Author response:

We thank the reviewers and the editor for providing a thorough and constructive review of the manuscript. Both reviewers indicated that the ordinal entropy production rate should be validated with a synthetic (surrogate) model of the calcium signal. We agree with this, and it will feature prominently in the revised version of the manuscript.

As described (see Author response image 1), we have already implemented a synthetic GCaMP model. In the simulations, the strength of the calcium intensity varies monotonically with the intensity of tFUS. Applying our D_sym measure to the synthetic data produces a monotonic dose-response for irreversibility. It does not reproduce the non-monotonic, moderate-dose peak that we found in our study. We report this preliminary finding here because it speaks directly to the reviewers’ central critique.

Below we provide details of the planned revision.

(1) A surrogate model of GCaMP’s response to tFUS does not reproduce the observed dose structure

We built a generative model of the measured calcium signal according to the underlying physics (Author response image 1A):

(1) A latent Poisson spike train whose rate during and after sonication is monotone in tFUS dose by construction, such that any non-monotonicity appearing downstream must follow from the measurement of irreversibility;

(2) Convolution with an indicator kernel having fast rise and slow decay, matching the kinetic asymmetry mentioned by the reviewers;

(3) A static saturating nonlinearity representing GCaMP6s saturation;

(4) Additive measurement noise applied after the nonlinearity, with standard deviation calibrated to the variance of the real data’s baseline periods;

(5) baseline z-scoring, windowing, and estimation of D_sym using the same parameters as the manuscript (m = 3, τ = 8, f_s = 32 Hz; 5 s baseline, 5 s stimulation, 10 s recovery).

We ran four separate versions of the simulation (with and without saturation, noise before and after the nonlinearity). Simulated GCaMP traces for the variant with saturation and post-nonlinearity noise are drawn in Author response image 1B.

Result. All four simulation variants produce ∆D_sym that increases monotonically with dose and is largest at the highest dose (Author response image 1C). During the recovery window, the difference between moderate (2.3 to 3.7 W/cm²) and maximum (7.4 W/cm²) dose is −0.297±0.066 for the full version with the nonlinearity, −0.222±0.070 without the saturating nonlinearity, and −0.006±0.003 for the two versions in which noise precedes the nonlinearity (mean ± SD across seeds). In contrast, the D_sym of the real data (reproduced for convenience in Author response image 1D) peaks at 3.7 W/cm².

Author response image 1.

A synthetic model of the calcium measurement chain does not reproduce the observed dose structure. (A) Generative model. A Poisson spike train whose rate is proportional to acoustic dose is convolved with a fast-rise, slow-decay indicator kernel, passed through a static saturating nonlinearity, corrupted by additive noise, and then analysed with the same windowing and ordinal estimator used in the manuscript. (B) Simulated traces at low, moderate, and high dose for the variant with saturation and post-nonlinearity noise. (C) Simulated change in irreversibility during recovery, for all four variants of the simulation. Every variant increases monotonically with dose. Markers denote the mean across 20 seeds (error bars show standard deviation). (D) Real data as shown in the manuscript: observed change in irreversibility during recovery for on-target sonication. Markers denote individual animals; boxes show the mean ± 1 SEM. Irreversibility peaks at moderate dose.

(1.1) Invariance of ordinal statistics

Ordinal pattern methods are invariant under any strictly monotonic pointwise transformation of the signal, because such a transformation preserves rank order. Two consequences follow, and we will state them in the Methods of the revision:

- The static component of the indicator nonlinearity, including saturation and the relationship between calcium concentration and fluorescence, cannot alter D_sym.

- The scaling of amplitude with dose does not affect D_sym.

Any potential confound of GCaMP kinetics on D_sym enters through the dynamics, which is what the simulations above address.

(1.2) Planned analyses

In the revision, we will add:

- A sweep over rise and decay constants, Hill coefficient, noise level, and spike statistics, reporting explicitly the region of parameter space, if any, in which a moderate-dose peak can be produced;

- Burstiness and adaptation in the latent source;

- Calibrating the amplitude of the simulated responses to the real data;

- Finite-sample bias and variability of D_sym as a function of window length, pattern count, and signal-to-noise ratio, addressing Reviewer 1’s questions about window duration and noise.

(2) Aspects of the data that argue against a confound of kinetics

Independent of simulation, certain results in the manuscript are not explainable from an effect of calcium indicator kinetics.

(2.1) Non-monotonic irreversibility is reported for off-target sonication – without an accompanying amplitude response. Off-target sonication produces no appreciable fluorescence response, meaning that there is no evoked transient whose kinetics could generate ordinal asymmetry, yet irreversibility still varies non-monotonically with dose. We will elaborate on the off-target EPR effects in the revision.

(2.2) The dose structure remains after adjustment for amplitude. Our nested model comparison shows that acoustic dose explains variance in D_sym beyond baseline irreversibility and the windowmatched change in fluorescence. This is inconsistent with the view of D_sym as a deterministic readout of the amplitude trajectory. In the revision, we will also include decay time constant as a covariate, as Reviewer 1 suggests.

(2.3) Prediction from the prestimulation window. Baseline D_sym is computed on a window that contains no stimulus and no evoked transient, and it predicts the subsequent calcium response after adjustment for baseline fluorescence and dose. The kinetics of a response that has not yet occurred cannot modulate the value of a quantity measured before it.

(3) Reviewer 2, point 3: variation in baseline EPR across nominal intensity

Reviewer 2 correctly observes that baseline irreversibility varies across nominal intensity by an amount comparable to the stimulation-induced change. This implies two candidate explanations, which we address separately.

Systematic differences between intensity groups. If trials at different nominal intensities were not evenly distributed through a recording session, then drifts in bleaching, arousal, or habituation would produce spurious baseline differences between intensity groups. We tested this directly in the 408 analysed CMT 2.5 Hz trials. Session position, defined as the index of a trial within its recording, does not differ across the six intensity levels: Kruskal-Wallis H(5) = 3.51, p = 0.62 on-target and H(5) = 1.70, p = 0.89 off-target, with mean session position varying only between 18.3 and 23.3 across intensities. Within individual recordings, the rank correlation between session position and intensity has a median of +0.03 and does not exceed 0.34 in magnitude in any of the 10 recordings. Acoustic intensity is therefore not confounded with position in the session, and we will report this in the revision.

Uncertainty larger than the plotted error bars. This is a fair concern and we will address it by reporting trial-resampling and pattern-count bootstrap confidence intervals for D_sym. We also note that our primary inference already includes baseline D_sym as a covariate in an ANCOVA specification.

(4) Reviewer 1, remaining points

Other conventional GCaMP metrics. The revision will add decay time constant and additional waveform descriptors to the amplitude-adjustment models described above in 2.2.

Stimulation periods of differing duration. We will report the dependence of D_sym on window length from the finite-sample bias analysis using synthetic data (see 1.2 above).

Lower signal-to-noise recordings. We will measure the dependence of D_sym on SNR with synthetic data.

The auditory confound. We will add a Discussion paragraph citing Sato et al. (2018), Guo et al. (2023) and Kop et al. (2024), stating that the responses analysed here may reflect an indirect auditory mechanism.

Figures and data

A stochastic thermodynamics framework for investigating neural responses to fo-cused ultrasound.

Distinct dose-response of temporal irreversibility and calcium amplitude.

Dose-response relationships for calcium amplitude and EPR.

Dissociation between irreversibility and calcium amplitude immediately after son-ication.

EPR cannot be explained by calcium amplitude.

Baseline irreversibility modulates EPR response during but not after stimulation.

Baseline EPR predicts GCaMP response beyond baseline GCaMP and intensity.

Per-animal ΔDsym dose-response: 2.5 Hz on-target.

Per-animal ΔDsym dose-response: 2.5 Hz off-target.

EPR and GCaMP dose-response at 20 Hz PRF.

Stimulation-window EPR cannot be fully explained by calcium amplitude.

Correlation between ΔDsym and ΔGCaMP across conditions and windows at 2.5 Hz.

Per-animal ΔGCaMP dose–response: 2.5 Hz on-target.

Per-animal ΔGCaMP dose–response: 2.5 Hz off-target.

Hierarchical mixed-model tests: EPR dose structure beyond baseline and calcium amplitude.

Per-animal ΔD_sym dose-response: 2.5 Hz on-target.

Per-animal ΔD_sym dose-response: 2.5 Hz off-target.

Correlation between ΔD_sym and ΔGCaMP across conditions and windows at 2.5 Hz.