Analysis of a lumped model of neocortex to study epileptiform ac

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1 of a lumped model of neocortex to study epileptiform activity Sid Visser Hil Meijer Stephan van Gils March 21, 2012

2 What is epilepsy? Pathology Neurological disorder, affecting 1% of world population Characterized by increased probability of recurring seizures Seizures The activity of braincells hypersynchronizes Observed as large oscillations

3 What is epilepsy? Pathology Neurological disorder, affecting 1% of world population Characterized by increased probability of recurring seizures Seizures The activity of braincells hypersynchronizes Observed as large oscillations

4 EEG during a seizure

5 Goal of this research Goals Study several models of neural activity in neocortex Identify correspondences between these models Extrapolate results from simpler models to more complex models Results so far A large, detailed model describing activity of individual neurons in both normal and epileptiform states A simple, lumped model that has shown to have some similarity [Visser et al., 2010]

6 Motivation of model Simulation on Youtube

7 Model Introduction Two excitatory populations: ẋ 1 (t) = µ 1 x 1 (t) F 1 (x 1 (t τ i )) + G 1 (x 2 (t τ e )) ẋ 2 (t) = µ 2 x 2 (t) F 2 (x 2 (t τ i )) + G 2 (x 1 (t τ e ))

8 Model Introduction Two excitatory populations: ẋ 1 (t) = µ 1 x 1 (t) F 1 (x 1 (t τ i )) + G 1 (x 2 (t τ e )) ẋ 2 (t) = µ 2 x 2 (t) F 2 (x 2 (t τ i )) + G 2 (x 1 (t τ e ))

9 Model Introduction Two excitatory populations: ẋ 1 (t) = µ 1 x 1 (t) F 1 (x 1 (t τ i )) + G 1 (x 2 (t τ e )) ẋ 2 (t) = µ 2 x 2 (t) F 2 (x 2 (t τ i )) + G 2 (x 1 (t τ e ))

10 Model Introduction Two excitatory populations: ẋ 1 (t) = µ 1 x 1 (t) F 1 (x 1 (t τ i )) + G 1 (x 2 (t τ e )) ẋ 2 (t) = µ 2 x 2 (t) F 2 (x 2 (t τ i )) + G 2 (x 1 (t τ e ))

11 Simplifications Introduction Fixed points and eigenvalues Linear stability Bifurcations First Lyapunov coefficient Make system symmetric: µ 1 = µ 2 := µ F 1 = F 2 := F G 1 = G 2 := G Choose F(x) and G(x) as rescalings of: S = (tanh(x a) tanh( a)) cosh 2 ( a)

12 Time rescaling Introduction Fixed points and eigenvalues Linear stability Bifurcations First Lyapunov coefficient Time is rescaled to non-dimensionalize system: ẋ 1 (t) = x 1 (t) α 1 S(β 1 x 1 (t τ 1 )) + α 2 S(β 2 x 2 (t τ 2 )) ẋ 2 (t) = x 2 (t) α 1 S(β 1 x 2 (t τ 1 )) + α 2 S(β 2 x 1 (t τ 2 )) Introduce vector notation: ẋ(t) = f(x t ), with x t C([ h, 0], R 2 ) and h = max(τ 1, τ 2 ). Main parameters of interest: representing connection strengths between populations.

13 Time rescaling Introduction Fixed points and eigenvalues Linear stability Bifurcations First Lyapunov coefficient Time is rescaled to non-dimensionalize system: ẋ 1 (t) = x 1 (t) α 1 S(β 1 x 1 (t τ 1 )) + α 2 S(β 2 x 2 (t τ 2 )) ẋ 2 (t) = x 2 (t) α 1 S(β 1 x 2 (t τ 1 )) + α 2 S(β 2 x 1 (t τ 2 )) Introduce vector notation: ẋ(t) = f(x t ), with x t C([ h, 0], R 2 ) and h = max(τ 1, τ 2 ). Main parameters of interest: representing connection strengths between populations.

14 Fixed points Introduction Fixed points and eigenvalues Linear stability Bifurcations First Lyapunov coefficient Corollary The origin is always a fixed point of the DDE. Theorem The DDE only has only symmetric fixed points, i.e. f(x) = 0 = x = (x, x ).

15 Linearization Introduction Fixed points and eigenvalues Linear stability Bifurcations First Lyapunov coefficient The DDE is linearized about an equilibrium: in which u 1 (t) = u 1 (t) k 1 u 1 (t τ 1 ) + k 2 u 2 (t τ 2 ), u 2 (t) = u 2 (t) k 1 u 2 (t τ 1 ) + k 2 u 1 (t τ 2 ), k 1 := α 1 β 1 S (β 1 x ) k 2 := α 2 β 2 S (β 2 x ). Note that k 1 and k 2 depend on the equilibrium at which we linearize.

16 Fixed points and eigenvalues Linear stability Bifurcations First Lyapunov coefficient Characteristic equation and eigenvalues Consider solutions of the form u(t) = e λt c, c R 2. Non-trivial solutions of the system correspond with (λ)c = 0, for: [ λ k1 e (λ) = λτ 1 k 2 e λτ ] 2 k 2 e λτ 2 λ k 1 e λτ 1 Non-trivial solutions exist when det (λ) = 0, or: (λ k 1 e λτ 1 + k 2 e λτ 2 ) (λ k }{{} 1 e λτ 1 k 2 e λτ 2 ) = 0. }{{} := +(λ) := (λ) This characteristic function has an infinite (but countable) number of roots.

17 Fixed points and eigenvalues Linear stability Bifurcations First Lyapunov coefficient Characteristic equation and eigenvalues Consider solutions of the form u(t) = e λt c, c R 2. Non-trivial solutions of the system correspond with (λ)c = 0, for: [ λ k1 e (λ) = λτ 1 k 2 e λτ ] 2 k 2 e λτ 2 λ k 1 e λτ 1 Non-trivial solutions exist when det (λ) = 0, or: (λ k 1 e λτ 1 + k 2 e λτ 2 ) (λ k }{{} 1 e λτ 1 k 2 e λτ 2 ) = 0. }{{} := +(λ) := (λ) This characteristic function has an infinite (but countable) number of roots.

18 Symmetries of solutions Fixed points and eigenvalues Linear stability Bifurcations First Lyapunov coefficient Theorem Roots of correspond to symmetric solutions, whereas roots of + relate to asymetric solutions. Proof. Let Z 2 act on C 2 so that 1 Z 2 acts as ξ(x, y) : (x, y) (y, x), then: { (λ)v = 0 (λ) = 0 ξv = v { (λ)v = 0 + (λ) = 0 ξv = v

19 Symmetries of solutions Fixed points and eigenvalues Linear stability Bifurcations First Lyapunov coefficient Theorem Roots of correspond to symmetric solutions, whereas roots of + relate to asymetric solutions. Proof. Let Z 2 act on C 2 so that 1 Z 2 acts as ξ(x, y) : (x, y) (y, x), then: { (λ)v = 0 (λ) = 0 ξv = v { (λ)v = 0 + (λ) = 0 ξv = v

20 Roots of (λ) Introduction Fixed points and eigenvalues Linear stability Bifurcations First Lyapunov coefficient Theorem For λ C 0 e λh and λ > C, with C C 0, (λ) > 0. Hence, (λ) has a zero free right half-plane [Bellman and Cooke, 1963].

21 Minimal Stability Region Fixed points and eigenvalues Linear stability Bifurcations First Lyapunov coefficient Theorem For k 1 + k 2 < 1 the system has no eigenvalues with positive real part.

22 Bifurcations Introduction Fixed points and eigenvalues Linear stability Bifurcations First Lyapunov coefficient Note: As in ODEs, an equilibrium of a DDE loses its stablity when either a single real eigenvalue passes through zero (fold/transcritical), or a pair of complex eigenvalues passes through the imaginary axis (Hopf).

23 Fold/transcritical bifurcation Fixed points and eigenvalues Linear stability Bifurcations First Lyapunov coefficient Theorem Equilibria undergo a fold or transcritical bifurcation on the lines 1 + k 1 ± k 2 = 0 in (k 1, k 2 )-space.

24 Hopf bifurcations Introduction Fixed points and eigenvalues Linear stability Bifurcations First Lyapunov coefficient Theorem Two curves, h + (ω) and h (ω), exist along which the linearized system has a pair of complex eigenvalues λ = ±iω. Proof. For now, we consider only + (iω) = 0; (iω) is similar. iω k 1 e iωτ 1 + k 2 e iωτ 2 = 0 [ ] [ ] [ ] cos(ωτ1 ) cos(ωτ 2 ) k1 1 = sin(ωτ 1 ) sin(ωτ 2 ) ω k 2

25 Hopf bifurcations Introduction Fixed points and eigenvalues Linear stability Bifurcations First Lyapunov coefficient Proof cntd. The matrix is invertible if det = sin(ω(τ 2 τ 1 )) 0, yielding: [ ] [ ] [ ] k1 1 sin(ωτ2 ) cos(ωτ = h + (ω) := 2 ) 1. sin(ω(τ 2 τ 1 )) sin(ωτ 1 ) cos(ωτ 1 ) ω k 2

26 Bifurcations in (k 1, k 2 )-space Fixed points and eigenvalues Linear stability Bifurcations First Lyapunov coefficient Very sensitive, complex branch structure Intersections correspond with codim-2 bifurcations. k 2 k 1 (τ 1 = 11.6, τ 2 = 20.3)

27 Returning branches Introduction Fixed points and eigenvalues Linear stability Bifurcations First Lyapunov coefficient Theorem For a given ω 0 > 0, (λ) has no roots λ = ±iω, ω > ω 0 inside the square k 1 + k 2 < 1 + ω0 2. Proof. First, assume iω is a root of + (a similar argument holds for ): 0 = 1 + iω + k 1 e iωτ 1 + k 2 e iωτ 2, 1 + iω k 1 k 2. Then, this root lies outside the designated square for ω > ω 0.

28 Revised stability region Fixed points and eigenvalues Linear stability Bifurcations First Lyapunov coefficient Conjecture For the considered parameters τ 1 and τ 2 the full stability region is bounded, connected and contains the origin.

29 The first Lyapunov coefficient Fixed points and eigenvalues Linear stability Bifurcations First Lyapunov coefficient Stability region partly bounded by curve of Hopfs The first Lyapunov coefficient is determined along the boundary Normal given by [Diekmann et al, 1995]: c 1 = 1 2 qt D 3 f(0)(φ, φ, φ) + q T D 2 f(0)(e 0 (0) 1 D 2 f(0)(φ, φ), φ) qt D 2 f(0)(e 2iω (2iω) 1 D 2 f(0)(φ, φ), φ).

30 The first Lyapunov coefficient Fixed points and eigenvalues Linear stability Bifurcations First Lyapunov coefficient Stability region partly bounded by curve of Hopfs The first Lyapunov coefficient is determined along the boundary Normal given by [Diekmann et al, 1995]: c 1 = 1 2 qt D 3 f(0)(φ, φ, φ) + q T D 2 f(0)(e 0 (0) 1 D 2 f(0)(φ, φ), φ) qt D 2 f(0)(e 2iω (2iω) 1 D 2 f(0)(φ, φ), φ).

31 Codim 2 bifurcations Introduction Fixed points and eigenvalues Linear stability Bifurcations First Lyapunov coefficient Generalized Hopf Lypunov coefficient passes through zero, Generalized Hopf bifurcation at boundary

32 Summary Introduction Fixed points and eigenvalues Linear stability Bifurcations First Lyapunov coefficient Equilibria Only symmetric equilibria Stability region identified Bifurcations of stability region Fold and transcritical bifurcations Hopf-bifurcations, both sub- and supercritical Zero-Hopf and Hopf-Hopf

33 Summary Introduction Fixed points and eigenvalues Linear stability Bifurcations First Lyapunov coefficient Equilibria Only symmetric equilibria Stability region identified Bifurcations of stability region Fold and transcritical bifurcations Hopf-bifurcations, both sub- and supercritical Zero-Hopf and Hopf-Hopf

34 Numerical bifurcation analysis Overview One parameter Two parameters One parameter Identify stable and unstable manifolds of non-trivial equilibrium and periodic solutions for varying α 2. Two parameters Continue boundaries of stable solutions in α 1 and α 2. Software DDE-BIFTOOL [Engelborghs et al., 2002] PDDE-CONT/Knut [Roose and Szalai, 2007]

35 Numerical bifurcation analysis Overview One parameter Two parameters One parameter Identify stable and unstable manifolds of non-trivial equilibrium and periodic solutions for varying α 2. Two parameters Continue boundaries of stable solutions in α 1 and α 2. Software DDE-BIFTOOL [Engelborghs et al., 2002] PDDE-CONT/Knut [Roose and Szalai, 2007]

36 Overview One parameter Two parameters Bifurcations in α A 2 B C D

37 Overview One parameter Two parameters Bifurcations in α 2 3 Bifurcation diagram II 2.5 I Maximal value 2 III 1.5 IV PD1 0 H1 B α 2 Detail I Detail II LPC4

38 Overview One parameter Two parameters Bifurcations in α 2 l k h Maximal value j g c i d f e α 2 a b

39 Fixed points in (k 1, k 2 )-space Overview One parameter Two parameters F1 B1 k H2 H3 H4 H1 Non-trivial Trivial k 1

40 Overview One parameter Two parameters Bifurcations of stable solutions in α 1 and α GH ZH2 ZH1 HH

41 Overview One parameter Two parameters Bifurcations of stable solutions in α 1 and α CP1 0.3 GH1 0.2 FF ZH2 ZH1 HH

42 Overview One parameter Two parameters Bifurcations of stable solutions in α 1 and α CP1 0.3 GH1 0.2 FF1 CP3 0.1 R1_1 0 ZH2 ZH1 HH

43 Complex bifurcation structure Overview One parameter Two parameters α 1 R2_1 FF2 R2_2 FF3 CP4 CP2 α 2

44 Comparison Introduction The bifurcation analysis is compared with a survey on the detailed model:

45 Introduction Comparison The bifurcation analysis is compared with a survey on the detailed model:

46 Summary and conclusions Summary We studied a simple, lumped model for neural activity Parameter regions of steady states and periodic solutions are identified The behavior of both models varies similarly for changes of parameters Regions of multistability are of interest for studying epilepsy

47 Future work Introduction Details Study the model s codim-2 bifurcations Proofs rather than conjectures Next steps Expand model (e.g. break symmetry) Parameter estimation

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