Brain Rhythms Sue Ann Campbell Department of Applied Mathematics - - PowerPoint PPT Presentation

Brain Rhythms

Sue Ann Campbell

Department of Applied Mathematics University of Waterloo

October 14, 2017

Outline

1

Biological Background

2

Mathematical Background

3

Modelling Rhythms

4

Summary

Sue Ann Campbell (Waterloo) FILOMACS 2 / 30

What are brain rhythms?

The electrical activity of the brain exhibits characteristic wave forms

What are brain rhythms?

The electrical activity of the brain exhibits characteristic wave forms These vary depending on the brain state and are characterized by their frequency

How do brain rhythms arise?

Need to understand a bit of physiology

How do brain rhythms arise?

A magnified slice of the brain

How do brain rhythms arise?

An individual neuron

How do brain rhythms arise?

Behaviour of individual neurons

How do brain rhythms arise?

Behaviour of brain - network of neurons

What is an oscillator?

Anything that varies periodically in time

What is an oscillator?

Anything that varies periodically in time Pendulum

What is an oscillator?

Anything that varies periodically in time Metronome

What is an oscillator?

Anything that varies periodically in time Firefly

What is an oscillator?

Anything that varies periodically in time

–3 –2 –1 1 2 3 θ 2 4 6 8 t

Neurons are oscillators

What happens when oscillators are connected?

Movies Brain slice: https://www.youtube.com/watch?v=t3TaMU_qXMc Fireflies: https://www.youtube.com/watch?v=ZGvtnE1Wy6U 32 metronomes:https://www.youtube.com/watch?v=5v5eBf2KwF8 2 metronomes: https://www.youtube.com/watch?v=yysnkY4WHyM

What happens when oscillators are connected?

Synchronization - oscillators all reach maximum at same time

What happens when oscillators are connected?

Synchronization - oscillators all reach maximum at same time Phase-locking - oscillators have fixed phase difference

Some Mathematics - Recursively Defined Sequences

Recall the Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, . . .

Some Mathematics - Recursively Defined Sequences

Recall the Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, . . . Can write a general formula/algorithm for generating the terms of this sequence x0 = 1 x1 = 1 xn = xn−1 + xn−2, n = 2, 3, 4, . . .

Some Mathematics - Recursively Defined Sequences

Recall the Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, . . . Can write a general formula/algorithm for generating the terms of this sequence x0 = 1 x1 = 1 xn = xn−1 + xn−2, n = 2, 3, 4, . . . A recursively defined sequence

Modelling With Recursively Defined Sequences

Let xn represent the value of some variable at time n Basic Idea: current value = (previous value) + (change)

Modelling With Recursively Defined Sequences

Let xn represent the value of some variable at time n Basic Idea: current value = (previous value) + (change) xn+1 = xn + ∆

Modelling With Recursively Defined Sequences

Let xn represent the value of some variable at time n Basic Idea: current value = (previous value) + (change) xn+1 = xn + ∆ Assumption: change in current value only depends on previous value

Modelling With Recursively Defined Sequences

Let xn represent the value of some variable at time n Basic Idea: current value = (previous value) + (change) xn+1 = xn + ∆ Assumption: change in current value only depends on previous value xn+1 = xn + g(xn)

Modelling With Recursively Defined Sequences

Let xn represent the value of some variable at time n Basic Idea: current value = (previous value) + (change) xn+1 = xn + ∆ Assumption: change in current value only depends on previous value xn+1 = xn + g(xn) Recursive formula for an unknown sequence x0, x1, x2, x3, ...

Simple Model

Assumption: Change is proportional to current value g(xn) = axn where a is a constant.

Simple Model

Assumption: Change is proportional to current value g(xn) = axn where a is a constant. Model: xn+1 = xn + axn = (1 + a)xn

Simple Model

Solving the Model Let starting value (time 0) be arbitrary: x0

Simple Model

Solving the Model Let starting value (time 0) be arbitrary: x0 Day 1: x1 = (1 + a)x0 = (1 + a)x0

Simple Model

Solving the Model Let starting value (time 0) be arbitrary: x0 Day 1: x1 = (1 + a)x0 = (1 + a)x0 Day 2: x2 = (1 + a)x1 = (1 + a)2x0

Simple Model

Solving the Model Let starting value (time 0) be arbitrary: x0 Day 1: x1 = (1 + a)x0 = (1 + a)x0 Day 2: x2 = (1 + a)x1 = (1 + a)2x0 Day n: xn = (1 + a)nx0

Simple Model

Solving the Model Let starting value (time 0) be arbitrary: x0 Day 1: x1 = (1 + a)x0 = (1 + a)x0 Day 2: x2 = (1 + a)x1 = (1 + a)2x0 Day n: xn = (1 + a)nx0 A geometric sequence

Simple Model

Solving the Model Let starting value (time 0) be arbitrary: x0 Day 1: x1 = (1 + a)x0 = (1 + a)x0 Day 2: x2 = (1 + a)x1 = (1 + a)2x0 Day n: xn = (1 + a)nx0 A geometric sequence What happens to population if a = 0? a > 0? −1 < a < 0?

Plotting the Solution

a = 0, x0 = 5 a = 0: Constant solution

Plotting the Solution

a > 0, x0 = 1 a > 0: Exponential Growth

Plotting the Solution

−1 < a < 0, x0 = 10 a < 0: Exponential Decay

Modelling Oscillators

Define the phase of the oscillator

Amplitude t (time) t +T t 0 t+∆ t

θ (phase)

∆t/T 2π∆ 2π 1 t/T

Modelling Oscillators with Recursive Sequences

Think of oscillator as angle of point moving on circle with fixed radius

x y θ

Change in θ in a small time interval θn+1 = θn + Ω

Two Coupled Oscillators

θ1,n+1 = θ1n + Ω + A sin(θ2n − θ1n) θ2,n+1 = θ2n + Ω + A sin(θ1n − θ2n) where A is a small number (|A| << 1).

Two Coupled Oscillators

θ1,n+1 = θ1n + Ω + A sin(θ2n − θ1n) θ2,n+1 = θ2n + Ω + A sin(θ1n − θ2n) where A is a small number (|A| << 1). Define phase difference φn = θ2n − θ1n φn+1 = φn − 2A sin(φn)

Two Coupled Oscillators

Simulations of model with A = 0.1 φ0 = 6, 5, 4, 3, 2, 1, 0 φn+1 = φn − 2A sin(φn)

Two Coupled Oscillators

Simulations of model with A = 0.1 φ0 = 6, 5, 4, 3, 2, 1, 0 φn+1 = φn − 2A sin(φn)

Two Coupled Oscillators

Simulations of model with A = 0.1 φ0 = 6, 5, 4, 3, 2, 1, 0 φn+1 = φn − 2A sin(φn)

Two Coupled Oscillators

Simulations of model with A = 0.1 φ0 = 6, 5, 4, 3, 2, 1, 0 φn+1 = φn − 2A sin(φn)

Two Coupled Oscillators

Simulations of model with A = 0.1 φ0 = 6, 5, 4, 3, 2, 1, 0 φn+1 = φn − 2A sin(φn)

Two Coupled Oscillators

Simulations of model with A = 0.1 φ0 = 6, 5, 4, 3, 2, 1, 0 φn+1 = φn − 2A sin(φn)

Two Coupled Oscillators

Simulations of model with A = 0.1 φ0 = 6, 5, 4, 3, 2, 1, 0 φn+1 = φn − 2A sin(φn)

Two Coupled Oscillators

Simulations of model with A = 0.1 φ0 = 6, 5, 4, 3, 2, 1, 0 φn+1 = φn − 2A sin(φn)

Two Coupled Oscillators

φn+1 = φn − 2A sin(φn) Special constant solutions: φn = φ∗, n = 1, 2, . . . (equilibrium solutions) Correspond to state the phase difference between the two oscillators is fixed (phase locking).

Two Coupled Oscillators

φn+1 = φn − 2A sin(φn) Special constant solutions: φn = φ∗, n = 1, 2, . . . (equilibrium solutions) Correspond to state the phase difference between the two oscillators is fixed (phase locking). Occur when sin(φ∗) = 0

Two Coupled Oscillators

φn+1 = φn − 2A sin(φn) Two possibilities φ = 0(2π): periodic solutions with two oscillators in-phase φ = π: periodic solutions with two oscillators anti-phase (one half period out of phase).

Two Coupled Oscillators

Two Coupled Oscillators

φn+1 = φn − 2A sin(φn) How the equilibrium solutions affect the behaviour depends on the sign of A.

Two Coupled Oscillators

φn+1 = φn − 2A sin(φn) How the equilibrium solutions affect the behaviour depends on the sign of A. A > 0

Two Coupled Oscillators

φn+1 = φn − 2A sin(φn) How the equilibrium solutions affect the behaviour depends on the sign of A. A < 0

Two Coupled Oscillators

φn+1 = φn − 2A sin(φn) How the equilibrium solutions affect the behaviour depends on the sign of A. A < 0

Two Coupled Oscillators - Different Frequencies

θ1,n+1 = Ω1 + A sin(θ2n − θ1n) θ2,n+1 = Ω2 + A sin(θ1n − θ2n) ⇓ φn+1 = ω + φn − 2A sin(φn)

Two Coupled Oscillators - Different Frequencies

θ1,n+1 = Ω1 + A sin(θ2n − θ1n) θ2,n+1 = Ω2 + A sin(θ1n − θ2n) ⇓ φn+1 = ω + φn − 2A sin(φn) Equilibrium solutions: φ∗ such that sin(φ∗) = ω.

Two Coupled Oscillators - Different Frequencies

θ1,n+1 = Ω1 + A sin(θ2n − θ1n) θ2,n+1 = Ω2 + A sin(θ1n − θ2n) ⇓ φn+1 = ω + φn − 2A sin(φn) Equilibrium solutions: φ∗ such that sin(φ∗) = ω. ω = 0: Two possibilities φ∗ = 0: in-phase φ∗ = π: anti-phase

Two Coupled Oscillators - Different Frequencies

θ1,n+1 = Ω1 + A sin(θ2n − θ1n) θ2,n+1 = Ω2 + A sin(θ1n − θ2n) ⇓ φn+1 = ω + φn − 2A sin(φn) Equilibrium solutions: φ∗ such that sin(φ∗) = ω. −1 < ω < 1 Two possibilities φ∗ close to in-phase φ∗: close to anti-phase

Two Coupled Oscillators - Different Frequencies

θ1,n+1 = Ω1 + A sin(θ2n − θ1n) θ2,n+1 = Ω2 + A sin(θ1n − θ2n) ⇓ φn+1 = ω + φn − 2A sin(φn) Equilibrium solutions: φ∗ such that sin(φ∗) = ω. ω < −1 or ω > 1: No equilibrium solutions

Two Coupled Oscillators - More Complicated Interaction

θ1,n+1 = Ω + A1 sin(θ2n − θ1n) + A2 sin(2(θ2n − θ1n)) + . . . θ2,n+1 = Ω + A1 sin(θ1n − θ2n) + A2 sin(2(θ1n − θ2n)) + . . . ⇓ φn+1 = φn − 2(A1 sin(φn) + A2 sin(2φn) + . . .) Still have equilibrium solutions at φ∗ = 0, π, but could have others as well.

Two Coupled Oscillators - More Complicated Interaction

θ1,n+1 = Ω + A1 sin(θ2n − θ1n) + A2 sin(2(θ2n − θ1n)) + . . . θ2,n+1 = Ω + A1 sin(θ1n − θ2n) + A2 sin(2(θ1n − θ2n)) + . . . ⇓ φn+1 = φn − 2(A1 sin(φn) + A2 sin(2φn) + . . .) Still have equilibrium solutions at φ∗ = 0, π, but could have others as well.

More Oscillators

θ1,n+1 = Ω + A12 sin(θ2n − θ1n) + A13 sin(θ3n − θ1n) + A14 sin(θ4n − θ1n) + θ2,n+1 = Ω + A12 sin(θ1n − θ2n) + A23 sin(θ3n − θ2n) + A24 sin(θ4n − θ2n) + θ3,n+1 = Ω + A13 sin(θ1n − θ3n) + A23 sin(θ2n − θ3n) + A34 sin(θ4n − θ3n) + θ4,n+1 = Ω + A14 sin(θ1n − θ4n) + A24 sin(θ2n − θ4n) + A34 sin(θ3n − θ4n) + . . . Many equilibrium solutions.

How Can Rhythms Arise?

Consider 6 identical oscillators. Synchronized

How Can Rhythms Arise?

Consider 6 identical oscillators. Phase-locked with phase difference 1/2 period.

How Can Rhythms Arise?

Consider 6 identical oscillators. Phase-locked with phase difference 1/3 period.

How Can Rhythms Arise?

Average - summed effect of all the oscillators

Summary

1

The brain is made up of many neurons.

2

Individual neurons act as oscillators - the electrical activity varies periodically in time.

3

Mathematical analysis shows that coupling such oscillators together can lead to synchronization or phase-locking.

4

Different types of phase-locking lead to different global (summed)

• scillations/frequencies.

5

This is one way brain rhythms may be produced.