A human brain is a col­lec­tion of cells that are mas­sively inter­con­nected. Assum­ing we knew how to build and pro­gram one, how much data would we need to recre­ate any par­tic­u­lar human? That is, what is the best esti­mates for the total size (in bytes) needed to store a Homo sapi­ens mind­state? In the fol­low­ing I’m going to try to work out an upper-bound esti­mate. A loose, gen­eral def­i­n­i­tion of “infor­ma­tion” is the num­ber of yes/no ques­tions needed to fully spec­ify an indi­vid­ual mem­ber of a given set of pos­si­bil­i­ties. “Twenty ques­tions” can, in the­ory, spec­ify the mem­ber of a set of 2²⁰=1 mil­lion pos­si­bil­i­ties. In gen­eral, if your set of pos­si­bil­i­ties has N pos­si­bil­i­ties:
log₂ N_{pos­si­bil­i­ties} = N_bits

Con­nec­tiv­ity Infor­ma­tion Content

There are about a 100 bil­lion neu­rons in the human brain. Each one makes, on aver­age, 10,000 con­nec­tions as inputs. How much infor­ma­tion is stored in the con­nec­tiv­ity graph of the neural net? How many ways can a sin­gle neu­ron make 10,000 con­nec­tions? Just

$$$ \left(\begin{array}{c} N\\m \end{array}\right) = \left(\frac{N!}{m! (N-m)!}\right) $$$

where $$N$$ is the num­ber of neu­rons and $$m$$ the num­ber of con­nec­tions. Given that each neu­ron is approx­i­mately inde­pen­dent, we then have for the entire brain a total num­ber of states:

$$$ \left(\begin{array}{c} N\\m \end{array}\right)^N = \left[\frac{N!}{m! (N-m)!}\right]^N $$$

The infor­ma­tion con­tent in bits is then

$$$\log_2 \left[ \frac{N!}{m!(N-m)!} \right]^N = N \left( N\log_2 N — m\log_2 m — (N-m)\log_2 (N-m) \right) $$$

Now the neural sys­tem is sparsely con­nected that is the num­ber of con­nec­tions, 10,000, is far less than the num­ber of neu­rons, 100 bil­lion, so $$N»m$$. This allows the sim­pli­fy­ing assumption

$$$ \begin{array}{c}
\log_2 (N-m) = \log_2 N \\
\log_2 N_{states} = Nm\log_2\frac{N}{m}
\end{array}
$$$

For the human brain: $$ 10^{11} \cdot 10^4 \cdot \log_2 10^7 = 23 \cdot 10^{15} $$ bits — That’s 3 petabytes! (about 2,000 mod­ern hard drives).

Now, actu­ally the brain is com­part­men­tal­ized and any given neu­ron can’t make a con­nec­tion to any of the other 100 bil­lion neu­rons. Instead, it can only con­nect to an aver­age sub­set of $$N_0$$ neu­rons. Let’s say only a mil­lion poten­tial part­ners are avail­able for any given neu­ron. The change is:

$$$\log_2 N_{\textrm{states}} = Nm\log_2\frac{N_0}{m} = 10^{11} \cdot 10^4 \cdot \log_2 100 = 7 \cdot 10^{15}$$$

1 petabyte, not much of a change, since the sheer num­ber of synapses involved sets the petabyte figure.

Synap­tic Weight Infor­ma­tion Content

The above cal­cu­la­tion naively assumes that each con­nec­tion is binary, just like a junc­tion. In real­ity, each con­nec­tion is a synapse that acts like a com­plex bio­chem­i­cal ampli­fier of elec­tronic sig­nals that holds lots of state vari­ables. If we assume each synapse has on aver­age σ state vari­ables with r bits of res­o­lu­tion, then we have the addi­tional infor­ma­tion source of $$ N\cdot m \cdot\sigma\cdot r$$.

An over­sim­pli­fied assump­tion is that there are only two vari­ables: the recep­tor num­ber and the size of the synapse, each vary­ing within a 6-bit range (a 64-fold range). This leads to the esti­mate of $$ 10^{15} \cdot 2\cdot 6 = 12 \textrm{petabits} = 1.5 \textrm{petabytes} $$ to store the synap­tic strengths in addi­tion to the con­nec­tiv­ity diagram.

There are cer­tainly more synap­tic state vari­ables than this. We can set an upper limit of per­haps 100 6-bit vari­ables aris­ing from vari­able local synap­tic pro­tein lev­els and mod­i­fi­ca­tion states (phos­pho­ry­la­tion, ubiq­ui­tiny­la­tion, etc.). This implies a much larger stor­age size of

$$$ 10^{15} \cdot 100\cdot 6 = 600 \textrm{petabits} = 75 \textrm{petabytes} $$$

There is ambi­gu­ity at how well such raw data com­presses, but it’s prob­a­bly safe to assume a infor­ma­tion den­sity of 1–100 petabytes to store a human mindstate.

Note that there are many other vari­ables, such as global pro­tein expres­sion lev­els and of course the tran­sient elec­tri­cal state of neu­rons but these vari­ables tend to scale per-neuron, not per-synapse, and so con­tribute a much smaller amount of infor­ma­tion that we can neglect for an order of mag­ni­tude analy­sis. (i.e. We have less than a giga­byte of genom­i­cally coded “human nature”.) Lim­it­ing Infor­ma­tion Den­sity of Mat­ter
Given the huge value of Avogadro’s num­ber (6.022 10²³), the infor­ma­tion den­sity of ordered mat­ter is poten­tially immense. Let us assume that there’s some means of encod­ing 1 bit per 1000 atoms (for met­als a cube roughly 3nm on side). How much infor­ma­tion can we hold in our hand?
(0.1 m / 3nm)³ = .3 10²³ bits, about 5 mil­lion petabytes, Enough to store 50,000–5 mil­lion mind­states.

This implies that sta­tic immor­tal­ity amounts to sev­eral mil­lion dol­lars of stor­age space¹, if only we knew how to tease the ghost from the skein and rehaunt another.

Jorge Luis Borges
*4 August 1899
†14 June 1986

“Life itself is a quo­ta­tion.“

“One con­cept cor­rupts and con­fuses the oth­ers. I am not speak­ing of the Evil whose lim­ited sphere is ethics; I am speak­ing of the infi­nite.“

“To fall in love is to cre­ate a reli­gion that has a fal­li­ble god.”

The Library of Babel was con­ceived by the inim­itable Jorge Luis Borges and stands as one of the great­est metaphors for com­bi­na­to­r­ial poten­tial ever penned.

A brief excerpt from the unnamed nar­ra­tor, a native of the library:

The uni­verse (which oth­ers call the Library) is com­posed of an indef­i­nite and per­haps infi­nite num­ber of hexag­o­nal gal­leries, with vast air shafts between, sur­rounded by very low rail­ings. From any of the hexa­gons one can see, inter­minably, the upper and lower floors. The dis­tri­b­u­tion of the gal­leries is invari­able. Twenty shelves, five long shelves per side, cover all the sides except two; their height, which is the dis­tance from floor to ceil­ing, scarcely exceeds that of a nor­mal book­case. One of the free sides leads to a nar­row hall­way which opens onto another gallery, iden­ti­cal to the first and to all the rest…

…there are five shelves for each of the hexagon’s walls; each shelf con­tains thirty-five books of uni­form for­mat; each book is of four hun­dred and ten pages; each page, of forty lines, each line, of some eighty let­ters which are black in color.

Based on this descrip­tion, it is pos­si­ble to cal­cu­late the size of the Library of Babel, pre­sum­ing that it is a finite uni­verse con­tain­ing every pos­si­ble book.

This is the nature of our Real­ity. It is but a tiny mote couched inside a sea of poten­tial­ity of math­e­mat­i­cal vastness.

The under­ly­ing asser­tion of most goofy new-age claims about quan­tum mechan­ics and the brain is that con­scious­ness is a quan­tum process. Of course, in a triv­ial sense it is quan­tum inso­far that every process in the phys­i­cal world seems to obey quan­tum mechan­ics. The exact claim is that some­thing “essen­tially quan­tum” is behind the phe­nom­e­non of con­scious­ness, that the com­pu­ta­tions of the brain actu­ally exploit uni­nu­itive quan­tum behav­iours that can­not be explained by a clas­si­cal physics pic­ture — the claim is that our brains are quan­tum computers.

You build a quan­tum com­puter by exploit­ing the fact that a sim­ple, per­fectly iso­lated phys­i­cal entity does not act like a tiny bil­liard, but rather as a complex-valued wave that isn’t in any par­tic­u­lar place at a given time, it’s spread out. We say that small sys­tems can be in “super­po­si­tions” of mul­ti­ple states. Now when the sys­tem inter­acts with the envi­ron­ment, by hit­ting a pho­ton from our lasers, say, it will “col­lapse” It should be noted that “col­lapse” is not a real a-priori phys­i­cal process, but only an appar­ent phe­nom­e­non. The mod­ern under­stand­ing is that col­lapse hap­pens when sys­tem of few degrees of free­dom inter­acts with one of very many (the envi­ron­ment) caus­ing the two to become “entan­gled” and forc­ing any given his­tory of the envi­ron­ment to “see” only one well defined state of the small sys­tem through a process called “deco­her­ence”. into one state, we will see the pho­ton bounc­ing off as though the par­ti­cle had been at one par­tic­u­lar place.

Quan­tum com­put­ers exploit super­po­si­tions by encod­ing data into the states of small par­ti­cles and allow­ing them to inter­act and evolve with each other iso­lated from the envi­ron­ment in such a way as to per­form a sim­ple cal­cu­la­tion with­out col­laps­ing until the very end when you mea­sure (read out) the answer. The advan­tage comes from the fact that every pos­si­ble his­tory of the sim­ple com­pu­ta­tion is per­formed in such a way that cer­tain par­al­lel algo­rithms can try all the com­bi­na­to­r­ial pos­si­bil­i­ties at once before being sum­moned to give an answer when they’re col­lapsed. A quan­tum com­puter with N bits is like clas­si­cal com­puter with 2^N bits.

The key require­ment is that the quan­tum com­puter not inter­act with the envi­ron­ment (stray light, cos­mic rays, etc.) dur­ing the dura­tion of the cal­cu­la­tion. The more com­plex the com­puter, the larger the num­ber of par­ti­cles needed to encode the data, the more exquis­itely sen­si­tive the com­pu­ta­tion becomes to out­side noise:

It is so dif­fi­cult to get more than a few par­ti­cles iso­lated long enough to per­form cal­cu­la­tions that after about 10 years of effort, the biggest quan­tum com­puter has about 8 qubits. The quan­tum brain hypoth­e­sis This hypoth­e­sis was first pro­posed by Roger Pen­rose in his book “The Emperor’s New Mind” and has since cap­tured a gen­er­a­tion of imag­i­na­tive the­o­rists who never both­ered to learn any­thing about the brain. says that there is some remark­able way that the pro­teins of our neu­rons could form an iso­lated net­work of qubits such that the brain could per­form quan­tum cal­cu­la­tions, and that the mys­te­ri­ous nature of con­scious­ness could be chalked up to the weird­ness of its quan­tum underpinnings.

There are two major rea­sons for why this quan­tum brain hypoth­e­sis is extremely doubtful:

1. Any time a stray par­ti­cle hits a quan­tum com­puter it col­lapses back into a world-entangled state, ruin­ing the com­pu­ta­tion before it’s fin­ished. The brain is a hot, dis­or­dered, mas­sively chaotic place with count­less par­ti­cles bounc­ing into every­thing a bil­lion times a sec­ond. The longest bio­log­i­cal quan­tum super­po­si­tion known hap­pens in chloro­phyll, and that lasts about a tril­lionth of a second.

A gen­eral, con­ser­v­a­tive cal­cu­la­tion about the sur­vival times of quan­tum states in the brain was done by Max Tegmark ¹, sug­gest­ing a tril­lionth of a sec­ond as the limit. Now, biol­ogy cer­tainly does exploit quan­tum effects at the timescales at which they hap­pen in cells. i.e. quan­tum effects influ­ence pho­to­syn­the­sis² and the elec­tron trans­port chain of res­pi­ra­tion. Nat­ural selec­tion can­not select for life that uses extremely short-lived phys­i­cal processes to per­form long-lived tasks.

2. Men­tal phe­nom­ena seem to be explic­a­ble solely in terms of the elec­tri­cal spikes neu­rons use to sig­nal with each other. i.e. In ani­mal exper­i­ments, we can see that the exter­nal infor­ma­tion of sight and sound is encoded in the fre­quency and pat­tern of these spikes in pop­u­la­tions of neu­rons. Con­trol­ling these elec­tri­cal sig­nals arti­fi­cially seems to influ­ence ani­mal behav­iour in a pre­dictable fashion.

We are still very much in igno­rance of the brain’s oper­a­tion, but not so much at the level of its bio­physics. More so in the level of detail about how these sig­nals work across the hun­dred bil­lion or so neu­rons of the brain, and how indi­vid­ual neu­rons alter their con­nec­tions and sen­si­tiv­i­ties over time to other neu­rons. The fastest of these processes hap­pen at the mil­lisec­ond timescale, mean­ing that any quan­tum process is much too fleet­ing to influ­ence the phe­nom­e­non we know to be directly involved in neural com­pu­ta­tion, by a fac­tor of at least 10⁹!

The con­sen­sus sci­en­tific opin­ion is that the brain acts as a mas­sively par­al­lel, sto­chas­tic,The one thing quan­tum mechan­ics does pro­vide is a kind of guar­an­tee of absolute, true “ran­dom­ness” at the level of the par­ti­cles that make up the brain. This in turn pro­vides a kind of basic guar­an­tee of non-predictability for human actions. and clas­si­cal com­puter. What­ever the “secret” to con­scious­ness is, it’s not quan­tum superposition.