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Using Morris-Lecar to Model Alzheimer’s Disease

BIOE 472: Neural Modeling

Tim Tang

Elena Kulikova

Marisol Montoya

That Other Guy

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Contents

Abstract ........................................................................................................................... 2 INTRODUCTION ................................................................................................................... 3

Alzheimer’s Disease ..................................................................................................... 3 Neural Modeling ........................................................................................................... 3 Morris-Lecar Model ..................................................................................................... 3

METHODS .......................................................................................................................... 4

Modeling Alzheimer’s ................................................................................................... 4 The Simulation ............................................................................................................. 5

RESULTS ........................................................................................................................... 5

Normal Neuron vs. AD Neuron .................................................................................... 5 Statistical Validation ..................................................................................................... 5 Input Current vs Equilibrium Potential ......................................................................... 5

DISCUSSION ...................................................................................................................... 7

Comparison of Normal vs Alzheimer’s ......................................................................... 7 Input Current vs Calcium Conductance ....................................................................... 7 Future Directions ......................................................................................................... 8

REFERENCES ..................................................................................................................... 8

Abstract

Alzheimer’s disease is a chronic form of dementia which primarily affects the aging population [1]. As the disease advances, it causes symptoms such as difficulty communicating, loss of spatial awareness, and mood swings [2]. In order to better understand the disease, the Morirs-Lecar model was applied to simulate the effect of Alzheimer’s on a neuron in

terms of spiking and equilibrium potential.

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Using Morris-Lecar to Model Alzheimer’s Disease

Tim Tang, Elena Kulikova, Marisol Montoya, Rostislav Tikhono

INTRODUCTION Alzheimer’s Disease Alzheimer’s disease (AD) is a chronic neurodegenerative disease which results in memory loss [1]. It is typically characterized by extracellular amyloid plaques, neurofibrillary tangles, and vascular and neuronal damage, and accounts for 60~80% of all dementia cases [3]. AD is especially prevalent in aging populations, and individuals over the age of 65 are considered “at risk” for acquiring the disease [3].

Neural Modeling In order to gain insight into physiological phenomena, engineers oftentimes implement the tools of modeling. Modeling can be profoundly intuitive by allowing experiments to be performed in otherwise physiologically difficult scenarios (frictionless environments, no recording noise, etc.). In neuroscience, neural modeling has proven to be a powerful tool in terms of understanding neuronal behavior, and allows for the collection of valuable preliminary data without performing wet lab experiments.

In the case of understanding Alzheimer’s Disease, the experiment conducted will use modeling techniques to generate simulated data of a neuron using the Morris-Lecar model, in hopes that the application of the model will yield insight into the physiological effect of AD on neurons and neural networks.

Morris-Lecar Model The Morris-Lecar model is a neuronal model developed by Catherine Morris and Harold Lecar [4]. This model, which focuses on calcium (Ca++) and potassium (K+) conductances, is a reduced form of the Hodgkin-Huxley model. This model depends upon three assumptions [5]:

1. The equations must be applied to a spatially iso-potential patch of membrane

2. The activation gates must rapidly follow changes in membrane potential

3. The dynamic probabilities can be approximated by first-order linear differential equations

The reason this model was chosen is primarily due to the amyloid plaques that form as a result of AD. These

Figure 1. Morris-Lecar model illustration [4]

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plaques contain amyloid β proteins (AβP), which result in an increase intraneuronal calcium, disrupting the calcium homeostasis [6]. It is important to note that the Morris-Lecar model in this experiment is strictly modeling the changes in calcium permea bility and conductance.

The governing equations for the Morris-Lecar model is as follows, and the circuit diagram for the model can be seen in Figures 1 and 2 [4]:

𝐶𝐶𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

= 𝐼𝐼 − 𝑔𝑔𝐿𝐿(𝑑𝑑 − 𝑑𝑑𝐿𝐿) − 𝑔𝑔𝐶𝐶𝐶𝐶𝑀𝑀𝑠𝑠𝑠𝑠(𝑑𝑑 − 𝑑𝑑𝐶𝐶𝐶𝐶)

− 𝑔𝑔𝐾𝐾𝑁𝑁(𝑑𝑑 − 𝑑𝑑𝐾𝐾) 𝑑𝑑𝑁𝑁𝑑𝑑𝑑𝑑

=𝑁𝑁𝑠𝑠𝑠𝑠 − 𝑁𝑁𝜏𝜏𝑁𝑁

𝑀𝑀𝑠𝑠𝑠𝑠 =12∗ �1 + tanh �

𝑑𝑑 − 𝑑𝑑1𝑑𝑑2

��

𝑁𝑁𝑠𝑠𝑠𝑠 =12∗ �1 + tanh �

𝑑𝑑 − 𝑑𝑑3𝑑𝑑4

��

𝜏𝜏𝑁𝑁 =1

𝜙𝜙 cosh �𝑉𝑉−𝑉𝑉32𝑉𝑉4

�

Where V is the membrane potential, M and N are recovery variables, I is the applied current, C is membrane capacitance, gL, gCa and gK are leak, calcium, and potassium conductances respectively, VL, VCa, and VK are leak, calcium, and potassium equilibrium potentials respectively, V1, V2, V3, and V4 are

tuning parameters for steady state and time constant, 𝜙𝜙 is the reference frequency, and 𝜏𝜏 is the time constant.

METHODS Modeling Alzheimer’s As explained previously, the Morris-Lecar model being used is only modeling AD in terms of a change in calcium conductance. The reason for this is to keep the initial model as simple as possible to maintain a relatively manageable scope and minimize the amount of independent variables in the experiment.

The reason calcium is being modified is due to the effect of AβPs changing the calcium permeability and conductance of the neuron. The literature values for the calcium conductance in individual affected by AD ranges from 40 ~ 4000 pico-Siemens [6]. Because this range is extraordinarily large, for the sake of this experiment, values of 30 ~ 60 pico-Siemens were used in order to simulate the effects of the early stage of AD onset. In order to model the range of conductances, a simple equation using the rand() function of MATLAB performed iterative pseudo-random simulations for neurons within the range of conductance values. This was doing as follows:

𝑔𝑔𝐶𝐶𝐶𝐶 = 𝐶𝐶𝑎𝑎𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 + 𝑟𝑟𝑎𝑎𝑟𝑟𝑑𝑑 ∗ (𝐶𝐶𝑎𝑎𝑐𝑐𝑒𝑒𝑒𝑒𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒 − 𝐶𝐶𝑎𝑎𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓)

Where gca is the calcium conductance used for the simulation, Caceiling is the max calcium conductance, and Cafloor is the minimum calcium conductance.

Figure 2. Morris-Lecar circuit diagram [4]

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The Simulation Using the aforementioned methods, 100 microamps of current was injected into the Morris-Lecar model using both nominal and AD affected calcium conductance values. Voltage plots, as well as phase diagrams for the voltage-nullcline, n-nullcline, and voltage trajectory were attained for each trial (Figure 4, 5).

The equilibriums were then averaged across 25 trials and compared to one another. After the simulation, some basic data analysis was done to quantitatively evaluate the statistical difference between the equilibrium values of the two groups.

Finally, an additional experiment was performed in modulating current input to the Morris-Lecar model neuron. The current ranged from 0 ~ 1200 microamps, increasing in increments of 50 microamps. This was then compared to the nominal and AD simulation results to see if the difference in the two models could be modeled simply in terms of input current.

RESULTS Normal Neuron vs. AD Neuron Looking at the voltage vs time graph and the phase diagram of the normal neuron, a clear spiking pattern can be seen. Over the 25 trials, the average equilibrium voltage was found to be -22.30 millivolts. An example of one of the normal simulations can be seen in Figure 4.

On the other hand, a clear distinction can be seen in the Alzheimer’s affected neuron. Using the same injected 100 microamp input current, the voltage vs time diagram shows an initial action potential spike, but no spiking afterward 50 milliseconds through the rest of the simulation. In addition, the equilibrium voltage was moved to 61.88 millivolts, a change of over 80 millivolts from baseline. The AD neuron simulation can be seen in Figure 5.

Statistical Validation

In addition to comparing the values of the normal and Alzheimer’s affected neuron simulations, statistical validation was also performed to demonstrate statistical significance between the two population groups. By performing a student’s t-test, the two populations were determined to be statistically significantly different with an α value of 4.032e-44 (Figure 3). Input Current vs Equilibrium Potential When varying the input current of the normal neuron, a change in the equilibrium potential similar to the normal vs AD neuron can be seen. With no current injection, an equilibrium

Figure 3. Histogram of equilibrium potentials

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Figure 4. Normal neuron voltage vs time and phase diagram

Figure 5. Alzheimer’s affected neuron voltage vs time and phase diagram

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potential of -60.85 millivolts can be observed, which is in the range of physiologically observed -60 ~ -70 mV range found in typical neurons [7].

When the input current is changed to 1200 microamps, however, the equilibrium potential is pushed to 66.14 millivolts, a potential greater than that observed in the AD affected neurons.

Throughout the range of input currents tested, spiking only occurred between 100 and 250 microamps of input current. All individual plots and MATLAB code are available upon request.

DISCUSSION Comparison of Normal vs Alzheimer’s A cursory analysis of the normal and AD affected neurons yields some preliminary insight into how Alzheimer’s disease might affect an individual. It appears that due to AβP increasing

calcium conductance, the excitation of the neuron is greatly affected by changing the neuron’s equilibrium potential and preventing it from repolarizing. This seems to be due to the change in conductance resulting in a greater intracellular calcium concentration, raising the equilibrium potential above the nominal range.

In addition, the trajectory and nullcline paths on the phase diagram are extremely different, but the reason for this is largely unclear. Further experiments and simulations would likely have to be performed in order to understand the significance of the phase diagram paths.

Input Current vs Calcium Conductance When comparing the current variance vs calcium conductance variance experiments, the results from the nominal values were nearly identical, which was to be expected, as there was no discernable difference between parameters other than the pseudo-

Figure 6. Comparison of nominal values for current and conductance variance simulations

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random factor used in the conductance varying experiment (Figure 6).

When comparing the AD affected neuron, even when the equilibrium potential was very similar, the phase diagrams showed vastly different Vm nullcline trajectories (Figure 7). It is currently unclear why the trajectories are different or whether or not it carries any physiological significance.

Future Directions As explained previously, the Morris-Lecar model used in this simulation was greatly simplified, and only modeled calcium conductance differences in the neuron. Future experiments should expand upon this model, building in complexity and detail in order to better simulate the effect of Alzheimer’s disease on neurons.

Additionally, it would be interesting to investigate the significance of the differences in phase diagrams, or to see how AD affects actual neural networks.

REFERENCES

[1] Burns, A; Iliffe, S (5 February 2009). "Alzheimer's disease.". BMJ (Clinical research ed.) 338: b158.

[2] "Dementia Fact sheet N°362". who.int. April 2012. 14 December 2014.

[3] "About Alzheimer's Disease: Symptoms". National Institute on Aging. Retrieved14 December 2014.

[4] Morris, Catherine; Lecar, Harold (July 1981), "Voltage Oscillations in the barnacle giant muscle fiber", Biophys J. 35 (1): 193–213

[5] Harold Lecar (2007) Morris-Lecar model. Scholarpedia, 2(10):1333., revision #91527

[6] Arispe, N., Rojas, E., & Pollard, H. B. (1993). Alzheimer disease

Figure 7. Comparison of Alzheimer's affected neuron for current and conductance variance simulations

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amyloid beta protein forms calcium channels in bilayer membranes: blockade by tromethamine and aluminum. Proceedings of the National Academy of Sciences of the United States of America, 90(2), 567–571.

[7] Purves D, Augustine GJ, Fitzpatrick D, et al., editors. Neuroscience. 2nd edition. Sunderland (MA): Sinauer Associates; 2001. Available from: http://www.ncbi.nlm.nih.gov/books/NBK10799