connecting epilepsy and alzheimer’s disease: a ...geza.kzoo.edu/~erdi/leckek/siam-lec.pdf ·...

Connecting Epilepsy and Alzheimer’sDisease: A Computational Modeling

FrameworkPeter Erdi

[email protected]

Henry R. Luce ProfessorCenter for Complex Systems Studies

Kalamazoo Collegehttp://people.kzoo.edu/ perdi/

andInstitue for Particle and Nuclear Physics, Wigner Research Centre, Hungarian Academy

of Sciences, Budapesthttp://cneuro.rmki.kfki.hu/

Collaborators

Takumi Matsuzawa, Kalamazoo College, USA

Siyuan Zhang, Kalamazoo College, USA,

Laszlo Zalanyi, Wigner Res. Cent. for Physics, Budapest, Hungary,

Tamas Kiss, Wigner Res. Cent for Physics, Budapest, Hungary

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Content

1. Alzheimer’s Disease and epilepsy: the big picture

2. Aβ effects on Synaptic Plasticity: Brief Summary of the

Experimental Background

3. A Phenomenological Model: the modified calcium control

hypothesis

4. Kinetic modeling of normal and Aβ modified synaptic plas-

ticity

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Alzheimer’s Disease and epilepsy: the big picture

Concentration-dependent

plasticity

Aβ cortico-hippocampal system

skeleton

network

theta-gamma

rhythm generator

synaptic plasticity network dysfunction

pre- and

postsynaptic

positive

feedback

memory

deficit

(mouse)

memory

deficit

(human)

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Synaptic plasticity: normal and pathological

Experimental background

Phenomenological model

Kinetic model


Many facts have been accumulated to support hypotheses that link the elevated levelof human amyloid precursor protein (hAPP) related β-amyloid (Aβ) to pre-clinical and clinical observations related to AD (Palop & Mucke 2009, 2010). The mostsignificant elements of a working hypothesis assume:

• Aβ alters hippocampal rhythmicity

• Aβ alters long term synaptic plasticity by several mechanisms, enhance long-termdepression (LTD) and impair long-term potentiation (LTP))

• Elevated Aβ implies neuronal dysfunction resulting from an impaired balance betweenpositive and negative feedback loops in modulation of synaptic transmission

• Non-convulsive, subclinical partial seizures worsen the memory and behavioral symp-toms in AD

• Antiepilectic drugs can reduce the deteriorating effects of epileptiform activity in AD

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Concentration-dependent

plasticity

Aβ cortico-hippocampal system

skeleton

network

theta-gamma

rhythm generator

synaptic plasticity network dysfunction

pre- and

postsynaptic

positive

feedback

memory

deficit

(mouse)

memory

deficit

(human)

Hypothetical causal chain toexplain the multiple and multileveleffects of Aβ: from alteredsynaptic plasticity via networkdysfunction to cognitive deficit.

Palop JJ, Mucke L.: Amyloid-beta-induced neuronaldysfunction in Alzheimer’sdisease: from synapses towardneural networks. Nat Neurosci.2010 Jul;13(7):812-8.

(A skeleton network of thehippcampal system generatesgamma and theta rhythms.)

Aβ concentration-dependentaltered synaptic plasticityimplies network dysfunctionincluding epileptiform activity

This activity contributes to cogni-tive deficit by positive feedbackcellular mechanisms.

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Aβ effects on Synaptic Plasticity: Brief Summary ofthe Experimental Background

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Summary of Aβ effects on synapticplasticity:(a)Aβ induces synaptic facilitationand occlusion of LTD.(b) Aβ induces mGlur- andNMDAR-dependent LTD facilita-tion.(c)Aβ induces LTP impairments,which is regulated by LTD path-ways.

Collected by Palop JJ, Mucke L.: Amyloid-beta-induced neuronal dysfunction in Alzheimer’s disease: from synapses toward neural networks.Nat Neurosci. 2010 Jul;13(7):812-8

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Synaptic plasticity models

Phenomenological models: from Hebb to realisticmodels

Phenomenological models: from Hebb to learningrules in Artificial Neural Network Models

Kinetic (”microscopic”) basis of realistic learningrules

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A Phenomenological Model: the modified calciumcontrol hypothesis

Shouval, H. Z., Bear, M. F., and Cooper, L. N. (2002). A unified theory of NMDA receptor-dependent bidirectional synaptic plasticity.

dWi (t)

dt= η([Ca2+ (t)])

(Ω([Ca2+ (t)]) − λWi (t)

). (1)

Wi: the synaptic weight of synapse i, the value of the functionΩ depends on calcium concentration, and determines both sign and magnitude of the change of synaptic strengthη is the learning rate, and also depends (typically monotonously increasingly) on calcium concentration,λ is a decay constant. To complete the model we need an equation which prescribes calcium dynamics.

d[Ca2+ (t)

]dt

= INMDA (t) − 1

τCa

[Ca2+ (t)

], (2)

A simple assumption is that the source of calcium depends on the NMDA current, as Eqn. 3 defines:

INMDA = P0GNMDA

[Ifθ (t) e

−tτf + Isθ (t) e

−tτs

]H (V ) (3)

where If and Is are the relative magnitude of the slow and fast component of the NMDAR current. If + Is = 1 is assumed. H(V ) is the generalform of the voltage dependence. θ = 0 if t < 0 and θ = 1 if t ≥ 0. P0 is the fraction of NMDARs in the closed state, and set to be 0.5.

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The whole lecture is focused about Ω function !

• How to incorporate Aβ effects into the phenonemenological

model?

• How to justify the form of the Ω by kinetic modeling?

• How to estimate the parameters of the kinetic model for the

Aβ-induced pathological case?

• How the kinetic model with the altered parameters reflects

the pathological synaptic plasticity?

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Ω([Ca

2+(t)])

=eβ2

([Ca2+(t)

]−α2

)1 + e

β2

([Ca2+(t)

]−α2

) − γeβ1

([Ca2+(t)

]−α1

)1 + e

β1

([Ca2+(t)

]−α1

) + γ (4)

Figure 1:

• The shape of the Ω function de-termines when LTP or LTD occurs(Shouval).

• The Ω (blue) and Ωnew (black andred) functions are plotted againstthe intracellular calcium concentra-tion.

• Ωnew with a pathological parame-ter set decreases the LTD thresholdand impairs LTP by weakening theLTP strength.

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A new Ω function was constructedto incorporate the effect of Aβ:

• the onset of LTP as a sigmoid func-tion with threshold set at α3

• ΩLTP and ΩLTD consists of twosigmoid functions capturing the on-set and the offset of the LTD pro-cess. Here the threshold parametersare functions of Aβ concentration.

• These two processes supposed to beequal in strength providing the pos-sibility to cancel each other, which isone possible way to eliminate LTP.

• The two processes are balanced butnot weighted equally, a synapse canbe potetiated three times strongerthan its basal level but can only beweakened to zero.

• To achieve this weighting in the nor-malized synaptic process model asigmoid function is composed withthe two competing processes withthe ability to set the basal synapticstrength level arbitrarily.

Ωnew([Ca2+

(t)], Aβ) =eβ(k1ΩLTP−k2ΩLTD)−ε

1 + eβ(k1ΩLTP−k2ΩLTD)−ε

ΩLTP([Ca2+

(t)]) =eβ3([Ca2+(t)]−α3)

1 + eβ3([Ca2+(t)]−α3)

ΩLTD([Ca2+

(t)], Aβ) =eβ1([Ca2+(t)]−α1(Aβ))

1 + eβ1([Ca2+(t)]−α1(Aβ))−

eβ2([Ca2+(t)]−α2(Aβ))

1 + eβ2([Ca2+(t)]−α2(Aβ))

(5)

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A Phenomenological Model: the modified calciumcontrol hypothesis.

Ωnew([Ca2+

(t)], Aβ) =eβ(k1ΩLTP−k2ΩLTD)−ε

1 + eβ(k1ΩLTP−k2ΩLTD)−ε

ΩLTP([Ca2+

(t)]) =eβ3([Ca2+(t)]−α3)

1 + eβ3([Ca2+(t)]−α3)

ΩLTD([Ca2+

(t)], Aβ) =eβ1([Ca2+(t)]−α1(Aβ))

1 + eβ1([Ca2+(t)]−α1(Aβ))−

eβ2([Ca2+(t)]−α2(Aβ))

1 + eβ2([Ca2+(t)]−α2(Aβ))

(6)

• The constructed function Ωnew alters synaptic plasticity.

• Ω in Eqn. 4 with Ωnew in Eqn. 6.

• LTP and LTD in Eqn. 6 are interpreted as activity of each process.

• α1,2,3 characterize calcium concentration when LTP and LTD processes are active.

• LTD process is active when the intracellular calcium concentration is between α1 and α2.

• When the calcium concentration is higher than α3, LTP process is active.

• k1,2 is activity coefficient for LTP and LTD respectively. If LTD process were blocked entirely, k2 would become zero. β and β1,2,3determine the steepness of the sigmoid functions. ε is related to the initial synaptic strength.

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A Phenomenological Model: the modified calciumcontrol hypothesis. Simulation results

Figure 2: Subthreshold LTD induction produces no change in the weight when Ωnew witha normal parameter set replaced Ω in Eqn. 1; however, it induced LTD for Ωnew with apathological parameter set.

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Figure 3: LTD induction protocol properly induced LTD for Ωnew with a normal andpathological parameter set.

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Figure 4: LTD induction protocol properly induced LTD for Ωnew with a normal andpathological parameter set. Block of the LTD process (k=0.01) properly prevented theLTD induction.

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Figure 5: LTP was induced for Ωnew with a normal and pathological parameter set. LTPwas observed for the normal, and the LTP was impaired for the pathological case. TheLTP was once again observed when the LTD process was blocked.

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Kinetic modeling of normal and Aβ modifiedsynaptic plasticity

Figure 6: Kinetic model of phosphorylation: D’Alcantara P, Schiffmann S and Swil-lens S (2003): Bidirectional synaptic plasticity as a consequence of interdependentCa2+-controlled phosphorylation and dephosphorylation pathways European Journal ofNeuroscience 17 2521–2528

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Parameter estimation

Figure 7: Aβ impairs the LTP and enhances LTD while it affects only LTD-related pathways. There are essentially 9 kinetic parameters

to solve for a steady-state solution of the Model III: Kd, nH , k1/k2, k3/k4, r1/r2, r3/r4, n2/n1, d1/d2, and p1/p2. Out of these

9 parameters, r1/r2, r3/r4, n2/n1, d1/d2, and p1/p2 are relevant parameters for dephosphorylation. To ease computation, we have only

focused on r3/r4, n2/n1, and p1/p2 to implement the effect of Aβ

Results: n2/n1=0.001734652 (Normal: 0.006), p1/p2=0.000428 (Normal: 0.001), r3/r4=40.37387632 (Normal: 100)

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Figure 8: Time-dependent receptor activity responses for different calcium signals

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Figure 9: Time responses of CaMKII, phosphatase, and AMPA-R activities to LTD induction protocol, expressed in percent of the respective

steady state activities obtained at [Ca2+]=0.07536microM. (Stimulation: [Ca2+]=0.28microM between t=0-50s)

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Figure 10: Time responses of CaMKII, phosphatase, and AMPA-R activities to subthreshold LTD induction protocol, expressedA in percent

of the respective steady state activities obtained at [Ca2+] = 0.07536microM. (Stimulation:[Ca2+] = 0.28microM between t = 0− 5s)

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Figure 11: Time responses of CaMKII, phosphatase, and AMPA-R activities to LTP induction protocol, expressedA in percent of the

respective steady state activities obtained at [Ca2+] = 0.07536microM . (Stimulation: [Ca2+]=1.0microM between t=0-5s)

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connecting epilepsy and alzheimer’s disease: a ...geza.kzoo.edu/~erdi/leckek/siam-lec.pdf ·...

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