Breathing Bangalore

In the suburbs of Bangalore, in one of the numerous buildings that house research and support facilities for nearly every major tech company in the world, scientists are working on understanding how you spread your attention when you navigate a web page.  A few of them had gathered in a conference room, listening as I described some of our work on how the brain controls movements of the eye.  

Gazing at the teak conference table, high back leather chairs, and sophisticated teleconferencing equipment, I considered the contrast: just a few streets away from this modern world where I was giving my talk, there were goats munching on a pile of refuse, and a small band of cows roaming happily against traffic.  A little farther, in the center of the city, there were scientists and engineers working on fundamental questions in the Indian Institute of Science, a major university on a beautiful wooded campus that housed, in addition to world class laboratories, large families of monkeys, bands of wild dogs, and bats the size of crows, all living freely, and from all indications, contently, alongside humans.

I think the most striking difference with anywhere else that I have visited is that people here seem to have an exceptional respect for life --- life of any form. Like most university campuses, this one also has large, impressive trees that dot the landscape.  But here, the human roads do not prevail.  Indeed, in many places the road has a large tree in the middle of it, with a trunk marked with a few reflectors, and the cars simply go around it.  At our university guest house, a sprawling hotel-like structure, there are a few places where the hallway turns at a strange angle.  Looking closer, I see that the building is bending around an old tree, and not the other way around.  This co-existence is on display with the wildlife that lives alongside us.  The faculty housing is in a wooded area, where monkeys also raise their families.  One morning, as we ate breakfast at the guest house, with the window open to let in the cool breeze, a family of macaque monkeys came to visit.  The mama-monkey took a piece of papaya from a table, and went over and fed her babies.

The weather is mild and pleasant; a pleasure to step outside and feel the sun and smell the trees.  But the university is an oasis.  The peace and quiet of the grounds are in stark contrast to the outside world.  As we step beyond the gates, we leave "jungle book" and enter the human world; with its crushing traffic of cars, motorized three-wheel rickshaws, and scooters, all communicating in the machine-made language of horns.

The human languages are myriad in India, but the main language, at least here in the south, is English.  The students tell me that they rely on English to talk to each other because each comes from a different part of India, with its own languages, and English is the only common tongue.  

The diversity of languages is complemented with the diversity of faiths.  In the mornings, I hear the Muslim call to prayer before sunrise, and then a few hours later, I see the Hindu temple as I walk to the university conference center.  On the steps of the center there is a familiar scene, one of the wild dogs napping in the sun.

Ask vs. Axe

A few months back, my administrative assistant was offered a wonderful new job and as a consequence, the department hired a replacement.  The new assistant is a capable, hardworking young lady.  A few days ago I noticed that she tends to use the word /axe/, instead of /ask/.  

I had heard this usage a number of times in Baltimore, particularly among African-Americans.  I wondered, is this a mispronunciation?  Perhaps something like /nuclear/ vs. /nucular/?  A bit of research made me understand that /axe/ has a long history in the English language, and is not a mispronunciation.

Oxford dictionary notes that /ask/ is the descendant of /ascian/, which in Old English means to demand, to seek from.  The alternative form of /ascian/ is /axian/, or in short form, /axe/.  Oxford notes its use in Chaucer: "I axe, why the fyfte man Was nought housband to the Samaritan?" (Wife's Prologue 1386), and "a man that ... cometh for to axe him of mercy." (The Parson's Tale 1386)  The book The Complete Works of Goeffrey Chaucer includes 5 passages where the word /axing/ is used.  The word /axe/ appeared in the first complete English translation of the Bible in 1535 by Miles Coverdale, who wrote: "Axe and it shal be giuen you," and "he axed for wrytinge tables."

According to Random House, “In American English, the /axe/ pronunciation was originally dominant in New England. The popularity of this pronunciation faded in the North early in the 19th century as it became more common in the South. Today the pronunciation is perceived in the US as either Southern or African-American. /axe/ is still found frequently in the South, and is a characteristic of some speech communities as far north as New Jersey, Pennsylvania, Illinois and Iowa.”

So /axe/ is a regional pronunciation, somewhat similar to the regional pronunciation variation of the word /idea/ and /idear/.

A life without memories

When he walked into the room, he looked a decade younger than 71; his face handsome, with few wrinkles.  I pulled out a chair and asked him to sit in front of a robotic contraption that earlier that morning we had set up in an examination room at the Clinical Research Center at MIT.  He sat calmly and avoided touching the contraption.  I pulled the robotic arm toward him and asked him to hold its handle.  He grabbed the handle and started moving it around, keeping his gaze on the handle.  I asked him to look up at a monitor, where he saw a little cursor moving around as he moved the robot’s handle.  The computer displayed a target box, and he moved the cursor into the box, at which point the computer animated it, producing an explosion. 

A smile came to his face.  He said: “You know, when I was a kid I liked to go bird hunting.” Exuberantly, he described the birds that he hunted, the guns that he owned, and the woods around his childhood home.  He continued doing the task, reaching while holding the robotic arm and making little explosions.  About five minutes later he said: “You know, when I was a kid I liked to go bird hunting.”  The exuberance was unabated.  He had no idea that a few minutes earlier he had told me that same story.

Memory without awareness

The day before, with my two graduate students Kurt Thoroughman and Maurice Smith, I had packed the robot and computers in the back of my wife’s station wagon and drove up from Baltimore to meet and examine Henry Molaison.  Henry, or as he is known to the scientific world ‘H.M.’, had suffered from debilitating seizures.  When he was 27, desperate for something that might help, he had agreed to an experimental procedure that surgically removed the hippocampus and amygdala from both sides of his cerebral cortex.  The surgery was successful, greatly reducing his seizures, but left him with a staggering deficit: an inability to form certain long-term memories.

Now, in the examination room, the robotic arm that Henry was moving started to produce forces, pushing his hand as he approached the target, making it so that he would miss-reach and not get those explosions.  But he kept practicing, and after a few minutes, a part of his brain that was not damaged learned how to generate the right motor commands so that his arm could compensate for those unusual forces.  Once again he got the explosions, and once again he excitedly told me of his childhood bird hunting days.

After about an hour of playing with the robotic arm, Henry left for lunch and an afternoon nap.  He returned about 4 hours later.  I said hello and ask him whether he remembered meeting me and playing with the robotic arm.  He said no.  I pushed the robot aside, showed him the exam chair, and asked him to sit down.  He sat down, but then something interesting happened: rather than avoiding the machine, the behavior of someone who has never done the task before, he grabbed its handle, brought it toward him, looked at the video monitor, and started to move the cursor toward the target. 

He had no awareness that he had seen me before, or that he had played with this robotic arm only hours earlier.  Yet, that experience had left two kinds of memories in his brain:  the memory of how to use the tool, and the memory that associated the sight of the tool, and the act of moving it, with a rewarding outcome.

He was not aware of it, but the sight of the strange tool, the robotic arm, was sufficient for him to want to hold it and move it around so that he could chase targets and get explosions.  Would he have voluntarily reached for the robot if, while playing with it, he had not had a pleasurable experience, recalling those childhood memories?  Probably not.  In 1911, Edouard Claparede, a physician in Geneva, described an amnesic woman much like Henry.  Claparede had wanted to test her memory, so he played a small joke on her: when he reached his hand out to shake hers, he had hidden a small tack in his palm.  When the patient shook his hand, she felt the sharp tack.  The next day when Claparede had approached her, she could not recall having seen him before.  However, when he reached out to shake her hand, she pulled away, despite being unable to say why she did not want to shake his hand. 

Henry could not remember the episode of having played with the robotic arm, but that act left a memory, associating the robot with a rewarding outcome.  The part of his brain that learned this value association was exhibiting its knowledge by reaching out to the robot and moving it in search of a target to explode. 

In addition to this value-action association, he also had a memory of how to use the contraption.  Those forces that he had practiced to overcome had left a different kind of memory: much like picking up a coke can that you expect to be full but is empty, Henry’s movements on this revisit had the ‘after-effects’ of the earlier experience.  The robot was not producing any forces, but he moved it as if he was expecting it to be producing those earlier forces.  When the forces were re-introduced, he moved the robot skillfully.

He did not have the ability to form memories of episodes of his life, but these two other intact forms of memory served him well.  For example, as he aged, he developed osteoporosis and required a walker to keep physically active.  With practice, he learned to use the walker skillfully.  Importantly, he would use the walker without being told to do so.  That is, he ‘knew’ that the walker was a useful tool that helped him get around.

Permanent present tense

What is it like to live a life with such a disability?  In a recent book titled “Permanent present tense”, Suzzane Corkin, a scientist who studied and cared for Henry for 46 years, describes his life in loving, exquisite detail.

In a photograph showing Henry with his mother at his 50^th birthday, he recognized his mother, but not himself.  While attending his 35^th high school reunion, he did not recognize anyone by sight or by name.  He could not recall any specific event in his life, even events from before his operation.  For example, he could not recall a single specific Christmas gift that his father had given him.  He remembered some of the facts that he had learned from the time before his operation, and the gist of the experienced events, but no recollection of any specific episodes.

Henry rarely spoke of being hungry or thirsty.  He never sought out food for himself; it was simply given to him by his caregivers.  In 1981, Corkin asked him to rate his hunger from 0 (famished) to 100 (absolutely full).  He consistently gave a rating of 50, whether he had just finished eating, or was about to eat.  One evening, Corkin played a small trick on him.  After Henry had finished eating and his tray had been taken away, the kitchen staff brought him another tray, with exactly the same meal.  Henry ate the second dinner, cleaning the plates, except for the salad.  He seemed unable to express a feeling of satiety.

When we were examining him, a caregiver mentioned that Henry rarely verbalized that anything might be wrong.  For example, if he had a tooth ache, he would rarely mention it.  Only by observing that he was deviating from his normal behavior during the day would the caregiver suspect that something was wrong.  The caregiver would then go through a list of things to see if they could find out what may be the problem.

Corkin tested Henry’s ability to perceive pain by using a hairdryer to project a spot of heat onto his skin.  The heat was not intense enough to burn the skin, but the idea was to test whether Henry could feel pain.  Corkin’s results showed that Henry not only could not discriminate normally between various levels of pain, he did not report any of the stimuli as painful, and never withdrew his arm.  It is possible that the inability to normally perceive pain, to know hunger or thirst, was related to his operation; perhaps it was associated with removal of the amygdala, as Corkin suggests.

Henry lived a life without keeping memories of the events, and without pain.  His father had passed away, and his mother, who had taken care of him for much of his life, was in a nursing home.  He kept two notes in his wallet that he had written to himself: “Dad’s gone”, “Mom’s in nursing home—is in good health.”

Henry died at the age of 82. 

References
Corkin S. (2013) Permanent present tense: the unforgettable life of the amnesic patient H.M., Basic Books.
Shadmehr R, Brandt J, Corkin S (1998) Time dependent motor memory processes in amnesic subjects. Journal of Neurophysiology 80:1590-1597.

Young neurons in an old brain

At the age of 88, my seemingly healthy and vigorous father suddenly died.  He had commanded an army, lived through a revolution, met kings and presidents, and through it all raised a family.  And then one night, all those memories, all those experiences, vanished.  Coming home from the funeral I realized that a library, stacked with history books, had burned to the ground. 

When we experience something, it can become a memory.  But what is this memory?  What is its neural substrate?  Can someday memories that we store in our brain be read out and stored in a machine?  Is there any hope that the library can be saved from the fire that consumes us as we die?

Standard model of memory

Our model of memory today is one of synaptic plasticity.  When we experience something, the neurons that are engaged by that experience produce electrical activity, and that activity can alter the strength of synapses that connect them to other neurons.  The electrical activity can also result in growth of new synapses.  Together, this altered strength of connectivity in an existing network of neurons is thought to be the basis of memory.  So in principle, if one could measure the strength of each synapse, and model the functional properties of each neuron, then one has a representation that approximates the state of brain of an individual.  The lifetime of memories and experience are within this representation. 

The problem, unfortunately, is that this concept of memory relies on the assumption that neurons themselves are fixed nodes, whereas the connections (that is, the synapses) are the changing components through which memories are stored.  This assumption, as it turns out, is false.  New neurons are born every day, and the human brain, even in old age, adds and subtracts nodes to the network.

Finding a neuron’s birthday

Between 1955 and 1963, there were numerous above ground tests of nuclear weapons.  With every explosion, the amount of isotope 14C was elevated in the atmosphere.  In 1963, there was a treaty that banned such tests, and since then the atmospheric level of 14C has declined because of uptake by plants.  This uptake takes place as 14C in the atmosphere reacts with oxygen to make CO2, which is then taken up by plants in photosynthesis. 

When we eat plants, or eat animals that feed on plants, the 14C is transferred to our body.  Once transferred to our body, 14C becomes part of the DNA of new born cells.  This happens when a cell divides and makes a copy of its chromosomes.  The copying process integrates the 14C into the newly made genome, making it so that by looking at the concentration of 14C in a cell’s DNA, and comparing it to the atmospheric DNA, one can tell when that cell was born. 

Kristy Spalding, Jonas Frisen, and their colleagues used this idea to find the birthday of neurons in the human brain.  In their study, they examined brains of people who had died between 2000 and 2012.  These people had had their brains preserved during autopsy, and so their brain could be studied. 

They focused their efforts on the neurons in the hippocampus region of the brain, a location that is critical for formation of new memories.  The hippocampus is the place in our brain where we form autobiographical memories, i.e., the kind of memories that describe places and people that we have met, events that have taken place in our life, etc.  Spalding and colleagues asked, how old are the neurons in the hippocampus of a person who was 30 years old when she died?  You might guess, well, the neuron is probably close to 30 years old.  But that assumes that all neurons are born soon after birth.  Strikingly, Spalding and colleagues found that the neurons were much younger than the person.

Neurons are much younger than the age of the person

The authors found that for a 20 year old, the average age of neurons in the hippocampus was 18.  For a 40 year old, the average age was 29.  For a 60 year old the average age was 37.  Remarkably, for an 80 year old, the average age of hippocampal neurons was 40!  

So the average neuron in the hippocampus of an 80 year old has been around only long enough to experience the last 40 years.  It cannot ‘remember’ anything from the first half of life, because it was not around to experience it.

Therefore, there is substantial neurogenesis throughout life in the hippocampus.  In fact, the rate of neurogenesis showed only a modest decline with aging.  They estimated that each day, 0.004% of the neurons in the dentate gyrus of the hippocampus die and are replaced with new ones.

Now it is possible that neurogenesis in the hippocampus is especially high, and other parts of the cerebral cortex may not have such a high turn-over.  But the relative youth of the neurons in the hippocampus raises a fundamental question:  what is memory if neurons are eliminated and replaced on a daily basis?

Richard Feynman, the celebrated physicist, during a lecture in 1955 to the National Academy of Sciences, described the basic problem:

“The radioactive phosphorus content of the cerebrum of the rat decreases to one half in a period of two weeks.  Now what does that mean?  It means that phosphorus that is in the brain of the rat, and also in mine, and yours, is not the same phosphorous as it was two weeks ago.  It means the atoms that are in the brain are being replaced: the ones that were there before have gone away.  So what is this mind of ours: what are these atoms with consciousness?  Last week’s potatoes! They now can remember what was going on in my mind a year ago, a mind which has long ago been replaced.  To note that the thing I call my individuality is only a pattern or a dance… The atoms come into my brain, dance a dance, and then go out--- there are always new atoms, but always doing the same dance, remembering what the dance was yesterday.”

The problem in neuroscience is to understand how to read this dance.  If we could, then in principle it should be possible to record and preserve our experiences, so that when we die, the library will remain standing.

References

Spalding KL et al. (2013) Dynamics of hippocampus neurogenesis in adult humans.  Cell 153:1219-1227.

Feynman RP (1988) What do you care what other people think? Further adventures of a curious character.  Bantam Books, page 244.

Rescuing the brain after a stroke

Neurons in the brain, like any other cell in our body, require oxygen to live.  They get this oxygen from the blood vessels that run nearby.  When there is a stroke, the cause is often a particle that has gotten stuck in a branch of an artery, blocking the flow of blood, producing ischemia. This loss of oxygen starts a cascade of events that culminate in the death of the neurons that live nearby.   In the last 3 years, there have been a couple of remarkable papers from a small laboratory in University of California Irvine that suggest a new and non-invasive way to fight this plumbing problem.

The connection between neurons and blood vessels

When someone touches your arm, the neurons in the arm area of your somatosensory cortex become highly active, producing what are called action potentials.  Action potentials are the only mechanism that neurons have to communicate with each other.  Generating an action potential requires energy, and this energy is supplied via the nutrients and oxygen that are carried by nearby blood vessels.  When neurons generate action potentials, support cells that monitor the neurons send signals to the cells that line the blood vessels, causing the vessels to locally enlarge.  This enlargement produces an increase in the blood volume and arrival of a greater amount of food and oxygen.   Indeed, this fact is the basis of a form of functional magnetic resonance imaging (fMRI) in which blood oxygenation levels are imaged and act as a proxy for activity in the nearby neurons.  So the neurons are in close contact with the vessels, and the vessels are the gardeners that provide the neurons with nutrients precisely when they need it.

Activating neurons in the hour after a stroke

When a blood vessel is blocked, the cells in the vicinity are deprived of their oxygen.  But blood vessels are not like branches on a tree where there is only one way to get to a spot.  Rather, they are a little like the highway system: there are multiple ways to get to a spot.  This is important because blocking a branch of an artery need not be catastrophic if a healthy branch could enlarge and supply some of the nutrients that are needed by the cells near the blocked branch of the artery.  But how can this be done?

In 2010, Christopher Lay and colleagues at University of California Irvine reported the results of an experiment that did just that, find a simple way to alert the healthy blood vessels to compensate for the blocked one.  In Lay et al. (2010), the authors first took a group of rats, anesthetized them, and then gave them a stroke in the base of the proximal middle cerebral artery (MCA).  They did this by tying a suture around a branch of MCA that supplies blood to the area of the rat’s somatosensory cortex which encodes sensory information from its whiskers.  This stopped the blood flow to that region, causing ischemia, and produced brain damage (called an infarct).  The next day, the rats were impaired in their ability to use their whiskers, and the somatosensory cortex showed clear signs of neural damage. 

They next took another group of rats and also gave them an MCA stroke, but rather than just letting them lie there, during the hour after the stroke they kept touching and moving their whisker (1sec of 5Hz deflections of a single whisker, once every 20 seconds).  Twenty four hours after the stroke, they tested the stimulated rats and found that the damage to the neural tissue was much less than in the non-stimulated rats.  Behavior, imaging, and neurophysiological investigation of the stimulated rats showed that by all measures touching the whisker seemed to have made a very significant difference.  

This positive effect happened only if the whisker was touched in the one hour or so after the stroke.  If the same touching was done at 3 hours, the effect was to worsen the stroke.  So there was a critical one hour time window after a stroke in which touching the body part (and presumably activating the neurons that reside in the affected cortex) seemed to dramatically reduce the damage normally caused by the stroke.  Stimulating the neurons in the stroke affected region seemed to provide them with a pathway to survival.

How could this have happened?  Further testing showed that blood reperfusion to the affected tissue was established via collateral flow from distal branches of the MCA (Lay et al. 2010). This reperfusion started at stimulation onset, and then grew gradually, reaching near normal levels at around 1.5 hours (Lay et al. 2011).  The reperfusion was absent in the non-stimulated animals.  It is possible that stimulating the whiskers immediately after the stroke had signaled a much larger blood vessel network than the nearby, blocked vessel.  In a control experiment, if a larger network of vessels was also blocked, then the stimulation made no difference.

One problem with these studies is that the rats were fairly young (in human terms, in their 20s).  People at that age do not usually have a stroke, and the brain is generally more plastic and forgiving at an early age.  So Lay and colleagues repeated their experiment in elderly rats, equivalent to around 60 year old humans (Lay et al. 2012).  They found that the stimulated elderly rats suffered an infarct that was much smaller than their control rats.  Stimulation was effective in the elderly as well as young.

Another problem with these studies is that the rats were anesthetized during the stroke and during the stimulation.  Of course, people are usually awake when they have a stroke.  Did the anesthesia play a critical role in the unusual success of the stimulation?  In a further study, the authors tried a new anesthetic that allowed them to occlude the MCA under anesthesia, but once that surgical procedure was completed and anesthesia removed, the animal could return to an awake state within minutes (Lay et al. 2013).  During this awake state they stimulated the whiskers and found recovery data similar to their previous results on deeply anesthetized animals.  The stimulation, and not the anesthesia, seemed to be the key factor.

These results are all from one laboratory, and need to be confirmed by other labs.  However, the results are tantalizing, as they suggest a stimulation based, non-invasive strategy during a critical period after stroke that may rescue the brain.

References

Lay, C. C., Davis, M. F., Chen-Bee, C. H., & Frostig, R. D. (2010). Mild sensory stimulation completely protects the adult rodent cortex from ischemic stroke. PloS one, 5(6), e11270.

Lay, C. C., Davis, M. F., Chen-Bee, C. H., & Frostig, R. D. (2011). Mild sensory stimulation reestablishes cortical function during the acute phase of ischemia. The Journal of Neuroscience, 31(32), 11495-11504.

Lay, C. C., Jacobs, N., Hancock, A. M., Zhou, Y., & Frostig, R. D. (2013). Early stimulation treatment provides complete sensory‐induced protection from ischemic stroke under isoflurane anesthesia. European Journal of Neuroscience, in press.

Boots

In the movie "Promised Land", there is a scene in which the main character, a corporate type, has been sent from New York to a small farming town to lease the land for gas drilling (played by Matt Damon).  After he arrives in town, he puts on his boots to go visit a farmer and make his proposition.  The boots are old, and not much to look at, but they are something that he has owned since his early days in a small town in Iowa.  In more ways than one, the boots are the only authentic thing about him.

                                                                 

It made me think of my old boots.  When I was 14, my first winter in America was approaching, and my American mom took me to the store to get some winter boots.  She, being a gentle soul, let me choose the one that I liked.  When my American dad saw them, he said that they were fine work boots, but they had no insulation and would not really do for the winter.  They were about ready to take them back, but I hemmed and hawed and said that I liked them and would be fine with them in the snow.  And so they let me keep them.

And indeed we were fine together.  With some bees wax, I made them waterproof and then took them on my little adventures: a week long hiking trip through the North Cascade Wilderness (where a bear tried to tear down the bag of food that we had hung from a tree), bucking bales on a hay farm (where I lasted only a single day, as the bales were 60+ pounds and I weighed barely twice as much), and oh so many fishing trips (where despite the loads of fish, we fried hot dogs for dinner).

This afternoon, after coming home from lab, I went down to the basement, pulled the boots out of the shoe rack, and put them on to do some lawn edging.  They are as comfortable as a pair of slippers, though aged a bit.  They are the oldest thing that I own, and still use.

A most unusual paper

In 1974, the Journal of Applied Behavior Analysis published a most unusual manuscript.  The journal received the manuscript on 25 October 1973, and published it without revision.  The manuscript contained not a single word of text, except for the title, name of the author, his affiliation, the subtitle "References", and a brief acknowledgement.  

There were no equations, no figures, and no references.  Essentially, the manuscript was a blank page, authored by Dennis Upper of Veteran's Administration Hospital of Brockton, Massachusetts.  

When the manuscript was published, the journal also published a reviewer's comments.  Here is what the reviewer had to say: 

"I have studied this manuscript very carefully with lemon juice and X-rays and have not detected a single flaw in either design or writing style.  I suggest it be published without revision.  Clearly it is the most concise manuscript I have ever seen --- yet is contains sufficient detail to allow other investigators to replicate Dr. Upper's failure.  In comparison with the other manuscripts I get from you containing all that complicated detail, this one was a pleasure to examine.  Surely we can find a place for this paper in the Journal --- perhaps on the edge of a blank page."


The paper was titled: "The unsuccessful self-treatment of a case of writer's block".  Since publication, it has been cited 29 times.

D Upper (1974) The unsuccessful self-treatment of a case of writer's block.  Journal of Applied Behavior Analysis 7(3):497.

A brief history of imaginary numbers

Some years back I sat with my son as he did his math homework.  Looking over his shoulder, I saw that he was working on the function 

After he finished plotting it, I thought, let’s try another one, 

I began by plotting the function for various integers of x and got the black dots in the graph below.  Then, naively, I connected the dots:

Looking at the plot, I realized that this could not possibly be right.  How could this function cross the x-axis?  That would mean that there are some values of x for which f(x) = 0.  But there were no such values.  After all, you could not raise a number to a power and get zero.  So what’s going on?

What is going on is that our function 

spends most of its time outside of my piece of paper, in an ‘imaginary’ world that lies above and below the plane of my paper.  The points that I had plotted in the figure (the black dots) are the points that I can see when this function crosses the plane of the paper.  The rest of the time the function is outside my plane.  Here is what our function really looks like:

In the above figure, the plane of the paper is colored green.  Our function is like a cork-screw, winding itself around the x-axis.  I wondered, how did humans discover that in addition to the “real” world (the plane of my paper), there must also exist an “imaginary” world?  What was the origin of the idea of imaginary numbers?

In a wonderful little book called “An imaginary tale: the story of square root of -1", Paul Nahin recounts the journey.  Surprisingly, the discovery has little to do with quadratic equations, and everything to do with cubics.

Roots of equations

In the 16^th century (and for a century or two after that), mathematicians were very much concerned with geometric meaning of equations.  So if you asked one what is the root of the following equation

he or she would think about it in terms of the function 

and ask where this function crosses the x-axis.  Here is what this function looks like:

Our quadratic function never crosses the x-axis, and so our 16^th century mathematician would respond by saying that the equation

 

is impossible because

 

never touches the x-axis.  That would be the end of the conversation.  Indeed, as Nahin explains, this is why the origin of imaginary numbers did not start with quadratic equations.  Rather, “impossible” numbers like

had their origin in cubic equations.

Depressed cubics

Scipione del Ferro was a 16^th century Italian mathematician working on cubic equations of the form:

(1)  

These are called depressed cubics because they are missing the quadratic x term.  His objective was to find the roots of this equation, which translates into finding the value or values of x for which this equation is true.  This means finding the value of x for which the function

 

crosses the x-axis.  A cubic function will always have at least one location at which it will cross the x-axis, so del Ferro knew that there must exist at least one value of x for which this equation is true.

He started by assuming that the solution could be written as the sum of two number, u and v: 

If we put this into our cubic equation we get:

(2)  

Expanding it we have:
(3)
We can pick u and v arbitrarily (as long as x = u + v), and so del Ferro picked u and v such that

This implies that the second term in Eq. (3) is zero, and so we have:

(4)  

To solve the above equation set

 

and so we have

Del Ferro knew how to solve quadratic equations.  We have

 

which means that:

(5)  

From Eq. (4) we had 

and so

(6)  

The way to understand Eqs. (5) and (6) is as follows: u can take on two values, one given by the plus term, and the other given by the minus term.  When u is given by the plus term, v is given by the minus term, and so on.  Now if we write the solution x = u + v, we end up with the expression:

(8)   

When p and q are positive the right side of Eq. (8) will become the third root of a negative number, which can be uncomfortable to deal with, and so let us re-write it by noting that

Using this we can re-write Eq. (8) as:

(9)  

del Ferro had found a solution to a cubic, something that had eluded man for 2000 years, ever since Babylonian times.  

This was a remarkable achievement indeed.  However, del Ferro knew that in his equation lied a deep mystery: when p and q were both positive his equation gave the correct answer, but when one or the other was negative, his equation gave an impossible answer.  He did not know why this formula seemed to fail in some cases.  This, it turns out, is the key mystery that led to discovery of imaginary numbers.

The impossible equation        

Consider the cubic equation

When we plot this equation, we have:  

The equation crosses the x-axis at x=2, and so 2 is one of the roots of this equation (indeed, 2 is the only real root).  Using del Ferro’s formula (Eq. 9) and a calculator we find that the rather hairy calculation produces an answer that is, remarkably, exactly 2.  So far so good.

Next, let us try the cubic

When we plot this equation, we have:  

We see that x=4 is a solution.  In fact, our cubic crosses the x-axis three times, and one of those times is at x=4 (this cubic has three real solutions).  But now let us try del Ferro’s formula.  From Eq. (8) we have:

(10)  
But if del Ferro’s formula is correct, then the following must be true:
(11)  

And so we arrive at the mystery: we know that x=4 is a solution to this cubic, and we know that del Ferro’s formula is correct.  Yet, when we use it, we get what appears to be an impossible equation (Eq. 11): we have two instances of a square root of a negative number, which at del Ferro’s time were thought to be meaningless, and yet when these two numbers add, they produce a real number!  How could that be?

It took another 50 years of thinking, and the result was a book entitled Algebra (1572), by Rafael Bombelli, a mathematician that received no college education.  He was the first to see that Eq. (11) required existence of a whole new set of numbers, called imaginary numbers.

He proposed that perhaps Eq. (11) is true because each of the third roots produce something that is partly real, and partly imaginary, and the sum causes the two imaginary parts to cancel, leaving only a real part.  That is, he proposed that:

(12)  

We proceed by cubing the two sides of Eq. (12):

 (13)  

To solve for a and b, we set:

 (14)  

And we find that the solution is a = 2, and b = 1.  So Bombelli showed that: 

 (15)  

And therefore Eq. (11) is true because the imaginary parts of the third roots cancel, leaving a real number. 

The origin of imaginary numbers was in cubic equations.  These equations always have at least one real root, clearly crossing the x-axis, yet del Ferro’s equation that was supposed to give that root instead gave an expression that included square root of negative numbers.  Bombelli showed that those “impossible number” were things that could be handled by introduction of what we now call imaginary numbers.  For that accomplishment, there is a crater on the moon named after Rafael Bombelli. 

Painful memories, and effortful actions

How does the brain evaluate a painful episode?  When you look back at an unpleasant episode of your life, how does your impression of it now relate to the actual experience that you had during the episode?

Surprisingly, when we recall a painful experience we seem not to evaluate it based on its duration, or its temporal integral, or its mean pain.  That is, it does not matter very much if one experience was on average more painful than another, nor does it matter that one experience was longer than another.  Rather, we seem to evaluate the totality of a painful experience using two factors: magnitude of the peak of the pain, and the magnitude of the pain as the episode ended.  Here, I will describe the basic experiments that led to these ideas, and then suggest a new interpretation of rather puzzling results regarding how the brain evaluates effort in simple motor control tasks.

Cold water bath

In 1993, Kahneman and colleagues asked 32 volunteers at University of California Berkeley to put both their hands in a cold water bath for 5 seconds.  Next, one hand was chosen at random and placed in cold water for 60 seconds (or 90).  After a brief rest period, the other hand was placed in cold water for 90 seconds (or 60).  In these two episodes the temperature of the water was the same for the first 60 seconds (21 degrees Centigrade).  However, in the last 30 seconds of the 90 second episode, the temperature was increased by 1.1 deg.  So in the 90 second episode one hand always experienced a longer period of discomfort, but the episode for that hand ended with slightly warmer water. 

During the time that their hand was in water the subjects used their other hand to adjust a knob to continuously indicate their discomfort.  As you would expect, the discomfort increased immediately as the hand was placed in the cold water, reached a peak at around 60 seconds, and then declined for the next 30 seconds. 

After the two episodes were completed, the subjects were told that they would need to put their hand in cold water one more time but that they could choose which episode they wanted.  The main dependent variable was the subject’s choice for this third episode.  Logically, no one should pick the episode that lasted 90 seconds.  But remarkably, most subjects (22 of 32, 69%) preferred to repeat the longer episode.  Indeed, most subjects indicated that the longer episode had caused less overall discomfort!

This suggested that when people evaluate painful episodes, what matters is not the duration, but rather the magnitude of the pain as the episode ended.  However, a potential confound with the cold water experiment is that we know that memory fades with time, and so perhaps evaluating the pain of an episode relies more on the ending because the memory of the early parts have faded.  Perhaps if the subjects were asked to remember the episode a few days later, they would not recall it the same way as a few minutes after the end of the episode.  Was this temporal decay the reason for the seemingly illogical choice?  To test for this, Kahneman and colleagues performed a new experiment.

The perceived pain of a medical procedure

Redelmeier and Kahneman (1996) asked patients that were undergoing colonoscopy (n=154) or lithotripsy (a procedure to destroy hardened masses, n=133) to give assessment of their pain by pointing to a scale at one minute intervals.  The colonoscopy lasted from 4-67 minutes, and the lithotripsy lasted from 18-51 min.  One hour after the procedure the patients were asked to judge the total amount of pain experienced using the same scale. 

To check for reliability of the evaluations, some of the patients were asked to recall the experience 6 months (colonoscopy) or 1 year (lithotripsy) later and again evaluate the total pain.  The retrospective ratings at 6 months and 1 year were correlated at r=0.77 and r=0.54 for the two groups.  For the colonoscopy group the ratings at 6 months had the same mean as at 1 hour, for the lithotripsy group the average ratings at 1 year were 15% higher than at 1 hour. 

In the colonoscopy procedure the pain intensity was higher at start than at end, whereas in the lithotripsy procedure pain intensity was low in the first few minutes and ended higher. 

Having collected these data, the investigators asked what aspect of the painful experience was a predictor of the immediate ratings at 1 hour, or the follow-up ratings at 6 months or 1 year.  Duration of the procedure was not a predictor of the immediate or follow-up ratings.  Rather, peak pain was the most powerful predictor of both ratings (r=0.6 for each), and end pain was the second most powerful predictor (r=0.4 for each).  These correlations held for both of the procedures.  The combination of the two factors increased the correlations to about 0.67 and 0.65 for immediate and follow-up ratings.

So people’s impression of the relative pain they endured during an episode remained fairly consistent at 1 hour and at many months after the episode.  Their impressions were predicted by two aspects of their actual experience: magnitude of the peak of the pain, and magnitude of the end pain.  Duration of the episode played little or no role.  

When we remember a painful episode, the most salient aspects of that episode seem to be the peak of the pain, and how it ended.  To improve our perception of a difficult episode, it may be more beneficial to prolong it and gradually reduce the pain, rather than shorten it and abruptly end the pain. 

Perception of effort

This idea of peak-end perception of pain may help us understand a rather puzzling result in the field of motor control.  One of the fundamental questions in motor control is how the brain evaluates effort.  The variables of interest are force and time, and the question is with regard to our perception of effort as a function of these variables.

In 2004, +Konrad Kording, +Daniel Wolpert and colleagues performed an experiment in which volunteers held a robotic arm and experienced a sinusoidal-like force profile of peak F and duration T.   Next, they experienced another force pattern of peak F’ and duration T’.  They then asked their volunteers which force they would like to experience again.  They were told that they should choose the force that required the least effort.  In this way, the investigators estimated indifference curves, i.e., curves along which the subjects were indifferent to changes in peak force and duration.

The rather unexpected result was that as the duration of a force pattern increased (beyond about 200ms), the indifference curve also increased.  This means that given a choice between some peak force and short duration, vs. the same peak force and longer duration, the subjects picked the longer duration!  

How could a longer duration of an effortful task be preferable to a shorter duration?

A close look at how the force patterns were produced provides a possible answer.  The forces were sinusoidal with a period that depended on T.  So as the duration increased, the rate at which the force changed decreased.  This means that for a longer duration force, the forces gradually came to an end, whereas for a short duration force, the forces rapidly came to an end.  People preferred the gradually ending force, despite the fact that they would be producing the forces for a longer amount of time.

The peak-end hypothesis of pain perception may have relevance to how the brain measures effort.

Acknowledgements: I am grateful to +Alaa Ahmed of University of Colorado for discussions regarding these ideas.

References

Kahneman D, Fredrickson BL, Schreiber CA, and Redelmeier DA (1993) When more pain is preferred to less: adding a better end. Psychological Science 4:401-405.

Kording KP, Fukinaga I, Howard IS, Ingram JN, and Wolpert DM (2004) A neuroeconomic approach to inferring utility functions in sensorimotor control.  PLoS Biology 2:e330.

Redelmeier DA, and Kahneman D (1996) Patients’ memories of painful medical treatments: real-time and retrospective evaluations of two minimally invasive procedures.  Pain 66:3-8.