__Computers in Neurophysiology__** **

Prepared by W.S. Rhode, Dept. of Neurophysiology

U.W. Medical School, Jan. 1976

The introduction of computers into data collection and analysis in neurophysiology occurred in earnest in the early 60's. A rapid-development in computer technology since then has made the minicomputer a common laboratory item due to the rapidly decreasing cost and increasing capability. Within a few years a computer with all the capabilities necessary for most laboratory control and data manipulation tasks will cost less than a $1000. This will bring it within the price range of nearly everyone.

In order to utilize the capabilities of these systems one must understand its characteristics. But before that one should understand why a computer has become a routine item in a neurophysiology lab. The development of electrodes, either surface, gross or microelectrodes, along with the necessary electronics to amplify and filter the physiological signals, resulted in a vast increase in the amount of data that could be collected. One technique which has been applied to analyzing the discharge pattern of neural discharges is to illuminate a point on an oscilloscope with the time of discharge encoded as the distance from a left-hand sync marker as illustrated in figure 1.

The exact time of occurrence of the discharge could be measured using a ruler and any of a number of simple statistical computations could be performed. This was extremely tedious and severely limited the productivity of the neural scientist.

__Background__
The development of a small computer which was designed for use by

biologists as a laboratory system to be connected up to equipment which is either

already present in the lab or special purpose devices which would be designed and "interfaced" to the computer, The computer was the 'LINC' or Laboratory Instrument Computer (1). This machine included many architecture innovations which were to become standard in the minicomputer industry in the years to follow. The prototype was built and demonstrated in 1962 at Lincoln Laboratory. It was promoted by NIH through an evaluation program which gave eleven LINCs to various groups including the Laboratory of Neurophysiology in 1963.

The goals of the development are given below since they are still applicable to neural science. "The goal of the development has been a machine which: (1) is small enough in scale so that the individual research worker or small laboratory group can assume complete responsibility for all aspects of administration, operation, programming, and maintenance; (2) provides direct, simple, effective means whereby the experimenter can control the machine from its console,, with immediate displays of data and results for viewing or photographing; (3) is fast enough for simple data processing "on-line" while the experiment is in progress, and logically powerful enough to permit more complex calculations later if required; (4) is flexible enough in physical arrangement and electrical characteristics to permit convenient interconnection with a variety of other laboratory apparatus -both analog and digital -- such as amplifiers, timers, transducers, plotters, special digital equipment, etc., while minimizing the need for complex intermediating

devices; and (5) includes features of design which facilitate the training of persons unfamiliar with the use of digital computers.

Computers
have grown in computational power by improved technology which allows basic
arithmetic and transfer operations to occur at very fast rates. The typical minicomputer can add two numbers
at a 500,000 to 10^{6} rate while larger (and more expensive) computers
can execute 10^{7} to 3 X 10^{7} instructions/sec. The size of the memory of the computer can
vary from a few thousand words to several million words. The average memory size has grown
dramatically over the last few years due to the decreasing cost. In 1965 a 32,000 word memory would have cost
$85,000; today it can be purchased for $2,000 with some indication that it will
cost one tenth that in another 5 years and be at least 10 times faster than the
1965 memory.

Speed and memory size are important characteristics for any computer since they can limit the applications which can be handled. For example, the control of a behavioral laboratory and recording of simple responses doesn't require a very powerful computer, while recording, digitizing (converting an analog signal to digital number) and analyzing 16 channels of EEG is a much more demanding task and graphical processing operations which require the graphical manipulation of images, e.g., neural morphology or the heart. These structures can be rotated on a graphical display to create an illusion of depth or a three dimensional object. This requires a reasonably powerful system to perform the computations though much of the graphics on a modern PC is in the graphics card.

There are a wide variety of systems available for use in laboratories today. The choice of systems is usually dictated by the number of dollars available. Many considerations do come into play: 1) the data rates of the experiment; 2) the knowledge of the user, e.g., can he interface his laboratory apparatus to the computer? What programming languages does he know? 3) the size of programs used to analyze data; 4) past experience and availability of computers; 5) local support for maintenance, programming, spare parts and peripherals, etc.

__What can computers contribute?__

The-primary question is ‘what is the goal of the neural sciences?’ In studying the central nervous system we would like to elucidate its operation at an electro-chemical level, anatomical level or physiological level, The explanation of the action of the brain or nervous system must be guided by some general principals (2):

1) the theory must be action oriented, that is one seeks to explain the interactions with the environment that the animal engages in; 2) perception is not only of ‘what' but is related to the environment in which interaction occurs, that is 'where'; 3) an adaptive system, such as the brain, must be able to correlate sensory data in a manner which facilitates the evolution of an internal model of the experience; 4) the organization must be hierarchical with adequate feedback loops to coordinate the many subsystems; and 5) the brain is a layered somatotopic computer.

These principles cover simple animal reactions, perception, memory and the adaptive nature of the organism. The investigative strategy for these principles varies with each. Often some combination is to be investigated which will demand a multidisciplinary approach. There is no one best way to investigate and explain these basic principles. Many approaches are used at each level.

One area which is being intensively pursued is sensory processes. For example, electrophvsiology, biochemistry, neuroanatomy, psychoacoustics, etc., have all been employed in pursuit of the explanation of how sound is encoded within the inner ear. In the course of the last few years the following techniques have all been used to determine the motion of the basilar membrane. (Motion of the cochlear sensory cells is in large part determined by the motion of this membrane.) Direct observations have included the use of the light microscope, the M8ssbauer effect, the capacitance probe, the laser interferometer, and fuzziness detection. Indirect observations of the motion of the basilar membrane have used auditory nerve recording, spiral ganglion recording, cochlear potentials, psychoacoustics, and mathematical models to infer the form of vibration. This is typical of any contemporary investigative area. Multi-disciplinary approaches often using a computer to assist in handling the large amounts of data are commonplace.

In the U.W. Neurophysiology Lab studies of the auditory nervous system have been performed for over 40 years. Averaged evoked response recordings and the determination of the vibratory characteristics of cochlear structures, along with microelectrode studies of all-or-none neural spike activity have benefited from the availability of a small laboratory computer. The computer system, which is based on a Classic LINC computer, has been used since 1963 for control and monitoring of stimulus parameters and data collection. While the LINC continues to be a useful tool, it nevertheless has some limiting features such as a small memory (2K), 12-bit word length, lack of direct memory access (DMA) capability and the lack of standard peripherals. In order to overcome these limitations, a new computer system (Harris 6024/5) has been purchased. A major effort is under way to enhance the experimental facilities with new and possibly unique stimulus generation and data collection equipment using PCs and DSPs.

The stimulus waveforms used to conduct research on the auditory system include sinusoids, trapezoids, pseudorandom noise, amplitude-and-frequency modulated sinusoids (AM and FM), swept frequencies, tone complexes, and biologically meaningful sounds such as speech. Digital stimulus generating equipment has been designed to produce these classes of stimuli and to be completely computer controlled.

For data collection, an event timer, a programmable analog-to-digital converter, and a digital spectrum analyzer have been designed to meet the demanding needs of research. Each device is designed for ease of use and includes data buffers which relax the requirements for rapid response by the computer system.

Besides stimulus generation and data collection, the data storage and analysis aspects of experimentation are equally important. A graphic display processor has been designed which includes complete buffering of the display yet provides dynamic capability to handle rapidly changing displays. An electrostatic printer/plotter provides hard copy graphic output which is so essential in data analysis. The primary data storage device is a tape cartridge unit with a 50 Mbit capacity, which is sufficient to handle most storage problems. The computer system has a Disc Monitor System which supports multiprogramming. Nearly all programming is done in Fortran eliminating the tedium of machine language coding and speeding program development.

__Stimulus Generation and Control__

The primary goal of the stimulus generation and control apparatus is to produce a wide class of auditory stimuli with proper fidelity and with complete computer control of the sequence of presentations. The apparatus includes a digital stimulus system, a pseudorandom noise generator and a variable length circulating buffer for the presentation of arbitrary waveforms.

__Digital Stimulus System__

This
system grew out of the need for a programmable sinewave oscillator with digital
control of stimulus presentation. While
a number of programmable frequency synthesizers appeared on the market in the
early 1970's, their principal limitation for auditory research is the harmonic
content of their output (greater than 0.1%). In 1972 we began the design of an
all-digital technique for waveform synthesis combined with electronic gating of
the signal, digital-timers, and a stimulus counter for stimulus sequence
control. The device is called the
Digital Stimulus System (3) or

A simplified hypothetical example of the technique for
synthesizing a sinewave using table look-up is shown in fig. 3A. The contents, F, of a 4-bit frequency register
are added to q,
the contents of the sine address register, which at time T equals F·T
modulo (2^{6} ) where T = 0,1,2...,¥, and 2^{6}=the
clock frequency. A table of 64 values of
the sine function is stored for each of the 64 possible values of q. The
value of F (1 to 15) determines the size of the step through the sine
table. If F = 1, then each of the 64
values of the sine (q)
are read out each second, whereas if F = 2 then every second value of the sine
function is read out twice a second. In
general it can be seen that the value of F is the frequency of the synthesized
sinewave. The result of synthesizing 4
and 8 Hz sinewaves is shown in fig. 3B.
The effect of quantizing a signal both in time and amplitude is obvious
and can be reduced by shortening the sample time (higher clock rate) and using
more bits to represent the signal.

The actual ^{19 }or 524,000 Hz. If the sine (q) was** **stored for each of the 2^{19} values of the sine address
register at an accuracy of 16 bits, an 8,000,000 bit memory would be
necessary. Only one-quarter of the sine
function need be stored and trigonometric identities can be used to further reduce
the size of the ROM to 16,000 bits. A
small sacrifice in accuracy is made to accomplish this savings in memory size;
the table is accurate to 1 part in 2^{15} for 2^{17} arguments
of q.
The distortion of the

be less than 0.01% for frequencies below 10 kHz; this distortion is due to the
digital-to-analog converter. The ^{-16}
to 2^{+16} , that is, any frequency up to 65,000 Hz can be
generated. The frequency is specified in
two parts, a 16-bit integer and a 16-bit fraction, and the initial phase angle
of the signal can be specified with 16-bit accuracy. Two or more systems can he interconnected to
produce Frequency Modulated signals, Amplitude Modulated signals or to
repetitively sweep a range of frequencies.

In
order to reduce the amount of energy at frequencies other than the one being
generated, the signal is turned on gradually.
That is, it is multiplied by a trapezoidal waveform which has a
programmable rise/fall time. A 16 x 16
bit digital multiplier is used to perform the multiplication. The advantage of this approach over the use
of an analog multiplier is that it doesn’t introduce any distortion when the
signal is fully on since the signal waveform is merely multiplied by a constant
value. The rise/fall time of the

The third major subsystem is the stimulus sequencer consisting of three digital timers, each with 4 presettable digits and a timing range of 1 msec. to 9999 sec., plus a 6-digit presettable counter which permits 1 to 999900 repetitions of a timing sequence. The timers and counters are interconnected so that sequences of stimuli can be generated as shown in fig. 5. A command from the computer starts the delay timer. When the delay timer has expired the repetition interval timer and stimulus duration timer are started. The repetition interval and the stimulus duration timers are retriggered if the stimulus counter is not at its final value when the repetition interval timer and the risetime delay counter expire.

The final step in the synthesis
process is the digital-to-analog conversion (fig. 4). This is the step which limits the accuracy of
waveform production. Theoretically, a
16-bit system should result in a system which has harmonic distortion in the
neighborhood of 2^{- 16} or -96 dB.

e present The system achieves about -80 dB distortion which is quite acceptable for most auditory experiments and is comparable to the best transducers for sound production. The limiting system component is the digital-to-analog converter or DAC. Any DAC can produce glitches or large undesirable steps in output when undergoing transitions in input addressing which involve a change in the state of a large number of bits, e.g., 001111111 to 01000000. A deglitcher amplifier is used to suppress these transients; this is a fast sample-and-hold circuit which maintains the output of the DAC at its previous value until the input address has had time to change and the output of the DAC has stabilized.

The philosophy incorporated into the design is to make most of the functions of the system capable of both manual and programmable control. This allows initial exploration of neural unit responsivity without the need for computer interaction. It is also useful for maintenance of the system.

__Pseudorandom Noise Generator__

Pseudorandom numbers are generated using a maximal length sequence generator (4). This device is useful because repeatable sequences are generated and the length of the sequence can be varied. This permits a reduction in the amount of computations necessary to recover the transfer function of the system being analyzed.

The basic idea is that a system's transfer function, the relation of its output to input, is recoverable by merely measuring its output, Y(jW), when the input, X(jW), to the system is white noise and the system is linear. When the input is white noise, X(jW) is equal to constant, K, and since Y(jW) = H(jW)·X(jW) it is obvious that the transfer function is H(jW) = Y(jW)/K.

__Waveform Generator__

In certain auditory experiments it is desirable to combine 3, 4 or more harmonics of a given frequency with specifiable amplitude and phase. The cost of building a separate sinewave synthesizer for each harmonic becomes prohibitive and has resulted in alternate methods of synthesizing harmonic complexes (5). One method of accomplishing this is to have a variable length circulating buffer which stores one cycle of the desired waveform. For example, if the 9th, 10th, llth and 13th harmonics of 100 Hz were to be combined, the basic or fundamental period of the complex wave would be 1/100 sec. or 10 msec. Therefore, a 10 msec sample of this waveform must be stored and the desired stimulus duration achieved by repeating the 10 msec waveform.

The maximum size of the buffer is to be 16,000 words. Therefore, at a sample rate of 64K, the maximum period can be 250 milliseconds. This period can be increased by increasing the length of the buffer or by decreasing the sample rate. This device can also be used as a simple buffer for delivering stimuli of arbitrary duration by merely transferring data from a disc file to the buffer such as speech, phonemes, or animal sounds.

__Other stimulus generation devices__

The previously mentioned devices are in a sense generic. They find application in investigating every sensory system at any level of the nervous system. The types of stimuli depend on the sense being explored but the 'electronics' is often nearly the same. The range of frequencies necessary to investigate the auditory system is from 10 to 100,000 Hertz while the tactile system doesn't require much more than 200 Hertz. The types of waveforms don't differ very much.

__Data Collection Facilities__

The principal data collected in our laboratory consist of trains of all-or-none unit spikes. Other frequently encountered data types include averaged evoked responses and continuous physiological signals such as EEG recordings, respiratory variables, etc. Several devices have been designed which are used to collect these separate data types including a multiple event timer, histogram binners, a programmable A/D system and a digital spectrum analyzer.

__Event Timer__

The conversion of the all-or-none unit activity to a discrete sequence of times of occurrence is by far the most important activity in the lab. Each time the voltage recorded from a microelectrode exceeds a specified threshold value the time is recorded as indicated in fig. 6. The event timer designed to perform this task resolves events by means of a time base which can be varied from 1 usec under program control. An 8.3 sec period can be timed with 1 msec resolution without overflow in the counter. The event timer can time up to 16 events and includes a 32-word buffer (a FIFO or First-In-First-Out buffer) to store event times. This relaxes the need for rapid response to timer events by the computer system.

The event timer can be turned on and off either by computer command or by a selectable sync (synchronization) pulse. The sync can arise in any of the three DSSs or an external source; one of the sources must be selected. The sync occurs whenever the signal is first turned on. The event timer is stopped by a terminate pulse which occurs whenever all the stimuli have been presented and the proper time has elapsed.

__Analog-to-Digital Conversion System__

The design objectives of the analog/digital converter system are simplicity of use and flexibility. It is a 16-channel system having sample rates as high as 100 kHz with 12-bit accuracy. The A/D inputs can be used single-ended, differential or pseudodifferential. The channels to be sampled are program selectable, as is the sample rate and number of samples to be taken. The device is connected to the computer on a DMA (direct memory access) channel so that the high sample rates can be accommodated by the system. No handling of the data is necessary except to move it out of its buffer in the computer's memory.

Three sampling techniques are available in the A/D system; 1) Quick-Scan in which conversion of each channel occurs as soon as the previous channel is converted until all selected channels have been sampled, the A/D then waits for the next clock pulse, 2) Uniform Interval in which the next channel is converted after a clock pulse occurs, and 3) Rapid Sample in which only channel 0 is sampled, permitting samples to be taken at a 160 kHz rate.

The Quick-Scan technique allows one to effectively sample all channels at the same instant in time. This is especially advantageous for low sample rates although it is desirable in general. The Uniform Interval mode is the common sampling technique. In the Uniform Interval mode, the rate is the channel sample rate multiplied by the number of channels to be sampled.

__Data Analysis__

Histograms

One of the most common techniques for analyzing distributions which are empirically derived is the histogram. The histogram displays the number of events in a range of the metric being used. For example, one could display the number of students with heights (or weights) as the abscissa of the display. For convenience one groups' this discrete data as a function of the independent variable. That is, the independent variable weight or height is discretized by letting a range of the variable be defined as a bin. The number of events which occur within a specified range determine the height of the bin.

f(x) = a binary process which has the value 1 at a discrete set of x's. The domain of f(x) can be viewed as all values on the real line. Due to the conditions imposed by the generation of f(x) often x is restricted to a domain of 0 to x max,

The computing of a histogram is basically a counting process. The domain is divided into a contiguous set of uniform intervals of length, Ax. Each of these intervals is called a bin.

Bin (i): = the ith interval of x.

__define__ an integer function

Then computing a histogram is defined as counting the number of events within each interval of the range of the histogram where the range is NAx. That is,

Bin (i) = Bin (i) + 1 when i-1 £

Note** **the 1st bin covers the range 0 __<__
x < Dx

the 2nd bin
covers the** **range Dx __<__
x < 2Dx

etc.

A random process is illustrated in the figure below with the resulting histogram.

The form of a histogram can vary from the bar graph presentation, to the outline of the histogram to the use of crosshatching to indicate two or more superimposed histograms.

There are several types of histograms used in various applications including post-stimulus time histograms, latency histograms, cycle histograms, etc., which will be discussed later.

Neuronal Spike Train

It is a working hypothesis (a) that there is a wealth of information about the structure and function of the nervous system which can be derived from a study of the timing of spike events; (b) that analysis of these signals can shed light on the mechanism of spike production within the observed cell, on the presynaptic input to the cell, and on the mechanisms by which the latter is transformed into a postsynaptic output; and (c) that observations of multiple units can reveal details of interconnections and functional interactions (6, 7).

We are primarily concerned with information processing by the nervous system. The use of some simple statistical measures allow a relatively concise characterization of the output of the neuron, which may be useful in the description, comparison and classification of nerve cells. The underlying neuronal processes have a degree of randomness. This gives rise to its characterization as a stochastic point process and the subsequent use of statistical methods of analysis. The methods of analysis will vary depending on the experimental circumstances and whether we are concerned with intra- or inter-neuronal phenomena.

In a neuronal spike train the interspike interval histogram serves as an estimate of the actual pdf. To construct the interval histogram the time axis is divided into m uniform intervals of length d. Each event (as shown in Fig. 6) is analyzed to determine which bin of the interval histogram it should be placed in by using the following relation.

* *N_{j}*. = *N_{j+1} if (j-1)d£T_{i} <jd where

N_{j}= number of events** **(spike intervals in the jth bin).

The total number of intervals are N where the number of spikes is N+l. This is due

to the fact that the first interval (0,T_{1}) is not
included in the analysis, only

the interspike
intervals. The ratio N_{j}/N is smoothed** **estimate of the pdf f(t).

Note: _{}

i.e., _{}

which is the probability that the duration of a random chosen interval lies between (j-1) and j6.

A
peak in the interval histogram shows a preferred periodicity in the
'train'. Certain neuronal systems will
show phase locking which is manifest by a multimodal appearance in the interval
histogram. The auditory periphery
demonstrates this at low frequencies as shown in figure 7.Note the** **importance of the proper bin width to
reveal the detail in the figure.

__Order-Dependent
Statistical Measures__

It is of interest to determine whether or not successive
interspike intervals are independent in the statistical sense and therefore
whether the spike train can be described as a realization of a renewal
process. The joint-interval histogram
was introduced by Rodieck, Kiang, & Gerstein (8) is displayed in the form
of a scatter diagram as shown in figure 8. Each pair of adjacent intervals in
the spike train is plotted as a point.
The (i-1)st interval is the distance along the abscissa while the i'th
interval is the distance along the ordinate.
If successive intervals are independent then the joint-interval
histogram will be symmetric about a 45^{0 }line through the origin.

Proof of
statistical independence requires that the joint probability be equal to the
product of the individual probabilities.
That is prob(T_{i}, T_{i+1},) = prob(T_{i}).prob
(T_{i+1}). In practice a
substitute is used for this test which involves computing the mean of each row
and each column in the JIH. The column
means are plotted vs row and the row means are plotted vs.** **column. If the intervals
are independent then the means will be parallel to the axes. This is considered necessary but not
sufficient for independence.

__Spike trains in the presence of stimuli__

Much
of the previous discussion involved characterizing a stochastic point process
on the basis of a spike train observed in the absence of any stimuli. That is only the spontaneous activity is
observed. In most neurophysiological
experiments different types of stimuli with various

patterning are used. The stimuli are
appropriate to the sensory modality or central nervous system function being
investigated. The stimuli could be
patterned repetitions of sounds, a spatially modulated video display, a
sequence of stepped increases in temperature, A random variation in angular
acceleration for vestibular stimulation, the application of a specified
concentration of chemical or various tactile stimuli.

The response would look more like figure 10 if the stimulus had been an acoustic click, that is an acoustic equivalent of an impulse which will result in information about the transfer function of the system under investigation.

The PST is an average of the spike train for a repeated stimulus. The acoustic click nay involve the repetition of 100's of clicks. A similar use of PST's obtained from the averaged velocity-modulated gamma-ray activity from a radioactive source required the averaging of 100,000 to 400,000 responses in order to obtain a reliable estimate of the systems impulse response. The gamma-rays arise as a. random process and produce spikes which are not very different than a neural spike and therefore can be analyzed in the same manner. In this instance knowledge of the meaning of the modulations in the PST permitted a recovery of the instantaneous velocity of the mechanical structure being investigated.

In general the bumps and wiggles of the PST reflect the underlying excitatory process and one must decide if they are significant or merely random fluctuations. The distribution of mean square deviations from mean bin level can be computed or a control case can be constructed using fictitious times of stimulus presentation and portions of a record where no actual stimulations were presented. In general meaningful features should have a width of several bins. Note one can always recompute the PST with a smaller bin width. Usually the response is obvious at all but threshold and subthreshold levels of stimulation.

The general analysis using PSTs can be used in other ways; one of which is the nth spike latency histogram. Usually this is used to determine the average time at which a neuron will discharge after the stimulus onset. Figure 11 could be the first spike latency for a neuron. The mean and variance of the distribution is usually calculated. The LH is useful for studying the travel time of spikes through the nervous system; This information helps to determine whether there are any synapses interposed between two recording sites as spikes require a significant amount of time to traverse each synapse.

Another very common analysis technique is the use of cycle histograms which have been called period or phase histograms. They are a type of PST where the synchronizing event is the zero crossing or initiation of a cycle of the stimulus. It is used to reveal phase-locking behavior in the system under investigation. Cycles could be based on diurnal patterns (1 day), traffic flow (hour, day, weeks, months, year), hamburgers eaten (day), etc. The resulting histogram allows calculation of the degree of response to some underlying variable which is the stimulus.

The
cycle histogram can be developed from the distribution of sample points events
around the unit circle as shown below.
The events x_{1}, x_{2}, x_{3} are transformed

to a sequence of events {q}, with values on the unit
circle [0, 2p]. The mean direction _{}of q_{1},q_{2},q_{3}, .... is defined to be the direction of the
resultant of the unit vectors _{}_{1},_{},... The Cartesian coordinates of P_{i } are (cosq_{i},sinq_{i})
so that the center of gravity of the points is _{} where

_{} _{}

let _{}

then R=n._{} is the length of the resultant and q is the solution of the
equations:

X=R.cos_{}, Y=R.sin_{}

i.e., dividing, _{}

or, _{}

__Additional Analysis__

Various
stimulus paradigms will require the systematic explorations of one or more
independent variables such as frequency, sound pressure level**, **the phase relation between two
signals, the contrast of a visual display, etc.
Some summary statistic is often used to analyze the behavior of a
neuron. A very common one is the number
of spikes discharge during some interval of the stimulus sequence. Graphs of the resulting metric can be
displayed as shown in figure 14. The
number of spikes recorded for each frequency and intensity of sound are shown. The resulting series of curves tell us what
region of the stimulus space the neural unit responds to. Many similar plots could be generated for any
measure we can compute.

The
methodology of neural unit data analysis is still evolving. Autocorrelation,
cross correlation, systems identification procedures are all of use. The problems of analysis are many as anyone
should expect in trying to investigate a system which contains 10^{10}
elements with several orders of magnitude greater connectivites. In trying to interpret the behavior of
population of neurons gross electrodes are often used. They record the average of many neurons
discharging together. Since there is
often a great deal of background noise -- electrical signal not related to the
stimulus being presented -- many responses to the stimulus are averaged together
to 'remove'** **the noise.

Fourier
analysis is often used to determine the spectral (frequency) characteristics of
physiological signals such as EEG,

__AER____ -- Averaged
Evoked Responses__

While the first recording of evoked potentials in mammals occurred in 1875 when they were recorded from the rabbit brain, it wasn't until the introduction of the electronic amplifier that the recording of brain potentials through the unopened skull was demonstrated by Hans Berger. The signal-to-noise ratio for evoked activity is low because it is obscured by the larger EEG activity. The method of choice for enhancing the SIN is to average the potentials which are regularly evoked by a repetitive stimulus.

The basic idea is very simple -- activity or electrical signals which are time locked to an event will sum according to a linear relation while incoherent signals, e.g., noise will sum in an RMS manner. This is the basis for SIN enhancement as n repetitions are added. If n signals, x(t), are added, S..= nx(t) while n segments of noise n(t) will grow as n. Therefore the S/N = n/ n = n. That is, adding 4 repetitions together will improve the S/N ratio by 2 or 6 dB. If 100 repetitions are necessary to obtain a noticeable evoked response; 400 responses would be necessary to improved S/N by a factor of 2 while 1600 responses would have to be averaged for a second factor of 2 improvement. The point is that the fastest improvement occurs for small n. If S/N isn't sufficient for some reasonable n then further improvements will come only at great expense. The additional problem that must be faced is whether the system under investigation will be constant (stable) over the averaging period.

The details of recording have been discussed in sufficient detail elsewhere (9). Whether non-polarizable electrodes are necessary is dependent on whether D.C. recording is necessary, on interference effects, and on the required ease of placement. Needle electrodes reduce skin surface potentials but exhibit an impedance which is inversely proportion to frequency.

In
order to facilitate the comparison of EEG's a standard electrode placement has
been proposed in 1947 and named the international 10-20 electrode
placement. Some

SER
stimuli often consist of electrical shocks to the median nerve at the wrist, or
to leg nerves. VER stimuli are often
flashes of white light while

If
short-latency _{3}, T_{4} (temporal) areas. Several studies find them to be maximum in
the vertex region (C_{Z}). This leads to two questions: 1) are they of
neural origin? and 2) are they generated in the primary auditory cortex? The use of scalp and subdural electrodes at a
point near the vertex demonstrate that the AERs have the same waveform and
latency at both sites with a similar waveform in T 4 support the conclusion
that the early

Another
issue in

The use of AERs is to correlate brain activity with sensory or motor behavior. We seek information on the timing; magnitude and location of neural events which take place in the brain during some sensory or behavioral sequence. Since scalp-recorded brain potentials provide a substantially degraded indication of intracranial processes, evidence of these events may be ambiguous or lacking.

Timing information is the most unequivocal data. The magnitude of neural activity are much less secure and the interpretation of it is ambiguous.

__Analysis__

The recorded brain potential V(t), can be represented as:

V(t) = E(t) + G(t)

where E(t): = the evoked response

and G(t): = the background EEG plus noise.

We want to characterize E(t), its variability and to describe the statistical features of the EEG.

In
sampling these signals we are subject to the same constraints as are applicable
to any signal analysis problem, i.e., the sampling rate must be at least twice
the highest frequency present in the signal.
The sample interval will be Dt and V(t) will be sampled at t_{i}=Dt.
i=1, ...,m.

The mean of V(t)

_{} (see fig. 15)

with the variance

_{}

The autocovariance can be evaluated as

_{}

which gives a measure of correlation between points on the
same record. A useful method of displaying averaged evoked
responses is shown in the following figure 16. The standard deviation, _{}), is the root-mean-square deviation from the mean and is
expressed in the same units as the data.
It is plotted above the averaged evoked potential _{}and shows the effect of increasing N from N=16 to 64 to
256. At N=256 S is small enough so that
we can trust the bumps and wiggles present in _{}. Remember the reduction in the standard error S is
proportional to _{}. A general guideline
for choosing N is to reduce the fluctuation in S to 10% of _{}, the average.

Since the major variance of the ensemble average is due to raw EEG an estimate of the standard deviation, S, of G(t) can be made by taking 1/2 the average peak-to-peak EEG amplitude. If it is 10 mV then N=100. In order to reduce it further by a factor of 2. N=400, and any improvement by a factor of 10 requires an increase in N by a factor of 100.

In
these arguments we have assumed that the time samples represented independent
processes, which is not true since _{}is correlated over time when an evoked response is
present. In the absence of information
on this temporal covariation, the independence assumption provides a
conservative estimate of accuracy. One
technique which will provide some capability in evaluating the characteristics
of the background activity is to remove the evoked response E(t) by alternately
adding and subtracting successive ensembles.
This is important only if access to a computer system capable of
performing variance calculations is not possible.

There
are several problems which are being addressed due to the availability of
computers for data analysis. They
include: 1) the detection of evoked potentials at psychophysical thresholds
which requires the application of statistical detection methods to the
evaluation of _{} and increase in the
RMS voltage or a change in the autocovariance could all signal a significant
change in the

__Fourier Analysis__

Fourier analysis allows one to identify the composition of a signal in terms of its frequency content. A signal which consists of a sinewave such as x(t) = A sin (2p.1000t) has all its energy at one frequency, 1000 Hertz. A square wave signal has energy not only at the fundamental frequency but at all the odd harmonics. Three common waveforms are shown below along with their Fourier representation.

The basic idea of Fourier analysis is that any arbitrarily varying time signal can be represented as the sum of a series of sine and cosine functions. That is:

_{}

(almost)regardless of the shape of x(t).

The
individual a_{i} and b_{i} coefficients must be determined by
multiplying x(t)** **by either the sine
or cosine** **term and integrating over
some portion of the signal x(t).

i.e. _{}

and similarly for the b_{i}.

Computer programs have been written to perform these analyses in an efficient manner utilizing the ingenious Fast Fourier Transform or FFT(10). There are many considerations in using these programs which one should be aware of.

The analysis of a 90 second section of two EEG channels is shown in Fig. 17. The 90 second record was analyzed as, 45 2-second records. This allows us to see how the spectral energy distribution changes as a function of time. This is a very common analysis technique used in many disciplines. Part of the original time record can be seen in the upper portion of Fig. 17.

Many other uses of Fourier analysis are made including the calculation of transfer functions and coherence functions.

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3. Rhode, W.S. and R.E. Olson (1975). A digital stimulus system. Monograph No. 2, Laboratory Computer Facility, Madison, Wisconsin.

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5. Wolf, V.R. and R.C. Bilger (1972). Generating complex waveforms. Behav. Res. Meth. and Instr., Vol. 4(5), 250-256.

6. Perkel**, **D.A., G.L. Gerstein**, **and
G.P. Moore. Neuronal Spike Trains and
Stochastic Point Processes** **I. The
Single Spike Train. Biophys. J. ** 7
**1967, pp. 391-418.

7. Perkel, D.H., A.L. Gerstein, G.P.
Moore. Neuronal spike trains and
stochastic point process II.
Simultaneous spike trains.
Biophys. J. __7__, 1967, pp.
419-440.

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9. Goff, W.R. (1974). Human Averaged Evoked Potentials: Procedures for Stimulating and Recording. In Bioelectric Recording Techniques, Part B. Ed. R.F. Thompson & M.M. Patterson, A.P.

10. Bergland, G.D. (1969). A Guided Tour of the Fast Fourier Transform,, IEEE Spectrum 6(7), 41-52.