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.
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Reflected Fractional Brownian Motion in 3D-Brain Shapes: Insights into the Distribution of Serotonergic Axons
The immediate neighborhood of virtually every brain neuron contains thin, meandering axons that release serotonin (5-HT). These axons, also referred to as serotonergic fibers, are present in nearly all studied nervous systems (both vertebrate and invertebrate) and appear to be a key component of biological neural networks. In the mammalian brain, they create dense meshworks that are macroscopically described by densities. It is not known how these densities arise from the trajectories of individual fibers, each of which resembles a unique random-walk path. This poses interesting theoretical questions, solving which will advance our understanding of brain plasticity and regeneration. For example, serotonin-associated psychedelics have recently been shown to promote global brain integration in depression [1], and serotonergic fibers are nearly unique in their ability to robustly regenerate in the adult mammalian brain [2].
We have recently introduced a conceptual framework that treats the serotonergic axons as “stochastic axons.” Stochastic axons are different from axons that support point-to-point connectivity (often studied with graph-theoretical methods) and require novel theoretical approaches. We have shown that serotonergic axons can be potentially modeled as paths of fractional Brownian motion (FBM) in the superdiffusive regime (with the Hurst exponent H > 0.5). Our supercomputing simulations demonstrate that particles driven by reflected FBM (rFBM) accumulate at the border enclosing the shape [3]. Likewise, serotonergic fibers tend to accumulate at the border of neural tissue, in addition to their general similarity to simulated FBM paths [4].
This work expands our previous simulations in 2D-brain-like shapes by considering the full 3D-geometry of the brain. This transition is not trivial and cannot be reduced to independent 2D-sections because increments of FBM trajectories exhibit long-range correlation. Supercomputing simulations of rFMB (H > 0.5) were performed in the reconstructed 3D-geometry of a mouse brain at embryonic day 17 (serotonergic fibers are already well developed at this age and begin to invade the cortical plate). The obtained results were compared to the actual distribution of fibers in the mouse brain. In addition, we obtained preliminary results by simulating rFBM with a region-dependent H. This next step in complexity presents challenges (e.g., it can be highly sensitive to mathematical specifications), but it is necessary for the predictive modeling of interior fiber densities in heterogenous brain tissue.
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- NSF-PAR ID:
- 10382028
- Date Published:
- Journal Name:
- Organization for Computational Neurosciences (OCNS)
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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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.more » « less
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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.more » « less
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