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

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Outline

Pathological oscillations within the basal ganglia

Mathematical model

Theoretical results

Numerical simulations

Conclusion

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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

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The basal ganglia

Cortex

Striatum

Substantia nigra

Thalamus

GPe STN

Excitatory

Inhibitory

Dopaminergic

Basal ganglia

Figure : Normal state

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The basal ganglia

Cortex

Striatum

Substantia nigra

Thalamus

GPe STN

Excitatory

Inhibitory

Dopaminergic

Basal ganglia

Figure : Parkinsonian state

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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.

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Pathological oscillations

Cortex

Striatum

Substantia nigra

Thalamus

GPe STN

Inhibitory

Inhibitrice

Dopaminergic

Figure : Possible involvement of the STN-GPe loop

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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.

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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.

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Firing rate modeling

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Pedunculopontine nucleus (PPN)

CortexStriatum

GPe STN

Excitatory

Inhibitory

PPN

Figure : Pedunculopontine nucleus : an external excitatory nucleus

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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.

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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

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Analysis in the absence of delays

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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.

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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).

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Robustness to delays

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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).

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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

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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.

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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.

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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).

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Numerical simulations

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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

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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

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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

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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).

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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

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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

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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.

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Conclusion

Thank you

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IntroductionFiring rate modelingAnalysis in the absence of delaysAnalysis in the presence of delaysNumerical simulationsConclusion