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Basal ganglia oscillations:
the role of delays and
external excitatory nuclei
Ihab Haidar
Supervised by:
William Pasillas-Lépine, Elena Panteley & Antoine Chaillet
Laboratoire des Signaux et Systèmes (LSS)-Supélec
5 June 2013
1 / 31
Outline
Pathological oscillations within the basal ganglia
Mathematical model
Theoretical results
Numerical simulations
Conclusion
2 / 31
The basal ganglia
Cortex
Thalamus
Subthalamic nucleus
Globus pallidus
In Parkinson's disease, the dopaminergic
neurons are destroyed in the substantia
nigra.
allows the communication
between neurones and is
involved in the motricity
control
Substantia
nigra
3 / 31
The basal ganglia
Cortex
Striatum
Substantia nigra
Thalamus
GPe STN
Excitatory
Inhibitory
Dopaminergic
Basal ganglia
Figure : Normal state
4 / 31
The basal ganglia
Cortex
Striatum
Substantia nigra
Thalamus
GPe STN
Excitatory
Inhibitory
Dopaminergic
Basal ganglia
Figure : Parkinsonian state
5 / 31
Neuron spikes and �ring rateA
B
C
Firing rate=number of spikes /unit of time
Dayan, P., and Abbott, L. (2001), “Computational and mathematical modelling of neural systems,” Theoretical neuroscience, MIT Press.
6 / 31
Pathological oscillations
Cortex
Striatum
Substantia nigra
Thalamus
GPe STN
Inhibitory
Inhibitrice
Dopaminergic
Figure : Possible involvement of the STN-GPe loop
7 / 31
The STN-GPe loop
A.L. Nevado-Holgado, J.R. Terry and R. Bogacz, Conditions for thegeneration of beta oscillations in the subthalamic nucleus-globus pallidusnetwork, The Journal of Neuroscience, vol. 30, no. 37, pp. 12340-12352, Sep.2010.
A. Pavlides, S.J. Hogan and R. Bogacz, Improved conditions for thegeneration of beta oscillations in the subthalamic nucleus-globus pallidusnetwork, European Journal of Neuroscience, vol. 36, pp. 2229-2239, 2012.
W. Pasillas-Lépine, Delay-induced oscillations in Wilson and Cowan's model :An analysis of the subthalamo-pallidal feedback loop in healthy andparkinsonian subjects, Biological Cybernetics, vol. 107 no. 3, pp. 289-308,2013.
8 / 31
The STN-GPe loop
τs ẋs = −xs + Ss
(− cgs xg (t − δgs ) + ccs Ctx
)τg ẋg = −xg + Sg
(csgxs(t − δsg )− cgg xg (t − δgg ) + cxgStr
)xs and xg represent the �ring rates of STN and GPe, respectively.
Ctx et Str describe the external inputs from the Cortex and Striatum,
respectively.
A.L. Nevado-Holgado, J.R. Terry and R. Bogacz, Conditions for the generation of beta
oscillations in the subthalamic nucleus-globus pallidus network, The Journal of Neuroscience,
vol. 30, no. 37, pp. 12340-12352, Sep. 2010.
9 / 31
Firing rate modeling
10 / 31
Pedunculopontine nucleus (PPN)
CortexStriatum
GPe STN
Excitatory
Inhibitory
PPN
Figure : Pedunculopontine nucleus : an external excitatory nucleus
11 / 31
Model
τs ẋs = Ss
(cps xp(t − δps )− cgs xg (t − δgs ) + us
)− xs
τg ẋg = Sg(csgxs(t − δsg )− cgg xg (t − δgg ) + ug
)− xg
τp ẋp = Sp(cspxs(t − δsp) + up
)− xp
(1)
xs , xg and xp represent the �ring rates of STN, GPe and PPN,respectively.
us , ug and up describe the external inputs from the Cortex and
Striatum.
12 / 31
Assumption 1
−3 −2 −1 0 1 2 30
0.2
0.4
0.6
0.8
1
Input
Act
ivat
ion
func
tion
S
i
σi=Max S′
i
Max Si
Min Si
Figure : Activation functions
13 / 31
Analysis in the absence of delays
14 / 31
Existence and uniqueness of equilibrium point
Theorem
Under Assumption 1, if
σpσscps c
sp ≤ 1 (2)
then the system (1) has a unique equilibrium point, for each constant
vector (u?s , u?g , u
?p) ∈ R3. Otherwise, there exists a constant vector
(u?s , u?g , u
?p) for which the system (1) has at least three distinct equilibria.
15 / 31
Global asymptotic stability
Proposition
Consider the system (1) and let Assumption 1 holds. Fix any constant
input vector u?, consider an equilibrium x? associated to these inputs. If
the conditions (2) and
σs (cps + c
gs ) + σgc
sg + σpc
sp < 2 (3)
are both satis�ed, then x? is globally asymptotically stable for (1).
16 / 31
Robustness to delays
17 / 31
Linearized system
By letting
e := x − x? and v := u − u?,the linearization arround x? is given by
τs ės = σ?s
(cps ep(t − δps )− cgs eg (t − δgs ) + vs
)− es
τg ėg = σ?g
(csges(t − δsg )− cgg eg (t − δgg ) + vg
)− eg
τp ėp = σ?p
(cspes(t − δsp) + vp
)− ep ,
(4)
where
σ?s := S′s(c
ps x
?p − cgs x?g + u?s ), σ?g := S ′g (csgx?s − cgg x?g + u?g )
σ?p := S′p(c
spx
?s + u
?p).
18 / 31
Two di�culties : two loops and irrational transfer functions
Hg(s)
Hs(s)
Hp(s)
cgse−δgss csge
−δsgs
cpse−δpss cspe
−δsps
++
++
++ +
+
Hsg(s)
Hsp(s)
vg
vp
vs es
With
Hs(s) =σ?s
τss + 1, Hp(s) =
σ?pτps + 1
and Hg (s) =σ?g
τg s + 1 + σ?gcgg e−δ
g
g s
19 / 31
Equivalent choice
Hsg
Hp
cpse−δpss cspe
−δsps
++
++
vp
vsHg
Hsp
cgse−δgss csge
−δsgs
++
++
vs
vg
Hsp :=Hs
1− cspcpsHpHse−(δsp+δps)s
Hsg :=Hs
1 + csgcgsHgHse−(δsg+δgs)s.
20 / 31
Cross-over frequency and delay margin
For a given transfer function G , the gain γG (ω) and phase ϕG (ω) arede�ned by
γG (ω) = 20 log10 |G (jω)| and ϕG (ω) = arg (G (jω)) .
Assume that G is a strictly proper transfer function and that γG is strictlydecreasing. If γG (0) > 0 we can de�ne ωG as the only frequency such that
γG (ωG ) = 0.
This frequency can be used to de�ne the delay margin ∆(G ) by the relation
∆(G ) =π − ϕG (ωG )
ωG.
21 / 31
Stability of the delayed feedback system
Letcp := c
ps c
sp , cg := c
gs c
sg ,
δp := δsp + δ
ps , δg := δ
sg + δ
gs
Theorem
Consider the delayed linearized system (4). Let u? ∈ R3 be any constantinput such that, for the equilibrium x? associated to these inputs, the
transfer functions Hg ,Hsp and Hsg are input-output stable. De�neH := cgHgHsp. Assume that the gain of H is strictly decreasing. For eachδp > 0, x
? is exponentially stable for the linearized system (4) if and only if
δg < ∆(H).
22 / 31
Numerical simulations
23 / 31
Parameter values
Si (x) =Bi
Bi + (Mi − Bi )e−4x, ∀ i ∈ {s, g , p} (5)
Parameter Value Description
Ms 300 spk/s STN Maximal �ring rate
Bs 17 spk/s Firing rate at rest for STN
Mg 400 spk/s GPe Maximal �ring rate
Bg 75 spk/s Firing rate at rest for GPe
Mp 300 spk/s PPN Maximal �ring rate
Bp 17 spk/s Firing rate at rest for PPN
24 / 31
Parameter values
Parameter Value Description
δsg 6 ms Delay from STN to GPe
δgs 6 ms Delay from GPe to STN
δgg 4 ms Internal self-inhibition delay in the GPe
τs 6 ms STN time constant
τg 14 ms GPe time constant
τp 6 ms PPN time constant
25 / 31
Parameter values
Parameter Healthy state Disease state
csg 14.3 15
cgs 1.5 14.3
cgg 6.6 12.3
us 0.01 0.03
ug 0.03 0.35
c ij = cij
H+ k
(c ij
D − c ijH)∀ i , j ∈ {s, g} (6)
where k is a parameter that describes the evolution of Parkinson's disease
26 / 31
Evaluation of the delay margin ∆(H)
cp
K
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
0.05
0.1
0.15
0.2
0.25
0.3
2
3
4
5
6
7
8
9
10
11
12x 10
−3
Figure : In�uence of cp and k on the delay margin ∆(H).
27 / 31
Temporal evolution of the nonlinear dynamics : k=0.25
−1.5 −1 −0.5 0 0.5 1 1.5−1.5
−1
−0.5
0
0.5
1
1.5
Re(H)
Im(H
)
Unit circleδ
g=0 ms
δg=12 ms
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
Time [s]
Firin
g r
ate
[sp
k/s]
EquilibriumSTNGPePPN
−1 0 1 2−1.5
−1
−0.5
0
0.5
1
1.5
Re(H)
Im(H
)
Unit circleδ
g=0 ms
δg=12 ms
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
Time [s]
Firin
g r
ate
[sp
k/s]
EquilibriumSTNGPePPN
cp = 0 :
cp = 0.1 :
Figure : In�uence on stability of the interconnection gain cspand cp
s, for k = 0.25.
On the left, the open-loop frequency-response is represented in a Nyquistdiagram. On the right, the temporal evolution of the system (1) is plotted.(a)-(b) : The system is simulated at A. (c)-(d) : The system is simulated at B
28 / 31
Temporal evolution of the nonlinear dynamics : k=0.15
−1 0 1 2−1.5
−1
−0.5
0
0.5
1
1.5
Re(H)
Im(H
)
Unit circleδ
g=0 ms
δg=12 ms
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
Time [s]
Firin
g r
ate
[sp
k/s]
EquilibriumSTNGPePPN
−1 0 1 2−1.5
−1
−0.5
0
0.5
1
1.5
Re(H)
Im(H
)
Unit circleδ
g=0 ms
δg=12 ms
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
Time [s]
Firin
g r
ate
[sp
k/s]
EquilibriumSTNGPePPN
cp = 0.3 :
cp = 0.6 :
Figure : In�uence on stability of the interconnection gain cspand cp
s, for k = 0.15.
On the left, the open-loop frequency-response is represented in a Nyquistdiagram. On the right, the temporal evolution of the system (1) is plotted.(a)-(b) : The system is simulated at A. (c)-(d) : The system is simulated at B
29 / 31
Conclusion
Theorem 1 shows the in�uence of the strengths of interconnection
between the STN and PPN on the multiplicity of equilibrium points
Theorem 2 shows how the transmission delays and the strengths of
interconnection between the STN and PPN can change stability of the
network and intervene in the modulation of pathological oscillations.
The consideration of external nuclei can shed additional light on how
the external inputs can a�ect the basal ganglia and thus lead to better
understanding of basal ganglia functioning.
30 / 31
Conclusion
Thank you
31 / 31
IntroductionFiring rate modelingAnalysis in the absence of delaysAnalysis in the presence of delaysNumerical simulationsConclusion