Deep brain microelectrode signal: $q$-statistical approach

Abstract

We characterize the amplitude statistics of intraoperative microelectrode recordings (MERs) obtained during deep brain stimulation (DBS) surgery in 46 patients with Parkinson's disease, using 184 recordings equally balanced between inside and outside the subthalamic nucleus (STN). The probability density of every recording is quantitatively well described by a $q$-Gaussian (grounded on a nonadditive entropic functional), $ρ(x) \propto [1 + β(q-1) x^2]^{-1/(q-1)}$, with $q > 1$ in all cases, reflecting persistent long-range temporal correlations inconsistent with Gaussian dynamics. Within the superstatistics framework, the slowly fluctuating local variance visible in the raw MER signals is a physical mechanism that directly generates the $q > 1$ form. Beyond individual fits, $q$ and $β$ collapse across all 184 recordings onto the single functional constraint $q = 3 - 1.85\,β^{-0.33}$ ($R \approx -0.91$), a reduction to one effective degree of freedom that is the quantitative hallmark of near-critical dynamics, previously identified in scale-free network growth and in acoustic precursors of material fracture. The index $q$ is statistically indistinguishable across the STN boundary ($\langle\bar{q}_\text{out}/\bar{q}_\text{in} \rangle = 1.03$), while the inverse-widthparameter shows a modest systematic difference ($\langle\barβ_\text{out}/\barβ_\text{in} \rangle = 1.18$). Since $q > 1$ is expected for any brain structure exhibiting long-range correlations, healthy or pathological, it is the tight $q(β)$ coupling, not $q > 1$ per se, that constitutes the candidate near-criticality signature of the parkinsonian cortico-basal-ganglia-thalamocortical loop.

Figures (4)

• Figure 1: Raw microelectrode recording amplitudes ($\mu$V) as a function of recording time (s) for a representative Parkinson's disease patient, following bandpass filtering (300--5000 Hz) and $z$-score normalisation. (a) Extra-STN recording; (b) intra-STN recording. The sustained amplitude elevation visible near the midpoint of panel (b) reflects the within-recording non-stationarity discussed in the text, consistent with slow variance fluctuations that physically generate the $q > 1$ form within the superstatistics framework.
• Figure 2: $q$-Gaussian fitting of MER amplitude distributions for two representative Parkinson's disease patients. The $q$-logarithmic transform $\ln_q[\rho(x)/\rho_0]$ is plotted against the squared normalised amplitude $x^2$, so that a $q$-Gaussian $\rho(x) = \rho_0\, e_q^{-\beta x^2}$ appears as a straight line with slope $-\beta$ (dashed lines). (a) Patient 4; (b) Patient 6. Blue symbols: intra-STN recordings; red symbols: extra-STN recordings. Fitted parameters $(q^{\star}, \beta^{\star})$ and the coefficient of determination $R^2$ for each fit are annotated within the panels..
• Figure 3: $q$ versus $1/\beta^{0.33}$ for 184 signals. The dashed straight line (black) is the linear fitting $(q=3-1.85/\beta^{0.33})$ of the outside (red circles) and inside (blue squares) datasets assuming $q$-Gaussian forms (notice that $q$-Gaussians are normalizable only if $q < 3$). The linear correlation coefficient $R \simeq - 0.912$ indicates a strong correlation between the values of $q$ and $\beta$. A monotonic behavior like this one characterizes criticality and has already been observed in the growth of asymptotically scale-free networksBritoSilvaTsallis2016NunesBritoSilvaTsallis2017CinardiRapisardaTsallis2020 and in the fracture of construction materialsGrecoTsallisRapisardaPluchinoFicheraContrafatto2020.
• Figure 4: Patient-wise ratios of mean fitted $q$-Gaussian parameters between extra-STN and intra-STN recordings. For each of the 46 patients ($x$-axis: patient index), green circles show the ratio $\bar{q}_{\text{out}}/\bar{q}_{\text{in}}$ of patient-averaged nonextensivity indices, and purple circles show the ratio $\bar{\beta}_{\text{out}}/\bar{\beta}_{\text{in}}$ of patient-averaged inverse-width parameters. The dashed horizontal line marks unity. Ensemble means are $\langle \bar{q}_{\text{out}}/\bar{q}_{\text{in}} \rangle = 1.03$ and $\langle \bar{\beta}_{\text{out}}/\bar{\beta}_{\text{in}} \rangle = 1.18$, confirming spatial uniformity of the nonextensivity index $q$ across the STN boundary, while the modest elevation of $\bar{\beta}_{\text{in}}$ reflects the well-documented increase in background spiking amplitude within the STN.