I. M. Appelpolscher, supervisor of the process control group of the Ideal Gas Company, has installed a 25 X 40 X 5-ft swimming pool in his backyard. The pool contains level and temperature sensors used with feedback controllers to maintain the pool level and temperature at desired values. Appelpolscher is satisfied with the level control system, but he feels that the addition of one or more feedforward controllers would help maintain the pool temperature more nearly constant. As a new member of the process control group, you have been selected to check Appelpolscher's mathematical analysis and to give your advice. The following information may or may not be pertinent to your analysis: (i) Appelpolscher is particular about cleanliness and thus has a high-capacity pump that continually recirculates the water through an activated charcoal filter. (ii) The pool is equipped with a natural gas-fired heater that adds heat to the pool at a rate Q(t) that is directly proportional to the output signal from the controller p(t) (iii) There is a leak in the pool, which Appelpolscher has determined is constant equal to F (volumetric flow rate). The liquidlevel control system adds water from the city supply system to maintain the level in the pool exactly at the specified level. The temperature of the water in the city system is T w, a variable. (iv) A significant amount of heat is lost by conduction to the surrounding ground, which has a constant, year-round temperature T 0 . Experimental tests by Appelpolscher showed that essentially all of the temperature drop between the pool and the ground occurred across the homogeneous layer of gravel that surrounded his pool. The gravel thickness is Lh, and the overall thermal conductivity is k 0 . (v) The main challenge to Appelpolscher's modeling ability was the heat loss term accounting for convection, conduction, radiation, and evaporation to the atmosphere. He determined that the heat losses per unit area of open water could be represented by where Tp = temperature of pool Ta = temperature of the air, a variable U = overall heat transfer coefficient Appelpolscher's detailed model included radiation losses and heat generation due to added chemicals, but he determined that these terms were negligible. (a) Draw a schematic diagram for the pool and all control equipment. Show all inputs and outputs, including all disturbance variables. (b) What additional variable(s) would have to be measured to add feedforward control to the existing pool temperature feedback controller? (c) Write a steady-state energy balance. How can you determine which of the disturbance variables you listed in part (a) are most/least likely to be important? (d) What recommendations concerning the prospects of adding feedforward control would you make to Appelpolscher?
Chapter 4: Neuron Structure and Function Electrical Signals in Neurons Property of excitability that gives neurons the ability to store, recall and distribute information Most neurons have a resting membrane potential of approximately -70mV During depolarization the charge difference between the inside and outside of the cell membrane decreases, membrane potential becomes less negative Either positively charged ions entering the cell or negatively charged ions moving out of the cell can make the inside of the cell membrane less negatively charged causing depolarization During hyperpolarization the membrane potential becomes more negative Either negatively charged ions entering the cell or positively charged ions moving out of the cell can make the inside of the cell membrane more negative causing hyperpolarization During repolarization the cell membrane returns to the resting membrane potential Goldman equation describes the resting membrane potential Three factors contribute to establishing membrane potential of the cell Distribution of ions across the plasma membrane Relative permeability of the membrane to these ions The charges on these ions Gated ion channels allow neurons to alter their membrane potentials If the membrane is not permeable to an ion, that ion does not contribute to the membrane potential If the membrane is highly permeable to an ion, that ion makes a large contribution to the membrane potential Equilibrium potential is the membrane potential at which the electrical and chemical gradients favoring the movement of a particular ion exactly balance each other and there is no net movement of that ion across the membrane As Na+ enters the cell, the inside of the cell becomes more and more positively charged until the membrane has depolarized In contrast, opening of K+ channels typically cause hyperpolarization It is changes in membrane permeability rather than measurable changes in ion concentration that cause the membrane potential to deviate from the resting membrane potential during electrical signals Signals in the Dendrites and Cell Body Binding of neurotransmitter to a specific ligand-gated receptor causes ion channels in the membrane to open or close, changing the permeability of the membrane Permeability alters the membrane potential and causes and electrical signal In dendrites and cell bodies of neurons the electrical signals are called graded potentials Graded potentials vary in magnitude Graded potentials vary in magnitude depending on the strength of the stimulus The amplitude of the graded potential directly reflects the strength of the incoming stimulus Graded potentials can either hyperpolarize or depolarize the cell, depending on the type of ion channel that is opened/closed Opening Na+ or Ca2+ channels will depolarize a neuron Opening K+ or Cl- channels will hyperpolarize a neuron Graded potentials are short-distance signals Graded potentials can travel through the cell but they decrease in strength as they get farther away from the opened ion channel Called conduction with decrement When neurotransmitter binds to a ligand-gated Na+ channel, the channel open and Na+ ions move into the cell Na+ entry causes a local depolarization in a small area of the membrane surrounding the opened channel Positive charge then spreads along the inside of the membrane causing depolarization Called electrotonic current spread Extent of this depolarization decreases as it moves farther and farther from the opened channels Action potentials are used to transmit information across distances of more than a few millimeters Action potentials are triggered by the net graded potential at the membrane of the axon hillock Graded potential causes the membrane potential at the axon hillock to depolarize beyond the threshold potential, the axon will “fire” an action potential If the membrane potential at the axon hillock does not reach the threshold potential the axon will not initiate an action potential Figure 4.7: Subthreshold and suprathreshold potentials: the resting potential of most neurons is around -70mV and the threshold potential is -55mV. A: Subthreshold graded potentials (less than +15mV) don’t trigger an action potential. B: graded potentials that are at or above the threshold potential (greater than +15mV) trigger an action potential Depolarizing graded potential is called an excitatory potential because it makes an action potential more likely to occur by bringing the membrane potential closer to threshold potential Hyperpolarizing graded potential makes an action potential less likely to occur, which is an inhibitory potential Graded potentials are integrated to trigger action potentials Graded potentials from different sites can interact with each other to influence the net change in membrane potential at the axon hillock Called spatial summation Figure 4.8: Spatial Summation Depolarization that occur at two slightly different times can also combine to determine the net change in membrane potential at the axon hillock Called temporal summation Figure 4.9: Temporal Summation Signals in the Axon Action potentials can be transmitted across long distances without degrading and differ from graded potentials in many respects Action potentials typically have three phases Depolarization phase o Triggered when the membrane potential at the axon hillock reaches threshold o Once reaches threshold, the adjacent axonal membrane quickly depolarizes Repolarization phase o Membrane potential rapidly returns to the resting membrane potential o Following repolarization membrane potential becomes even more negative than the resting membrane potential o May approach the K+ equilibrium potential After-hyperpolarization phase o Typically lasts between 2-15 msec o Membrane returns to the resting membrane potential Voltage-gated channels shape action potential Opening and closing of voltage-gated ion channels cause the characteristic phases of the action potential Changes in membrane potential change the shape of voltage-gated ion channels, allowing ions to move across the membrane Model used as an example: giant axon of the squid Squid giant axon sends signals from the central nervous system to the muscle of the squid’s mantle cavity, thus is part of an invertebrate motor neuron Opening of voltage-gated Na+ channels initiates depolarization phase of the action potential Opening of voltage-gated K+ channels initiates repolarization phase in squid giant axon When membrane potential at the axon hillock approaches the threshold potential voltage-gated Na+ channels in the axon hillock begin to open, changing the permeability of the membrane to Na+ ions, allowing Na+ ions to move across the membrane Probability of a given voltage-gated Na+ channel being open depends on the size of the graded potential An excitatory graded potential that depolarizes the membrane toward the threshold potential increases the probability that a voltage-gated Na+ channel will be open At the threshold potential more voltage-gated Na+ channels will be open tan when the axon hillock is at the resting membrane potential increasing permeability of the membrane to Na+ Na+ influx from the first voltage-gated channels to open in response to the graded potential further depolarizes the local region of the membrane further increasing the probability that voltage-gated Na+ channels will open o causing even more voltage-gated Na+ channels to open o further increasing the permeability of the membrane allowing even more Na+ ions to enter the cell Action potentials generally occur in the axon not in the cell body or dendrites of a neuron Voltage-gated Na+ channels close, terminating the depolarization phase of the action potential Threshold depolarization of the membrane at the axon hillock increases probability that voltage- gated K+ channels will open When voltage-gated K+ channels open the permeability of the membrane to K+ ions increases K+ ions leave the cell in response to their electrochemical driving force Making the intracellular side of the membrane more negative Causes repolarization phase of the action potential Voltage-gated Na+ channels have two gates To open, the Na+ channel undergoes a conformation change that opens an activation gate allowing Na+ ions to move across the membrane Opening of activation gate increases permeability of the membrane to Na+ As Na+ enters the cell more and more voltage-gated Na+ channels open and the axonal membrane potential rapidly becomes less negative, depolarizing the cell toward the equilibrium potential for Na+ As the membrane potential approaches the equilibrium potential for Na+ the electrochemical gradient that acts as a driving force for Na+ movement decreases and Na+ entry slows Meanwhile time-dependent conformational change occurs in the channel, closing the inactivation gate With the inactivation gate closed, no more Na+ can enter the cell, terminating the depolarization phase of the action potential In response to changes in the membrane potential caused by the actions of the voltage-gated K+ channels, the inactivation gate resets and the channel returns to its initial conformation ready to initiate another action potential Action potentials transmit signals across long distances Action potentials are often described as all or nothing phenomena because once an action potential has been initiated it always proceeds to conclusion Action potential in one part of the axon triggers other action potentials in other action potentials in adjacent areas of the axonal membrane In neurons the first action potential at the axon hillock causes another action potential farther down the axon and so on down to the axon terminal The last axon potential at the axon terminal is identical to the first action potential at the axon hillock Action potentials can be conducted across long distances without decaying Vertebrate motor neurons are myelinated Axons of vertebrate motor neurons are wrapped in an insulating layer of myelin Specialized lipid-rich cells (Schwann cells) form the myelin sheath by wrapping in a spiral pattern around the axon of the neuron Several Schwann cells may wrap long axons separated by areas of exposed axonal membrane called nodes of Ranvier that contain high densities of voltage-gated channels Myelinated regions of the axons are the internodes Action potentials only occur in the nodes of Ranvier Action potential appears to jump from node to node along the axon Electrotonic currents can travel farther with less degradation trough the internodes than through an equivalent region of unmyelinated axon Electrontonic current spread is much faster than generating an action potential Figure 4.14: Structure of the myelin sheath Axons conduct action potentials unidirectionally Action potentials occur only in the downstream direction Absolute refractory period prevents backward transmission of action potentials Also prevents summation of action potentials The absolute and relative refractory periods prevent retrograde transmission of action potentials Figure 4.10: the phases of a typical action potential. A: changes in membrane potential during an action potential. B: changes in membrane permeability during and action potential Action potential frequency carries information Action potentials carry information by changing frequency rather than amplitude Figure 4.15: Frequency of action potentials Signals Across the Synapse Cell that transmits the signal is referred to as the presynaptic cell Cell receiving the signal is called postsynaptic cell Space between the presynaptic and postsynaptic cell is referred to as the synaptic cleft Synapse between a motor neuron and a skeletal muscle cell is termed the neuromuscular junction Terminology Excitable cell: cell that is capable of producing an action potential Neuron: specialized cell in the nervous system that communicates using chemical and electrical signals, many, but not all, neurons are excitable cells that generate action potentials Motor neuron: neuron that transmits signals from the central nervous system to skeletal muscles Dendrites: branching extensions of a neuronal cell body that carry signals toward the cell body Cell body: Also called soma; cell body of a neuron, containing the nucleus Axon hillock: junction between the cell body and axon of a neuron. In many neurons, the axon hillock is the site of action potential initiation, acting as the trigger zone for the neuron Axon: projection of the cell body of a neuron that is involved in carrying information, usually in the form of action potentials, from the cell body to the axon terminal Action potential: relatively large-amplitude, rapid change in the membrane potential of an excitable cell as a result of the opening and closing of voltage-gated ion channels; involved in transmitting signals across long distances in the nervous system Myelin sheath: insulating wrappings of vertebrate axons that are composed of multiple layers of glial cell plasma membrane. Invertebrate axons have analogous wrappings, but they are not generally termed a myelin sheath Axon terminal: distal end of an axon that forms a synapse with an effector cell or neuron Synapse: junction between a neuron and another neuron or effector cell; consists of a presynaptic cell, the synaptic cleft, and a postsynaptic cell Depolarization (phase): change in the membrane potential of a cell from its normally negative resting membrane potential to a more positive value; a relative increase in the positive charge on the inside of the cell membrane Hyperpolarization: change in the membrane potential of a cell from its normally negative resting membrane potential to a more negative value; a relative increase in the negative charge on the inside of the cell membrane Graded potentials: changes in the cell membrane potential of a cell that vary in magnitude with the stimulus intensity; results from the opening and closing of ion channels Threshold potential: the critical value of the membrane potential in an excitable cell to which the membrane must be depolarized in order for an action potential to be initiated Excitatory potential: change in the membrane potential in an excitable cell that increases the probability of action potential initiation in that cell Inhibitory potential: change in the membrane potential that decreases the probability of action potential initiation in an excitable cell Spatial summation: process by which graded potentials at different points in the membrane (occurring at the same time) combine to influence the net graded potential of a cell Temporal summation: process by which graded potentials occurring at slightly different times combine to influence the net graded potential of the cell Repolarization (phase): return of the membrane potential of a cell toward the resting membrane potential following a depolarization of hyperpolarization After-hyperpolarization (phase): prolonged hyperpolarization following an action potential Absolute refractory period: period during and immediately following an action potential in which an excitable cell cannot generate another action potential, no matter how strong the stimulus Relative refractory period: period immediately following the absolute refractory period in which an excitable cell will generate an action potential only if exposed to a suprathreshold stimulus Voltage-gated ion channel: membrane protein containing an aqueous pore that can be opened in response to changes in membrane potential Activation gate: one of the two gates that open and close voltage-gated sodium channels Inactivation gate: one of the two gates that open and close voltage-gated sodium channels Schwann cell: type of glial cell in the vertebrates that forms the myelin sheath around axons in the peripheral nervous system. Glial cells: group of several types of cells that provide structural and metabolic support to neurons Node of Ranvier: gap of exposed axonal membrane between two regions of myelin sheath Salutatory conduction: mode of conduction of action potentials in myelinated axons in which action potentials appear to jump from one node of Ranvier to the next Presynaptic cell: neuron that transmits a signal across a synapse to a postsynaptic cell Postsynaptic cell: cell (either neuron or effector) that receives a signal from a presynaptic cell across a synapse Synaptic cleft: extracellular space between a presynaptic cell and postsynaptic cell at a synapse Neuromuscular junction: synapse between a motor neuron and a skeletal muscle cell Synaptic vesicles: neurotransmitter-containing vesicles that release neurotransmitter into a synapse Acetylcholine: neurotransmitter found in most animal species in many types of neurons, including motor neurons and the autonomic ganglia of vertebrates