Health of candidates
Dr. Jay Maron, physicist
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Head and eye stability

You can judge the health of a brain by observing the stability of the head and eyes. The more stable they are, the better you can think. If you watch a Shaolin monk you will see that their heads and eyes are always under control. What follows is a physics discussion of the geometry of the neck vertebrae, the brain, the ears, and the eyes, with the purpose of providing the tools required to assess mental health.

Atlas vertebra

Axis vertebra

Head motion

The atlas and axis vertebra functions like a telescope altazimuth mount.

Altazimuth telescope mount
Keck telescope altazimuth mount


The Atlas-Skull joint controls pitch and the Axis-Atlas joint controls yaw. Roll is controlled collectively by all the neck vertebrae.

The atlas vertebra is at the center of the head and your eyes and ears are at the same level as the atlas vertebra.

Suboccipital muscles
Suboccipital muscles

Head balance

The center of mass of the skull is slightly forward of the contact point between the skull and the atlas vertebra. If the muscles in your neck were to relax then your head would pitch forward. If your spine muscles were to relax then your torso would pitch forward. The muscles in the back of your neck and spine act reflexively to prevent you from falling forward. This motion is coordinated with the breathing cycle.

The spine needs gravity to calibrate its curvature. If you spend more than a month in weightlessness then the function of your spine degrades.

Breathing cycle

Video of the breathing cycle

Bruce Lee: When the opponent expands I contract, when he contracts I expand, and when there is an opportunity, I do not hit, it hits all by itself.

Breathing is coordinated with skeletal motion to minimize energy expenditure. Motion cycles between the following two columns.

Inhale                     Exhale
Diaphram contracts         Diaphragm expands
Abdominals expand          Abdominals contract
Gut squashed by diaphram   Gut expands
External intercostals      Internal intercostals
Ribcage expands            Ribcage contracts
Pelvic floor expands       Pelvic floor contracts
Spine muscles contract     Spine muscles release
Head rises                 Head descends
Arms out                   Arms in
Arms rotate thumbs up      Arms rotate thumbs down
Elbows rotate out          Elbows rotate in
Open hand                  Form fist
Feet rotate outward        Feet rotate inward
Knees apart                Knees together
Lower back arches          Lower back sags
Hips rotate forward        Hips rotate back
Daydream                   Focus
Rebalance                  Exertion
High moment of inertia     Low moment of inertia
Discard angular momentum   Discard pressure
Tongue makes "U" shape     Tongue makes flat shape, such as for the letter "L"
Spread fingers into a fan  Bring fingers together like a fin
When you are in action, your breathing adjusts to support the timing of your skeletal motion, and if it has any spare time, it sucks in as much air as possible.

When you are relaxing, your breathing adjusts to minimize energy, coordinate cycles, and smooth transitions.

Motions that are bilaterally symmetric tend to have a natural relationship with the breathing cycle.

Total lung volume          =  4 Liters
Tidal volume               =  Volume of air exhaled during passive breathing
                           =  .5 Liters
Residual volume            =  Volume of air in the lungs after forceful exhalation
                           =  1.2 Liters
Inspiratory reserve volume =  Volume of air that can be inhaled beyond the volume
                              associated with passive breating.
The less air in your lungs the less pressure and the greater the stability.

David Blaine: Breathing techniques

Balance is cyclic in nature. You are never perfectly still.

Glen Levy.   At time=3:00 there is a visualization of the weight of his feet on the ground, where you can see the weight oscillate between the left and right foot. When he is on one foot the weight oscillates between the front and back of his foot.

Bird lungs

Air flow in a bird lung is uni-directional, allowing them to extract twice as much oxygen from the air as mamallian lungs.


Phase angle between two waves
Waves in phase
Waves out of phse

If two waves have the same wavelength then we can define a phase angle between them.

Energy loss is minimized if the elements of the breathing cycle are in phase.

If a set of oscillators are mechanically connected so that energy can travel between them, the phases tend to synchronize.

Video: phase locking of a set of metronomes


External intercostals
External intercostals
External intercostals

Internal intercostals
Internal intercostals

Innermost intercostals
Innermost intercostals
Innermost intercostals

The external intercostals expand the ribcage and the internal intercostals contract it.

Youtube: Intercostals

Pelvic diaphragm

The pelvic diaphragm works in opposition to the thoracic diaphragm. When the thoracid diaphragm contracts the pelvic diaphragm expands and vice versa. This moves the gut cyclically up and down.

Walk cycle

Walking motion naturally cycles back and forth between the following two columns.

Left foot down                   Right foot down
Left arm up                      Right arm up
Body rotates right               Body rotates left
Right hand rotates thumbs-up     Right hand rotates thumbs-down
Left hand rotates thumbs-down    Left hand rotates thumbs-up

The walk cycle is independent of the breathing cycle.

Motions that are bilaterally asymmetric tend to have a natural relationship with the walk cycle.

Video:    Ant walk cycle     Spider walk cycle

Insects don't have enough neurons to calculate balance and so they walk with a sequence of alternating tripods.

Axis cycle

Head rotates left                   Head rotates right
Right hand rotates thumbs-up        Right hand rotates thumbs-down
Left hand rotates thumbs-down       Left hand rotates thumbs-up
Right foot rotates right            Right foot rotates left
Left foot rotates right             Left foot rotates left
Right arm rotates out               Right arm rotates in
Left arm rotates in                 Left arm rotates out
Jaw pivots right                    Jaw pivots left
Tongue pivots right                 Tongue pivots left
Eyes pivot right                    Eyes pivot left
Head rolls right                    Head rolls left
Shoulders rotate right              Shoulders rotate left

The axis cycle is independent of the breathing cycle.

Motions that are bilaterally asymmetric tend to have a natural relationship with the axis cycle.

The structure of the axis angle is determined by conservation of angular momentum.

Wrist cycle

Right wrist rotates thumbs up     Right wrist rotates thumbs down
Left wrist rotates thumbs down    Left wrist rotates thumbs up
Head rotates left                 Head rotates right

Quadrupole cycle

Knees out        Knees in
Inhale           Exhale

Energy conservation

A spring contains compression energy if compressed and tension energy if stretched. Your muscles also contain energy when compressed or stretched.

M  =  Mass
V  =  Velocity
g  =  Gravity constant  =  9.8 meters/second^2
H  =  Height
X  =  Displacement of a spring
K  =  Spring constant
F  =  Force on a spring
   =  -K X
Ek =  Kinetic energy        =  .5 M V^2
Eg =  Gravitational energy  =   M g H
Es =  Spring energy         =  .5 K X^2
Et =  Thermal energy
E  =  Total energy
   =  Ek + Eg + Es + Et
Total energy is conserved.

Youtube: Park Min Young.   This video is an example of transforming energy between the forms {kinetic, gravitational, pressure, tension} while minimizing losses. To do this one must move with fluidity. One of the focuses is to stablize the motion of the right hand.


The spine consists of a set of curves that function as shock absorbers.

Cervial vertebrae
Cervical vertebrae

Lumbar vertebrae

Atomic force microscopy

Atomic force microscope
Optics table

The principal technical hurdle to atomic force microscopy is vibration isolation. Multiple layers of isolation are used.


The muscles of the back are continuously connected between the shoulder blades and the hips, to coordinate motion of the limbs.

Ulna and tibia


In your forearm, the ulna is the large bone and the radius is the small bone. The forearm should ideally rotate around the large bone.

The ulna connects to your hand on the pinky side and the radius connects to the thumb side. Your hand should rotate about the pivot point where your ulna connects to your wrist.

In your lower leg the tibia is the large bone and the fibula is the small bone.

The tibia connects to your foot at the big toe side and the fibula connects at the little toe site.

Buddha Palm

In sumo one strikes with an open palm. The force should be channeled through the ulna with minimal tension in the hand.

The ear

The timpanic membrane converts air pressure waves into mechanical motion of the ear bones. The ear bones amplify the signal and transmit it to the stapes bone, which is connected to the oval window of the cochlea. Vibrations in the oval window cause vibrations in the fluid of the cochlea, where they are converted into neural signals and interpreted in the brain.

If a sound wave in air encounters water then 1/30 of the sound energy is transmitted to the water and the rest reflects back into the air. If sound waves were transmitted directly from the air to the fluid of the cochlea then they would suffer this loss. The ear bones function to improve the transmission efficiency from the air to the fluid of the cochlea.

The tympanic membrane has 13 times the area of the oval window, and so the signal is amplified by a factor of around 13.

As pressure waves travel along the cochlea the cochlea narrows. The narrower the cochlea, the higher the frequency range it is sensitive to. Low frequencies are detected at the beginning of the cochlea and high frequencies are detected at the end.

If the sound level is too loud then the muscles of the middle ear shut down the motion of the ear bones. This is the "acoustic reflex" (Wikipedia).

The function of the ear bones was first explained by Helmholtz.

The cochlea


A microphone records sound pressure as a function of time and a seismometer records displacement as a function of time. Your ears don't work anything like this. Your ears function instead detect the frequency spectrum, analogous to a spectrum of light.

Cross-section of the cochlea showing the organ of Corti

There are 20000 hairs arranged along the length of the cochlea, each tuned to a different frequency. Each hair functions as a resonator.

High frequencies are detected at the start of the cochlea and low frequencies are detected

The perceived loudness depends on the duration of the note. For notes less than .2 seconds the loudness is proportional to duration and for notes more than .2 seconds the loudness is independent of duration. This suggests that the cochlea functions like a resonator, because it takes time for a resonance to activate.

If the duration of a note is much longer than 1 second then our attention fades and the note seems less loud.

Our ability to resolve frequency depends on the sharpness of the resonators in the cochlea. The brain provides active feedback to sharpen the resonance and suppress resonances at nearby frequencies.

If there are two sounds with different frequencies, then if the frequencies are too close to each other they will interfere with each other in the cochlea, and if they are far enough apart they can be sensed independently. The frequency width for interference is on the order of a perfect fifth.

Noise tend to obstruct our ability to resolve pitch.

Nerve signals travel both from the cochlea to the brain and from the brain to the cochlea. The brain provides active feedback to refine the function of the cochlea.

There are nerves that travel directly back and forth between your ears for stereo processing.

The semicircular canals of the cochlea are a gyroscope. Rotating your head causes fluid to flow in the canals, which is detected by hair cells. The function of the gyroscope and the function of the auditory system are connected.

The vestibule and the saccule are hardened objects used to detect linear acceleration. WHen you accelerate these objects are displaced, which is detected by hair cells.

Your ears are at the center of your skull, aligned with the pivot point that connects your skull to the top of your spine. The ear is involved in calculating balance.

Basilar membrane

The basilar membrane functions like a harp or piano. It is a strip running the length of the cochlea, narrow at the end closest to the ear and wide at the end farthest from the ear, like a necktie. It is also stiffer the closer it is to the narrow end. The resonant frequency at any particular point along the basilar membrane increases with stiffness and decreases with width, giving it a frequency range that varies from high to low as you traverse from the narrow to the wide end. Siffness is controlled with muscle tension.

The lower the frequency of the wave, the further it propagates along the basilar membrane. High-frequency waves diminish before they get to the wide end.

The fact that low frequencies propagate further along the basilar membrane is analogous to the fact that low-frequency pitches more easily pass through walls than high-frequency pitches. A low-frequency pitch has more time to move the wall for a given sound pressure.

Youtube: Basilar membrane

Helmholtz was the first to characterize the function of the basilar membrane

Absolute pitch

One out of every 10000 people has "absolute pitch", where for example you can tell if a pitch is higher, lower, or equal to 440 Hertz. Everone else has "relative pitch", where pitch ratios can be sensed but not absolute pitch. This suggests that there is no fixed place on the basilar membrane that corresponds to 440 Hertz.

If you don't have absolute pitch it is difficult to acquire it. From Wikipedia: "There are no reported cases of an adult obtaining absolute pitch ability through musical training; adults who possess relative pitch, but who do not already have absolute pitch, can learn pseudo-absolute pitch, and become able to identify notes in a way that superficially resembles absolute pitch. Moreover, training pseudo-absolute pitch requires considerable motivation, time, and effort, and learning is not retained without constant practice and reinforcement."

Frequency and time

The ear is sensitive to both pitch and time. Pitch is measured by position along the basilar membrane and time is measured by differences beween neural signals. For high-frequency pitches we are more sensitive to frequency and for low-frequency pitches we are more sensitive to time.


Fennec fox

The pinna is the outer part of the ear that collects sound and helps in determing its directionality. All human pinna are unique in shape and if the shape were to change it would affect your ability to determine the direction of sound.

A large pinna can amplify sound by 10 to 15 decibels.

Haas effect for echos

Suppose a sound pulse arrives at your ear and an echo arrives a time T later. If T < 30 ms then you don't notice the echo and if T > 30 ms you notice the echo.

The distance a sound wave travels in a time of 30 ms is 10 meters. A concert hall has to be smaller than this to not sound like it has echos.

To do echolocation you have to train your ears to be sensitive to intervals less than 30 ms.

Bats use high frequencies for echolocation because they diffract less than low frequencies and hence give better resolution.

Ear training

Anguy   Keep your elbow high.  You want your back doing the hard labor.
        You're holding.  Never hold.
Arya    What?
Anguy   Your muscles tense up when you hold.  Pull the string back to the center
        of your chin and release.  Never hold.
Arya    But I have to aim.
Anguy   Never aim.
Arya    Never aim?
Anguy   Your eye knows where it wants the arrow to go.  Trust your eye.

Speed of sound

Bruce Lee: Experiments indicate that auditory cues, when occurring close to the athlete, are responded to more quickly than visual ones. Make use of auditory clues together with visual clues, if possible. Remember, however, the focus of attention on general movement produces faster action than focus on hearing or seeing the cue.

Bruce Lee: You hear the bird chirping? If you don't hear the bird you cannot hear your opponent.

Neuron             100
Sound in air       343         At 20 Celsius
Sound in water    1482
Light        300000000

Time in milliseconds:

.000003  Time for light to cross a 10 meter orchestra
    .2   Electric synapse. These synapses are 2-way and they do not amplify signals
    .7   Time for a water pressure wave to travel 1 meter through your body
   2     Chemical synapse. These synapses are 1-way and they can amplify signals
   1     Time for a neural signal to travel 10 cm, the size of a brain
  10     Time for a neural signal to travel from your fingers to your brain
   3     Time for sound to travel 1 meter, the distance to an adjacent musician
   7     Period of a 130 Hertz wave. This is the frequency of a viola C string
  30     Time for sound to travel across a 10 meter orchestra.
  62     Time between notes in "Flight of the Bumblebee"
For an orchestra to have good timing it must use visual cues. Sound isn't fast enough. This is especially true at the rear of the viola section amidst the cacophony of winds and brass.

The Europa Galante uses precise visual cues.

Pressure waves in your muscles deliver information 15 times faster than neurons.

Listen down

When listening to an orchestra one's attention most easily falls on the high-frequency instruments. Practice listening to the low-frequency instruments, especially the cellos and basses. They control the long-term temporal coherence.

Bruce Lee: You hear the bird chirping? If you don't hear the bird you cannot hear your opponent.

If you can't hear the violas you can't hear the chord. Practice listening to the middle note of the chord.

One should also practicing listening to instruments at minimal volume. Loud volume obstructs our ability to resolve pitch.

Listen to silence

The lower the sound intensity, the more sensitive we are to pitch. Practice listening to music at minimal volume.

Listen ahead

Anticipate the pitch in your mind before you play it. The cochlea has active feedback from the brain and this helps harness it.

Develop fast reactions to adjust the pitch to be in tune with the rest of the orchestra.

Frequency sensitivity of the human ear

Frequency   Wavelength
 (Hertz)     (meters)

   20        15        Lower limit of human frequency sensitivity
   41         8.3      Lowest-frequency string on a string bass or bass guitar
   65         2.52     Lowest-frequency string on a cello
  131         2.52     Lowest-frequency string on a viola
  440          .75     The A-string on a violin
  660          .75     The E-string on a violin (highest-frequency string)
20000          .016    Upper limit of human hearing



Visual resolution

Visual acuity is measured by determining the smallest letter that you can resolve and then calculating the angle. 20/20 vision corresponds to an angle of .0015 radians or .086 degrees. For example, if you have 20/20 vision and are reading letters at a distance of 1 meter,

Height of the letter   =  Y  =       =  .0015 meter
Distance to the letter =  X  =       = 1      meter
Resolution angle       =  A  =  Y/X  =  .0015 radians        For small angles, sin(A) ≈ A
To convert the resolution angle into visual acuity or lens strength,
Resolution    Visual   Correcting lens
for letters   acuity     (diopters)

 .0015        20/20          0
 .0030        20/40         -1
 .0060        20/80         -2
 .011         20/150        -3
 .025         20/300        -4
 .030         20/400        -5
 .038         20/500        -6


A lens focuses incoming light onto a single point on the retina. The focal power of a lens depends on its thickness.

Distance from the lens to the target       =  X
Distance from the lens to the focal point  =  L
Lens focal length                          =  F
Lens focal power                           =  D  =  F-1  (diopters)

Lens equation:  F-1  =  X-1 + L-1

If   X ≫ L   then   L ≈ F
We henceforth assume L=F.

The eye uses both the cornea and lens to focus light. The lens focal power can be adjusted by the eye muscles and the cornea focal power is fixed. For the eye,

Distance from lens to retina     =  F  =.0017 meter
Focal power of the lens          =  Dl =  20  diopters
Focal power of cornea            =  Dc =  40  diopters
Focal power of the lens + cornea =  D  =  F-1  =  Dl + Dc  =  60 diopters
Accomodation as a function of age

The "Amplitude of accomodation" is the change in diopters of the lens as it goes from minimum focus to maximum focus. As you age your lenses lose their ability to change shape. The above figure shows the amplitude of accomodation as a function of age, where the "B" curve is the mean and the "A" and "C" curves are one standard deviation below and above the mean.

Video: Eye focus



Nearsightedness is corrected with a diverging lens (negative diopters) and farsightness is corrected with a converging lens (positive diopters). Reading glasses have focusing power of between +1 and +3 diopters. Glasses for nearsightedness typically range from -1 to -6 diopters.


An imperfect lens fails to focus light onto a point. There are various kinds of distortion.

Spherical aberration
Spherical aberration
Spherical aberration
Chromatic aberration

Barrel distortion
Pincushion distortion

Lenses that are radially symmetric tend to perform well at the image center and less well off-center. For barrel and pincushion distortion this can be corrected with software (electronic for a camera and neural for the eye). Optical astigmatism and coma can be corrected with multiple lenses but this isn't an option with the eye. These off-center distortions tend to be unimportant for the eye because the eye only attempts to obtain high resolution at the image center, at the fovea.

If the eye is not radially symmetric the distortion is called "astigmatism", and can be corrected with a compensating lens that is also radially asymmetric. These lenses have the shape of a rugby ball.


In 1855 Helmholtz published the theory of eye focus. When viewing a far object, the circularly arranged ciliary muscle relaxes allowing the lens zonules and suspensory ligaments to pull on the lens, flattening it. The source of the tension is the pressure that the vitreous and aqueous humours exert outwards onto the sclera. When viewing a near object, the ciliary muscles contract (resisting the outward pressure on the sclera) causing the lens zonules to slacken which allows the lens to spring back into a thicker, more convex, form.


Aperture size = Wavelength
Aperture size = 5 * Wavelength
Aperture size = 4 * Wavelength

Laser spot
Intensity as a function of radius for a laser spot

A wave passing through an aperture is diffracted, blurring the image.

W  =  Wavelength of a wave (meters)
D  =  Size of an aperture  (meters)
A  =  Characteristic diffraction angle of a wave passing through the aperture
   ~  W/D   if W << D
   ~  1     if W >= D
If the wavelength is larger than the aperture then the wave is strongly diffracted and energy propagates in all directions. If W/D >> 1 then the pattern approaches a limit.

All waves diffract, including sound and light. Light passing through your pupil is diffracted and this sets the limit of the resolution of the eye. For a person with 20/20 vision,

Wavelength of green light        =  W  = 5.5⋅10-7 meters
Diameter of a human pupil        =  D  =  .005   meters
Characteristic diffraction angle =  A  =  .00011 radians  =  W/D
Resolution for parallel lines          =  .0003  radians
Resolution for letters                 =  .0015  radians
Resolution for faces                   =  .006   radians
20/40 vision corresponds to doubling these angles.

A person with 20/20 vision can distinguish parallel lines that are spaced by an angle of .0003 radians, about 3 times the diffraction limit. Text can be resolved down to an angle of .0015 radians.

If you are looking at a screen that is .2 meters from your eyes, the minimum resolvable pixel size is

Pixel size  =  .0003 * .2  =  .00006 meters  =  .06 mm.
The colossal squid is up to 14 meters long, has eyes up to 27 cm in diameter, and inhabits the ocean at depths of up to 2 km. It has large eyes for their light-gathering power in the dark ocean.

The record for human acuity is 20/8 and for eagles it is 20/2.

PHET simulation on wave diffraction and interference

Depth perception

There are two ways to measure parallax: "without background" and "with background". The presence of a background improves the precision that is possible.

Without background:

With background:

The binocular reflex rotates the eyes so that they converge at the same distance.

Ocular dominance: Two-thirds of the population is right-eye dominant and one-third is left-eye dominant.

Depth can be perceived with parallax, which uses the finite spacing between the eyes.

Depth perception can also be achieved with motion, which requires only one eye.


9-blade iris
Pentacon 2.8/135 lens with 15-blade iris

The iris controls the diameter of the aperture. The iris has a diameter of 11-13 mm and the pupil ranges in diameter from 2 to 8 mm.

Detecting direction with the ear

Sound is strongly diffracted by the ear. For example,

Wavelength of a 440 Hertz sound  =  .8 meters
Aperture of the ear              =  .01 meters
Wavelength / Aperture            =  80
Since Wavelength/Aperture > 1, the wave is strongly diffracted and it is impossible to use a "sound lens" to sense direction.

The distance between our ears is 20 cm, which corresponds to a wave frequency of 1700 Hertz. Waves below this frequency diffract strongly around our head and waves above this frequency diffract weakly. We can sense the direction of a high-frequency wave by using the loudness difference between our ears. This works for frequencies larger than 1700 Hertz.

For waves with a frequency less than 1700 Hertz the wavelength is larger than your head and you can sense direction from the difference in phase arriving at each ear. This works if the wavelength is smaller than 1700 Hertz.

The resolution of the human ear for sensing direction is around 15 degrees.

Color vision

Spectrum of red, green, and blue cone cells
Dashed line: Brightness sensitivity as a function of frequency
Chlorophyl spectrum

Many birds, amphibians, reptiles, and insects can see 4 colors (tetrachromat). Mammals originally could see 4 colors and then they lost 2 of them. Most mammals can see 2 colors (green and blue) and humans are one of the few mammals that can see 3 colors.

Mantis shrimp
Mantis shrimp
Tetrachromat (4 color receptors)

The eyes of a Mantis shrimp have 12 color channels, including UV, and they are sensitive to linear and circular polarization. Each eye is trinocular, for a total of 6 channels for depth perception.


Our perception of visual brightness is logarithmic, analogous to decibels for sound. Brightness is measured in Watts/meter^2. The limit of human sensitivity is around 10^(-10) Watts/meter^2. Uranus is at the edge of visibility and Neptune is too faint to be seen.

                  Brightness    Magnitude

Sun                  1360         -26.7
Full Moon            2.6e-3       -12.7
Mars                 3.1e-7        -2.9
Jupiter              3.1e-7        -2.9
Sirius               1.2e-7        -1.5     Brightest star
Saturn               3.4e-8         -.5
Uranus               1.6e-10        5.3     Discovered 1781
Human eye limit      1e-10          6
Neptune              1.6e-11        7.8     Discovered 1846
Keck 10-meter limit  1e-19         28       Limit of the Keck 10-meter telescope
Hubble limit         1e-20         31       2.4 meter space telescope
Webb limit           1e-21         33       6.5 meter space telescope

Astronomers use a logarithmic unit of brightness called the "Magnitude".
Magnitude  =  -19.2  -  2.5*LogBaseTen(Brightness)

Brightness  =  2.16e-8 * 10^(-Magnitude/2.5)

The range of human brightness sensitivity is

Range  =  (Brightness of the sun)  /  (Minimum detectable brightness)
       ~  (1000 Watts/meter^2)     /  (1e-10 Watts/meter^2)
       ~   1e12
The range of human loudness sensitivity is
Range  =  (Maximum loudness without discomfort)  /  (Minimum detectable loudness)
       ~  (10 Pascals)^2  /  (.00002 Pascals)^2
       ~  2.5e11
Ears and eyes both have a dynamic range of around 10^12 for energy density.

Visible spectrum

                                    Wavelength (nanometers)

Red edge of the visible spectrum            750
Green light                                 555
Blue edge of the visible spectrum           400
The frequency range of vision is less than one octave. The resolution of the human eye for detecing color wavelength is around 1 percent of the visual spectrum (1 percent of an octave), similar to the resolution for sensing pitch.
Human brightness sensitivity

We estimate the minimum number of photons per second that the eye can detect.

W  =  Wavelength of a photon of light
   =  5.55e-7 meters for a green photon
C  =  Speed of light
   =  3.00e8 meters/second
F  =  Frequency of a photon of light
   =  5.4e14 Hertz for a green photon
h  =  Planck constant
   =  6.62e-34 Joule seconds
E  =  Energy of a photon
   =  h F
   =  3.6e-19 Joules for a green photon
D  =  Diameter of the pupil
   =  .005 meters
A  =  Cross-sectional area of the pupil
   =  2e-5 meters^2
B  =  Brightness in Watts/meter^2
   =  10^(-10) Watts/meter^2 for the limit of human sensitivity
N  =  Photons per second passing through the pupil at the limit of human sensitivity
   =  B A / E
   =  5600

The limit of human sensitivity is around 5600 photons/second.
The senses

Brightness sensitivity
1 million colors
Mapping the 1D spectrum to a 3D {Red,Green,Blue} coordinate

                     Width       Min            Max           Max/Min    Pixels

Audio frequency      .006      20 Hertz        20000 Hertz       1000     1200
Audio loudness                 .00002 Pascals   10 Pascals     500000
Audio angle          .1
Visible frequency    .01       4e14 Hartz      7e14 Hertz        1.75      100
Visible intensity              5e-11 W/m^2     100 W/m^2         2e12
Visible colors (RGB)               -               -                        e7
Visual angle         .0003     .0003 radians     2 radians       7000
Force                .02       .1 grams         500 kg        5000000
Time                 .1        .1 seconds     100000 seconds  1000000
Temperature         1 Kelvin   250 Kelvin        320 Kelvin       1.3

Vestibular system

Semicircular canals
Semicircular canal

The semicircular canals are .8 mm in diameter.

Vestibulo-Ocular reflex

The eyes detect head movement from the vestibular system and use it to stabilize the image.

Eye Saccades

The eye moves in sudden jumps. It will be stable for an interval and then it will make a discontinuous jump before returning to stability. The jumps are called "saccades". Saccades are analogous to Earthquakes. When the eye is held stable tension builds up until a saccade occurs.

Youtube: Eye saccades in slow motion


Visual information crosses at the optic chiasm before being assembled at the rear of the brain.

The motor cortex is in front of the somatic cortex.

Optic chiasm

Information from the eyes crosses at the optic chiasm.


Corpus callosum

The corpus callosum connects the two brain hemispheres. It is tangibly larger and more plastic in musicians.

Endocrine glands
Pineal gland

Energy dissipation

Power at maximum exertion       = 1500 Watts
Power used by the body at rest  =  100 Watts
Power used by the brain         =   20 Watts

Motor cortex

Motor cortex
Motor cortex
Motor and somatic cortex

The motor cortex is in front of the somatic cortex.

Information from the eyes passes through the motor cortex before being assembled at the rear of the brain. The motor cortex is an image-stabilization system for the eyes. Visual input requires neural processing before it can be interpreted.

In the motor cortex, proceeding from the center to the edge of the brain corresponds to proceeding from the feet to the head of the body. It represents a stack of reference frames starting from the ground and proceeding upward.

Cerebrospinal fluid

Composition of the brain
Distribution of cerebrospinal fluid

The brain produces 500 mL of cerebrospinal fluid per day and at any given time there is 100-160 mL present.


Cell membrane

Cell membranes assemble spontaneously from phospholipid molecules. They are mechanically flexibible due ot the ability of the phospholipids to rearrange themselves.

Cell membranes pass fat-soluble molecules and block water-soluble molecules. Proteins can move molecules across the membrane.

Ion pump

A membrane has ion channels that passively permit ion flow, and ion pumps that actively transport ions. Most ion channels are permeable only to specific types of ions. Ion channels can be modulated by either the membrane voltage or by chemicals.

The sodium-potassium pump generates a membrane voltage of around 70 mVolts, with the cell interior being negative.

In each cycle of the sodium-potassium pump, 3 sodium ions move outward and 2 potassium ions move inward. The pump requires hours to establish equilibrium. The pump is powered by ATP and the voltage gradient it produces provides a power source for other ion pumps.

In each cycle of the sodium-calcium pump, 3 sodium ions move inward and 1 calcium ion moves outward. This pump is powered by the membrane potential and doesn't require ATP.

An ion channel for a given ion does not pass larger ions, and most ion channels are specific for one ion. For example, most potassium channels are characterized by 1000:1 selectivity ratio for potassium over sodium, though potassium and sodium ions have the same charge and differ only slightly in radius. The pore is small enough so that ions must pass in single-file.

An action potential involves the opening and closing of ion channels and doesn't involve ion pumps. If the ion pumps are turned off by removing their energy source, or by adding an inhibitor such as ouabain, an axon can still fire hundreds of thousands of action potentials before the amplitudes begin to decay significantly.

The chloride ion is not actively pumped and takes on an equilibrium concentration given by the membrane potential.

Potassium channel protein with a potassium ion in the center

Resting potential
Skeletal muscle cells  = -95 mV
Smooth muscle cells    = -60 mV
Astroglia (Glia cells) = -85 mV +- 5 mV
Neurons                = -65 mV +- 5 mV
Red blood cells        =  -8 mV
Photoreceptor cells    = -40 mV


Microscope image

Signals travel from the cell body outward along an axon, jump to the dendrite of another neuron at a synapse, then travel inward along the dendrite to the cell body of the new neuron.

A neuron has at most one axon but the axon can branch hundreds of times. A neuron can have multiple dendrites. There are, however, many exceptions to these rules: neurons that lack dendrites, neurons that have no axon, synapses that connect an axon to another axon or a dendrite to another dendrite, etc. In certain sensory neurons (pseudounipolar neurons), such as those for touch and warmth, the electrical impulse travels along an axon from the periphery to the cell body, and from the cell body to the spinal cord along another branch of the same axon.

The longest axons in the human body are those of the sciatic nerve, which run from the base of the spinal cord to the big toe of each foot. The diameter of axons is also variable. Most individual axons are microscopic in diameter (typically about one micrometer across)

Brain neurons               = 100   billion
Brain neurons (cerebrum)    =  16.3 billion
Brain neurons (cerebellum)  =  69   billion
Brian glia cells            = 100   billion
Brain synapes               = 100   trillion
Neuron volume / Glia volume =   1.0

Neuron speed (with myelin)  = 100   m/s
Neuron speed (no myelin)    =   2   m/s
Axons for motor muscles     = 100   m/s            (16 um diameter)
Axons for sensory muscles   =  10   m/s            ( 8 um diameter)

Size of brain               =  15      cm       =  15000 neurons across
Distance between neurons    =  10      μm
Axon diameter (large)       =  20      μm
Axon diameter (small)       =   1      μm
Membrane thickness          =    .0075 μm
Chemical synapse gap        =    .020  μm
Electric synapse gap        =    .0035 μm
Node of Ranvier diameter    =   1.5    μm +- .5 μm
Node of Ranvier spacing     =1000      μm        (Distance between adjacent nodes)
Axon max size in humans     = 106      μm
Dendrite max size in humans =1000      μm
Electric synapse diameter   =    .0016 μm
Electric synapse length     =    .0075 μm

Neuron body ion channels    =   1      μm-2
Axon hillock ion channels   = 150      μm-2
Myelin ion channels         =  25      μm-2
Node of Ranvier ion channels=5000      μm-2   (Between 2000 and 12000 μm-2)

Brain neuron density        =    .0010 μm-3
Brain synapse density       =   1.0    μm-3

Chemical synapse time       =   2.0    ms
Electric synapse time       =    .2    ms
Sodium action potential     =   1      ms      (Duration)
Calcium action potential    = 100      ms      (Duration)
Sodium-Potassium pump time  = 107      ms      (Hours)  (Time to reach equilibrium)

Spines per dendrite         =1000
Sodium ratio                =   9        (Exterior concentration / interior concentration)
Potassium ratio             =  20        (Interior concentration / exterior concentration)
K+ current / Na+ current    =  20        (Current across membrane in resting state)

Typical membrane potential  =   -.07 Volts     (The cell interior is negative)
Sodium reversal potential   =   +.10 Volts
Potassium reversal potential=   -.90 Volts
Chloride reversal potential =   -.07 Volts     (Same as resting potential)
Membrane breakdown voltage  =    .2  Volts
Breakdown field (air)       =   3    Volts/μm
Breakdown field (membrane)  =  27    Volts/μm
Breakdown field (vacuum)    =  30    Volts/μm
Breakdown field (water)     =  68    Volts/μm
Membrane capacitance        =   2    uF/cm2

Max action potential rate   = 100    seconds-1

Axon diam. / Nerve diam.    =    .6        Nerve diameter corresponds to axon plus myelin sheath


1)  Unipolar neuron. Axon and dendrite emerging from same process.
2)  Bipolar neuron. Axon and dendrite on opposite ends.
3)  Multipolar neuron. One axon and many dendrites.
4)  Anaxonic. The axon can't be distingished from the dendrites.

Axons connect to the cell body through the axon hillock. The axon hillock is the last site in the cell where membrane potentials propagated from synaptic inputs are summated before being transmitted to the axon.

Action potential

If the voltage across the membrane exceeds the threshold, voltage-gated sodium ion channels open and sodium rushes into the cell, accelerating the voltage rise. When the voltage reaches its peak the sodium channels close and potassium channels open, restoring the potential to its resting state.

If the voltage change is too small to cross the threshold, the potassium current exceeds the sodium current and the voltage returns to its normal resting value.

After the action potential fires the axon enters a refractory state, which is responsible for the unidirectional propagation of action potentials along axons. At any given moment, the patch of axon behind the actively spiking part is refractory, but the patch in front, not having been activated recently, is capable of being stimulated by the depolarization from the action potential.


A myelin coating increases the speed of signals. Myelinated axons are known as nerve fibers.

Signal propatation in myelinated axons is called "saltatory conduction", where the signal jumps rapidly from one node of Ranvier to the next.

In the central nervous system (CNS), myelin is produced by oligodendroglia cells. Schwann cells form myelin in the peripheral nervous system (PNS). Schwann cells can also make a thin covering for an axon which does not consist of myelin (in the PNS). A peripheral nerve fiber consists of an axon, myelin sheath, Schwann cells and its endoneurium. There are no endoneurium and Schwann cells in the central nervous system.

In myelinated axons, ionic currents are confined to the nodes of Ranvier and far fewer ions leak across the membrane than in unmyelinated axons, saving metabolic energy.

Myelin decreases capacitance and increases electrical resistance across the cell membrane.

Myelin permits large organisms to exist by enabling fast communication between distant body parts.

When a peripheral nerve fiber is severed, the myelin sheath provides a track along which regrowth can occur. However, the myelin layer does not ensure a perfect regeneration of the nerve fiber. Some regenerated nerve fibers do not find the correct muscle fibers, and some damaged motor neurons of the peripheral nervous system die without regrowth. Unmyelinated fibers and myelinated axons of the mammalian central nervous system do not regenerate.

Chemical synapse

When an axon signal reaches a synapse, calcium channels open and calcium flows into the cell. Vesicles, which store neutrotransmitters, then open and release neurotransmitters into the synaptic gap. The neurotransmitters diffuse across the synaptic gap and bind to the target cell, triggering an action potential in the target cell.

Synapses are usually located at the terminals of axons but they can also be located at junctions along the axon ("in passing" synapses). A single axon with all its branches can innervate multiple parts of the brain and generate thousands of synaptic terminals.

A chemical synapse can amplify signals and an electric synapse cannot.

                   Time    Spacing
                   (ms)     (nm)
Chemical synapse    2        30
Electric synapse     .2       3.5
Lipid vesicle
Neurotransmitters are stored in lipid vesicles.

Lipid vesicles

Electric synapse

Electric synapses are faster than chenical synapses but they can't amply signals like chemical synapses.

In an electric synapse, signals jump from the membrane of one cell to another through a connexon joint. A connexon joint is a tunable iris composed of 6 connexin proteins.

The response is always the same sign as the source. For example, depolarization of the pre-synaptic membrane always induces depolarization in the post-synaptic membrane, and vice versa for hyperpolarization.

The response in the postsynaptic neuron is in general smaller in amplitude than the source. The amount of attenuation of the signal is due to the membrane resistance of the presynaptic and postsynaptic neurons.

Because electrical synapses do not involve neurotransmitters, electrical neurotransmission is less flexible than chemical neurotransmission.

Long-term changes can be seen in electrical synapses. For example, changes in electrical synapses in the retina are seen during light and dark adaptations of the retina.

Glial cells

Astrocytes in blue

Astrocytes in blue
Astrocytes in blue

Astrocytes          Provide nutrients to neurons
Microglial cell     Cleanup
Oligodendrocyte     Add myelin to axons in the central nervous system
Schwann cell        Add myelin to axons in the peripheral nervous system

Gial cells perform functions such as:

Surround neurons and hold them in place
Supply nutrients and oxygen to neurons
Supply nutrients and oxygen to neurons
Destroy pathogens and remove dead neurons
Regulate the clearance of neurotransmitters from the synaptic cleft
Release gliotransmitters such as ATP, which modulate synaptic function.

Glial cells are known to be capable of mitosis whereas neurons usually are not.

In the brain "gray matter" is mostly neurons and "white matter" is mostly glial cells.

In the cerebral cortex the distribution of glia types is:

Oligodendrocytes   .756
Astrocytes         .173
Microglia          .065

                  Neurons   Glia
                  (109)     (109)
Cerebral cortex    16.3     60.8
Cerebellum         69.0     16.0


Nerve bundles


In a supercomputer the time to multiply two numbers is much shorter than the communication time with memory. Brains are the reverse. Signal speed is faster than computation. A neural signal travels 20 cm (all the way across the brain) during the time of one chemical synapse.

                CPUs   Flops   Devices         Cycles/second    Devices * Cycles/second

Brain            1      .1     1014 synapses       102                1016
Supercomputer   106    1016    106   CPUs          1010               1016
Flops = Floating point operations per second.

Human neurons are as small as physics will allow. If they were smaller then they would be close enough for signals to jump between them even without synapses.


Neurons do not undergo cell division. In most cases, neurons are generated by special types of stem cells. Astrocytes are star-shaped glial cells that have also been observed to turn into neurons by virtue of the stem cell characteristic pluripotency. In humans, neurogenesis largely ceases during adulthood; but in two brain areas, the hippocampus and olfactory bulb, there is strong evidence for generation of substantial numbers of new neurons.

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