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Anatomical Connectivity from Tractography-free Diffusion Tensor Imaging and Its Application to EEG Brain Imagin

Joonas Lahtinen

arXiv:2609.31410v1math.NAcs.NAmath.DSmath.PR

TL;DR

Kalman-filter EEG source imaging uses random-walk evolution, while DTI offers anatomical connectivity information without tractography. This paper incorporates raw DTI-derived probabilistic connections into KF and SKF evolution models; in synthetic evoked-potential experiments, DTI-based estimates show more spatial spread and neighboring-region correlation, with localization outcomes varying by scenario.

  • Problem

    Random-walk evolution models lack prior connectivity information needed to represent spatial jumps between connected brain regions, while tractography-based connectivity workflows are time-consuming and can be sensitive to noise and tracking choices.

    Source: §1 · p. 2, §2.4 · p. 4

  • Method

    The paper derives probabilistic connections between anatomical regions from raw DTI diffusion and incorporates the resulting transition model into KF and SKF.

    Source: §1 · p. 2, §2.4 · pp. 5–6

  • Results

    DTI-based evolution makes estimates more spatially widespread and increases correlation between neighboring anatomical regions; in the auditory scenario, DTI-evolution methods correctly shift activity between insulae, and DTI-KF most accurately localizes the final component.

    Source: §4.3 · p. 11, §5 · p. 15

  • Takeaways & Limitations

    In the tested synthetic scenarios, DTI-based evolution can capture deep sources and constrain estimated activity toward the correct or nearest anatomical region, although localization is not guaranteed beyond the insula in the auditory estimates.

    Source: §4.2 · p. 9, §4.3 · p. 11

    Solving the DTI-conductivity model remains time-consuming, and the reported evaluation uses synthetic somatosensory and auditory evoked potentials.

    Source: §5 · p. 15

Abstract

from arXiv · show

This proof-of-concept paper demonstrates how the Kalman filter, as a brain imaging method, can be enhanced by incorporating raw diffusion tensor imaging (DTI) data without tractography, in addition to EEG recordings. An efficient algorithm is designed to streamline the otherwise tedious process of inferring connectivity between brain regions from DTI data. The DTI-based evolution model is applied to the Kalman filter and the Standardized Kalman filter, and these are compared with their conventional counterparts using the random walk evolution model. The numerical experiments use synthetic somatosensory and auditory evoked potentials. The results show that the DTI-modelled Kalman filter can obtain deep activity in the brainstem and thalamus. Moreover, the connection model limits the estimated spreads to the correct or nearest anatomical brain region.

1. Introduction

DTI maps neural connections by using direction-dependent water diffusion, but noisy, blurry images make those measurements uncertain. The paper incorporates DTI-derived probabilistic connections into Kalman-filter evolution models to address spatial jumps that random-walk models cannot represent without prior connectivity.

1. Introduction

  • In organized fibrous tissues, water diffuses more freely along fibers than perpendicular to them, providing DTI with a basis for mapping tissue structure.The degree of anisotropy reflects how directionally constrained diffusion is.Source: §1 · p. 1
  • DTI maps neural connections from water diffusion along nerve fibers, although its images are inherently blurry and noisy.The paper proposes treating uncertain measurements probabilistically, in keeping with the certainty the imaging provides.Source: §1 · p. 2
  • Random-walk evolution models struggle to represent spatial jumps between connected brain regions when EEG cannot detect signals traveling between them.The paper identifies prior knowledge of possible connections as necessary to model these jumps.Source: §1 · p. 2
  • The paper uses raw DTI-derived information to establish probabilistic anatomical connections and turn flow proportions into evolution models for KF and SKF.It asks whether DTI can be used without preprocessing and tractography, and whether connectivity-based evolution can benefit EEG source imaging.Source: §1 · p. 2

2. Methods

The methods estimate diffusion tensors from diffusion-weighted measurements, then use Kalman filtering to update brain-state estimates over time. The paper derives DTI-based transitions from probabilistic diffusion between anatomical regions and applies the resulting model to KF and SKF.

2.1. Diffusion Tensor Imaging

  • Diffusion tensors are fit from diffusion-weighted image measurements in six directions using the Stejskal–Tanner equation and least squares.The measurements are organized into a linear system; the signal intensity, baseline intensity, and diffusion-weighting factor are defined in the text.Source: §2.1 · p. 3

2.2. Kalman filter

  • The Kalman filter recursively updates brain-state estimates from EEG observations, using a transition matrix that is either identity for random-walk evolution or DTI-derived.The observation model assumes Gaussian noise, and the state evolves linearly with Gaussian process noise.Source: §2.2 · pp. 3–4

2.3. Standardized Kalman filter

  • SKF expresses estimates as time-varying Z-scores; with a static transition matrix for KF, its corresponding transition matrix is necessarily time-varying.The process-noise covariance is transformed using the standardization matrix.Source: §2.3 · p. 4

2.4. DTI diffusion process

  • The DTI diffusion model treats the spatially varying tensor as inducing a Riemannian metric, using geodesic distance to estimate signal-travel probabilities.Those probabilities yield hitting probabilities between facing region surfaces, avoiding tractography's point estimates; the model assumes low curvature for paths from thalamus to neocortex or between neocortical regions.Source: §2.4 · pp. 4–5
  • A shooting algorithm reduces geodesic computation by optimizing path midpoints, checking suspiciously slow segments, and correcting outliers.The method uses 12 iteration steps and reduces outliers every fourth step; standard pathfinding alternatives were considered too computationally taxing for the tensor format and size.Source: §2.4 · pp. 5–6
  • The transition-matrix connection from one region to another sums voxel-to-voxel hitting probabilities and is scaled so each column sums to one, assuming no activity loss.The probabilities are computed between opposing region surfaces, from source-surface voxels to target-surface voxels.Source: §2.4 · p. 6

3. Experiments

The experiments use synthetic somatosensory and auditory evoked-potential scenarios to examine DTI-based evolution models alongside conventional KF and SKF. They assess source localization, activity spread, amplitude time courses, and— for the auditory scenario—time delays and cross-correlation between paired regions.

3. Experiments

  • The synthetic SEP scenario uses Gaussian pulses at 14, 16, 20, and 22 ms, with deep brainstem or thalamic activity and additional cortical contributors at 20 and 22 ms.The analysis considers localization and relative cortical and deep activity amplitudes, using qualitative brain-model estimates and normalized amplitude time series.Source: §3 · p. 7
  • The synthetic AEP scenario alternates causal activity between distant regions across hemispheres, from the left insula through temporal gyri to the right hemisphere.The analysis also examines delays and cross-correlation between paired regional time courses to assess whether distinct estimates are overly correlated.Source: §3 · p. 7
  • The study compares KF and SKF using random-walk evolution with DTI-KF and DTI-SKF using DTI-based evolution.The inversion is evaluated on synthetic somatosensory and auditory evoked-potential scenarios.Source: §3 · pp. 7–8

4. Results

The optimal-path test showed exponentially decreasing path error as iterations progressed. In synthetic somatosensory evoked potentials, DTI-modelled filters recovered deep sources, with DTI-SKF producing the smallest EMD across examined regions.

4.1. Optimal path test

  • For the highly nonlinear test path, Euclidean path error decreased exponentially with iterations; Figure 2 tracks the reduction through 65 iterations.Figure 2 also presents the solution after 100 iterations.Source: Figure 2 · p. 8, §4.1 · p. 8

4.2. Synthesized Somatosensory evoked potentials

  • KF detected only cortical activity; SKF localized deep activity satisfactorily but spread cortical estimates and placed later thalamic components too anterior.Source: §4.2 · pp. 8–9
  • DTI-KF and DTI-SKF detected deep sources; DTI-SKF cortical estimates were extremely focal and nearly accurate, while its thalamic spread fell between DTI-KF and SKF.DTI-SKF's estimation tail indicated posterior-lateral thalamic contribution more clearly than DTI-KF.Source: §4.2 · p. 9
  • DTI-KF yielded the most balanced cortical and deep estimates, while the other methods' time-course curves were nearly the same.The EMD analysis cautions that early deep-region spikes may reflect unwanted activity anywhere in source space, not necessarily deep activity.Source: §4.2 · p. 9
  • DTI-SKF showed the smallest EMD across examined regions; DTI-KF had nearly zero brainstem EMD but the largest thalamic EMD spike.The passage identifies total EMD as the more important measure of total estimation error.Source: §4.2 · p. 9

DTI-KF DTI-SKF

In the synthesized auditory experiment, DTI-based estimates reduce mislocalization relative to random-walk methods, though they do not always peak at the active locations. The figures compare the methods’ source projections, regional time courses, and Earth Mover’s Distances.

DTI-KF DTI-SKF

  • Figure 3 contrasts KF and SKF using random-walk evolution with their counterparts using a DTI-based transition model across four SEP events.The random-walk methods appear in the upper block and DTI-based methods in the lower block.Source: Figure 3 · p. 10

4.3. Synthesized auditory evoked potential

  • DTI-evolution methods correctly shift activity from left to right, and DTI-KF localizes the last component most accurately.Random-walk methods keep the peak on the left during the rightward shift; all methods detect the last component on the correct side.Source: §4.3 · p. 11, Figure 5 · p. 12
  • Hemisphere-wise estimates follow the corresponding true trajectories, but random-walk KF and SKF estimates correlate nearly perfectly and cannot localize beyond the correct hemisphere.The DTI model separates regional estimates, but ideal localization beyond the insula is not guaranteed.Source: §4.3 · p. 11, Figure 6 · p. 13
  • The DTI model significantly drops EMD values, and mislocalized activity appears close to active regions even when it does not peak at the correct locations.Figure 7 reports EMD by region and hemisphere, plus summed total EMD.Source: §4.3 · p. 11, Figure 7 · p. 14

5. Discussion

The discussion presents a tractography-free, probabilistic DTI connectivity model for Kalman-filter EEG source imaging. Its estimates stay nearer true sources in complex auditory potentials, while computation remains a clinical-use constraint.

5. Discussion

  • The connectivity-based model limits estimates from spreading far from true sources and slightly distinguishes nearby active regions; the authors find it more beneficial for KF than SKF.The discussion attributes SKF’s broader estimates to its low-resolution topography.Source: §5 · p. 15
  • Raw tractography-free DTI data form a probability-based map of inter-region flow, which is incorporated into the brain-activity transition matrix.The DTI model represents transition as instantaneous decay at the starting region and equal activity gain in target regions.Source: §5 · p. 15
  • Solving the DTI-conductivity model remains time-consuming, constraining clinical applicability pending further work.The authors identify DTI preprocessing and more efficient geodesic computations as future research topics.Source: §5 · p. 15

Appendix A. Fokker-Planck equation on Riemann manifolds and diffusion tensor–metric connection

The appendix derives a Riemann-manifold Fokker–Planck representation from a time-homogeneous stochastic process, then connects driftless diffusion to a metric through the diffusion tensor.

Appendix A. Fokker-Planck equation on Riemann manifolds and diffusion tensor–metric connection

  • The derivation starts from a time-homogeneous stochastic process and uses the Fréchet derivative for a smooth compactly supported function.The passages introduce a test function h and its Fréchet derivative.Source: Appendix A · p. 16
  • The derivation uses the Chapman–Kolmogorov identity, a second-order Taylor expansion, and integration by parts to obtain a Fokker–Planck representation.The derivation begins with a smooth compactly supported function h.Source: Appendix A · pp. 16–17
  • With drift set to zero and g_ji defined using the pseudoinverse of σ_kiσ_kj, the process follows a standard Wiener process on a flat space with holes where σ(x)=0.The passage identifies σ^Tσ as the contravariant metric of (M,g).Source: Appendix A · p. 17

Appendix B. Convergence of the shooting algorithm for geodesics

The appendix analyzes the shooting algorithm’s approximation of geodesics, relating its subdivision schedule to an error bound involving geodesic acceleration.

Appendix B. Convergence of the shooting algorithm for geodesics

  • The shooting method provides a linear approximation for a path γ, with a Lagrange error bound for the Taylor series.The stated bound is expressed in terms of the maximum geodesic acceleration and K=(sqrt(8n+1)-1)/2.Source: Appendix B · p. 17
  • The algorithm halves every subinterval after (n^2+n)/2 steps.The passage gives this subdivision schedule as part of the convergence argument.Source: Appendix B · p. 17
  • The bound’s acceleration term is the maximum norm of the geodesic’s second derivative, and its constant K is specified as (sqrt(8n+1)-1)/2.The appendix then expresses the bound in terms of n, but the supplied passage does not show that expression.Source: Appendix B · p. 17

Appendix C. Derivation of the hitting probability formula

The appendix derives a hitting-probability formula by expressing probability around a geodesic in local cylindrical coordinates and marginalizing directions orthogonal to the path.

Appendix C. Derivation of the hitting probability formula

  • The resulting random process is a Wiener process, and the change-of-variables result is extended to general continuous paths using Eq. (14) and r0=h/2.The extension follows the observed change of variables in Eq. (C.9).Source: Appendix C · pp. 17, 19
  • The local coordinate construction uses a rotationally symmetric cylindrical system around the path, and assumes unit-speed parameterization when the path is geodesic.The passage states that for a geodesic, the path speed is identically 1.Source: Appendix C · p. 18
  • The probability distribution is represented in a tube-like cylindrical coordinate system around the geodesic, then marginalized over variables orthogonal to the path.This yields the probability of the path’s radial coordinate lying within the tube.Source: Appendix C · p. 18
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