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Chapter 10
Appendix

10.1  Appendix A - The Psychophysical Laws

     As a result of a huge amount of experimental work over the last 150 or more years, there have been delineated a few relationships between physical stimuli in the world outside the human nervous system and subjective sensory experience of these stimuli which have achieved the status of "laws". These psychophysical laws have a rather different standing from purely physical laws (which describe physical phenomena such as the falling of apples and the movement of planets) because the psychological phenomena on which psychophysical laws are based are much less precise and harder to measure than physical phenomena. Nevertheless, such psychophysical laws as are currently accepted should probably be included in the facts that would have to be explained by a theory of consciousness. The most basic of the so-called psychophysical laws are stated as follows.

10.1.1  Weber's Law

     If X and X + ∆X are the magnitudes of two stimuli that can just be distinguished from each other (i.e. that are "just noticeably different"), then Weber's Law says that:

where Θ is a constant for each sensory attribute.

     In words, the bigger a sensory stimulus, the bigger has to be the change in that stimulus before any difference can be noticed subjectively.

     This empirical generalization was first stated by Weber in 1834, to describe both his own data on the discrimination of weights lefted simultaneously by both hands and on the visual discrimination of the length of lines, and the findings of others on auditory discrimination of the frequency of tones. It was named Weber's Law by Fechner in 1860, though Fechner noted that in some cases, the data available by that time were actually better fit by the equation

where X₀ is a small constant.

     In the auditory system, for example, a large number of empirical studies have established that Weber's Law holds accurately for the discrimination of the level of Gaussian noise, but not for the amplitude of a pure tone (Laming, 1986).

10.1.2  Fechner's Logarithmic Law

     From Weber's Law and certain additional assumptions - for example the assumption that sensation differences corresponding to just noticeable differences in sensation (sensation jnd's) are equal, Fechner deduced his famous logarithmic law:

where Ψ is a sensation and Φ is its stimulus. To state this law in words, a sensation is proportional to the logarithm of its stimulus. Fechner's law immediately became the subject of enormous controversy and was eventually excoriated by none other than that paragon of psychological virtue William James. However to this day it still forms the basis of the universally used decibel scale of loudness, which relates perceived loudness in decibels (dB) to the logarithm of the ratio between the measured sound intensity, I and a reference sound intensity, I_ref:

     The use of the decibel scale reflects the fact that, roughly speaking, a sound only doubles in perceived loudness when the physical intensity of the sound increases ten-fold⁶. This means that the auditory system of humans can deal with a huge range of sound intensity without being overwhelmed. It also places a very useful empirical constraint on any hypothesis concerning the electromagnetic pattern which constitutes the sensation of a sound.

10.1.3  Stevens' Power Law

     While Fechner measured sensation by means of the just noticeable difference or jnd, S.S. Stevens employed units defined by ratio and interval scales⁷ and concluded that The Psychophysical Law is not a logarithmic law but a power law (e.g. Stevens, 1957 ^[280]). Stevens suggested that, for example, loudness L, is a power function of the physical intensity of the sound I, such that

where k is a constant that depends on the subject and the units used. In other words, the perceived loudness of a given sound is proportional to its physical intensity raised to the power of 0.3.

     To generalize, this equation can be written as:

where Ψ is the magnitude of a sensation in psychological units, Φ is the magnitude of the stimulus in physical units and n is an exponent that varies in magnitude with the dimension being scaled - e.g. n is 0.3 for loudness, 0.3 - 0.5 for intensity of light, 1.0 for taste and so on.

     Stevens' work led to the rise of what some have called "the new psychophysics" (Gescheider, 1997 ^[111]) and dozens of investigators have confirmed his power law. However, it will be apparent that none of these so-called psychophysical laws measures the quantity of sensation in absolute terms. All measurements of sensation are couched in terms of the relationship of the sensation of interest to other sensations evoked by other stimuli. In fact the results obtained tend to depend on the measurement method used and on the environmental conditions when the measurements are made. There are many exceptions to the power law (Gescheider, 1997 ^[111]).

     In summary, psychophysics is a huge and complicated field of study. For our present purposes, it is perhaps sufficient to state that (a) the measurement of subjective sensations is fraught with difficulty and (b) it is relatively rare for the magnitude of a measured sensation to covary on a one to one basis with the magnitude of its physical stimulus.

10.2  Appendix B - A Beginner's Guide to Cellular Neurophysiology

10.2.1  The Resting Membrane Potential

     The central fact of cellular neurophysiology is that the inside of nerve cells is slightly electrically negative with respect to the outside. This situation arises because the chemical composition of the intracellular environment is different from the chemical composition of the extracellular environment. Relatively speaking, there is a high concentration potassium ions inside the cell and a low concentration outside. Since the membrane separating the inside contents of the cell from the outside fluid is somewhat permeable to potassium ions under normal conditions, these ions tend to diffuse out of the cell (simply because there is a natural tendency for any particle in a watery environment to diffuse from areas where there is a lot of it to areas where there isn't). As positively charged potassium ions leave the cell, the intracellular environment becomes more and more electrically negative. Opposite charges attract, so eventually this intracellular negativity becomes strong enough to prevent any more potassium ions leaving the cell. The force of the concentration gradient influencing the ions to leave the cell is now equalled by the opposing force of the electrical gradient influencing them to go back in. The intracellular voltage at which this situation obtains is called the equilibrium potential for potassium and is about -90 mV.

     Most neurons sit for most of the time at an intracellular resting membrane potential that is slightly less negative than the equilibrium potential for potassium. However there are various ways in which the resting membrane potential of neurons can suddenly be changed.

10.2.2  Action potentials

     Firstly, the neuron can fire an action potential. Action potentials involve sodium ions. The concentration situation for sodium ions (which also carry one positive charge) is the opposite of that for potassium ions i.e. there is a lot more sodium outside the cell than inside. Most of the time the cell membrane is completely impermeable to sodium ions, but there do exist sodium channels (i.e. protein-lined pores in the lipid membrane which can open to let sodium ions flow through them) which are voltage dependent. When the inside of the cell for one reason or another becomes less negative, these voltage dependent channels suddenly open and sodium rushes into the cell. Because the concentration gradient for sodium is opposite to that for potassium, the equilibrium potential for sodium is +58 mV (as opposed to -90 mV for potassium). So if the sodium channels stayed open for long enough, the influx of sodium would continue until the inside of the cell was +58 mV with respect to the outside. Before that happens however, the changing internal voltage closes the sodium channels and opens a separate set of potassium channels. Whenever membrane channels open, ions flow through them in such a direction as to bring the intracellular potential towards the equilibrium potential for the ion involved, so potassium ions now flow out of the cell again until the resting membrane potential is restored.

     This whole sequence of events takes only a few milliseconds. Thus an action potential consists of a sharp spike of intracellular positivity. Because of the characteristics of the sodium channels involved, action potentials are "all-or-none" events. Once the threshold membrane voltage is reached (about -45 mV) they either happen or don't happen, with no shades of gray. As such, action potentials provide neurons with what is essentially a digital coding capability; on or off, yes or no, 0 or 1.

     Action potentials have two more important characteristics. They propagate down axons (a) very fast and (b) non-decrementally. Thus they are well adapted for conveying information over large distances in a remarkably reliable fashion.

10.2.3  Excitatory and inhibitory postsynaptic potentials (epsps and ipsps)

     The other main way in which the resting membrane potential of neurons can be altered is by means of postsynaptic potentials. These can be either excitatory (i.e. tending to bring the neuron towards the threshold for firing an action potential) or inhibitory (i.e. tending to keep the neuron away from the threshold for firing an action potential).

     The gap between neurons where information passes from one to the other is called a synapse. Synapses usually involve the axon of one neuron (which is called the presynaptic neuron when speaking of this particular synapse) making a contact on one of the dendrites of another neuron, (which is called the postsynaptic neuron). When the action potential in the presynaptic axon reaches the axon terminal, it causes the release of vesicles of chemical neurotransmitter into the gap, or synaptic cleft. The neurotransmitter diffuses across the gap and interacts with transmitter receptors in the membrane of the postsynaptic cell. There are a number of different neurotransmitters in different kinds of neurons (one transmitter per neuron) and each kind of neurotransmitter has its own kinds of receptors, which are coupled to specific ion channels in the membrane of the postsynaptic cell. Interaction of a transmitter with its receptor opens (or occasionally closes) the ion channel associated with the receptor and ions then flow through the channel in such a way as to bring the internal voltage of the postsynaptic cell towards the equilibrium potential for the ion in question. The voltage changes brought about by these ion flows are called postsynaptic potentials.

     Postsynaptic potentials differ from action potentials in one very important way. Postsynaptic potentials are not all-or-none events like action potentials - they are graded. This means that if more neurotransmitter has been released (as a result of more action potentials reaching the presynaptic terminal) the postsynaptic potential will be bigger and longer-lasting. If some of the postsynaptic receptors have been blocked by a drug, the postsynaptic potential will be smaller and shorter-lasting. If two postsynaptic potentials are evoked close together, in either time or space, they summate or add together to have a greater effect on the postsynaptic cell. This latter feature is very important, since it is rare that one epsp alone can bring a postsynaptic cell to the action potential threshold or one ipsp prevent firing. So while action potentials are essentially digital events, postsynaptic potentials are analogue ones. This becomes important in the generation of extracellular field potentials, which can be measured by EEG recordings.

     The genesis of extracellular field potentials is described in Appendix C.

10.3  Appendix C - Generation of Extracellular Field Potentials

     When neurotransmitter released by the nerve terminal acts on its postsynaptic receptors, ion channels open and positive ions flow into the postsynaptic cell, as described in Appendix B. The removal of these positive ions from the extracellular fluid leaves a measurable negativity in the area around the synapse. Therefore while an excitatory postsynaptic potential is happening, an extracellular electrode positioned near the synapse measures a negative-going synaptic potential. At the same time, because of the influx of positive ions in the synaptic region of the postsynaptic cell, a flow of current out of the cell in the non-synaptic regions is set up. If an electrode is placed outside a region of the cell that is not near the synapse, it records this outward current. Therefore the same synaptic potential recorded in these extrasynaptic regions is positive-going.

     This change in sign of extracellular field potentials depending on the location of the recording electrode in relation to the synapse contrasts with the situation with intracellular recording, (where the recording electrode is inserted into the cell) when the postsynaptic potential is positive-going as a result of the influx of positive ions no matter where in the cell it is recorded.

     In order to understand more about the generation of field potentials, knowledge of one physical law is needed. This is Ohm's Law, which can be stated as

where V is voltage, I is current and R is resistance.

     Electrophysiological potentials are measured as the voltage drop across a particular resistance. With intracellular recording, the relevant resistance is that of the cell membrane. With extracellular recording it is the resistance of the extracellular fluid, which is usually very low compared to the resistance of the cell membrane. According to Ohm's Law, a given flow of current will produce a larger voltage across a large resistance than it will across a small resistance. Therefore, any given flow of ionic current produces a much bigger intracellular potential (i.e. voltage) than it does an extracellular potential. In fact in order for extracellular potentials to reach appreciable size, the field epsps from a number of spatially lined up cells have to summate. This means that the synapses have to be active synchronously. A high local extracellular resistance is also important in generation of relatively large field potentials.

10.3.1  Field potentials and the EEG

     The EEG as measured by electrodes on the scalp is the sum total of all the field potentials that are generated within some unknown distance of the recording electrode. Clearly, those field potentials generated closest to the recording site will have the biggest influence on the waveforms recorded. In fact it is generally thought that only field potentials generated in the cerebral cortex are close enough to the scalp to be recordable by EEG electrodes at all. The case for this has never really been proven however. Certainly the first 10 milliseconds-worth of the auditory evoked potential is believed to be generated in the brainstem; but it is true that the amplitude of these waveforms is very small in comparison to the general EEG (i.e. fractions of a microvolt compared to about 50 microvolts for the total EEG).

     The major difficulty about the interpretation of EEG recordings is that sorting out the cellular correlates of electrical activity which can only be measured many cm away from its site of generation is no easy matter. Even determining the general area in which a particular waveform in an evoked potential is generated is not straightforward. Calculating the detailed spatiotemporal configuration of an electromagnetic pattern at its generation site from measurements taken some distance away from that site is formidably difficult. This is known in mathematical and engineering circles as the inverse problem. Progress on description of the electromagnetic patterns covarying with conscious experience awaits an acceptable solution of the inverse problem for electroencephalographic measurements.

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