Dynamical network motifs: building blocks of complex dynamics in biological networks

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Dynamical network motifsbuilding blocks of
complex dynamics in biological networks

• Valentin Zhigulin
• Department of Physics, Caltech, and
• Institute for Nonlinear Science, UCSD

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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

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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

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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?

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Simple dynamical model
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Hopfield (attractor) networks

• Symmetric connectivity ? fixed point attractors
• Memories (patterns) are stored in synaptic
  weights
• Current paradigm for the models of working
  memory

There is no spatio-temporal dynamics in the model
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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

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Further simplifications of the model
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Dynamics in a simple circuit

• Single, input-dependent attractor
• Robust, reproducible dynamics
• Fast convergence regardless of initial conditions

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LLE lt 0
LLE 0
LLE gt 0
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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 104p
  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 LLEgt0), periodic (at least one LLE0 and no
  LLEgt0) or fixed point (all LLEslt0) dynamics
• Calculate F (fraction of an ensemble for each
  type of dynamics) as a function of p

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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

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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

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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 2n2
• 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 motifs 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

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Motifs with periodic dynamics (LLE0)
3 nodes, 3 links -
4 nodes, 5 links -
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Motifs with chaotic dynamics (LLEgt0)
5 nodes 9 links
7 nodes 10 links
8 nodes 11 links
6 nodes 10 links
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Appearance of dynamical motifs in random networks
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Appearance of dynamical motifs II
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Prediction of the transition in random networks
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Appearance of chaotic motifs
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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

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2D model of a spatially distributed network
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Dynamics in the 2D model

• Only 1 of networks with ?2 exhibit chaotic
  dynamics
• 99 of networks with ?10 exhibit chaotic
  dynamics
• Calculations show that chaotic motifs are absent
  in networks with local connectivity (?2) and
  present in non-local networks (?10)
• Hence local clustering of connections plays an
  important role in defining dynamical properties
  of a network

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Number of motifs in spatial networks
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Computations in a model of a cortical microcircuit
Maass, Natschläger, Markram, Neural Comp., 2002
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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.
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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)