Network-driven synchronisation of phase-coupled oscillators

Transcription

1 Network-driven synchronisation of phase-coupled oscillators University of Exeter CEMPS Exeter, 4 th Feb 2014

2 Outline Motivation The Classical Kuramoto Model

3 A normal EEG Fp1 AVG Fp2 AVG F3 AVG F4 AVG C3 AVG C4 AVG P3 AVG P4 AVG O1 AVG O2 AVG F7 AVG F8 AVG T3 AVG T4 AVG T5 AVG T6 AVG Fz AVG Cz AVG Pz AVG Scale

4 EEG recording of a generalised seizure Fp1 Fp2 F7 F3 Fz F4 F8 A1 T3 C3 Cz Electrode C4 T4 A2 T5 P3 Pz P4 T6 O1 O2 Poly1 Poly2 Poly Time (ms) x 10 4

5 EEG recording of a generalised seizure Fp1 Fp2 F7 F3 Fz F4 F8 A1 T3 C3 Cz Electrode C4 T4 A2 T5 P3 Pz P4 T6 O1 O2 Poly1 Poly2 Poly Time (ms) x 10 4

6 Motivation Epilepsy - A Network Problem!

7 phase and moto net- 5). tatively in isotropic ted by the anatomializing on the ampliity in the hem to the nnectivity the funcronization entioned nizability chard and scopically tural cons of averes appears tradition ilson and an, 1975; 8). Neural a rhythm, transition t al., 1974; 999; David of two or dam et al., 01; David networks al., 2010). ass model scriptions ral models token, the where N e and N i are the numbers c of excitatory and inhibitory Motivation neurons. By this averaging, E and The I represent Classical the mean Kuramoto firing rates Model of dt N l = 1 d h EE wk = vk + sin( wl wk) all excitatory and inhibitory neurons, Networks respectively, of Kuramoto of the neural Models For the sake of legibility, however, we here refer to (2) also as population in question, i.e., that at node Application k. to EEG Data the Kuramoto model. Within that population, P every E k k neuron receives Cinput ki E I from all As mentioned above, the effect of increasing h in the isotropic other neurons of the population. Furthermore, the excitatory units case is to increase the phase synchrony amongst the oscillators. individually receive constant external inputs p n, whose average is 1 N Suppose the coupling is weak (i.e., smaller than the critical value, e given by P = N p e n = 1 n. The c EI sum of all cinputs IE is (instantaneously) or h h integrated in time when it exceeds some threshold u. This thresholding is realized by means of a sigmoid function S. Without loss of c c, then the oscillators phases disperse, whereas for strong coupling h h the oscillators become synchronous, i.e., the phases are locked at fixed differences. In the intermediate case h h generality we here chose S[x] = (1 + e x ) 1 ; we note that, in general, c, clusters of synchronous oscillators may emerge. However, many other I k the thresholds may differ between excitatory and inhibitory units 1. oscillators, whose natural ORIGINAL frequencies RESEARCH ARTICLE are at the tails of the distribution, are not locked into a cluster. doi: /fninf In other words, as h increases, the In consequence, the mean NEUROINFORMATICS firing rates of the neural populations can published: 15 July 2011 be cast in the following dynamical system c II interaction functions overcome the dispersion of natural frequencies d On the influence of amplitude on the connectivity between dt E E S a c E phases c I P v n resulting in a transition from incoherence, to partial and then = + E( EE IE ue + ) FIGURE 1 Network of Wilson Cowan models. At each node k a neural full synchronization (Acebron et al., 2005; Breakspear et al., 2010). population containing excitatory and inhibitory units (E d Andreas Daffertshofer* and Bernadette C. M. van Wijk dt I I S a c E c I k and I k, respectively) = yields + self-sustained I( oscillations. EI II uother I) nodes are connected to the excitatory Linking neural mass models to phase oscillators unit by means of ΣC kl E l. Note that this (mean-field) coupling is scaled by a Research Institute MOVE, VU University Amsterdam, Amsterdam, Netherlands When deriving the Kuramoto network from the Wilson Cowan oscillator network, the major ingredient is to average every oscillator over scalar h see Eq. 1 for details. The characteristics of this dynamical system range from a mere Edited by: In recent studies, functional connectivities have been reported to display characteristics of fixed-point relaxation to Olaf limit Sporns, Indiana cycle University, oscillations USA (self-sustained one cycle when assuming that its amplitude and phase change slowly complex networks that have been suggested concur with those of the underlying structural, oscillations) depending on Reviewed parameter by: settings (Wilson i.e., anatomical, and Cowan, as compared to the oscillator s frequency. That is, time-dependent oscillators. The Kuramoto networks. Do functional networks always agree with structural ones? In all Joana R. B. model Cabral, Universitat and its Pompeu link to the here-discussed 1972), in particular on the Fabra, choice Spain of the external input generality, P. That this input question can amplitude be answered and with phase no : are for fixed, instance, the a system fully synchronized is integrated state over one period network of Wilson Cowan models will be briefly sketched in the Juergen Kurths, Humboldt Universität, would imply isotropic homogeneous functional connections irrespective of the real underlying is usually chosen at random. In the current study, we restrict all to remove all harmonic oscillations, and, subsequently, amplitude and following two sub-sections. Germany structure. A proper inference of structure from function and vice versa requires more than a parameter values to the regime *Correspondence: within which the dynamics sole focus on displays phase are again considered to be time-dependent (Guckenheimer and phase synchronization. We show that functional connectivity critically depends Andreas Daffertshofer, Research self-sustained oscillations; see Appendix. on amplitude variations, which Holmes, implies 1990) that, in general, this procedure phase patterns is also should referred be analyzed to as a combination of Kuramoto network of phase Institute MOVE, oscillators VU University To combine Wilson Cowan Amsterdam, models Van der Boechorststraat in a network, 9, conjunction different with populations are now connected a.daffertshofer@vu.nl via their excitatory units of phase by oscillators. virtue of mation (Haken, 1974). As shown in more detail in the Appendix, the the corresponding rotating amplitude. wave approximation We discuss this issue and slowly by comparing varying the amplitude phase approxi- The collective behavior 1081 of BT Amsterdam, a network Netherlands. of oscillators, synchronization whose states patterns are of interconnected Wilson Cowan models vis-à-vis Kuramoto networks captured by a single scalar phase w For the interconnected Wilson Cowan models we derive analytically how the sum of all E l in the dynamics of E k each, can, in first approximation, k (see Figure 1). connectivity The dynamics phase dynamics of the system (1) can in this way be approximated as between phases explicitly depends on the generating oscillators amplitudes. be represented by the set of N coupled differential Eq. at node k then becomes In consequence, the link between neurophysiological N studies and computational models d h always requires the incorporation of the amplitude dynamics. Supplementing synchronization Ncharacteristics by amplitude patterns, as captured by, e.g., spectral power in M/EEG recordings, d will certainly aid our understanding of the relation between structural and functional organizations dt E E S a c E c I P h N CE dt N CaS Rl wk = vk + ( 0) N kl E x Ek, sin wl wk d h ( ) w 2 l = 1 Rk k = vk + Dkl sin( wl wk) (2) dt k = k + NE k k E + k + kl l i = 1 EE IE u in neural networks at large. l N h = 1 Keywords: connectivity, phase synchronization, Kuramoto N Ca 3 ( 0) kl ES 16 network, Wilson Cowan x Ek, model, RkRl( ( cee + 3c IE ) sin( wl wk) amplitude dependency d That is, the k-th oscillator, with natural frequency v l = 1 k, adjusts its Iphase k = Iaccording k + S ai( ceiek ciiik I) dt to input from other u (1) INTRODUCTION oscillators through a pair-wise + 2c discriminating between frequency and phase-locked activity EEcIE cos( w (Baker l wk)) phase interaction function The interplay sin(w between l wstructural k ). The and connectivity functional brain matrix = networks has et al., 1999; Mima and Hallett, 1999; Salinas and Sejnowski, 2001; In D kl words, is again all Wilson Cowan scaled by become overall a oscillators, popular coupling topic of located research strength, at in recent nodes h. years. As l in will It the is currently be with Fries, S 2005; and Womelsdorf S referring et al., to 2007). the It first is usually and assumed third derivative that of the network drive the change believed of the that firing the topologies rate of structural the excitatory and functional units networks in sigmoid amplitude function or power S. variations The parameter take place on long x ( 0) sketched below, h serves as a bifurcation parameter in that small Ek, time is given scales when by various empirical systems may disagree (Sporns and Kötter, 2004) compared to the phase dynamics and are therefore considered negligible. The coupling that does, or does not, yield synchrony N between E k. The values connectivity of h yield is a given network but systematic by the behavior analyses real-valued tackling that essentially matrix this issue are C kl agrees few that and has far with between. 0 h vanishing diagonal elements, In a combined i.e., Cneural mass and graph theoretical model of electroencephalographic signals, it was found that patterns of functional whether this assumption is valid, and by this, tackle if a sole focus oscillators hence 0exclusively 0 depends on the phase. Here we ask kk = 0. That connectivity matrix ( ) ( ) ( ) xek ae c Ek c Ik ue Pk l is scaled via the overall coupling strength h. It is important to note N CE ( 0 the entirely uncoupled case (i.e., the phases are not synchronized), ), = + + EE IE kl l = 1 whereas h larger than a certain critical value h connectivity are influenced by but c causes the phases not identical to those of on phase really covers all functional characteristics of networks. In that to the synchronize. C connectivity The the frequencies matrix corresponding is here v k structural are always distributed Helmut level identified (Ponten according Schmidt et al., as 2010). the to In this the present ( 0) ( study 0) we describe the dynamics of neural populations Ubiquitousness of the Kuramoto Model

8 The Kuramoto Model The model: θ j = ω j + K N N sin(θ n θ j ), j = {1,..., N} n=1 The order parameter r defines the degree of synchronicity : re iψ = 1 N N e iθn = θ j = ω j + rk sin(ψ θ j ), n=1 Thermodynamic limit (N ), g(ω) = 1 π e (Ω ω)2 : r = π rk ( ) ( )) /2 (rk) 2 (rk) 2 (I 2 e (rk)2 0 + I 1 = F (rk) 2 2

9 The Kuramoto Model Synchronisation sets in at critical coupling strength K c : r, f(rk) K=2 K=1.5 1 r K=1 K= r K C K 3

10 The Kuramoto model

11 Directed Networks of Kuramoto Populations :

12 Directed Networks of Kuramoto Populations : The full model with adjacency matrix Â: θ p j = ω p j + K p N N P sin(θn p θ p j ) + n=1 s=1 A s,p N N sin(θn s θ p j ), n=1 define local order parameters: = r p e iψp = 1 N N e iθp n, n=1 θ p j P = ω p j + r p K p sin(ψ p θ p j ) + r s A s,p sin(ψ s θ p j ). s=1

13 Directed Networks of Kuramoto Populations : under the assumption that ψ p = ψ s p, s we obtain r p = F ( K p r p + ) P A s,p r s, p = {1,..., P}. s=1 linearisation about (r 1,..., r P ) = (0,..., 0) yields ( ) Kp 1 r p + 1 K c K c P m=1 A s,p r s = 0. nontrivial solutions exist if ( ) Â det + ˆK = 0, K p,p = K p 1, K c K c K s,p = 0 if s p

14 Directed Networks of Kuramoto Populations : rewriting  = C ˆρ yields a critical value for the global coupling parameter C: ( ) Cc det ˆρ + ˆK = 0. K c if ˆρ is triangular, ( ) Cc ( ) det ˆρ + ˆK = det ˆK 0 K c

15 2 populations

16 cycles

17 a loop of 3 populations plus one extra connection r 1 = F (r 1 K + r 2 C + r 3 C) r 2 = F (r 2 K + r 1 C) r 3 = F (r 3 K + r 2 C) 1 r K = C 1.8 2

18 a hierarchical network of 3: r 1 = F (r 1 K + r 2 C + r 3 C) r 2 = F (r 2 K) r 3 = F (r 3 K + r 2 C) 1 r K = C 1.8 2

19 Larger Networks

20 Larger Networks Network-driven synchronisation of phase-coupled oscillators

21 Larger Networks Helmut Schmidt

22 Motivation The Classical Kuramoto Model Larger Networks

23 Directed Networks from EEG Data To compute (delayed) cross-correlation between time-series x i and x j we follow Bialonski & Lehnertz (2013): with ρ i,j = max τ ξ(x i, x j )(τ) = { } ξ(x i, x j )(τ), ξ(xi, x i )(0)ξ(x j, x j )(0) { T τ t=1 x i(t + τ)x j (t) if τ 0 T t=1+τ x i(t + τ)x j (t) if τ < 0. Fp1 AVG Fp2 AVG F3 AVG F4 AVG C3 AVG C4 AVG P3 AVG P4 AVG O1 AVG O2 AVG F7 AVG F8 AVG T3 AVG T4 AVG T5 AVG T6 AVG Fz AVG Cz AVG Pz AVG Scale

24 Motivation The Classical Kuramoto Model Networks of Kuramoto Models Directed Networks from EEG Data

25 Critical Coupling Constant C c critical coupling constant controls patients ** * 0.2 low high delta theta beta gamma alpha alpha

26 Seizure Onset Zones P 2 P 3 P 1 P 4 P 5 P 6 P 7

27 Seizure Onset Zones