The geometry of spontaneous spiking in neuronal networks

Transcription

1 The geometry of spontaneous spiking in neuronal networks Georgi Medvedev Drexel University Mathemacs Colloquium Rensselaer Polytechnic Instute March 4, 203 Work par=ally supported by NSF through grant DMS 09367

2 Network dynamics = local dynamics+coupling 7/3/3 2

3 LaNce Dynamical Systems u t = Δu + Fu ' u i, j = γ u i, j + u i+, j + u i, j + u i, j+ 4u i, j + Fu i, j γ = h 2 Coupled map lances Kaneko 80 s 7/3/3 3

4 The Adjacency Matrix Adjacency matrix A =! # # # # " $ & & & & % 7/3/3 4

5 The pixel picture y x Lovasz, Large networks and graph limits, 203 5

6 The Kuramoto model φ i t = ω + n i+k j=i k sin φ j φ i, i [n] cf. Kuramoto, ω = 0, φ i+ φ i = const q- twisted states: φ i = qi + j mod n, q, j Z Wiley, Strogatz, Girvan 2006, Girnyk, Hasler, Maistrenko 202 7/3/3 6

7 The con=nuum limit φ i t = ω + n i+k j=i k sin φ j φ i, i [n] t φ x,t = ω + W x, ysin φy,t φx,t dy Wiley, Strogatz, Girvan 2006, Girnyk, Hasler, Maistrenko 202, Grinshpan, Kaliuzhnyi- Verbovetski, Medvedev 203 7/3/3 7

8 Chimera states t φ x,t = ω + Gx ysin φy,t φx,t+α dy Gx = exp{ κ x } Kuramoto, Babogtokh 2002, Abrams, Strogatz /3/3 8

9 Power grids and consensus protocols Power system model Dorfler, Bullo SICON 202 M i φ i + Di φ i = ω i + a ij sin φ j φ i δ ij, i [n] j i Consensus protocol Medvedev, SICON 202 φ i = a ij t φ j φ i +ξ i t, i [n] j i 7/3/3 9

10 Ac=vity paberns in Locus Coeruleus neurons correlate with cogni=ve performance Usher et al, Science, 999; Brown et al, J. Comp. Neurosci., /3/3 0

11 The conductance- based model of the LC neuron G.S. Medvedev and S. Zhuravytska, J. Nonlinear Sci., 202 7/3/3

12 7/3/3 2 N i v n v n n w v v d g n v I Cv i i i i i i j N i ij i i ion i,2,...,,, = = + + = = τ σ The coupled network of LC neurons

13 Incorpora=ng connec=vity 7/3/ i i i i i v v g v v g v + + = +,...,,,... 2 N v v v V glv V = =

14 The graph Laplacian Images downloaded from simonsfounda=on.org/ ij Lv = b U = x i x j 2 = x T Lx cf. Erica Klarrich, Network solu=ons, simonsfounda=on.org/ 7/3/3 4

15 Algebraic graph theory: Define matrices associated with graphs. Understand graphs using methods of linear algebra and matrix analysis. Biggs, Algebraic graph theory Spectral graph theory: Relate structural proper=es of graphs to the eigenvalues of matrices associated with the graphs. Chung, Spectral graph theory 7/3/3 5

16 7/3/ L = D A = " # $ $ $ $ % & ' ' ' ' " # $ $ $ $ % & ' ' ' ' = " # $ $ $ $ % & ' ' ' ' = H Alterna=vely, using coboundary matrix, Graph Laplacian H H L T = Adjacency matrix A = ! " # # # # $ % & & & & Matrices

17 Eigenvalues of L = H T H = λ λ 2... λ n λ = 0, =,,, Algebraic connec=vity AC cf. Fiedler,973 λ 2 > 0 Large AC implies good connec=vity Example: λ 2 P n n 2 0, n 7/3/3 7

18 Random Walk on G X 0, X, X 2,..., X i,... The rate of convergence to the sta=onary distribu=on is determined by λ 2 Example: λ 2 P n n 2 0, n 7/3/3 8

19 Expanders are sparse well- connected graphs Hoory, Linial, Wigderson, Bull. Am. Math. Soc., 2006 {G n } : λ 2 G n α > 0 n Pinsker 973 Margulis 988 Lubotzky, Phillips, and Sarnak 988 Reingold, Vadhan, and Wigderson /3/3 9

20 Random graphs are good expanders Random d- regular graphs: P{λ 2 G n d 2 d ε} = o n ε > 0 Friedman, 2008 λ 2 G n d 2 d + o n The Alon- Boppana inequality 7/3/3 20

21 Effec=ve resistance R ij Image downloaded from simonsfounda=on.org/ Total effec=ve resistance RG = n n i= 2 λ i L RG = i< j R ij Gosh, Boyd, Saberi, Minimizing effec=ve resistance of a graph, SIAM Review, 2008

22 Weak coupling => the rate slows down Intermediate coupling => clusters and waves Strong coupling => synchroniza=on 7/3/3 22

23 The firing rate plot 7/3/3 23

24 The center manifold reduc=on The original system The reduced system z = z 2 + σ 2 w Rewrite as a gradient system, z = U 'z+ σ 2 w Uz = z 3 z3 7/3/3 24

25 The reduced coupled system! # # # # " z z 2... z n $! & # & # & = # # & # % " z 2 z z 2 n $ & & & γlz +σw & & % z = f z γ Lz +σ W, z R n 7/3/3 25

26 The energy z = Uz+ σ 2 W, z R n Uz = γ 2 zt Lz + Φz Excitable energy local Φz = N Fz i, Fξ = 2 3 +ξ 3 ξ 3 i= Energy of connec=ons global γ 2 zt Lz = γ 2 Hz, Hz 7/3/3 Image downloaded from simonsfounda=on.org/

27 The =me and loca=on of the first exit z = Uz+σ 2 w Large devia=on es=mates U * :=Uz * = min z D Uz A lim σ 0 P zτ z * < δ = B τ exp σ 2 U * Freidlin, Wentzell, Random Perturba=ons of Dynamical Systems 7/3/3 27

28 The exit problem for the coupled system z = Uz+ σ 2 W, z R n U * :=Uz * = min z D Uz A lim σ 0 P zτ z * < δ = Bτ exp σ 2 U * 7/3/3 28

29 The minimiza=on problem z = Uz+ σ 2 W U γ z min z D 7/3/3 29

30 The minimiza=on problem: weak coupling N γ 2 3 U z = Hz, Hz + F z, F ξ = + ξ ξ 2 i i= 3 3 Minima of Φ z: ξ 2 ξ 3 ξ =,, =,, =,, 7/3/3 30

31 The weak coupling regime Use the Implicit Func=on Theorem φ0 = n, d dγ φ i γ γ=0 = l i 7/3/3 3

32 The weak coupling regime => firing rate min U γ z D i = 4 3 +γ degv i +Oγ 2 # Eτ expσ 2 4 % $ 3 +γ volg & ' 7/3/3 32

33 For weak coupling the connec=vity is not important 7/3/3 33

34 Strong coupling => synchroniza=on Uz = γ 2 Hz, Hz + Φz Hz, Hz min ker H=span{,,..,} γ 2 λ 2 L n = span{,,...,} = minu γ z D 7/3/3 34

35 For strong coupling the connec=vity is important 7/3/3 35

36 Intermediate coupling => clusters 7/3/3 36

37 Transi=on to synchrony 7/3/3 37

38 The geometry of synchroniza=on z = f z γlz, z R n n n = span{,,...,} R n z = η,ζ R R n, η = n n T z, ξ= Hz 7/3/3 38

39 Slow- fast system ξ = γ 2 ˆLξ +σ H W +... n η = f η+ σ n w n Reduced matrix ˆL = HL H + R n n, ˆL > 0, λ ˆL = λ 2 L Medvedev, SICON 202 7/3/3 39

40 Fast subsystem: Synchroniza=on ξ = γ 2 ˆLξ +σ H W Expected value Covariance matrix mt = Eξt Vt = E[ξt mtξt mt T ] Synchronizability Stochas=c stability robustness to noise m = γ 2 ˆLm V = γ 2 ˆLV +V ˆL+σ 2 I n 7/3/3 40

41 Fast subsystem: Synchroniza=on Expected value Covariance matrix mt = Eξt Vt = E[ξt mtξt mt T ] Synchronizability Stochas=c stability mt = m0exp{-γλ 2 Gt} 0 Tr Vt σ 2 2γ κg κg = Tr{ ˆL H H T } cycle subspace of G, effec=ve resistance of G cf. Medvedev, SICON 202 7/3/3 4

42 Slow subsystem: escape from the poten=al well ξ = γ 2 ˆLξ +σ H W +... n η = f η+ σ n w n # τ exp 2nΔU & %, ΔU =U U0 = 4 $ σ 2 ' 3 7/3/3 42

43 Mixed- mode oscilla=ons 6 2 n vt 0 Frequency, % v t Number of small oscillations M., Hitczenko, The Poincare map of randomly perturbed periodic mo=on, J. Nonlin. Sci., accepted

44 Why beta cells exhibit spontaneous spiking in isola=on and regular burs=ng in electrically coupled islets of Langerhans? One cell Network Scenario A Scenario B Sherman, Rinzel, Sherman 88; Sherman, Rinzel 9 M., Zhuravytska, Biol. Cyberne=cs, 202

45 Conclusions The varia=onal interpreta=on of pabern forma=on in coupled networks Bifurca=ons in stochas=c system Slow- fast analysis of synchroniza=on: role of network topology 7/3/3 45

46 Related papers G.S. Medvedev and S. Zhuravytska, The geometry of spontaneous spiking in neuronal networks, J. Nonlinear Sci., 202 G.S. Medvedev and S. Zhuravytska, Shaping burs=ng by electrical coupling and noise, Biol. Cyberne=cs, vol. 06, pp , 202. G.S. Medvedev, Stochas=c stability of con=nuous =me consensus protocols, SIAM J. Control Op=m., Vol. 50, No. 4, pp , /3/3 46