Connectome-Specific Harmonic Waves

Se­len Ata­soy’s new the­ory of brain func­tion called Con­nec­tion-Spe­cific Har­monic Waves (CSHW) could be a gi­ant step for­ward in our un­der­stand­ing of con­scious­ness. To put this in per­spec­tive I will first re­view what we know so far about how minds work.

Ar­tifi­cial Neu­ral Networks

Much of the most re­cent progress in ma­chine learn­ing (ML) is in the realm of image recog­ni­tion. That’s be­cause Ar­tifi­cial Neu­ral Net­works (ANNs) are un­usu­ally good at image pro­cess­ing. Much of the re­cent progress in ma­chine learn­ing (ML) has been ap­ply­ing ar­tifi­cial neu­ral net­works to new prob­lem do­mains.

This has been made pos­si­ble by a sin­gle in­ven­tion, Graph­i­cal Pro­cess­ing Units (GPUs). A GPU is a com­puter chip that gen­er­ates the images in graph­i­cally-in­ten­sive videogames by perform­ing ma­trix alge­bra. Train­ing an ANN is ma­trix alge­bra too so the fastest way to train an ANN is with a GPU. This has al­lowed ANN hard­ware to out­pace than Moore’s Law, if just for a mo­ment.

The sim­plest kind of neu­ral net­work is a feed for­ward neu­ral net­work (FFNN). An FFNN has lay­ers of nodes with con­nec­tions be­tween them. You train a neu­ral net­work with the back­prop­a­ga­tion al­gorithm. The im­por­tant things to take away from the back­prop­a­ga­tion al­gorithm is it’s re­cur­sive. There­fore neu­ral net­works pos­sess a frac­tal ar­chi­tec­ture. This is im­por­tant and we’ll get back to it later.

Be­sides FFNNs, the other kind of neu­ral net­work is called a re­cur­rent neu­ral net­work (RNN). This is a neu­ral net­work with cyclic feed­back loops, which cre­ates hys­tere­sis (short-term mem­ory). RNNs have seen suc­cess in nat­u­ral lan­guage pro­cess­ing but are not used in the flashy new ad­vance­ments such as self-driv­ing cars. That’s be­cause RNNs are more com­pli­cated than FFNNs. We un­der­stand RNNs less well than FFNNs. RNNs are sim­ple enough to use on very short time scales but we don’t know how to scale them up to long time scales.

Due to our re­li­ance on FFNNs over RNNs, we don’t know how to get an ANN to han­dle time-se­ries data. I know this be­cause I run a startup that uses ML to pro­cess time se­ries data. Self-driv­ing cars use FFNNs. FFNNs’ in­abil­ity to pro­cess time se­ries data was a con­tribut­ing fac­tor to the Uber self-crash­ing car.

We’ll get back to RNNs later too.


Your brain is a neu­ral net­work. There are two CSHW-re­lated differ­ences be­tween your brain’s neu­rons and ANN neu­rons.

  • ANN neu­rons pro­duce float­ing point out­put. Biolog­i­cal neu­rons can’t con­trol the am­pli­tude (voltage) of their ac­tion po­ten­tial. It’s bi­nary all-or-noth­ing. In­stead, the fre­quency of a neu­ron’s ac­tion po­ten­tials in­creases with the neu­ron’s net stim­u­lus. Neu­rons in ANNs do not mod­u­late the fre­quency of their out­put.

  • The hu­man brain ex­hibits va­ri­ety of differ­ent large-scale har­monic neu­ral os­cilla­tions (brain­waves) cor­re­spond­ing to differ­ent men­tal states like wake­ful­ness, sleep phases and REM sleep. ANNs do not ex­hibit these.

We know how in­di­vi­d­ual neu­rons in the hu­man brain work. We know what differ­ent re­gions of the brain do be­cause we ob­serve how hu­man be­hav­ior changes when differ­ent brain re­gions are dam­aged. But we don’t know how neu­rons work to­gether to cre­ate these brain re­gions. CSHW sug­gests an an­swer to this ques­tion.

At the same time, global workspace the­ory (GWT) offers an ob­serv­able defi­ni­tion of con­scious­ness[1]. Ba­si­cally it’s the idea that con­scious­ness is the thing your whole mind is think­ing about. Un­der the hood, one of your brain’s parts net­works broad­casts it­self to the rest of your brain and this be­comes the thing you’re think­ing about. GWT is well-sup­ported by both psy­cholog­i­cal ex­per­i­ments and con­tem­pla­tive tra­di­tion but we don’t know how the brain does it. Our ANNs have no global workspace.

To sum­ma­rize.

  1. We know what neu­rons are.

  2. We can simu­late ar­tifi­cial neu­rons to ac­com­plish sim­ple tasks at a speed com­pet­i­tive with hu­man be­ings. This is the best method of writ­ing soft­ware to do things like image recog­ni­tion and play­ing go.

  3. We know what each part of the hu­man brain does.

  4. We don’t know to build a ma­chine to ac­com­plish these com­pli­cated tasks us­ing ar­tifi­cial neu­rons.

In short, our ANNs are fast, scal­able and par­alleliz­able. Our ANNs can solve prob­lems where con­cep­tual com­plex­ity is very small. How­ever, our ANNs have trou­ble han­dling con­cep­tual com­plex­ity and time-se­ries data. We don’t know how to make the in­duc­tive step of putting our sim­ple neu­ral net­works to­gether into a larger in­tel­li­gence. They can pat­tern-match but they can’t strate­gize. And they lack con­scious­ness. This might not be a co­in­ci­dence.

If brains pos­sess a frac­tal ar­chi­tec­ture then we’re miss­ing the hi­er­ar­chi­cal in­duc­tive step.


In 2016, Se­len Ata­soy pub­lished a pa­per in Na­ture Com­mu­ni­ca­tions ti­tled “Hu­man brain net­works func­tion in con­nec­tome-spe­cific har­monic waves”. Here’s the most im­por­tant sen­tence.

[E]igen­de­com­po­si­tion of the Laplace op­er­a­tor...can pre­dict the col­lec­tive dy­nam­ics of hu­man cor­ti­cal ac­tivity at the macro­scopic scale.

―Ata­soy, S., Don­nelly, I. & Pear­son, J. Hu­man brain net­works func­tion in con­nec­tome-spe­cific har­monic waves. Nat Com­mun 7, 10340 (2016) doi:10.1038/​ncomms10340

“[E]igen­de­com­po­si­tion of the Laplace op­er­a­tor” means find­ing the har­mon­ics of the con­nec­tome. Un­stated in this sen­tence is the pos­si­bil­ity that eigen­de­com­po­si­tion of the Laplace op­er­a­tor can pre­dict col­lec­tive dy­nam­ics on ar­bi­trary scales. In case that doesn’t make sense, here’s a crash course on acous­tics.

Every sound wave can be bro­ken down into the su­per­po­si­tion of res­o­nant fre­quen­cies or har­mon­ics[2]. This forms a ba­sis for sound waves os­cillat­ing through the ge­om­e­try. This isn’t limited to sound waves. Any kind of wave can be bro­ken down this way. Differ­ent har­mon­ics have differ­ent fre­quen­cies. Higher-fre­quency har­mon­ics os­cillate faster and prop­a­gate shorter dis­tances. Lower fre­quency har­mon­ics os­cillate slower and prop­a­gate farther dis­tances.

This kind of res­o­nance hap­pens when­ever waves bounds through a solid struc­ture, such as sound waves through a vi­o­lin or x-rays through a crys­tal. Se­len Ata­soy and her lab have con­firmed res­o­nance of brain­waves bounc­ing through the con­nec­tome. The neu­rons in a sin­gle func­tional re­gion of the brain (a re­gion we’ve ob­served to do some­thing im­por­tant) res­onate to­gether. She calls this a “state net­work”.

So what?

When you press mid­dle C on a pi­ano it’s not just mid­dle C that vibrates. The other C strings will vibrate too, es­pe­cially those clos­est to mid­dle C. That’s be­cause from left to right the strings for each oc­tave are half as long as the pre­vi­ous. In­te­ger mul­ti­ples like this pro­duce res­o­nance.

Every note on the pi­ano has a par­tic­u­lar res­o­nance with each other keys. Some pairs of notes are highly res­o­nant with each other. Other pairs have low res­o­nance[3]. It de­pends on the ra­tio of one fre­quency to an­other.

CSHW is a sim­ple, el­e­gant way to co­or­di­nate many differ­ent sub-net­works into a hu­man brain. When differ­ent net­works are out of phase with each other the in­puts of one turn into static for the other, which is math­e­mat­i­cally equiv­a­lent to tun­ing out a ra­dio.

This could ex­plain what med­i­ta­tion does.


Con­ven­tion­ally sci­en­tific in­for­ma­tion on med­i­ta­tion is hard-to-come-by be­cause:

  1. Psy­chol­ogy as a sci­ence is less than a hun­dred years old. We’ve barely re­cov­ered from the be­hav­iorist over­re­ac­tion to Freud.

  2. Govern­ments sup­press re­search on psychedelics. We lost sci­en­tific re­search into med­i­ta­tion amidst the col­lat­eral dam­age.

  3. Govern­ment fund­ing for psy­cholog­i­cal re­search re­volves around cur­ing dis­eases. Med­i­ta­tion is for healthy peo­ple to get bet­ter.

  4. MRIs are ex­pen­sive.

  5. Long-term med­i­ta­tion takes se­ri­ous ded­i­ca­tion ev­ery day for decades. Figur­ing out what does and doesn’t work takes cen­turies.

  6. The only in­tel­lec­tual tra­di­tions to record this knowl­edge in use­ful form ex­ist out­side the Western in­tel­lec­tual tra­di­tion.

Eastern monks and yo­gis have been ex­per­i­ment­ing with med­i­ta­tion, com­par­ing their re­sults and iter­at­ing this tech­nol­ogy in an un­bro­ken dharma for sev­eral mil­len­nia. Our un­der­stand­ing of med­i­ta­tion is like nat­u­ral­ism in the time of Dar­win ex­cept this time re­li­gion has all the data.


Two and a half thou­sand years ago lived an In­dian prince named Sid­dhartha. He mas­tered the already an­cient Hindu yo­gic tech­niques. He med­i­tated for sev­eral years. Then one day, while med­i­tat­ing un­der a tree, he saw the truth of re­al­ity. We don’t know ex­actly what this means. But we do know he got there via med­i­ta­tion, it freed Sid­dhartha of dukkha and it was per­ma­nent. This state is called en­light­en­ment.

Then Sid­dhartha es­tab­lished a monas­tic or­der to pass down his dis­cov­er­ies and im­prove upon them. This or­ga­ni­za­tion evolved into the world re­li­gion of Bud­dhism. Some sects have moved on from med­i­ta­tion. Others have been im­prov­ing upon Sid­dhartha’s tech­niques up to the pre­sent day. But all of them share the ob­jec­tive of recre­at­ing the en­light­en­ment state of mind Sid­dhartha achieved so long ago.

You’re prob­a­bly won­der­ing how we can ver­ify this. The Dalai Lama likes sci­ence so he helped con­vince some as­cetic yo­gis to come down from the moun­tains and fly across the world and sub­mit to brain scans while med­i­tat­ing and while at rest[4]. Some im­por­tant dis­cov­er­ies stand out.

  1. The yo­gis had re­duced gala­vanic skin re­sponse in an­ti­ci­pa­tion of phys­i­cal pain.

  2. The yo­gis’ de­fault mode net­work did not ac­ti­vate dur­ing wake­ful rest.

  3. The yo­gis were in a con­stant state of gamma wave ac­tivity[5].

Dis­cov­ery (1) is sug­ges­tive of re­duced dukkha. Dis­cov­ery (2) is rele­vant to cy­ber­net­ics, which we’ll get to later. Dis­cov­ery (3) might be an ob­jec­tive met­ric we could use to mea­sure en­light­en­ment states.

I’ve repli­cated these re­sults my­self by achiev­ing med­i­ta­tive states where my de­fault mode net­work stops mak­ing noise. This hap­pens around the 30 minute mark, af­ter ac­cess med­i­ta­tion and mus­cle spasms. Time spend in these states re­duces my dukkha. I wouldn’t be sur­prised if I’ve also in­creased my gamma wave ac­tivity for min­utes at a time. I can’t af­ford an fMRI to ver­ify any of this ob­jec­tively, but my ex­pe­rience is typ­i­cal[6] for med­i­ta­tors on a path to en­light­en­ment.

If gamma waves are re­lated to en­light­en­ment then we can fi­nally ground en­light­en­ment in the ma­te­rial uni­verse.

Un­der CSHW, gamma waves are when ev­ery­thing in your brain is liter­ally in sync with ev­ery­thing else. If CSHW and GWT are both true and en­light­en­ment equals con­tin­u­ous gamma waves then to­gether they would ex­plain the mean­ing of the weird sub­jec­tive de­scrip­tions of en­light­en­ment peo­ple give like “my mind is big­ger”[7]. Your con­scious­ness re­ally is big­ger in­stead of be­ing frac­tured like a split brain pa­tient.


For a mind to in­ter­act in­tel­li­gently with its en­vi­ron­ment the mind has to in­clude a sim­plified model of its en­vi­ron­ment. Even a ther­mos does this when it de­cides whether to keep your drink hot or cold.

Your con­scious­ness lives on the con­nec­tome and never in­ter­acts with re­al­ity di­rectly. In­stead, your con­scious­ness in­ter­acts with the sim­plified model of re­al­ity cre­ated by your mind. Your Self and your Other are both men­tal con­structs.

If CSHW is true and en­light­en­ment equals gamma waves then in an en­light­en­ment state the Self and the Other would be plugged into one an­other. Un­der nor­mal cir­cum­stances your Self and bits of your mind’s rep­re­sen­ta­tion of the ex­ter­nal world may be out of res­o­nance. This is easy to un­der­stand if you’ve been closed off parts of your­self in re­ac­tion to abuse.

In this way, CSHW may go a long way to­wards ex­plain­ing anattā and its ces­sa­tion. See, ev­ery­one who at­tains en­light­en­ment does so through one of the three marks of ex­is­tence. The three marks are an­iccā (im­per­ma­nence), dukkha (un­satis­fac­tori­ness or suffer­ing), and anattā (non-self). To un­der­stand one of them is to un­der­stand all of them. In other words, they’re three differ­ent ways of get­ting at the same Truth.

The Truth is that ev­ery­thing you ex­pe­rience is a con­struc­tion of your mind. But “see­ing the Truth” doesn’t mean un­der­stand­ing this in­tel­lec­tu­ally. “See­ing the Truth” means get­ting your var­i­ous state net­works into res­o­nance. You can do this by sit­ting still (or walk­ing calmly or do­ing sim­ple chores) and clear­ing your mind. If your mute your sen­sory in­puts and de­fault mode net­work long enough then even­tu­ally all your state net­works will sync up like a room­full of pen­du­lum clocks.

Th­ese tra­di­tions pre­sent ev­i­dence CSHW is an im­por­tant part of how the brain solves its challenge of co­or­di­nat­ing neu­ral net­work sub­sys­tems.

A pop­u­lar sec­u­lar med­i­ta­tion man­ual Mas­ter­ing the Core Teach­ings of the Bud­dha: An Unusu­ally Hard­core Dharma Book by Daniel M. In­gram “Dharma Dan” ap­proaches en­light­en­ment via vipas­sana med­i­ta­tion. Vi­pas­sana is the tech­nique of pay­ing close at­ten­tion to what’s hap­pen­ing in your mind. Dharma Dan em­pha­sizes pay­ing at­ten­tion to in­di­vi­d­ual high fre­quency brain­waves. Ac­cord­ing to There­vada the­ory, if you use vipas­sana to look at it your con­scious ex­pe­rience at a high enough time re­s­olu­tion your con­scious ex­pe­rience breaks down into dis­crete frames or os­cilla­tions. This pro­cess also gen­er­ates in­sight with leads to en­light­en­ment.

From the per­spec­tive of CSHW what’s go­ing on in high fre­quency vipas­sana is you’re di­rect­ing your global workspace to a sin­gle high fre­quency os­cilla­tion in­stead of jump­ing around from one sig­nal source to an­other. Since ev­ery net­work is always work­ing to­wards an­ti­ci­pat­ing its own in­puts this nat­u­rally leads to in­creased res­o­nance as each state net­work syncs its in­ter­nal clock with the tar­get of vipas­sana at­ten­tion.

The fMRI data cor­rob­o­rat­ing this (the Ti­be­tan yo­gis from ear­lier) is based on lov­ingkind­ness med­i­ta­tion, not vipas­sana, but the prin­ci­ple still ap­plies. All con­tem­pla­tive tra­di­tions fo­cus con­scious­ness on a sin­gle ob­ject[8] for a long time. No mat­ter what the ob­ject is, even­tu­ally this should lead to in­creased res­o­nance, which ex­plains how con­tem­pla­tive tech­niques as differ­ent as kasina fire med­i­ta­tion can pro­duce such similar out­comes.

CSHW es­tab­lishes a math­e­mat­i­cal foun­da­tion for why anger and ha­tred are uni­ver­sally up­rooted across the var­i­ous con­tem­pla­tive tra­di­tions. Anger and ha­tred do not ex­ist in iso­la­tion. They are felt to­ward your mind’s con­cep­tu­al­iza­tion of some­thing that isn’t you. The dis­tinc­tion be­tween your­self and the other is premised off of idea that you are sep­a­rate from it. But your men­tal model of the Other is liter­ally part of your brain. If your brain is in to­tal res­o­nance then you can’t feel sep­a­ra­tion from the Other. En­light­ened in­di­vi­d­u­als don’t feel anger or ha­tred be­cause that would be an a pri­ori con­tra­dic­tion. Similarly, anger and ha­tred are ob­sta­cles to en­light­en­ment.


Our ANNs can scale to ar­bi­trar­ily large in­put/​out­put di­men­sion­al­ity be­cause they’re frac­tal struc­tures in two di­rec­tions: in­put/​out­put di­men­sion­al­ity and num­ber of hid­den lay­ers. You can cut an ANN in half along ei­ther of these di­rec­tions and get two smaller neu­ral net­works.

This is a spe­cial case of a gen­eral prin­ci­ple. An in­for­ma­tion pro­cess­ing sys­tem can scale to ar­bi­trar­ily com­plex prob­lems if and only if the sys­tem is struc­tured frac­tally. Sys­tems with­out a frac­tal struc­ture will even­tu­ally en­counter a com­pu­ta­tional cliff.

We’ve hit this cliff with our ANNs. Our ANNs scale well in the afor­men­tioned di­rec­tions for which they pos­sess frac­tal ge­om­e­try. But their hi­er­ar­chi­cal struc­ture is non-frac­tal, so they don’t work on hi­er­ar­chi­cal con­cep­tual prob­lems. Our brains are bet­ter than our ANNs when it comes to strate­gic rea­son­ing.

CSHW sug­gests a frame­work for how to co­or­di­nate ANNs hi­er­ar­chi­cally. Low fre­quency waves prop­a­gate farther than high fre­quency waves. So when­ever you go up an or­der of mag­ni­tude in phys­i­cal scale the wave­length in­creases of the rele­vant brain­waves in­crease. CSHW works the same on ev­ery scale of ob­ser­va­tion...all the way down to in­di­vi­d­ual neu­rons. Re­mem­ber how biolog­i­cal neu­rons send a pulse of ac­tion po­ten­tials in­stead of mod­u­lat­ing voltage? That could be the base case to our frac­tal in­duc­tion.

The in­di­vi­d­ual com­po­nents should be easy to build out of RNNs. They could be scaled with CSHW in­duc­tively to larger phys­i­cal di­men­sions and time di­men­sions. This could au­to­mat­i­cally solve the other prob­lem of how to build RNNs that work for large time scales thereby putting us a gi­ant step for­ward to­wards build­ing an ar­tifi­cial gen­eral in­tel­li­gence.

  1. I’m us­ing “con­scious­ness” to re­fer to the global workspace in GWT. I mean to im­ply noth­ing meta­phys­i­cal with the term. ↩︎

  2. In real-world ap­pli­ca­tions this re­sults in re­con­struc­tion er­rors, es­pe­cially for square waves. This is ad­dressed in Ata­soy’s pa­per. ↩︎

  3. Pi­ano strings differ­ing by ex­actly the golden ra­tio have min­i­mal res­o­nance. ↩︎

  4. You can find this re­search and more in the book Altered Traits by Daniel Gole­man, a fas­ci­nat­ing book on what sci­ence knows on the long-term effects of se­ri­ous med­i­ta­tion. It was pub­lished in 2017 so it con­tains up-to-date in­for­ma­tion. How­ever, many of the stud­ies are un­repli­cated. Con­sid­er­ing the his­tor­i­cal ob­sta­cles to this re­search, we’re lucky to have any­thing at all. ↩︎

  5. It frus­trates me that sci­en­tists haven’t con­ducted the same ex­per­i­ments on Zen mas­ters who live in cities and have a for­mal sys­tem for cer­tify­ing who has be­come en­light­ened. I want to know if fMRI scans cor­re­late with dharma trans­mis­sion. ↩︎

  6. For a first-per­son ac­count of what it’s like to fol­low this path to its con­clu­sion I recom­mend Hard­core Zen: Punk Rock, Mon­ster Movies, & the Truth About Real­ity by Brad Warner and The Science of En­light­en­ment: How Med­i­ta­tion Works by Shinzen Young. ↩︎

  7. This de­scrip­tion comes from a young woman who stum­bled into stream en­try (a sec­u­lar name for en­light­en­ment) out­side of any med­i­ta­tive tra­di­tion. If the woman comes from a Chris­tian tra­di­tion (as this woman did) the ex­pe­rience can be con­fus­ing. It is be­lieved a small num­ber of ran­dom peo­ple stum­ble into en­light­en­ment un­pre­dictably. ↩︎

  8. Ex­cept non­d­ual tra­di­tions like Zen. They aban­don the med­i­ta­tion tar­get and shoot straight to­wards en­light­en­ment. ↩︎