complexity

## What is complexity?

complexity (wordnet, noun)  
: complexity, complexness -- (the quality of being intricate and compounded;)
: disordered vs ordered complexity  
: elaborateness, elaboration, intricacy, **involution**

involution (wn, noun)  
: (marked by elaborately complex detail)  
: exponentiation, involution -- (the process of raising a quantity to some assigned power)  
: involution, enfolding -- (the action of **enfolding** something)

enfold (wn, verb)
: envelop, enfold, enwrap, wrap, enclose -- (enclose or enfold completely with or as if with a covering; "Fog enveloped the house")

Note:

complexity is generally not well defined...

Quotes from Alan Perlis, a former CS dept chair at a leading university in "Epigrams on Programming", for Association for Computing Machinery's (ACM) SIGPLAN journal

> 25. One can only display complex information in the mind. Like seeing, movement or flow or alteration of view is more important than the static picture, no matter how lovely.

> 58. Fools ignore complexity. Pragmatists suffer it. Some can avoid it. Geniuses remove it.

> 105. You can’t communicate complexity, only an awareness of it.

> 31. Simplicity does not precede complexity, but follows it.

---

## Complex forms

<div><img src="figs/IMG_0214_complexity_sea-lilly_f8297c6.jpg" width="400px"><figcaption>sea lilly, complex symmetry</figcaption></div>

<div><img src="figs/IMG_0215_complexity_jellyfish_13b5982.jpg" width="400px"><figcaption>jellyfish symmetry</figcaption></div>

Note:

* see also the great book *On Growth and Form* 1917 by D'Arcy Wentworth Thompson (1860–1948). 
* available in public domain :) https://archive.org/details/ongrowthform00thom

Computers, pattern, chaos, and beauty. Clifford A. Pickover St Martins Press NY 1990
Pattern, symmetry, beauty
Genesis Equations (or biological feedback forms) chp 8
p. 96
> Fig. 8.1 Intricate biological shapes. The physical world contains complicated patterns formed from simple shapes through the repeated application of dynamic procedures. Shown here is a sea-lilly.

p. 103
> Fig. 8.8 A variety of simple, symmetrical natural forms (jellyfish).

Computers, pattern, chaos, and beauty. Clifford A. Pickover St Martins Press NY 1990
Pattern, symmetry, beauty
Genesis Equations (or biological feedback forms) chp 8

* feedback: part of a systems output returns to the input. An iteration or recursion in math (repeating a set of operations), like f(f(x)) composing a function with itself.

* self squaring example: `z -> z**2 + µ`
* Gaston Julia (1893-1978), feedback with complex values for z gives striking results  `z-> z**2 + µ`

complex 
: if real numbers are points on a line, then complex numbers represent points on a plane. An orthogonal space
: if z is complex, then z = a + ib, where i=sqrt(-1)
: complex coined by Carl Gauss (1832) numbers with a real and imaginary component

* complexity of non-linear systems investigated by interating functions like z_n = f(z_n-1, µ); n = 1,2,3...
* for some z0 the sequence may diverge (increasingly large), others the iteraction converges to small number or can hop between sets of numbers (limit cycles) or display apparently random behavior (chaos)

video of self-reflected microetching creative visualization of the human brain  
[how it was made](http://www.gregadunn.com/self-reflected/how-self-reflected-was-made/)

## What is form?

Overview for the noun "form" in wordnet has numerous forms, totalling sixteen! Here are some of them:

form (wn, noun)  
: form, shape, pattern -- (a perceptual structure;)  
: shape, form, configuration, contour, conformation -- (any spatial attributes (especially as defined by outline);  
: shape, form -- (the spatial arrangement of something as distinct from its substance;)  
: form, shape, cast -- (the visual appearance of something or someone;) 
: kind, sort, form, variety -- (a category of things distinguished by some common characteristic or quality;)  
: form, variant, strain, var. -- ((biology) a group of organisms within a species that differ in trivial ways from similar groups;)
: form -- (a particular mode in which something is manifested;)  
: phase, form -- ((physical chemistry) a distinct state of matter in a system; matter that is identical in chemical composition and physical state and separated from other material by the phase boundary;)

</div>

Note:

form (wn, verb)  
: form, organize, organise -- (create (as an entity);)  
: form, take form, take shape, spring -- (develop into a distinctive entity;)

---

## Spirals in nature

Note:

Computers, pattern, chaos, and beauty. Clifford A. Pickover St Martins Press NY 1990

Synthesizing Nature
p. 211, 212

> Fig 12.14. Examples of spiral in nature. Several shells of various ammonites (extinct shellfish) are shown

the logarithmic spiral: `r = ke^a*theta`
nautilus shells and sea shells, animal horns, seeds of plants like sunflower and daisy, scales of a pinecone, spiralling fibers at apex of the heart

---

## Complexity from temporal feedback

<div><img src="figs/IMG_0218_complexity_fractals_self-similarity_ef56035.jpg" width="500px">
<figcaption>from *Computers, pattern, chaos, and beauty*, C. Pickover St Martins Press NY 1990
</figcaption></div>

Note:

* feedback: part of a systems output returns to the input. An iteration or recursion in math (repeating a set of operations), like f(f(x)) composing a function with itself.
* Iteration. Interaction. **Time**

Computers, pattern, chaos, and beauty. Clifford A. Pickover St Martins Press NY 1990
Genesis equations, Feedback
p. 100
> Fig. 8.5 Feedback plot. The equation used was z = z^5 + µ.

* self squaring example: z -> z**2 + µ
* Gaston Julia (1893-1978), feedback with complex values for z gives striking results  z-> z**2 + µ

complex 
: if real numbers are points on a line, then complex numbers represent points on a plane. An orthogonal space  
: if z is complex, then z = a + ib, where i=sqrt(-1)  
: 'complex' coined by Carl Gauss (1832) for numbers with a real and "imaginary" component  
: Euler's (astonishing) identity-- or `e^ix = cos(x) + isin(x)`, when `x=π` then `1 + e^iπ = 0` !

* complexity of non-linear systems investigated by interating functions like `z_n = f(z_n-1, µ); n = 1,2,3...`
* for some z0 the sequence may diverge (increasingly large), others the iteraction converges to small number or can hop between sets of numbers (limit cycles) or display apparently random behavior (chaos)

---

## Complexity: living systems

<div><img src="figs/IMG_0211_human_system_complexity_ffb404a.jpg" width="300px"><figcaption>a human system</figcaption></div>

<div><img src="figs/IMG_0217_human_brain_electronic_circuit_684bf0b.jpg" width="400px"><figcaption>human brain and an electronic circuit</figcaption></div>

Note:

Computers, pattern, chaos, and beauty. Clifford A. Pickover St Martins Press NY 1990
Synthesizing Nature
p. 194 
human, Gray?

Synthesizing Nature
p. 13
> Fig. 2.5 The human brain and an electronic circuit.

p. 103
> Fig. 8.8 A variety of simple, symmetrical natural forms (jellyfish).

---

## A complex system

...could be

: a set of many interacting objects  
: a combination of objects whose interactions give rise to 
system whose behavior is greater than the sum of its parts

## What is complex?

complex (wordnet, adj)  
: (complicated in **structure**; consisting of interconnected parts;)
: (difficult to analyze or understand;)

Note:

"he's more complex than he seems on the surface"

complex (wn, noun)  
: a whole structure made up of interconnected or related structures  
: a compound described in terms of the central atom to which other atoms are bound or coordinated

## What is structure?

structure (wn, noun)  
: (the complex composition of knowledge as elements and their combinations;)  
: structure, anatomical structure, complex body part, bodily structure, body structure -- (a particular complex anatomical part of a living thing and its construction and arrangement;)

Note:

not just a bunch of things lumped together, but a combination of things staving off entropy. The second law of thermodynamics

## Classifying structure

Note:

from Computers, pattern, chaos, and beauty. Clifford A. Pickover St Martins Press NY 1990
p. 60
> Fig. 4.14 Classification of the shapes of nature and math. Self-similar structures, when magnified, look like the original shape. "Random" self-similar structures are self-similar only in the statisitcal sense so even though they don't repeat their pattern exactly, they clearly have the same look at different magnifications. Speech waveforms, at sentence time scales, are statisically self-similar.

---

## What is a system?

system (wn, noun)  
: (a procedure or process for obtaining an objective;)   
: (a group of independent but interrelated elements comprising a unified whole;)  
: ((physical chemistry) a sample of matter in which substances in different phases are in equilibrium;)   
: (a group of physiologically or anatomically related organs or parts; "the body has a system of organs for digestion")  
: (the living body considered as made up of interdependent components forming a unified whole;)

Note:

some (funny) wordnet quotes for system:  
"a vast system of production and distribution and consumption keep the country going"... *production and consumption of useful content or vast waste?* -jba ;)
"in a static system oil cannot be replaced by water on a surface"; 
"exercise helped him get the alcohol out of his system"

## Entropy

: from Greek word for transformation  
: S = *k*ln(W)  
: joules per kelvin

Schrodinger introduced the term negative entropy in 1944 in his book 'What is life?' [Schrödinger:1944].

Léon Nicolas Brillouin, then shortened it to negentropy in 1953 [Brillouin:1953].

[Brillouin:1953]: Brillouin, Leon: (1953) "Negentropy Principle of Information", J. of Applied Physics, v. 24(9), pp. 1152–1163

Note:

: second law of themrodynamics, isolated systems always go towards thermodynamic equilibrium, a state with maximum entropy and never decreasing entropy over time 
: W is the probability distribution for the possible microstates that can explain the observed system's macrostate, the position and momentums of the systems objects  
: *k* is Boltzmann constant, *k*b = 1.380x10^-23^ J/K. One of the seven defining constants of SI base units, is energy divided by temperature, same dimension as entropy  
: Ludwig Boltzmann's entropy formula carved on his gravestone <https://en.wikipedia.org/wiki/Boltzmann%27s_entropy_formula>  
: non-isolated systems like an organism can decrease entropy, as long as the entropy of the environment increases by at least the same amount, thus total entropy remains constant. Thus entropy can decrease in a system, as long as total entropy of Universe does not  
: entropy as a function of state of the system, entropic change determined by initial and final states. A reversible process does not result in entropic change, irreversible processes always increase entropy

>Information is classically defined as reduction of uncertainty: the more numerous the alternatives that are ruled out, the greater the reduction of uncertainty, and thus the greater the information. (Tononi [3], p. 217)

In information theory negentropy is a measure of distance to normality. Measures difference between a given distribution and Gaussian distribution with same mean and variance.

Why? Because the normal or Gaussian distribution is the distribution with the highest entropy. Thus the difference means that negentropy is always positive.

let J be negentropy, then Smax is the differential entropy of the Gaussian density with same mean and variance as p_x and Sp_x is the

`J = Smax - Sp_x`

`Sp_x = H(X) = -∑Pxlogb(Px)`

The logarithm base is usually 2 for bits, *e* for nats, or 10 for bans.

Information entropy is used in independent components analysis. Is the average level of information, information content, surprise, or uncertainity in all of a variable's possible outcomes.

For a pair of two-sided coin tosses, the information entropy in bits is base2log of the number of possible outcomes (2^2). `S=log2(2^2) = 2`.

C. Shannon's "fundamental problem of communication" is for a receiver to identify what data was generated by the source based on the signal it receives through a channel.

Low probability events produced by a data source carry more information that a high probability value. Surprise. Redundancy.

For thermodynamics, like a Carnot heat engine, entropy can be thought of as a measure of thermal energy per temperature unit not availabe for doing work. The amount of reversible heat divided temperature. Room temperature entropy decreases as it 'inputs' energy into a glass holding ice water. The ice water glass 'system' has an increase in entropy greater than then loss of the room temperature for this room + glass+ water 'universe'.

For statistical mechanics, the Gibbs entropy formula, entropy is a logarithmic measure of the number of states with significant probability of being occupied.

`S = -*k*∑p_i_∙ln(p_i_)`

von Neumann entropy, extending entropy into the quantum domain:

`S = -*k*∑Tr(ρlog(ρ))` ρ is the probability density matrix and Tr is trace op

Shannon entropy, from C. Shannon 1948 on information theory regarding amount of information in transmitted messages. Attenutation in phone-line signals. Amount of information that is missing before reception.

```
H(X) = -∑p(xi)∙log(p(xi))
H = -∑p_i_∙ln(p_i_)
```

If each message is equally probable, then Shannon entropy would be number of binary (bits) yes/no questions required to determine content of the message.

Nat or nit is a unit of information or entropy, based on natural logarithms, powers of *e* instead of base2 which define the bit. One nat is 1/ln(2) bits ==1.44. One nat is the information content of a uniform distribution on interval [0,e].

If entropy is measured in units of k per nat (natural unit of information), normalizing the Boltzmann constant to 1 then the thermodynamic entropy is measured is nats giving the Boltzmann entropy formula or Boltzmann-Planck equation:

`H=*k*log(W)`; where W is the number of real microstates corresponding the system (e.g. gas particles) macrostate. Relation between entropy and n ways entities of a thermodynamic system can be arranged. For a system where microstates do not have equal probabilities, Gibbs entropy is more appropriate generalization. W from 'Wahrscheinlichkeit', German word for probability.

---

## Examples

* sand dunes, avalanches
* chemical waves, chaos 
* life. biological complexity, seashells

Note:

---

## Brain complexity

Tensional and compressive forces result in cerebral folding.

<div><img src="figs/2020-06-01-231346.png" width="400px"><figcaption>Tallinen:2016, Fig1 
</figcaption></div>

<div><img src="figs/2020-06-01-231854.png" width="400px"><figcaption>Tallinen2016, Fig3a</figcaption></div>

Note:

* Tensegrity, tensional forces of long cellular processes (axon bundles), Felleman and Van Essen 1991 monkey neocortical/visual wiring map [^Felleman:1991]

[^Felleman:1991]: Distributed hierarchical processing in the primate cerebral cortex. pmid:1822724

Takes between 22 – 26 weeks of gestation (154 – 182d. or 5.1 – 6 mo.) before fissures and gyri start forming in the human brain (Tallinen Nature Physicis 2016). Between 33–37 weeks the convolutions take on the complexity seen in the newborn human brain (7.7–8.6 mo.).

[^Tallinen:2016]: On the growth and form of cortical convolutions. http://dx.doi.org/10.1038/nphys3632

---

## Cephalopod eyes

<div><img src="figs/IMG_0219_cephalopod_nautilis_drawing_owen_96238b7.jpg" width="400px"><figcaption>Nautilis drawking, R. Owen</figcaption></div>

<div><img src="figs/IMG_0220_cephalopod_octopus_eye_f4c01c2.jpg" width="400px"><figcaption>J. Allman, *Evolving Brains* p81</figcaption></div>

Note:

* divergence from vertebrate lineage over 550mya
* still many neurons (500,000,000), two-thirds of which are in the arms [^Hochner:2006]

[^Hochner:2006]: Hochner B, Shomrat T, Fiorito G., The octopus: a model for a comparative analysis of the evolution of learning and memory mechanisms. Biol Bull. 2006 Jun;210(3):308-317.

Evolving Brains. John Morgan Allman
Eyes, Noses, and Brains
p. 80,81
> A dissected nautilus drawn by Richard Owen. Half the shell has been removed to reveal the chambers.

> The structure of the octopus eye. based on the studies of JZ Young. The visual images is inverted by the lens. The neural representation of the retinal image is reinverted by the dorsoventral crossing of the myriad optic nerves shich project topographically onto the optic lobe. The upper diagram illsutrates a section throgh the retinal photoreceptors. The photrecpeive memberanes are located in the microvilli, which are oriented in the horizontal or vertical plane, and which probalby analyse the place of prolarized light. The axons of the photrecpeors form the optic nerves. Theus theo octupus eye projects directlhy hto the brain, unlikes the vertebrae retinal phoorecpetors which connect to a searies of neurons whithin the reintal itself before connecting via the optic neve to the brain.

---

## Genome size does not predict nervous system complexity

organism   | # of genes | # of base pairs | # of neurons | development time (young adult)  
---------- | ---------- | ---- | ------------ | -------------------------  
*Caenorhabditis elegans* (nematode) | ~19,000    | ~97 million | 302          | 8 hrs  
*Drosophila melanogaster* (fruit fly)      | ~15,000    | ~120 million | ~250,000 | 7–11 days  
*Danio rerio* (zebrafish)      | ~24,000    | ~1.5 billion | ~10,000,000 | 30 days  
*Xenopus tropicalis* (frog) | ~21,000 | ~1.7 billion |  ~16,000,000 | 4 months   
Mouse      | ~25,000    | ~3.5 billion | ~71,000,000 | 2-3 months  
Human      | ~20,000    | ~3.5 billion | ~100,000,000,000 | 18 years  
African elephant | ~20,000    | ~3.1 billion | ~267,000,000,000 | 18 years

</div>

Note:

Number of genes is not related to nervous system complexity or size. The nematode c. elegans has just 302 neurons, and yet its genome contains virtually as many genes as a humans. An african elephant brain weighs 3 times more than a human brain and has 3 times the number of neurons.

Even number of base pairs: Paris japonica (white, star like flower) has 150 billion base pairs of DNA (50x larger than that of a human haploid genome)

The largest brains are those of sperm whales, weighing about 8 kg (18 lb). An elephant's brain weighs just over 5 kg (11 lb), a bottlenose dolphin's 1.5 to 1.7 kg (3.3 to 3.7 lb), whereas a human brain is around 1.3 to 1.5 kg (2.9 to 3.3 lb). Brain size tends to vary according to body size.

animal | n neurons | n synapse
--- | --- | ---
roundworm | 302 | 7500
jellyfish | 5600 | 
sea slug | 18,000 | 
amphioxus | 20,000 | 
larval zebrafish | 100000 |
fruitfly | 250,000 | <10,000,000
ant | 250,000 |
honey bee | 960,000 |
cockroach | 1,000,000 |
guppy | 4,300,000 |
frog | 16,000,000 |
zebra finch | 131,000,000 |
brown rat | 200,000,000 | 4.48x10^11 
red junglefowl | 221,00,000 |
ferret | 404,000,000 |
gray squirrel | 453,660,000 |
octopus | 500,000,000 |

sources for n neurons

* http://faculty.washington.edu/chudler/facts.html  
* https://en.wikipedia.org/wiki/List_of_animals_by_number_of_neurons

E. Coli K-12: 4,639,221 bp, 4377 genes  
Naegleria gruberi (unicellular free living eukaryote organism) 41 x 10^6 bp 15,727 genes

<https://www.xenbase.org/anatomy/intro.do>
xenopus genome project <http://viewer.shigen.info/xenopus/>

X. tropicalis develops faster with a higher optimal temperature (25-30°C vs 16-22°C), and has a diploid, smaller genome (instead of tetraploid) compared with X. laevis. Still has big eggs (0.8mm vs 1.2mm), large brood size (500-2000+) and large (but smaller) adults (4-5cm vs 10cm) as X. laevis  
: gastrulation at 7-8hrs, tailbud at 20hrs and tadpole at 36hrs. Reproductive adults at 12mo.  
: https://www.xenbase.org/anatomy/static/intro/xenopus_life_cycle_small.png

xenopus hatchlings  
: can detect light intensity (pineal gland, then with eyes a few days later) Foster and Roberts 1982)  
: discriminate touch (skin is excitable with cardiac-like all or none APs for a primitive noxious stimuli response like Cnidarians (Mackie 1970). Body surface innervated by touch neurons to trigeminal gangli and spinal cord (Roberts 1980)  
: lateral line neuromasts caudal to eyes for water current responses by swimming into them (Roberts 2009)  
: have few thousand functioning central neurons (populations of 20-150) on each side [Roberts:2010]

[Roberts:2010]: Roberts A, Li WC, Soffe SR. How neurons generate behavior in a hatchling amphibian tadpole: an outline. Front Behav Neurosci. 2010 Jun 24;4:16. doi:10.3389/fnbeh.2010.00016. PMID: 20631854; PMCID: PMC2903309.

Dictyostelium discoideum  
: 34 Mb haploid genome, six chromosomes, 12500 genes  
: soil dwelling amoeba, slime mold  
: eukaryote transitioning between unicelular to multicellular slugs to fruiting body  
: found in soil, moist leaf litter. diet is bacteria like E. Coli  
: bacteria secretion of folic acid attracts the myxamoebae, which divide by mitosis during the vegetative stage why consuming bacteria   
: mostly asexual lifecycle-- vegetative, aggregation, migration, culmination  
: starvation during aggregation causes the myxamoebae to make glycoproteins that help cell-cell adhehsion and adenylyl cyclase which makes cAMP. cAMP works as a chemotactic signal, attracting neighboring amoebae then form a motile pseudoplasmodium, a slug up to 2-4mm long and 100000 cells  
: https://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Cell-Shape-Dynamics-From-Waves-to-Migration-pcbi.1002392.s007.ogv/180px--Cell-Shape-Dynamics-From-Waves-to-Migration-pcbi.1002392.s007.ogv.jpg  
: 5μm spores and amoebas to aggregated multicellular slugs of 1mm length  
: sexual reproduction also possible. three mating types with three strains having different gene combinations that specify the three different sexes, can only mate with the two different sexes <https://doi.org/10.1126%2Fscience.1197423> u<https://pubmed.ncbi.nlm.nih.gov/21148389>
: https://en.wikipedia.org/wiki/Dictyostelium_discoideum
: during chemotaxis, cAMP and amoeba movements occur every six minutes with amoebae moving toward concentration gradient for 60s before stopping. Oscillations in groups of cells results with propogating spiral waves of varying cAMP concentrations  http://www.whydomath.org/Reading_Room_Material/ian_stewart/2000_11.html
: may exhibit food husbandry or 'farming' behavior! https://doi.org/10.1038%2Fnature09668

* Drosophila 7-11 days (28-34degs C)
* zebrafish 3-4 days juvenile swimming and visual behavior. young adult at 3 mo. full adult at 6 mo.
* genome sizes at http://www.biology-pages.info/G/GenomeSizes.html

What about **cerebral cortical neuron number**? 
* generally humans thought to have highest neocortical neuron number (and therefore greatest amount of cortical links, inter-connects)
* but cetaceans approach and even rival humans in these measures... dolphins, killer whales, blue whale
* indeed a study in the past decade revealed the long-finned pilot whale as having more neocortical neurons than human.
    - Globicephala melas; latin "globe" and greek kephale "head"
    - up to 6.7 m long, 2300kg. second larges delphinid next to the *Orcinus orca*

- very social, small long term social units of 8-12 individual, aggregates of 100s to 1000s of individuals observed. usually small familial pods.
    - gestation 12 - 16 months, calives are up to 2.0 m at birth and 75kg
    - sex maturity for females and males at eight and 12 years respectively
    - <https://en.wikipedia.org/wiki/Long-finned_pilot_whale>

<https://en.wikipedia.org/wiki/List_of_animals_by_number_of_neurons#cite_note-80>
African elephant 	5,600,000,000 	Pallium (cortex) 	Loxodonta africana[^Herculano-Houzel2014]  
Chimpanzee 	7,400,000,000 Pallium (cortex) 	Pan troglodytes[^Collins2016]  
Bottlenose dolphin 	12,700,000,000 Pallium (cortex) 	Tursiops truncatus[^Herculano-Houzel2019]  
Human 	21,000,000,000 Pallium (cortex) 	Homo sapiens[^Herculano-Houzel2015]  
Blue whale 	15,000,000,000 Pallium (cortex) 	Balaenoptera musculus[^Herculano-Houzel2019]  
Long-finned pilot whale 	37,200,000,000 Pallium (cortex) 	Globicephala melas[^Mortensen2014]  
Killer whale 	43,100,000,000 Pallium (cortex) 	Orcinus orca[^Herculano-Houzel2019]

[^Collins2016]: Collins CE, Turner EC, Sawyer EK, Reed JL, Young NA, Flaherty DK, et al. Cortical cell and neuron density estimates in one chimpanzee hemisphere. Proc Natl Acad Sci U S A. (2016). 113:740–5. doi:10.1073/pnas.1524208113

[^Herculano-Houzel2014]: Herculano-Houzel S, Avelino-de-Souza K, Neves K, Porfírio J, Messeder D, Mattos Feijó L, et al. The elephant brain in numbers. Front Neuroanat. (2014). 8:46. doi:10.3389/fnana.2014.00046

[^Mortensen2014]: Mortensen HS, Pakkenberg B, Dam M, Dietz R, Sonne C, Mikkelsen B, et al. Quantitative relationships in delphinid neocortex. Front Neuroanat. (2014). 8:132. doi:10.3389/fnana.2014.00132

[^Herculano-Houzel2019]: Herculano-Houzel S. Longevity and sexual maturity vary across species with number of cortical neurons, and humans are no exception. J Comp Neurol. (2019). 527:1689–1705. doi:10.1002/cne.24564

[^Herculano-Houzel2015]: Herculano-Houzel S, Catania K, Manger PR, Kaas JH. Mammalian brains are made of these: A dataset of the numbers and densities of neuronal and nonneuronal cells in the brain of glires, primates, scandentia, eulipotyphlans, afrotherians and artiodactyls, and their relationship with body mass. Brain Behav Evol. (2015). 86:145–63. doi:10.1159/000437413