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Mathematical modeling for resting state functional connectivity of cortical and sub-cortical networks

Mathematical modeling studies for brain activity at rest have mainly focused on networks of cortical areas. However, cortical areas are connected to sub-cortical areas, such as the thalamus and the hippocampus, whose contribution to the resting state activity cannot be ignored. Further, modeling for diseases such as stroke, in which the focus is usually in a sub-cortical region, requires taking sub-cortical networks into consideration.

Here, we use the structural connectivity (SC) matrix of 83 areas (68 cortical, 15 sub-cortical) from 10 subjects, obtained using diffusion spectrum imaging, to model the activity of these regions. Each node of this network is modeled using the dynamical mean field model of Wong and Wang [1, 2] involving a single stochastic differential equation for the population firing rate. In the absence of coupling, each area is in the spontaneous state of low firing rate. Next, we simulate deterministic dynamical equations for the first and second order statistical moments of this stochastic system and obtain simulated functional connectivity (sFC). We then vary three coupling parameters - wc, wsc and wcsc - multiplying the SC strengths between (a) cortical areas, (b) sub-cortical areas and (c) cortical and sub-cortical areas respectively and compare the sFC with the empirical functional connectivity (FC) during resting state. We also calculate the bifurcation diagram of the network as a function of these three parameters.

In this three-dimensional parameter space we obtain the maximum correlation between FC and sFC values for both cortical and sub-cortical areas near the bifurcation surface where the spontaneous state loses stability with higher values of wc and wcsc compared to wsc. Interestingly, in this region, while the sFC for cortical areas correlates well with the corresponding SC values, it does not do so for sub-cortical areas. This suggests that the functional connectivity between sub-cortical areas during the resting state is influenced more by their anatomical connections with the cortical areas than between themselves. It also validates our choice of using three coupling parameters instead of just one global coupling parameter. We aim to use this study to model the resting state functional connectivity in patients of stroke.


  1. Wong K-F, Wang X-J: A recurrent network mechanism of time integration in perceptual decisions. J Neurosci. 2006, 26 (4): 1314-1328. 10.1523/JNEUROSCI.3733-05.2006.

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  2. Deco G, Ponce-Alvarez A, Mantini D, Romani GL, Hagmann P, Corbetta M: Resting-state functional connectivity emerges from structurally and dynamically shaped slow linear fluctuations. submitted for publication

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Correspondence to Mohit H Adhikari.

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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Adhikari, M.H., Griffa, A., Hagmann, P. et al. Mathematical modeling for resting state functional connectivity of cortical and sub-cortical networks. BMC Neurosci 14 (Suppl 1), P101 (2013).

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  • Firing Rate
  • Functional Connectivity
  • Cortical Area
  • Bifurcation Diagram
  • Coupling Parameter