Electrical resistivity and conductivity
Electrical resistivity and its converse, electrical conductivity, is a fundamental property of a material that quantifies how it resists or conducts the flow of electric current. A low resistivity indicates a material that allows the flow of electric current. Resistivity is represented by the Greek letter ρ; the SI unit of electrical resistivity is the ohm-metre. For example, if a 1 m × 1 m × 1 m solid cube of material has sheet contacts on two opposite faces, the resistance between these contacts is 1 Ω the resistivity of the material is 1 Ω⋅m. Electrical conductivity or specific conductance is the reciprocal of electrical resistivity, it represents a material's ability to conduct electric current. It is signified by the Greek letter σ, but κ and γ are sometimes used; the SI unit of electrical conductivity is siemens per metre. In an ideal case, cross-section and physical composition of the examined material are uniform across the sample, the electric field and current density are both parallel and constant everywhere.
Many resistors and conductors do in fact have a uniform cross section with a uniform flow of electric current, are made of a single material, so that this is a good model. When this is the case, the electrical resistivity ρ can be calculated by: ρ = R A ℓ, where R is the electrical resistance of a uniform specimen of the material ℓ is the length of the specimen A is the cross-sectional area of the specimenBoth resistance and resistivity describe how difficult it is to make electrical current flow through a material, but unlike resistance, resistivity is an intrinsic property; this means that all pure copper wires, irrespective of their shape and size, have the same resistivity, but a long, thin copper wire has a much larger resistance than a thick, short copper wire. Every material has its own characteristic resistivity. For example, rubber has a far larger resistivity than copper. In a hydraulic analogy, passing current through a high-resistivity material is like pushing water through a pipe full of sand—while passing current through a low-resistivity material is like pushing water through an empty pipe.
If the pipes are the same size and shape, the pipe full of sand has higher resistance to flow. Resistance, however, is not determined by the presence or absence of sand, it depends on the length and width of the pipe: short or wide pipes have lower resistance than narrow or long pipes. The above equation can be transposed to get Pouillet's law: R = ρ ℓ A; the resistance of a given material is proportional to the length, but inversely proportional to the cross-sectional area. Thus resistivity can be expressed using the SI unit "ohm metre" (i.e ohms divided by metres and multiplied by square metres }. For example, if A = 1 m2 ℓ = 1 m the resistance of this element in ohms is numerically equal to the resistivity of the material it is made of in Ω⋅m. Conductivity, σ, is the inverse of resistivity: σ = 1 ρ. Conductivity has SI units of "siemens per metre". For less ideal cases, such as more complicated geometry, or when the current and electric field vary in different parts of the material, it is necessary to use a more general expression in which the resistivity at a particular point is defined as the ratio of the electric field to the density of the current it creates at that point: ρ = E J, where ρ is the resistivity of the conductor material, E is the magnitude of the electric field, J is the magnitude of the current density,in which E and J are inside the conductor.
Conductivity is the inverse of resistivity. Here, it is given by: σ = 1 ρ = J E. For example, rubber is a material with large ρ and small σ—because a large electric field in rubber makes no current flow through it. On the other hand, copper is a material with small ρ and large σ—because a small electric field pulls a lot of current through it; as shown below, this expression simplifies to a single number when the electric field and current density are constant in the material. When the resistivity of a material has a directional component, the most general definition of resistivity must be used, it starts from the tensor-vector form of Ohm's law which relates the electric field inside a material to the electric current flow. This equation is general, meaning it is valid in all cases, including those mentioned above. However, this definition is the most complicated, so it is only directly used in anisotropic cases, where the more simple definitions cannot be applied. If the material is not anisotropic
A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions represent physical quantities, the derivatives represent their rates of change, the equation defines a relationship between the two; because such relations are common, differential equations play a prominent role in many disciplines including engineering, physics and biology. In pure mathematics, differential equations are studied from several different perspectives concerned with their solutions—the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas. If a closed-form expression for the solution is not available, the solution may be numerically approximated using computers; the theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy. Differential equations first came into existence with the invention of calculus by Newton and Leibniz.
In Chapter 2 of his 1671 work Methodus fluxionum et Serierum Infinitarum, Isaac Newton listed three kinds of differential equations: d y d x = f d y d x = f x 1 ∂ y ∂ x 1 + x 2 ∂ y ∂ x 2 = y He solves these examples and others using infinite series and discusses the non-uniqueness of solutions. Jacob Bernoulli proposed the Bernoulli differential equation in 1695; this is an ordinary differential equation of the form y ′ + P y = Q y n for which the following year Leibniz obtained solutions by simplifying it. The problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, Joseph-Louis Lagrange. In 1746, d’Alembert discovered the one-dimensional wave equation, within ten years Euler discovered the three-dimensional wave equation; the Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point.
Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. In 1822, Fourier published his work on heat flow in Théorie analytique de la chaleur, in which he based his reasoning on Newton's law of cooling, that the flow of heat between two adjacent molecules is proportional to the small difference of their temperatures. Contained in this book was Fourier's proposal of his heat equation for conductive diffusion of heat; this partial differential equation is now taught to every student of mathematical physics. For example, in classical mechanics, the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow these variables to be expressed dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation may be solved explicitly. An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance.
The ball's acceleration towards the ground is the acceleration due to gravity minus the acceleration due to air resistance. Gravity is considered constant, air resistance may be modeled as proportional to the ball's velocity; this means that the ball's acceleration, a derivative of its velocity, depends on the velocity. Finding the velocity as a function of time involves solving a differential equation and verifying its validity. Differential equations can be divided into several types. Apart from describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution. Used distinctions include whether the equation is: Ordinary/Partial, Linear/Non-linear, Homogeneous/Inhomogeneous; this list is far from exhaustive. An ordinary differential equation is an equation containing an unknown function of one real or complex variable x, its derivatives, some
In geometry the orientation, angular position, or attitude of an object such as a line, plane or rigid body is part of the description of how it is placed in the space it occupies. Namely, it is the imaginary rotation, needed to move the object from a reference placement to its current placement. A rotation may not be enough to reach the current placement, it may be necessary to add an imaginary translation, called the object's location. The location and orientation together describe how the object is placed in space; the above-mentioned imaginary rotation and translation may be thought to occur in any order, as the orientation of an object does not change when it translates, its location does not change when it rotates. Euler's rotation theorem shows that in three dimensions any orientation can be reached with a single rotation around a fixed axis; this gives one common way of representing the orientation using an axis–angle representation. Other used methods include rotation quaternions, Euler angles, or rotation matrices.
More specialist uses include Miller indices in crystallography and dip in geology and grade on maps and signs. The orientation is given relative to a frame of reference specified by a Cartesian coordinate system. In general the position and orientation in space of a rigid body are defined as the position and orientation, relative to the main reference frame, of another reference frame, fixed relative to the body, hence translates and rotates with it. At least three independent values are needed to describe the orientation of this local frame. Three other values are All the points of the body change their position during a rotation except for those lying on the rotation axis. If the rigid body has rotational symmetry not all orientations are distinguishable, except by observing how the orientation evolves in time from a known starting orientation. For example, the orientation in space of a line, line segment, or vector can be specified with only two values, for example two direction cosines. Another example is the position of a point on the earth described using the orientation of a line joining it with the earth's center, measured using the two angles of longitude and latitude.
The orientation of a plane can be described with two values as well, for instance by specifying the orientation of a line normal to that plane, or by using the strike and dip angles. Further details about the mathematical methods to represent the orientation of rigid bodies and planes in three dimensions are given in the following sections. In two dimensions the orientation of any object is given by a single value: the angle through which it has rotated. There is only one fixed point about which the rotation takes place. Several methods to describe orientations of a rigid body in three dimensions have been developed, they are summarized in the following sections. The first attempt to represent an orientation was owed to Leonhard Euler, he imagined three reference frames that could rotate one around the other, realized that by starting with a fixed reference frame and performing three rotations, he could get any other reference frame in the space. The values of these three rotations are called Euler angles.
These are three angles known as yaw and roll, Navigation angles and Cardan angles. Mathematically they constitute a set of six possibilities inside the twelve possible sets of Euler angles, the ordering being the one best used for describing the orientation of a vehicle such as an airplane. In aerospace engineering they are referred to as Euler angles. Euler realized that the composition of two rotations is equivalent to a single rotation about a different fixed axis. Therefore, the composition of the former three angles has to be equal to only one rotation, whose axis was complicated to calculate until matrices were developed. Based on this fact he introduced a vectorial way to describe any rotation, with a vector on the rotation axis and module equal to the value of the angle. Therefore, any orientation can be represented by a rotation vector that leads to it from the reference frame; when used to represent an orientation, the rotation vector is called orientation vector, or attitude vector.
A similar method, called axis–angle representation, describes a rotation or orientation using a unit vector aligned with the rotation axis, a separate value to indicate the angle. With the introduction of matrices, the Euler theorems were rewritten; the rotations were described by orthogonal matrices referred to as rotation matrices or direction cosine matrices. When used to represent an orientation, a rotation matrix is called orientation matrix, or attitude matrix; the above-mentioned Euler vector is the eigenvector of a rotation matrix. The product of two rotation matrices is the composition of rotations. Therefore, as before, the orientation can be given as the rotation from the initial frame to achieve the frame that we want to describe; the configuration space of a non-symmetrical object in n-dimensional space is SO × Rn. Orientation may be visualized by attaching a basis of tangent vectors to an object; the direction in which each vector points determines its orientation. Another way to describe rotations is using rotation quaternions called versors.
They are equivalent to rotation matrices and rotation vectors. With respect to rotation vectors, they can be more converted to and from matri
Membrane potential is the difference in electric potential between the interior and the exterior of a biological cell. With respect to the exterior of the cell, typical values of membrane potential given in millivolts, range from –40 mV to –80 mV. All animal cells are surrounded by a membrane composed of a lipid bilayer with proteins embedded in it; the membrane serves as a diffusion barrier to the movement of ions. Transmembrane proteins known as ion transporter or ion pump proteins push ions across the membrane and establish concentration gradients across the membrane, ion channels allow ions to move across the membrane down those concentration gradients. Ion pumps and ion channels are electrically equivalent to a set of batteries and resistors inserted in the membrane, therefore create a voltage between the two sides of the membrane. All plasma membranes have an electrical potential across them, with the inside negative with respect to the outside; the membrane potential has two basic functions.
First, it allows a cell to function as a battery, providing power to operate a variety of "molecular devices" embedded in the membrane. Second, in electrically excitable cells such as neurons and muscle cells, it is used for transmitting signals between different parts of a cell. Signals are generated by opening or closing of ion channels at one point in the membrane, producing a local change in the membrane potential; this change in the electric field can be affected by either adjacent or more distant ion channels in the membrane. Those ion channels can open or close as a result of the potential change, reproducing the signal. In non-excitable cells, in excitable cells in their baseline states, the membrane potential is held at a stable value, called the resting potential. For neurons, typical values of the resting potential range from –70 to –80 millivolts; the opening and closing of ion channels can induce a departure from the resting potential. This is called a depolarization if the interior voltage becomes less negative, or a hyperpolarization if the interior voltage becomes more negative.
In excitable cells, a sufficiently large depolarization can evoke an action potential, in which the membrane potential changes and for a short time reversing its polarity. Action potentials are generated by the activation of certain voltage-gated ion channels. In neurons, the factors that influence the membrane potential are diverse, they include numerous types of ion channels, some of which are chemically gated and some of which are voltage-gated. Because voltage-gated ion channels are controlled by the membrane potential, while the membrane potential itself is influenced by these same ion channels, feedback loops that allow for complex temporal dynamics arise, including oscillations and regenerative events such as action potentials; the membrane potential in a cell derives from two factors: electrical force and diffusion. Electrical force arises from the mutual attraction between particles with opposite electrical charges and the mutual repulsion between particles with the same type of charge.
Diffusion arises from the statistical tendency of particles to redistribute from regions where they are concentrated to regions where the concentration is low. Voltage, synonymous with difference in electrical potential, is the ability to drive an electric current across a resistance. Indeed, the simplest definition of a voltage is given by Ohm's law: V=IR, where V is voltage, I is current and R is resistance. If a voltage source such as a battery is placed in an electrical circuit, the higher the voltage of the source the greater the amount of current that it will drive across the available resistance; the functional significance of voltage lies only in potential differences between two points in a circuit. The idea of a voltage at a single point is meaningless, it is conventional in electronics to assign a voltage of zero to some arbitrarily chosen element of the circuit, assign voltages for other elements measured relative to that zero point. There is no significance in which element is chosen as the zero point—the function of a circuit depends only on the differences not on voltages per se.
However, in most cases and by convention, the zero level is most assigned to the portion of a circuit, in contact with ground. The same principle applies to voltage in cell biology. In electrically active tissue, the potential difference between any two points can be measured by inserting an electrode at each point, for example one inside and one outside the cell, connecting both electrodes to the leads of what is in essence a specialized voltmeter. By convention, the zero potential value is assigned to the outside of the cell and the sign of the potential difference between the outside and the inside is determined by the potential of the inside relative to the outside zero. In mathematical terms, the definition of voltage begins with the concept of an electric field E, a vector field assigning a magnitude and direction to each point in space. In many situations, the electric field is a conservative field, which means that it can be expressed as the gradient of a scalar function V, that is, E = –∇V.
This scalar field V is referred to as the voltage distribution. Note that the definition allows for an arbitrary constant of integration—this is why absolute values of voltage are not meaningful. In general, electric fields can be treated as
The voltage clamp is an experimental method used by electrophysiologists to measure the ion currents through the membranes of excitable cells, such as neurons, while holding the membrane voltage at a set level. A basic voltage clamp will iteratively measure the membrane potential, change the membrane potential to a desired value by adding the necessary current; this "clamps" the cell membrane at a desired constant voltage, allowing the voltage clamp to record what currents are delivered. Because the currents applied to the cell must be equal to the current going across the cell membrane at the set voltage, the recorded currents indicate how the cell reacts to changes in membrane potential. Cell membranes of excitable cells contain many different kinds of ion channels, some of which are voltage-gated; the voltage clamp allows the membrane voltage to be manipulated independently of the ionic currents, allowing the current-voltage relationships of membrane channels to be studied. The concept of the voltage clamp is attributed to Kenneth Cole and George Marmont in the spring of 1947.
They began to apply a current. Cole discovered that it was possible to use two electrodes and a feedback circuit to keep the cell's membrane potential at a level set by the experimenter. Cole developed the voltage clamp technique before the era of microelectrodes, so his two electrodes consisted of fine wires twisted around an insulating rod; because this type of electrode could be inserted into only the largest cells, early electrophysiological experiments were conducted exclusively on squid axons. Squids squirt jets of water when they need to move as when escaping a predator. To make this escape as fast as possible, they have an axon; the squid giant axon was the first preparation that could be used to voltage clamp a transmembrane current, it was the basis of Hodgkin and Huxley's pioneering experiments on the properties of the action potential. Alan Hodgkin realized that, to understand ion flux across the membrane, it was necessary to eliminate differences in membrane potential. Using experiments with the voltage clamp and Andrew Huxley published 5 papers in the summer of 1952 describing how ionic currents give rise to the action potential.
The final paper proposed the Hodgkin–Huxley model which mathematically describes the action potential. The use of voltage clamps in their experiments to study and model the action potential in detail has laid the foundation for electrophysiology; the voltage clamp is a current generator. Transmembrane voltage is recorded through a "voltage electrode", relative to ground, a "current electrode" passes current into the cell; the experimenter sets a "holding voltage", or "command potential", the voltage clamp uses negative feedback to maintain the cell at this voltage. The electrodes are connected to an amplifier, which measures membrane potential and feeds the signal into a feedback amplifier; this amplifier gets an input from the signal generator that determines the command potential, it subtracts the membrane potential from the command potential, magnifies any difference, sends an output to the current electrode. Whenever the cell deviates from the holding voltage, the operational amplifier generates an "error signal", the difference between the command potential and the actual voltage of the cell.
The feedback circuit passes current into the cell to reduce the error signal to zero. Thus, the clamp circuit produces a current opposite to the ionic current; the two-electrode voltage clamp technique is used to study properties of membrane proteins ion channels. Researchers use this method most to investigate membrane structures expressed in Xenopus oocytes; the large size of these oocytes allows for easy manipulability. The TEVC method utilizes two low-resistance pipettes, one sensing voltage and the other injecting current; the microelectrodes are filled with conductive solution and inserted into the cell to artificially control membrane potential. The membrane acts as a dielectric as well as a resistor, while the fluids on either side of the membrane function as capacitors; the microelectrodes compare the membrane potential against a command voltage, giving an accurate reproduction of the currents flowing across the membrane. Current readings can be used to analyze the electrical response of the cell to different applications.
This technique is favored over single-microelectrode clamp or other voltage clamp techniques when conditions call for resolving large currents. The high current-passing capacity of the two-electrode clamp makes it possible to clamp large currents that are impossible to control with single-electrode patch techniques; the two-electrode system is desirable for its fast clamp settling time and low noise. However, TEVC is limited in use with regard to cell size, it is more difficult to use with small cells. Additionally, TEVC method is limited in that the transmitter of current must be contained in the pipette, it is not possible to manipulate the intracellular fluid while clamping, possible using patch clamp techniques. Another disadvantage involves "space clamp" issues. Cole's voltage clamp used a long wire. TEVC microelectrodes can provide only a spatial point source of current that may not uniformly affect all parts of an irregularly shaped cell; the dual-cell voltage clamp technique is a specialized variation of the two electrode voltage clamp, is on
The FitzHugh–Nagumo model, named after Richard FitzHugh who suggested the system in 1961 and J. Nagumo et al. who created the equivalent circuit the following year, describes a prototype of an excitable system. The FHN Model is an example of a relaxation oscillator because, if the external stimulus I ext exceeds a certain threshold value, the system will exhibit a characteristic excursion in phase space, before the variables v and w relax back to their rest values; this behaviour is typical for spike generations in a neuron after stimulation by an external input current. The equations for this dynamical system read v ˙ = v − v 3 3 − w + I e x t τ w ˙ = v + a − b w; the dynamics of this system can be nicely described by zapping between the left and right branch of the cubic nullcline. The FitzHugh–Nagumo model is a simplified version of the Hodgkin–Huxley model which models in a detailed manner activation and deactivation dynamics of a spiking neuron. In the original papers of FitzHugh, this model was called Bonhoeffer–van der Pol oscillator because it contains the van der Pol oscillator as a special case for a = b = 0.
The equivalent circuit was suggested by Jin-ichi Nagumo, Suguru Arimoto, Shuji Yoshizawa. FitzHugh–Nagumo model on Scholarpedia Interactive FitzHugh-Nagumo. Java applet, includes parameters can be changed at any time. Interactive FitzHugh–Nagumo in 1D. Java applet to simulate 1D waves propagating in a ring. Parameters can be changed at any time. Interactive FitzHugh–Nagumo in 2D. Java applet to simulate 2D waves including spiral waves. Parameters can be changed at any time. Java applet for two coupled FHN systems Options include time delayed coupling, self-feedback, noise induced excursions, data export to file. Source code available
The lipid bilayer is a thin polar membrane made of two layers of lipid molecules. These membranes are flat sheets; the cell membranes of all organisms and many viruses are made of a lipid bilayer, as are the nuclear membrane surrounding the cell nucleus, other membranes surrounding sub-cellular structures. The lipid bilayer is the barrier that keeps ions and other molecules where they are needed and prevents them from diffusing into areas where they should not be. Lipid bilayers are ideally suited to this role though they are only a few nanometers in width, they are impermeable to most water-soluble molecules. Bilayers are impermeable to ions, which allows cells to regulate salt concentrations and pH by transporting ions across their membranes using proteins called ion pumps. Biological bilayers are composed of amphiphilic phospholipids that have a hydrophilic phosphate head and a hydrophobic tail consisting of two fatty acid chains. Phospholipids with certain head groups can alter the surface chemistry of a bilayer and can, for example, serve as signals as well as "anchors" for other molecules in the membranes of cells.
Just like the heads, the tails of lipids can affect membrane properties, for instance by determining the phase of the bilayer. The bilayer can adopt a solid gel phase state at lower temperatures but undergo phase transition to a fluid state at higher temperatures, the chemical properties of the lipids' tails influence at which temperature this happens; the packing of lipids within the bilayer affects its mechanical properties, including its resistance to stretching and bending. Many of these properties have been studied with the use of artificial "model" bilayers produced in a lab. Vesicles made by model bilayers have been used clinically to deliver drugs. Biological membranes include several types of molecules other than phospholipids. A important example in animal cells is cholesterol, which helps strengthen the bilayer and decrease its permeability. Cholesterol helps regulate the activity of certain integral membrane proteins. Integral membrane proteins function when incorporated into a lipid bilayer, they are held to lipid bilayer with the help of an annular lipid shell.
Because bilayers define the boundaries of the cell and its compartments, these membrane proteins are involved in many intra- and inter-cellular signaling processes. Certain kinds of membrane proteins are involved in the process of fusing two bilayers together; this fusion allows the joining of two distinct structures as in the fertilization of an egg by sperm or the entry of a virus into a cell. Because lipid bilayers are quite fragile and invisible in a traditional microscope, they are a challenge to study. Experiments on bilayers require advanced techniques like electron microscopy and atomic force microscopy; when phospholipids are exposed to water, they self-assemble into a two-layered sheet with the hydrophobic tails pointing toward the center of the sheet. This arrangement results in two "leaflets"; the center of this bilayer contains no water and excludes molecules like sugars or salts that dissolve in water. The assembly process is driven by interactions between hydrophobic molecules. An increase in interactions between hydrophobic molecules allows water molecules to bond more with each other, increasing the entropy of the system.
This complex process includes non-covalent interactions such as van der Waals forces and hydrogen bonds. The lipid bilayer is thin compared to its lateral dimensions. If a typical mammalian cell were magnified to the size of a watermelon, the lipid bilayer making up the plasma membrane would be about as thick as a piece of office paper. Despite being only a few nanometers thick, the bilayer is composed of several distinct chemical regions across its cross-section; these regions and their interactions with the surrounding water have been characterized over the past several decades with x-ray reflectometry, neutron scattering and nuclear magnetic resonance techniques. The first region on either side of the bilayer is the hydrophilic headgroup; this portion of the membrane is hydrated and is around 0.8-0.9 nm thick. In phospholipid bilayers the phosphate group is located within this hydrated region 0.5 nm outside the hydrophobic core. In some cases, the hydrated region can extend much further, for instance in lipids with a large protein or long sugar chain grafted to the head.
One common example of such a modification in nature is the lipopolysaccharide coat on a bacterial outer membrane, which helps retain a water layer around the bacterium to prevent dehydration. Next to the hydrated region is an intermediate region, only hydrated; this boundary layer is 0.3 nm thick. Within this short distance, the water concentration drops from 2M on the headgroup side to nearly zero on the tail side; the hydrophobic core of the bilayer is 3-4 nm thick, but this value varies with chain length and chemistry. Core thickness varies with temperature, in particular near a phase transition. In many occurring bilayers, the compositions of the inner and outer membrane leaflets are different. In human red blood cells, the inner leaflet is composed of phosphatidylethanolamine, phosphatidylserine and phosphatidylinositol and its phosphorylated derivatives. By contrast, the outer leaflet is based on phosphatidylcholine, sphingomyelin and a variety of gly