Visual evoked potentials (VEPs) may be analyzed by examination of the morphology of their components, such as negative (N) and positive (P) peaks. However, methods that rely on component identification may be unreliable when dealing with responses of complex and variable morphology; therefore, objective methods are also useful. One potentially useful measure of the VEP is the correlation dimension. Its relevance to the visual system was investigated by examining its behavior when applied to the transient VEP in response to a range of chromatic contrasts (42%, two times psychophysical threshold, at psychophysical threshold) and to the visually unevoked response (zero contrast). Tests of nonlinearity (e.g., surrogate testing) were conducted. The correlation dimension was found to be negatively correlated with a stimulus property (chromatic contrast) and a known linear measure (the Fourier-derived VEP amplitude). It was also found to be related to visibility and perception of the stimulus such that the dimension reached a maximum for most of the participants at psychophysical threshold. The latter suggests that the correlation dimension may be useful as a diagnostic parameter to estimate psychophysical threshold and may find application in the objective screening and monitoring of congenital and acquired color vision deficiencies, with or without associated disease processes.

*x*= 0.305,

*y*= 0.310) and luminance (20 cd/m

^{2}). The grating stimuli were squares with sides subtending an angle of 5°. At maximum contrast, the colors were of CIE chromaticity coordinates

*x*= 0.380,

*y*= 0.270 (magenta), and

*x*= 0.230,

*y*= 0.350 (cyan), the L-, M-, and S-cone contrasts were 0.16, −0.40, and 0.07, respectively, and the pooled root mean square (rms) cone contrast theoretically produced by the stimulus at maximum contrast was 0.25 (McKeefry, Parry, & Murray, 2003). In the present study, chromatic contrast level was expressed as a percentage of maximum available chromatic contrast. The spatial, temporal, and chromatic parameters of the stimuli were chosen to stimulate the L–M chromatic contrast system preferentially (Kulikowski, McKeefry, & Robson, 1997; McKeefry, Russell, Murray, & Kulikowski, 1996; Mullen, 1985; Murray, Parry, Carden, & Kulikowski, 1987; Rabin, Switkes, Crognale, Schneck, & Adams, 1994; Suttle & Harding, 1999) while minimizing the effects of chromatic aberration (Flitcroft, 1989).

_{color1}/ (L

_{color1}+ L

_{color2})) was varied by the participant. In 9 of the 11 participants, VEPs were recorded at their individual isoluminance color ratios. The remaining two participants were tested at the average isoluminance color ratio of a larger group of participants because their VEPs were originally recorded as part of a different experiment. The use of an average isoluminance value for those two participants may introduce luminance cues into their responses but this is not relevant to the question of whether VEPs may be described by the correlation dimension, so they are included in that part of our analysis. Data from these two participants, however, were not used to investigate correlations between the correlation dimension, Fourier power, and chromatic contrast.

*τ*= 4 ms and 6 ms) were used and the correlation dimension,

*D*

_{2}, was estimated for five embedding dimensions,

*m*= 1, 2, 3, 4, and 5. The

*τ*values were chosen based on an exploration of the reconstructed pathways in three-dimensional space arising from different values of

*τ*. The final values of

*τ*were selected so that (1) the reconstructed pathway would not be collapsed or aligned along the diagonal of three-dimensional phase space (Grassberger and Procaccia, 1983b) and (2) the smallest ripples in the VEP time series would not be excluded from the reconstructed phase space trajectory of the VEP. Apart from these considerations, the choice of one delay time over another should not greatly affect the estimated correlation dimension (Mazaraki, 1997; Takens, 1981).

*r*

_{min}) and the logarithm of the largest distance (represented by

*r*

_{max}) were computed. A series of ‘bins’ was then created to record the correlation sum,

*C*(

*r*), which is the normalized number of pairs of points with a separation distance of less than a specified distance

*r*. The process of depositing counts of data into bins is analogous to recording counts of the occurrence of events within data in a frequency histogram. In this study, 64 bins (an arbitrary number) were used and the width of each bin was set to (

*r*

_{max}−

*r*

_{min}) / 64. Thus, from first to last, the separation distances used in the analysis were defined by the series

*r*

_{1},

*r*

_{2},…

*r*

_{n}, where the radius

*r*

_{n}=

*r*

_{min}+

*n*(

*r*

_{max}−

*r*

_{min}) / 64, where

*n*= 1 to 64.

*D*

_{2}can be approximated by

*D*

_{2}∼ log(

*C*(

*r*))/log(

*r*) (Grassberger & Procaccia, 1983a, 1983b) as outlined in 1. Thus,

*D*

_{2}was calculated as the slope, dlog(

*C*(

*r*))/dlog(

*r*), of the linear scaling portion of the plot of log(

*C*(

*r*)) versus log(

*r*). In this study, the slope of the scaling portion of the plot was calculated by determining the slopes of

*i*consecutive local portions of the plots, with each portion of the plot consisting of

*k*consecutive points (

*k*= 6, 12), and finding the maximum slope out of all the calculated slopes for each

*k.*The slope for each

*i*th portion was determined to be the slope of the line of best fit (least squares method) drawn through the points that were part of the

*i*th portion of the plot (for example, see Figure 2,

*i*= slope number).

*C*(

*r*)) versus log(

*r*) and the objective of estimating slope from a region of stable linear scaling. Such a region on the plot should contain successive local slopes of approximately equal magnitude and create a high plateau in a plot of local slope as a function of slope number (for example, see Figure 3). This is because progressively increasing slopes will occur over a range of small log(

*r*) such that the scarcity of pairs of points separated by <

*r*may result in any slope taken from this region being an inaccurate reflection of the phase-space filling properties of the trajectory. On the other hand, progressively decreasing slopes will occur as log(

*r*) becomes so large that C(

*r*) will contain almost all of the pairs of the points in the trajectory so will gradually form a plateau in the plot of log(

*C*(

*r*)) versus log(

*r*) until every pair of points in the trajectory is included. Previous researchers have used the “middle-third” of the curve of log(

*C*(

*r*)) versus log(

*r*) in order to avoid calculating slope from regions of too small and too large log(

*r*), but this method can also result in error if it straddles two kinds of straight-line behaviors (noise and deterministic chaos) (Henry et al., 2001). On exploration of the data, the local maximum slope was found to lie within a range of local slopes of equal magnitude that lay between regions of progressively increasing and progressively decreasing slopes so the maximum slope was used as an indicator of

*D*

_{2}. In the present study, because the aim is to determine the relevance of the parameter to the functioning of the visual system by comparing across stimulus and perceptual conditions and not to estimate the absolute number of dynamical variables of the system, provided the method used is constant for all time series estimated, this method of

*D*

_{2}estimation from the slope will permit comparison between the

*D*

_{2}'s of the VEPs. In total, four estimates of

*D*

_{2}were made for each embedding dimension as two

*τ*values and two

*k*values were used.

*m,*is the upper bound of the correlation dimension,

*D*

_{2}, that may be estimated from the reconstructed data. Thus, as

*m*is increased,

*D*

_{2}should also increase until

*D*

_{2}reaches a plateau value. A plateau could be achieved in two different ways; (1) a plateau representing the true dimension of the reconstructed path (a solid sphere would appear to be a two-dimensional object if embedded in a two-dimensional space but remains a three-dimensional object when embedded in any higher dimensional space), (2) a plateau that is an artefact of the finite data size (as the embedding dimension is increased, more and more data points are needed to fill the space). For a finite data set comprising

*N*= 10

^{3}points, a plateau value of

*D*

_{2}≥ 2 log

_{10}

*N*= 6 should be discounted as an artefact of the finite data size (Eckmann & Ruelle, 1992), and so to avoid this ‘false plateau’ we have restricted the dimension of the embedding space to

*m*≤ 5. In the analysis below,

*D*

_{2}was plotted as a function of

*m*for

*m*= 1, 2, 3, 4, and 5, and in cases where the plot appeared to plateau, the

*D*

_{2}values for

*m*= 5 were taken as estimates of the true dimension of the reconstructed path. Where

*D*

_{2}did not appear to plateau with increasing embedding dimension, it was regarded as a result indistinguishable from uncorrelated noise. In the case where

*D*

_{2}plateaued, a further test using surrogate data was employed to differentiate between a deterministic response and a correlated noise response (Theiler & Rapp, 1996).

*t*-test. Nonlinearity of the 0% chromatic contrast responses, where there was no evoked potential in response to chromatic modulations, was similarly tested by comparison with surrogates of the 0% data. If both the experimental time series data and the surrogate data sets do not yield statistically significantly different correlation dimensions, then the possibility that the VEP is a linear system in combination with random noise may not be excluded. If the estimated correlation dimension of the data is found to be close to 5, the upper bound for reliable estimates of the fractal dimension, then the possibility that the data set represents random noise may not be excluded. If the slope of the plot of

*D*

_{2}as a function of

*m*is not found to plateau but continues to increase with increasing

*m,*then again the possibility that the data set represents random noise may not be ruled out. To determine whether the function reached a plateau, a plateau index was defined as

*D*

_{2}(5) −

*D*

_{2}(4), where values in parentheses are embedding dimensions of 4 and 5. An index of at least 0.3 (arbitrarily chosen) was taken to indicate no plateau.

*m*= 5 was compared with the Fourier power of the VEP and the chromatic contrast of the stimulus to look for associations. Because one of the contrast levels selected is a multiple of psychophysical threshold, an association between the correlation dimension of the VEP and perception of the stimulus was also considered.

*C*(

*r*) versus log

*r,*and

*m*versus

*D*

_{2}for

*τ*= 4, 6, and

*k*= 6, 12. The fractal dimension for the dynamical system described by this VEP was eventually estimated at 2.41. It can be seen that, unlike the original VEP data, the surrogate data's corresponding

*D*

_{2}versus

*m*plot does not plateau and that the mean correlation dimension at

*m*= 5 is 4.23.

*m*= 5 (see Figure 6). Figure 6 shows box plots depicting these data for all of the VEPs in response to chromatic contrast, all the VEPs in response to 0% chromatic contrast and their respective surrogate data. An index of ≥0.3 (arbitrarily chosen) was taken as an indication that the function failed to plateau. The VEP and 0% data do plateau, as indicated by mean indices of 0.18 (SD 0.11) and 0.18 (SD 0.06), respectively. The surrogate data do not plateau, as indicated by mean indices of 0.95 (SD 0.34) and 0.83 (SD 0.33), respectively. A paired

*t*-test comparison between the VEP, 0%, and their respective surrogate data sets, for both the parameters of plateau index and the correlation dimension (at

*m*= 5), resulted in statistically significant differences between the VEP and surrogate data sets (

*p*< 0.0001) for both parameters. The correlation dimensions of the VEPs at

*m*= 5 could therefore be regarded as the fractal dimension of the embedded time series, but this was not the case for surrogate data.

*N*= 9). The data points that belong to each individual participant have been joined by lines. Separate lines have been used for each participant. In Figures 8A, 8B, and 8C, correlation dimension, Fourier power, and latency as a function of chromatic contrast of the stimulus are presented. In Figures 8D and 8E, points depicting fractal dimension as a function of Fourier power and latency are depicted with dotted lines joining the data points in order of the rank of the powers for Figure 8D and in order of the ranks of chromatic contrast for Figure 8E.

*df*= 26,

*p*= 0.007 (two-tailed)). However, this relationship was significant in only two out of the nine participants (Spearman's Rho). There was a significant positive relationship between the N-peak latency and the correlation dimension of the VEPs overall for the group (Spearman's Rho: 0.41,

*df*= 23,

*p*= 0.040 (two-tailed)) and was also strong for each of the participants (Spearman's Rho: 1.0). There was a significant negative relationship between the chromatic contrast of the stimulus and the correlation dimension of the VEP (Spearman's Rho: −0.889,

*df*= 27,

*p*< 0.0001 (two-tailed)), as shown in Figure 8A. The relationship was significant in seven out of the nine participants (Spearman's Rho).

*W*(coefficient of concordance) was calculated. For correlation dimension as a function of power, Kendall's

*W*was 0.30 and the asymptotic significance was 0.09, indicating that there was little to no agreement in the ordering of correlation dimension as a function of power across participants. The nonsignificant asymptotic significance value indicates that there is a random distribution of ranks of the correlation dimension as a function of power. For correlation dimension as a function of latency, Kendall's

*W*was 0.88 and the significance was 0.0004, indicating a high agreement in the ordering of correlation dimension as a function of latency. For correlation dimension as a function of chromatic contrast, Kendall's

*W*was 0.790 and the asymptotic significance was 0.001, indicating that there was high agreement in the ordering of correlation dimension as a function of chromatic contrast across participants. The significant asymptotic significance value indicates that there is a nonrandom distribution of ranks of the correlation dimension as a function of chromatic contrast.

*m*dimensions, then each reconstructed phase space coordinate

*X*

_{i}is an

*m*component vector which is obtained from the time series

*y*(

*t*

_{1}),

*y*(

*t*

_{2}),… by the prescription

*X*

_{i}= (

*y*(

*t*

_{i}),

*y*(

*t*

_{i}+

*τ*),

*y*(

*t*

_{i}+ 2

*τ*),…,

*y*(

*t*

_{i}+ (

*m*− 1)

*τ*). Here

*m*is called the embedding dimension,

*τ*is a constant known as the delay time, and the index

*i*denotes ordering in time.

*m*component vector. The comb is then slid along the data to create a series of phase space points that makes up the reconstructed phase space trajectory.

*D*

_{2}, to characterize the phase space filling properties of attractors.

^{1}It is obtained by covering the set with boxes of a given size (

*r*) and then computing the probability

*p*

_{i}(

*r*) (equivalent to the relative frequency in sufficiently large data sets) of having a point of the set in the

*i*th such box. The correlation dimension is defined by

_{i}

*p*

_{i}(

*r*)

^{2}is the probability of finding a pair of points in a box of size

*r*. The so-called Grassberger–Procaccia algorithm (Grassberger & Procaccia, 1983a, 1983b) provides a computationally efficient way to implement this measurement. For small values of

*r,*the probability of having a pair of points in a box of size

*r*is equal to the probability of having a pair of points with separation distance less than

*r*. For sufficiently large data sets, of number

*N,*this latter probability is given by the correlation sum (also termed the correlation integral)

*θ*is the Heaviside function (a discontinuous step function which has a value of either 0 or 1 and may be defined as

*θ*(

*x*) = 0 for

*x*< 0, and as

*θ*(

*x*) = 1 for

*x*> 0) which acts as a counter of the number of pairs of points with separation <

*r*when combined with ∣

*X*

_{i}–

*X*

_{j}∣ (the separation distance between two points on the attractor,

*X*

_{i}and

*X*

_{j}). The multiplier 1/

*N*

^{2}is included to normalize the count by the number of pairs of points on the attractor (without double counting).

*r*), the correlation sum grows like a power such that

*r*=

*l*and

*D*

_{2}=

*v*.) The “∼” symbol in this expression is used to indicate that this is not an exact equality but is a scaling relation that is expected to be valid for sufficiently large

*N*and small

*r.*After taking logarithms of each side of the scaling relation and rearranging terms, we have the result

*D*

_{2}is deduced from the slope of the straight line scaling region in a plot of log(

*C*(

*r*)) versus log(

*r*). The scaling relation is a type of dimensional measurement that in this case describes how a reconstructed phase space trajectory fills a given phase space. Thus, the Grassberger–Procaccia algorithm (1983a, 1983b) has been widely used to measure the correlation dimension of reconstructed phase space trajectories from time series. In these applications, the time series is embedded at increasing embedding dimensions

*m*until the measured

*D*

_{2}reaches a plateau, as revealed in a plot of

*D*

_{2}versus

*m*. One of the most significant limitations in these applications is that the time series needs to be large enough to exhibit genuine scaling behavior at large embeddings. A rule of thumb (Eckmann & Ruelle, 1992) is that the upper limit on a reliable measurement of

*D*

_{2}is given by

*D*

_{2}= 2 log

_{10}

*N*. Thus, any plateau in the plot of

*D*

_{2}versus

*m*at values of

*D*

_{2}higher than this bound should be discounted as artefacts of the finite data size.