To obtain reliable transient auditory evoked potentials (AEPs) from EEGs recorded using high stimulus rate (HSR) paradigm, it is critical to design the stimulus sequences of appropriate frequency properties. Traditionally, the individual stimulus events in a stimulus sequence occur only at discrete time points dependent on the sampling frequency of the recording system and the duration of stimulus sequence. This dependency likely causes the implementation of suboptimal stimulus sequences, sacrificing the reliability of resulting AEPs. In this paper, we explicate the use of continuous-time stimulus sequence for HSR paradigm, which is independent of the discrete electroencephalogram (EEG) recording system. We employ simulation studies to examine the applicability of the continuous-time stimulus sequences and the impacts of sampling frequency on AEPs in traditional studies using discrete-time design. Results from these studies show that the continuous-time sequences can offer better frequency properties and improve the reliability of recovered AEPs. Furthermore, we find that the errors in the recovered AEPs depend critically on the sampling frequencies of experimental systems, and their relationship can be fitted using a reciprocal function. As such, our study contributes to the literature by demonstrating the applicability and advantages of continuous-time stimulus sequences for HSR paradigm and by revealing the relationship between the reliability of AEPs and sampling frequencies of the experimental systems when discrete-time stimulus sequences are used in traditional manner for the HSR paradigm.

In studying the auditory evoked potentials (AEPs), high stimulus-rate (HSR) paradigm featuring shorter and irregular interstimulus intervals (ISIs) has been proposed by Delgado and Özdamar [

As shown by Jewett et al. [

In reality, different recording systems may not always be operated at the frequency exactly identical to that chosen for stimulus sweep design. When a recording system works at a different frequency, the timing of the onsets of stimulus events in the discrete-time stimulus sweep is to be resampled. It is unclear about the impacts of the resampling and actual AD rate on the deconvolution performance. Moreover, many optimization methods can only deal with continuous variables for the convenience of being exposed to certain mathematic operations [

To address these critical questions, in this paper, we derived the frequency representation of a continuous-time impulse sequence to solve the deconvolution problem for the HSR paradigm. Using simulated EEGs (based on real AEP data and simulated noises) and four optimized continuous-time stimulus sequences, we demonstrate the applicability and the advantages of continuous-time stimulus sequences for HSR paradigm. We also illustrate the relationship between the AD rate for discretizing a continuous-time sequence and the errors in terms of temporal locations of stimulus impulse, frequency properties of discretized stimulus sequence, and the deconvolved AEP as compared to the ground truth AEP.

Under discrete HSR condition, the observed sweep-response

Note that the circulant convolution is adopted in (

Equation (

From (

To address this issue, Jewett et al. [

Knowing the frequency property of a stimulus sequence can assist the assessment of the noise attenuation performance of the inverse filter for the deconvolution problem [

In this section, we derive the continuous-time convolution relationship between the stimulus sequence and transient response, and the estimation of transient response in a general form.

Similar to the

Since all the variables (except the error term

The transient response

Distortion of the estimated transient response is introduced in the second term of the right hand side of (

Equation (

A stimulus sequence

In this continuous-time form of stimulus sequence, the

(a) A schematic diagram of a stimulus loop that contains 17 individual stimulus impulses to indicate the repetitive presentations in an experiment. (b) A sweep of stimulus sequence with impulses occurring at

This is a continuous function in the frequency domain [

The Fourier transform of (

Equation (

Based on (

Equation (

Equation (

As shown in (

Figure

To further evaluate the overall quality of the stimulus sequence in (

In this section, we generate continuous-time stimulus sequences to be used for examining the impact of AD rates on the performance of inverse filtering in solving the transient AEPs. Using various optimization methods [

Four optimized stimulus impulse sequences obtained using differential evolution algorithm.

Sequence |
Stimulus interval ( |
---|---|

Seq 1 |
27.21, 22.34, 27.16, 21.40, 21.43, 23.26, 27.18, 24.38, 26.19, 21.48, 27.16, 24.32, 27.05, 23.57, 24.02, 26.30, 27.04, 21.40, 23.41, 27.21, 22.16, 24.62, 21.47, 22.37, 22.54, 27.19, 27.21, 21.40, 21.40, 21.49, 27.21, 22.57, 25.52, 25.15, 27.21, 21.40, 21.40, 27.21, 24.92, 21.40, 25.93, 27.21, 27.21, 21.43, 21.74, 22.23, 27.20, 26.74, 27.21, 21.40, 24.39, 21.49, 24.38, 27.21, 22.54, 27.20, 21.40, 21.86, 27.21, 21.97, 22.59, 25.09, 27.15, 21.48, 26.27 |

| |

Seq 2 |
23.49, 21.91, 27.96, 27.96, 21.79, 21.79, 24.88, 27.96, 24.36, 21.79, 26.27, 23.13, 26.82, 27.96, 21.79, 27.96, 23.67, 22.83, 27.96, 21.79, 22.40, 27.96, 27.96, 21.91, 21.79, 27.96, 23.33, 27.96, 21.79, 27.96, 21.91, 27.94, 26.77, 21.79, 24.64, 27.25, 21.79, 24.89, 27.96, 25.00 |

| |

Seq 3 |
28.38, 21.90, 28.38, 23.20, 21.90, 28.38, 28.24, 21.90, 25.00, 28.38, 21.90, 28.38, 28.38, 22.24, 25.10, 28.38, 28.38, 24.72, 21.90, 28.37, 25.12, 27.13, 23.68, 21.90, 21.90, 28.38, 21.90, 24.35, 22.14, 28.24, 28.38, 23.78, 22.29, 28.38, 22.07, 25.35, 21.90, 28.38, 25.00, 21.90 |

| |

Seq 4 |
26.40, 27.75, 21.66, 27.90, 21.94, 27.91, 21.66, 22.67, 23.90, 27.64, 27.91, 21.88, 21.66, 21.66, 27.91, 27.91, 21.66, 27.03, 27.42, 26.29, 21.66, 25.65, 27.90, 21.66, 27.89, 25.77, 21.66, 21.66, 26.10, 27.91, 21.66, 24.73, 27.91, 21.66, 27.90, 24.01, 21.66, 27.90, 21.66, 23.63 |

These four sequences are given in the form of ISI-series which can be expressed as

In actual experiments, the recorded EEGs (including transient AEPs and noises) are band-limited signals. They are digitized in time and amplitude according to Nyquist-Shannon sampling theorem. In this study, we use a real AEP signal previously measured using CLAD method [

The interstimulus interval of continuous-time stimulus sequence can be discretized at required temporal resolution. The discretization will introduce round off errors no matter how high is sampling frequency

The discretized temporal location

The normalized root-mean-square error is defined as an overall measure of the errors caused by the temporal discretization of the continuous-time stimulus impulse sequence

To examine the error introduced by the temporal discretization of four stimulus sequences in Table

The graphic representation of the relationship between

As seen above, the temporal discretization introduces errors of the temporal location of the stimulus impulses, causing the difference between

Using the same four dataset in Table

The graphic representation of the relationship between

Comparison with respect to the

Comparison with respect to the

Figure

Here we use one stimulus sequence to exemplify the differences caused by discretization at two different AD rates:

Figure

Although the continuous-time stimulus sequence

Likewise, the inverse Fourier transform of

To quantify the difference between the estimated and the true solutions for both continuous and discrete stimulus sequence, we can define root mean square error for

By replacing

To examine the relationship between AD rate and

The errors between estimated and true solution at various AD rates for four sequences.

Seq. ID | SNR |
AD rate (kHz) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Inf | 1 | 2 | 3 | 4 | 5 | 7 | 9 | 11 | 15 | 20 | 25 | ||

Seq 1 | 9.5 | 2.02 | 20.53 | 12.65 | 10.31 | 8.85 | 7.87 | 6.67 | 5.94 | 5.29 | 4.58 | 3.96 | 3.54 |

0 | 8.24 | 68.01 | 43.29 | 35.02 | 29.79 | 26.66 | 22.55 | 20.02 | 17.89 | 15.50 | 13.41 | 12.00 | |

−6.0 | 18.51 | 141.81 | 91.21 | 73.72 | 62.67 | 56.15 | 47.48 | 42.10 | 37.68 | 32.63 | 28.22 | 25.27 | |

| |||||||||||||

Seq 2 | 9.5 | 4.19 | 31.52 | 21.39 | 17.91 | 15.40 | 14.05 | 11.72 | 10.26 | 9.40 | 8.02 | 6.98 | 6.43 |

0 | 9.52 | 78.32 | 52.98 | 44.67 | 38.33 | 35.15 | 29.23 | 25.55 | 23.44 | 19.99 | 17.44 | 6.23 | |

−6.0 | 18.14 | 149.64 | 101.20 | 85.47 | 73.32 | 67.31 | 55.94 | 48.87 | 44.83 | 38.25 | 33.39 | 31.15 | |

| |||||||||||||

Seq 3 | 9.5 | 5.02 | 27.32 | 18.70 | 15.48 | 13.67 | 11.78 | 10.07 | 8.83 | 8.00 | 6.85 | 5.96 | 5.32 |

0 | 10.60 | 63.02 | 42.47 | 35.49 | 31.81 | 26.83 | 23.06 | 20.21 | 18.27 | 15.66 | 13.63 | 12.17 | |

−6.0 | 19.13 | 117.29 | 78.61 | 65.94 | 59.41 | 49.73 | 42.81 | 37.53 | 33.89 | 29.07 | 25.29 | 22.57 | |

| |||||||||||||

Seq 4 | 9.5 | 3.48 | 21.60 | 15.11 | 12.64 | 10.88 | 9.63 | 8.31 | 7.35 | 6.52 | 5.63 | 4.85 | 4.37 |

0 | 10.79 | 67.96 | 47.93 | 39.82 | 34.17 | 30.32 | 26.14 | 23.13 | 20.61 | 17.75 | 15.30 | 13.76 | |

−6.0 | 22.53 | 139.85 | 98.80 | 81.91 | 70.28 | 62.39 | 53.76 | 47.56 | 42.44 | 36.51 | 31.48 | 28.31 |

The errors between estimated and true solution at various AD rates (1–25 kHz) as well as the case of continuous-time sequence (tick “c” in the horizontal coordinate). Four sequences are examined under three SNR conditions (see the legend).

To gain more straightforward understanding of above results, we choose to present the resulting AEPs from different SNR conditions and at two representative AD rates. Figure

Comparison of transient AEPs solved by CLAD method at different SNR conditions and at two representative discretization frequencies. (a) AEPs (the original in dashed blue, the one recovered based on continuous-time stimulus sequence in thin red, and the one recovered based on discrete-time stimulus sequence in black). The discretization frequency (AD rate)

Solving the transient AEPs in frequency domain under HSR paradigm has been investigated and implemented recently using discrete-time stimulus sequences (e.g., in [

This paper answers the first question by explicating the frequency presentation of continuous-time stimulus sequences and its use in solving the transient AEP using deconvolution algorithm. As detailed before, the repetitive continuous-time stimulus sequences have discrete frequencies in the Fourier domain and as such the convolution model and deconvolution algorithm can be similarly presented as for the discrete-time time system. In practice, the continuous-time stimulus sequences can be approximated using very high temporal resolution discrete-time stimulus sequences. For example, the temporal resolution for the stimulus sequences in Table

To conclude, this study demonstrates the applicability and advantages of continuous-time stimulus sequences for AEP studies using HSR paradigm and reveals the reciprocal relationship between the errors in recovered AEPs and the sampling frequencies of the experimental systems when discrete-time stimulus sequences are used in traditional manner for the HSR paradigm.

The authors wish to thank Professor Sung-Phil Kim as well as the anonymous reviewers for useful comments on an earlier version of this paper. This work was supported by the National Natural Science Foundation of China (no. 61172033 and no. 61271154) and by Talent Introduction Foundation (2009) for the Institution of Higher Education in Guangdong.