Introduction to Electrophysiological Methods and Instrumentation

By Franklin Bretschneider and Jan R. de Weille

Introduction to Electrophysiological Methods and Instrumentation, Second Edition covers all topics of interest to electrophysiologists, neuroscientists and neurophysiologists, from the reliable penetration of cells and the behavior and function of the equipment, to the mathematical tools available for analyzing data. It discusses the pros and cons of techniques and methods used in electrophysiology and how to avoid pitfalls. Although the basics of electrophysiological techniques remain the principal purpose of this second edition, it now integrates several current developments, including, amongst others, automated recording for high throughput screening and multimodal recordings to correlate electrical activity with other physiological parameters collected by optical means.

This book provides the electrophysiologist with the tools needed to understand his or her equipment and how to acquire and analyze low-voltage biological signals.

• Introduces possibilities and solutions, along with the problems, pitfalls, and artefacts of equipment and electrodes
• Discusses the particulars of recording from brain tissue slices, oocytes and planar bilayers
• Describes optical methods pertinent to electrophysiological practice
• Presents the fundamentals of signal processing of analogue signals, spike trains and single channel recordings, along with procedures for signal recording and processing
• Includes appendices on electrical safety and foundations of useful mathematical tools

• Language: English
• Publisher: Academic Press
• Release date: Nov 13, 2018
• ISBN: 9780128142110

Franklin Bretschneider

Dr. Bretschneider is a neurobiologist, currently as guest researcher at Utrecht University (Utrecht, The Netherlands). His research over the years has focused on the field of sensory physiology of aquatic animals, and teaching on sensory physiology and electrophysiology. His current lab interest is in bat research, with recording methods and instrumentation as a main focus.

Book preview

Introduction to Electrophysiological Methods and Instrumentation

Second Edition

Franklin Bretschneider

Utrecht University, Utrecht, The Netherlands

Jan de Weille

University of Montpellier, Montpellier, France

Table of Contents

Cover image

Title page

Copyright

Preface

Acknowledgments

Summary

Chapter 1. Electrical Quantities and Their Relations

Electric Charge, Current, and Potential

Resistance

Capacitance

Magnetism

Self-Inductance

Direct and Alternating Current; Frequency

Reactance

Chapter 2. Electrical and Electronic Circuits; Measurements

Current and Voltage Sources

Components, Unwanted Properties

Unwanted Properties, Impedance

Cables

Circuits, Schematics, and Kirchhoff's Laws

Compositions of Similar Components; Attenuators

Practical Voltage Sources and Current Sources

Voltage and Current Measurement

Composition of Unequal Components: Filters

Integration and Differentiation

LC Filters

Chapter 3. Electronic Devices

Active Elements

Semiconductors

Diodes and Transistors

Other Semiconductor Types

Field-Effect Transistors

Ion-Sensitive Field-Effect Transistors

Amplifiers, Gain, Decibels, and Saturation

Noise, Hum Interference, and Grounding

Differential Amplifiers, Block Diagrams

Operational Amplifiers, Feedback

Electronic Filters

Chapter 4. Electronic and Electrophysiological Instrumentation

Electrophysiological Preamplifiers

Amplifier for Extracellular Recording

Testing Amplifiers

Amplifier for Intracellular Recording

Patch-Clamp Amplifier

Two-Electrode Voltage-Clamp Amplifier

Measurement of Membrane Capacitance in Voltage-Clamp

Power Supplies

Signal Generators

Electronic Voltmeters

Electrometers

Oscilloscopes

Important Properties of Oscilloscopes

Chart Recorders

Digital Electronics, Logic

A/D and D/A Conversions

Transducers

Digital Signals; Aliasing

Computers and Programming

Chapter 5. Electro(de) Chemistry

Electrolytes

The Metal–Electrolyte Interface

Capacitance of Polarized Electrodes

Faradaic Processes: The Redox Couple

Practical Electrodes

Electrochemical Cells, Measuring Electrodes

The Ag/AgCl Electrode

Liquid Junction Potentials

Membrane Potentials

Derivation of the Equilibrium Potential

The Reversal Potential

Ion Selectivity

Electrodes Sensitive to H+ and Other Ions

The Glass Micropipette

Patch Pipettes

The Semipermeable Patch

Ground Electrodes

Chapter 6. Volume Conduction: Electric Fields in Electrolyte Solutions

Homogeneous Electric Field

The Monopole Field

The Dipole Field and Current Source Density Analysis

Multipole Fields and Current Source Density

Chapter 7. The Analysis Toolkit

Introduction

Systems Analysis

Convolution

The Laplace Transform

The Fourier Transform

Odd and Even Functions

Linearity

Analog-To-Digital and Digital-To-Analog Conversions

Signal Windowing

Digital Signal Processing

Autocorrelation

Cross-correlation

The Discrete Fourier Transform

The Detection of Signals of Known Shape

Digital Filters

Fourier Filters and Noncausal Filters

Nonlinear Systems Analysis

Chapter 8. Recording of Electrophysiological Signals

The Intracellular Recording

The Extracellular Recording

The Electrocardiogram

The Electroencephalogram

Other Surface Recording Techniques

Recording of Secretory Events

Recording From Brain Slices

Planar Lipid Bilayers

Automation

Voltage-Clamp

Alternative Essays

Impedance and Metabolism

Optimizing Screening Data Acquisition

Telemetry

Chapter 9. Analysis of Electrophysiological Signals

Analysis of Action Potential Signals

Population Spike and Gross Activity

Single-Unit Activity

Uncertainty and Ambiguity in Spike Series

Interval Histogram

Poisson Processes

The Gamma Distribution

The Mathematics of Random Point Processes

Markov Chains

Time Series Analysis: Spike Rate, Interval Series, and Instantaneous Frequency

Spike Frequency or Rate

Interval Series and Instantaneous Frequency

Dot Display

Stimulus-Response Characteristics: The PSTH

Terminology: The Hodgkin and Huxley Channel

Analysis of Macroscopic (Whole-Cell) Currents

The Current to Voltage (I/V) Curve

Leak Subtraction by Extrapolation

Leak Subtraction by Prepulses: The P/N Method

Noise Analysis: Estimating the Single-Channel Conductance From Whole-Cell or Large Patch Recordings

Noise Analysis: Estimating Channel Kinetics

Analysis of Microscopic (Unitary) Currents

Calculating Dwell-Time Histograms From Markov Chains

Estimating Model Parameters From Whole-Cell and Patch-Clamp Data

Chapter 10. Microscopy and Optical Methods in Electrophysiology

Refraction

Diffraction

Image Formation in the Bright-Field Microscope

Fluorescence Microscopy

Multiphoton Microscopy

Optical Fibers

Optical Single-Channel Recording

Opsins, Light-Sensitive Actuators

Voltage-Sensitive Fluorescent Probes

Optrodes

Appendix

Index

Copyright

Academic Press is an imprint of Elsevier

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Background image:

In vitro culture of neocortex cells from the rat brain. Labelling: Red: anti-neurofilament antibody for neurones. Green: anti-glial fibrillary acidic protein antibody for astrocytes. Blue: Hoechst 33342 for nuclei.

White graphs:

Front cover right: Instrumentation; A patch-clamp amplifier simplified to its iconic essence. Front cover left: Recording; Whole-cell currents from a moto-neuron of the spinal cord of a rat embryo in response to voltage-clamp pulses. Back cover: Analysis; A current to voltage plot using the data from the middle panel

Purple graphs:

Front cover: Recording; Synaptic currents recorded from rat neocortex neurones. Back cover: Analysis; The frequency spectrum of a long stretch of synaptic current data. The spectrum is then used for the detection of synaptic events (marked by the ticks in the right panel above the current trace).

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Notices

Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

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Preface

All living cells use electricity for some function or another. Even bacteria communicate with each other through electricity and have been shown to elicit electrical spikes of millisecond duration, uncannily akin spiking by our own neurones. Mitochondria, which are thought to be of bacterial origin, use electricity to fuel the phosphorylation of ADP into the energy carrier ATP. Perhaps viruses are the sole creatures that are electrically inert, even if some viruses contain DNA coding for ion channels. The sensitive plant or touch-me-not (Mimosa pudica) closes its leaflets upon mechanical stimulation. An action potential carried by potassium ions propagates slowly away from the site of stimulation causing other leaflets to close as well. The Venus flytrap, Dionaea muscipula, fires a couple of action potentials before closing the trap over an insect within a fraction of a second. Plants do not merely use electricity for exotic behavior but more generally rely on ion channels for volume regulation, most notably that of guard cells and signaling, involving photosynthesis and chloroplast movement. Electrophysiological signals are the fastest in living nature: The brain resolves time differences of less than 20  μs between the arrival of sound in the left and right ears in real time to determine the direction the sound is coming from. In addition to fast signaling, electrical processes are implied in the sensitive detection of weak signals from the environment. Certain fish bestowed with electroreceptors, sense organs for electricity, react to voltages as small as 1  μV across their body wall.

Ever since the archetypical protozoan ion channel came into being, evolution has engendered a wealth of ion channels with diverse functions, modes of operation, and selectivities. The human genome harbors hundreds of genes coding for (units of) ion channels of which more than 200 are destined for the plasma membrane alone. An equally impressive number of ion transporters add to the diversity of electrical signals in living cells. The study of all those channels and transporters in isolation or in their physiological context is a valid goal by itself and is much the realm of electrophysiology. Electrical signals can equally be used as tokens of health and disease, for diagnostic purposes as in clinical medicine. The electrocardiogram (ECG) and the electroencephalogram (EEG) are standard techniques used in that sense. Even if it might be conceivable to routinely operate equipment for diagnostic purposes without the slightest inkling of its inner workings as if it were a kitchen appliance, such lack of comprehension is certainly not to be recommended in the context of research. Whenever we try to measure something, we cannot prevent perturbing the object of observation to some extent. This is of course a general problem in science, but it appears most acute in the practice of electrophysiology, where it seems to be far much easier to record artifacts than the real events. The best way to dominate these problems is knowledge of the potential interactions between equipment and object under study. This is to justify Instrumentation in the title of the book. The Methods aspect of the book partly encompasses instrumentation, since selecting the right equipment for the job is part of a method to record particular events. Other methods apply to the subsequent analysis and presentation of the recorded data. Introduction alludes to our target group, which are graduate students who wish for a solid and up-to-date basis in electrophysiology. It is specifically destined for readers without a formal training in electronics, signal analysis, or electrochemistry and serves as a thorough, yet easy-to-digest introduction that should lead all the way up from a first recognition of principles to the understanding and the routine application of the various methods.

In the early days of electrophysiological recording, amplifiers and other tools were often built by the physiologists themselves. Nowadays, many types of instruments for recording, processing, and stimulation, versatile and almost perfect, can be delivered off the shelf. Despite the streamlined technology and the many computer algorithms available for filtering or postprocessing of the signals, all students of electrophysiology should gain proper insight in the working principles of their principal tools, specifically the vital stages like preamplifiers and electrodes that are connected to the living preparation under study. In planning experiments with the concomitant purchase of instruments, one has to know the possibilities to choose from and their consequences for the validity of the measurements. Since most of these instruments depend heavily on electronic circuitry, introductory electronics takes a significant part in this book. The chapters concerning it are preceded by a summary of the basics of electricity theory. An important source of artifacts in electrophysiological records concerns the chemistry of electrodes. Electrodes are not inert objects but exchange ions with the medium possibly creating an oxidative, reductive, or hyperosmotic environment. They may induce the formation of capacitive double layers that are a blessing in some situations and a nuisance in others. The description of electrode properties could be the subject of an entire volume of which our book resumes the principal aspects.

The second edition contains several updates that we will not enumerate at length here. We just mention a few. Notably, the chapter that overviews some of the classical techniques of recording electrophysiological data includes several updates since the first edition. Although electrophysiology is still the field of predilection of skilled personnel, several companies make efforts to shift it to a push-the-button exercise. Their efforts are possibly sparked by the huge interest that the pharmaceutical industry has in the development of automated electrophysiology due to the compulsory electrophysiology tests imposed by the FDA and the EMA. In the wake of these developments, several unrelated on-a-chip applications have appeared that we now discuss as well in the second edition. The chapter about optical methods in electrophysiology is new to the second edition too. Of course, every electrophysiologist has to know how to use a microscope. In addition, several fluorescent dyes and engineered proteins have become available with which cells can be manipulated and examined optically.

Acknowledgments

This book is the product of years of experience and cooperation with many colleagues and students. In the first place, we would like to thank our teacher and colleague Dr R.C. Peters, who laid the foundation of the first edition of this book and encouraged one of us (FB) to extend, improve, and publish it over the many years of our cooperation. We owe him many contributions and suggestions. Dr P.F.M. Teunis gave valuable comment and kindly provided the statistical data pertaining to gamma distributions. Many more people provided valuable comment on the first draft, among them Dr A.C. Laan, Mr W.J.G. Loos, Mr R.J. Loots, Mr A.A.C. Schönhage, Mr R. van Weerden, and several anonymous referees. We also acknowledge the smooth cooperation of Ms K. Anderson and Ms N. Farra and other people at Elsevier. Finally, we would like to thank all our students for explicit or implicit contributions and for their patience with the earlier versions of this book.

Summary

Introduction to Electrophysiological Methods and Instrumentation covers all topics of interest to electrophysiologists, neuroscientists, and neurophysiologists, from the reliable penetration of cells, the behavior and function of the equipment, to the mathematical tools available for analyzing data. It discusses the pros and cons of techniques and methods used in electrophysiology and how to avoid their pitfalls.

Particularly in an era where off-the-shelf solutions are available that in some cases reach high levels of automation requiring less and less intervention of the user, it is important for the electrophysiologist to understand how his or her equipment manages the acquisitions and analyses of low-voltage biological signals. Introduction to Electrophysiological Methods and Instrumentation addresses this need. The book presents the basics of the passive and active electronic components and circuitry used in apparatuses such as (voltage-clamp) amplifiers, addressing the strong points of modern semiconductors, and the limitations inherent to even the highest-tech equipment. It concisely describes the theoretical background of the biological phenomena. The book includes a very useful tutorial in electronics, which will introduce students and physiologists to the important basics of electronic engineering needed to understand the function of electrophysiological setups. The vast terrain of signal analysis is dealt with in a way that is valuable to both the uninitiated and the expert. For example, the utility of convolutions and (Fourier, Pascal) transformations in signal detection, conditioning, and analysis is presented both in an easy to grasp graphical form and in a more rigorous mathematical way.

Although the basics of electrophysiological techniques remain the principal purpose of this second edition, it now integrates several current developments. These developments are, among others, automated recording for high-throughput screening and multimodal recordings to correlate electrical activity with other physiological parameters that are often, but not exclusively, collected by optical means. A discussion on signal transducers and relevant aspects of microscopy has been included for this reason.

Chapter 1

Electrical Quantities and Their Relations

Abstract

Electrophysiology is the field of research on electrical processes in living creatures. This includes everything from current flow through wires to signals in our instruments to electrochemical processes, in living cells and organs, and in the electrodes we use to study them. So, although most of the vast field of electricity theory is outside the scope of this book, we need to deal with a handful of quantities that play important parts, such as charge, voltage (potential), current, resistance, impedance, and especially their complicated changes in time. We start with the main definitions and the relationships between these quantities.

Keywords

Capacitance; Charge; Current; Inductance; Magnetism; Potential; Resistance

Outline

Electric Charge, Current, and Potential

Resistance

Capacitance

Magnetism

Self-Inductance

Direct and Alternating Current; Frequency

Reactance

Electrophysiology is the field of research on electrical processes in living creatures. This includes everything from current flow through wires to signals in our instruments to electrochemical processes, in living cells and organs, and in the electrodes we use to study them. So, although most of the vast field of electricity theory is outside the scope of this book, we need to deal with a handful of quantities that play important parts, such as charge, voltage (potential), current, resistance, impedance, and especially their complicated changes in time. We start with the main definitions and the relationships between these quantities.

Electric Charge, Current, and Potential

The basic quantity is the electric charge, buried in the atomic nucleus as what we call a positive electric charge and in the electrons surrounding it, which we call negative charge. The unit of electric charge (abbreviated Q) is the coulomb (abbreviated C), defined in (macroscopic) electric circuits in the 18th century.

The underlying fundamental constant, found much later (in 1909, by R. Millikan), is the elementary charge, the charge of one electron, which amounts to 1.6021  ×  10−¹⁹  C. Since this quantum of electricity is so small, most electric phenomena we will describe may be considered as continuous rather than discrete quantities.

By the nature of atoms, most substances, and indeed most materials in daily life, are neutral. Obviously, this does not mean that they have no charges at all, but that (1) the number of positive charges equals the number of negative charges and (2) the opposite charges are so close together that they are not noticeable on a macroscopic scale. This means that a number of substances can be teased to release electricity, e.g., by rubbing them together. This was indeed the way electricity was discovered in antiquity and was examined more systematically from the 18th century on. Many science museums are the proud owners of the large static electricity generators invented by among others van Marum and Wimshurst. These machines generated rather high voltages (around 50  kV), but at very low current strengths (1  μA), and so were not of much practical use.

Nowadays, most sources of electric energy are electrodynamic, made by rotating machines such as the generators in our power plants, in cars, and on bicycles.

These machines can produce almost any voltage and current needed, usually as alternating current (AC) that can be transformed into lower or higher voltages as desired. In addition, electrochemical processes, found originally by Galvani and Volta, are employed in the arrays of galvanic cells we call batteries and accumulators. Both forms of source deliver the electrical energy at lower voltages (say, 12  V) but allow far larger currents to be drawn (hundreds of amperes in the case of a car battery).

This brings us to the most important quantities to describe electrical phenomena: the unit of tension, the volt (V), and the unit of current, the ampere (A). Note that the correct spelling of the units is in lowercase letters when spelled out, but abbreviated as a single capital. In Anglo-Saxon countries, tension is often called voltage. Both units have practical values, that is, it is perfectly normal to have circuits under a tension of 1  V or carrying 1  A in the lab or even at home. The definitions are derived from other fundamental physical quantities:

Charge (Q): 1coulomb is defined as the charge of 6.2415×10¹⁸ electrons.

Current (I): 1ampere is a current that transports 1coulomb of charge per second.

An overview of electrical quantities, their units, and symbols is given in the Appendix.

The origin of the definition of tension, or potential difference, is a bit more intricate. The electrical forces that act on charges (or charged objects) depend not only on the field strength but also on the distance traveled. Thus, the electrical potential (abbreviated as U) is defined in terms of the amount of energy, or work (abbreviation W), involved in the movement of the charge from a certain point in the electric field to infinity (where the electrical forces are zero by definition). If one does not move to infinity, but from one point in the field to another point, less energy is involved. This is called the potential difference between the two points. Where to choose the two points will be a matter of practical, quantitative discussion.

Electrophysiologists measure what they call the membrane potential by sticking one electrode into a cell. Formally, then, the reference electrode should be placed at infinity, where the potential is defined to be zero. In practice, however, the potential difference between inside and just outside the cell is measured. For this purpose, the potential just outside the cell can be considered to be sufficiently close to zero. This is because the membrane has a resistance that is many orders of magnitude higher than the fluids inside and outside the cell.

Other circumstances, however, change this view radically: Many electrophysiological quantities are recorded entirely outside the cells, such as electrocardiogram, electroencephalogram, and signals from nerves and muscles. In this case the potential outside the cell cannot be considered zero! Instead, the potential difference between two extracellular points constitutes the whole signal. Nevertheless, the potential difference across the cell membrane is called potential in the long tradition of electrophysiology. The unit of tension, or potential difference, is the volt.

Tension (U): 1  volt is the tension between two points that causes 1  joule (J) of work (W) to be involved in carrying 1  coulomb of charge from one point to the other.

We use involved because the energy is either necessary for or liberated by the movement, depending on the direction.

Resistance

The concept of resistance stems directly from these fundamental quantities: If a certain current flows through an object as a consequence of a tension applied to this object, it exhibits the phenomenon of resistance, which is defined as the ratio of voltage to current.

These relations are remembered better in the form of equations:

The latter law is known as Ohm's Law and is familiar to all people who handle electrical processes. It is often seen in the two other forms, depending on which is the unknown quantity:

This means that, knowing any two quantities, Ohm's Law gives you the third one. This is used very frequently. In electrophysiology, for instance, one needs to calculate electrode resistances from the voltage that develops when feeding a constant current through the electrode, membrane resistances from measured current values together with the clamping voltage, and so on.

Resistance is the property of an object, such as a micropipette or a cell membrane. Solids, such as copper, and fluids, such as water, also have resistance, but the value depends on the dimensions of the body or water column. The resistance per unit of matter is called specific resistance or resistivity. The dimension is Ωm (ohm meter). In electrochemistry, where the small unit system (cgs system) is still used frequently, the unit of resistivity is the Ωcm. As a guideline, fresh water has a resistivity of about 1  kΩcm (10  Ωm) and seawater about 25  Ωcm (0.25  Ωm). Obviously, metals are better conductors, i.e., they have far lower resistivity values: in the order of 10−⁵  Ωcm.

The dimension ohm meter may seem odd at first but is easily explained since the resistance is proportional to the length of a water column and inversely proportional to the cross section, which is width × height, or the square of the diameter. So, it is actually a simplification of Ωcm²/cm.

Other, related quantities we have mentioned already are power and energy, or work. The quantity energy (symbol W) has a unit called the joule (J). The related, often more interesting, quantity of energy per unit of time is called power (symbol P) and has the unit watt (W). So, the performance of loudspeakers, car motors, and stoves is expressed in W. The longer they are used, the more energy is spent (which must be paid), but power is the best characteristic. Electrical power depends on voltage, current, and, through Ohm's Law, resistance:

and so on. Work is simply power times time:

and so on. The latter equation is known as Joule's Law.

Capacitance

The quantity to be discussed next is capacitance. This is the ability to store electric charge associated with a voltage. Now what is meant with store?

The phenomenon shows up, either wanted or not, when two conducting wires, or bodies in general, are brought close together. If one of the conductors carries a positive charge and the other one a negative charge, a (relatively high) voltage exists between the two. When brought closely together, however, the electric fields influence each other, thereby partially neutralizing the effect. (If equal positive and negative charges would coincide exactly, or have the same center of gravity, the net result would be zero charge or neutrality. This is why atoms, in general, are neutral.) In other words, by bringing two conductors together, the voltage decreases. Therefore, the charge is partially hidden or stored. The shorter the distance, and the larger the surface area, the more charge can be stored.

Note that this works only if the two conductors are separated by a very good insulator, such as a vacuum or dry air. Otherwise, a current would neutralize the charges. Other good insulators are glass, most ceramics, and plastics. Note also that charge storage is different from what we saw with resistors: A voltage exists across a resistor only as long as a current is flowing through it. The moment the current stops, the voltage will be zero. A capacitance behaves differently. This can be seen by comparing electric quantities with hydraulic ones. An amount of water is the analogue of an electric charge, a flow of water is the analogue of an electric current, and a water level corresponds to an electric voltage. Capacitance is an analogue of a vat or water butt. When water flows in a vat, the water level builds up slowly, depending on the total amount of water poured in. In a small vat, a certain water level is reached with a smaller amount of water than in a large vat. The larger vat is said to have a larger storage capacity.

In the same way, a capacitor is a vat for electric charge, and the word capacitance is derived directly from this analogy. The unit of capacitance (symbol C) is the farad (symbol F, after Faraday).

Capacitance (C): 1  farad is the storage capacity that causes a tension of 1  V to arise by transferring 1  coulomb of charge or

Check the following derived equations:

The capacitance exhibited by two conductors depends on their distance and hence on the form of the objects. Wires, spheres, and irregular shapes have part of the surface area closer and part farther from the other conductor. So computing capacitances can be complicated, but for two parallel plates, it is easy:

(1.1)

Here, A is the surface area (l × w for a rectangle, π r² for a circle), and d is the distance between the plates. Note that the substance between the plates, called the dielectricum, plays an important part. This property depends on the material used and is called dielectric constant, symbol ε. It consists of two parts: ε0 is a physical constant called the absolute permittivity of free space, or absolute dielectric constant, and has a value of 8.854  ×  10−¹²  F/m. The second part, εr, is called the relative permeability or relative dielectric constant, often dielectric constant for short, and is determined by the material between the plates. By definition, the vacuum has a dielectric constant of unity, air has a value only slightly higher (1.00058), but other insulators have more marked effects on the capacitance. The list below gives approximate values (the dielectric constant is temperature dependent and often also frequency dependent):

Apparently, the dielectric medium influences the electric field in the gap between the plates. The relatively high value of water is due to the dipole form of the water molecules, together with their mobility in liquid water. Thus, a large part of an applied electric field disappears in the reorientation of water molecules. But since even the purest water conducts electricity a bit, water is not suited as a dielectric.

Magnetism

Magnetic processes, like electrical ones, are known since antiquity. The intimate relations between the two were found much later, however: only about two centuries ago.

The main relation is the following: An electric current induces a magnetic field, but the converse is not true; a magnetic field does not induce an electric current, only a changing magnetic field generates an electric current.

Because of this relation, magnetic quantities are often expressed in electrical units. A useful measure of magnetic processes is the magnetic flux density (or magnetic inductance or inductance for short), which has a unit called tesla.

Inductance: 1  tesla (T) is the inductance that exerts a force (F) of 1  newton (N) at 1  meter (m) distance from a wire carrying a current of 1  A or

The term flux density can be seen by the following relation:

which means that, integrated over one square meter, a magnetic induction corresponds to an amount of energy.

Self-Inductance

The interaction of electric current and magnetic field leads to the notion of self-inductance in the following way: a changing electric current induces a changing magnetic field, but a changing magnetic field induces a current again. Since this induced current is in the opposite direction, the net effect is that the current that flows is lower than it would be without self-inductance. The unit of self-inductance (symbol L) is henry (symbol H). Since self-inductance arises by changing current strength, the description of the process involves time and follows from the following relation:

In words, the induced voltage is proportional to the change of current strength dI/dt and the magnitude of the self-inductance L (and has the opposite polarity). Hence,

Self-inductance: 1  henry is the self-inductance that causes a tension of 1  volt to develop when the current strength changes (minus) 1  ampere per second.

Apparently, self-inductance is a fundamental property of any conductor carrying current. Nevertheless, self-inductance is strongest when a long wire is wound into a coil, also called an inductor or solenoid. By joining many turns of wire together, the magnetic fields caused by one and the same current are added. Adding a core of iron, ferrite, or any other ferromagnetic material enhances the magnetic induction still further. So, the magnetic properties of iron can be expressed in a quantity similar to the dielectric constant in storing electric charge. This is the relative magnetic permeability and has the symbol μr. The vacuum has a permeability of unity, and many substances have values close to 1. Materials that have a (slightly) smaller value, such as water, are called diamagnetic (μr about 0.99999); they are repelled by a magnetic field (like with the famous experiment in which a frog is hovering in a very strong magnetic field). Materials with a slightly higher permeability (about 1.01) are known as paramagnetic. In fact, all materials show diamagnetism to some degree, but in paramagnetic materials the latter property dominates the first. Far more conspicuous are the so-called ferromagnetic materials that have permeability values far higher than unity. The best known is iron, of course, but some iron oxides (ferrites), nickel, and chrome are also ferromagnetic. The table below shows approximate values for a few well-known materials:

The highest values are from metal alloys specifically designed to have extremely high μr. Since μr depends on the magnetic field strength, the values in the table are approximate maximum values. Because of this behavior, ferromagnetic materials are used as core materials in electromagnets, transformers, etc. and to store information, such as on computer disks. Iron is suited only for low frequencies (50  Hz–50  kHz), whereas ferrites are useful for both low and higher frequencies (up to about 30  MHz). At still higher frequencies, self-inductance is getting so dominant that no core is needed. This is exploited in radio and television sets but does not play a role in electrophysiology. Note, however, that coils made to block (high-frequency) radio waves from an electrophysiological setup must use ferrites as core materials to be effective.

Direct and Alternating Current; Frequency

An electric current that flows in a certain direction and does never change direction is called a direct current, abbreviated DC (or dc). In a more strict sense, a DC is a current that is constant over time. The tension (or voltage) associated with a direct current is called a DC voltage. Static electricity, such as that caused by the charge built into sticky photo albums, and the voltages of galvanic cells (batteries and accumulators) are examples of DC voltages. In science, the term is used in the more strict sense of an absolutely constant voltage. A battery or other power supply is said to deliver a DC voltage. For the batteries, this is only approximately true, since the actual voltage depends on the current drawn from it (the load), and in most types the voltage decreases slowly by exhaustion during use. Nevertheless, the notion of a DC is useful in contrast with an alternating current, or AC, which is a current changing direction all the time, usually on a regular basis. In many countries the mains voltage, for example, changes direction 50 times per second (in parts of the Americas and Japan 60). Strictly speaking, the mains voltage changes direction 100 (or 120) times per second: first from plus to minus, then from minus to plus again. The convention, however, is to count the number of repetitions, or cycles per second, called the frequency. The unit of per second is the hertz (Hz). 1  Hz  =  1 cycle/s.

In electrophysiology, one distinguishes frequency from rate: A sinusoidal or other continuous waveform, such as a sound wave or a modulated light intensity, is said to have a frequency (in Hz), whereas a pulse train, such as a train of spikes (nerve impulses), is said to have a certain rate. This distinction is made for an important reason: If a frequency is changed, all parts of the process are accelerated or slowed down. A receiving apparatus, such as an amplifier, must be adjusted to allow the changed frequency to pass. Pulses, such as nerve spikes, however, have the same shape, and hence the same speed the voltage is changing with, irrespective of their rate. This has consequences for the recording apparatus (amplifiers): see Chapter 3. To underline the difference, a separate unit called the adrian (after the pioneer electrophysiologist Lord Edgar D. Adrian) has been proposed but did not catch on among electrophysiologists. For rates, the unit is usually notated simply as /sec, sec−¹, or sp/sec. The difference is illustrated in Fig. 1.1. The sinusoidal shape is considered to be the basic alternating current. There are several reasons to do this. In the first place, a sine wave arises in electric generators, such the bicycle dynamo, by rotating a magnet in a coil, or alternatively rotating a coil in a magnet.

Figure 1.1  Parameters of (sine) frequency (left) and (pulse) rate (right).

Thus, most electric power distributed in society is made up of a sine wave at 50  Hz (in America 60  Hz). Secondly, it can be shown mathematically that, upon transforming a signal waveform to the frequency domain, a sine is the building element having only a single frequency. Other waveforms, such as square wave, sawtooth, or the tone of an organ pipe, can be considered as combinations of sine waves at different frequencies.

A square wave with a frequency of 100  Hz for instance has components at 300, 500, 700  Hz, and so on. These higher frequencies are always a multiple of the basic, or fundamental frequency, and are called harmonics, overtones, or partials. Note that the fundamental frequency, or fundamental for short, is also called the first harmonic (so the first overtone is the second harmonic).

Actually a sine wave should be called a sinusoidally changing voltage rather than a wave, since the word wave means the spatial spreading of disturbances in a medium, whereas our sine wave is only present at the output terminal of an instrument. Nevertheless, it is a common habit to speak of sine and square waves. We prefer the term signal.

A sine signal is a function of time, and so the voltage at any moment can be described by the following function:

Here, Ut is the voltage at any moment t, Umax is the maximum voltage, usually called the amplitude, and ω is the angular frequency, i.e., the number of radians per second. The angular frequency is related to the frequency (f; the number of cycles per second) by:

When connected to a resistor, an alternating voltage will give rise to an alternating current

.

Reactance

Things get more complicated when we apply alternating voltages and currents to parts that are not simply resistors. For example, we have seen earlier that applying a DC voltage to a capacitor will lead to charging, but once the capacitor is charged to the full input voltage, no current will flow: The two conductors are separated by an insulator, allowing no current to flow between them. When one applies an AC voltage to a capacitor, however, a current (an alternating current) will flow and will keep flowing as long as the AC voltage is applied. Because the input voltage changes polarity many times per second, the capacitance will be charged positively, then negatively, then positively again, and so on. Thus, despite the insulator, a current seems to flow through it.

Note that, although no charge can flow through the insulating layer, the AC current flow through a capacitance is very real. It is not difficult to understand that the magnitude of the current that flows will depend on the capacitance value: A large capacitor stores a larger charge at a given voltage and will sustain a larger current when reverse-charged many times per second. Thus, a 1  pF