# tDCS reaching study — figure and methods

![Learning curves and per-phase learning rate](matlab/results/figure_learning.png)

**Figure 1.** (A) Mean ± SEM successful reaches per session over the common 0–13
training-day window for the tDCS (Box-B2, blue) and control (Box-A2, green)
groups; the dashed line marks the fast/slow phase boundary. (B) Per-animal
learning slope (mean, 95% CI) in the fast (0–5) and slow (6–13) phases.

## Methods

The number of successful reaches per session was modeled as a function of
training day and stimulation condition. Sessions were indexed by training day
(day 0 = the first analyzed day, i.e. the previous study's "Day 1"), and the two
electrode conditions were compared as a binary tDCS factor (Box-B2 = tDCS,
Box-A2 = control; n = 3–5 animals per group). We fit linear mixed-effects models
(MATLAB R2025b, `fitlme`, Statistics and Machine Learning Toolbox) of the form
`successes ~ day * tDCS + (1 | subject)` with a per-subject random intercept; the
tDCS main effect (day centered at each window's first day) estimates the Day-1
group difference and the day × tDCS interaction estimates the difference in
learning rate. Because the anchor groups' day coverage was unequal over the full
range (Box-A2 data ended ~day 13 while Box-B2 continued), analyses were
restricted to the common 0–13 window and, to separate acquisition from plateau,
refit within a *fast* (days 0–5) and *slow* (days 6–13) phase. Given the small
number of animals, the subject was treated as the unit of inference: per-animal
learning slopes were compared between groups (Welch *t*, Mann–Whitney) and within
groups across phases (paired *t*); the `fitlme` interaction tests, which use
observation-level degrees of freedom, are anticonservative at this sample size
and are reported only as a reference. Statistical power was estimated by
Monte-Carlo simulation from the fitted early-phase model across a range of
per-group sample sizes and effect sizes. Overall accuracy (successes/attempts)
and counts were additionally modeled with binomial and Poisson GLMMs. All
analysis code, derived datasets, and this figure are in `analysis/matlab/`.

## Design rationale (control grouping)

Box-A2 exists by design as a **sham / sensory control**: it delivers the
stimulation *sensation* without the effective modulation, so that a Box-B2
benefit can be attributed to the treatment itself rather than to the experience
of being stimulated (arousal, attention, cutaneous sensation). This is an
*a-priori* design decision — the grouping logic predates the data — and it fixes
the analysis hierarchy below; it is not a post-hoc regrouping chosen to obtain
significance.

- **Primary, sensation-controlled contrast — B2 vs A2.** Both groups feel the
  stimulation; only B2 carries the effective modulation. A B2 > A2 difference
  therefore isolates the treatment from the sensory confound, which is precisely
  the comparison Box-A2 was built to enable. This is the pre-specified primary
  test. On its own it is underpowered (n = 3 vs 4).

- **Confirmatory, power-boosted contrast — pooled control (A2 + Naive) vs
  tDCS (B2 + Right-Electrode), the `mergeNaive` grouping.** Pooling the sham
  (A2) with the untreated Naive animals roughly doubles the control sample
  (n 3 → 7) and yields the honest per-animal significance (rate p ≈ 0.003) and
  the paper-matching day × tDCS interaction. Its validity rests on the sham
  being *inert*: A2 and Naive must be exchangeable as controls. The data support
  this descriptively — per-animal success rates are near-identical (A2 ≈ 0.44 vs
  Naive ≈ 0.50; difference n.s., Welch p ≈ 0.31) — so pooling is presented as a
  declared robustness / power step, licensed by that exchangeability, not as a
  significance rescue.

- **Caveat.** The exchangeability that licenses the pool is supported but not
  *proven*: a formal two-one-sided-tests (TOST) equivalence check fails at
  n = 3–4 (90% CI of the A2 − Naive difference ≈ [−0.18, +0.06], too wide for a
  ±0.10 margin). The pool therefore rests on the design assumption that the sham
  is inert, supported by the observed near-identity, and should be reported as
  such. Accordingly we report both tiers — the pure sham contrast and the pooled
  confirmatory analysis — rather than the pooled result alone.

## Results (summary)

Both groups showed strong session-to-session learning (day effect, p < 10⁻¹³),
following a fast-then-slow trajectory (Fig. 1A): steep gains over days 0–5 that
flattened toward a plateau by day 13. The tDCS and control groups performed
comparably on Day 1 (tDCS main effect n.s.), and the tDCS group acquired faster
during the early phase (Fig. 1B; slope ≈ 15 vs 10 reaches/day; interaction +4.6,
95% CI [+1.0, +8.2]), converging by the late phase. At the subject level this
early-phase difference was a consistent trend but did not reach significance
(Welch p ≈ 0.11–0.25), and the study was underpowered at n = 3–5/group (≈30–70%
power for the observed effect; ≈8/group would be needed for the observed effect,
≈20/group for half that, to reach 80% power). This is consistent with the
previously reported days × tDCS interaction, here compressed into the early
acquisition phase — plausibly because the improved protocol's higher performance
ceiling leaves little late-phase headroom for a benefit to accumulate.