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What would be the conduction speed of A-alpha fibers, were they unmyelinated?

What would be the conduction speed of A-alpha fibers, were they unmyelinated?


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It's known (Hursh 1939) that myelinated axons exhibit the behavior $v = 6d$, where $v$ is the propagation speed [m/s] and $d$ is the axon diameter [μm].

A related relation for unmyelinated axons is (Malmivuo & Plonsey, eq. 2.1):

$$v = sqrt{frac{i_ ext{Na max}}{r_i c_m^2 V_mathrm{th}}}$$

I want to approximate the propagation speed that A-alpha nerve fibers would exhibit, if they were unmyelinated. How could I find values for the constants in the above equation? Is there an easier way to conduct these calculations, or has there already been a study done on this?

For the (myelinated) A-alpha fibers, d = 13-20 μm and v = 80-120 m/s according to Wikipedia.


The provided relation for unmyelinated axons only holds up to certain diameter value of the axon and is valid only as long as the sodium conductance iNa is uniformly distributed along the cross-section of the axon. This explains why non-myelinated fibers are so thin, being only 0.2-1.5 μm in diameter.

In axons having a larger diameter the axon potential can only move between Ranvier nodes. If this fiber gets unmyelinated the axon potential just stops propagating, due to several reasons:

  1. Sodium concentration through the complete axon surface is too high so that it diffuses internally along the axon and reduces the concentration gradient in the direction of AP propagation.

  2. AP is propagated not only along the length of axon but also laterally, that leads to its flatenning over the axon surface, decreasing its amplitude etc.

Unfortunately I couldn't find the reprint of the original publication in Biofizika referenced by the webpage you provide the link to, so I cannot investigate the constraints of their model for AP propagation.

In order to estimate the propagation velocity in unmyelinated axons of large diameter I would take some theoretical papers investigating the properties of nerve fibers undergoing demyelination. Z. J. Koles and M. Rasminsky "A computer simulation of conduction in demyelinated nerve fibres", J Physiol. 1972 December; 227(2): 351-364. seems to be one of the earliest publications on this topic and it is freely available online.

In this paper the authors tried to simulate the conductance of demyelinated motor axon with diameter of 10μm and 5μm myelin sheath. They gradually decreases the amount of myelin width until 2.7% of the initial value where they saw the abolishment of AP propagation. The propagation time was 12.5 times higher than in normally myelinated fibers (for the distance where AP could travel). If we consider the initial propagation velocity being 80-120 m/s then after demyelination it is reduced down to 6-10 m/s and fade after a short travel.


Chemical signaling by diffusion

Faster signaling by chemical transport

1 is the dividing line between diffusion-dominated and transport-dominated signaling. With D for small molecules

10 -6 cm 2 s -1 (100 m 2 s -1 ) and intracellular transport processes reaching up to 100 m s -1 (Table I), diffusion can dominate in organisms below

1 m (bacterial size). However, for even small molecules traveling at the slow rates of axonal transport (1 m s -1 when in motion: Roy et al., 2007) over distances of centimeters, Pe is much greater than 1, and transport becomes by comparison a "rapid conduction" modality (e.g. Mignot et al 2007).

Voltage-based passive signaling


What is myelin?

The evolution of a character is better appreciated if examples of convergent emergence of the same character are available for comparison. Three instances are known among invertebrates of the evolution of axonal sheaths possessing the functional properties and many of the structural properties of vertebrate myelin. Comparison of these invertebrate myelins raises the question of what structural features must a sheath possess in order to produce the two principal functional characteristics of impulse speed enhancement and energy savings. This essay reviews the features recognized by early workers as pertaining to myelin in vertebrate and invertebrate alike: osmiophilia, negative birefringence and saltatory conduction. It then examines common features revealed by the advent of electron microscopy: multiplicity of lipid membranes, condensation of those membranes, specialized marginal seals, and nodes. Next it examines the robustness of these features as essential components of a speed-enhancing sheath. Features that are not entirely essential for speed enhancement include membrane compaction, spiral wrapping of layers, glial cell involvement, non-active axonal membrane, and even nodes and perinodal sealing. This permissiveness is discussed in relation to the possible evolutionary origin of myelin.


Musculoskeletal System: Nerve Tissue

What is an example of afferent and efferent neurons communicating directly?

- neurons produce signals that travel quickly to other cells.

Axon Hillock
> hillock membrane sensitivity?

- extensive branching at the end of the axon.

- most common type of neuron

Processes (Dendrites and axons)?
> where do they start? Orientation?

Processes (Dendrites and axons)?

What percentage are they of the CNS cells?

- 90% of cells in CNS are neuroglia

What do they help form? How?

Regulatory function in Brain?

- their perivascular feet form an important part of the "blood-brain barrier". By separating neurons from capillaries.

- help regulate brain tissue intercellular fluid by selectively absorbing ions and nutrients from blood and by removing waste.

- concentrate in areas of infection, trauma or stroke.

What is formed by?
> In CNS?
> In PNS?

- Formed:
> By Oligodendrocytes in CNS
> By Schwann cells in the PNS.

Electrical potentials?
> how is it measured?

What kind of membranes do they have (polarized or depolarized)

K+ Leaky Channels? Function?

- Na+/K+ pumps 3out:2in ratio
> pumps work continuously, require 1 ATP per cycle.

- Large, can't escape and lend to negative charge inside of the cell.

- K+ leak channels let some K+ follow its concentration gradient out of the cell, which makes membrane potential more negative.

Axon Hillock? Compared to cell body?
> What does it ion channels make it?

On Axon? Where on the axon?

- high density of voltage-gated Na+ and K+ challenge vs much less on cell body.
> trigger zone for initiating an action potential.

- voltage-gated Na+ and K+ channels. Clustered at Nodes of Ranvier.

When at rest, what are voltage and ligand
channels like?

- get weaker the farther they spread

Action on the membrane potential?

Action on the membrane potential?

- Cl- flowing into cell or K+ leaving cell

Temporal Summation?
> Result?

2. When threshold reached, opens voltage-gated Na+ channels.

3. Voltage-gated Na+ channels open fast and incoming Na+ further depolarizes membrane, opening more of these channels (positive feedback). Voltage-gated K+ channels slowly start to open.

4. Depolarization peaks at +35mV. Na+ channels close as membrane becomes positive

5. Voltage-gated K+ channels fully opened. K+ outflow repolarizes membrane.

6. Since volt-gated K+ channels close slowly, hyperpolarizes slightly below resting potential.


Contents

Ultimately, conduction velocities are specific to each individual and depend largely on an axon's diameter and the degree to which that axon is myelinated, but the majority of 'normal' individuals fall within defined ranges. [1]

Nerve impulses are extremely slow compared to the speed of electricity, where the electric field can propagate with a speed on the order of 50–99% of the speed of light however, it is very fast compared to the speed of blood flow, with some myelinated neurons conducting at speeds up to 120 m/s (432 km/h or 275 mph).

Motor fiber types
Type Erlanger-Gasser
Classification
Diameter Myelin Conduction velocity Associated muscle fibers
α 13–20 μm Yes 80–120 m/s Extrafusal muscle fibers
γ 5–8 μm Yes 4–24 m/s [2] [3] Intrafusal muscle fibers

Different sensory receptors are innervated by different types of nerve fibers. Proprioceptors are innervated by type Ia, Ib and II sensory fibers, mechanoreceptors by type II and III sensory fibers, and nociceptors and thermoreceptors by type III and IV sensory fibers.

Sensory fiber types
Type Erlanger-Gasser
Classification
Diameter Myelin Conduction velocity Associated sensory receptors
Ia 13–20 μm Yes 80–120 m/s [4] Responsible for proprioception
Ib 13–20 μm Yes 80–120 m/s Golgi tendon organ
II 6–12 μm Yes 33–75 m/s Secondary receptors of muscle spindle
All cutaneous mechanoreceptors
III 1–5 μm Thin 3–30 m/s Free nerve endings of touch and pressure
Nociceptors of neospinothalamic tract
Cold thermoreceptors
IV C 0.2–1.5 μm No 0.5–2.0 m/s Nociceptors of paleospinothalamic tract
Warmth receptors
Autonomic efferent fibre types
Type Erlanger-Gasser
Classification
Diameter Myelin Conduction velocity
preganglionic fibers B 1–5 μm Yes 3–15 m/s
postganglionic fibers C 0.2–1.5 μm No 0.5–2.0 m/s
Peripheral Nerves
Nerve Conduction velocity [5] [6]
Median Sensory 45–70 m/s
Median Motor 49–64 m/s
Ulnar Sensory 48–74 m/s
Ulnar Motor 49+ m/s
Peroneal Motor 44+ m/s
Tibial Motor 41+ m/s
Sural Sensory 46–64 m/s

Normal impulses in peripheral nerves of the legs travel at 40–45 m/s, and 50–65 m/s in peripheral nerves of the arms. [7] Largely generalized, normal conduction velocities for any given nerve will be in the range of 50–60 m/s. [8]

Nerve conduction studies Edit

Nerve Conduction Velocity is just one of many measurements commonly made during a nerve conduction study (NCS). The purpose of these studies is to determine whether nerve damage is present and how severe that damage may be.

Nerve conduction studies are performed as follows: [8]

  • Two electrodes are attached to the subject's skin over the nerve being tested.
  • Electrical impulses are sent through one electrode to stimulate the nerve.
  • The second electrode records the impulse sent through the nerve as a result of stimulation.
  • The time difference between stimulation from the first electrode and pick-up by the downstream electrode is known as the latency. Nerve conduction latencies are typically on the order of milliseconds.

Although conduction velocity itself is not directly measured, calculating conduction velocities from NCS measurements is trivial. The distance between the stimulating and receiving electrodes is divided by the impulse latency, resulting in conduction velocity. NCV = conduction distance / (proximal latency-distal latency)

Many times, Needle EMG is also performed on subjects at the same time as other NCS procedures because they aid in detecting whether muscles are functioning properly in response to stimuli sent via their connecting nerves. [8] EMG is the most important component of electrodiagnosis of motor neuron diseases as it often leads to the identification of motor neuron involvement before clinical evidence can be seen. [9]

Micromachined 3D electrode arrays Edit

Typically, the electrodes used in an EMG are stuck to the skin over a thin layer of gel/paste. [8] This allows for better conduction between electrode and skin. However, as these electrodes do not pierce the skin, there are impedances that result in erroneous readings, high noise levels, and low spatial resolution in readings. [10]

To address these problems, new devices are being developed, such as 3-dimensional electrode arrays. These are MEMS devices that consist of arrays of metal micro-towers capable of penetrating the outer layers of skin, thus reducing impedance. [10]

Compared with traditional wet electrodes, multi-electrode arrays offer the following: [10]

  • Electrodes are about 1/10 the size of standard wet surface electrodes
  • Arrays of electrodes can be created and scaled to cover areas of almost any size
  • Reduced impedance
  • Improved signal power
  • Higher amplitude signals
  • Allow better real-time nerve impulse tracking

Anthropometric and other individualized factors Edit

Baseline nerve conduction measurements are different for everyone, as they are dependent upon the individual's age, sex, local temperatures, and other anthropometric factors such as hand size and height. [5] [11] It is important to understand the effect of these various factors on the normal values for nerve conduction measurements to aid in identifying abnormal nerve conduction study results. The ability to predict normal values in the context of an individual's anthropometric characteristics increases the sensitivities and specificities of electrodiagnostic procedures. [5]

Age Edit

Normal 'adult' values for conduction velocities are typically reached by age 4. Conduction velocities in newborns and toddlers tend to be about half the adult values. [1]

Nerve conduction studies performed on healthy adults revealed that age is negatively associated with the sensory amplitude measures of the Median, Ulnar, and Sural nerves. Negative associations were also found between age and the conduction velocities and latencies in the Median sensory, Median motor, and Ulnar sensory nerves. However, conduction velocity of the Sural nerve is not associated with age. In general, conduction velocities in the upper extremities decrease by about 1 m/s for every 10 years of age. [5]

Sex Edit

Sural nerve conduction amplitude is significantly smaller in females than males, and the latency of impulses is longer in females, thus a slower conduction velocity. [5]

Other nerves have not been shown to exhibit any gender biases. [ citation needed ]

Temperature Edit

In general, the conduction velocities of most motor and sensory nerves are positively and linearly associated with body temperature (low temperatures slow nerve conduction velocity and higher temperatures increase conduction velocity). [1]

Conduction velocities in the Sural nerve seem to exhibit an especially strong correlation with the local temperature of the nerve. [5]

Height Edit

Conduction velocities in both the Median sensory and Ulnar sensory nerves are negatively related to an individual's height, which likely accounts for the fact that, among most of the adult population, conduction velocities between the wrist and digits of an individual's hand decrease by 0.5 m/s for each inch increase in height. [5] As a direct consequence, impulse latencies within the Median, Ulnar, and Sural nerves increases with height. [5]

The correlation between height and the amplitude of impulses in the sensory nerves is negative. [5]

Hand factors Edit

Circumference of the index finger appears to be negatively associated with conduction amplitudes in the Median and Ulnar nerves. In addition, people with larger wrist ratios (anterior-posterior diameter : medial-lateral diameter) have lower Median nerve latencies and faster conduction velocities. [5]

Medical conditions Edit

Amyotrophic lateral sclerosis (ALS) Edit

Amyotrophic Lateral Sclerosis (ALS) aka 'Lou Gehrig's disease' is a progressive and inevitably fatal neurodegenerative disease affecting the motor neurons. [9] Because ALS shares many symptoms with other neurodegenerative diseases, it can be difficult to diagnose properly. The best method of establishing a confident diagnosis is via electrodiagnostic evaluation. To be specific, motor nerve conduction studies of the Median, Ulnar, and peroneal muscles should be performed, as well as sensory nerve conduction studies of the Ulnar and Sural nerves. [9]

In patients with ALS, it has been shown that distal motor latencies and slowing of conduction velocity worsened as the severity of their muscle weakness increased. Both symptoms are consistent with the axonal degeneration occurring in ALS patients. [9]

Carpal tunnel syndrome Edit

Carpal tunnel syndrome (CTS) is a form of nerve compression syndrome caused by the compression of the median nerve at the wrist. Typical symptoms include numbness, tingling, burning pains, or weakness in the hand. [12] [13] CTS is another condition for which electrodiagnostic testing is valuable. [12] [14] However, before subjecting a patient to nerve conduction studies, both Tinel's test and Phalen's test should be performed. If both results are negative, it is very unlikely that the patient has CTS, and further testing is unnecessary. [13]

Carpal tunnel syndrome presents in each individual to different extents. Measurements of nerve conduction velocity are critical to determining the degree of severity. [14] [15] These levels of severity are categorized as: [12] [13]

  • Mild CTS: Prolonged sensory latencies, very slight decrease in conduction velocity. No suspected axonal degeneration.
  • Moderate CTS: Abnormal sensory conduction velocities and reduced motor conduction velocities. No suspected axonal degeneration.
  • Severe CTS: Absence of sensory responses and prolonged motor latencies (reduced motor conduction velocities).
  • Extreme CTS: Absence of both sensory and motor responses.

One common electrodiagnostic measurement includes the difference between sensory nerve conduction velocities in the pinkie finger and index finger. In most instances of CTS, symptoms will not present until this difference is greater than 8 m/s. [12] [13]

Guillain–Barré syndrome Edit

Guillain–Barré syndrome (GBS) is a peripheral neuropathy involving the degeneration of myelin sheathing and/or nerves that innervate the head, body, and limbs. [7] This degeneration is due to an autoimmune response typically initiated by various infections.

Two primary classifications exist: demyelinating (Schwann cell damage) and axonal (direct nerve fiber damage). [7] [16] Each of these then branches into additional sub-classifications depending on the exact manifestation. In all cases, however, the condition results in weakness or paralysis of limbs, the potentially fatal paralysis of respiratory muscles, or a combination of these effects. [7]

The disease can progress very rapidly once symptoms present (severe damage can occur within as little as a day). [7] Because electrodiagnosis is one of the fastest and most direct methods of determining the presence of the illness and its proper classification, nerve conduction studies are extremely important. [16] Without proper electrodiagnostic assessment, GBS is commonly misdiagnosed as Polio, West Nile virus, Tick paralysis, various Toxic neuropathies, CIDP, Transverse myelitis, or Hysterical paralysis. [7] Two sets of nerve conduction studies should allow for proper diagnosis of Guillain–Barré syndrome. It is recommended that these be performed within the first 2 weeks of symptom presentation and again sometime between 3 and 8 weeks. [16]

Electrodiagnostic findings that may implicate GBS include: [6] [7] [16]

  • Complete conduction blocks
  • Abnormal or absent F waves
  • Attenuated compound muscle action potential amplitudes
  • Prolonged motor neuron latencies
  • Severely slowed conduction velocities (sometimes below 20 m/s)

Lambert-Eaton myasthenic syndrome Edit

Lambert–Eaton myasthenic syndrome (LEMS) is an autoimmune disease in which auto-antibodies are directed against voltage-gated calcium channels at presynaptic nerve terminals. Here, the antibodies inhibit the release of neurotransmitters, resulting in muscle weakness and autonomic dysfunctions. [17]

Nerve conduction studies performed on the Ulnar motor and sensory, Median motor and sensory, Tibial motor, and Peroneal motor nerves in patients with LEMS have shown that the conduction velocity across these nerves is actually normal. However, the amplitudes of the compound motor action potentials may be reduced by up to 55%, and the duration of these action potentials decreased by up to 47%. [17]

Peripheral diabetic neuropathy Edit

At least half the population with diabetes mellitus is also affected with diabetic neuropathy, causing numbness and weakness in the peripheral limbs. [18] Studies have shown that the Rho/Rho-kinase signaling pathway is more active in individuals with diabetes and that this signaling activity occurs mainly in the nodes of Ranvier and Schmidt-Lanterman incisures. [18] Therefore, over-activity of the Rho/Rho-kinase signaling pathway may inhibit nerve conduction.

Motor nerve conduction velocity studies revealed that conductance in diabetic rats was about 30% lower than that of the non-diabetic control group. In addition, activity along the Schmidt-Lanterman incisures was non-continuous and non-linear in the diabetic group, but linear and continuous in the control. These deficiencies were eliminated after the administration of Fasudil to the diabetic group, implying that it may be a potential treatment. [18]


What would be the conduction speed of A-alpha fibers, were they unmyelinated? - Biology

A- represents negatively charged proteins.

Inside a cell considerable calcium is sequestered. For example, from the endoplasmic reticulum of muscle cells Ca++ is released when the cell is stimulated.

Cells have ionic pumps that maintain concentration gradients. The ions themselves can "diffuse" in or out of the cell through specialized protein channels.

Reading:
George B. Benedek & Felix M. H. Villars, Physics With Illustrative Examples from Medicine and Biology, Vol 3: Electricity and Magnetism, Addison-Wesley, Reading, Massachusetts (1979).

Howard C. Berg, Random Walks in Biology, Princeton Univ. Press (1983). A statistical physics look at the diffusion-drift development that leads to the Nernst potential (p. 141). Berg is well-known for his "Life at low Reynolds number" essay: see p. 75 of the book.

Bertil Hille, Ion Channels of Excitable Membranes, Sinauer Associates, 814 pp., (2001)

Fick's First Law
Consider diffusion flux J along one dimension: where the last form is the gradient in 3D.
In these equations J is a flux [a vector, particles/(area-sec)] and concentration is C at point x. D is the diffusion coefficient, and has dimensions of cm^2/sec. Notice the minus sign! A positive concentration gradient leads to a negative direction for diffusion. See figure below. Imagine we're considering a concentration gradient for K+=potassium ions across a cell's membrane.

Flux as current: Fick's 1st Law tells us about a diffusive flux of particles, charged or uncharged. For example, glucose is an uncharged particle in solution, and is subject to Fick's Law just as well as charged K+ and Cl-, but glucose flux is NOT a current! A flux of anions (+) is a positive current in the same direction as the flux, while a flux of chloride cations (-) is a current in the opposite direction.

The diffusion of charged particles (in the case we're considering, of K+) will set up an E field which will oppose the diffusion flux, and in fact will set up a voltage difference across the membrane.

Consider also that charged ions going in and out of a cell are going in and out of a fairly confined space, and the charge accumulation or deficit can be enough to generate a significant E field across the membrane. Even outside the cell space is rather confined, with the spacing between cells again able to be measured in angstroms, leaving "outside" not the same as a resevoir.

Drift of charged particles in an E field. In a material, charged particles will "drift" with a velocity proportional to their mobility &mu, their charge, and the strength of the E field:

If the charged particles were in a plasma they would move under the influence of F=ma, accelerating, but here in a material the particle reach a "terminal velocity". (Demo with corn syrup, where a ball bearing falls faster under the influence of gravity than a marble of the same size.)
mobility &mu is a property of a particle in a material (in this case aqueous electrolyte, and has units cm/(sec N ) .

The flux due to drift in the E field will be proportional to the concentration of the ions.
(which can be morphed to Ohm's Law, and, again, compare to F = ma)

We can now combine the diffusion flux and the drift flux in a steady-state version of KCL (flux as current with explicit account taken of area normal to current flow).

We need to relate diffusion constant D to mobility &mu : Einstein found
(in metals), where gas constant R and Boltzmann constant k are related by
R = kN, where N is the number of molecules in question. (ref: Van Vlack, Elements of Materials Science 2nd Ed. Addison-Wesley, 1964. pp 105, 98). As a result (considering other material factors too), when temperature rises, diffusion coefficient D always increases while mobility &mu increases for non-metals and decreases for metals. Further information: in F. Reif, Statistical Physics McGraw-Hill, New York, (1967), page 337 shows that, in general, .

Rewriting the flux equation, we have

where you can see that temperature T is still involved but mobility &mu has cancelled out. Later, when different ionic species have their own mobility, various &mu's will survive in a multi-ion formula.

Definition of potential difference = voltage. Now we need to remind you that
Since this integral is "conservative" you can go along any path from gnd to point P (in our case, from outside to inside the cell, across the membrane). Therefore integrate the flux balance equation to end up computing voltage. We will be grounding the extracellular space, called OUT in the integral.

Remembering log(X) - log(Y) = log(X/Y), and computing that, at room temperature, kT/q = 25 mV we have
,the Nernst equation for a singly-charged positive ionic species at room temperature. Consider a ratio of internal to external potassium of 10:1, we find that V K = -58 mV, which turns out to be what is measured.

E field strength across the membrane: 70 mv divided by 10 Angstroms

= 100M V/meter. near the breakdown strength.

What happens to the Nernst potential if calcium instead of potassium is considered? Ans: Calcium is Ca++, a doubly charged ion, in solution. Therefore substitute 2q in the kT/q term of the Nernst equation. The Nernst voltage is reduced by a factor of 2! Think of it this way: in the same E field a Ca++ ion will experience twice the force as a K+ ion. Therefore half the field strength would be needed to exert the same force on Ca++.

What happens if chloride ion is considered in a Nernst potential calculation? There are two ways to think about that question: (1) Since chloride ion has the opposite sign of K, then all other things being equal, the sign of the answer for the Nernst potential should be opposite to that of potassium. (2) Since chloride has a higher concentration outside than inside the cell, then the sign of the answer should be the same as potassium.

Where does the sign change come in when considering negatively charged ions? Consider the diffusion flux. If chloride Cl- concentration is greater OUT than IN the direction of diffusion of chloride ions from OUT to IN (left to right below). But because chloride has a negative charge, the direction of the chloride diffusion current will be opposite, from right to left.

Therefore the diffusion current term in the flux balance will have the opposite sign for a negatively charged ion.

Now consider the drift flux due to the electric field of charge separation. In our K+ equation the term q was +e, where e is the magnitude of the charge on an electron. Now q becomes -e for the chloride ion. But the electric field changes direction too, because negative charges instead of positive charges have moved into position to block the further diffusion of chloride ions. You can write the drift equation as

and you see it will have the same sign as the potassium version. Thus the only effective sign change occurs in the diffusion flux term, and we will expect a sign change in the answer if we're considering Cl- to have the same concentration gradient as K+.

Given the concentration gradient of sodium Na+, what will be the sign of the "sodium equilibrium potential" (Nernst potential considering sodium alone)? Because sodium concentration is higher outside the cell than in, it's Nernst potential will be positive, and will follow the same Nernst equation logarithmic law as potassium.

Protein as pumps and channels for ions. We have been working with mobilities of ions as factors in their Nernst potentials. In fact, most ions move relatively freely inside and outside the cells it's at the membrane barrier that mobility becomes important. We understand that ionic imbalances are maintained by pumps, in the form of proteins in the cell membrane, in somewhat the same way that an air conditioner in a window helps maintain a temperature gradient from inside to outside the house. Proteins also form channels for specific ions, and the permeability (a more common term for ionic mobility in the membrane) of a channel can be modulated by synaptic activity or transmembrane voltage. It is possible to record, by means of a patch clamp electrode isolating a small section of membrane, the signals of individual channels. See below, from www.ccs.fau.edu/

Reading Bertil Hille, Ion Channels in Excitable Membranes, 3rd Ed., Sinauer Associates (2001).

The 1991 Nobel Prize in Physiology or Medicine went to Bert Sakmann and Erwin Neher for their work on patch clamping.

EXAMPLE: Say the concentration of K+ and Cl- is the same C(x) everywhere and that the mobility of K+ > mobility Cl-. What is an expression for the transmembrane voltage? ANS: You need to add up the fluxes of K+ and Cl-. We already have the K+ flux:

and the Cl- flux sum will be
because the Cl- diffusion flux is in the opposite direction, but the electrical flux is the same direction. Add the two contributions up, sum to zero, separate variables and see

now integrate from OUT to IN, as before, and obtain
, which gives the potassium-only answer from before if &mu Cl = 0.

Another problem: Now assume the concentrations differences of sodium and potassium are equal and opposite across a membrane. If the mobilities of Na and K are equal, then the transmembrane voltage will be zero. In fact the voltage is negative, in the direction of the K equilibrium potential. Assume the mobility of K > mobility of Na ions. What will be the voltage difference? Note that if C(x) is the potassium concentration function, then the Na concentration function is C(-x). Furthermore . How far can you get?
A better approach, when the concentration gradients of the ionic species differ: compute independently the Nernst potential of each ionic species. Then use superposition and compute the total of the respective ionic mobilities, for a weighted sum:

where N is the total number of ionic species, and Vj is the Nernst potential of the jth species.
Example. Assume the concentration gradients of Na and K are equal and opposite, +58 and -58mV. Suppose at rest the mobility of K is 3 times greater than the mobility of Na. Then Vinside = 0.75*(-58) + 0.25*58 = -29mV.

Increase in Na+ permeability during excitation: In the resting state of a nerve cell sodium mobility across the membrane is much lower than potassium, and the cell maintains a negative voltage. When a nerve or muscle cell is stimulated by synaptic transmission, the mobility (or channel conductance, or permeability) for sodium transiently increases to a value greater than that for potassium and the cell internal voltage "spikes" above zero volts for about a millisecond. (image below from http://www.bio.psu.edu/Courses/Fall2002/Biol142/neurons/neurons.html)

The following assumes passive conduction of voltage changes down an axon or dendrite. Consider a cylindrical tube as a model for a dendrite or axon process. The wall of the tube will be a high resistance membrane and the inside of the tube will be low resistance axoplasm.

Say the inside of the tube has resistance/length = r-in, as shown below.


How will r-in depend on cable diameter? Consider the material property we called rho in strain gauge development: here it will appear again, as specific axial resistance of "axoplasm", and its value is about 100 &Omega--cm. Therefore r-in = rho/area.

Conductance g-in = 1/r-in will increase as the square of the diameter.

Now consider the leakage current im going out of the membrane. Per "compartment" of length we have a picture like,

If you differentiate again and substitute the above equation in, you have

where the minus signs cancel out.

Now on to the time domain: Consider that there is capacitance in the membrane:

the membrane current is now to be expressed as:

How are r-in, r-m and c-m calculated from physical properties of the cell and its membrane? Recall from the strain gauge lecture that resistance R, in ohms, for a rod of length L and cross-section A is

where rho is the resistivity of the material of the rod. The units of resistivity are Ohm-cm. Resistance per unit length, by a "dimensional analysis," is therefore

where d is the diameter of the cable under consideration. Therefore knowing radius r allows calculation of resistance r-in.

It is known that resistivity of axoplasm is 100 Ohm-meter (From Neuron to Brain , p. 141)
Compare this to the resistivity of metal, like copper: about 10^-8 ohm-m!
Axoplasm is the same order as crystal silicon resistivity.

What's the resistance of a 1 cm long axon, diameter 10 microns? about 10^10 Ohms!

Next, consider membrane as a sheet of material specified by capacitance/cm^2 Cm, and by conductance/cm2 Gm. Conductance has units of mhos. Why use conductance? It's proportional to the area of the membrane under consideration. For a cable of diameter d, the circumference is pi·d. So pi·d*1cm is the area of a unit length (cm) of membrane. Capacitance per unit length c-m is then Cm·pi·d and conductance per unit length is Gm·pi·d. therefore

Both capacitance per unit length and resistance per unit length can therefore be calculated from material properties of membrane.
Membrane has 1 muF /cm^2 and a RESISTANCE of about 2000 Ohms/cm^2
These factors allow calculation of time constant and length constant of membrane.
A passive voltage change will decay toward zero with length constant lambda.

Solutions to the cable equation: See D. J. Aidley, The Physiology of Excitable Cells, page 50 ff and
B. Katz, Nerve, Muscle and Synapse, Oxford Univ Press (1970)

Myelination significantly increases Rm and therefore increases the length constant of a axon, typically from 10 to 2000 microns! The nodes of Ranvier are closer than one length constant. A myelin wrap can also reduce membrane capacitance (remember capacitors in series?) so the effective time constant of the membrane is about the same the result is faster propagation of an action potential in a myelinated axon.

Calculating the speed of conduction down a cable:

From ICHIJI TASAKI, "ON THE CONDUCTION VELOCITY OF NONMYELINATED NERVE FIBERS," Journal of Integrative Neuroscience, Vol. 3, No. 2 (2004) 115𤩬.

The cable equation is a PDE run matlab function pdex4.m to see a related example. For both time and position x, V can vary. a 3-D plot is needed.
C:MatlabR12ToolboxMatlabdemos


Lesson Explainer: Nerve Cells Biology

In this explainer, we will learn how to describe the structure of different types of nerve cells and outline their functions.

The picture below is one of the first clear images of a neuron hand drawn by Santiago Ramon y Cajal, the father of modern neuroscience. In his drawing, you can see that neurons are separate, individual cells. Prior to these drawings by Ramon y Cajal, most scientists of the time believed the nervous system was a network of continuous fibers. This is because in the late 1800s, while microscopy could allow observation of the brain cells, it did not have the capability to capture these observations outside of the observer’s eye. This is why these drawings of neurons are so special! They made it obvious that neurons were individual specialized cells that seemed to communicate across a small “gap” (what we now call the synapse).

Neurons are the main signaling unit of the human nervous system. You may recall that neurons are specialized cells that transmit nerve impulses and are found throughout the central and peripheral nervous systems. By best estimates, the adult human brain contains about 86 billion neurons. It would take you over 3‎ ‎000 years to count them all!

While neurons are the main signaling unit, they are highly specialized and, therefore, need support cells to function. Neurons are supported by the other main cell type found in the human nervous system, glial cells (or glia). Unlike neurons, glia do not produce electrical impulses, and because of this, they were thought not to be critically important in nervous system function. However, glial cells, which are also called neuroglia, accomplish multiple key functions: they provide structural framework, form a barrier with blood vessels, insulate neurons, monitor and clean up the environment, and help maintain the health of neurons by repairing damaged parts and providing nutrients. This support by the neuroglia is very important since neurons cannot undergo mitosis like most other body cells. This is because neurons lack centrioles, which are important organelles in cell division.

Key Term: Neuron

A neuron is a specialized cell that transmits nerve impulses.

Key Term: Glia (Neuroglia)

Glia are nonneuronal cells that provide support to neurons.

Since neurons are the main signaling unit of the human nervous system, most neurons share the same basic anatomy. A neuron has five common features: the dendrites, cell body, axon, myelin sheath, and axon terminals. These anatomical structures are illustrated below in Figure 2. The transmission of electrical impulses involves each of these features, starting with the dendrites.

The dendrites receive chemical signals from other neurons at the level of contact formed with other neurons, called synapses. At synapses, the chemical signal, called neurotransmitter, generates an electrical signal, which is then conducted by dendrites to the cell body of the neuron. The impulses travel from the dendrites to the cell body to be integrated with other signal inputs received by the other dendrites. After integration and processing in the cell body, the electrical signal travels down the axon toward its final destination. Electrical signals never travel back from the terminals to the soma.

Some neurons are wrapped in a fatty coating called myelin sheath, which helps increase the conduction of the electrical signal down the axon. Finally, to pass the electrical impulse to another neuron or muscle cell, the axon terminals convert the impulse into chemical signals and release them across a small gap. Reception of the chemical signals by the dendrites of another neuron or the muscles helps sustain the sequence of information.


What is C fibers?

Click to see complete answer. In this way, what are a delta and C fibers?

C-nerve fibers are unmyelinated. A-alpha nerve fibers carry information related to proprioception (muscle sense). A-beta nerve fibers carry information related to touch. A-delta nerve fibers carry information related to pain and temperature. C-nerve fibers carry information related to pain, temperature and itch.

Subsequently, question is, what are the two types of pain fibers? Each of the spinal nerves emerging from the spinal cord through the space between two vertebrae consists of two types of fibres: sensory fibres, which come from the dorsal root of the nerve, and motor fibres, which come from its ventral root.

In this regard, what are a fibers?

Type A fibers: They are myelinated. They have a diameter of 1.5-20 micron. Their speed of conduction is 4-120 m/sec, which shows that they have a really fast conduction of impulse. Examples of type A fibers are skeletomotor fibers, fusimotor fibers and afferent fibers to skin.

What are cardiac C fibers?

These fibers are thought to act mainly as tension receptors within the cardiac wall (38), and they are stimulated mechanically when cardiac filling pressure [left ventricular (LV) end-diastolic pressure (LVEDP)] increases in the setting of volume expansion.


6 AXON CALIBER

In addition to myelin and the axon sub-domains described above, another major determinant of conduction velocity along myelinated axons is the morphology of the axon itself, and in particular its cross-sectional size, or caliber. Axons in the CNS vary in cross-sectional size (caliber) by over 100-fold in diameter or 10,000-fold in area (Perge, Niven, Mugnaini, Balasubramanian, & Sterling, 2012 ). Although not an absolute relationship in vivo, in general CNS axons under 0.5 μm in diameter are unmyelinated, and those over that value myelinated, with some myelinated axons in the vertebrate CNS exceeding 50 μm in diameter (Klingseisen & Lyons, 2017 Perge et al., 2012 ). Axon caliber is in and of itself a signal that can regulate both myelin sheath formation and growth (Bechler, Byrne, & Ffrench-Constant, 2015 Lee et al., 2012 Mayoral, Etxeberria, Shen, & Chan, 2018 ), and in the PNS at least, myelin in turn reciprocates and supports the continued growth of axons in caliber (Sherman et al., 2012 ). Once again such bidirectional interactions indicate that adaptive changes to either axons or myelin are likely interdependent.

In myelinated axons conduction velocity increases proportionally with increasing axon diameter (Waxman & Bennett, 1972 ), because larger diameter leads to a reduced axial resistance and an increase in the inward ionic current, due to larger surface area. In addition to regulating conduction velocity, axons of different caliber have also been proposed to have distinct capacities to sustain high-frequency firing of APs (Perge et al., 2012 ) (Figure 1 ), although this remains to be experimentally validated. Axons can also exhibit notable changes in caliber along their length (Greenberg, Leitao, Trogadis, & Stevens, 1990 ) and such variation has also been predicted to have the capacity to change conduction velocity along axons (Goldstein & Rall, 1974 ). In the avian auditory system, differences in axonal diameter between axon branches of the same neuron, but of different length, has been associated with tuning of conduction velocities (in conjunction with specific patterns of myelination) to allow simultaneous AP arrival at target neurons (Seidl & Rubel, 2016 ). In addition, it is well-known that there are conspicuous constrictions in axon calibre at the PNP region, which have also been predicted to regulate conduction velocity (Halter & Clark, 1993 ).

Recent studies have indicated that like many other aspects of the myelinated axon, axon diameter is also adaptable and can change in response to neuronal activity. In one super-resolution-based study of cultured unmyelinated axons, it was found that both synaptic boutons and axons rapidly increase in diameter following high-frequency neuronal stimulation (Chéreau, Saraceno, Angibaud, Cattaert, & Nägerl, 2017 ). Interestingly, the observed increase in the synaptic bouton size was temporary and predicted to decrease AP conduction velocity, whereas the longer-term increase in axon caliber was predicted to speed up AP propagation. In vivo data supporting the premise that neuronal activity can regulate caliber along myelinated axons has recently come from findings that axon diameter increases with the onset of hearing and that blocking auditory input resulted in axons growing to smaller diameters (Sinclair et al., 2017 ). However, changes to myelination were also observed with both hearing onset and sensory deprivation, highlighting yet again the need to experimentally disentangle how specific forms of activity might coordinately or differentially affect axon caliber, myelination, and axon domains along myelinated axons and in intact circuits.


Nerve Conduction

A s the result of the experimental demonstrations carried out by Luigi Galvani and his followers , the electrical nature of the nerve-muscle function was unveiled. However, a direct proof could only be made when scientists could be able to measure or to detect the natural electrical currents generated in the nervous and muscular cells. Galvani did not have the technology to measure these currents, because they were too small. Electroscopes , the devices used in that time, were not sensitive enough. As a result, the study of bioelectricity almost disappeared from the scientific scene until 1827.

In 1826, Johannes Müller (1801-1858), a noted German psychologist and physiologist, proposed his theory of "specific nerve energies", which stated that different nerves (optical, auditive, etc.) carried a kind of "code", which identified their origin to the brain. His position was based on vitalism, a philosophical doctrine which affirmed that life was characterized by an intrinsic "vital energy". However, it was important as the beginning of a whole new school of neurophysiological thought, which eventually would refute vitalism as a valid concept in biology.

The stage for the revolutionary discoveries on nerve function which would be made in the next decades was set against a background of continuous advances in anatomical knowledge about the nervous system. In 1836, Robert Remak described myelinated and unmyelinated axons. In the next year Jan Purkyne described cerebellar cells and identified the neuron nucleus and processes. Again in 1838, both he and Remak suggested that the nerve fibers and nerve cells are joined (i.e., the nerve fiber or axons is a process arising from the nerve cell). In 1839, Theodor Schwann proposed the cell theory, i.e., that the nervous system is composed of individual neural cells.

Jan Purkyne and Theodor Schwann

Then, in 1848-9, half a century after Galvani's discovery, and thanks to the invention of the galvanometer (made two decades earlier), the Swiss-German scientist Emil Heinrich Du Bois-Reymond (1818-1896), professor of physiology in Berlin and a disciple and sucessor of Johannes Müller, was able to use a sensitive apparatus developed by him to detect what he called " action current " in the frog's nerve. It was named this way because Du Bois-Reymond noted a small negative variation of the resting electrical potential in metallic electrodes connecting the nerve to a galvanometer only when the stimulation of the nerve (mechanical or electrical) would elicit a response from the muscle. He was able to demonstrate that this phenomenon of "negative variation" also occurs in striated muscle and is the primary cause of muscular contraction.

Emil Du Bois-Reymond and his
interpretation of electric fields
in the nerve-muscle plaque .

The action current (later called action potential) was discovered to be a kind of "electrical impulse wave" which propagated at a fixed and relatively slow speed along the nerve fiber. In 1852, Hermann von Helmholtz (1821-1894) was able to measure the speed of frog nerve impulses, and determined it to be about 27 meters/second. Du Bois-Reymond contributions, published in his book " Untersuchungen über thierische Elektricität ". ("Researches on Animal Electricity") in 1848, created the field of scientific electrophysiology. The work of both men served to refute the view by their teacher, Johannes Müller, that the nerve impulse was an example of a vital function that could never be measured experimentally, and their collaboration with a remarkable group of physiologists, composed by Carl Ludwig and Ernst von Brücke was very important to reduce physiology to applied physics and chemistry, a trend that has dominated physiology and medicine ever since. They had " sworn to each other to validate the basic truth that in an organism no other forces have any effect than the common physiochemical ones. . . . "

The impressive technological advances in the second half of the 19th century would advance the new field of electrophysiology.