The Treatment of Bipolar Disorder: Modeling Lithiums Effect on - PowerPoint PPT Presentation

The Treatment of Bipolar Disorder: Modeling Lithium’s Effect on Neuronal Bursting Rosa Rossi-Goldthorpe Bowdoin College January 27th, 2019 NCUWM 2019

The Biology The Model and the Math Lithium in the Model Lithium Treatment Table of Contents The Biology The Model and the Math Lithium in the Model Lithium Treatment 2 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Table of Contents The Biology The Model and the Math Lithium in the Model Lithium Treatment 3 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Bipolar Disorder • Mental illness characterized by “unusual shifts in mood, energy, activity levels, and the ability to carry out day-to-day tasks” (NIMH) • Extreme mood “episodes” tend to be between two extreme mood-states: mania and depression (NIMH) • The transition between extreme moods is not cyclical; both internal and environmental stressors can trigger a mood swing 4 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Lithium as a Mood Stabilizer • Lithium is the most commonly prescribed drug for bipolar disorder, the only widely used mood-stabilizer • Newer mood stabilizers are not typically as effective (Thies-Flechtner, 1996) • Competing hypotheses about biological mechanism of bipolar and how lithium corrects it • Lithium treatment tailored to individual 5 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Proposed Mechanism Potential Genetic Mechanism High IP 3 Excess Ca 2+ High NT Release Gene Mutation Gene mutation associated with high risk of bipolar disorder causes elevated Ca 2+ levels(Psychiatric Genomics Consortium, 2011) Proposed Lithium Mechanism Inhibits IP 3 R action Lowers Ca 2+ Lithium Decrease NT 6 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Neurons and Neurotransmitters (NT) Neurons: • Type of cell in the nervous system • Carries information throughout the nervous system by chemical and electrical signals (NINDS) • Electrical impulse to chemical signal to electrical impulse Neurotransmitters: • Chemical messengers that transmit information to a neuron • Norepinephrine implicated in euphoria/grandiosity, defining characteristics of mania (Goodwin, 1974) 7 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Action Potentials (AP) • APs are electrical impulses caused by membrane depolarization (Bear, 84) • Arrival of AP in terminal triggers release of NT(Bear, 122) 8 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Model Hypothesis • Burst-frequency determines amount of NT released (Wilkins, 232) Assume high NT release causes mania. And lithium treat- ment works by lowering intracellular calcium which lowers burst frequency. This decreases NT release which stabi- lizes mood. ? Lowers Ca 2+ Decrease burst freq. Lithium Decrease NT 9 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Table of Contents The Biology The Model and the Math Lithium in the Model Lithium Treatment 10 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Morris-Lecar Model dV dt = 1 5 6 ( I − g ca M ∞ ( V − V ca ) − g k W ( V − V k ) − g l ( V − V l )) C dW = φ ( W ∞ − W ) dt τ W 1 3 V − V 1 5 46 M ∞ ( V ) = 1 + tanh 2 V 2 1 3 V − V 3 5 46 W ∞ ( V ) = 1 + tanh 2 V 4 11 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Bursting in 2-D ML Model What is a“burst”? • “a rapid cluster of action potentials followed by a brief pause” (Bear, 106) • 2-D model, every action potential represents a burst (Williams, 2013) 1 1 Hayashi, 2016. Orange by Rosa 12 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Nernst Equation for Calcium I Ca = g Ca M ∞ ( V − V Ca ) Reversal potential is the membrane voltage at electochemical equilibrium (Bear, 68) 3 [ Ca 2+ ] out V Ca = RT 4 2 F ln [ Ca 2+ ] in So, if intracellular Ca 2+ concentration, [ Ca 2+ ] in , increases, the V Ca would decrease 13 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Table of Contents The Biology The Model and the Math Lithium in the Model Lithium Treatment 14 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Lithium Addition to Morris-Lecar Model How does the addition of lithium change the model? ? Lowers Ca 2+ Decrease burst freq. Lithium Decrease NT If bipolar disorder is caused by abnormal intracellular lev- els of calcium, where intracellular calcium is elevated, we would expect the reversal potential of calcium, V Ca , to be lower than the average value.

The Biology The Model and the Math Lithium in the Model Lithium Treatment Lithium Addition to Morris-Lecar Model How does the addition of lithium change the model? ? Lowers Ca 2+ Decrease burst freq. Lithium Decrease NT ? Increase V Ca If bipolar disorder is caused by abnormal intracellular lev- els of calcium, where intracellular calcium is elevated, we would expect the reversal potential of calcium, V Ca , to be lower than the average value. 15 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Bifurcations Using V ca as the bifurcation parameter 16 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Bifurcations Emergence of Limit Cycles, V Ca = 85 . 6 T ≈ 73 . 85; f ≈ . 0135, for stable LC 17 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Frequency of Limit Cycle 0.015 0.0145 0.014 0.0135 0.013 0.0125 0.012 0.0115 0.011 0 20 40 60 80 100 120 140 160 180 200 18 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Response Function for Lithium to Vca V ca = α tanh( . 003( L ( t ) − 1000)) + β ; α and β are scaling factors to account for physiological differences 130 120 110 100 90 80 70 60 0 200 400 600 800 1000 1200 1400 1600 1800 2000 19 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Table of Contents The Biology The Model and the Math Lithium in the Model Lithium Treatment 20 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Lithium Concentration in Bloodstream Differential equation for exponential decay: dL = − α L dt L (0) = L 0 ln(2) = α 36 Solution Flow: L 0 e − α t φ t ( L 0 ) = L n = φ τ ( L n − 1 ) + κ 21 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Equilibrium for Lithium Concentration FDA has the standard lithium dosage at 300 mg every 8 hours ( τ = 8, κ = 300): 2200 L * 2000 1800 1600 1400 1200 1000 800 600 400 t * 200 0 0 2 4 6 8 10 12 10 8 Where ( L ∗ ) and when ( t ∗ ) does L stabilize? 22 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Equilibrium for Lithium Concentration FDA has the standard lithium dosage at 300 mg every 8 hours ( τ = 8, κ = 300): L ∗ 2097 . 1 mg ≈ t ∗ 13 days ≈ 2200 L * 2000 1800 1600 1400 1200 1000 800 600 400 t * 200 0 0 2 4 6 8 10 12 10 8 23 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Numerical Simulations First 36 hours of Lithium, 5 doses 1400 1200 1000 800 600 400 200 0 2 4 6 8 10 12 14 16 18 10 7 24 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Numerical Simulations 20 10 0 V -10 -20 -30 -40 0 100 200 300 400 500 600 700 800 900 1000 t 25 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Numerical Simulations 1400 1200 1000 800 600 400 200 0 2 4 6 8 10 12 14 16 18 10 7 26 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Numerical Simulations 20 10 0 V -10 -20 -30 -40 0 100 200 300 400 500 600 700 800 900 1000 t 27 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Numerical Simulations 1400 1200 1000 800 600 400 200 0 2 4 6 8 10 12 14 16 18 10 7 28 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Numerical Simulations 40 0.5 0.45 30 0.4 20 0.35 10 0.3 0 V 0.25 -10 0.2 -20 0.15 -30 0.1 -40 0.05 -50 0 0 100 200 300 400 500 600 700 800 900 1000 -50 -40 -30 -20 -10 0 10 20 30 40 t 29 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Numerical Simulations 16 days of Lithium treatment 2200 2000 1800 1600 1400 1200 1000 800 600 400 200 0 2 4 6 8 10 12 14 10 8 30 / 38

The Biology The Model and the Math Lithium in the Model Lithium Treatment Numerical Simulations 50 0.6 40 0.5 30 20 0.4 10 0.3 V 0 -10 0.2 -20 -30 0.1 -40 0 -50 -50 -40 -30 -20 -10 0 10 20 30 40 50 0 100 200 300 400 500 600 700 800 900 1000 t 31 / 38