The Growth Curve and an Associated Closed-Form Expression

The growth curve or growth function may be defined as the relationship between the stimulation current and the estimate of neural activation as quantified with an ECAP; the threshold is defined as the intercept of the linear portion of the curve with the x-axis (Adenis et al., 2018). A substantially linear response is seen above threshold for growth curves acquired in the spine (Parker et al., 2012) with ideally no neural response apparent below threshold. A hypothetical example of such a growth curve is shown in Figure 2, Curve A. Here, the ECAP_amp is plotted as a function of the stimulation current (I_stim). Below threshold (picked arbitrarily at 4 mA), no ECAP is observed. Above threshold, the ECAP_amp grows linearly at 15 μV/mA. The entire growth curve may be described completely with just two parameters: the x-axis intercept (I_thr), and the slope (S_resp) of the suprathreshold component that represents the extent of neural activation.

Illustrative ECAP growth curves. Curve A shows an “ideal” case where no neural activation is present up to a threshold I_thr, at which point the neural response grows linearly with a slope S_resp. Curve B shows a more representative model that includes curvature, σ, near neural threshold, and misclassification of stimulation artifact as neural response.

Two important differences exist between the ideal growth curve described above and those observed clinically, however. First, there is a substantial curvilinear component near threshold; neural activation does not instantaneously transition from zero to linear growth once a threshold is crossed. Second, a non-physiologic component of the ECAP estimate that grows linearly with increasing stimulation current – generally attributable to misclassification of stimulation artifact as ECAP – may manifest below threshold. The extent of this latter effect depends on the degree by which the signal chain rejects artifact and preserves neural response. Both these effects are shown in another hypothetical example in Figure 2, Curve B. First, a smooth transition from no neural activation to the linear response modeled with S_resp is introduced by means of parameter σ (described below) which is set in this example at 0.3 mA. Second, the growth curve consists of the “pure” neural activation of Figure 2, Curve A but also includes contribution from stimulation artifact misclassified as ECAP. Here, the artifact grows linearly with a slope (S_art) of 0.5 μV/mA. An offset N of 2 μV is also included.

For analysis purposes, then, a five-parameter equation is introduced that captures the contribution of both stimulation artifact and the underlying neural signal with associated curvature near threshold to the overall growth curve. Such an equation is shown here:

As described above, S_resp models the rate of growth of the response in the linear region, while S_art relates to rate of growth of the artifact with current. N captures the contribution of residual noise. The neural growth curve component R(I, I_thr, σ) is modeled as follows:

The shape of R(I, I_thr, σ) relates to the cumulative distribution of fiber thresholds in the dorsal columns, while I_thr and σ characterize the spreading of current between the stimulating electrodes and the dorsal column fibers. The utility of these equations lies with the potential to gain insight into the underlying neural electrophysiology and associated phenomena by analysis of the constituent components driving the morphology of the growth curve.