## Abstract

Objective

An important problem in systems neuroscience is the relation between complex structural and functional brain networks. Here we use simulations of a simple dynamic process based upon the susceptible–infected–susceptible (SIS) model of infection dynamics on an empirical structural brain network to investigate the extent to which the functional interactions between any two brain areas depend upon (i) the presence of a direct structural connection; and (ii) the degree product of the two areas in the structural network.

Methods

For the structural brain network, we used a 78 × 78 matrix representing known anatomical connections between brain regions at the level of the AAL atlas (Gong et al., 2009). On this structural network we simulated brain dynamics using a model derived from the study of epidemic processes on networks. Analogous to the SIS model, each vertex/brain region could be in one of two states (inactive/active) with two parameters β and δ determining the transition probabilities. First, the phase transition between the fully inactive and partially active state was investigated as a function of β and δ. Second, the statistical interdependencies between time series of node states were determined (close to and far away from the critical state) with two measures: (i) functional connectivity based upon the correlation coefficient of integrated activation time series; and (ii) effective connectivity based upon conditional co-activation at different time intervals.

Results

We find a phase transition between an inactive and a partially active state for a critical ratio τ = β/δ of the transition rates in agreement with the theory of SIS models. Slightly above the critical threshold, node activity increases with degree, also in line with epidemic theory. The functional, but not the effective connectivity matrix closely resembled the underlying structural matrix. Both functional connectivity and, to a lesser extent, effective connectivity were higher for connected as compared to disconnected (i.e.: not directly connected) nodes. Effective connectivity scaled with the degree product. For functional connectivity, a weaker scaling relation was only observed for disconnected node pairs. For random networks with the same degree distribution as the original structural network, similar patterns were seen, but the scaling exponent was significantly decreased especially for effective connectivity.

Conclusions

Even with a very simple dynamical model it can be shown that functional relations between nodes of a realistic anatomical network display clear patterns if the system is studied near the critical transition. The detailed nature of these patterns depends on the properties of the functional or effective connectivity measure that is used. While the strength of functional interactions between any two nodes clearly depends upon the presence or absence of a direct connection, this study has shown that the degree product of the nodes also plays a large role in explaining interaction strength, especially for disconnected nodes and in combination with an effective connectivity measure. The influence of degree product on node interaction strength probably reflects the presence of large numbers of indirect connections.

An important problem in systems neuroscience is the relation between complex structural and functional brain networks. Here we use simulations of a simple dynamic process based upon the susceptible–infected–susceptible (SIS) model of infection dynamics on an empirical structural brain network to investigate the extent to which the functional interactions between any two brain areas depend upon (i) the presence of a direct structural connection; and (ii) the degree product of the two areas in the structural network.

Methods

For the structural brain network, we used a 78 × 78 matrix representing known anatomical connections between brain regions at the level of the AAL atlas (Gong et al., 2009). On this structural network we simulated brain dynamics using a model derived from the study of epidemic processes on networks. Analogous to the SIS model, each vertex/brain region could be in one of two states (inactive/active) with two parameters β and δ determining the transition probabilities. First, the phase transition between the fully inactive and partially active state was investigated as a function of β and δ. Second, the statistical interdependencies between time series of node states were determined (close to and far away from the critical state) with two measures: (i) functional connectivity based upon the correlation coefficient of integrated activation time series; and (ii) effective connectivity based upon conditional co-activation at different time intervals.

Results

We find a phase transition between an inactive and a partially active state for a critical ratio τ = β/δ of the transition rates in agreement with the theory of SIS models. Slightly above the critical threshold, node activity increases with degree, also in line with epidemic theory. The functional, but not the effective connectivity matrix closely resembled the underlying structural matrix. Both functional connectivity and, to a lesser extent, effective connectivity were higher for connected as compared to disconnected (i.e.: not directly connected) nodes. Effective connectivity scaled with the degree product. For functional connectivity, a weaker scaling relation was only observed for disconnected node pairs. For random networks with the same degree distribution as the original structural network, similar patterns were seen, but the scaling exponent was significantly decreased especially for effective connectivity.

Conclusions

Even with a very simple dynamical model it can be shown that functional relations between nodes of a realistic anatomical network display clear patterns if the system is studied near the critical transition. The detailed nature of these patterns depends on the properties of the functional or effective connectivity measure that is used. While the strength of functional interactions between any two nodes clearly depends upon the presence or absence of a direct connection, this study has shown that the degree product of the nodes also plays a large role in explaining interaction strength, especially for disconnected nodes and in combination with an effective connectivity measure. The influence of degree product on node interaction strength probably reflects the presence of large numbers of indirect connections.

Original language | English |
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Pages (from-to) | 149-160 |

Number of pages | 12 |

Journal | International Journal of Psychophysiology |

Volume | 103 |

DOIs | |

Publication status | Published - 2016 |

## Keywords

- Graph theory
- Brain dynamics
- SIS model
- Phase transition
- Functional connectivity
- Effective connectivity