Collective dynamics in sparse networks

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1 Parma 22 p. Collective dynamics in sparse networks Simona Olmi Istituto dei Sistemi Complessi - CNR - Firenze Istituto Nazionale di Fisica Nucleare - Sezione di Firenze Centro interdipartimentale di Studi sulle Dinamiche Complesse - CSDC - Firenze

2 Parma 22 p. 2 Summary The dynamics of sparse random networks is investigated for Leaky Integrate-and-Fire (LIF) neuron models phase-oscillators that cannot behave chaotically under any forcing Stuart- Landau oscillators units that may become chaotic as a result of a periodic forcing Chaotic maps units that are chaotic by themselves In all cases few tens of random quenched connections are sufficient to substain a nontrivial macroscopic dynamics The microscopic evolution is extensive The underlying dynamical rule is non additive due to the examined topology

3 Parma 22 p. 3 Introduction Consider a composite system described by a joint state x n = x x 2... x n Additivity A many-body physical observable Q(x n ) is said to be additive with respect to the two subsystems with states x m = x x 2... x m and x n m = x m+ x m+2... x n if Q(x n ) = Q(x m ) + Q(x n m ) The definition can be generalized to any partitions of the joint state x n in two or more subsystems. Extensivity A joint observable Q(x n ) is extensive if the ratio Q(x n )/n reaches a constant in the limit n In most cases encountered in physics additivity does imply estensivity but extensivity additivity? H. Touchette, Physica A, 35, 22: 84-88

scales as N 2/3 The mutual interactions between the subsystems vanish in the limit N as N /3 The overall system is additive On the left: Random

4 Parma 22 p. 4 Graph Bipartitioning On the right: Nearest Neighbour A volume in a regular lattice scales as N A surface that divides the system in two subsystems scales as N 2/3 The mutual interactions between the subsystems vanish in the limit N as N /3 The overall system is additive On the left: Random Network We want to identify the two minimally-connected components in a random sparse network of size N and constant connectivity K The number of interconnections between the two subsystems is proportional to N whenever K > 2 log 2 [ W. Liao, PRL, 59, 625 (987)] Therefore the system is non-additive

5 Leaky integrate-and-fire model Linear integration combined with reset = formal spike event Equation for the membrane potential v, with threshold Θ and reset R : v = I v v = I( e t ) + v e t If I > Θ Repetitive Firing, Supra-Threshold If I < Θ Silent Neuron, Below Threshold In networks: at reset a pulse is sent to other neurons Θ α=3 α=3 v F(t) 5 F(t) = α 2 t e -αt time R 2 time Parma 22 p. 5

6 Parma 22 p. 6 Pulse coupled network We investigate a sparse random network, characterized by a connectivity K which remains fixed for increasing system size. Each unit is pulse-coupled as follows v j = I v j + g KX X ǫ ji S(t t (k) i ), j =,..., N K i=,( j) k= with the pulse shape given by S(t) = α 2 texp( αt); the connectivity matrix ǫ ji = if an incoming connection from i to j is present, otherwise (quenched disorder). More formally we can rewrite the dynamics as v j = I v j + g K E j(t), j =,..., N the field E j (t) is due to the (linear) super-position of all the past pulses The field evolution (in between consecutive spikes) is given by Ë j (t) + 2αĖj(t) + α 2 E j (t) = The effect of a pulse emitted at time t is Ė j (t + ) = Ėj(t ) + α2 /N

7 Bifurcation Diagram Two different macroscopic regimes are observed: the Asynchronous State, for small K, characterized by an incoherent dynamics of the neurons in the network the coherent Partially Synchronized regime, above the critical value Kc 9 where the mean field E(t) exhibits collective periodic oscillations A finite and small connectivity suffices to sustain a macroscopic motion 3 Emin, Emax N= N=5 N= 2 K=2 K=2 4 E(t) = P (t) = P 2 PN i= PN i= Ei (t), Pi (t), Pi (t) = E i + αei I =.3, g =.2, α = 9 N N 2 3 E K Parma 22 p. 7

8 Parma 22 p. 8 Lyapunov Analysis The microscopic dynamics, both below and above K c, is characterized by extensive high-dimensional chaos since the spectra of the Lyapunov exponents (LEs) collapse onto one another if they are plotted versus i/n like in chains with nearest-neighbour coupling (see fig. (a)) the Kaplan- Yorke dimension D KY N the convergence occurs also for the largest LEs (see fig. (b)), unlike the massively (globally) connected case where the largest LEs vanish in the limit N λ i.7.2 (a) -.4. i/n.2 λ max.8.7 N (b) 25 5

9 Parma 22 p. 9 Conclusions I Macroscopic dynamics is regular Asynchronous State: fixed point Partial Synchronization: periodic behavior Microscopic dynamics is chaotic the Kaplan- Yorke dimension D KY N In all the examined models, we have observed that Collective oscillations arise upon increasing K Collective motion is generic and robust and coexists with a microscopically chaotic and extensive dynamics The existence of a limit Lyapunov spectrum in regular lattices is the natural consequence of the additivity of the dynamics. In sparse networks the evolution is non-additive, so the extensive nature of the microscopic evolution is due to more subtle properties OPEN PROBLEM S. Luccioli, S. Olmi, A. Politi, A. Torcini, PRL, 22

10 Parma 22 p. Conclusions II What does it happen in a small world arrangement? K = 2N s + k R, where k R is the number of random links per node and N s is the maximal distance of connections between neighbouring sites a lower number of long-range connections may be necessary, although the overall number of links is larger the network structure plays a nontrivial role in determining the number of links that can sustain a macroscopic motion.4 N=4 sparse network N= nearest neighbours σ h.2 K=2N s +k R, N s =2 2 K K c

11 Parma 22 p. Posizioni Marie Curie Il progetto europeo FP7 Marie Curie - Initial Training Network Neural Engineering Transformative Technologies (NETT) che coinvolge 7 Universitá Europee ed Istituti di Ricerca, e aziende quali Partner associati, offre subito 8 posizioni di dottorato e 2 posizioni Post-Doc Per l Italia fa parte del Network l Istituto dei Sistemi Complessi del CNR Coordinatore Alessandro Torcini Tematica: Neuroscienze Computazionali Laurea in Matematica, Fisica o Ingegneria Sedi di Lavoro: Gran Bretagna, Olanda, Portogallo, Spagna Salari Altamente Remunerativi Scadenza: Entro GIUGNO - LUGLIO 22 per maggiori informazioni:

12 Parma 22 p. 2 Extensive Behaviour At variance with mean-field models, E i (t) fluctuates no matter how large is the network, as it is the sum of a finite number of contributions. To determine its variability we calculated the eigenvalues of the covariance matrix C ij C ij = δe i (t)δe j (t) δe i (t) δe j (t), δe i (t) = E i (t) E(t) the eigenvalues collapse onto one another if they are plotted versus i/n confirming the extensivity of microscopic fluctuations since E i (t) is the sum of K contributions, its variance is expected to be of order /K Kµ i 2 Kµ i 2.5 i/n i/n

13 Coupled Logistic Maps I The dynamics on a network of N coupled logistic maps is defined as x n+ (i) = ( g)f(x n (i)) + gh n (i),, j =,..., N with the internal dynamics given by f(x) = ax( x); g is the coupling strength and the local field reads as h n (i) = NX S ij f(x n (j)) K j=.4 σ h.5 N=5 N= N=2 N=4 σ h N /2 σh 2 = h2 n h n 2, h n average field.2 2 K 6.7 h n+ a = 3.9, g =., K c K K c.65.7 h n Parma 22 p. 3

14 Parma 22 p. 4 Coupled Logistic Maps II.4 λ i (a) (b) Kµ i (c) (d) i/n.5 i/n (a) LEs spectra for K = and N = 2 5 (b) LEs spectra for K = 8 and N = 2 5 Covariance exponents spectra for (c) K = and N = ; (d) K = and N =

15 Stuart-Landau Oscillators The dynamics on a network of N Stuart-Landau oscillators is defined as w i = wi ( + ic2 ) wi 2 wi + g( + ic )(Wi wi ) where wi is a complex variable, and Wi =.6 2 = hw 2 i hw i2, σw (c) K =, (d) K = c = 2, c2 = 3, g =.47 K PN j= Sij wj denotes the local field.4 σw (a) W (b) w.2.2. N= N= w.5 λi 2 K K λi (c) (d).2 - N= N=2 N=4-2. i/n 2 N=4 N=8 N= i/n Parma 22 p. 5

16 Parma 22 p. 6 Fully coupled network F(t) 5 F(t) = α 2 t e -αt α=3 α=3 For fully coupled networks the membrane potentials v displays only regular solutions: periodic or quasi-periodic 2 time Depending on the shape of the pulse (value of α) : Excitatory Coupling - g > Low α Splay State Larger α Partially Synchronized State α Fully Synchronized State Inhibitory Coupling - g < Low α Fully Synchronized State Larger α Several Synchronized Clusters α Splay State

17 Parma 22 p. 7 Splay State Splay States are collective solutions emerging in Homogeneous Networks of N neurons the dynamics of each neuron is periodic the field E(t) is constant (fixed point) the interspike time interval (ISI) of each neuron is T the ISI of the network is T/N - constant firing rate the dynamics of the network is Asynchronous v k.4.2 Neuron Index time time Abbott - van Vreeswiijk, PRE (993) -- Zillmer et al. PRE (27)

18 Partially Synchronized State α 2 5 Partial Synchronization 5 Splay State g Partial Synchronization is a collective dynamics emerging in Excitatory Homogeneous Networks for sufficiently narrow pulses the dynamics of each neuron is quasi periodic - two frequencies the firing rate of the network and the field E(t) are periodic the quasi-periodic motions of the single neurons are arranged (quasi-synchronized) in such a way to give rise to a collective periodic field E(t) van Vreeswiijk, PRE (996) - Mohanty, Politi EPL (26) This peculiar collective behaviour has been recently discovered by Rosenblum and Pikovsky PRL (27) in a system of nonlinearly coupled oscillators Parma 22 p. 8

Stability of the splay state in pulse-coupled networks

Krakow August 2013 p. Stability of the splay state in pulse-coupled networks S. Olmi, A. Politi, and A. Torcini http://neuro.fi.isc.cnr.it/ Istituto dei Sistemi Complessi - CNR - Firenze Istituto Nazionale