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 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) ] 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

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

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

Dynamics in a simple circuit Single, input-dependent attractor Robust, reproducible dynamics Fast convergence regardless of initial conditions time

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 ~10 4 ·p networks with random connections (~ pN 2 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

Dynamical transition in the ensemble Similar transitions had been observed in models of genetic networks [Glass and Hill, Europhys. Lett ] and ‘balanced’ neural networks [van Vreeswijk and Sompolinsky, Science ]

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 ~2 n 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 p l, we are only interested in motifs with small number of links

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 nodes 11 links 6 nodes 10 links 7 nodes 10 links

Appearance of dynamical motifs in random networks

Appearance of dynamical motifs II

Prediction of the transition in random networks

Appearance of chaotic motifs 1.Under-sampling 2. 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

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 Maass, Natschläger, Markram, Neural Comp., 2002

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)