^{*}

Edited by: Pedro Antonio Valdes-Sosa, Joint China Cuba Lab for Frontiers Research in Translational Neurotechnology, Cuba

Reviewed by: Yasser Iturria Medina, Montreal Neurological Institute and Hospital, Canada; Jiaojian Wang, University of Electronic Science and Technology of China, China

*Correspondence: Marieke Musegaas

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

We consider the problem of computing the influence of a neuronal structure in a brain network. Abraham et al. (

In this paper we consider the problem of computing the influence of a neuronal structure in a brain network. The aim of this paper is to improve upon the techniques underlying the methodology proposed by Abraham et al. (

Abraham et al. (

In this paper we introduce an alternative coalitional game which in our opinion has several advantages. First of all, by satisfying superadditivity the game is more intuitive from a game theoretical point of view. Secondly, using the Shapley value of this game as an alternative rating it allows to directly specifying relative influence of neuronal structures. We apply our alternative rating model to the brain networks considered by Kötter et al. (

A brain network is a

^{1}

Note that (

Note that SCC({1, 2, 3},

A ^{A}) corresponding to a brain network (

for all ^{A} is defined by the number of strongly connected components in its induced subgraph.

Alternatively, we define the ^{A}) corresponding to (

for all ^{A} is defined by the number of ordered pairs (

A basic property for coalitional games is superadditivity. A coalitional game is called called

for all ^{A}). This is illustrated in the following example.

^{A}) and (^{A}) is presented below.

{1} | {2} | {3} | {4} | {1, 2} | {1, 3} | {1, 4} | {2, 3} | {2, 4} | {3, 4} | {1, 2, 3} | {1, 2, 4} | {1, 3, 4} | {2, 3, 4} | {1, 2, 3, 4} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

^{A}( |
1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 1 | 1 |

^{A}( |
0 | 0 | 0 | 0 | 2 | 0 | 0 | 1 | 1 | 1 | 4 | 4 | 1 | 6 | 12 |

Note that (^{A}) is not superadditive since, e.g.,

It is readily checked that (^{A}) is superadditive. △

In contrast to the coalitional game (^{A}), we show in the following proposition that the brain network game (^{A}) does satisfy superadditivity.

^{A})

^{A}(^{A}(^{A}(

For showing this, let

The

where _{S} is such that all marginal contributions are weighted adequately to obtain an efficient allocation of the worth of the grand coalition.

In the context of coalitional games corresponding to brain networks, the Shapley value can be interpreted as a measure for the influence of a neuronal structure. Abraham et al. (^{A}) as a rating for the neuronal structures in a brain network. Similarly, we consider the Shapley value Φ(^{A}) as a rating.

^{A}) and (^{A}) of Example 2.2. The Shapley rating Φ(^{A}) is given by^{2}

while the Shapley rating Φ(^{A}) is given by

both determining a ranking (2, 3, 4, 1) or (2, 4, 3, 1) (there is a tie for the second highest ranking). We note that a lower Shapley rating in ^{A} indicates a higher influence in a brain network. On the contrary, a higher Shapley rating in ^{A} indicates a higher influence.

Since a Shapley rating in ^{A} can be negative, as is the case in this example, it is not possible to determine the relative influence of two vertices on the basis of Φ(^{A}). On the other hand, a Shapley rating in ^{A} can not be negative by definition because of superadditivity. Therefore, using Φ(^{A}), we can say that the influence of vertex 2 in the brain network (

A common problem in the analysis of brain networks is the fact that it is not known whether some specific connections (arcs) are present or not [cf. Kötter and Stephan (

We assume that each possible arc (_{ij} ∈ [0, 1]. Clearly, for each present arc we set _{ij} = 1 and for each absent arc we set _{ij} = 0. All probabilities are summarized into a vector ^{p}) in which the worth of a coalition equals the expected (in the probabilistic sense) number of ordered pairs for which there exists a directed path in its induced subgraph. Without providing the exact mathematical formulations the following example illustrates how to explicitly determine the coalitional values in a stochastic brain network game.

_{14} and _{31}, respectively. The complete corresponding vector

( |
(1, 2) | (1, 3) | (1, 4) | (2, 1) | (2, 3) | (2, 4) | (3, 1) | (3, 2) | (3, 4) | (4, 1) | (4, 2) | (4, 3) |
---|---|---|---|---|---|---|---|---|---|---|---|---|

_{ij} |
1 | 0 | _{14} |
1 | 1 | 0 | _{31} |
0 | 1 | 0 | 1 | 0 |

In total there are four possible brain networks. These different brain networks are illustrated above and the corresponding probabilities for those networks are _{14}_{31}, (1 − _{14})_{31}, _{14}(1 − _{31}) and (1 − _{14})(1 − _{31}) for (a), (b), (c), and (d), respectively.

The expected number of ordered pairs for which there exists a directed path in the induced subgraph of coalition {1, 3, 4} is computed by taking the following weighted average

The worth of every coalition is presented below.

{1} | {2} | {3} | {4} | {1, 2} | {1, 3} | {1, 4} | {2, 3} | {2, 4} | {3, 4} | {1, 2, 3} | {1, 2, 4} | {1, 3, 4} | {2, 3, 4} | {1, 2, 3, 4} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

^{p}( |
0 | 0 | 0 | 0 | 2 | _{31} |
_{14} |
1 | 1 | 1 | 4+2_{31} |
4+2_{14} |
1+_{14}+_{31} |
6 | 12 |

The Shapley rating of the game (^{p}) is given by

For example, if

with corresponding ranking (2, 4, 3, 1). △

In this section we apply the Shapley rating based on the brain network game (^{A}) to the two large-scale brain networks considered by Kötter et al. (

The first large-scale brain network is the macaque visual cortex with thirty neuronal structures as illustrated in Figure 1 of Kötter et al. (^{A}) and (^{A}) can be found below in (a) and (b), respectively.

^{A}) |
^{A}) |
||||
---|---|---|---|---|---|

1. | V4 | 1. | V4 | ||

2. | FEF | 2. | FEF | ||

3. | 46 | 3. | Vp | ||

4. | V2 | 4. | V2 | ||

5. | Vp | 5. | 46 |

Note that both ratings agree on the top 5; only with respect to the positions 3 and 5 there are some minor differences.

The entire Shapley rating Φ(^{A}) of the macaque visual cortex can be found in Figure

The second large-scale brain network is the macaque prefrontal cortex with twelve neuronal structures as illustrated in Figure 3A of Kötter et al. (^{p}) and the corresponding Shapley rating Φ(^{p}). The ranking based on the Shapley rating Φ(^{p}) can be found below.

1. | 9 |

2. | 24 |

3. | 12 |

4. | 10 |

5. | 46 |

6. | 25 |

7. | 11 |

8. | 8B |

9. | 13 |

10. | 8A |

11. | 45 |

12. | 14 |

MM is the first author and the corresponding author. MM was present at all processes: the early research process, the programming process and the writing process. BD and PB contributed to the early research process and later on to the process of commenting on the work written by MM.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

^{1}This instance of a brain network is also used in Example 1 in Section 3.1 of Moretti (

^{2}Because of a mistake in the worth of ^{A}({1, 2, 3}), the Shapley value is incorrectly stated by Moretti (