Identifiability Analysis and Noninvasive Online Estimation of the First-Order Neural Activation Dynamics in the Brain With Closed-Loop Transcranial Magnetic Stimulation

Abstract

Background: Neurons demonstrate very distinct nonlinear activation dynamics, influenced by the neuron type, morphology, ion channel expression, and various other factors. The measurement of the activation dynamics can identify the neural target of stimulation and detect deviations, e.g., for diagnosis. This paper describes a tool for closed-loop sequential parameter estimation (SPE) of the activation dynamics through transcranial magnetic stimulation (TMS). The proposed SPE method operates in real time, selects ideal stimulus parameters, detects and processes the response, and concurrently estimates the input-output (IO) curve and the first-order approximation of the activated neural target. Objective: To develop a method for concurrent SPE of the first-order activation dynamics and IO curve with closed-loop TMS. Method: First, identifiability of an integrated model of the first-order neural activation dynamics and IO curve is assessed, demonstrating that at least two IO curves need to be acquired with different pulse widths. Then, a two-stage SPE method is proposed. It estimates the IO curve by using Fisher information matrix (FIM) optimization in the first stage and subsequently estimates the membrane time constant as well as the coupling gain in the second stage. The procedure continues in a sequential manner until a stopping rule is satisfied. Results: The results of 73 simulation cases confirm the satisfactory estimation of the membrane time constant and coupling gain with average absolute relative errors (AREs) of 6.2% and 5.3%, respectively, with an average of 344 pulses (172 pulses for each IO curve or pulse width). The method estimates the IO curves' lower and upper plateaus, mid-point, and slope with average AREs of 0.2%, 0.7%, 0.9%, and 14.5%, respectively. The conventional time constant estimation method based on the strength-duration (S-D) curve leads to 33.3% ARE, which is 27.0% larger than 6.2% ARE obtained through the proposed real-time FIM-based SPE method in this paper. Conclusions: SPE of the activation dynamics requires acquiring at least two IO curves with different pulse widths, which needs a controllable TMS (cTMS) device with adjustable pulse duration. Significance: The proposed SPE method enhances the cTMS functionality, which can contribute novel insights in research and clinical studies.

Closed-Loop Estimation of Neurostimulation Strength-Duration Curve Using Fisher Information Optimization and Comparison With Uniform and Random Methods

“…The stopping rule is arbitrarily defined to satisfy the convergence criterion (6) with ϵ = 0.001 for T = 5 successive times. As discussed in [31], [34]- [36], there is a trade-off between the estimation accuracy (i.e., ϵ and T values) and the number of samples in a successful termination. Reducing the convergence tolerance and increasing the successive times parameter would improve the estimation, however, more SD data are needed to meet the stopping rule.…”

“…In [31], [34]- [36], the FIM optimization was developed for closed-loop estimation of the neural input-output curve and activation dynamics including the membrane time constant and coupling gain. No paper or study has been published on the FIM-based closed-loop SD curve estimation, and comparison with the uniform and random methods.…”

“…1. Weiss [27] and Lapicque [28] SD curve models have widely been utilized in the literature, [1], [17], [29]–[31]. Weiss proposed the hyperbolic form of the SD curve, I = b (1 + c/d ), where I is the threshold current, b is the rheobase current, c is the chronaxie, and d is the pulse width, and Lapicque proposed the exponential SD curve in the form of I = b/ (1 − exp (− d/c )), [32].…”

“…In the existing literature, the pulse widths are chosen randomly and/or uniformly distributed, in an open-loop framework, prior to the experiments [1], [17], [26], [31], [33]. There is no systematic method or result to determine how many data points are enough to estimate a SD curve with a desired level of accuracy.…”