快猫短视频

JIM Collins didn鈥檛 enjoy trying to push old people over, but someone had to
do it. He鈥檚 been looking for a way to work out how steady people are on their
feet, so that he can pick out those who are at risk of falling before they
injure themselves. Collins isn鈥檛 a doctor, or even a physiotherapist. He鈥檚 a
physicist, and now he and others have come up with a mathematical model that
describes how our muscles keep our ever-moving skeleton upright.

It鈥檚 proved a tough problem. To tackle it, the physicists have drawn on
mathematical methods that were developed to probe mysteries such as the
structure of the Universe, turbulence in fluids and the science of combustion.
But it鈥檚 been worth the effort because the practical benefits of this work could
be immense. They include helping missions into space, and testing whether people
are fit to control delicate or dangerous machinery. Most importantly of all, it
could help keep frail old people on their feet, and save the billions of dollars
that healthcare systems spend every year on patching people up after they fall
over.

The great leap forward came when Collins, a researcher in the Boston
University Center for BioDynamics, decided that simply pushing people over was
never going to reveal the subtle dynamics of balance. So with his colleague
Carlo De Luca he decided to devise a gentler approach. Rather than giving their
subjects a shove to see how they recover, the researchers did precisely the
opposite. They told them to try to stand still, and measured how the centre of
pressure of their feet鈥攖he points on the soles where the body鈥檚 weight is
supported鈥攎oves as they stand on an instrument-laden platform. Collins
calls it a 鈥渟ouped-up bathroom scale鈥�, and it can measure the vertical,
horizontal and rotational forces applied by your feet.

The first thing the researchers noted was that everyone sways. It doesn鈥檛
matter whether you are feeble or fit, the fact is that you have never stood
stock still in your life. Your skeleton is like a teetering column of pencils
balanced on a finger-tip. What鈥檚 more, it鈥檚 continually pulled off-balance
whenever your muscles twitch, when you inhale or exhale, as your heart beats, or
even if you are brushed by a gentle breeze. No wonder Collins and De Luca saw
the foot鈥檚 centre of pressure wriggling around like a demented eel.

But there was something familiar about the path it traced out. As soon as
Collins saw it, he realised that it was just like Brownian motion, the random
walk traced out by a dust mote battered by the molecules in air. The similarity
was so strong that the researchers decided to analyse their standing data by
borrowing the techniques used in the study of Brownian motion: statistical
mechanics.

This yielded a set of complex graphs showing how people鈥檚 balance shifts as
they attempt to stand still. Collins has identified three distinct things going
on from these graphs. For a couple of seconds, 鈥渋ndividuals more or less drift,鈥�
Collins says. Then, every few seconds, there is a corrective response. From the
graph, Collins could see that something was sporadically responding to the
random drift. He believes this comes from a reflex response: as we drift, our
muscle tension changes, and the reflex attempts to change it back. It isn鈥檛 a
very precise mechanism, however. 鈥淭he point that this correction mechanism
brings you back to also has a random drift,鈥� Collins says. The graph revealed
that a more accurate method of staying upright kicks in every now and
then鈥攑robably our response to visual cues or our balance organs. This
pulls us back into a fully upright position. And then we start to drift
again.

This process of drift, slight correction and full correction never stops. As
a result, we are constantly shifting our posture and standing in new ways: your
body explores many different stable positions while standing still. This might
all sound rather uncontrolled, but it really is how perfectly healthy people
keep themselves upright. 鈥淚t鈥檚 a characteristic of the everyday operation of the
system,鈥� Collins says. He believes that it simplifies the sensory information
and physical effort needed to remain upright.

His analysis of how people stand still鈥攖he researchers call it 鈥渜uiet
standing鈥濃�攖urns out to be a good way of characterising the balance of a
wide range of subjects. Collins and his team studied astronauts before and after
shuttle flights, and people with balance disorders. They were able to discern
features on the graphs that were caused by physiological abnormalities of their
subjects, such as stiff joints or an impaired balance system. That was a good
start, and allowed Collins to recognise patterns that could sound the alarm. But
he wanted to go a step further and be able to predict what the graph will look
like for someone with a particular physical problem, or if they are given a
gentle nudge. And for that, he needed a proper physical model.

Fortunately, Carson Chow, a physicist working at the University of Colorado,
Boulder, was looking for a new challenge. 鈥淚 wanted to go into biology,鈥� he
recalls. He heard about Collins鈥檚 postural research, and offered his expertise
in statistical mechanics. Soon afterwards, he was settled in Boston, working on
finding ways to explain the details of the random walk. 鈥淚 really didn鈥檛 have
any opinion about it,鈥� Chow says. 鈥淭o me it was just another problem.鈥�

But Chow quickly became engrossed. When he looked at the graphs he was struck
by the details of drift and correction he saw鈥攊t looked similar to
something he had seen in his previous work on statistical mechanics. A point on
an infinitely long string that鈥檚 being buffeted by random blows moves in
precisely the same way. So Chow suggested modelling the human body as an
infinitely long string鈥攁n object with an infinite number of joints. The
number of joints in the body is not infinite, but it鈥檚 pretty large and for a
physicist this leap from a lot to infinite is not so very far.

You can visualise Chow鈥檚 model as a strand of spaghetti immersed in a hot,
viscous liquid. The thermal jiggling of the molecules in the liquid batters each
point in the spaghetti strand in the same way that it would toss a suspended
dust particle into Brownian motion. This models the action of the twitching
muscles that make our posture both drift and return. Chow also added one more
constraint: a piece of elastic attached to the string that stops it moving too
far.

Chow is the first to admit that his model is a drastic simplification of a
standing human. But it has one big thing going for it. The equations are
solvable, and the solutions work. Using standard mathematical tools, Chow is
able to accurately reproduce Collins鈥檚 empirical data. And because he can do
things like vary the string鈥檚 stiffness, his model does what Collins was aiming
for: it can simulate the standing patterns that might be produced by patients
with problems such as seized joints. Comparing patterns generated by different
versions of the model with patterns from patients provides a useful means of
diagnosis. So far, Collins has done tests on patients suffering from Parkinson鈥檚
disease as well as elderly patients who are prone to falling.

Even better, Chow had also discovered a way to predict a person鈥檚 response to
a push or a knock. A mathematical trick called the fluctuation-dissipation
theorem can take the small, natural fluctuations of a system and use them to
predict its response to a bigger push. So, after standing a patient on Collins鈥檚
souped-up scale for a few minutes, and applying the fluctuation-dissipation
theorem to the data they collect, the researchers can tell how that person will
fare if they suffer a knock or a dizzy spell. For frail patients, it鈥檚 a more
acceptable way of assessing balance than giving them a shove.

Collins is working with Lewis Lipsitz of the Harvard Medical School in
Boston, carrying out a clinical study on the technique. They believe that this
will bring real benefits by identifying older people who are prone to falling.
鈥淔alling, among the elderly in particular, is a major problem in the US and
around the world,鈥� Collins says. It can lead to severe injury requiring
hospitalisation, or even death. Once identified, these individuals could improve
their balance through physiotherapy or t鈥檃i chi. Those at most risk could also
be provided with padded clothing.

But Chow鈥檚 model, successful as it is at predicting when someone is prone to
losing their balance, can鈥檛 actually model what happens when they fall. The task
of going that one step further has been taken on by David Hochberg, a
theoretical physicist working at the European Space Agency鈥檚 Laboratory for
Space Astrophysics and Fundamental Physics in Spain. Hochberg鈥檚 previous work
had been involved with such esoterica as wormholes and elementary particles, but
like Chow he鈥檚 always on the lookout for a new challenge. One of his interests
is complex physical systems that are driven by noise, such as the behaviour of
the stock market, the boiling of water or the movements of a flame. Like human
posture, these systems are hard to describe exactly with solvable mathematical
equations.

Hochberg was looking for a problem to help one of his postdoc researchers,
Francisco Alonso-S谩nchez, become familiar with a mathematical technique
called the renormalisation group. This technique can often do a wonderful job of
simplifying the analysis of complex systems. Browsing the public collection of
preprints on the Los Alamos website, he came across Collins and Chow鈥檚 paper. It
was just the sort of thing he was after to help Alonso-S谩nchez learn.
鈥淏ut as one thing led to another we found that the problem was even more
interesting than I thought,鈥� Hochberg says.

Starting with Collins and Chow鈥檚 linear equations that model people swaying,
Hochberg and Alonso-S谩nchez tweaked them the simplest way they could
think of that would enable the swaying spaghetti to fall over. 鈥淲e added in a
little non-linear term,鈥� Hochberg says. This brings in the idea that, beyond a
certain point in the swaying, small additional movements could have large
effects on posture. It was also designed to take into account the body鈥檚
asymmetry: the effects of rocking backwards are different to the effects of
rocking the same distance forwards. 鈥淭he rest of it is just battering this model
over the head with the renormalisation group,鈥� Hochberg says.

This small change makes the spaghetti equation turn nasty, and it can鈥檛 be
solved exactly. But Hochberg鈥檚 skill with the renormalisation group avoids the
need for detailed, messy analysis and allows the researchers to identify the
conditions under which a 鈥減hase transition鈥� occurs. That might mark the change
from water into steam or, as in this case, from standing upright to toppling
over. Hochberg and Alonso-S谩nchez鈥檚 model can analyse such everyday
problems as stumbling or tripping while running for a bus. 鈥淕iven that we do see
relatively large perturbations in real life, I think this is a pretty
significant development,鈥� Collins says.

Alonso-S谩nchez and Hochberg have already thought up ways to put their
technique to practical use. They are working on the design of a new posture
measurement tool that would measure the body at several different points. This
would allow them to construct a more accurate model of standing that could take
into account the motion of all of the body鈥檚 joints. They are also looking at
how their equations could be applied to building robots that can stand and walk
without toppling over.

Another idea is to modify their model for low-gravity regimes, to predict how
astronauts鈥� posture might respond to the microgravity of a trip to Mars, and
what walking might be like when they get there. As yet, however, they are still
working out how to include gravity in the equations.

Alonso-S谩nchez is also looking at the possibility of using his
equations to test people to see if they are fit to operate dangerous machinery
or fly an airliner. He hopes to use a device like Collins鈥檚 super-scale to
record tiny postural sways, and then translate the results into a test that can
tell if an individual is too tired, drunk or affected by drugs or disease to
work safely.

Standing still seems mundane, yet it has proved to be remarkably complicated.
Only the most sophisticated tools of modern mathematical physics are proving up
to the task of understanding it . But now that the science of standing is on a
firm mathematical footing, it鈥檚 really going places.