IntroductionAnalysis
Numerical continuationConclusions
Analysis of a lumped model of neocortex to studyepileptiform activity
Sid Visser Hil Meijer Stephan van Gils
March 21, 2012
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
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
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
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
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
EEG during a seizure
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
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 complexmodels
Results so far
A large, detailed model describing activity of individualneurons in both normal and epileptiform states
A simple, lumped model that has shown to have somesimilarity [Visser et al., 2010]
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Motivation of model
Simulation on Youtube
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Model
Two excitatory populations:
x1(t) = −µ1x1(t)−F1(x1(t − τi )) + G1(x2(t − τe))
x2(t) = −µ2x2(t)−F2(x2(t − τi )) + G2(x1(t − τe))
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Model
Two excitatory populations:
x1(t) = −µ1x1(t)−F1(x1(t − τi )) + G1(x2(t − τe))
x2(t) = −µ2x2(t)−F2(x2(t − τi )) + G2(x1(t − τe))
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Model
Two excitatory populations:
x1(t) = −µ1x1(t)−F1(x1(t − τi )) + G1(x2(t − τe))
x2(t) = −µ2x2(t)−F2(x2(t − τi )) + G2(x1(t − τe))
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Model
Two excitatory populations:
x1(t) = −µ1x1(t)−F1(x1(t − τi )) + G1(x2(t − τe))
x2(t) = −µ2x2(t)−F2(x2(t − τi )) + G2(x1(t − τe))
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Fixed points and eigenvaluesLinear stabilityBifurcationsFirst Lyapunov coefficient
Simplifications
Make system symmetric:
µ1 = µ2 := µ F1 = F2 := F G1 = G2 := G
Choose F(x) and G(x) as rescalings of:
S = (tanh(x − a)− tanh(−a)) cosh2(−a)
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Fixed points and eigenvaluesLinear stabilityBifurcationsFirst Lyapunov coefficient
Time rescaling
Time is rescaled to non-dimensionalize system:
x1(t) = −x1(t)− α1S(β1x1(t − τ1)) + α2S(β2x2(t − τ2))
x2(t) = −x2(t)− α1S(β1x2(t − τ1)) + α2S(β2x1(t − τ2))
Introduce vector notation:
x(t) = f(xt), with xt ∈ C ([−h, 0],R2) and h = max(τ1, τ2).
Main parameters of interest: representing connection strengthsbetween populations.
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Fixed points and eigenvaluesLinear stabilityBifurcationsFirst Lyapunov coefficient
Time rescaling
Time is rescaled to non-dimensionalize system:
x1(t) = −x1(t)− α1S(β1x1(t − τ1)) + α2S(β2x2(t − τ2))
x2(t) = −x2(t)− α1S(β1x2(t − τ1)) + α2S(β2x1(t − τ2))
Introduce vector notation:
x(t) = f(xt), with xt ∈ C ([−h, 0],R2) and h = max(τ1, τ2).
Main parameters of interest: representing connection strengthsbetween populations.
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Fixed points and eigenvaluesLinear stabilityBifurcationsFirst Lyapunov coefficient
Fixed points
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∗).
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Fixed points and eigenvaluesLinear stabilityBifurcationsFirst Lyapunov coefficient
Linearization
The DDE is linearized about an equilibrium:
u1(t) = −u1(t)− k1u1(t − τ1) + k2u2(t − τ2),
u2(t) = −u2(t)− k1u2(t − τ1) + k2u1(t − τ2),
in which
k1 := α1β1S ′(β1x∗) k2 := α2β2S ′(β2x∗).
Note that k1 and k2 depend on the equilibrium at which welinearize.
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Fixed points and eigenvaluesLinear stabilityBifurcationsFirst Lyapunov coefficient
Characteristic equation and eigenvalues
Consider solutions of the form u(t) = eλtc, c ∈ R2. Non-trivialsolutions of the system correspond with ∆(λ)c = 0, for:
∆(λ) =
[λ+ 1 + k1e−λτ1 −k2e−λτ2
−k2e−λτ2 λ+ 1 + k1e−λτ1
]Non-trivial solutions exist when det ∆(λ) = 0, or:
(λ+ 1 + k1e−λτ1 + k2e−λτ2)︸ ︷︷ ︸:=∆+(λ)
(λ+ 1 + k1e−λτ1 − k2e−λτ2)︸ ︷︷ ︸:=∆−(λ)
= 0.
This characteristic function has an infinite (but countable) numberof roots.
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Fixed points and eigenvaluesLinear stabilityBifurcationsFirst Lyapunov coefficient
Characteristic equation and eigenvalues
Consider solutions of the form u(t) = eλtc, c ∈ R2. Non-trivialsolutions of the system correspond with ∆(λ)c = 0, for:
∆(λ) =
[λ+ 1 + k1e−λτ1 −k2e−λτ2
−k2e−λτ2 λ+ 1 + k1e−λτ1
]Non-trivial solutions exist when det ∆(λ) = 0, or:
(λ+ 1 + k1e−λτ1 + k2e−λτ2)︸ ︷︷ ︸:=∆+(λ)
(λ+ 1 + k1e−λτ1 − k2e−λτ2)︸ ︷︷ ︸:=∆−(λ)
= 0.
This characteristic function has an infinite (but countable) numberof roots.
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Fixed points and eigenvaluesLinear stabilityBifurcationsFirst Lyapunov coefficient
Symmetries of solutions
Theorem
Roots of ∆− correspond to symmetric solutions, whereas roots of∆+ relate to asymetric solutions.
Proof.
Let Z2 act on C2 so that −1 ∈ Z2 acts as ξ(x , y) : (x , y) 7→ (y , x),then:
∆−(λ) = 0⇔
{∆(λ)v = 0
ξv = v
∆+(λ) = 0⇔
{∆(λ)v = 0
ξv = −v
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Fixed points and eigenvaluesLinear stabilityBifurcationsFirst Lyapunov coefficient
Symmetries of solutions
Theorem
Roots of ∆− correspond to symmetric solutions, whereas roots of∆+ relate to asymetric solutions.
Proof.
Let Z2 act on C2 so that −1 ∈ Z2 acts as ξ(x , y) : (x , y) 7→ (y , x),then:
∆−(λ) = 0⇔
{∆(λ)v = 0
ξv = v
∆+(λ) = 0⇔
{∆(λ)v = 0
ξv = −v
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Fixed points and eigenvaluesLinear stabilityBifurcationsFirst Lyapunov coefficient
Roots of |∆(λ)|
Theorem
For |λ| ≥ C0|e−λh| and |λ| > C , with C ≥ C0, |∆(λ)| > 0. Hence,∆(λ) has a zero free right half-plane [Bellman and Cooke, 1963].
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Fixed points and eigenvaluesLinear stabilityBifurcationsFirst Lyapunov coefficient
Minimal Stability Region
Theorem
For |k1|+ |k2| < 1 the system has no eigenvalues with positive realpart.
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Fixed points and eigenvaluesLinear stabilityBifurcationsFirst Lyapunov coefficient
Bifurcations
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 imaginaryaxis (Hopf).
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Fixed points and eigenvaluesLinear stabilityBifurcationsFirst Lyapunov coefficient
Fold/transcritical bifurcation
Theorem
Equilibria undergo a fold or transcritical bifurcation on the lines1 + k1 ± k2 = 0 in (k1, k2)-space.
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Fixed points and eigenvaluesLinear stabilityBifurcationsFirst Lyapunov coefficient
Hopf bifurcations
Theorem
Two curves, h+(ω) and h−(ω), exist along which the linearizedsystem has a pair of complex eigenvalues λ = ±iω.
Proof.
For now, we consider only ∆+(iω) = 0; ∆−(iω) is similar.
iω + 1 + k1e−iωτ1 + k2e−iωτ2 = 0[cos(ωτ1) cos(ωτ2)sin(ωτ1) sin(ωτ2)
] [k1
k2
]=
[−1ω
]
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Fixed points and eigenvaluesLinear stabilityBifurcationsFirst Lyapunov coefficient
Hopf bifurcations
Proof cntd.
The matrix is invertible if det = sin(ω(τ2 − τ1)) 6= 0, yielding:[k1
k2
]= h+(ω) :=
−1
sin(ω(τ2 − τ1))
[sin(ωτ2) cos(ωτ2)− sin(ωτ1) − cos(ωτ1)
] [1ω
].
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Fixed points and eigenvaluesLinear stabilityBifurcationsFirst Lyapunov coefficient
Bifurcations in (k1, k2)-space
Very sensitive, complex branch structure
Intersections correspond with codim-2 bifurcations.
k1
k2
(τ1 = 11.6, τ2 = 20.3)
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Fixed points and eigenvaluesLinear stabilityBifurcationsFirst Lyapunov coefficient
Returning branches
Theorem
For a given ω0 > 0, ∆(λ) has no roots λ = ±iω, ω > ω0 inside the
square |k1|+ |k2| <√
1 + ω20.
Proof.
First, assume iω is a root of ∆+ (a similar argument holds for ∆−):
0 = |1 + iω + k1e−iωτ1 + k2e−iωτ2 |,≥ |1 + iω| − |k1| − |k2|.
Then, this root lies outside the designated square for ω > ω0.
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Fixed points and eigenvaluesLinear stabilityBifurcationsFirst Lyapunov coefficient
Revised stability region
Conjecture
For the considered parameters τ1 and τ2 the full stability region isbounded, connected and contains the origin.
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Fixed points and eigenvaluesLinear stabilityBifurcationsFirst Lyapunov coefficient
The first Lyapunov coefficient
Stability region partly bounded by curve of Hopfs
The first Lyapunov coefficient is determined along theboundary
Normal given by [Diekmann et al, 1995]:
c1 =1
2qTD3f(0)(φ, φ, φ)
+ qTD2f(0)(e0·∆(0)−1D2f(0)(φ, φ), φ)
+1
2qTD2f(0)(e2iω·∆(2iω)−1D2f(0)(φ, φ), φ).
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Fixed points and eigenvaluesLinear stabilityBifurcationsFirst Lyapunov coefficient
The first Lyapunov coefficient
Stability region partly bounded by curve of Hopfs
The first Lyapunov coefficient is determined along theboundary
Normal given by [Diekmann et al, 1995]:
c1 =1
2qTD3f(0)(φ, φ, φ)
+ qTD2f(0)(e0·∆(0)−1D2f(0)(φ, φ), φ)
+1
2qTD2f(0)(e2iω·∆(2iω)−1D2f(0)(φ, φ), φ).
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Fixed points and eigenvaluesLinear stabilityBifurcationsFirst Lyapunov coefficient
Codim 2 bifurcations
Generalized Hopf
Lypunov coefficient passes through zero,
Generalized Hopf bifurcation at boundary
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Fixed points and eigenvaluesLinear stabilityBifurcationsFirst Lyapunov coefficient
Summary
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
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Fixed points and eigenvaluesLinear stabilityBifurcationsFirst Lyapunov coefficient
Summary
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
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
OverviewOne parameterTwo parameters
Numerical bifurcation analysis
One parameter
Identify stable and unstable manifolds of non-trivial equilibriumand periodic solutions for varying α2.
Two parameters
Continue boundaries of stable solutions in α1 and α2.
Software
DDE-BIFTOOL [Engelborghs et al., 2002]
PDDE-CONT/Knut [Roose and Szalai, 2007]
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
OverviewOne parameterTwo parameters
Numerical bifurcation analysis
One parameter
Identify stable and unstable manifolds of non-trivial equilibriumand periodic solutions for varying α2.
Two parameters
Continue boundaries of stable solutions in α1 and α2.
Software
DDE-BIFTOOL [Engelborghs et al., 2002]
PDDE-CONT/Knut [Roose and Szalai, 2007]
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
OverviewOne parameterTwo parameters
Bifurcations in α2
100 200−0.1
0
0.1
0.2
0 100 2001.4
1.6
1.8
2
0 100 200−1
0
1
2
3
0
0
100 200−1
0
1
2
3
A B
C D
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
OverviewOne parameterTwo parameters
Bifurcations in α2
0.521 0.5221.3
1.4
1.5
0.5962 0.59642.476
2.478
2.48
2.482
0.46 0.465 0.471.4
1.5
1.6
1.7
0.61 0.615 0.622.5
2.55
2.6
0.4 0.5 0.6 0.7 0.8 0.9 1−0.5
0
0.5
1
1.5
2
2.5
3
I
II
III
IV
α2
Max
imal
val
ue
PD1
PD3
LPC2
LPC1
PD2
PD4
F1
H2 H3
LPC3
LPC4
H1 B1
Detail I Detail II
Bifurcation diagram
Detail III Detail IV
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
OverviewOne parameterTwo parameters
Bifurcations in α2
Max
imal
val
ue
α2
a
e
b
f
g
h
j
l k
ic d
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
OverviewOne parameterTwo parameters
Fixed points in (k1, k2)-space
0 0.1 0.2 0.3 0.40
0.4
0.8
1.2
1.6
Trivial
k1
k2
H1
Non-trivial
H4
B1
H3H2
F1
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
OverviewOne parameterTwo parameters
Bifurcations of stable solutions in α1 and α2
0.4 0.5 0.6 0.7 0.82
1
0
0.1
0.2
0.3
0.4
GH1
HH1
ZH1ZH2
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
OverviewOne parameterTwo parameters
Bifurcations of stable solutions in α1 and α2
0.4 0.5 0.6 0.7 0.82
1
0
0.1
0.2
0.3
0.4
GH1
FF1
HH1
ZH1ZH2
CP1
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
OverviewOne parameterTwo parameters
Bifurcations of stable solutions in α1 and α2
0.4 0.5 0.6 0.7 0.82
1
0
0.1
0.2
0.3
0.4
GH1
FF1
R1_1
HH1
ZH1ZH2
CP1
CP3
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
OverviewOne parameterTwo parameters
Complex bifurcation structure
α2
α1
FF2
R2_1
R2_2
FF3
CP2
CP4
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Comparison
The bifurcation analysis is compared with a survey on the detailedmodel:
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Comparison
The bifurcation analysis is compared with a survey on the detailedmodel:
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Summary and conclusions
Summary
We studied a simple, lumped model for neural activity
Parameter regions of steady states and periodic solutions areidentified
Conclusions
The behavior of both models varies similarly for changes ofparameters
Regions of multistability are of interest for studying epilepsy
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity
IntroductionAnalysis
Numerical continuationConclusions
Future work
Details
Study the model’s codim-2 bifurcations
Proofs rather than conjectures
Next steps
Expand model (e.g. break symmetry)
Parameter estimation
Sid Visser, Hil Meijer, Stephan van Gils Analysis of a lumped model of neocortex to study epileptiform activity