Friday, December 20, 2013

Temporal Basis Functions and the Finite Impulse Response (FIR)

We neuroscientists tend to make a lot of assumptions about the brain: First, that everybody has one; second, that the more your neurons fire, the more blood will flow to those neurons; and third, that from the first two assumptions we can create sophisticated, impressive-sounding models and theories about the brain that explain everything, from consciousness and being to schizophrenia and Nutella addiction. Not that we aren't humble about it.

However, one of the most tenuous, sometimes ridiculous notions we have is that this blood flow, this continuous lifestream attempting to quench the brain's unslakable thirst with each pulse, is the same, acts the same, looks the same in every nook and cranny, every bump and crevice, every gyrus and sulcus of cortical and subcortical real estate. Why would we maintain such a clearly psychotic fantasy in the first place? Assumptions, madame, pure assumptions.

A word or two is in order about the origins of this one assumption in particular. In the beginning, a band of intrepid scientists observed that blood flow changes in primary sensory areas resulted in a gamma-shaped curve that peaks around five seconds followed by a long-lasting decline and undershoot around the thirty-second mark; at high field strengths, sometimes a small initial undershoot can also be observed. The results were so consistent in these areas, and the effects so strong, that these observations were soon cast into a model called the canonical hemodynamic response function (HRF), which has been widely used ever since.

And, for the most part, it has done pretty well. Typically a young, ambitious researcher, fueled by nothing more than the lust for knowledge and fame, for girls and gold and hazelnut spread - possibly even a swimming pool with all of the above mixed in there - will carry out his analyses using a whole-brain approach; that is, where the onset of each condition in each voxel is convolved with the hemodynamic response function. The height of the HRF is then estimated for each condition and averaged across trials, and contrasts can be carried out on these estimated heights. So far, so good; and plenty of high-quality research has been done using just this technique.

However, this is only a single parameter we are estimating for each condition; and if you happen to be a nerd, or a masochist, or a person who likes to do things the hard way - in other words, if you are in academia - you will doubtless be curious about other parameters you can harvest from your data. Sure enough, likeminded nerds have made these options available in the form of Finite Impulse Response (FIR) functions, which do not make any assumptions about the shape of the neural activity following a stimulus onset, and can therefore provide more detailed, flexible information about what is going on. All you do is specify that you wish to use that type of basis function, as well as the length of the interval you are interested in and how many time points you wish to estimate. Usually, FIR analyses are most interpretable when specified over a timecourse that doesn't include significant overlap with another condition, and when the timecourse is partitioned into units that are the same length as the time it takes to acquire each scan.

For now, I will focus on how to do this in AFNI. AFNI's version of FIR functions includes several ways to estimate each timepoint after a specified onset, including TENT, CSLPIN, and SIN basis functions, although for most purposes you will use TENT. (The TENT function uses a piecewise linear spline method to estimate brain activity at each timepoint; the details of this method are pretty complicated, which is a classy way of saying that I don't completely understand it.) The TENT function takes in three arguments: The beginning and ending times relative to the onset of a stimulus, or whatever timepoint you are interested in; and how many parameters to estimate. If b is the beginning and c is the end, with n being the number of parameters to estimate, then the time interval is (c-b)/(n-1). For example, if I wanted to estimate 5 parameters within a window from 0-8 seconds after a specified time, then the time interval between each point would be (8-0)/(5-1) = 2.

Although this would be usually implemented with event-related designs, I'll stick with AFNI's dataset #6, which uses a block design. Note that some of the shapes of the HRF can look pretty funky, compared to what you would expect when convolving with a canonical HRF; this should increase the already profound feelings of despair you have about cognitive neuroscience, and spur you to search for fields of more meaningful endeavor, such as specializing in pottery, martial arts, or poetry slams; specifically, slamming the country of Chile.

Here is what the code would look like for a typical FIR analysis in AFNI, assuming (har!) that you have already analyzed dataset #6:


setenv subj = 'FT'

3dDeconvolve -input pb.$subj.r*.scale+orig.HEAD
-polort 3
-num_stimts 2
-stim_times 1 stimuli/AV1_vis.txt 'TENT(0,20,11)'
-stim_label 1 Vrel
-stim_times 2 stimuli/AV2_aud.txt 'TENT(0,20,11)'
-stim_label 2 Arel
-x1D X.xmat.1D -xjpeg X.jpg
-cbucket cstats.$subj

More information coming soon, but right now I have a bus to catch back to warm, sunny Minnesota! At least that's how my online travel agent, Amir, described it before I paid him five thousand dollars to arrange my trip and give me advice on how to woo the locals. Apparently, bright orange ski parkas are in style. Ladies?

Saturday, November 30, 2013

Introduction to Reinforcement Learning Models

Someone very near and dear to me just sent me a picture of herself cuddled up on the couch in her pajamas with an Argentinian Tegu. That's right lady, I said Tegu. The second coming of Sodom and Gomorrah - you heard it here first, folks! I mean, I know it's the twenty-first century and all, but what the heck.

Looks like I'll be pushing her to buy that lucrative life insurance policy much earlier than planned!

Anyway, I think that little paroxysm of righteous anger provides an appropriate transition into our discussion of reinforcement learning. Previously we talked about how a simple model can simulate an organism processing a stimulus, such as a tone, and begin to associate that with rewards or lack of rewards, which in turn leads to either greater levels of dopamine firing, or depressed levels of dopamine firing. Over time, dopamine firing begins to respond to the conditioned stimulus itself instead of the reward as it becomes more tightly linked to receiving the reward in the near future. This phenomenon is so strong and reliable across all species, it can even be observed in the humble sea slug Aplysia, which is one ugly sucker if I've ever seen one. Probably wouldn't stop her from cuddling up with that monstrosity, though!

Anyway, that only describes one form of learning - to wit, classical conditioning. (Do you think I am putting on airs when I use a phrase like "to wit"? She thinks that I do; but then again, she also has passionate, perverted predilections for cold-blooded wildlife.) Obviously, any animal in the food chain - even the ones she associates with - can be classically conditioned to almost anything. Much more interesting is operant conditioning, in which an individual has to make certain choices, or actions, and then evaluate the consequences of those choices. Kind of like hugging reptiles! Oh hey, she probably thinks, let's see if hugging this lizard - this pebbly-skinned, fork-tongued, unblinking beast - results in some kind of reward, like gold coins shooting out of my mouth. In operant conditioning parlance, the rush of gold coins flowing out of one's orifice would be a reinforcer, which increases the probability of that action in the future; while a negative event, such as being fatally bitten by the reptile - which pretty much any sane person would expect to happen - would be a punisher, which decreases the probability of that action in the future.

The classically conditioned responses, in other words, serve the function of a critic which monitors for stimuli and reliably-predicted reinforcers or punishers following those stimuli, while operant conditioning can be thought of as an actor role, where choices are made and the results evaluated against what was expected. Sutton and Barto, a pair of researchers considerably less sanguinary than Hodgkin and Huxley, were among the first to propose and refine this model, assigning the critic role to the ventral striatum and the actor role to the dorsal striatum. So, that's where they are; if you want to find the actor component of reinforcement learning, for example, just grab a flashlight and examine the dorsal striatum inside someone's skull, and, hey presto! there it is. I won't tell you what it looks like.

However, we can form some abstract idea about what the actor component looks like by simulating it in Matlab. No, just in case you were wondering, this won't help you hook up with Komodo Dragons! It will, however, refine our understanding of how reinforcement learning works, by building upon the classical conditioning architecture we discussed previously. In this case, weights are still updated, but now we have two actions to choose from, which results in four combinations: either one or the other, both at the same time, or neither. In this example, only doing action 1 will lead to a reward, and this gets learned right quick by the simulation. As before, a surface map of delta shows the reward signal being transferred from the actual reward itself to the action associated with that reward, and a plot of the vectors shows action 1 clearly dominating over action 2. The following code will help you visualize these plots, and see how tweaking parameters such as the discount factor and learning rate affect delta and the action weights. But it won't help you get those gold coins, will it?

close all

numTrials = 200;
numSteps = 100;
weights = zeros(100,200); %Array of weights from steps 1-100, initialized to zero

discFactor = 0.995; %Discounting factor
learnRate = 0.3; %Learning Rate
delta = zeros(100,200); %Empty vector
V = []; %Empty vector
x = [zeros(1,19) ones(1,81)];

r = zeros(100,200); %Reward vector, which will be populated with 1's whenever a reward occurs (in this case, when action1 == 1 and action2 == 0)


for idx = 1:numTrials
    for t = 1:numSteps-1
        if t==20
            as1=x(t)*W1; %Compute action signals at time step 20 within each trial
            ap1 =  exp(as1)/(exp(as1)+exp(as2)); %Softmax function to calculate probability associated with each action
            ap2 =  exp(as2)/(exp(as1)+exp(as2));
            if n<(idx)=1;
            if n<ap2                a2(idx)=1;
            if a1(idx)==1 && a2(idx)==0 %Only deliver reward when action1 ==1 and action2 ==0
        V(t,idx) = x(t).*weights(t, idx);
        V(t+1,idx) = x(t+1).*weights(t+1, idx);
        delta(t+1,idx) = r(t+1,idx) + discFactor.*V(t+1,idx) - V(t,idx);
        weights(t, idx+1) = weights(t, idx)+learnRate.*x(t).*delta(t+1,idx);
        W1 = W1 + learnRate*delta(t+1,idx)*a1(idx);
        W2 = W2 + learnRate*delta(t+1,idx)*a2(idx);
    w1Vect(idx) = W1;
    w2Vect(idx) = W2;


set(gcf, 'renderer', 'zbuffer') %Can prevent crashes associated with surf command

hold on
plot(w2Vect, 'r')



Oh, and one more thing that gets my running tights in a twist - people who don't like Bach. Who the Heiligenstadt Testament doesn't like Bach? Philistines, pederasts, and pompous, nattering, Miley Cyrus-cunnilating nitwits, that's who! I get the impression that most people have this image of Bach as some bewigged fogey dithering around in a musty church somewhere improvising fugues on an organ, when in fact he wrote some of the most hot-blooded, sphincter-tightening, spiritually liberating music ever composed. He was also, clearly, one of the godfathers of modern metal; listen, for example, to the guitar riffs starting at 6:38.

...Now excuse me while I clean up some of the coins off the floor...

Friday, November 29, 2013

Master's Recital Music Videos

Since there are several haters out there who doubt that I can play piano, here, finally, is video evidence from a recent recital. In case you're confused, I'm the tall guy at the keyboard wearing all black.

Any mistakes, ensemble slipups, or counting errors are solely my fault, and in no way reflect on Sonja. (I said I could play; I didn't say anything about playing well.) Make sure to buy a bunch of hand lotion and Kleenex before listening to these masterpieces.

Thursday, November 28, 2013

Reminiscences of SFN 2013

In the past two weeks since I have returned from San Diego, I still find myself struck with lucid recollections during my hypopompic states, the memories of my travels twinned with hallucinations no less beautiful and no less real, were I to try distinguish the two.

When I close my eyes I can still feel the pitch-perfect weather against my skin; I can hear the purl of fountainwater underneath the lush gardens and exquisite statuary of Balboa Park; I can see those silvery, sunwarmed beaches, speckled with families and surfers and oceangoers of all stripes and ages, the terns overhead lazily riding the wind thermals and the surf below gently lapping at the shores. The man is not to be envied who does not find his spirit refreshed and invigorated by the scintillating waves of the ocean, his eros not aroused by the sight of brown-skinned beauties emerging from the sea with beads of saltwater clinging to their skin and their delicate pink toes sinking into the argentate sand; no, he is not to be envied who does not find some spark of religious awe kindled by the sight of the sun bleeding slowly into the horizon and replaced by the pale disc of the moon, pasted in that inky firmament like some ghostly wafer, overlooking the dark abyss of water out there past men's knowing, where stars are drowning and whales ferry their vast souls through the black and seamless sea.

It is well to be surrounded by such sights and sounds, as, set against the backdrop of a conference devoted to science, the conscious mind is all the more appreciative of the particulars and the practicalities of what he believes, secretly or openly, to be the healthiest, the most rigorous, the most downright of human endeavors - that of scientific inquiry. Aristotle once claimed that the twin peaks of human pleasure consist of one, sexual intercourse, and two, thinking; and once one has felt the slow-burning satisfaction of scientific experimentation, of hypotheses proposed and tested, of results surprising one in the most unexpected of ways, what right man would believe otherwise? And after a day of lively discussion and heated debate, after filling one's cup with as many poster sessions and workshops and talks as one can handle and drinking it to the lees, then one encounters the night; and, the mind still reeling from the heady fumes of science, the senses attuned to all the nuances that weren't there before, walking past the garish lighting of the restaurants and pubs of the Gaslamp district, brazen hussies with their sultry strolls and minimalist vesture calling out to each other in the darkness, steroid-inflated bouncers guarding the doors of nightclubs exuding faint rumbles of bass punctuated by shrieks and laughter - and it is here that one becomes aware of certain beauties and lusts and terrors and menace that until now were only thinly hidden. The juxtaposition of such different modes of experience makes each of them in turn that much more powerful, more savory, more piquant.

During my days at the conference inside the convention center, therefore, I expected all of the hobgoblins and ecstasies of the nights before to melt away like snow in sunshine; but even here there is an element of the surreal. Within the bowels of the convention center, several football fields long, were rows upon rows of posters, almost beyond reckoning; here is one person surrounded by intrigued colleagues, gesturing expressively with his hands, his face beaming; there is another over there who could not be more different, all alone, head down and sullenly gazing at the floor, one hand holding the opposite forearm, a perfect picture of dejection. Wind your way through the exhibition section where companies are hawking their wares, photos showing how results look before and after the application of their device, mechanical contraptions demonstrating how the latest stereotaxic equipment drills into any location without any error, no fuss, no muss. You would expect the vendors to be much more animated, to act like some sort of scientific carnival barker; but unfortunately, they sit around, this one checking her phone, that one with a saturnine expression pasted on his face and a toothpick affixed to the side of his mouth.

Upstairs through the pavilio, and enter the ballroom, where chairs are stacked in rows as neatly and ceremoniously as gravestones in a cemetery. Far away at the front of the ballroom a speaker is at the podium, a miniscule dot at this distance, but whose person is projected on several large screens hanging from the ceiling, the amount of exposure beyond the most egomaniacal totalitarian's wet dream. After a couple of hours of talks and results and diagrams of models, out the door again into the hallway, past several smaller conference rooms packed with listeners. Along my way I reach down to pick up a discarded pamphlet off the floor from the American Association for the Advancement of Science; inside, it laments that more than half of the United States population still believes in psychic phenomena such as ESP and seances. "Fifty-seven percent of American adults believe in phenomena unsupported by any evidence whatsoever," it says. "It would be better to get that number closer to zero." It then lists several resources and initiatives to educate the population to think scientifically. The younger the age at which they can stage an intervention, it seems, the better.

While I can appreciate the sentiment, part of me thinks that this feeling is misguided. I have several close colleagues who would be horrified to wake up to a world denuded of superstitions and myths; so satisfying is the sense of superiority they feel in mocking those who still hold groundless beliefs, and the repercussions so minimal, that to take that away from them would be to take away their chief joy. Conversations would dry up, bereft of the usual potent feelings of solidarity and indignation, and the only ties that used to bind them to other like-minded individuals would dissolve. No, clearly a world free of people believing crazy shit would be a catastrophe. I think that a more reasonable goal would be to get the number of persons down to about five or ten percent; that way, the Association can still claim no small measure of success in their crusade, and there will still be plenty of eccentrics left over to insult, belittle, and marginalize.

After making the rounds at all of the talks, I go back to the poster session, where new posters have been pinned up on the boards, some of them still reeking the stench of hot ink fresh off the printer. Wander around, and you begin to notice how some individuals tend to dominate the conversation surrounding a poster and poins out all the experimental flaws with a minimum of decency. To counteract this, I usually leave in one or two glaringly obvious errors in any poster I present or any paper I submit; that way, one can more easily comment on it and feel as though they have done something useful, usually leaving all of the other material alone.

That being said, however, still be aware that there is much research out there which is smoke and mirrors; having been in the game for quite some time, I can provide a short list of words and phrases that should immediately set off alarm bells in your head: neuroscience; significant; brain; rat; human; monkey; hypothesis; anterior cingulate; activation; voxels; cake; "game-changer"; default poop network. Beware the siren song of these words that charm the ear and bewitch the mind; they are beautiful but treacherous ondines who, given the chance, will wrap their briny arms around you, dragging you down to your death in the bottomless sea.

Saturday, November 9, 2013

Society for Neuroscience 2013!

Hey guys,

This afternoon, I will be leaving Indiana for temperate, sunwarmed San Diego! If you get a chance, be sure to stop by my poster and say hi!

Poster time: Monday, November 11th, 1:00pm-5:00pm
Poster board number: KKK25, Halls B-H

Friday, November 8, 2013

Master's Recital: Debussy, Bach, and Shostakovich

For those of you in Bloomington, I will be accompanying for a cellist's master's recital here at the Jacobs School of Music. The program features a sonata by Debussy so unbelievably colorful and vivid, your synesthesia will go haywire; some of the clearest, most soul-refreshing chamber music by Bach; and everyone's favorite, Shostakovich's colossal cello concerto no. 1, which, in classical music terms, is known as a bodice-ripper.

To bring you this, we've put in countless hours of arduous, painstaking practice, endured invective, censorship, and misunderstanding from the most trenchant of critics, and gone through long nights of rehearsal punctuated by hours of bickering and quarreling, torn sheet music and thrown metronomes - but always followed by tearful reconciliation. Has it been worth it? Heck yes. But then again, I'll let you all be the judge of that.

When: 7:00pm, Friday, November 8th
Where: Recital Hall, ground floor of Merrill Hall (1201 E 10th St)
Who: Andy Jahn, Piano; Sonja Kraus, Cello

===== Program =====

Debussy: Sonata for Cello and Piano
I. Prologue: Sostenuto e molto risoluto
II. Sérénade
III. Finale: Animé: Léger et nerveux

Bach: Gamba Sonata No. 1 in G Major for Cello and Piano
I. Adagio non troppo
II. Allegro, ma non tanto
III. Andante quasi Lento
IV. Allegro moderato

Shostakovich: Cello Concerto No. 1 in E-Flat Major, Op. 107
I. Allegretto
II. Moderato
III. Cadenza
IV. Allegro con moto

Thursday, November 7, 2013

Computational Models of Reinforcement Learning: An Introduction

The process of learning what is good for us, and what is bad for us, is incredibly complex; but the rudiments have been outlined, and we can gain some insight by starting with the basic building blocks of what is known as reinforcement learning. During reinforcement learning, we come to associate certain actions with specific outcomes - push one button and get a piece of cake; push another button, and receive a blast of voltage to your nipples. Through experience we begin to flesh out a mental picture of what decisions are likely to lead to certain events; and, though merely observing someone else we can learn about what to do, or what not to do, even in the absence of reinforcers or punishers.

Before we get there, however, let's approach our subject from an even more basic form of learning - classical conditioning. In this case, no actions are needed; one merely observes a stimulus, such as a tone of a certain frequency, and learns that it predicts a specific outcome, such as the arrival of food. In this case, the tone is the conditioned stimulus, the food is the unconditioned stimulus, and salivating in response to the tone, after enough pairings between the tone and the food, becomes the conditioned response.

Let me give an example from my dating history. You may find this particular story I am about to relate to be way, way too much information; but if you've been reading for this long, I assume that we're on close enough terms that divulging such graphic details of my personal life will, far from driving us apart, bring us closer together by allowing us to bond over our shared humanity.

So. Onions. I am - or I used to be - indifferent to them. All I could say about them was that they had a smooth, eely texture when fried in oil; that they released a pungent aroma when sliced, diced, and crushed; and that their flavor was particularly sharp. Other than that, I had nothing else to say about them. Onions were onions.

But one day - never mind when - I began to see a girl who absolutely loved onions. Onions were inseparable from any dish she made; and so close was the association between her mood and the amount of onions she put into her cooking - casseroles, curries, tartlets, you name it - that, were I to witness her eating an entire onion in the raw, I would assume her to be in the seventh heaven.

My little onion, I used to call her, as a sign of my undying affection; and whenever we made love, we would first scatter onion shavings upon the bed, or the grass, or the movie theater seat, as a ritual to consecrate the beautiful, sacred act that was about to be made manifest. And when she would part the pillowy gates of her mouth and cleave her lips to mine, that pregnant moment filled with an anticipation so poignant you could hardly bear it, I would inhale deeply, feeling the overpowering, acrid smell of onions run over single one of my nose hairs and driving my olfactory bulbs insensate with desire.

"Darling, do you love me?" she would ask, breathing heavily, the odorous waves of onion wafting across the thin slit of air between us and mooring within my nostrils.

"Yes, my little onion," I would reply. "Yes; yes; a thousand times yes!"

Such was our love, then; and you can hardly imagine my shock and desolation when, several years into this relationship of onions and unadulterated bliss, some knave, jealous of our happiness no doubt, took it upon himself to poison one of her onions, and kill her! My bereavement was only slightly assuaged by the fact that she had, only a few days before, took out an extremely lucrative life insurance policy, having named me as the sole beneficiary.

After three painful, soul-searching days of mourning, however, I eventually gained the strength to renew my courtships with several other desirable young ladies. Yet, while throughout this period I continued to seduce innumerable women and live a Byronesque lifestyle of aristocratic excess, I couldn't help feeling some conspicuous lack, some defect in any affair, any tryst I willingly thrust myself into. At first I blamed the girls themselves: this one with long, lithe arms, but perhaps a shade too willowy; this one with a bold, intriguing personality, but perhaps a bit too pert for my taste; and yet another, sloe-eyed, with beautiful brown irises, but which, upon closer inspection, revealed the slightest of discrepancies in the size of one pupil compared to the other. Not having taken myself for a very discriminating fellow before my relationship with Mary, that light of my life, that fire of my loins - in other words, that onion chick I was talking about earlier, in case you couldn't tell - I found myself at a total loss.

While ruminating over my sudden change in amorous tastes, one day I found myself absentmindedly skimming the menu at a local bistro; and then - mirabile dictu! - I saw the item French onion soup inconspicuously nestled under the Appetizers section. Feeling my pulse quicken, I followed my instincts and ordered the soup, aware that I had hit upon the answer to my problem. Soon after, a disembodied hand placed the soup in front of me; and, slowly, meaningfully, I gazed down into the thick brown liquid. I braced myself, inhaled deeply, and somewhere in my brain a key unlocked the overflowing warehouse of my desire. Memories came flooding back; memories of Mary; memories of onion; and, most of all, memories of that pungent, acidic smell crushed out from the shavings underneath our bodies.

Having solved the puzzle, I now embark on a new chapter of my life; and nowhere do I go now without my peeler, and without my paring knife!

This story wonderfully illustrates some of the key components of classical conditioning. First, an unconditioned stimulus - Mary - elicited an unconditioned response from me - feelings of arousal. Because of Mary's repeated pairings with onions, the onions became a conditioned stimulus that signified an upcoming session of especially gratifying hanky-panky, and eventually by themselves elicited the conditioned response of arousal.

In psychological terms, this process of learning is called the "critic" part of learning; a stimulus signified some kind of upcoming reward, and over time a person learns this associations, eventually beginning to shift their usual feelings of pleasure and excitement from the reward itself to the stimulus signifying the reward. The critic evaluates how reliable the association is, and, depending on the individual, associations can be learned relatively slowly, or relatively quickly.

Let's focus on a landmark Science paper by Schultz, Dayan, & Montague (1997). This paper mathematically modeled different phases of reinforcement learning, and outlined several equations that can simulate how much an organism will response to the conditioned stimulus and to the reward itself. The following Matlab code implements equations 3 and 4 from the paper, using 200 trials and 100 timesteps within each trial. The weights are updated on each trial, and the prediction error, represented by delta, will become increasingly larger and move close to the time of the presentation of the conditioned stimulus. Note in the following figure from the Schultz et al paper, that when an organism has been conditioned to expect a reward at a certain time, the omission of that reward will lead to a large negative deflection in the prediction error signal.

Similar surface maps can be generated using the following code; I suggest adjusting the learning rate and discount factor parameters to see how they affect the error prediction signal, and also the administration of the reward at different times. Building up this intuition will be critical in understanding more advanced models of reinforcement learning, in which outcomes are contingent upon particular actions. And don't forget to keep eating those onions!

numTrials = 200;
numSteps = 100;
weights = zeros(100,200); %Array of weights from steps 1-100, initialized to zero

discFactor = 0.995; %Discounting factor
learnRate = 0.3; %Learning Rate
delta = zeros(100,200);
V = []; %Empty vector, sum of all future rewards
x = [zeros(1,19) ones(1,81)]; %Presentation of conditioning stimulus
r = zeros(100,200); %Reward

for idx = 1:numTrials
    for t = 1:numSteps-1   
        V(t,idx) = x(t).*weights(t, idx);
        V(t+1,idx) = x(t+1).*weights(t+1, idx);
        delta(t+1,idx) = r(t+1,idx) + discFactor.*V(t+1,idx) - V(t,idx);
        weights(t, idx+1) = weights(t, idx)+learnRate.*x(t).*delta(t+1,idx);


Friday, November 1, 2013

Indianapolis Monumental Marathon: Participant Tracking

Fellow brainbloggers, cognitive neuroscientists, and endurance sport enthusiasts,

Tomorrow I will be competing in the Indianapolis Monumental Marathon. Conditions at start time are low forties and partly cloudy, with a high chance of lactic acid, extreme muscle fatigue, and reduced sperm count. My goals: Break the two-hours-and-thirty-minutes barrier; place in the top ten; and to bring the heat on all these pasty-faced, weekend-warrior poltroons and send y'all back to Ireland.

If you would like to get updates on my race, simply sign up here and enter my name (First name: Andrew; Last name: Jahn). In a way, it's just like watching a horror film, or witnessing a particularly nasty breakup - all of the thrill, none of the danger. The things I do for you guys.

Thursday, October 31, 2013

Andy's Brain Blog Advice Column: Should You Go to Graduate School?

Around this time of year legions of students will submit their graduate school applications; and, if I close my eyes, I can almost hear the click of the mouse buttons, the plastic staccato of keyboards filling in address information and reflective essays, the soft, almost inaudible squelching of eyeball saccades in their sockets as they gather information about potential laboratories to toil in and advisors to meet. So vivid is the imagination of these sounds, so powerful are the memories of my experience, that part of me can't help but feel a rill of nostalgia flutter down my spine, and possibly, somewhere, deep down, even a twinge of envy. I remember, as a young man, the heady experience of the application process: The shivers of expectation; the slow-burning, months-long buildup of excitement; the thrill of embarking upon an adventure of continuing to do work that you loved, but with new people to meet, new places to discover, and new worlds to conquer. For those about to undertake this journey, I say - Good fortune to you.

However, even in these times of expectation and excitement, I cannot refrain from advising caution; for I once knew a man in a similar situation, who, at the height of his powers, tried his hand at graduate studies; but, rather than augmenting his already considerable gifts, led to the most horrific of decays. So great a man was he, that to think of him is to think of an empire falling. This may smack of hyperbole; but the great promise of his early years, followed by the precipitous decline upon his entry to graduate school, do suggest the tragic dimensions of which I speak.

In his youth he was a hot-blooded hedonist, snatching at all pleasures as he could, carelessly, almost impulsively, like a shipwrecked sailor grasping at driftwood. During these years his life was one of wild debauch, filled with wagers and duels, wine-soaked bacchanalias and abducted women. Endowed with Herculean stamina and the unchained libido of a thousand-and-three Don Juans, every muscle, every sinew, every fiber of his being, was directed at vaulting his pleasure to its highest pinnacle and beyond. A dark aura of raw sexuality exuded from his being; the wellsprings were perennial which fueled his twisted desires. He wouldn't have known an excess if he saw one - his lusts were of such depravity they would have eclipsed even de Sade's darkest fantasies.

The nonstop orgies of his early years eventually petered out, however, and one morning he awoke to find himself in extreme want. Abandoned by his mistress, his fortune squandered, he eventually decided that applying to graduate school would be the best option; after all, styling himself a freethinker and an intellectual, the pursuits of business and politics seemed inadequate, even vulgar. A life of the mind, he concluded, was the only one for him, and thus did he eschew the red and the black in favor of the white labcoat of the researcher.

Among any other trade this man would have been happy, motivated and fulfilled, perfectly at home among the elegant rakes of any other era; but ambition denied withered him; his incessant studies dried up the springs of his energy; and melancholy marked him for her own. Instead of a life of health, vigor, and adventure, now he whittled away his days in a dreary, windowless room performing the most perfunctory and mind-numbing of tasks. Instead of using his masculine touch to awaken hundreds of young maidens into womanhood, now he could only practice a crippled eros that repeatedly failed to take wing. Poverty, alcoholism, and overwork became the staples of his life; his last years were clouded by religious mania; and, misunderstood and forgotten, he spent his final days in utter squalor, dying much as he foresaw - like a poisoned rat in a hole.

Limerick Intermezzo

There was a young man from Stamboul,
Who soliloquized thus to his tool:
"You took all my wealth
And you ruined my health,
And now you won't pee, you old fool."

My friend's story, though extreme, represents the experience of no small number of graduate students. It is not uncommon for the typical graduate to spend the prime of life in an environment he detests, doing work he abominates, with the energy that should go into the flower instead remaining in the leaves and stem. Frustration, disappointment, and monotony become his bywords. The great expectations he begins with, the intoxicating freedom of his new schedule, are all too quickly transformed into feelings of ennui and despair; the hot blood that once coursed through his veins gradually congeals into cold slime. He criticizes his program, his field, his advisors, all the while oblivious to the fact that he is a willing coauthor of his own misery. He manages to project a certain nonchalance, he gets along agreeably enough with his friends, but his most private moments - if not spent in a haze of wine or the arms of some debauched wench - are torture.

And yet - I have known a few individuals who persevere even under the most sordid of circumstances, who, even in the face of the most formidable of challenges, manage to live bravely, even joyfully. They are impervious to the most depressing of environments and the most hateful of colleagues. For these resilient few, their passion lifts them above the waves that would drown the merely indifferent; the iron in their souls allows them to withstand blows that would crush the weaker-willed. (I do not count myself among their number, but then again, I have never had the desire; I have been more than able to make up for any defects of personality or intelligence though flattery, intimidation, bribery, and blackmail.)

Let he who is considering graduate school, therefore, take stock of his weaknesses, and of his strengths; let him calculate the risks; let him understand that persisting in anything that leaves him feeling enervated and worthless is not the sign of some tragic hero, but the mark of a fool - it is the first step on the path to spiritual suicide.

If, by chance, he does have many years of happiness, let him rejoice in them all; yet let him remember the days of darkness, for they shall be many.

Wednesday, October 23, 2013

Mathematically Describing Neuronal Connections in the Brain

Students in my classes have started to catch on to the fact that I tend to dress up for lectures I am particularly excited about. For a topic I'm indifferent toward or lukewarm about, I wear my standard dress shirt, slacks, and dress shoes. To prepare for a subject I like a little bit more, that's when I throw on a sports jacket, and possibly a nice belt. But get me all hot and bothered, and that's when I break out...the ties.

Not surprisingly, then, I wear a three-piece suit when talking about neurons. These sexy little suckers act as the basic cells of communication throughout your brain and throughout your nervous system, relaying electrical transmissions all the way down your axons to the synaptic gap, those terminal buttons precariously poised on the precipice of a protoplasmic kiss, until finally, excruciatingly, those tiny vesicles of chemical bliss burst from their vile durance, recrudescent, crushing out the last throb of the last ecstasy man or monster has ever known.

...Let me catch my breath...Where was I? Oh yes - neurons. Besides their role in transmitting electrical and chemical signals throughout the brain, they also exist in astonishingly high numbers, with somewhere on the order of tens of billions of neurons packed into a single brain. On top of this, each one can share hundreds or thousands of connections with other neurons, leading to a staggering number of potential synaptic connections. The mind boggles.

To provide the full mathematical treatment of understanding neurons, we are joined again by Keith "The Rookie" Bartley, whose interest in synaptic connections was recently piqued by an introductory cognitive science course. Along the way Keith touches on mathematics, the Turing Test, Friends, oatmeal, the uncanny valley, and where genitals are represented in the brain, providing a theoretical basis for why foot massages can lead to greater chances for successful coitus.


As a TA for an Introduction to Cognitive Science course, one of our instructors briefly discussed the concept of Block's "Aunt Bertha Machine" and how the Turing test alone required more bits of memory than there are atoms in the universe. (If you aren't familiar with Ned Block's work click here).  Below is a section of one of his slides:

Volume: (15*10^9 light-years)^3 = (15*10^9*10^16 meters)^3
Density: 1 bit per (10^-35 meters)^3
Total storage capacity: 10^184 bits < 10^200 bits < 2^670 bits
Critical Turing Test length: 670 bits < 670 characters < 140 words < 1 minute

Difficult as it is for some people to conceptualize and subsequently deal with such induced feelings of simultaneous intelligence, stupidity, and insignificance pervading recapitulations of their own life's meaning, I would like to warn those people that the following information about the capacity of the human brain is likely to do much worse. READ RESPONSIBLY.

For the human brain, the possible number of combinations and permutations of neural connections has been purported to vastly exceed the number of elementary particles in the Universe. Consider for a moment that the brain has 85,000,000,000 neurons, we'll round that up to the previously estimated 100,000,000,000 for hypothetical simplicity, each with a capacity for up to 10,000 synaptic "connections". 

1! = 1
10! = 3,628,800
100! = 9.33 x 10^157
1000! = 4.02 x 10^2,567
***This is where Google's calculator starts to report infinity***
For bigger factorials, we'll have to use Stirling's approximation.

So using Stirling's approximation....

10,000,000,000! ~ 2.33 x 10^95,657,055,186
100,000,000,000! ~ 3.75 x 10^1,056,570,551,815

But wait, each of the 100 billion neurons can have 10,000 synaptic connections, so…

10^11 * 10,000 = 1e26

So rather...

100,000,000,000,000,000,000,000,000! ~ REALLY BIG NUMBER

But Keith! Come on, this looks like a gross misrepresentation of the limits of human cognition?

Our theories about the brain are much more modular in scope, but at the same time, distributed enough to adapt. These two points function as a much better descriptor of the networked brain. It's reasonable to see, in our post-Scopes trial times, that humans' brains developed around how they are utilized in their day-to-day lives, and perhaps more importantly, the phenomenal ability to constantly adapt, even in the wake of extreme trauma. The development and existence of a single "grandmother neuron", is misrepresentative of our degree of sustainability in the brain. Neurons for your grandmother have to be very distributed, so that when some of your brains cells die off, as they often do, it's important you don't forget the old woman that squeezes your cheeks when she sees you, lest you punch her in the face for assault. Jennifer Aniston would no longer instill memories of how much time you wasted watching reruns of Friends on TBS every afternoon for 4 years, and Halle Berry would no longer remind you of how Hollywood reduced one of the greatest antiheroines in the history of DC Comics to a mannequin in spandex with a speech impediment. 

At the same time, however, that three pounds of oatmeal between your ears still retains a relative degree of modularity in regional function, which is a reason why your brain is compartmentalized in various folds (gyri) and crevices (sulci). When you have an itch from what is in fact a really small bug bite, your desired area to itch is very distributed across skin because the signals in the brain are themselves both distributed and modular. A diagram often used to demonstrate this modularity is the cortical homunculus.

Straight out of the backwoods of the uncanny valley, this diagram demonstrates the relative intensity and location that each section along your somatosensory cortex corresponds with on your body. As for the proximity of feet relative to your genitals, well that might just explain a lot about some guys now wouldn't it.