In a column posted a few days ago (November 1) I mentioned that my friend John Evans, a Cambridge (England) mathematician, has developed a general formula for estimating biocomplexity. It is quite simple, using only two variables: the number of units in a system, and the number of connections (interactions) each unit has with other units in the system. Today, in fact, biologists publish ‘interactomes” with furry ball figures that illustrate the number of proteins in a given cell and the number of interactions each protein has with other proteins. The concept of complex interactomes has become embedded in systems biology. 

    John’s formula is simple: C (complexity) = logN * (1 + 2logZ) where N is the number of units and Z is the average number of interactions. (If you would like to see his logic, you can read the paper cited below.) In the paper, we tested the formula by calculating C for the nervous systems of animals ranging from nematodes (N = 302 neurons, Z ~ 10) through insects and frogs and finally a series of mammalian brains. 

    What I proposed is that it might be an interesting exercise to see how well the complexity calculation predicts our perception of animal intelligence, so I challenged readers to use their intuition to put a set of mammalian species into order from most to least intelligent. Ten readers immediately replied, which was sufficient for our purposes. (If you would like to see all ten lists, they are in the Comments at the end of the earlier column.)    I put together a consensus list simply by adding up the value of the placements, then putting the animals in order from lowest to highest totals. For instance, if all ten readers put humans first, the total would be 10, if chimps were number 2 in everyone’s lists, their total would be 20, and so on. Here is the consensus list and totals for each animal:

1. Human 10

2. Chimpanzee         24

3. Dolphin 33

4. Gorilla 34

5. Elephant 51

6. Horse 62

7. Dog 69

8. Cat 75

9. Rat 92

10. Opossum 100

11. Mouse 107

    Now I will present several other rankings that are based on the variables we need to use in the calculation. This is followed by a ranked list calculated from the complexity formula itself, and finally a list which was normalized to take into account a third variable called encephalization quotient which I will explain later. We will compare that list to the consensus list to see how well the formula fits our expectations, and then pose a question for discussion. 

    One thing to make clear is that we will take Z to be a constant for all the mammalian species. It is estimated that each cortical neuron connects with around 1000 other neurons, so the second term in the formula is 1 + 2log1000) or 7. (In simpler organisms Z is much smaller. For instance, in nematodes Z ~ 10.)

    The first list is ranked according to brain weight, and of course the elephant comes out on top, with humans and dolphins tied for second. (This list includes a rhesus monkey which I forget to include in my earlier column):

Brain weight (grams)

Elephant 4200
Dolphin 1350
Human 1350
Horse 510
Gorilla 480
Chimpanzee     380
Rhesus 88
Dog 64
Cat 25
Opossum     7.6
Rat 2
Mouse 0.3

The next list ranks the animals according to the number of cortical neurons estimated to be present in the brain of each species. In this list, humans and elephants are in a virtual tie for first place, with ~11 billion cortical neurons, followed by chimps, dolphins and gorillas:

Cortical neurons (millions)

Human 11500

Elephant 11000
Chimpanzee     6200
Dolphin 5800
Gorilla 4300
Horse 1200
Rhesus 480
Dog 610
Cat 300
Opossum         27
Rat 15
Mouse 4

The next list shows the order given by John’s formula. Again, humans and elephants are close due to the fact that they have the same number of neurons:

Ranking according to calculated complexity

Human 70

Elephant 70
Chimp/dolphin     68 (tied)
Gorilla 68
Horse 64
Monkey 61
Cat 58
Dog 57
Opossum 52
Rat 50
Mouse 46

It doesn’t seem reasonable that humans and elephants are so close in the rankings, and in fact in the consensus list elephants are ranked fifth, below gorillas. Are we missing something? Maybe we can do better by incorporating the encephalization quotient (EQ). When the amount of brain tissue in a series of animals is plotted against size, from mice to elephants, there is a roughly linear relationship.  However, the value for some animals lies significantly above the line, while others are well below the line. Humans come out on top of the EQ ranking, followed by dolphins, chimps and gorillas. Here is our list according to EQ:

Ranked by EQ

Human 7.6

Dolphin 5.3
Chimpanzee 2.4
Monkey 2.1
Gorilla 1.6
Elephant 1.3
Dog 1.2
Cat 1.0
Horse 0.9
Mouse 0.5
Rat 0.4
Opossum 0.2 

The way I think about EQ is that an animal like an elephant, with an EQ of 1.3, needs a greater absolute number of neurons to serve the much larger number of cells in their bodies, but these neurons are not necessarily given over to intelligence. Humans, with the highest EQ of all (7.6) have developed larger brains in relation to body size because our evolutionary pathway happened to select for whatever it is that we call intelligence, which apparently requires more brain tissue devoted to that function. 

We can use relative EQ to correct for the effect of body size by normalizing against the human EQ. The complexity equation then becomes:

C = log(N*EQa/EQh) * (1 + 2logZ), where EQa is the animal EQ and EQh is the human EQ, taken to be 7.6.

Normalized complexity compared to the consensus list

Human                70    Human

Dolphin                 67    Chimpanzee

Chimp                 65    Dolphin

Elephant              64    Gorilla

Gorilla                 63     Elephant

Monkey/Horse    57 (tied)    Horse

Cat                       53    Dog

Dog                     52    Cat

Opossum/rat        41 (tied)    Rat

Mouse                 38    Opossum


Well, that’s pretty amazing! The consensus list and the calculated list are very similar, with no animal farther away than a single rank inversion between the two lists. 

What does it all mean? I have a couple of suggestions. The first is that a certain level of complexity is required for higher nervous functions, just as my Mac iBook is much more complex than the Apple IIe I purchased back in 1981. It is interesting that all five animals with complexity values of 63 and above are self-aware, at least according to the following test. If you glue a round red dot to the forehead of a chimpanzee (or a dolphin, as demonstrated by Lori Marino) and let the animal see itself in a mirror, it will react to the presence of the dot. All animals with complexity of 57 and below are unable to understand that the image in the mirror is in fact themselves, and they pay no attention to the dot. 

The second point is that a chimpanzee, although self-aware, cannot come close to what we recognize as human intelligence. It seems that a complexity value of 70 or above is essential, that is, 11 billion neurons, each with 1000 connections to other neurons, and an EQ of 7.6.

A caveat: I am not a neurobiologist, and am uncritically taking all of the parameters used in the calculation out of the literature.  If you read the literature, you can probably find more sophisticated theories of the relationship between neuroanatomy and self-awareness, intelligence and the conscious state. 

Now, for those readers who enjoy Scientific Blogging, here is the question: Is there a minimal complexity required for the phenomena of self-awareness and consciousness? If so, how can we arrive at a quantitative estimate of that complexity?

Reference: Deamer DW, Evans J. 2006. Numerical analysis of biocomplexity. In Life As We Know It. J Seckbach, ed. p 201 - 12. New York: Springer.