dynamical network motifs: building blocks of complex dynamics in biological networks
DESCRIPTION
Dynamical network motifs: building blocks of complex dynamics in biological networks. Valentin Zhigulin Department of Physics, Caltech, and Institute for Nonlinear Science, UCSD. Spatio-temporal dynamics in biological networks. Periodic oscillations in cell-cycle regulatory network - PowerPoint PPT PresentationTRANSCRIPT
Dynamical network motifs:
building blocks of complex dynamics in biological networks
Valentin ZhigulinDepartment of Physics, Caltech, and
Institute for Nonlinear Science, UCSD
Spatio-temporal dynamics in biological networks Periodic oscillations in cell-cycle regulatory network
Periodic rhythms in the brain
Chaotic neural activity in models of cortical networks
Chaotic dynamics of populations’ sizes in food webs
Chaos in chemical reactions
Challenges for understanding of these dynamics Strong influence of networks’ structure on their dynamic
It induces long term, connectivity-dependent spatio-temporal correlations which present formidable problem for theoretical treatment
These correlations are hard to deal with because connectivity is in general not symmetric, hence dynamics is non-Hamiltonian
Dynamical mean field theory may allow one to solve such a problem in the limit of infinite-size networks [Sompolinsky et al, Phys.Rev.Lett. 61 (1988) 259-262]
However, DMF theory is not applicable to the study of realistic networks with non-uniform connectivity and a relatively small size
Questions
How can we understand the influence of networks’ structure on their dynamics?
Can we predict dynamics in networks from the topology of their connectivity?
Simple dynamical model
k
j
xF
ix
xxFxdt
dx
ik
ij
i
i
ijjij
ikkikii
i
nodeby excitation ofstrength
nodeby inhibition ofstrength
nstimulatio external ofstrength
tynonlineari sigmoid )(
node of activity''
)(
Hopfield (‘attractor’) networks
Symmetric connectivity → fixed point attractors Memories (patterns) are stored in synaptic weights Current paradigm for the models of ‘working memory’
jiijjiij ;
There is no spatio-temporal dynamics in the model
Networks with random connectivity For most biological networks exact connectivity is not
known
As a null hypothesis, let us first consider dynamics in networks with random (non-symmetric) connectivity
Spatio-temporal dynamics is now possible
Depending on connectivity, periodic, chaotic and fixed point attractors can be observed in such networks
Further simplifications of the model
graph theofmatrix adjacency
inhibition ofstrength 5
field excitatorymean 1
tynonlineari sigmoid )(
node of activity''
)(
ij
i
ijjiji
i
A
xF
ix
xAFxdt
dx
Dynamics in a simple circuit
Single, input-dependent attractor Robust, reproducible dynamics Fast convergence regardless of
initial conditions
00.2
0.40.6
0.8
00.2
0.40.6
0.8
00.20.40.6
0.8
00.2
0.40.6
0.8
10 20 30 40 50
0.2
0.4
0.6
0.8
time
20 40 60 80 100
0.2
0.4
0.6
0.8
1
10 20 30 40 50
0.2
0.4
0.6
0.8
1
20 40 60 80 100
0.2
0.4
0.6
0.8
1
LLE ≈ 0
LLE > 0
LLE < 0
Studying dynamics in large random networks Consider a network of N nodes with some probability p of node-to-
node connections
For each p generate an ensemble of ~104·p networks with random connections (~ pN2 links)
Simulate dynamics in each networks for 100 random initial conditions to account for possibility of multiple attractors in the network
In each simulation calculate LLE and thus classify each network as having chaotic (at least one LLE>0), periodic (at least one LLE≈0 and no LLE>0) or fixed point (all LLEs<0) dynamics
Calculate F (fraction of an ensemble for each type of dynamics) as a function of p
0.00 0.02 0.04 0.06 0.08 0.10
0.0
0.2
0.4
0.6
0.8
1.0
chaotic
F
p
periodic
Dynamical transition in the ensemble
Similar transitions had been observed in models of genetic networks [Glass and Hill, Europhys. Lett. 41 599] and ‘balanced’ neural networks [van Vreeswijk and Sompolinsky, Science 274 1724]
0.00 0.02 0.04 0.06 0.08 0.10
0.0
0.2
0.4
0.6
0.8
1.0
dynamicchaotic
F
p
periodic
Hypothesis about the nature of the transition As more and more links are added to the network,
structures with non-trivial dynamics start to form
At first, subnetworks with periodic dynamics and then subnetworks with chaotic dynamics appear
The transition may be interpreted as a proliferation of dynamical motifs – smallest dynamical subnetworks
Testing the hypothesis Strategy:
Identify dynamical motifs - minimal subnetworks with non-trivial dynamics Estimate their abundance in large random networks
Roadblocks: Number of all possible directed networks growth with their size n as ~2n
2
Rest of the network can influence motifs’ dynamics
Simplifications: We can estimate the number of active elements in the rest of the network and make
sure that they do not suppress motif’s dynamics Number of non-isomorphic directed networks grows much slower:
Since the probability to find a motif with l links in a random network is proportional to pl, we are only interested in motifs with small number of links
N 3 4 5 6 7 8
NAll 512. 65536. 3.36 107 6.87 1010 5.63 1014 1.84 1019
NNI 16. 218. 9608. 1.54 106 8.82 108 1.79 1012
Motifs with periodic dynamics (LLE≈0)
3 nodes, 3 links -
4 nodes, 5 links -
Motifs with chaotic dynamics (LLE>0) 5 nodes
9 links
8 nodes11 links
6 nodes10 links
7 nodes10 links
Appearance of dynamical motifs in random networks
Appearance of dynamical motifs II
Prediction of the transition in random networks
0.00 0.02 0.04 0.06 0.08 0.10
0.0
0.2
0.4
0.6
0.8
1.0
periodic motifs
dynamic networks
chaotic networks
F
p
periodic networks
Appearance of chaotic motifs
0.00 0.02 0.04 0.06 0.08 0.10
0.0
0.2
0.4
0.6
0.8
1.0
dynamic networks
chaotic networks
periodic networks
chaotic motifs
periodic motifs
F
p
1. Under-sampling2. Over-counting
How to avoid chaos ?
Dynamics in many real networks are not chaotic
Networks with connectivity that minimizes the number of chaotic motifs would avoid chaos
For example, brains are not wired randomly, but have spatial structure and distance-dependent connectivity
Spatial structure of the network may help to avoid chaotic dynamics
2D model of a spatially distributed network
2.5
5
7.5
10
2.5
5
7.5
10
0
0.2
0.4
0.6
0.8
2.5
5
7.5
10
)/exp(~)( 22 ijij ddp
Dynamics in the 2D model
Only 1% of networks with λ=2 exhibit chaotic dynamics
99% of networks with λ=10 exhibit chaotic dynamics
Calculations show that chaotic motifs are absent in networks with local connectivity (λ=2) and present in non-local networks (λ=10)
Hence local clustering of connections plays an important role in defining dynamical properties of a network
Number of motifs in spatial networks
Computations in a model of a cortical microcircuit
)/exp(~)( 22 ijij ddp
Maass, Natschläger, Markram, Neural Comp., 2002nscomputatio useful2
ncomputatio no chaos, 8
dynamics no 1
Take-home message
Calculations of abundance of dynamical motifs in networks with
different structures allows one to study and control dynamics in
these networks by choosing connectivity that maximizes the
probability of motifs with desirable dynamics and minimizes
probability of motifs with unacceptable dynamics.
This approach can be viewed as one of the ways to solve an
inverse problem of inferring network connectivity from its
dynamics.
Acknowledgements
Misha Rabinovich (INLS UCSD) Gilles Laurent (CNS & Biology, Caltech) Michael Cross (Physics, Caltech)
Ramon Huerta (INLS UCSD) Mitya Chklovskii (Cold Spring Harbor Laboratory) Brendan McKay (Australian National University)