Do Brain Networks Evolve by Maximizing Their Information Flow Capacity?

Abstract

We propose a working hypothesis supported by numerical simulations that brain networks evolve based on the principle of the maximization of their internal information flow capacity. We find that synchronous behavior and capacity of information flow of the evolved networks reproduce well the same behaviors observed in the brain dynamical networks of Caenorhabditis elegans and humans, networks of Hindmarsh-Rose neurons with graphs given by these brain networks. We make a strong case to verify our hypothesis by showing that the neural networks with the closest graph distance to the brain networks of Caenorhabditis elegans and humans are the Hindmarsh-Rose neural networks evolved with coupling strengths that maximize information flow capacity. Surprisingly, we find that global neural synchronization levels decrease during brain evolution, reflecting on an underlying global no Hebbian-like evolution process, which is driven by no Hebbian-like learning behaviors for some of the clusters during evolution, and Hebbian-like learning rules for clusters where neurons increase their synchronization.

Fig 1. Results for the global synchronization and information flow capacity properties for the C.elegans and averaged human BDNs.

Parameter space for the global synchronization ρ in panel (A) and for the upper bound for MIR, I _c, in panel (B) for the C.elegans BDN. Panels (C) and (D) are similar but for the averaged global synchronization ⟨ρ⟩₆ and averaged upper bound for MIR, ⟨I _c⟩₆ for the six human BDNs. Here, g _n is the chemical and g _l the electrical coupling of Eq (2). The regions between the left vertical axes and white dotted lines are replotted in Fig 2 in a finer resolution.

Fig 2. Magnification for the global synchronization and information flow capacity properties of Fig 1.

Panel (A): Parameter space for ρ and panel (B) for I _c for the C.elegans BDN. Panel (C): Similarly for ⟨ρ⟩₆ and panel (D) for the averaged upper bound for MIR, ⟨I _c⟩₆, of the six human BDNs. Here, g _n is the chemical and g _l the electrical coupling of Eq (2).

Fig 3. Results for the global and local synchronization, and information flow capacity properties for the evolved networks of Materials and Methods, Subsection A Model for Brain Network Evolution Based on the Maximization of Information Flow Capacity.

Panel (A): Parameter space for the synchronization ⟨ρ⟩₅. Panel (B): Parameter space for the averaged upper bound for MIR, ⟨mMIR⟩₅, from the five realizations of a network of 60 neurons with six, equally sized, small-world clusters. Case 𝓐 of high synchronization and low information flow capacity is denoted by ▲ and case 𝓑 of low synchronization and high information flow capacity by ●. Panels (C) to (H) are plots for the local synchronization ρ _ci of the six communities of the C.elegans brain network. To be compared with panel (A). Here, g _n is the chemical and g _l the electrical coupling of Eq (2).

Fig 4. Structural properties and normalized Laplacian spectra of the brain networks considered in this study.

Panels (A) to (D): Plot of the pdf of the normalized degrees ${\bar{k}}_i$ (panel (A)), plot of the clustering coefficient $CC({\bar{k}}_i)$ (panel (B)), plot of the average normalized degree $k_{nn}({\bar{k}}_i)$ of the neighbors of nodes with normalized degree ${\bar{k}}_i$ (panel (C)) and the network with its distinct clusters and communities given by different colors for the case 𝓐 of high synchronization and low mMIR of the model for brain network evolution of 60 neurons and 6 clusters. Panels (E) to (H): Same as in panels (A) to (D) but for the case 𝓑 of low synchronization and high mMIR of the same model. Panels (I) to (L): Same as in panels (A) to (D) but for the C.elegans brain network. Panels (M) to (P): Same as in panels (A) to (D) but for the human subject A₁ brain network. In the plots of the second column, we show with blue dashed lines the exponential dependence of CC(${\bar{k}}_i$) to ${\bar{k}}_i$ to guide the eye, where ${\bar{k}}_i$ is the normalized degree. It is defined as ${\bar{k}}_i=k_i/k_{max}$, where k _i is the node degree and k _max is the largest node degree in the network. For the first row, k _max = 7, for the second k _max = 8, for the third k _max = 76 and for the fourth, k _max = 87. Panel (Q): Normalized Laplacian spectra of the brain network of C.elegans (solid curve) and of the averaged over the six human subjects (dashed curve). Panel (R): Similarly for the brain network of case 𝓐 of high synchronization and low information flow capacity (solid curve) and, for case 𝓑 of low synchronization and high information flow capacity (dashed curve) of the model for brain network evolution.

Fig 5. Spectral distances between the network topology of the C.elegans, humans and evolved networks and, global synchronization and information flow capacity properties for an evolved BDN of 120 neurons.

Panel (A): Parameter space for the spectral distance D between the C.elegans brain network and the averaged model for brain network evolution of 60 neurons and 6 small-world clusters and, panel (B) similarly for the spectral distance between the averaged brain network of the six human subjects and the same averaged network created by our brain network evolution. ▲ denotes case 𝓐 and ● case 𝓑, both explained in Materials and Methods, Subsection Brain Network Evolution Promotes Global no Hebbian-like and, Local Hebbian-like and no Hebbian-like Evolution Learning Processes. Both panels to be compared with Fig 3(B). Panel (C): Parameter space for the synchronization ρ and panel (D) for the mMIR of evolved networks using our brain network evolution process with 120 neurons. Panels (E), (F) are similar to (A), (B) for the spectral distance between the C.elegans, averaged brain network of the six human subjects and the model for brain network evolution of 120 neurons. Panels (E), (F) to be compared with panel (D). Here, g _n is the chemical and g _l the electrical coupling of Eq (2).

Fig 6. Brain network evolution promotes Hebbian-like and no Hebbian-like processes and modular organization in the brain dynamical networks.

Panels (A), (B): Global synchronization ρ and mMIR respectively for cases 𝓐 and 𝓑. θs are the slopes of the two dashed blue lines which are fitted to the black curves to demonstrate the decrease of the global neural synchrony during brain network evolution. Panels (C), (D) show how the average pair-wise synchronization ⟨ρ _ij⟩_cl of the clusters c _l, l = 1, …, 6 changes during evolution, for cases 𝓐, 𝓑 respectively. Panels (E), (F) show the pair-wise neural synchronization level ρ _ij of the finally evolved BDNs for the same cases. The horizontal axes (Number of links) correspond to the added links during brain network evolution that lead to the increase of the information flow capacity at each step. The caption of panel (D) for the black curves is the same as in panel (C). These results are for the studied model of brain network evolution with 60 neurons and 6 small-world clusters.