Lecture notes on Biophysics

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Introduction to biophysics Bert Kappen, Department of Biophysics Radboud University Nijmegen February 7, 2008 1Contents 1 Preface 4 2 Introduction to neurons and the brain 5 2.1 Nerve cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 The nervous system . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Some features of the cortex . . . . . . . . . . . . . . . . . . . 12 2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3 Electrical properties of cells 19 3.1 Ion channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 The Nernst equation . . . . . . . . . . . . . . . . . . . . . . . 21 3.3 The Goldman equation . . . . . . . . . . . . . . . . . . . . . . 23 3.4 The Nernst-Planck equation . . . . . . . . . . . . . . . . . . . 24 3.5 The Hodgkin-Katz experiments . . . . . . . . . . . . . . . . . 27 + 3.5.1 The role of K . . . . . . . . . . . . . . . . . . . . . . 27 + 3.5.2 The role of Na . . . . . . . . . . . . . . . . . . . . . . 28 3.5.3 Permeability changes during action potential . . . . . . 30 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4 The Hodgkin-Huxley model of action potentials 36 4.1 The voltage clamp technique . . . . . . . . . . . . . . . . . . . 36 4.2 Two types of voltage dependent ionic currents . . . . . . . . . 36 4.3 The Hodgkin-Huxley model . . . . . . . . . . . . . . . . . . . 43 + 4.3.1 The K conductance . . . . . . . . . . . . . . . . . . . 48 + 4.3.2 The Na conductance . . . . . . . . . . . . . . . . . . 50 4.3.3 Action potentials . . . . . . . . . . . . . . . . . . . . . 51 4.4 Spike propagation . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.4.1 Passive current ow . . . . . . . . . . . . . . . . . . . . 53 4.4.2 Spike propagation . . . . . . . . . . . . . . . . . . . . . 57 4.4.3 Myelin . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 25 Synapses 67 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.2 Chemical synapses . . . . . . . . . . . . . . . . . . . . . . . . 67 5.3 The post-synaptic potential . . . . . . . . . . . . . . . . . . . 68 5.4 Stochastic PSPs . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.5 Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.6 Long term potentiation . . . . . . . . . . . . . . . . . . . . . . 77 5.7 Hebbian learning . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.7.1 Ocular dominance . . . . . . . . . . . . . . . . . . . . . 79 5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6 Perceptrons 82 6.1 Threshold units . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.2 Linear separation . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.3 Perceptron learning rule . . . . . . . . . . . . . . . . . . . . . 84 6.3.1 Convergence of Perceptron rule . . . . . . . . . . . . . 86 6.4 Linear units . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.4.1 Gradient descent learning . . . . . . . . . . . . . . . . 89 6.4.2 The value of  . . . . . . . . . . . . . . . . . . . . . . . 90 6.5 Non-linear units . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.6 Multi-layered perceptrons . . . . . . . . . . . . . . . . . . . . 92 6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 31 Preface This introductory course on biophysics introduces the principles of electrical excitability of cell membranes, which form the basis of all information pro- cessing in the nervous system. The course covers some of the classical results, such as cellular membranes, ionic currents, equilibrium behavior and action potentials. The course is intended for physics students and will therefore have an emphasis on physical modeling. Section 2 is an introductory chapter, where I will give an overview of some of the basic anatomical properties of the nervous system and of nerve cells and discuss the spiking behavior of nerve cells and their functional relevance. In section 3, I will discuss the stationary behavior of the cell, such as the relation between ionic concentrations inside and outside the cell, the ionic currents and the membrane potential. In section 4, I will discuss the mechanism for action potential generation, spike propagation, linear cable theory and the role of myelin. In section 5, I will discuss synapses and some aspects of learning. In section 6, I will give a brief introduction to a class of popular neural networks, the (multi-layered) perceptrons. Bert Kappen Nijmegen, January 2007 42 Introduction to neurons and the brain Perhaps the major reason that neuro science is such an exciting eld is the wealth of fundamental questions about the human brain (and the rest of the nervous system) that remain unanswered. Such understanding entails unrav- eling the interconnections of large numbers of nerve cells, that are organized into systems and subsystems. The fact that cells are the basic element of living organisms was recog- nized early in the nineteenth century. It was not until well into the twen- tieth century, however, that neuro scientists agreed that nervous tissue, like all other organs is made up of these fundamental units. Santiago Ram on y Cajal argued persuasively that nerve cells are discrete entities and that they communicate with one another by means of specialized contacts called synapses. The human brain is estimated to contain 100 billion neurons and several times as many supporting cells, called neuroglial cells. 2.1 Nerve cells In most respects, the structure of neurons resembles that of other cells. Each cell has a cell body containing a nucleus, endoplasmic reticulum, ribosomes, Golgi apparatus, mitochondria, and other organelles that are essential to the function of all cells (see g. 2). Speci c for nerve cells, is their dendritic structure (see g. 3. The dendrites (together with the cell body) provide sites for the synaptic contacts made by the terminals of other nerve cells and can thus be regarded as specialized for receiving information. The number of inputs that a particular neuron receives depends on the complexity of its dendrite and can range from 1 to about 100.000. The information from the inputs that impinge on the dendrites is 'read out' at the border of the cell body and the axon. The axon is an extension that may reach from a few hundred micrometers to a meter. Typical axons in the brain a a few millimeters long. Axons in the spinal cord are about a meter long. The axon carries electrical signals over such distances through action potentials, a self-generating electrical wave that propagates from the cell body to the end of the axon. The information encoded by action potentials is passed on to the next cell by means of synaptic transmission. The arrival of the action potential causes the release of neurotransmitters, which in turn modify the electrical proper- ties of the post-synaptic cell. The net e ect is a change of the membrane 5Figure 1: The brain consists of network of neurons (Ram on y Cajal, 1910). Shown is one of the original Golgi stain images of rat cortex. Only a small fraction of neurons are stained with this technique. 6Figure 2: The major light and electron microscopical features of neurons. A) Diagram of nerve cells and their component parts. B) Axon initial seg- ment (blue) entering a myelin sheath (gold). C) Terminal boutons (blue) loaded with synaptic vesicles (arrowheads) forming synapses (arrows) with a dendrite (purple). D) Transverse section of axons (blue) ensheathed by the processes of oligodendrocytes (gold). E) Apical dendrites (purple) of cortical pyramidal cells. F) Nerve cell bodies (purple) occupied by large round nu- clei. G) Portion of a myelinated axon (blue) illustrating the intervals between adjacent segments of myelin (gold) referred to as nodes of Ranvier (arrows). 7Figure 3: Cells stained with silver salts (Golgi stain). indicates axon. a) muscle cell b-d) retinal cells e) Cortical pyramidal cell f) Cerebellar Purkinje cell potential of the post-synaptic cell. It is thought, that glia cells do not play a primary role in information processing in the brain. The di erent types of glia cells have two impor- tant functions. The astrocytes maintain in a variety of ways the appropriate chemical environment for the nerve cells. The oligodendrocytes or Schwann cells lay down a laminated wrapping called myelin around some, but not all, axons, which has important e ects on the speed of action potential propaga- tion. 2.2 The nervous system The nervous system is traditionally divided into a central and peripheral component (see g. 4). The peripheral system contains the sensory neurons, which receive information from the outside world, and the motor neurons, that connect to muscles and glands. Sensory information is processed in 8Figure 4: The major components of the nervous system and their functional relationships. A) The CNS (brain and spinal cord) and the PNS (spinal and cranial nerves). B) The peripheral nervous system receives sensory input and outputs motor commands. The central nervous system provides the 'mapping' from sensory input to motor output. the brain, with the ultimate goal to generate the appropriate motor actions. Nerve cells that carry information toward the central nervous system are called a erent neurons, nerve cells that carry information away from the brain are called e erent neurons. Nerve cells that only participate in the local aspects of a circuit are called inter-neurons. The simple spinal re ex circuit in g. 5 illustrates this terminology. The central nervous system is usually considered to include seven basic parts (see g. 6): the spinal cord; the medulla, the pons and the midbrain (collectively called the brainstem); the cerebellum; the diencephalon and the cerebral hemispheres (collectively called the forebrain). The thalamus relays information to the cerebral cortex from other parts of the brain. Specialized substructures of the thalamus are engaged in motor functions and reproduction and hormone secretion. The brainstem contains structures, such as the superior colliculus that is involved in eye movement. The major function of the cerebellum is coordination of motor activity, pos- ture and equilibrium. Like the cerebral cortex, the cerebellum is covered by a thin cortex. Another important area of the central nervous system is the hippocampus which is thought to be involved in the storage of episodic memories. It is not visible in g. 6, since it is located centrally. 9Figure 5: A simple re ex circuit, the knee-jerk response, illustrates several points about the functional organization of neural circuits. Stimulation of a muscle stretch receptor initiates action potentials that travel centrally along the a erent axons of the sensory neurons. This information stimulates spinal motor neurons by means of synaptic contacts. The action potentials gen- erated in motor neurons travel peripherally in e erent axons, giving rise to muscle contraction. Bottom) Relative frequency of action potentials (in- dicated by individual vertical lines). Notice the modulatory e ect of the interneuron. 10Figure 6: A) The terms anterior, posterior, superior, and inferior refer to the long axis of the body. B) The major planes of section used in cutting or imaging the brain. C) The subdivisions and components of the central nervous system. 11Figure 7: Structure of the human neocortex. A) summary of the cellular composition of the six layers of the neocortex. B) Based on variations in thickness, cell density and other histological features of the six neo-cortical laminae, the neocortex can be divided into areas (Brodmann 1909). These anatomical distinctions have later been shown to relate to di erent functions. Red indicates the primary motor cortex, blue the primary somatic sensory cortex, green the primary auditory cortex and yellow the primary visual cortex. All other Brodmann areas are considered association cortex. 2.3 Some features of the cortex The cerebral hemispheres, also called the cerebral cortex are two convoluted sheets of neural tissue of about 2 mm thickness and spreads over about 2 11 dm each. The sheets are connected through the corpus callosum (800 million bers). The cortical sheet contains six layers that can be identi ed anatomically ( g. 7a). This structure of six layers is remarkably uniform through the cortex. Local di erences has lead to the classi cation of the cortex into cortical areas (see g. 7b). The cortical tissue consists for about 80 % of pyramidal cells ( g. 3) and the remainder are so called inter-neurons. There are two types of pyramidal neurons, the upper pyramidal neurons lying in layers II and III and the lower pyramidal neurons which we nd mainly in layer V. Both receive their input signals from stellate cells, which are inter-neurons lying in layer IV. The lower pyramidal neurons are output neurons; their axons make contact with the thalamus. The upper pyramidal neurons make distant connections with the pyramidal cells of other cortical areas. The six layer structure is schematically 12Figure 8: Canonical neo-cortical circuitry. Green arrows indicate outputs to the major targets of each of the neo-cortical layers in humans; orange arrow indicates thalamic input (primarily to layer IV); purple arrows indicate input from other cortical areas: and blue arrows indicate input from the brainstem to each layer. drawn in g. 8. Neurons in the sensory parts of the cortex, such as the visual, auditory or somatic sensory cortex, respond selectively to stimuli from the outside world. This gives rise to the notion of a receptive eld of a neuron, which is the collection of all stimuli that elicit an electrical response in that neuron. In Fig. 9 we see an example of a somatosensory receptive eld. The use of micro- electrodes to record action potential activity for di erent stimuli, provides a cell-by-cell analysis of the receptive eld of each cell and the organization of topographic maps. In Fig. 10, we see an example of a visual receptive eld. Some neurons in the visual cortex respond selectively to the orientation of a light bar. Each neuron has its preferred orientation. Nearby pyramidal cells can make direct excitatory synaptic connections or indirect inhibitory connections by connecting to an inhibitory interneu- ron, which in turn connects to another pyramidal cell. The probability of connection is very high for nearby pyramidal neurons and drops o at about 30m. Therefore, neurons within a cortical column, which is a cross-section of the cortical sheet of about this diameter, display strongly correlated ac- 13Figure 9: Single-unit electrophysiological recording from cortical pyramidal neuron, showing the ring pattern in response to a speci c peripheral stim- ulus. A) Typical experimental set-up. B) De ning neuronal receptive elds. Figure 10: Neurons in the visual cortex respond selectively to oriented edges. A) An anesthetized cat focuses on a screen, where images can be projected; an extracellular electrode records the responses of neurons in the visual cortex. B) Neurons in the visual cortex typically respond vigorously to a bar of light oriented at a particular angle and weakly (or not at all) to other orientations. 14Figure 11: A) Ocular dominance stripes in LGN and layer IV primary visual cortex. B) Pattern of ocular dominance columns in human striate cortex. tivity. The result is that nearby neurons have similar functional roles. An example is ocular dominance given in g. 11. The lateral geniculate nucleus (LGN) receives inputs from both eyes, but this information is segregated in separate layers. In many species, including most primates, the inputs from the two eyes remain segregated in the ocular dominance columns of layer IV, the primary cortical target of LGN axons. Layer IV neurons send their axons to other cortical layers; it is at this stage that the information from the two eyes converges onto individual neurons. Such correlated activity can also be measured in vivo. Fig. 12 shows that neurons in the same column have identical orientation preference. Neurons in nearby columns have similar orientation preference. Thus, this part of visual cortex displays a topographical map, meaning that stimulus features (in this case the orientation) are mapped continuously onto the spatial location in the cortex. Cortical maps are found throughout the sensory cortices and motor cor- tex. Fig. 13 shows that nearby neurons in the auditory cortex respond prefer- entially to nearby frequencies. Typically, maps are deformed representations, that use more neurons to represent important regions. In g. 14 shows the example of the somatotopic order in the human primary somatic sensory cortex. 15Figure 12: Columnar organization of orientation selectivity in the monkey striate cortex. Vertical electrode penetrations encounter neurons with the same preferred orientations, whereas oblique penetrations show a systematic change in orientation across the cortical surface. Figure 13: The human auditory cortex. A) Diagram showing the brain in left lateral view. The primary auditory cortex (A1) is shown in blue. B) The primary auditory cortex has a tonotopic organization. 16Figure 14: Somatotopic order in the human primary somatic sensory cor- tex. A) approximate region of human cortex from which electrical activity is recorded following mechanosensory stimulation of di erent parts of the body. B)Somatotopic representation of the whole body. C) Cartoon of the homunculus constructed on the basis of such mapping. The amount of so- matic sensory cortex devoted to hands and face is much larger than the relative amount of body surface in these regions. 172.4 Summary Although the human brain is often discussed as if it were a single organ, it contains a large number of systems and subsystems. Various types of neurons in these systems are assembled into interconnected circuits that relay and process the electrical signals that are the basis of all neural functions. Sensory components of the nervous system supply information to the central nervous system about the internal and external environment. The integrated e ects of central processing are eventually translated into action by the motor components. The material in this chapter is largely based on 1. 2.5 Exercises 1. Propose a neuron with its input dendritic tree connected to the retina, such that the neuron has the receptive eld property as observed in gure 10. 2. Think about a neural network that may cause the occular dominance patterns observed in gure 11. (a) Consider the strenght and sign of the forward connections from the eyes to the cortex. (b) The lateral connections within the cortex are typically of the Mex- ican hat type: short range excitatory connections and long range inhibitory connections. Explain their role. 3. Suppose you were a neuron and you could only communicate with your fellow neurons through the emission of action potentials. How would you do it? Describe two ways and discuss their respective advantage and disadvantages. 18Figure 15: Recording passive and active electrical signals in a nerve cell. 3 Electrical properties of cells Nerve cells generate electrical signals that transmit information. Neurons are not good conductors of electricity, but have evolved elaborate mecha- nisms for generating electrical signals based on the ow of ions across their membranes. Ordinarily, neurons generate a negative potential, called the resting membrane potential , that can be measured by intracellular record- ing. The action potential is a short spike in the membrane potential, making the membrane potential temporarily positive. Action potentials are propa- gated along the length of axons and are the fundamental electrical signal of neurons. Generation of both the resting potential and the action potential can be understood in terms of the nerve cell's selective permeability to dif- ferent ions and the relative concentrations of these ions inside and outside the cell. The best way to observe an action potential is to use an intracellular microelectrode to record directly the electrical potential across the neuronal membrane ( g. 15). Two micro-electrodes are inserted into a neuron, one of these measures the membrane potential while the other injects current into the neuron. Inserting the voltage-measuring microelectrode into the neuron reveals a negative potential, the resting membrane potential. Typical values are -60-80 mV. Injecting current through the current-passing microelectrode alters the neuronal membrane potential. Hyper-polarizing current pulses decrease the membrane potential and produce only passive changes in the membrane potential. Depolarizing currents increase the membrane potential. 19Figure 16: Ion pumps and ion channels are responsible for ionic movements across neuronal membranes. Small currents evoke a passive response. Currents that exceed a threshold value, evoke an action potential. Action potentials are active responses in the sense that they are generated by changes in the permeability of the neuronal membrane. 3.1 Ion channels Electrical potentials are generated across the membranes of neurons (in fact of all cells) because (1) there are di erence in the concentrations of speci c ions across nerve cell membranes and (2) the membranes are selectively per- meable to some of these ions ( g. 16). The ion concentration gradients are established by proteins known as ion pumps, which actively move ions into or out of cells against their concentration gradients. The selective permeability of membranes is due largely to ion channels, proteins that allow only certain kinds of ions to cross the membrane in the direction of their concentration gradients. Thus, channels and pumps basically work against each other, and in so doing they generate cellular electricity. Membrane channels can open or close in response to changes in their di- rect vicinity, such as a change in the membrane potential, changes in the concentration of neurotransmitters, or sensory input. For instance, hair cells in the cochlea (inner ear) mechanically deform in response to sound, and this 20

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