Nervous System Theory: An Introductory Study

By K. N. Leibovic

Nervous System Theory: An Introductory Study focuses on the nervous system theory, stressing the means for understanding the nature of the biological system rather than the elaboration of mathematical theories. This book begins with a discussion on single-cell responses, followed by a discussion of sensory information processing that leads into models of perceptual processes and their possible neural bases. This text concludes with some general principles and theoretical investigations relating to units that make up a nervous system, through a sensory pathway and central structures. The peripheral stimuli that explain the operations of the brain are also described. This publication is a good reference for neurologists, medical practitioners, and researchers conducting work on the nervous system theory.

• Language: English
• Publisher: Academic Press
• Release date: Oct 22, 2013
• ISBN: 9781483267319

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CONCLUSION

Chapter 1

INTRODUCTION

Publisher Summary

The nerve cell is distinguished from other cells by having a number of long, fibrous processes emanating from the cell body and by its ability to generate and transmit signals. Of these the axonal signals are of constant amplitude and variable frequency, whereas the dendritic ones decay in amplitude as they spread along the fiber. This chapter discusses the resting potential and the transmission of signals in axons and dendrites. It discusses the synaptic communication and some reduced models of neuron function. It is found that different problems necessitate the formulation of different models. For example, in the treatment of the resting potential, the ionic flows are divided into components due to diffusion and the electric potential, whereas in the treatment of the cable properties of a nerve fiber, all the flows are considered as electric current dependent on conductance, capacitance, and potential. Again in the treatment of active membranes, the ionic flows are separated into species components, each with its own conductance and driving force expressed in terms of potentials.

The nerve cell is distinguished from other cells by having a number of long, fibrous processes emanating from the cell body and by its ability to generate and transmit signals. Of these the axonal signals are of constant amplitude and variable frequency, whereas the dendritic ones, as a rule, decay in amplitude as they spread along the fiber. In both cases the signals are in the form of changes of membrane potential. In the resting state the potential inside the cell is some 50–90 mV below that outside the cell. This potential difference is regulated by differences of ionic concentration.

The cell membrane is 50–100 Å thick and separates two aqueous solutions of which the one outside is rather more electroconductive than the one inside. The sodium and chloride ion concentrations in the external medium are about 10 times as high as those inside the cell, whereas the potassium ion concentration inside the cell is about 30 times that outside. The permeability of the cell membrane is low but the K+ and CI− ions move through it much more readily than the Na+ ions, whose normal leakage into the cell is counteracted by the metabolically driven sodium pump.

If the ionic permeabilities of the membrane are fixed, then a change of polarization produced at some point on a fiber will spread and decay as it is conducted electrotonically, as in a passive, leaky cable. In an active fiber, on the other hand, the ionic permeabilities depend on the state of polarization of the membrane, and a pulse of depolarization can be regenerated as it is conducted along the fiber. Thus, in an axon, when the resting potential of the initial segment is raised above a certain threshold value, the depolarization is rapidly amplified, producing a spike, or action potential, which travels along the axon with constant amplitude. This is based on the following mechanism (Hodgkin and Huxley, 1952). The depolarization of the membrane increases Na+ permeability, whereupon external Na+ ions rapidly enter the cell and amplify the initial depolarization. With only a brief delay, however, the K+ permeability is increased and K+ flows out of the cell, while the Na+ permeability is reduced to its former value, returning the membrane potential toward the resting level. As the spike develops, the membrane immediately ahead is depolarized and the process is repeated along the length of the fiber, ensuring the propagation of the action potential. The process can be compared to conduction along a leaky cable with repetitive boosting. But during and shortly after the action potential, the fiber goes through a refractory period in which the generation of new impulses is suppressed. In nonmyelinated axons the action potential travels continuously along the fiber. Myelinated axons are surrounded by insulating sheaths with periodic interruptions, the so-called nodes of Ranvier, and the action potential jumps from node to node and conduction is more rapid.

Some neuron characteristics are illustrated quantitatively by the following typical figures. The internal resistivity is of the same order of magnitude as, though up to three times higher than, the external resistivity, both being within the range of a few tens to a few hundreds of ohm centimeters. The axonal membrane resistance is of the order of a few thousand ohms per square centimeter and membrane capacitance is typically 1 μF/cm². During the nerve impulse in a squid giant axon, some 3-4 × 10x−12 mole of Na+ ions may be taken up, and the same amount of K+ ions may be lost, per square centimeter of fiber. This represents only a small fraction of the ionic content of the squid axon—in the case of K+ it is of the order of 10−6—and the original concentrations are restored quickly by metabolic activity.

Although the system consisting of membrane, external medium, and internal medium is electrically neutral on the macroscopic scale, the inside of the cell membrane carries a slight negative charge which is balanced by an equal positive charge on the outside. This capacitative charge is of the order of 6 × 10−8 C/cm², corresponding to a capacitance of 1 μF and a potential difference of, say, 60 mV. From this it can be seen that relatively large changes of potential can arise from quite small charge displacements. For example, a 20% change of potential would involve a change of 1.2 × 10−8 C/cm², corresponding to a displacement of some 7.5 × 10¹⁰ univalent ions.

Although the foregoing figures are fairly representative, it should be remembered that the properties of neurons vary widely. For example, fiber lengths may vary between several microns and a meter or more, fiber diameters between a micron or less and as much as a millimeter. In general, the larger the diameter, the higher the conduction velocity and the shorter the spike duration and refractory period. The variables are, however, inter-related in a complex fashion. Conduction velocity, for example, depends among other factors on the rate at which Na+ permeability increases and then decreases, the rate and timing with respect to Na+ of the K+ permeability changes, and the spike amplitude and the ionic mobilities within the fiber.

The axon represents the output line of the neuron. It usually ends in a number of branches making contact with the dendrites or cell body of other neurons. The dendrites represent input lines, and the numerous signals impinging on them are integrated within the dendritic tree and the cell soma. Although dendrite responses are not as well understood as the axonal potential, they clearly form a very important part of neuron function. The extent and the ramifications of a dendritic tree are often most impressive in relation to the rest of the cell. In many dendrites there may only be electrotonic conduction, but in others spikes are generated, and clearly involve an active membrane.

Communication between neurons can take place through chemical transmitters at synaptic junctions, through direct electrical transmission at tight junctions, or through changes in the ionic composition of the intercellular space produced by activity of neighboring cells. The chemical transmitter is stored in synaptic vesicles and released across the synaptic gap in quantal fashion. The ionic permeabilities of the postsynaptic membrane are altered selectively, depending on the transmitter and the membrane properties. As a result, the postsynaptic membrane may be either depolarized or hyperpolarized.

Properties of nerve cells have been described extensively as, for example, in the following publications, where additional references can be found: Cole (1968); Eccles (1964); Hodgkin and Huxley (1952); Katz (1966); Lorente de Nó (1947); Mountcastle (1968); Quarton et al. (1967); Ruch et al. (1965).

The topics mentioned in this introduction will be taken up in more detail in the following chapters, starting with the resting potential, going on to the transmission of signals in axons and dendrites, and concluding with synaptic communication and some reduced models of neuron function. It will be found that different problems necessitate the formulation of different models. For example, in the treatment of the resting potential the ionic flows are divided into components due to diffusion and the electric potential, whereas in the treatment of the cable properties of a nerve fiber all the flows are considered as electric current dependent on conductance, capacitance, and potential. Again in the treatment of active membranes the ionic flows are separated into species components, each with its own conductance and driving force expressed in terms of potentials. Clearly there is a connection between different formulations, but the moral of all this is that it is necessary to maintain some flexibility in formulating a model. This is dictated by convenience of the mathematical treatment of experimental results, and it is developed further in the final chapters of this part of the book.

REFERENCES

COLE, K.S.Membranes, Ions and Impulses.. Berkeley, California: Univ. of California Press, 1968.

ECCLES, J.C.The Physiology of Synapses.. New York: Springer-Verlag, 1964.

HODGKIN, A.L., HUXLEY, A.F. J. Physiol.. 1952; 116:449–506. [117, 500–544.].

KATZ, B.Nerve, Muscle and Synapse.. New York: McGraw-Hill, 1966.

LORENTE DE NÓ, R. A Study of Nerve Physiology. Stud. Rockefeller Inst. Med. Res.. 1947; 131:132.

MOUNTCASTLE, V.B., eds. Medical Physiology,, II. St. Louis, Missouri: Mosby, 1968.

QUARTON, G.C., MELNECHUK, R., SCHMITT, F.O., eds. The Neurosciences.. Rockefeller Univ. Press, New York, 1967.

RUCH, TC., PATTON, H.D., WOODBURY, J.W., TOWE, A.L.Neurophysiology.. Philadelphia, Pennsylvania: Saunders, 1965.

Chapter 2

THE RESTING POTENTIAL

Publisher Summary

The resting potential of the nerve cell is because of the differences of ionic concentration, which are maintained on the two sides of the membrane by metabolic processes. If undisturbed, the system remains in a steady state, but it is not in thermodynamic equilibrium. When a molecule is dissociated into ions in a solution, and the latter is divided by a semipermeable membrane into two compartments with unequal ionic concentrations, then an electric potential difference exists between the compartments. This chapter discusses how a relationship between potential and concentration differences at equilibrium can be derived. It discusses the exchange of ions across a membrane as a more accurate derivation of the resting potential must take into account that several species are involved in ion exchange across the membranes and as it must include the different permeabilities of the membrane to the various ions.

As already mentioned, the resting potential of the nerve cell is due to the differences of ionic concentration which are maintained on the two sides of the membrane by metabolic processes. If undisturbed, the system remains in a steady state, but it is not in thermodynamic equilibrium. Nevertheless, the following argument gives reasonable agreement with experimental data, for the departure from equilibrium is not too great. In addition, it is worth comparing the equilibrium treatment with the subsequent one, which takes into account ionic exchanges across the membrane.

When a molecule is dissociated into ions in a solution, and the latter is divided by a semipermeable membrane into two compartments with unequal ionic concentrations, then an electric potential difference exists between the compartments. The relationship between potential and concentration differences at equilibrium can be derived as follows (Moelwyn-Hughes, 1966).

Consider two compartments 1 and 2 at different electric potentials ϕk k = 1, 2. Suppose a small quantity of material δn moles carrying an electric charge δnzF is transferred reversibly from compartment 2 to compartment 1 at constant temperature T and pressure p. F is the Faraday constant and z is the valence of the material. In general, the internal energy U, entropy S, and volume V per mole of the substance will be different in the two compartments. The change of internal energy δn ΔU of the transferred material will be due to a change of heat energy δnT ΔS, a change of work energy -δnp Δ(pV), and a change of electric energy -δnzf Δϕ, where ΔX = X2 - X1 is the difference of a quantity X between the two compartments. Thus,

that is,

(2.1)

where G = H - TS is the Gibbs free energy. If the two compartments are separated by a semipermeable membrane, which allows passage of the material through it, then a reversible transfer implies that the compartments are in equilibrium. Hence equation (2.1) holds only at equilibrium and, conversely, at equilibrium the total energy of the substance on the two sides of the membrane must be equalized and hence (2.1) is true. When we have to distinguish between different component materials, then equation (2.1) is written

(2.2)

where μt is the chemical potential, or the Gibbs free energy per mole of component t. The quantity μt + zt Ft and is called the eletrochemical potential.

Now, in an ideal solution at constant temperature and pressure, the difference in the Gibbs free energy of a solute t which is present on two sides of a partition in concentrations Ct1 and Ct2, respectively, is given by

(2.3)

Hence, from (2.2) and (2.3), we obtain the Nernst equation:

(2.4)

It should be noted that equation (2.4) can be valid only when the membrane is permeable to the ionic species t. If there are only an ion and its counterion and the membrane is equally permeable to both, then the concentrations on the two sides will equalize as a result of passive diffusion. If the membrane is completely impermeable, then the concentrations and potentials on the two sides can clearly have any values, since there need be no electrochemical equilibrium between them, as implied by equation (2.2).

As an example of (2.4) consider a representative 30 : 1 ratio of potassium concentrations between the inside and outside of a neuron. Substituting R = 8·3136 × 10⁷ ergs/°C/mole, F = 96,489 C/mole, T = 290°K, and z = +1 in equation (2.4) gives

ϕ(inside) - ϕ(outside) = −85 mV approximately.

Indeed, it is found experimentally that this is quite close to the actual value of the membrane potential difference, the inside of the cell being usually between 50 and 90 mV negative with respect to the outside.

A more accurate derivation of the resting potential must take into account that several species are involved in ion exchange across the membranes, and it must include the different permeabilities of the membrane to the various ions. Thus, consider the exchange of ions across a membrane under the following assumptions.

1. The only forces acting are concentration and voltage gradients within and close to the membrane. Within the solutions separated by the membrane the concentrations and potentials are constant.

2. Individual ions move through the membrane independently of each other, and the fluxes due to diffusion and the electric field are additive.

Let the fluxes of ion species t due to diffusion and potential gradient be JtDt Jtϕ respectively. Then

where Dt is the diffusion coefficient, ct the concentration, ηt the mobility, zt the valence, e the electronic charge, and ϕ the potential. The mobility is related to the other parameters through Einstein’s equation

(2.5)

where k is Boltzmann’s constant, F Faraday’s constant, and R the gas constant. Hence the total flux of ion species t across the membrane is given by the Nernst–Planck equation:

(2.6a)

or

(2.6b)

At equilibrium Jt = 0; hence, in this special case,

that is,

This is the same result as in (2.4). The suffices i, e refer to the inside and outside (exterior) of the membrane, respectively, and ϕt is the equilibrium potential for ion species t. In this situation the ionic flow due to diffusion is just balanced by the flow due to the electric potential.

In a neuron the Na+, K+, and CI− ions are the ones primarily involved in flow through the membrane. The following additional assumptions will now be made.

3. The sum of the ionic currents in the steady state is zero.

4. The membrane is homogeneous.

5. The electric field inside the membrane is constant.

6. The ionic concentrations at the membrane interfaces are proportional to the concentrations in the solutions in contact with the membrane.

Let d be the membrane thickness, y the distance of a point within the membrane from the inner surface, and ϕm = ϕi - ϕe the membrane potential difference. Then from assumption 5

(2.7)

This is the potential to which each ion is subject at the point y within the membrane. If it, is the steady-state current carried by ion species t from the inside to the outside of the cell, then the flux of that species is

(2.8)

Here Jt is a scalar quantity, since only the y direction is involved and, moreover, for purely radial flow without sources or sinks, it is independent of y. From (2.6)–(2.8) we have

Integrating and applying the boundary conditions, we obtain

(2.9)

This holds for each ionic species, and since the net current is zero in the steady state,

(2.10)

Moreover, from assumption 6, if the proportionality constants inside and outside the membrane are equal,

where βNa is the proportionality constant and [Na]e, [Na]i denote the external and internal Na+ concentrations. Similar expressions hold for CeK, CiK CeCl CiCl From (2.9) and (2.10), then, follows the Goldman equation for the resting potential ϕr.

(2.11)

where Pt = βt Dt; t = K, Na, Cl. This equation accurately predicts ϕr for suitable Pt values. It also can be used to evaluate the dominant terms contributing to ϕr.

We might ask how the sodium pump enters into the foregoing model. The answer is that the sodium and potassium exchange which it effects is at the expense of an energy source outside the system under consideration. Thus it may be thought of simply as equivalent to a Na+ source outside and a Na+ sink inside the cell, and similarly for K+. While the steady-state ionic concentrations are thus maintained inside and outside the cell, the forces involved in the ionic fluxes are only due to the electric and diffusion potentials. As stated before, an underlying assumption is that the sodium pump is not electrogenic. There is, however, some evidence to the contrary (Kerkut and Thomas, 1965) which may have to be included in a more rigorous treatment. Nevertheless, for most purposes (2.11) is in good agreement with experimental results (Ruch et al., 1965).

REFERENCES

KERKUT, G.A., THOMAS, R.C. Comp. Biochem. Physiol.. 1965; 14:167–183.

MOELWYN-HUGHES, E.A.A Short Course of Physical Chemistry.. New York: American Elsevier, 1966.

RUCH, T.C., PATTON, H.D., WOODBURY, J.W., TOWE, A.L.Neurophysiology.. Philadephia, Pennsylvania: Saunders,