# Rainer Engelken

It is fascinating to me how neurons coordinate their activity in collective dynamics to encode and process information in the brain. I am exploring chaotic dynamics of neural tissue from a dynamical system perspective. This is rooted in the mathematical field of ergodic theory. Novel numerical and analytical techniques from this field allow us to characterize the phase space organization of large neural circuit models. As it is hard to visualize and imagine a 1000D entangled attractor living in a 10000D phase space, we use small networks for illustration and as a more intuitive approach to our scientific questions. I would love to depict the rich spatio-temporal chaos of large networks, let me know, if you have an idea how to do that!

Here the spike onset rapidness is lower, therefore the strange attractor fills more of the phase space. Each neuron is modelled by a differential equation describing the temporal evolution of its membrane potential. Whenever a neuron generates an action potential, connected neurons receive a pulse. We map the membrane potential to a phase variable and thus obtain a network of pulse-coupled oscillators. The wave is in Coyote Buttes North.

Example code in julia (julialang.org) for rapidness=1 under GNU
GPLv3:

A,ϕ,l=0.<[0 0 0;1 0 1;0 1 0],randn(3),[]

for s=1:7^7

m,j=findmax(ϕ)

ϕ+=(π/2-m)

ϕ[A[:,j]],ϕ[j]=atan(tan(ϕ[A[:,j]])-1),-π/2

j==1 && append!(l,ϕ[2:3])

end

plot(l[1:2:end],l[2:2:end],".k",markersize=.1);axis("off")

This shows a Poincaré section through the chaotic strange attractor of a three neurons network. Whenever one specific neuron fires an action potential, the values of the other two neurons are stored. This results in a 2D section through the 3D phase space. Hot colors indicate high density. We found a dramatic influence of single cell dynamics on the collective dynamics: Increasing the action potential onset rapidness leads to decreasing chaos, Kaplan-Yorke-attractor dimension and Kolmogorov-Sinai-entropy production, which vanishes at a critical value. In large networks, the low attractor dimensionality would not be revealed by a 2D section.

The Fiery Furnace is a labyrinth of narrow sandstone canyons and fins at Arches National Park.