Abstract
This paper discusses some of the practical limitations and issues, which exist for the input–output (IO) slope curve estimation (SCE) in neural, brain and spinal, stimulation techniques. The drawbacks of the SCE techniques by using existing uniform sampling and Fisher-information-based optimal IO curve estimation (FO-IOCE) methods are elaborated. A novel IO SCE technique is proposed with a modified sampling strategy and stopping rule which improve the SCE performance compared to these methods. The effectiveness of the proposed IO SCE is tested on 1000 simulation runs in transcranial magnetic stimulation (TMS), with a realistic model of motor evoked potentials (MEPs). The results show that the proposed IO SCE method successfully satisfies the stopping rule, before reaching the maximum number of TMS pulses in 79.5% of runs, while the estimation based on the uniform sampling technique never converges and satisfies the stopping rule. At the time of successful termination, the proposed IO SCE method decreases the 95th percentile (mean value in the parentheses) of the absolute relative estimation errors (AREs) of the slope curve parameters up to 7.45% (2.2%), with only 18 additional pulses on average compared to that of the FO-IOCE technique. It also decreases the 95th percentile (mean value in the parentheses) of the AREs of the IO slope curve parameters up to 59.33% (16.71%), compared to that of the uniform sampling method. The proposed IO SCE also identifies the peak slope with higher accuracy, with the 95th percentile (mean value in the parentheses) of AREs reduced by up to 9.96% (2.01%) compared to that of the FO-IOCE method, and by up to 46.29% (13.13%) compared to that of the uniform sampling method.
Competing Interest Statement
The authors have declared no competing interest.
Footnotes
The source code is available at [42]
Abbreviations & Nomenclatures
- TM
- Transcranial magnetic stimulation
- IO
- Input–output
- MEP
- Motor evoked potential
- FIM
- Fisher information matrix
- FO-IOCE
- Fisher-information-based optimal IO curve estimation
- SCE
- Slope curve estimation
- ARE
- Absolute relative estimation error
- x
- TMS pulse strength normalized between 0 and 1, x = 1 means 100% pulse strength
- xmin
- Minimum TMS pulse strength. It is presumed to be 0 in this work
- xmax
- Maximum TMS pulse strength. It is presumed to be 100% in this work
- y(x)
- (Reference) IO curve
- y′ (x)
- (Reference) IO slope curve, y′ = dy/dx
- yl
- (Reference) low-side plateau of IO curve
- yh
- (Reference) high-side plateau of IO curve
- m
- (Reference) midpoint of IO curve
- x*
- (Reference) TMS pulse strength which targets the midpoint of the IO curve, i.e., x* = m
- s
- (Reference) slope parameter
- θ
- (Reference) parameter vector θ = [θ1θ2θ3θ4]⊤ = [ylyhm s]⊤
- Parameter vector used for FIM optimization
- nbase
- Number of baseline samples
- n
- Number of TMS pulses
- N
- Total number of samples N = nbase + n
- N0
- Size of the data set for the initial curve fitting. It is equal to the number of baseline samples plus the first three initial TMS pulses, N0 = nbase + 3
- Nmax
- Maximum number of TMS pulses
- IO data set used for curve fitting
- xN
- All TMS stimuli,
- Estimation of θ after the stimulation of the n-th TMS pulse
- Estimation of θl, l = 1, 2, 3, 4, after the stimulation of the n − th TMS pulse
- Lower bound of the estimate of θ, i.e., estimated parameters cannot be lower than values in
- Upper bound of the estimate of θ, i.e., estimated parameters cannot be greater than values in
- ŷn
- Estimation of the IO curve after the stimulation of the n − th TMS pulse
- ϵl
- Convergence tolerance
- T
- Number of successive times the convergence criterion must be satisfied in stopping rule
- Number of TMS pulses satisfying the stopping rule for the first time
- Number of TMS pulses satisfying the stopping rule for the second time. The proposed IO SCE method successfully stops at this point
- Relative estimation error of θl, l = 1, 2, 3, 4, after the stimulation of the n−th TMS pulse
- Relative estimation error of IO curve derivative at x*, after the stimulation of the n −th TMS pulse