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Abstract When recording the movement of individual animals, cells or molecules one will often observe changes in their diffusive behaviour at certain points in time along their trajectory. In order to capture the different diffusive modes assembled in such heterogeneous trajectories it becomes necessary to segment them by determining these change-points. Such a change-point detection can be challenging for conventional statistical methods, especially when the changes are subtle. We here applyBayesian Deep Learningto obtain point-wise estimates of not only the anomalous diffusion exponent but also the uncertainties in these predictions from a single anomalous diffusion trajectory generated according to four theoretical models of anomalous diffusion. We show that we are able to achieve an accuracy similar to single-mode (without change-points) predictions as well as a well calibrated uncertainty predictions of this accuracy. Additionally, we find that the predicted uncertainties feature interesting behaviour at the change-points leading us to examine the capabilities of these predictions for change-point detection. While the series of predicted uncertainties on their own are not sufficient to improve change-point detection, they do lead to a performance boost when applied in combination with the predicted anomalous diffusion exponents. 

Abstract We introduce the stochastic process of incremental multifractional Brownian motion (IMFBM), which locally behaves like fractional Brownian motion with a given local Hurst exponent and diffusivity. When these parameters change as function of time the process responds to the evolution gradually: only new increments are governed by the new parameters, while still retaining a power-law dependence on the past of the process. We obtain the mean squared displacement and correlations of IMFBM which are given by elementary formulas. We also provide a comparison with simulations and introduce estimation methods for IMFBM. This mathematically simple process is useful in the description of anomalous diffusion dynamics in changing environments, e.g. in viscoelastic systems, or when an actively moving particle changes its degree of persistence or its mobility. 

Neural networks in adult vertebrate brains are physically embedded in meshworks of thin, functionally active axons (fibers) that originate in the brainstem. As these fibers weave through neural tissue, releasing serotonin (5-HT) with glutamate and other neurotransmitters, they produce a dense matrix macroscopically described by regional fiber densities. This matrix is fundamentally associated with neuroplasticity, with implications for mental disorders and artificial neural networks. We have recently shown that its self-organization strongly depends on the stochastic properties of single fibers and their interaction with the three-dimensional (3D) geometry of the brain. Specifically, the trajectories of individual fibers can be described as paths of reflected fractional Brownian motion (FBM) [1, 2]. We are currently using transgenic, in vitro [3], and other experimental approaches to guide further modeling efforts and to motivate the development of the FBM theory itself [4]. In a major extension of our previous studies, we used supercomputing to simulate 960 fibers in a complex, 3D-dimensional shape constructed from serial sections of the late-embryonic mouse brain (at E17.5) [5]. The fibers were modeled as paths of reflected FBM (H = 0.8) which interacted with pial and ventricular borders. The simulated densities were compared to the actual regional fiber densities in a recently published comprehensive map. Strong similarities were found in the forebrain and midbrain. This study demonstrates that regional “serotonergic” fiber densities can achieve a substantial degree of self-organization with no biological guiding signals, with implications for neurodevelopment, neuroplasticity, and brain evolution. Support: NSF-BMBF CRCNS (NSF #2112862 to SJ & TV; BMBF #STAXS to RM). References: [1] Janušonis et al. (2020) Front. Comput. Neurosci. 14: 56; [2] Vojta et al. (2020) Phys. Rev. E 102: 032108; [3] Hingorani et al. (2022) Front. Neurosci. 16: 994735; [4] Wang et al. (2023) arXiv 2303.01551; [5] Janušonis et al. (2023) bioRxiv 10.1101/2023.03.19.533385. 

In vertebrate brains, virtually all neural circuits operate inside a dense matrix of axons (fibers) that have a strongly stochastic character. These fibers originate in the brainstem raphe region, produce highly tortuous trajectories, and release serotonin (5-hydroxytryptamine, 5-HT), with other neurotransmitters. They can robustly regenerate in the adult mammalian brain and appear to support neuroplasticity [1], with implications for mental disorders [2] and artificial neural networks [3]. The self-organization of this “serotonergic” matrix remains poorly understood. In our previous study, we have shown that serotonergic fibers can be modeled as paths of fractional Brownian motion (FBM), a continuous-time stochastic process. FBM is parametrized by the Hurst index, which defines three distinctive regimes: subdiffusion (H < 0.5), normal diffusion (H = 0.5), and superdiffusion (H > 0.5). In two-dimensional (2D) shapes based on the adult mouse brain, simulated FBM-fibers (with H = 0.8) have produced regional distributions similar to those of the actual serotonergic fibers [4]. However, increments of superdiffusive FBM trajectories have long-range positive correlations, which implies that a fiber path in one 2D-section depends on its history in other sections. In a major extension of this study, we used a supercomputing simulation to generate 960 fibers in a complex, three-dimensional shape based on the late-embryonic mouse brain (at embryonic day 17.5). The fibers were modeled as paths of reflected FBM with H = 0.8. The reflection was caused by natural neuroanatomical borders such as the pia and ventricles. The resultant regional densities were compared to the actual fiber densities in the corresponding neuroanatomically-defined regions, based on a recently published comprehensive map [5]. The relative simulated densities showed strong similarities to the actual densities in the telencephalon, diencephalon, and mesencephalon. The current simulation does not include tissue heterogeneities, but it can be further improved with novel models of multifractional FBM, such as the one introduced by our group [6]. The study demonstrates that serotonergic fiber densities can be strongly influenced by the geometry of the brain, with implications for neurodevelopment, neuroplasticity, and brain evolution. Acknowledgements: This research was funded by an NSF-BMBF CRCNS grant (NSF #2112862 to SJ & TV; BMBF #STAXS to RM). References: 1. Lesch KP, Waider J. Serotonin in the modulation of neural plasticity and networks: implications for neurodevelopmental disorders. Neuron. 2012, 76, 175-191. 2. Daws RE, Timmermann C, Giribaldi B, et al. Increased global integration in the brain after psilocybin therapy for depression. Nat. Med. 2022, 28, 844-851. 3. Lee C, Zhang Z, Janušonis S. Brain serotonergic fibers suggest anomalous diffusion-based dropout in artificial neural networks. Front. Neurosci. 2022, 16, 949934. 4. Janušonis S, Detering N, Metzler R, Vojta T. Serotonergic axons as fractional Brownian motion paths: Insights Into the self-organization of regional densities. Front. Comput. Neurosci. 2020, 14, 56. 5. Awasthi JR, Tamada K, Overton ETN, Takumi T. Comprehensive topographical map of the serotonergic fibers in the male mouse brain. J. Comp. Neurol. 2021, 529, 1391-1429. 6. Wang W, Balcerek M, Burnecki K, et al. Memory-multi-fractional Brownian motion with continuous correlation. arXiv. 2023, 2303.01551. 

The self-organization of the brain matrix of serotonergic axons (fibers) remains an unsolved problem in neuroscience. The regional densities of this matrix have major implications for neuroplasticity, tissue regeneration, and the understanding of mental disorders, but the trajectories of its fibers are strongly stochastic and require novel conceptual and analytical approaches. In a major extension to our previous studies, we used a supercomputing simulation to model around one thousand serotonergic fibers as paths of superdiffusive fractional Brownian motion (FBM), a continuous-time stochastic process. The fibers produced long walks in a complex, three-dimensional shape based on the mouse brain and reflected at the outer (pial) and inner (ventricular) boundaries. The resultant regional densities were compared to the actual fiber densities in the corresponding neuroanatomically-defined regions. The relative densities showed strong qualitative similarities in the forebrain and midbrain, demonstrating the predictive potential of stochastic modeling in this system. The current simulation does not respect tissue heterogeneities but can be further improved with novel models of multifractional FBM. The study demonstrates that serotonergic fiber densities can be strongly influenced by the geometry of the brain, with implications for brain development, plasticity, and evolution. 

Abstract The stochastic trajectories of molecules in living cells, as well as the dynamics in many other complex systems, often exhibit memory in their path over long periods of time. In addition, these systems can show dynamic heterogeneities due to which the motion changes along the trajectories. Such effects manifest themselves as spatiotemporal correlations. Despite the broad occurrence of heterogeneous complex systems in nature, their analysis is still quite poorly understood and tools to model them are largely missing. We contribute to tackling this problem by employing an integral representation of Mandelbrot’s fractional Brownian motion that is compliant with varying motion parameters while maintaining long memory. Two types of switching fractional Brownian motion are analysed, with transitions arising from a Markovian stochastic process and scale-free intermittent processes. We obtain simple formulas for classical statistics of the processes, namely the mean squared displacement and the power spectral density. Further, a method to identify switching fractional Brownian motion based on the distribution of displacements is described. A validation of the model is given for experimental measurements of the motion of quantum dots in the cytoplasm of live mammalian cells that were obtained by single-particle tracking.