Connecting Epilepsy and Alzheimer s Disease: A Computational Modeling Framework

Transcription

1 Connecting Epilepsy and Alzheimer s Disease: A Computational Modeling Framework Péter Érdi perdi@kzoo.edu Henry R. Luce Professor Center for Complex Systems Studies Kalamazoo College perdi/ and Institue for Particle and Nuclear Physics, Wigner Research Centre, Hungarian Academy of Sciences, Budapest

2 Collaborators Takumi Matsuzawa, Kalamazoo College, USA Siyuan Zhang, Kalamazoo College, USA, László Zalányi, Wigner Res. Cent. for Physics, Budapest, Hungary, Tamás Kiss, Wigner Res. Cent for Physics, Budapest, Hungary 2

3 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 plasticity 3

4 Alzheimer s Disease and epilepsy: the big picture cortico-hippocampal system Aβ synaptic plasticity network dysfunction skeleton network pre- and postsynaptic positive feedback memory deficit (mouse) theta-gamma rhythm generator Concentration-dependent plasticity memory deficit (human) 4

5 Synaptic plasticity: normal and pathological Experimental background Phenomenological model Kinetic model

6 Alzheimer s Disease and epilepsy: the big picture cortico-hippocampal system Aβ synaptic plasticity network dysfunction Hypothetical causal chain to explain the multiple and multilevel effects of Aβ: from altered synaptic plasticity via network dysfunction to cognitive deficit. skeleton network pre- and postsynaptic positive feedback Palop JJ, Mucke L.: Amyloidbeta-induced neuronal dysfunction in Alzheimer s disease: from synapses toward neural networks. Nat Neurosci Jul;13(7): memory deficit (mouse) (A skeleton network of the hippcampal system generates gamma and theta rhythms.) theta-gamma rhythm generator Concentration-dependent plasticity memory deficit (human) Aβ concentration-dependent altered synaptic plasticity implies network dysfunction including epileptiform activity This activity contributes to cognitive deficit by positive feedback cellular mechanisms. 6

7 Aβ effects on Synaptic Plasticity: Brief Summary of the Experimental Background 8 Summary of Aβ effects on synaptic plasticity: (a)aβ induces synaptic facilitation and occlusion of LTD. (b) Aβ induces mglur- and NMDAR-dependent LTD facilitation. (c)aβ induces LTP impairments, which is regulated by LTD pathways. Collected by Palop JJ, Mucke L.: Amyloid-beta-induced neuronal dysfunction in Alzheimer s disease: from synapses toward neural networks. Nat Neurosci Jul;13(7):

8 A Phenomenological Model: the modified calcium control hypothesis Shouval, H. Z., Bear, M. F., and Cooper, L. N. (2002). A unified theory of NMDA receptor-dependent bidirectional synaptic plasticity. dw i (t) dt = η([ca 2+ (t)]) ( Ω([Ca 2+ (t)]) λw i (t) ). (1) W i : the synaptic weight of synapse i, the value of the function Ω depends on calcium concentration, and determines both sign and magnitude of the change of synaptic strength η 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 [ Ca 2+ (t) ] dt = I NMDA (t) 1 τ Ca [ Ca 2+ (t) ], (2) A simple assumption is that the source of calcium depends on the NMDA current, as Eqn. 3 defines: [ I NMDA = P 0 G NMDA I f θ (t) e t τ f ] + I s θ (t) e t τs H (V ) (3) where I f and Is are the relative magnitude of the slow and fast component of the NMDAR current. I f + Is = 1 is assumed. H(V ) is the general form of the voltage dependence. θ = 0 if t < 0 and θ = 1 if t 0. P 0 is the fraction of NMDARs in the closed state, and set to be

9 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? 9

10 A Phenomenological Model: the modified calcium control hypothesis ([ ([ Ω Ca 2+ ]) (t) = eβ 2 Ca 2+ ) (t) ] α 2 ([ 1 + e β 2 Ca 2+ ] ) γ (t) α 2 e β 1 ([Ca 2+ ) (t) ] α 1 ([ Ca 2+ ) + γ (4) (t) ] α e β 1 The shape of the Ω function determines when LTP or LTD occurs (Shouval). The Ω (blue) and Ωnew (black and red) functions are plotted against the intracellular calcium concentration. Ωnew with a pathological parameter set decreases the LTD threshold and impairs LTP by weakening the LTP strength. Figure 1: 10

11 A Phenomenological Model: the modified calcium control hypothesis A new Ω function was constructed to incorporate the effect of Aβ: the onset of LTP as a sigmoid function with threshold set at α 3 Ω LTP and Ω LTD consists of two sigmoid functions capturing the onset and the offset of the LTD process. Here the threshold parameters are functions of Aβ concentration. These two processes supposed to be equal in strength providing the possibility to cancel each other, which is one possible way to eliminate LTP. The two processes are balanced but not weighted equally, a synapse can be potetiated three times stronger than its basal level but can only be weakened to zero. To achieve this weighting in the normalized synaptic process model a sigmoid function is composed with the two competing processes with the ability to set the basal synaptic strength level arbitrarily. Ωnew([Ca 2+ (t)], Aβ) = eβ(k 1 Ω LTP k 2 Ω LTD ) ɛ 1 + e β(k 1 Ω LTP k 2 Ω LTD ) ɛ Ω LTP ([Ca 2+ (t)]) = eβ 3 ([Ca2+ (t)] α 3 ) Ω LTD ([Ca 2+ (t)], Aβ) = 1 + e β 3 ([Ca2+ (t)] α 3 ) eβ 1 ([Ca2+ (t)] α 1 (Aβ)) 1 + e β 1 ([Ca2+ (t)] α 1 (Aβ)) e β 2 ([Ca 2+ (t)] α 2 (Aβ)) 1 + e β 2 ([Ca2+ (t)] α 2 (Aβ)) (5) 11

12 A Phenomenological Model: the modified calcium control hypothesis. Ωnew([Ca 2+ (t)], Aβ) = eβ(k 1 Ω LTP k 2 Ω LTD ) ɛ 1 + e β(k 1 Ω LTP k 2 Ω LTD ) ɛ Ω LTP ([Ca 2+ (t)]) = eβ 3 ([Ca2+ (t)] α 3 ) 1 + e β 3 ([Ca2+ (t)] α 3 ) (6) Ω LTD ([Ca 2+ (t)], Aβ) = eβ 1 ([Ca2+ (t)] α 1 (Aβ)) 1 + e β 1 ([Ca2+ (t)] α 1 (Aβ)) e β 2 ([Ca 2+ (t)] α 2 (Aβ)) 1 + e β 2 ([Ca2+ (t)] α 2 (Aβ)) The constructed function Ωnew alters synaptic plasticity. Ω in Eqn. 4 with Ωnew in Eqn. 6. LT P and LT D in Eqn. 6 are interpreted as activity of each process. α 1,2,3 characterize calcium concentration when LTP and LTD processes are active. LTD process is active when the intracellular calcium concentration is between α 1 and α 2. When the calcium concentration is higher than α 3, LTP process is active. k 1,2 is activity coefficient for LTP and LTD respectively. If LTD process were blocked entirely, k 2 would become zero. β and β 1,2,3 determine the steepness of the sigmoid functions. ɛ is related to the initial synaptic strength. 12

13 A Phenomenological Model: the modified calcium control hypothesis. Simulation results Figure 2: Subthreshold LTD induction produces no change in the weight when Ω new with a normal parameter set replaced Ω in Eqn. 1; however, it induced LTD for Ω new with a pathological parameter set. 13

14 A Phenomenological Model: the modified calcium control hypothesis. Simulation results Figure 3: LTD induction protocol properly induced LTD for Ω new with a normal and pathological parameter set. 14

15 A Phenomenological Model: the modified calcium control hypothesis. Simulation results Figure 4: LTD induction protocol properly induced LTD for Ω new with a normal and pathological parameter set. Block of the LTD process (k=0.01) properly prevented the LTD induction. 15

16 A Phenomenological Model: the modified calcium control hypothesis. Simulation results Figure 5: LTP was induced for Ω new with a normal and pathological parameter set. LTP was observed for the normal, and the LTP was impaired for the pathological case. The LTP was once again observed when the LTD process was blocked. 16

17 Kinetic modeling of normal and Aβ modified synaptic plasticity Figure 6: Kinetic model of phosphorylation: D Alcantara P, Schiffmann S and Swillens S (2003): Bidirectional synaptic plasticity as a consequence of interdependent Ca2+-controlled phosphorylation and dephosphorylation pathways European Journal of Neuroscience

18 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: K d, n H, k 1 /k 2, k 3 /k 4, r 1 /r 2, r 3 /r 4, n 2 /n 1, d 1 /d 2, and p 1 /p 2. Out of these 9 parameters, r 1 /r 2, r 3 /r 4, n 2 /n 1, d 1 /d 2, and p 1 /p 2 are relevant parameters for dephosphorylation. To ease computation, we have only focused on r 3 /r 4, n 2 /n 1, and p 1 /p 2 to implement the effect of Aβ Results: n2/n1= (Normal: 0.006), p1/p2= (Normal: 0.001), r3/r4= (Normal: 100) 18

19 Figure 8: Time-dependent receptor activity responses for different calcium signals 19

20 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+]= microM. (Stimulation: [Ca2+]=0.28microM between t=0-50s) 20

21 Figure 10: Time responses of CaMKII, phosphatase, and AMPA-R activities to subthreshold LTD induction protocol, expressedâ in percent of the respective steady state activities obtained at [Ca 2 +] = microM. (Stimulation:[Ca2+] = 0.28microM between t = 0 5s) 21

22 Figure 11: Time responses of CaMKII, phosphatase, and AMPA-R activities to LTP induction protocol, expressedâ in percent of the respective steady state activities obtained at [Ca2+] = microM. (Stimulation: [Ca2+]=1.0microM between t=0-5s) 22