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Reaction-diffusion models in weighted and directed connectomes [1]

['Oliver Schmitt', 'Medical School Hamburg', 'University Of Applied Sciences', 'Hamburg', 'University Of Rostock', 'Department Of Anatomy', 'Rostock', 'Christian Nitzsche', 'Peter Eipert', 'Vishnu Prathapan']

Date: 2022-11 

GM and MM in an embedded pathway of a subconnectome

The central pathways of mechanosensitivity originate from the central processes of pseudounipolar neurons in the dorsal root ganglia outside the spinal cord. The connections of the first cervical segments of this pathways were filtered from the complete connectome and both sides of the central nervous system were selected to compute the weighted, directed and bilateral adjacency matrix (Fig 1). These segments manifest particularly severe changes in multiple sclerosis [90, 91]. A symmetric graph representation of this matrix is shown in Fig 2. The source of the mechanosensitive pathway starts in the peripheral nervous system from the mechanoreceptors of the subepidermal layers of the skin or from internal organs. In multiple sclerosis, the mechanosensitive projection exhibits particularly pronounced alterations at specific sites in the spinal cord, primarily causing common neurological symptoms [92]. We consider the central projection from the dorsal root ganglion (first neuron) to the ipsilateral cuneate nucleus (second neuron) then to the right ventrolateral thalamic nucleus (third neuron) with termination in the right somatosensory cortex (terminal neuron). This mechanosensitive pathway is embedded in the bilateral network shown in Fig 2. The first three cervical spinal cord segments were used in the RD modeling.

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TIFF original image Download: Fig 1. The weighted adjacency matrix of a bilateral mechanosensory subconnectome. The weighted adjacency matrix of the spinal cord, brainstem, diencephalic and cortical connectivity of the mechanosensory pathways. The last character of the area abbreviation indicates the side of the hemisphere: L: left hemisphere, R: right hemisphere. AGl: Lateral agranular prefrontal cortex, AGm: Medial agranular prefrontal cortex, CERC: Cerebellar cortex, Cu: Cuneate nucleus, DCeN: Cerebellar nuclei, DRGC1: Dorsal root ganglion of cervical segment 1, DRGC2: Dorsal root ganglion of cervical segment 2, DRGC3: Dorsal root ganglion of cervical segment 3, Gr: Gracile nucleus principal part, ILN: Intralaminar nuclei, IO: Inferior olive, LTNG: Lateral thalamic nuclear group, mPFC: Medial prefrontal cortex, Pn: Pontine nuclei, PTG: Posterior group, S1: Primary somatosensory cortex, S2: Secondary somatosensory cortex, VL: Ventrolateral thalamic nucleus, VNT: Ventral thalamus, VPL: Ventral posterolateral thalamic nucleus. https://doi.org/10.1371/journal.pcbi.1010507.g001

Functionally similar regions can be reconstructed based on their dynamic properties in both the GM and MM models. By comparing the coherent dynamics between MM and GM processes, we want to find out to what extent these two models produce similar results in the same connectome. If similar dynamical behavior of functionally similar regions is obtained in different models, this indicates the reproducibility of a result by another model and thus a certain stability of the dynamics independent of a specific model. Furthermore, we can consider to what extent the similar results of both models can be explained by the specific connectivity rather than by minor changes in parameter settings of the models. We expect that functionally intensive or densely interconnected areas of somatosensory and somatomotor cortical areas will exhibit similar diffusion dynamics and therefore form greater synchronization of concentrations outputs of diffusion functions. This can be tested by comparing pairs of regions with large coherence using a cross-correlation analysis. The somatosensory regions constitute a set of nodes with common dynamical behavior. A GM and a MM process was applied to the mechanosensory subconnectome. The cross-correlation matrix of co-activations has been determined and analyzed by spectral clustering to obtain groups of regions that share similar synchronous behavior. The left and right hemispheric primary and secondary somatosensory regions build such a group of regions in both RD models (Figs 13 and 14). Moreover, the MM model allows a separation of primary and secondary motoric regions as well as primary and secondary somatosensory regions (Fig 14). PPT PowerPoint slide

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TIFF original image Download: Fig 13. Result of clustering the cross-correlation matrix of a GM process. The somatosensory regions constitute a cluster (green rectangle). https://doi.org/10.1371/journal.pcbi.1010507.g013 PPT PowerPoint slide

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TIFF original image Download: Fig 14. Result of clustering the cross-correlation matrix of a MM process. The somatosensory regions are contained in the same cluster like in the GM model. https://doi.org/10.1371/journal.pcbi.1010507.g014

The Wilson Cowan model, like the GM and MM models, leads to stable oscillations with coherent patterns. The neural mass model of Wilson and Cowan (WC) [98–100] was used (“WC-simulation” in neuroVIISAS [101]) to obtain an visual impression of the dynamics (Fig BO in S1 Text) which results from the same network as used for the GM and MM models (Figs 1 and 2). We used the parameters for a limit cycle oscillation of a single Wilson-Cowan oscillator: a E : 1.2, a I : 2.0, c EE : 5.0, c II : 1.0, c IE : 6.0, c EI : 10.0, θE: 2.0, θI: 3.5, η: 20.0, P: 0.25, E 0 : 0.1, I 0 : 0.1, time steps: 1000, time step: 0.1. After about 100 steps the WC model gives rise to stable oscillation within the interconnected regions. The similarity of functions is largest within the peaks of oscillations. The Kuramoto index over all regions displays a regular and stable course. The DRG regions without input connections show a small initialization peak and after about 100 steps they show, as expected, a flat line. Principally, the WC model generates strongly regular oscillations which exhibit some differences when comparing with GM and MM models. The MM model produces much more irregular functions of concentrations with an obvious lower synchronized behavior as can be seen in the smaller Kuramoto indices. The GM model generates more regular oscillations with damping of amplitudes.

The GM and MM models produce different dynamics in degree preserving surrogate networks. The same number of nodes and connections were used by generating Erdös-Rényi (uniform distributed edges) [102], Watts-Strogatz (small-world) [103], Barabasi-Albert (scale-free) [104], Ozik-Hunt-Ott (small-world) [105], rewiring [106], Klemm-Eguiluz (growing scale-free) [107] and multifractal (cluster coefficient) [108] randomized networks to investigate the effects of structural changes of a network to GM and MM models. The GM model is able to produce regular and stable oscillatory functions within nearly all random networks with the exception of the rewiring network (Fig BH in S1 Text). The cross-correlation matrices display large interregional correlations of function similarities. In the case of the rewiring network the Kuramoto index shows strong changes and functions in early stages of the iterations are damped. Interestingly, the Watts-Strogatz model generates a relatively homogeneous network with regard to edge distribution. However, a chessboard like pattern of large and small cross-correlations can be seen in the cross-correlation matrix. The MM model generates much more irregular oscillations in the different random networks. An obvious feature is the slight overlapping low pass oscillation which is missing in the rewiring network. The rewiring network appears to generate more regularity of the oscillating functions. The limit cycles int phase diagram are lying closer together indicating more similarity of waves (Fig BI in S1 Text).

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[1] Url: https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1010507

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