrate some dimensions of memory. The sample below is highly selective. Some important dimensions are omitted, including level, location ("engram), "representational complexity, "development, and emotion. When appropriate, for each dimension (definition 1), the relevant magnitude (definition 2) will be noted.

1. Time. This is per definition an essential dimension of memory. In humans, memories can last anywhere between 100 and 109 s, depending on whether they are sensory, short-term, long-term, or practically permanent ("consolidation, "phase, "taxonomy; "collective memories may extend over 1012s). This immediately implies that it is naive to expect to find a master solution to the mechanisms of memory. A 'cognitive beat' is 10-2-10-1s. Whether a "stimulus triggers the formation of a memory or not is determined within 10-1-100s ("attention, "perception, "phase). "Working memory lasts 101-102s. Cellular consolidation of long-term memory takes place over 102-104s; system consolidation in the mammalian brain requires 103-106s and possibly even more (Dudai 1996).All this means that critical events in the biology of "acquisition should be addressed by biophysics (Dudai 1997&). Practically, most of the current research on acquisition actually addresses events that take place long after initial critical decisions have been made, and some even forgotten. This is especially true for molecular studies, which deal with processes and mechanisms in the 102-105s range. These are hence expected to tap after-effects of initial encoding and registration, consolidation, and "homeostasis.

2. Size of the neural machinery. The number of neurons expected to encode a relatively simple defensive reflex in "Aplysia is 102-103. The minimal number of cortical neurons needed to reliably encode and transmit physiologically meaningful information is estimated to be <102 (e.g. Shadlen and Newsome 1998). However, real-life engrams in the mammalian brain are expected to be distributed over much larger numbers of neurons, reaching even 107-108, depending on the complexity of the representation (upper estimates are based on "functional neuroimaging; for some estimates from cellular physiology, see Hurlbert and Derrington 1993).4

3. Capacity. This is discussed separately ("capacity).

4. Depth. Introspection suggests that we devote different amounts of mental resources to the acquisition of different types of information. This intuition is supported by systematic research, which shows that the depth and extent of encoding varies among types of tasks and situations, and further, that this 'level of processing' of the acquired information has much to do with the robustness of retention and subsequent "retrievability (Craik and Lockhart 1972). For example, in verbal tasks, phonetic processing is considered shallower than semantic processing, and semantic encoding commonly results in better retrieval than phonetic encoding. The same applies to depth of processing in sensory "perception and in mastering "skill. 'Depth' is a dimension of memory that has so far been influential in human memory research more than in animal research.

5. Types. The "zeitgeist portrays about a dozen types of memory systems in the mammalian brain (e.g. Milner et al. 1998; "declarative memory, "taxonomy). All these systems are expected to have evolved in response to specific needs and phylogenetic pressures. Although some basic molecular components are shared by multiple systems ("CREB, "immediate early genes, "ion channels, "receptors), the heterogeneity of types renders it again reasonable to conclude that there is no master solution to the mechanisms of memory. 6. Subjective dimensions. The effectiveness of a stimulus is a function of the interaction of the stimulus dimensions ("context included) with the "subject's state. The latter can be measured by subjective dimensions such as arousal, "attention, intention, awareness, familiarity, and expectancy (e.g. Boring 1935; Berman et al. 1998; "algorithm, "declarative memory, "learning, "surprise).

"Reductionists are possessed by the search for elementary dimensions. It is useful though to remember that emergent properties are also important dimensions; and, further, that the brain in fact accommodates all the perceived and inferred dimensions within it, and in that respect is omnipotent and dimensionless. This was crisply perceived by Dickinson (1896): 'The Brain—is wider than the Sky/For—put them side by side—/The one the other will contain/With one—and You—beside—/The Brain is deeper than the sea— /For—hold them—Blue to Blue—/The one the other will absorb—/As Sponges—Buckets—do—/The Brain is just the weight of God—/For—Heft them—Pound for Pound—/And they will differ—if they do—/As Syllable from Sound—'.

Selected associations: Cell assembly, Level, Reduction, Stimulus, Synapse

'Definitions 1 and 3 are different formulations of the same idea, safe that in practice, the 'measures' in definition 1 are not necessarily independent. Broader, more sophisticated definitions of 'dimension' have been developed in the exact sciences, but they far exceed the scope of this discussion (Mandelbrot 1977; Greene 1999).

2This argument echoes the Panglossian paradigm, which trusts that natural selection is an optimizing agent (see *paradigm). It is prudent to take into account the possibility that brain and memory systems are what they are not only because of adaptation but also because of accumulative structural and functional constraints. This caveat, however, is unlikely to nullify the subsequent assumption in the text concerning the limitations of our brain.

3Intuition' refers to a fast, mostly innately predisposed system of tacit knowledge about the world.

4The potential distinction between the minimal number of neurons obligatory for representing an item and the actual number engaged in representing that item is another issue, the discussion of which exceeds the scope of the present discussion.

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