Fractal Dimension Analysis of Transient Visual Evoked Potentials: Optimisation and Applications

Purpose The visual evoked potential (VEP) provides a time series signal response to an external visual stimulus at the location of the visual cortex. The major VEP signal components, peak latency and amplitude, may be affected by disease processes. Additionally, the VEP contains fine detailed and non-periodic structure, of presently unclear relevance to normal function, which may be quantified using the fractal dimension. The purpose of this study is to provide a systematic investigation of the key parameters in the measurement of the fractal dimension of VEPs, to develop an optimal analysis protocol for application. Methods VEP time series were mathematically transformed using delay time, τ, and embedding dimension, m, parameters. The fractal dimension of the transformed data was obtained from a scaling analysis based on straight line fits to the numbers of pairs of points with separation less than r versus log(r) in the transformed space. Optimal τ, m, and scaling analysis were obtained by comparing the consistency of results using different sampling frequencies. The optimised method was then piloted on samples of normal and abnormal VEPs. Results Consistent fractal dimension estimates were obtained using τ = 4 ms, designating the fractal dimension = D2 of the time series based on embedding dimension m = 7 (for 3606 Hz and 5000 Hz), m = 6 (for 1803 Hz) and m = 5 (for 1000Hz), and estimating D2 for each embedding dimension as the steepest slope of the linear scaling region in the plot of log(C(r)) vs log(r) provided the scaling region occurred within the middle third of the plot. Piloting revealed that fractal dimensions were higher from the sampled abnormal than normal achromatic VEPs in adults (p = 0.02). Variances of fractal dimension were higher from the abnormal than normal chromatic VEPs in children (p = 0.01). Conclusions A useful analysis protocol to assess the fractal dimension of transformed VEPs has been developed

Range of values for <i>τ</i> for VEPs recorded at >1000 Hz such that reconstructed phase space trajectories were not stretched along the diagonal such that fluctuations in the trajectory were evident.

<p>Range of values for <i>τ</i> for VEPs recorded at >1000 Hz such that reconstructed phase space trajectories were not stretched along the diagonal such that fluctuations in the trajectory were evident.</p

<i>D</i>_<i>2</i> estimated using a delay time of 4.4 ms of VEPs in response to binocular magenta-cyan stimulation in children with normal and abnormal visual systems.

<p><i>D</i>_<i>2</i> estimated using a delay time of 4.4 ms of VEPs in response to binocular magenta-cyan stimulation in children with normal and abnormal visual systems.</p

Children’s VEPs in response to magenta-cyan pattern-onset offset grating stimuli analysed using the optimised protocol.

<p>(a) shows the children’s VEPs drawn from children with normal visual systems. (b) shows the abnormal VEPs, with the average of the three lowest <i>D</i>_<i>2</i> VEPs in bold blue, the average of the three highest <i>D</i>_<i>2</i> VEPs in bold red, and the remainder of VEPs in bold black. Individual VEPs, which were the average of two recordings, are presented in grey. To illustrate the morphology of abnormal VEPs which tended towards the extremes of low and high <i>D</i>_<i>2</i> more clearly, the three abnormal VEPs with the lowest and highest <i>D</i>_<i>2</i> are shown in (c) and (d) respectively. The group averaged responses are in bold. Individual responses, which were the average of two recordings, are presented as the thinner lines.</p

Group mean (SD) of VEP component amplitude and latency data.

<p>Group mean (SD) of VEP component amplitude and latency data.</p

Reconstructed phase space trajectories of a VEP recorded with 3606 Hz sampling frequency embedded in 3 dimensional phase space using different values of <i>τ</i>.

<p>From top left to bottom right, <i>τ</i> = 1, 3, 6, 9, 13, 16, 19, 22.</p

Depiction of the optimised analysis method.

<p>Top panel, example VEP. Intermediate panels show plots of log(<i>C</i>(<i>r</i>)) vs log(<i>r</i>) and plots of running average slope of log(<i>C</i>(<i>r</i>)) vs log(<i>r</i>) as a function of their ordered calculation for m = 1 to 7. The red crosses indicate the data derived from the middle third of the plot log(<i>C</i>(<i>r</i>)) vs log(<i>r</i>). The final panel is a plot of <i>D</i>_<i>2</i> for m = 1 to 7: the ceiling value at m = 7 is an indicator of the fractal dimension of the system.</p