Modeling altered synaptic plasticity due to Aβ effectsTakumi Matsuzawa

1. Project OverviewDiverse lines of evidence suggest that amyloid-β (Aβ) peptides have a causal role in

its pathogenesis of Alzheimer’s disease. The pathological accumulation of oligomeric Aβ assemblies depresses excitatory transmission at the synaptic level. NMDAR activation may cause either LTP or LTD, depending on intracellular calcium rise in the dendritic spines. Pathologically, the accumulation of Aβ impairs LTP, and enhances LTD; however, the mechanism has not yet been fully understood.

We adopt a phenomenological approach based on calcium control hypothesis. We assume that the change of synaptic efficacy is fully determined by the intercellular calcium concentration [Ca2+]. It postulates that a large calcium transient induces LTP, whereas moderate increase in the calcium concentration induces LTD. This assumption has received significant experimental support by now. We will first study how the calcium control hypothesis induces LTP and LTD. In the calcium control hypothesis, this depends on the Ω function in Eq. (1). The mathematical details are explained more in Section 3 Calcium Control Hypothesis.

Our second goal is to make connections between phenomenological results and biophysical models of Altzheimer. Palop and Mucke reported accumulation of Aβ suppresses excitatory synaptic transmission, and facilitates LTD after subthreshold LTD inductions (Figure 1), inducing spine shrinkage and synaptic loss. High Aβ concentration activates NMDARs, followed by influx of calcium current. The mechanism underlying the Aβ-induced LTD has not yet fully understood. Palop and Mucke proposed possible mechanisms (Figure 2). This will be helpful in making assumptions about the dynamics of the calcium. The biggest problem in the project may be how the Aβ concentration is related to the dynamics of calcium.

Figure 1. Aβ-induced basal synaptic depression

Figure 2. Presynaptic and postsynaptic regulation of synaptic transmission by Aβ (a) Hypothetical relationship between Aβ level and synaptic activity. Intermediate levels of Aβ enhance synaptic activity presynaptically, whereas abnormally high or low levels of Aβ impair synaptic activity by inducing postsynaptic depression or reducing presynaptic efficacy, respectively. (b) Within a physiological range, small increases in Aβ primarily facilitate presynaptic functions, resulting in synaptic potentiation. (c) At abnormally high levels, Aβ enhances LTD-related mechanisms, resulting in postsynaptic depression and loss of dendritic spine. Retrieved from the paper by Palop and Mucke

Thus, the goals of the project consist of two parts. 1 To study how the calcium control hypothesis induces LTP and LTD, and how the

calcium-dependent Ω function affects to the synaptic efficacy2 To relate the dynamics of calcium with concentration of Aβ biophysically

(Is the relationship between the dynamics of calcium and the Aβ concentration consistent with the mechanism proposed by Palop and Mucke? How do we confirm?)

2. ProblemsAlready, there are two big problems to be answered.1 How should we assume the relationship between the Aβ concentration and the

dynamics of calcium? How should we rewrite Eq. (3)?- Helpful papers: - Jorge J. Palop and Lennart Mucke. Amyloid- Induced Neuronal Dysfunction in β

Alzheimer’s Disease: From Synapses toward Neural Networkshttp://www.ncbi.nlm.nih.gov/pmc/articles/PMC3072750/

- Any experimental data to confirm the relationship?2 What should be the shape of the Ω function and η function in Eq. (1)?

- Shouval constructed the calcium control hypothesis with the following Ω function and η function.

→ This can be answered by studying how the shapes of the Ω function affect the synaptic efficacy.

Figure 3 The Ω function and η function in Shouval’s paper in 2002. A) The function: when [Ca2+] < θd, the synaptic weight vector stays at the basal level; when θd < [Ca2+] < θp,, the synaptic weight is reduced (LTD); for θp < [Ca2+], the synaptic weight is increased (LTP). (B) The learning rate as a function of intracellular calcium.

3 Equations of the Ω function and η function?- Helpful information:http://icwww.epfl.ch/~gerstner/SPNM/node74.html

4 Values of parameters?- If, Is, GNMDA, and H(V) in Eq. 3

3. Topics to be studied and MethodsSteps Topics Duration

1Simulation Set Up How the shape of Ω affects when LTP and LTD occurs (Calcium dependent model and Ω function) Spring Break

2 Study how dynamics of calcium depends on NMDA (Dynamics of Calcium) 2 weeks

3 Study how NMDA affects when LTP and LTD occurs (Calcium dependent plasticity) 2 weeks

4 Relate Aβ concentration and NMDA biophysically 2 weeks

The calcium control hypothesis will be applied throughout this project (See Calcium Control Hypothesis). Through the step 1, deeper understanding about the calcium dependent model should be achieved. MATLAB will be used to solve differential equations explained in Calcium Control Hypothesis, and to plot results.

Detailed dynamics of calcium by aggregation of Aβ is still unknown. Yet, it is believed that Aβ has direct effect on the NMDARs. The following paper will be reviewed with detail since it discusses Aβ and NMDARs from biological aspects.

- Danysz, W. & Parsons. C.G. Alzheimer’s disease, b-amyloid, glutamate, NMDA receptors and memantine : searching for the connections. British Journal of Pharmacology. 2012. Retrieved from http://onlinelibrary.wiley.com/store/10.1111/j.1476-5381.2012.02057.x/asset/j.1476-5381.2012.02057.x.pdf?v=1&t=i7labzzl&s=b7a14ba69c50a4241ecc518b62d7f1b4ebb99c28

These steps should confirm the result by Palop and Mucke (Figure 1). Our model should confirm long-term (~20min) basal synaptic depression after subthreshold LTD induction. Hopefully, the model implies the mechanism of Aβ-induced basal synaptic depression at some extent, connecting the phenomenological model (the LTD proposed by calcium control hypothesis) and biological model (Aβ aggregation → interaction between Aβ and NMDARs → Dynamics of Calcium).

4. Calcium Control HypothesisAβ-induced synaptic depression may result from an initial increase in synaptic

activation of NMDARs by glutamate, followed by excessive influx of calcium ions. This causes synaptic NMDAR desensitization, leading to memory loss.

A simple implementation of the calcium control hypothesis is given by the equation

Here, Wi is the synaptic weight of synapse i, η is the learning rate with dependence on calcium concentration. Ω determines the sign of the magnitude of the change of the synaptic strength. λ is a decay constant. As mentioned, dynamics of calcium depends on NMDA current. A simple assumption is that NMDARs are the primary sources of calcium. Then, the change of calcium can be expressed as calcium current through NMDARs and a decay term.

The time constant τca is 50 ms according to the paper by Shouval, Bear, and Cooper. Calcium current through IMDARs occurs as a result of activation of IMDARs by increased concentration of Aβ. Shouval, Bear, and Cooper used NMDAR voltage dependence, ligand-binding kinetics, and calcium dynamics to assume dynamics of calcium current is governed by the following equation.

where H(V) summarizes the voltage dependence as described by Jahr and Stevens. P0 = 0.5 is the fraction of NMDARs in the closed state that shift to the open state after each presynaptic spike. θ(t) is the zero if t < 0 and one if t ≥ 0. The temporal dynamics are assumed to be the sum of a fast (τf = 50 ms) and a slow (τs = 200 ms) exponential. 

5. Important References[1] Palop, J., and Mucke, L. Amyloid- Induced Neuronal Dysfunction in Alzheimer’s β

Disease: From Synapses toward Neural Networks. Retrieved from:http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3072750/

[2] Shouval, H. Z., Bear, M. F., and Cooper, L. N. (2002). A unified theory of NMDA receptor dependent bidirectional synaptic plasticity. Proc. Natl. Acad. Sci. USA, 99:10831-6. Retrieved from: http://www.pnas.org/content/99/16/10831.full.pdf

[3] Castellani GC, Quinlan EM, Cooper LN and Shouva HZ: A biophysical model of bidirectional synaptic plasticity: Dependence on AMPA and NMDA receptors. Proc. Natl. Acad. Sci. USa, 98:12772-7.

Papers containing experimental data:[4] Li S. Soluble oligomers of amyloid protein facilitate hippocampal long-term β

depression by disrupting neuronal glutamate uptake. Neuron. 2009;62:788–801 Retrieved from http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2702854/?report=reader

[5] Hsieh, H. AMPAR removal underlies A -induced synaptic depression and dendritic βspine loss. Neuron. 2006;52:831–843. Retrieved from: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1850952/?report=reader Basic modeling concepts: Calcium dependent models of bidirectional synaptic plasticity