Gap Junctions
Figure 7.13 Diagram of a region of membrane containing gap junctions, based on electron microscope images and X-ray diffraction data. Each connexon is composed of six gap-junction proteins, called connexins, arranged hexagonally. Connexons in apposed membranes meet in the intercellular space to form the gap junction. Alberts et al., 1994, Fig. 19-15. Figure 7.13 Diagram of a region of membrane containing gap junctions, based on electron microscope images and X-ray diffraction data. Each...
The Pupil Light Reflex
The control of pupil size is yet another way in which the eye can adjust to varying levels of light intensity. While the adjustment of pupil size accounts for much less of visual adaptation than those mechanisms described earlier, it is nonetheless an important control mechanism. The size of the pupil of the eye is determined by a balance between constricting and dilating mechanisms. Pupil constriction is caused by contraction of the circularly arranged pupillary constrictor muscle, which is...
Summary
To the novice, these models may appear as a bunch of numbers and equations pulled magically out of a hat. To summarize and help give some perspective on the structure of these models, in Table 4.6 is presented an overview of the currents and gating variables used in each of the four models presented above. Table 4.6 Summary of currents for different ionic models.
The HodgkinHuxley Equations
The traveling pulse for the Hodgkin-Huxley equations must be computed numerically in one of two ways. The simplest way is to simulate the partial differential equation on a long one-dimensional spatial domain, or one can use the technique of shooting. In fact, shooting was used by Hodgkin and Huxley in their 1952 paper to demonstrate that the Hodgkin-Huxley equations support a traveling wave solution. Shooting is also the method by which a rigorous proof of the existence of traveling waves has...
CapillaryAlveolar Transport
To understand something about the transport of a gas across the capillary wall into the alveolar space, we begin with the simplest possible model. We suppose that a gas such as oxygen or carbon dioxide is dissolved in blood at some concentration U uniformly across the cross-section of the capillary. The blood is flowing along a capillary that is bounded by alveolar air space. The partial pressure of the gas in the alveolar space, Pg, is taken to be constant. Consider a segment of the capillary,...
Fluid Absorption
The primary function of the gastrointestinal tract is to absorb nutrients from the mix of food and liquid that moves through it. To accomplish this, the absorptive surface of the intestines consists of many folds and bends called valvulae conniventes, which increase the surface area of the absorptive mucosa about threefold. Located over the entire surface of the mucosa of the small intestine are millions of villi, which project about 1 mm from the surface of the mucosa and enhance the...
Leukocytes
The leukocytes white blood cells are the mobile units of the body's immune system. There are six types of white blood cells normally found in the blood. These are the neutrophils, eosinophils, basophils, monocytes, lymphocytes, and plasma cells. The neutrophils, eosinophils, and basophils are called granulocytes, or in clinical terminology, polymorphonuclear PMN cells, because they have a granular appearance and have multiple nuclei. The normal adult human has about 7000 white blood cells per...
The Discrete Bistable Equation
The discrete bistable equation is the system of equations 9.27 where f V has typical bistable form, as, for example, 9.2 or 9.3 . The study of the discrete bistable equation is substantially more difficult than that of the continuous version 9.1 . While the discrete bistable equation looks like a finite difference approximation of the continuous bistable equation, solutions of the two have significantly different behavior. It is a highly nontrivial matter to prove that traveling wave solutions...
CarrierMediated Transport
Some substances are insoluble in the cell membrane and yet pass through by a process called carrier-mediated transport. It is also called facilitated diffusion in many physiology books, although we prefer to reserve this expression for the process described in the previous section. Carrier-mediated transport is the means by which some sugars cross the cell membrane to provide an energy source for the cell. For example, glucose, the most important of the sugars, combines with a carrier protein...
The Microcirculation and Filtration
The purpose of the circulatory system is to provide nutrients to and remove waste products from the cellular interstitium. To do so requires continuous filtration of the interstitium. This filtration is accomplished primarily at the level of capillaries, as fluid moves out of the capillaries at the arteriole end and back into the capillaries at the venous end. The efflux or influx of fluid from or into the capillaries is determined by the local pressure differences across the capillary wall. In...
PhasePlane Analysis
The -cell model can be simplified by ignoring the dynamics of m and h, thus removing the time dependence of the Ca2 current Rinzel and Lee, 1986 . The simplified model Figure 6.2 Bursting oscillations in the -cell model, calculated using the parameter values in Table 6.1. Figure 6.2 Bursting oscillations in the -cell model, calculated using the parameter values in Table 6.1. -Ica V - gKn4 f V - Vk - gL V - Vl , 6.6 This separates the -cell model into a fast subsystem the V and n equations and a...
Sinoatrial Node
The primary function of SA nodal cells is to initiate a regular heartbeat. They have little contractile function and therefore little calcium. The most widely used model of action potential behavior for SA nodal cells is due to Yanagihara et al. 1980 . As with all cardiac cell models, the YNI model is of Hodgkin-Huxley type. The YNI model includes four time-dependent currents. These are the sodium current INa, which is a fast inward current, and the potassium current IK, both of which are...
Piecewise Linear FitzHughNagumo Equations
The dispersion curve in Fig. 9.11 was found for the FitzHugh-Nagumo system 9.48 - 9.49 with piecewise linear functions 9.52 and 9.53 . The calculation is similar to that for the traveling pulse Rinzel and Keller, 1973 . Since this system is piecewise linear, we can express its solution as the sum of three exponentials, on the interval 0 lt f lt f1, and as on the interval fi lt f lt f2, where v cwf. We also assume that v gt a on the interval f i lt f lt f2. The numbers i i, 2, 3, are roots of...
Fitting Data
Suppose a muscle in tetanus is held at a fixed tension until it reaches its isometric length, and then the tension is suddenly decreased and held fixed at a lower value. A typical result is shown in Fig. 18.9A, where we plot the muscle length against time. As soon as the tension is reduced, the muscle length decreases plotted in the vertical direction as the elastic element contracts. After a transition period during which the length exhibits small oscillations, the muscle decreases in length...
Appendix Linear Systems Theory
One of the most widely used tools in the study of the visual system is linear systems analysis. Here we have assumed that the basic tools of linear function theory, such as Fourier transforms, delta functions, and the convolution theorem, are familiar to readers. There are numerous books that provide the necessary background, for example, Papoulis 1962 . The essential idea of linear systems theory is that for any linear differential equation L u f t , where L is a time-autonomous differential...
Lateral Inhibition A Quantitative Model
A more detailed model of receptor horizontal cell interactions was constructed by Krausz and Naka 1980 , and the model parameters were determined by fitting to experimental data from the catfish retina. The model is depicted in Fig. 22.13. In this model, receptor and horizontal cells are assumed to form continuous sheets, within which voltage spreads continuously. The coupling coefficient for voltage spread in the sheet of receptors differs from that in the horizontal cell sheet. The receptors...
Voltage and Time Dependence of Conductances
The key step to sorting out the dynamics of the conductances came from the development of the voltage clamp. A voltage clamp fixes the membrane potential, usually by a rapid step from one voltage to another, and then measures the current that must be supplied in order to hold the voltage constant. Since the supplied current must equal the transmembrane current, the voltage clamp provides a way to measure the transient transmembrane current that results. The crucial point is that the voltage can...
Muscle
Muscle cells resemble nerve cells in their ability to conduct action potentials along their membrane surfaces. In addition, however, muscle cells have the ability to translate the electrical signal into a mechanical contraction, which enables the muscle cell to perform work. There are three types of muscle cells, namely skeletal muscle, which moves the bones of the skeleton at the joints cardiac muscle, whose contraction enables the heart to pump blood and smooth muscle, which is located in the...
A Simple Crossbridge Model The Huxley Model
To formulate a mathematical model describing crossbridge interactions in a sarcomere, we suppose that a crossbridge can bind to an actin binding site at position x, where x measures the distance along the thin filament to a binding site from the crossbridge, and x 0 corresponds to the position in which the bound crossbridge exerts no force during the power stroke on the thin filament Fig. 18.11 . Crossbridges can be bound to a binding site with x gt 0, in which case they exert a contractile...
Oxygen Uptake
The chemistry for the absorption of oxygen by hemoglobin has a similar, but nonlinear, effect. We take a simple model for the chemistry of hemoglobin discussed in Section 16.2.1 , namely Of course, there are more detailed models of hemoglobin chemistry, but the qualitative behavior is affected little by these details. We write the conservation equations as vdW Do2 oo2P02 - W 4k-2Y - 4k2 W4, 17.17 where W 02 , Y Hb 02 4 , Hb , and D02 is the oxygen exchange rate constant. The last of these...
The Circulatory System
The circulatory system forms a closed loop for the flow of blood that carries oxygen from the lungs to the tissues of the body and carries carbon dioxide from the tissues back to the lungs Figs. 15.1 and 15.2 . Because it is a closed loop system, there are two pumps to overcome the resistance and maintain a constant flow. The left heart receives oxygen-rich blood from the lungs and pumps this blood into the systemic arteries. The systemic arteries form a tree of progressively smaller vessels,...
Leukocyte Chemotaxis
Leukocytes crawl about in tissue by putting out pseudopodal extensions by which they adhere to the fibrous matrix of the tissue. In uniform chemical concentrations of chemical stimulus, their motion is that of a persistent random walk. At random times they undergo random changes in direction. The persistence time, the average time between changes of direction, is on the order of a few minutes, and the speed of migration is on the order of 2-20 m min. One important question is how leukocytes are...
Singular Perturbation Theory
The fast branch of the dispersion curve can be found for a general FitzHugh-Nagumo system in the limit e 0 using singular perturbation theory. A periodic wave consists of an alternating series of upjumps and downjumps, separated by regions of outer dynamics. The phase portrait for a periodic wave train is sketched in Fig. 9.12. To be Figure 9.11 Dispersion curves for the piecewise linear FitzHugh-Nagumo equations shown for e 0.1 and 0.01. The dashed curve shows the singular perturbation...
Insulin Oscillations with Intermediate Frequency
Insulin also oscillates with a period of about 10-20 minutes Figs. 19.11 and 19.12 . Because these oscillations occur in islets and the isolated pancreas, it appears that unlike the ultradian oscillations, they are caused by a mechanism intrinsic to the pancreatic islets. These oscillations also occur in the experimental perifusion system depicted in Fig. 19.16. A thin layer of insulin-secreting -cells is sandwiched between beads and exposed to the flow of a solution, the perifusate, with flow...
Drugs and Toxins
Many drugs act by blocking a specific ion channel. There are numerous specific channel blockers, such as sodium channel blockers, potassium channel blockers, calcium channel blockers, and so on. In fact, the discovery of site-specific and channel-specific blockers has been of tremendous benefit to the experimental study of ion channels. Examples of important channel blockers include verapamil calcium-channel blocker , quinidine, sotolol, nicotine, DDT, various barbiturates potassium-channel...
The Cell Membrane
The cell membrane provides a boundary separating the internal workings of the cell from its external environment. More importantly, it is selectively permeable, permitting the free passage of some materials and restricting the passage of others, thus regulating the passage of materials into and out of the cell. It consists of a double layer a bilayer of phospholipid molecules about 7.5 nm 75 A thick Fig. 2.1 . The term lipid is used to specify a category of water-insoluble, energy rich...
Enzyme Kinetics
To see where some of the more complicated reaction schemes come from, we consider a reaction that is catalyzed by an enzyme. Enzymes are catalysts generally proteins that help convert other molecules called substrates into products, but they themselves are not changed by the reaction. Their most important features are catalytic power, specificity, and regulation. Enzymes accelerate the conversion of substrate into product by lowering the free energy of activation of the reaction. For example,...
The QuasiSteadyState Approximation
An alternative analysis of an enzymatic reaction was proposed by Briggs and Haldane 1925 , and their analysis is now the basis for most present-day descriptions of enzyme reactions. Briggs and Haldane assumed that the rates of formation and breakdown of the complex were essentially equal at all times except perhaps at the beginning of the reaction, as the complex is filling up . Thus, dc dt should be approximately zero. With this approximation, it is relatively simple to determine the velocity...
The FitzHughNagumo Equations
To understand the structure of a traveling pulse it is helpful first to study traveling pulse solutions in the FitzHugh-Nagumo equations e e2 d 2 f V'W , 9.48 where e is assumed to be a small positive number. Without any loss of generality, space has been scaled so that the diffusion coefficient of v is e2. It is important to realize that this does not imply anything about the magnitude of the physical diffusion coefficient. We are simply scaling the space variable so that in the new coordinate...
Cardiac Output
During a heartbeat cycle, the pressure and volume of the heart change in a highly specific way, shown in Fig. 15.5. Notice that the pressure-volume loops are of rectangular Figure 15.5 Experimental data of the pressure-volume relationship during the heartbeat cycle in the denervated left ventricle of the dog. Sagawa et al., 1978, Fig. 11.4. A Three beats from different end-diastolic volumes and against different arterial pressures are shown in solid lines, with the broken lines representing the...
Jop
Since F N0 is a monotone decreasing function of N0, 16.37 is guaranteed to have a unique solution. In fact, the solution is a monotone decreasing function of the Figure 16.6 Plot of left- and right-hand sides of 16.37 for three different values of f and for F N jN and X 10. parameter 3, indicating that at higher death rates, the cell population drops while the production of cells increases. An illustration of these facts is provided by the graph in Fig. 16.6, where the two curves F N and 1 N-ox...
IntensityResponse Curves and the NakaRushton Equation
Light adaptation in photoreceptors is elegantly summarized by intensity-response curves Fig. 22.3 , a set of curves that repays careful consideration. For a fixed background light level I0, we consider the response to a family of superimposed light steps. For each light step, the photoreceptor membrane potential shows a large transient response, followed by a slower change to the steady-state level, as the photoreceptor adapts to the maintained stimulus. For instance, in response to a step...
Monodomain reduction
Equation 11.26 can be reduced to a monodomain equation for the membrane potential in one special case. Notice that V Vi a ae 1 aeV V it , 11.36 so that the balance of transmembrane currents becomes Cm'dF lion V a, ai ae 1aeVV V ai ai ae 1ii. 11.37 Here we see that there is possibly a contribution to the transmembrane current from the divergence of the total current. We know that V-it 0, so this source term is zero if the matrix ai ai ae 1 is proportional to a constant multiple of the identity...
The Mechanisms of Calcium Release 531 IP3 Receptors
Although the two-pool model reproduces experimental data extremely well, both qualitatively and quantitatively, recent experimental evidence indicates that the role of Ca2 is more complicated than was assumed in this model. In the two-pool model Ca2 stimulates its own release thus the term cp Kp cp in 5.5 , while the flow of Ca2 from the internal store is terminated when the concentration of Ca2 in the internal store becomes too low thus the term cf Kf cf . However, it now appears that not only...
A Simple Circulatory System
Figure 15.6 Schematic diagram of the simplest circulation model, with a single-chambered heart and a single loop. Figure 15.6 Schematic diagram of the simplest circulation model, with a single-chambered heart and a single loop. To illustrate how all the above pieces fit together to give a model of the circulatory system, consider a simple circulatory system with one loop and a single-chambered heart Fig. 15.6 . To begin with, suppose we have only a heart and a resistive closed loop. For the...
Appendix Math Background
It is certain that some of the mathematical concepts and tools that we routinely invoke here are not familiar to all of our readers. In this first chapter alone, we have Figure 1.9 Solution of the Goldbeter-Lefever model with v 200, n 120. used nondimensionalization, phase-plane analysis, linear stability analysis, bifurcation theory, and asymptotic analysis, all the while assuming that these are familiar to the reader. The purpose of this appendix is to give a brief guide to those techniques...
Pulsatile Insulin Secretion
Hormones secreted from cells in the pancreas are responsible for the control of glucose, amino acids, and other molecules that are necessary for metabolism. The pancreas contains a large number of secretory cells, grouped into about one million islets of Langerhans consisting of approximately 2,500 cells each. There are three principal Figure 19.9 Pulsatile secretion of LH, FSH, and testosterone in men. Berne and Levy, 1993, Fig. 48-15, p. 912. Figure 19.9 Pulsatile secretion of LH, FSH, and...
The activation sequence
The usefulness of the eikonal-curvature becomes apparent when one attempts to compute the activation sequence. The activation sequence is the spatial and temporal sequence in which the medium is activated by a wave initiated by the SA node. Determination of the activation sequence is complicated by a number of features. First, as mentioned above, the medium is strongly anisotropic. Further, the fiber orientation in myocardial tissue varies through the thickness of the tissue, rotating...
Waves with Curvature
Wave fronts in two- or three-dimensional media are not expected to be plane waves. They are typically initiated at a specific location, and so might be circular in shape. Additionally, the medium may be structurally inhomogeneous or have a nonsimple geometry, all of which introduce curvature into the wave front. It is known that curvature plays an important role in the propagation of a wave front in an excitable medium. A physical explanation makes this clear. Suppose that a circular wave front...
Appendix Transform Methods
To follow all of the calculations and complete all the exercises in this chapter, you will need to know about Fourier and Laplace transforms, generalized functions and the delta function, Green's functions, as well as some aspects of complex variable theory, including contour integration and the residue theorem. If you have made it this far into this book, then you are probably familiar with these classic techniques. However, should you need a reference for these techniques, there are many...
Parabolic Bursting
Another well-studied example of bursting is found in the Aplysia R-15 neuron Fig. 6.1E . Analysis of a detailed model by Plant 1981 shows that the mathematical structure of this bursting oscillator is different from that in the 0-cell model Rinzel and Lee, 1987 . The 0-cell model has two fast variables, one slow variable, bistability, and a hysteresis loop. At the end of a burst, a homoclinic bifurcation is crossed, leading to an increasing period through the burst. Plant's model, on the other...
The Electrocardiogram 1411 The Scalar ECG
One of the oldest and most important tools for evaluating the status of the heart and the cardiac conduction system is the electrocardiogram ECG . It has been known since 1877, when the first ECG recording was made, that the action potential of the heart generates an electrical potential field that can be measured on the body surface. When an action potential is spreading through cardiac tissue, there is a wave front surface across which the membrane potential experiences a sharp increase....
Cardiac Propagation
Cardiac cells perform two functions in that they are both excitable and contractile. They are excitable, enabling action potentials to propagate, and the action potential causes the cells to contract, thereby enabling the pumping of blood. The electrical activity of the heart is initiated in a collection of cells known as the sinoatrial node SA node located just below the superior vena cava on the right atrium. The cells in the SA node are autonomous oscillators. The action potential that is...
Glucose Oscillations
An additional feature of E. coli is that lactose is not utilized when there is adequate glucose. The mechanism for this control is as follows preceding the promoter region of the lac operon where the RNA polymerase must bind to begin transcription, there is another region, called a CAP site catabolic gene activator protein , which can be bound by a dimeric molecule CAP. CAP by itself has no influence on transcription unless it is bound to cyclic AMP cAMP , but when CAP is bound to cAMP the...
The shallowwater approximation
In the shallow-water approximation, we assume that the wavelengths of the waves on the basilar membrane are greater than the depth of the cochlea. As a consequence, we assume that an 0 for all n gt N, for some integer N such that Nl L 1. Since the sum over n includes only those terms with nl L 1, it follows that for each term in the sum, tanh 2nnl L can be approximated by the lowest-order term in its Taylor expansion. Thus, tanh 2nnl L 2nnl L, and so the sum becomes - i n v an 2nn 2 e2ML. 23.40...
Critical Size of a Pacemaker
The SA node is a small clump of self-oscillatory cells in a sea of excitable but nonoscilla-tory cells whose function is to initiate the cardiac action potential. SA nodal cells have no contractile function and therefore no contractile machinery. Thus, when viewed in terms of contractile efficiency, SA nodal cells are a detriment to contraction and a waste of important cardiac wall space. On the other hand, the SA node cannot be made too small because presumably, it would not be able to...
Spiral Waves in Xenopus
In 1991, it was discovered by Lechleiter and Clapham and their coworkers that intracellular Ca2 waves in immature Xenopus oocytes showed remarkable spatiotemporal Figure 12.4 Speed-curvature equation for the piecewise linear two-pool model. Sneyd and Atri, 1993, Fig. 4. Figure 12.4 Speed-curvature equation for the piecewise linear two-pool model. Sneyd and Atri, 1993, Fig. 4. organization. By loading the oocytes with a Ca2 -sensitive dye, releasing IP3, and observing Ca2 release patterns with a...
Quantal Nature of Synaptic Transmission
Chemical synapses are typically small and inaccessible, crowded together in very large numbers in the brain. However, neurons also make synapses with skeletal muscle cells, and these are usually much easier to isolate and study. For this reason, a great deal of the early experimental and theoretical work on synaptic transmission was performed on the neuromuscular junction, where the axon of a motorneuron forms a chemical synapse with a skeletal muscle fiber. The response of the muscle cell to a...
Intercellular Communication
For multicellular organisms to form and operate, cellular behavior must be vastly more complex than what is seen on the single-cell level. Cells must not only regulate their own growth and behavior, they must also communicate and interact with their neighbors to ensure the correct behavior of the entire organism. Intercellular communication occurs in a variety of ways, ranging from hormonal communication on the level of the entire body to localized interactions between individual cells. Our...
Webers Law and Contrast Detection
One of the basic features of the retina is light adaptation, the ability to adapt to varying levels of background light. Over a wide range of light levels, the sensitivity of the retina is observed to be approximately inversely proportional to the background light level. This fact is known as Weber's law, or the Weber-Fechner law. There are three common definitions of sensitivity. It can mean psychophysical sensitivity, defined as 1 threshold, where the threshold is the minimal stimulus...