Self organised small-world networks in brain architecture

By Mirjam Bruinsma, Maarten van den Ende, Nigel van Herwijnen and Thomas van der Veen

Overview

We model one of the most complex systems: the human brain

Great number of neurons and enormous number of dendrons

Overview

Self organised network:

  • Mashed rat brains
  • Kept alive in petridishes
  • Small-world networks grow naturally

Model

Donald Olding Hebb (1904 - 1985)

Model

Ivan Petrovitsj Pavlov (1849 - 1936)

Model

Random graph:

  • Number of nodes
  • Relatively high number of edges
  • Assign random activation value

    $$ [-1, 1] $$

Model

Iteratively update graph:

  1. Apply logistic map
  2. Randomly select pivot
  3. Find candidate and outcast
  4. Rewire if possible

Model

1 - Apply logistic map

  • Each nodes activation value is updated according to the equation below

    $$ x^i_{n+1} = (1- \epsilon) \cdot f(x^j_n)+\frac{\epsilon}{M_i}\sum_{j \in B(i)}f(x^j_n) $$

    Where

    $$ f(x) = 1 - a x^2 $$

    Epsilon is the coupling parameter

    a is a constant in the logistic map

Model

1 - Apply logistic map

In [3]:
plt.scatter(a,e)
plt.title("Parameter space")
plt.xlabel("a")
plt.ylabel("epsilon")
plt.grid()
plt.show()

Model

2 - Randomly select pivot


Randomly select one node in the network

Model

3 - Find candidate and outcast


  • Find candidate node

From all other units, select the one whose activation value is closest to the pivot’s.

  • Find outcast node

From all connected nodes, select the one whose activation value is furthest from the pivot's.

Model

4 - Rewire if possible


  • Remove edge from pivot to outcast and place edge from pivot to candidate
  • Expceptions:
    • No outcast available
    • Pivot and candidate are already connected
In [4]:
init_notebook_mode(connected=True)
figure = build_fig()          
iplot(figure, validate = False)

building figure

Metrics and Concepts


  • Clustering Coefficient (CC)
  • Phase transition
  • Phase diagram

Results

CC - #edges

Results

CC - #edges

Results

CC - #edges

Results

CC - iters

In [5]:
plotly.tools.embed("https://plot.ly/~mirbruin/0/#/")
Out[5]:

Final network

In [6]:
plotly.tools.embed("https://plot.ly/~maarten.w.j/672")
Out[6]:

Conclusion

  • Self organizing behaviour
  • Effect of parameter values
  • Possible phase transitions and different regimes

Further research

  • Explore phase diagram
  • Explore other metrics
    • Characterictic path length
    • Modularity
  • Explore larger structures
  • Explore other mappings