F

Magnetic Fields and Nuclear Orientation

• When an external magnetic field is applied, hydrogen nuclei align either parallel or antiparallel to the direction of the magnetic field.
• The magnetization vector $M$ represents the total magnetic field resulting from the excess of protons.
• Larmor frequency is the frequency of the transient motion reported from the magnetic moments.
• Protons exhibit a tumbling motion around the external magnetic field $B_0$, similar to a spinning top in Earth's gravitational field.
• The rotation frequency ($w<em>0$) is described by the equation $w</em>0 = γB_0$, where γ is the gyromagnetic ratio, a constant value for each atom.
• As the strength of the magnetic field increases, the Larmor frequency also increases.
• Rotation frequencies at 1.0 T and 1.5 T fields are 42 MHz and 63 MHz, respectively, which fall within the radio frequency range.

Stimulation and Resonance

• To detect magnetization $M$, it must be displaced from the external field $B_0$ .
• This is achieved by applying a radio frequency (RF) pulse, with a frequency equal to the Larmor frequency of the protons, perpendicular to $B_0$.
• The protons absorb energy from this RF pulse.
• Magnetization follows a spiral trajectory and moves away from its initial position. The extent of this movement is proportional to the duration or strength of the applied radio frequency pulse.
• The condition for these frequencies is essential for coordination and for the nuclei to surpass the energy threshold $dE$.
• Magnetization $M$ deviates from its initial position parallel to $B<em>0$, and it can be resolved into two components: longitudinal magnetization $M</em>z$ (parallel to $B<em>0$) and transverse magnetization $M</em>{xy}$ (antiparallel to $B_0$).
• After a 90° pulse, only transverse magnetization remains.
• A 180° pulse reverses the magnetization vector completely, causing the perpendicular magnetization $M_{xy}$ to become zero.

Relaxation

• Nuclei return to their initial state by precessing around the external magnetic field and transferring excess energy to other nuclei. This process is called relaxation.
• Relaxation occurs after a delay following stimulation; it requires the nucleus to transfer excess energy to its surroundings.
• The duration of relaxation depends on the physical characteristics of the tissue.

Quantum Level

• Before the application of an RF pulse, magnetic moments of particles are random, and their components in the xy-plane cancel out.
• Magnetization exists only along the z-axis, parallel to the external magnetic field $B_0$.
• The RF pulse causes the magnetic moments to rotate in phase, resulting in a component in the plane perpendicular to $B_0$. This shifts the magnetization from its initial position.
• Analogy: Passengers randomly positioned on a ship suddenly move to one side, causing the ship to drift.
• A 90° pulse sets the z-component of the $M$ vector to zero, and the magnetization rotates in the xy-plane.
• This moving magnetic field induces a potential difference at the ends of a coil, producing an electric current, much like in television antennas.

Free Induction Decay (FID)

• Magnetization rotates in the xy-plane; however, this magnetization rapidly decays as magnetic moments lose phase coherence and return to random orientations.
• This decay results in a signal known as free induction decay (FID).
• Longitudinal magnetization recovers rapidly.
• RF energy absorbed by nuclei is emitted through interaction with the environment, allowing them to return to their initial equilibrium state. The signal waveform is sinusoidal.

Relaxation Times

Time T1

• After RF radiation stops, magnetization returns to its original position, parallel to the initial field $B<em>0$. This is characterized by time constants $T</em>1$ and $T_2$.
• $T_1$ characterizes the recovery rate of magnetization $M$.
• After $5 Imes T_1$, relaxation is complete.
• $T_1$, or spin-lattice relaxation time, is the time required for longitudinal magnetization to recover to 63% of its initial value $M$.
• The magnetic moment reaches 86% of its maximum value in $2 Imes T<em>1$ and 95% in $3 Imes T</em>1$.
• $T_1$ reflects the rate of energy transfer from excited nuclei to the environment and varies for different tissue types.
• $T_1$ is longer in stronger magnetic fields.
• $T_1$ depends on the size of the tissue molecule and the environment surrounding the proton.
• Small water molecules move rapidly, reducing their chance of energy emission through interaction; thus, clean water and cerebrospinal fluid have long $T_1$ values.
• Large fat molecules in dense atomic grids have short $T_1$ times.

Time T2

• The re-bending of longitudinal magnetization is accompanied by the decay of transverse magnetization, which occurs much faster than longitudinal magnetization decay.
• Magnetic moments lose phase due to interactions of their magnetic fields, causing the spinning transverse magnetization to spread out.
• $T<em>2$, or spin-spin relaxation time, is the time required for transverse magnetization to decrease by 37% of its maximum value $M</em>a$.
• $T_2$ also depends on tissue type. It is short in solids, where magnetic moments are exposed to varying local magnetic fields, and longer in fluids, where random molecular motion reduces these changes.

Time T2*

• Local inhomogeneities in the external magnetic field impact transverse magnetic relaxation.
• The total transverse magnetic relaxation is expressed as $T_2*$.
• Inhomogeneities in the static magnetic field increase the rate of total transverse relaxation.
• The actual longitudinal magnetization recovers faster than expected based on tissue characteristics alone.
• The value of $T<em>2*$ is usually smaller than $T</em>2$.

Magnetic Resonance Imaging

Gradient Fields

• A perfectly homogeneous magnetic field cannot provide spatial information because all nuclei have the same frequency and signal.
• To obtain position-dependent responses, gradient fields are superimposed on the homogeneous field.
• Gradient fields vary the magnetic field in a specified direction and are used to select, prepare, and detect magnetic resonance signals.
• A linear gradient magnetic coil consists of two coils spaced apart with current flowing in opposite directions to create equal and opposite fields.
• One coil increases the static magnetic field, while the other decreases it. At the midpoint, the fields cancel out, leaving only the effect of the external magnetic field $B_0$.
• Modern MRI systems use conductive coils to generate graded fields in three axes, allowing selection of anatomical planes (coronal, lateral, or sagittal) using appropriate gradient fields.

Section Selection

• In an inhomogeneous magnetic field, nuclei tune at different frequencies ($w<em>0 = γB</em>0$).
• Applying an RF pulse of frequency $w_0$ tunes protons at the midpoint of the coils.
• If the pulse has a frequency spectrum $dw$, it tunes not only a plane but a slice $Dz$.
• The thickness of the slice can be determined by changing $dw$ or by altering the slope of the gradient coil while keeping $dw$ constant.

BASIC TECHNIQUES FOR MAGNETIC RESONANCE SIGNAL PRODUCTION (SEQUENCES)

General Aspects

• Image quality in MRI depends on pulse sequences and parameters like proton density and tissue relaxation times ($T<em>1$, $T</em>2$, $T_2*$).
• TR (repetition time) is the time interval between two consecutive 90° pulses.
• TE (echo time) is the time between pulse application and signal detection.

Basic Sequences

Spin-Echo Sequence

• A 90° excitation pulse rotates the magnetic moment vector in the xy-plane.
• The sequence includes one or more 180° reset pulses to generate an echo.
• FID depends mainly on time $T_2$.

Saturation Recovery Sequence

• This sequence has repeated 90° pulses.
• Longitudinal magnetization is zero after the first 90° pulse, leading to saturation.
• The next 90° pulse is applied only when longitudinal magnetization returns to its initial position, based on the tissue's $T_1$.

Inversion Recovery Pulse Sequence

• A 180° pulse inverts the magnetic vector.
• A second 90° pulse is applied before $T_1$ relaxation completes to detect transverse magnetization.
• A disadvantage is the long time required for image production.

MRI INFORMATION

Signal and Imaging Detail

• The MRI signal depends on proton density (concentration of 1H nuclei) and magnetic recovery times ($T<em>1$, $T</em>2$, $T_2*$).
• Tissues consist of cells with organic (lipids, amino acids, carbohydrates), water, and inorganic components.
• The signal originates from hydrogen nuclei in free and bound water and moving lipids.
• Pathological changes can include morphological changes, alterations in tissue composition/function, and changes in intra- and extra-cellular substances.
• Spatial resolution in clinical MRI systems (0.5-1mm) is adequate for detecting morphological changes.
• Changes in proton density and magnetic recovery times affect contrast and allow for the detection of structural and functional tissue changes.
• Electromagnetic noise can degrade image quality, so protocols should maximize useful signal and minimize noise.
• The signal-to-noise ratio (S/N) should be as high as possible (>50).

Contrast

• Contrast is a crucial parameter for image quality. There must be sufficient signal differentiation between neighboring tissues.
• Maximize contrast using configured pulse sequences and special techniques like fat suppression, signal suppression of normal structures, and contrast agents.
• Relative contrast should be >10% for minimum discriminability.

Signal-to-Noise Ratio (S/N)

• Proportional to:

  • The square of the static magnetic field intensity $H<em>0$: $[H</em>0^2]$
  • The density of hydrogen nuclei in water and fat molecules: $[ρ(H)]$
  • The density of enhancing contrast agents: $[ρ(H_{enh})]$
  • The slice thickness: $[Δz]$
  • The dimensions of the image pixels: $[Δx Imes Δy]$
  • The square root of the number of sampling repetitions: $[NEX^{1/2}]$
  • The square root of the number of phase encoding projections: $[N_y^{1/2}]$
  • The square root of the number of frequency encoding projections: $[N_x^{1/2}]$
  • The geometric characteristics of coils

• Inversely proportional to:

  • The square root of the RF reception bandwidth: $[BW^{1/2}]$
  • The speed of movement of the imaged tissue.
  • The density of suppressive contrast agents: $[ρ(H_{sat})]$

• Depends on:

  • The type of sequence used (spin echo, saturation recovery, etc.).
  • Quantitative parameters of the sequence (TR, TE, flip angle).
  • Intrinsic tissue characteristics ($T<em>1$, $T</em>2$, $T_2*$).
  • Type and flow rates of body fluids.

• The above dependencies hold inversely for spatial resolution.

• Signal-to-noise ratio competes with spatial resolution.

Signal Intensities and Imaging Conventions

• In spin echo, signal intensity of a pixel (S) is:
  $S<em>{pixel} = [1 - e^{-\frac{TR}{T</em>1}}] \cdot [e^{-\frac{TE}{T_2}}] \cdot ρ(H)$
• (a): Increasing exponential function related to longitudinal magnetization $M_z$.
• (b): Decreasing exponential function related to transverse magnetization $M_{xy}$.
• (c): Increasing linear function related to the density of hydrogen nuclei.
• $T<em>1$ values increase slightly with the static magnetic field intensity. For a 1T field, soft tissue $T</em>1$ values range from 400 ms (liver) to 800 ms (gray matter) and 4000 ms for CSF.
• $T<em>2$ values do not depend on the static magnetic field intensity. $T</em>2$ values range from 40 ms (liver) to 100 ms (gray matter) and 2000 ms for CSF.
• Contrast can be controlled through appropriate selection of TR and TE values.

T1-weighted Images

• Choosing a TR=400 ms and a TE=20 ms enhances term (a) and minimizes term (b), influencing contrast based on the TR/$T_1$ parameter.
• Tissues with long $T_1$ (e.g., CSF, bone) appear black.
• Tissues with average $T_1$ (e.g., liver, muscle, white matter) appear gray.
• Tissues with short $T_1$ (e.g., fat) appear white.

T2-weighted Images

• Choosing a TR=3000 ms and a TE=100 ms enhances term (b) and minimizes term (a), influencing contrast based on the TE/$T_2$ parameter.
• Tissues with high $T_2$ (e.g., CSF, bladder) appear white.
• Tissues with average $T_2$ appear gray.
• Tissues with short $T_2$ (e.g., bone) appear black.

Proton Density Weighted Images

• Choosing a TR=6000 ms and a TE=20 ms enhances term (c) and minimizes terms (a) and (b), influencing contrast based on proton density.
• Tissues with high PD appear white.
• Tissues with average PD appear gray.
• Tissues with low PD appear black.

Optics and Vision

Light Production Mechanisms

• Light Sources
• Photometry

Geometrical Optics

Physical Mechanism of Vision

Microscopes

• Matter consists of atoms (X).
• Electrons near the nucleus have low energy.
• The tables list the number of electrons for different elements.

Light Production Mechanism

• Light is an electromagnetic wave that carries energy and is produced during the de-excitation of atoms or molecules in excited states.
• Atoms emit EM radiation (energy) in discrete quantities called quanta or photons, each with energy $E = hv$, where:
  • $v$: EM radiation frequency
  • $h$: Planck’s constant
• The emitted photon has energy $hv = E<em>{in} - E</em>f$

Light Production Mechanisms - Excitation

The main mechanisms of excitation of the atoms or molecules of the matter are:

• Heating to a high temperature (rapid collisions between the particles lead to excitation)
• Electrical discharge (collisions between moving ions and atoms/molecules)
• Light absorption

Emission Spectra

• Linear emission spectra are seen in monoatomic gases, where energy levels are quantized.
• Band spectra are seen in polyatomic gases where many energy levels are present.
• Continuous spectra are seen in hot liquids and solids where interactions are strong and there are no clear energy levels.

Absorption Spectra

Materials absorb the wavelengths they emit (reversal of spectral lines).

Fluorescence and Phosphorescence

• Fluorescence: Production of light (photons) from instant de-excitation of the excited atom at the ground level either directly, or through an intermediate metastable level (de-excitation time <10ns)
• Phosphorescence: Production of light (photons) from delayed de-excitation of the atom, from the excited state to the ground level, through a metastable state (de-excitation time »1ms)

Stimulated Emission - LASER

• Excited atoms trapped in a metastable state can be de-excited if they interact with a photon of exactly the same energy as the photon produced during their de-excitation.
• LASER: Light Amplification by Stimulated Emission of Radiation
• LASER beam characteristics:
  • purely monochromatic (single energy) beam
  • consists of photons in phase
  • propagates in a straight line without diverging
  • large energy concentration on a small surface area

Photometry

• The eye is sensitive to light, but cannot objectively compare two light sources.
• Luminous flux (lm): $\Phi = \frac{dE}{dt} = \frac{energy \ emitted \ to \ all \ directions}{time\ dt}$
• Luminous intensity (cd): $I = \frac{d\Phi}{d\Omega}$
  • For point-source: $I = \frac{\Phi}{\Omega} = \frac{\Phi}{4\pi}$
  • For light beam: $I = \frac{d\Phi}{dS} \ (lm/m^2)$
• Lux (for surfaces): $I = \frac{d\Phi<em>E}{dS</em>E}$

Photometry Organs

• Photographic plates: Tarnishing is proportional to exposure to light (exposure – illumination X time)
• Photomultipliers: The generated current at the output is proportional to the illumination at the input.
• Photocells: Crystal diodes (pn or pr) or crystal triodes (pnp or npn). The electric current produced is proportional to the lighting they receive.
• Photoresistors: The change in resistance is proportional to the illumination

Geometrical Optics

• Geometrical optics(GO) examines the rays propagation, reflection, diffusion and refraction of light within various materials
• In OG we use the concept of ray, which is the straight line of propagation of the E/M wave. Many rays form a beam.
• Three types of beams:
  • Parallel
  • Divergent
  • Convergent

Light Propagation

• Light propagates in a straight line.
• The speed of light in a vacuum is c=300,000 km/s=3x10^8 m/s.
• C=λ·v, where λ: wavelength and v: frequency
• The speed of light propagation in transparent materials is slower than in a vacuum
• Refractive index (n): n=c0/c, where c0: speed in vacuum and c: speed in the material
• Shadow and penumbra behind a non-transparent object is a result of the rectilinear propagation of light

Reflection

• Reflection is the change of the direction and the orientation of a light ray that falls on a surface that is characterized as reflective surface, or reflector
• Reflection on a flat reflector:
  1. Incident ray, reflected ray, and transverse to the incidence point axis, all on the same plane
  2. Incidence angle (α) = reflection angle (β)

Image Formation

The determination of the image is based on the drawing of four rays, which, approximately, pass through the same point. These rays are called principal rays. In practice, only two are sufficient.

• A ray parallel to the optical axis, after reflection in a concave reflector, always passes through the focal point E. For a convex reflector, we take the extension of the reflected ray.
• A ray passing through the focal point is reflected parallel to the optical axis for a concave reflector, while for a convex reflector its extension is reflected.
• A ray passing through the center of curvature O follows the same path reflected.
• A ray that meets the top of the reflector is reflected at an angle equal to its angle to the optical axis.

Diffusion

• Diffusion is observed when light falls on non-reflective surfaces (e.g. on a rough surface, on dust grains, water vapor, etc.)
• When the dimensions of the scattering particles are larger than the wavelength λ of the incident radiation, the scattered light is white, while when they are smaller the scattering is more intense for short-wavelength radiation (Rayleigh's Law): $I = A/λ^4$
• Question: How is the color of the sky explained?

Refraction

• When a beam of monochromatic light hits the interface between two media, the beam is deflected from its original path.
• The lines of light propagation are straight in both mediums forming an angle with an apex above the dividing surface.
• Laws of refraction:
  1. The incident and refracted rays lie in the same plane (plane of refraction) which is perpendicular to the refracting surface.
  2. Snell’s law: The quotient of the sines of the angles of incidence and refraction is constant (independent of the angle of incidence) and is equal to the quotient of the refractive indices, $\frac{sin \alpha}{sin \beta} = \frac{n<em>2}{n</em>1}$

Prisms

• The angle of deflection of a refracted beam depends on the refractive indices (n1, n2) of the two media and on the angle of incidence (α). Refractive indices depend on the frequency of incident light.
• A prism is a transparent object that consists of two or more non- parallel flat surfaces that form an oblique angle between them.

Lenses

• Lenses are transparent objects of small thickness defined by two spherical surfaces or one spherical and one flat. Lenses are divided into two categories:
  • Converging: thin at the edges-fat in the middle
  • Diverging: fat at the edges-thin in the middle
• A parallel beam incident on a converging lens emerges converged, while if it strikes a diverging lens it emerges diverged.
• For a converging lens with refractive index n:
  $\frac{1}{f} = (n-1)(\frac{1}{R<em>1} - \frac{1}{R</em>2})$
  $m = \frac{\alpha}{\beta} = \frac{A'B'}{AB}$

Lenses - Ray Diagrams

• We draw the three main rays
• A ray parallel to the optical axis, after being refracted by the thin lens, always passes through the focus of a converging lens. For divergent we take the extension of the refracted ray
• A ray that passes through the center of the lens d deviates from its straight path, because in the region of the center the two surfaces are parallel, since the lens is thin
• A ray passing through the first focal point, after meeting the lens, is refracted parallel to the principal axis.

Lens Aberrations

The above applies to thin lenses which are incident by monochromatic rays with directions close to the main axis of the lenses. If any of these are not true, deviations, known as aberrations, are observed.

Vision

Vision Physical Mechanism

Vision takes place in three stages:

1. The eye focuses the external image on the retina
2. Special cells create (photoreceptors) and transmit (optic nerve) the stimulus to the brain
3. The visual region of the occipital lobe processes the stimuli and "perceives" the image

Eye Capabilities

1. Has a sharp wide-angle lens
2. Has a highly efficient focusing system (20cm-infinity)
3. Has a sensitive and automatically adjustable diaphragm
4. Works satisfactorily in a wide lighting region (sunny day lighting/dark day lighting = 10^10)
5. Image is taken by two eyes and thus a sense of depth is created
6. Has a mechanism for automatic control of intraocular pressure and maintaining the shape of the eye
7. It is protected by bone and fat which are shock-absorbing materials
8. The muscles of the eye give the possibility for orientation of the eye in any direction within the visual field
9. It has an automatic cleaning and polishing system of the corneal lens that frees it from scratches (opening and closing of the eyelids) which can work independently for each eye for communication with the opposite sex

Focusing Mechanism of the Eye

• The cornea and lens of the eye are responsible for focusing external images onto the retina.
• The refractive indices (IR) of all parts of the eye as well as the curvature of the cornea are constant for a particular person (the IR of the lens and the curvature of the cornea can vary from person to person)
• The ability of the eye to focus is due to the ability of the lens of the eye to change curvature.

Stimulus Creation

• Photoreceptors (rods and cones) are special cells responsible for creating the stimulus. The stimulus is an action potential produced through photochemical processes."
• Low-frequency photons (infrared) do not have enough energy to trigger an action potential, while high-frequency photons (ultraviolet) are absorbed before reaching the retina.
• Cones and rods are present on the entire surface of the retina (except the blind spot), but the process of vision takes place mainly in a small area, the fovea centralis of the macula lutea.

Stimulus Creation - Cones & Rods

• The cones are 6.5 X 10^6 per eye and are used for vision in a bright environment
  • In the fovea centralis: 1 cone→1 optic fiber elsewhere: many cones →1 optic fiber
• The rods are 120 X 10^6 per eye and are used for vision in the dark
  • Everywhere: many rods→1 optic fiber photosensitivity spatial distribution

Eye Parameters

• Iris pupil diameter 3-8 mm (16-114 mm²). Surface ratio 7. Light intensity ratio 1010
• Adaptation time: 5 sec at high intensity and 300 sec in darkness.
• A distant object is focused 30.6 mm from the cornea. The globe is 24 mm long.
• Focusing is performed by the lens.

Macula Lutea & Fovea Centralis

• Macula lutea – fovea centralis(0.3 mm diameter) : Sharp vision
• An object with a vertical diameter of 0.3 cm at a distance of 10 m forms on the retina an image of 0.1 mm height
• Different points are perceived as one when their images are formed in the same or adjacent retinal cells.
• The eye cannot distinguish objects that their images in the central fovea are distant less than 2 µm.
• Rayleigh Criterion:
  $\lambda<em>m: wavelength\ at\ the\ eye$$l: iris-retina\ distance$$a: iris\ diameter$ for \lambdao = 0.55\mum: r=2 \mum

Refractive Errors

• Hyperopia: eye with a short axis or cornea ofsmall curvature (image behind the retina)
• Presbyopia: reduction in the ability of the eye to adapt (focus) (the image in front of the retina)
• Astigmatism: uneven corneal curvature in different directions

Classic Microscope

• Magnifying power of a microscope M is defined as the ratio of the angle (θ) subtended by the image to angle θ’ subtended at the eye by the object (25cm minimum eye-object image)
• M=θ/θ’ , (M<1000)

Electronic Microscope

The image is produced by bombarding electrons on the object to be displayed. The various regions of the object cause different attenuation. These differences are illustrated on a fluorescent screen. Magnification can reach up to 400,000. Classic and electronic microscope similarities

Photoconductors

• Optical fibers consist of a transparent material of low refractive index, surrounded by a material of high refractive index.
• A photoconductor usually consists of several optical fibers and has the ability to bend.

Bioelectricity

Overview

• Bioelectricity refers to the electrical activity that occurs within living organisms, particularly at the cellular and molecular levels. It involves the generation, transmission, and response to electrical signals within biological systems.
• The nervous system is responsible for creating, transmitting, and decoding these electrical signals.
• The existence of electrical activity in the human tissue is associated with the concept of 'life,' while its absence is linked to 'death.’
• The absence of electrical activity in the brain is considered satisfactory evidence of a person's death ('brain death'), allowing for the removal of organs for transplantation.
• A complete understanding of the functioning of the nervous system has not yet been achieved, although significant progress has been made in recent years.

The Nervous System (NS)

The nervous system is divided into:

• Central NS: includes the brain and spinal cord
• Peripheral NS:
  • Somatic NS:
    • Some of the peripheral nerves transmit sensory information (such as temperature, touch, pain, etc.) to the brain or spinal cord, while others carry instructions from the brain or spinal cord to the skeletal muscles.
  • Autonomic NS: controls smooth muscles and glands, and its function is involuntary.
    Cardiac muscle, Smooth muscles, and Skeletal muscle are all controlled by the Autonomic and Somatic nervous systems

Nerve Cell or Neuron

The neuron is a specialized cell consisting of:

• Cell body: contains the cell nucleus
• Nerve axon (or axon):
  • consists of axoplasm surrounded by the cell membrane
  • diameter 1-20 μm and length up to 1 m
  • originates from the cell body and can branch
  • is responsible for the propagation/transfer of electrical signals.
• Dendrites: originate from the cell body (shorter and thinner than the axon)
• Synapses: connections between the dendrites of different neurons or neuron/muscle connections. Communication between neurons is attributed to synapses.
• Schwann cells: surround the nerve axon, forming multiple layers of myelin.
• Nodes of Ranvier: gaps of 1 μm in length in the myelin layer surrounding the axon (one every 1 mm of the axon).

Characteristics of Axon

The nerve axon imitates a cable consisting of a cylindrical insulating membrane of small thickness (cell membrane) containing conductive material (axoplasm), separating it from the conductive material of the extracellular space.

Physical Characteristics of Axon

• The resistance of 1 cm of axon filament is equal to the resistance of the thinnest copper wire (diameter: 8 μm) with a length of 70 km!!!
• It is therefore surprising how nature has constructed a perfect communication system using a material that an electronic engineer would consider an excellent insulator.

Resting Potential of Axon

• Every nerve cell at rest exhibits a potential difference (Vin-Vout) between the internal and external surfaces of the cell membrane known as the resting potential (Vrest).
• Convention: the potential of the extracellular space is taken as Vout = 0.
• The Vrest of neurons is -90 mV i.e. the cell interior is at a lower potential
• The Vrest is not due to the difference in negative-positive charges, as both the axoplasm and extracellular space appear neutral.
• The Vrest is established due to the different concentrations of Na+, K+, and Cl- ions within the axoplasm and extracellular space in association with the selective permeability of the neuron's cell membrane to those ions.

Nernst Equation

• If Vrest=Veq for a specific ion X, then the [X]out and [X]in do not change (there is equilibrium)
• If Vrest≠Veq for a specific ion, then ions tend to move in or out of the cell (passive flow) to change the membrane potential to Veq (there is not equilibrium)
• The equilibrium potential (Veq) for an ion X with different concentrations in and out of the nerve cell membrane is described by the Nernst equation:
  $V<em>{eq} = \frac{RT}{zF} ln\frac{[X]</em>{out}}{[X]_{in}}$
  where, R is the gas constant, T is the absolute temperature, z is the valence of the ion, F is Faraday's constant, and [X]out and [X]in represent the ion X concentrations outside and inside the cell, respectively.

NOTE: ln is natural log

Equilibrium Axon Membrane Potentials

• For axon of a nerve cell, calculate Veq for ions Na+, K+ and Cl- since we know ions’ concentrations:
  CinNa+= 12 mole m-3
  CoutNa+= 145 mole m-3
  CinK+ = 155 mole m-3
  CoutK+ = 4 mole m-3
  CinCl- = 4 mole m-3
  CoutCl- = 120 mole m-3
• Using Nernst equation, we may estimate:
  Veq Na+ = +66 mV
  Veq K+ = -98 mV
  Veq Cl- = -90 mV
  Remember Vrest =-90 mV

State of Equilibrium

• The equilibrium potential for Cl- ions (Veq Cl- = -90 mV) is equal to the resting membrane potential (Vrest = -90mV), indicating equilibrium for Cl-.
• However, there is no equilibrium for Na+ (Veq Na+ = +66 mV) and K+ (Veq K+ = -98 mV).
• Thus, Na+ ions tend to flow into the axoplasm, and K+ ions move out (passive flow of Na+ and K+ ions).
• However, changes in ion concentrations do not occur (the Vrest is preserved at -90 mV).
• This is due to an active transport mechanism of Na+ and K+ (energetic flow) against the respective gradients, known as the Na-K pump
• The Na-K pump maintains constant ion concentrations and, consequently, an unchanged resting potential, at the expense of energy (given that membrane permeability remains constant)
• If an external factor alters membrane permeability abruptly, then the Vin-Vout abruptly changes to other than Vrest value

Sodium-Potassium Pump Function

Passive and active flow of Na+ and K+ ions