Mathematical analysis of the equations describing the processes in NNs can establish any dependencies between quantitative network characteristics, the information capacity of the network, and the probab- ilities of recognition and retention of information. It has been proposed that electromyographic (EMG) patterns can be analyzed and classified by NNs [11] where the standard BP algorithm is used for decom- posing surface EMG signals into their constituent action potentials (APs) and their firing patterns [12].
A system such as this may help a physician in diagnosing time-behavior changes in the EMG.
The need for a knowledge-based system using NNs for evoked potential recognition was described by Bruha and Madhavan [13]. In this chapter, the authors used syntax pattern-recognition algorithms as a first step, while a second step included a two-layer perceptron to process the list of numerical features produced by the first step.
Myoelectric signals (MES) also have been analyzed by NNs [14]. A discrete Hopfield network was used to calculate the time-series parameters for a moving-average MES. It was demonstrated that this network was capable of producing the same time-series parameters as those produced by a conventional sequential least-squares algorithm. In the same paper, a second implementation of a two-layered perceptron was used for comparison. The features used were a time-series parameter and the signal power in order to train the perceptron on four separate arm functions, and again, the network performed well.
Moving averages have been simulated for nonlinear processes by the use of NNs [15]. The results obtained were comparable with those of linear adaptive techniques.
Moody et al. [16] used an adaptive approach in analyzing visual evoked potentials. This method is based on spectral analysis that results in spectral peaks of uniform width in the frequency domain. Tunable data windows were used. Specifically, the modified Bessel functions Io−sin h, the gaussian, and the cosine- taper windows are compared. The modified Bessel function window proved to be superior in classifying normal and abnormal populations.
Pulse-transmission NNs — networks that consist of neurons that communicates with other neurons via pulses rather than numbers — also have been modeled [7,17]. This kind of network is much more realistic, since, in biological systems, action potentials are the means of neuronal communication. Dayhoff [18] has developed a pulse-transmission network that can temporally integrate arriving signals and also display some resistance to temporal noise.
Another method is optimal filtering, which is a variation of the traditional matched filter in noise [19].
This has the advantage of separating even overlapping waveforms. It also carries the disadvantage that the needed knowledge of the noise power spectral density and the Fourier transform of the spikes might not be always available.
Principal-components analysis also has been used. Here the incoming spike waveforms are represented by templates given by eigenvectors from their average autocorrelation functions [20]. The authors found that two or three eigenvectors are enough to account for 90% of the total energy of the signal. This way each spike can be represented by the coordinates of its projection onto the eigenvector space. These coordinates are the only information needed for spike classification, which is further done by clustering techniques.
7.1.1 Multineuronal Activity Analysis
When dealing with single- or multineuron activities, the practice is to determine how many neurons (or units) are involved in the “spike train” evoked by some sensory stimulation. Each spike in a spike train
represents an action potential elicited by a neuron in close proximity to the recording electrode. These action potentials have different amplitudes, latencies, and shape configurations and, when superimposed on each other, create a complex waveform — a composite spike. The dilemma that many scientists face is how to decompose these composite potentials into their constituents and how to assess the question of how many neurons their electrode is recording from. One of the most widely used methods is window discrimination, in which different thresholds are set, above which the activity of any given neuron is assigned, according to amplitude. Peak detection techniques also have been used [21]. These methods perform well if the number of neurons is very small and the spikes are well separated. Statistical methods of different complexity also have been used [22–24] involving the time intervals between spikes. Each spike is assigned a unique instant of time so that a spike train can be described by a process of time points corresponding to the times where the action potential had occurred. Processes such as these are called point processes, since they are characterized by only one number. Given this, a spike train can be treated as a stochastic point process that may or may not be stationary. In the former case, its statistics do not vary in the time of observation [25]. In the second case, when nonstationarity is assumed, any kind of statistical analysis becomes formidable.
Correlations between spike trains of neurons can be found because of many factors, but mostly because of excitatory or inhibitory interactions between them or due to a common input to both. Simulations on each possibility have been conducted in the past [26]. In our research, when recording from the optic tectum of the frog, the problem is the reversed situation of the one given above. That is, we have the recorded signal with noise superimposed, and we have to decompose it to its constituents so that we can make inferences on the neuronal circuitry involved. What one might do would be to set the minimal requirements on a neural network, which could behave the same way as the vast number of neurons that could have resulted in a neural spike train similar to the one recorded.
This is a very difficult problem to attack with no unique solution. A method that has attracted atten- tion is the one developed by Gerstein et al. [22,27]. This technique detects various functional groups in the recorded data by the use of the so-called gravitational clustering method. Although promising, the analysis becomes cumbersome due to the many possible subgroups of neurons firing in synchrony.
Temporal patterns of neuronal activity also have been studied with great interest. Some computational methods have been developed for the detection of favored temporal patterns [28–31]. My group also has been involved in the analysis of complex waveforms by the development of a novel method, the ST-scan method [32]. This method is based on well-known tomographic techniques, statistical analysis, and template matching. The method proved to be very sensitive to even small variations of the waveforms due to the fact that many orientations of them are considered, as it is done in tomographic imaging.
Each histogram represents the number of times a stroke vector at a specific orientation is cutting the composite waveform positioned at the center of the window. These histograms were then fed to a NN for categorization. The histograms are statistical representations of individual action potentials. The NN therefore must be able to learn to recognize histograms by categorizing them with the action potential waveform that they represent. The NN also must be able to recognize any histogram as belonging to one of the “learned” patterns or not belonging to any of them [33]. In analyzing the ST-histograms, the NN must act as an “adaptive demultiplexer.” That is, given a set of inputs, the network must determ- ine the correspondingly correct output. This is a categorization procedure performed by a perceptron, originally described by Rosenblatt [34]. In analyzing the ST-histograms, the preprocessing is done by a perceptron, and the error is found either by an LMS algorithm [35] or by an ALOPEX algorithm [36–38].
7.1.2 Visual Evoked Potentials
Visual evoked potentials (VEPs) have been used in the clinical environment as a diagnostic tool for many decades. Stochastic analysis of experimental recordings of VEPs may yield useful information that is not well understood in its original form. Such information may provide a good diagnostic criterion in differentiating normal subjects from subjects with neurological diseases as well as provide an index of the progress of diseases.
These potentials are embedded in noise. Averaging is then used in order to improve the signal-to-noise (S/N) ratio. When analyzing these potentials, several methods have been used, such as spectral analysis of their properties [39], adaptive filtering techniques [40,41], and some signal enhancers, again based on adaptive processes [42]. In this latter method, no a priori knowledge of the signal is needed. The adaptive signal enhancer consists of a bank of adaptive filters, the output of which is shown to be a minimum mean-square error estimate of the signal.
If we assume that the VEP represents a composite of many action potentials and that each one of these action potentials propagates to the point of the VEP recording, the only differences between the various action potentials are their amplitudes and time delays. The conformational changes observed in the VEP waveforms of normal individuals and defected subjects can then be attributed to an asynchrony in the arrival of these action potentials at a focal point (integration site) in the visual cortex [36]. One can simulate this process by simulating action potentials and trying to fit them to normal and abnormal VEPs with NNs.
Action potentials were simulated using methods similar to those of Moore and Ramon [43] and Bell and Cook [44]. Briefly, preprocessing of the VEP waveforms is done first by smoothing a five-point filter that performs a weighted averaging over its neighboring points
S(n)= [F(n−2)+2F(n−1)+3F(n)+2F(n+1)+F(n+2)]/9 (7.1) The individual signals vjare modulated so that at the VEP recording site each vj has been changed in amplitude am(j)and in phase ph(j). The amplitude change represents the propagation decay, and the phases represent the propagation delays of signals according to the equation
vj(i)=am(j)ãAP[i−ph(j)] (7.2)
For a specific choice of am(j)and ph(j), j=1, 2,. . ., N , the simulated VEP can be found by
VEP=b+k N
j=1
vjα (7.3)
where k is a scaling factor, b is a d.c. component, andαis a constant [37,38].
The ALOPEX process was used again in order to adjust the parameters (amplitude and phase) so that the cost function reaches a minimum and therefore the calculated waveform coincides with the experimental one.
The modified ALOPEX equation is given by
pi(n)=pi(n−1)+γ pi(n)R(n)+àri(n) (7.4) where pi(n)are the parameters at iteration n, andγ andàare scaling factors of the deterministic and random components, respectively, which are adjusted so that at the beginningγ is small andàis large. As the number of iterations increases,γ increases whileàdecreases. The cost function R is monitored until convergence has been achieved at least 80% or until a preset number of iterations has been covered.
The results obtained show a good separation between normal and abnormal VEPs. This separation is based on an indexλ, which is defined as the ratio of two summations, namely, the summation of amplitudes whose ph(i)is less than 256 msec and the summation of amplitudes whose ph(i)is greater than 256 msec. A large value of l indicates an abnormal VEP, while a small l indicates a normal VEP.
The convergences of the process for a normal and an abnormal VEP are shown in Figure 7.1 and Figure 7.2, respectively, at different iteration numbers. The main assumption here is that in normal individuals, the action potentials all arrive at a focal point in the cortex in resonance, while in abnormal subjects there exists as asynchrony of the arrival times. Maier and colleagues [45] noted the importance of
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(a) (b)
(c)
FIGURE 7.1 Normal VEP. (a) The fitting at the beginning of the process, (b) after 500 iterations, and (c) after 1000 iterations. Only one action potential is repeated 1000 times. The x-axis is×10 msec; the y-axis is in millivolts.
source localization of VEPs in humans in studying the perceptual behavior in humans. The optimization procedure used in this section can help that task, since individual neuronal responses are optimally isolated.
One of the interesting points is that signals (action potentials) with different delay times result in composite signals of different forms. Thus the reverse solution of this problem, that is, extracting individual signals from a composite measurement, can help resolve the problem of signal source localization. The multiunit recordings presented in the early sections of this paper fall in the same category with that of decomposing VEPs. This analysis might provide insight as to how neurons communicate.