Scaling of differentiation in networks: nervous systems, organisms, ant colonies, ecosystems, businesses, universities, cities, electronic circuits, and Legos.

Nodes in networks are often of different types, and in this sense networks are differentiated. Here we examine the relationship between network differentiation and network size in networks under economic or natural selective pressure, such as electronic circuits (networks of electronic components), Legos (networks of Lego pieces), businesses (networks of employees), universities (networks of faculty), organisms (networks of cells), ant colonies (networks of ants), and nervous systems (networks of neurons). For each of these we find that (i) differentiation increases with network size, and (ii) the relationship is consistent with a power law. These results are explained by a hypothesis that, because nodes are costly to build and maintain in such "selected networks", network size is optimized, and from this the power-law relationship may be derived. The scaling exponent depends on the particular kind of network, and is determined by the degree to which nodes are used in a combinatorial fashion to carry out network-level functions. We find that networks under natural selection (organisms, ant colonies, and nervous systems) have much higher combinatorial abilities than the networks for which human ingenuity is involved (electronic circuits, Legos, businesses, and universities). A distinct but related optimization hypothesis may be used to explain scaling of differentiation in competitive networks (networks where the nodes themselves, rather than the entire network, are under selective pressure) such as ecosystems (networks of organisms).

Evidence that appetitive responses for dehydration and food-deprivation are learned.

Rats do not seek water when cellularly dehydrated until they are about 4 weeks of age. This lack of appetitive 'seeking' behavior in young rats differs from their precocious ingestive responses such as an increased intake of solutions infused into their mouths when they are dehydrated as young as 2 days of age. Using video analysis of appetitive behavior in a structured environment, we document this early absence of appetitive responding and the subsequent acquisition of dehydration-elicited appetitive behavior. Weaning age pups were separated into four conditions: (i) experienced, dehydrated; (ii) experienced, nondehydrated; (iii) inexperienced, dehydrated; and (iv) inexperienced, nondehydrated. 'Experienced' rats received a dehydration and drinking experience prior to the test, and 'dehydrated' rats were dehydrated (by injection of a salt load) at the time of test. At the test, all water and food was removed from the test cages, eliminating the confounding of appetitive and consummatory measures. Despite the fact that pups in all conditions had experience with water and had previously drunk, only the 'experienced' pups differentially sought water when dehydrated. Parallel experiments with food deprivation produced similar results. Pups did not exhibit food-seeking behavior when food-deprived unless they had previous experience with food deprivation and eating. The appetitive 'seeking' behavior for feeding also appears to be learned. Directed appetitive behavior in general may thus be acquired.

Universal scaling laws for hierarchical complexity in languages, organisms, behaviors and other combinatorial systems.

There are many complex systems in nature where components, or "words", are combined together to make expressions, or "sentences". Such combinatorial systems include: (1) human language, where sentences are composed of words; (2) bird vocalization, where songs are built from syllables; (3) organisms, where organism-expressions (e.g. the tonsil) are made out of cells; (4) behavioral repertoire, where mammalian behavior consists of a temporal arrangement of muscle contractions; (5) universities, where student academic degrees are comprised of departmental concentrations; and (6) electronic devices, where the device's actions are implemented via strings of button-presses. My central aim here is to discover how combinatorial systems accommodate greater numbers of expressions; that is, what changes do combinatorial systems undergo when they "say more things?" Are there general laws characterizing the properties of combinatorial systems as the number of expressions increases? If so, what are they? My main result is that, in all the kinds of combinatorial system mentioned above, there appear to be general laws describing how combinatorial systems change as they become more expressive. In particular, in each of these cases, increase in expression complexity (i.e. number of expressions the combinatorial system allows) is achieved, at least in part, by increasing the number of component types. Each kind of system follows one of two kinds of scaling law. In the first kind of scaling law, expression complexity increase is carried out exclusively by increasing the number of component types; the number of components per expression (i.e. the expression length) remains invariant. This applies to human language over history, bird vocalization, organisms in phylogeny and ontogeny, and universities. In the second kind of scaling law, expression complexity is accomplished by increasing in a law-like manner both the number of component types and the expression length. This applies to two cases of the ontogeny of language-the development of words and sentences, and the development of phonemes and morphemes-and to mammalian behavior. By treating these diverse systems as combinatorial systems we, in addition to elucidating general principles underlying such systems, gain insight into each kind of system mentioned.

'Perceiving the present' as a framework for ecological explanations of the misperception of projected angle and angular size.

An implicit, underlying assumption of most Helmholtzian/Bayesian approaches to perception is the hypothesis that the scene an observer perceives is the probable source of the proximal stimulus. There is, however, a nontrivial latency (on the order of 100 ms) between the time of a proximal stimulus and the time a visual percept is elicited. It seems plausible that it would be advantageous for an observer to have, at any time t, a percept representative of what is out there at that very time t, not a percept of the recent past. If this is so, it implies a modification to the implicit hypothesis underlying most existing probabilistic approaches to perception: the new hypothesis is that, given the proximal stimulus, the scene an observer perceives is the probable scene present at the time of the percept. That is, the hypothesis is that what an observer perceives is not the probable source of the proximal stimulus, but the probable way the probable source will be when the percept actually occurs. A model of an observer's typical movements in the world is developed, and it is shown that projected angles are perceived in a way consistent with the way the probable source will project to the eye after a small time period of forward movement by the observer. The predicted and actual direction of projected-angle misperception is sometimes toward 90 degrees and sometimes away from 90 degrees, depending on whether the probable source angle is lying in a plane parallel or perpendicular to the probable direction of motion, respectively. The perception of angular size for lines in a figure with cues they are lying in a plane perpendicular to the direction of motion is also shown to fit the predictions of the model.

Principles underlying mammalian neocortical scaling.

The neocortex undergoes a complex transformation from mouse to whale. Whereas synapse density remains the same, neuron density decreases as a function of gray matter volume to the power of around -1/3, total convoluted surface area increases as a function of gray matter volume to the power of around 8/9, and white matter volume disproportionately increases as a function of gray matter volume to the power of around 4/3. These phylogenetic scaling relationships (including others such as neuron number, neocortex thickness, soma radius, and number of cortical areas) are clues to understanding the principles driving neocortex organization, but there is currently no theory that can explain why these neocortical quantities scale as they do. Here I present a two-part model that explains these neocortical allometric scaling laws. The first part of the model is a special case of the physico-mathematical model recently put forward to explain the quarter power scaling laws in biology. It states that the neocortex is a space-filling neural network through which materials are efficiently transported, and that synapse sizes do not vary as a function of gray matter volume. The second part of the model states that the neocortex is economically organized into functionally specialized areas whose extent of area-interconnectedness does not vary as a function of gray matter volume. The model predicts, among other things, that the number of areas and the soma radius increase as a function of gray matter volume to the power of 1/3 and 1/9, respectively, and empirical support is demonstrated for each. Also, the scaling relationships imply that, although the percentage of the total number of neurons to which a neuron connects falls as a function of gray matter volume with exponent -1/3, the network diameter of the neocortex is invariant at around two. Finally, I discuss how a similar approach may have promise in explaining the scaling relationships for the brain and other organs as a function of body mass.

The economy of the shape of limbed animals.

A simple, high-level wire-minimization model appears to drive the relationship between animal limb number and body-to-limb proportion in some animals across at least seven phyla: annelids, arthropods, cnidarians, echinoderms, molluscs, tardigrades and vertebrates. Given an animal's body-to-limb proportion, the model enables one to estimate the animal's number of limbs, and vice versa. Informally, the model states that a limbed animal's large-scale morphology is set so as to maximize its number of limbs subject to the constraint that there is not a more economical shape which reaches out to the same places. A consequence of animals conforming to the model is that their large-scale morphology is "minimally wired." Just as wire minimization is important in artificial information processing devices, it is hypothesized that one reason why animals' large-scale morphologies conform to a save-wire principle is to minimize the system-wide information processing times.

Thirst modulates a perception.

Does thirst make you more likely to think you see water? Tales of thirsty desert travelers and oasis mirages are consistent with our intuitions that appetitive state can influence what we see in the world. Yet there has been surprisingly little scrutiny of this appetitive modulation of perception. We tested whether dehydrated subjects would be biased towards perceptions of transparency, a common property of water. We found that thirsty subjects have a greater tendency to perceive transparency in ambiguous stimuli, revealing an ecologically appropriate modulation of the visual system by a basic appetitive motive.

Modeling the large-scale geometry of human coronary arteries.

Two principles suffice to model the large-scale geometry of normal human coronary arterial networks. The first principle states that artery diameters are set to minimize the power required to distribute blood through the network. The second principle states that arterial tree geometries are set to globally minimize the lumen volume. Given only the coordinates of an arterial tree's source and "leaves", the model predicts the nature of the network connecting the source to the leaves. Measurements were made of the actual geometries of arterial trees from postmortem healthy human coronary arteriograms. The tree geometries predicted by the model look qualitatively similar to the actual tree geometries and have volumes that are within a few percent of those of the actual tree geometries. Human coronary arteries are therefore within a few percent of perfect global volume optimality. A possible mechanism for this near-perfect global volume optimality is suggested. Also, the model performs best under the assumption that the flow is not entirely steady and laminar.