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MathToolsfor Neuroscience

Greg Ilana

Today:

•Introduction

•Equations are your friends

•<break>

•Remember Calculus?

The two questions:

What am I going to learn?

Why should I care?

a foreign language: Math

Because you will be tested on it.

Why Is Important for a Neuroscientist to Learn Math?

To calculate stuff

To prove stuff

To understand

to express, to describe, to communicatejust like a language

Math is a language

nouns:

verbs:

clauses:

pronouns:

sentences:

3, π, ∞cat, truth, transcendence

x, y, her, him, somestuff

+, ∫ dxrun, conjure

3x2the gray cat

God is dead. E = mc2

But I’m an American…

Precise, Unambiguous Expression

Universal and Stable

Truth-Preserving Manipulation

…why can’t you all just speak English?

Math is a special language

a foreign language?

Why do we care?

Analysis

Description

Why do we care?

Statistics

Description

Why do we care?

Statistics

Modeling

Who is the course for?

Survey

Review

What are the Goals?

Intuition & Comfort

Broad Introduction

Solid Statistics

Read Any Paper

Propose Novel Analyses

What are the Goals?

II. Probability and Statistics

I. The Basics

III. Advanced Topics

But I’m Not A Systems Neuroscientist

Molecular Genetics

Cognitive Neuroscience

Technical Stuff

Website

Technical Stuff

mathtools.stanford.edu

Technical Stuff

Problem Sets

mathtools.stanford.edu

Technical Stuff

Problem Sets

mathtools.stanford.edu

Survey & Sign Up Sheet

Lecture Notes

Feedback

Equations are Your Friends

Your Pet Equation

How to Speak Math

Why Speak Math?

Precise Expression

Universal and Stable

Truth-Preserving Manipulation

What’s in an Equation, Really?

It’s a statement.

It’s an ‘is’ statement.

17 - 3 5 = 2

What’s in an Equation, Really?

It’s an ‘is’ statement.

17 - 3 5 ‘is’ 2

What’s in an Equation, Really?

Three Types of ‘is’

Three Types of Equations

Equivalence

Evaluation

Description

Three Types of Equations

Equivalence

Evaluation

Description

Evaluation

‘is the numerical value’

€

17 − 5 × 3 = 2

€

−e iπ =1

Three Types of Equations

Equivalence

Evaluation - ‘is the numerical value’

Description

Three Types of Equations

Equivalence

Evaluation - ‘is the numerical value’

Description

Equivalence

‘is equivalent to’ ‘can be rewritten as’

€

2x + 3x = 5x

sin x∫ =−cosx

Three Types of Equations

Equivalence - ‘can be rewritten’

Evaluation - ‘has the numerical value’

Description

Three Types of Equations

Equivalence - ‘can be rewritten’

Evaluation - ‘has the numerical value’

Description

Description

‘is defined as’ ‘has the form’

€

y = mx + b

€

V =4

3πr3

Three Types of Equations

Equivalence - ‘can be rewritten’

Evaluation - ‘has the numerical value’

Description - ‘has the form’

Manipulation (get a geek)

Arithmetic (get a calculator)

Science (get a clue)

MATLAB

Mathematica

Three Types of Equations

Equivalence - ‘can be rewritten’

Evaluation - ‘has the numerical value’

Description - ‘has the form’

Manipulation (get a geek)

Arithmetic (get a calculator)

Science (get a clue)

MATLAB

Mathematica

More on Descriptive Equations

Functions and Relations

Metrics and Statistics

What are Functions?

Mappings from Input to Outputs

€

y = f (x)

What are Functions?

Mappings from Input to Outputs

€

y = f (x)

€

x

€

y

€

fdoseresponsecontrast firing ratetim

efrustration

What are Functions?

€

y = m(x) + b

€

y = sin(x)

€

V =4

3πr3

€

y = f x( )

x

y

f(x)

x

y

f(x)

x

y

f(x)

f(50) = ?1246 63.081

Functions come in all flavors

Functions come in all flavors

Functions come in all flavors

QuickTime™ and aMPEG-4 Video decompressor

are needed to see this picture.

Types of Equations

Equivalence - Manipulation (a la Mathematica)

Evaluation - Arithmetic (a la MATLAB)

Description - Science

Functions - relating input to output

Metrics(e.g. sinusoidal oscillation)

(e.g. 17 - 5 * 3 = 2 )

(e.g. 2x + 3x = 5x )

Types of Equations

Equivalence - Manipulation (a la Mathematica)

Evaluation - Arithmetic (a la MATLAB)

Description - Science

Functions - relating input to output

Metrics(e.g. sinusoidal oscillation)

(e.g. 17 - 5 * 3 = 2 )

(e.g. 2x + 3x = 5x )

What are Metrics?

Measures of a Quantity of Interest

€

σ 2 =x i − mean( )

2

Ni=1

N

∑

Often formulaic ‘something-ness’

‘fat-ness’

Types of Equations

Equivalence - Manipulation (a la Mathematica)

Evaluation - Arithmetic (a la MATLAB)

Description - Science

Functions - relating input to output

Metrics - formulaic ‘something-ness’

(e.g. sinusoidal oscillation)

(e.g. variance and mean)

(e.g. 17 - 5 * 3 = 2 )

(e.g. 2x + 3x = 5x )

How To Read an Equation

I. Consider the Context

Consider the Context

• Don’t look at the equation

• Anticipate the content

• What are we trying to describe?

How To Read an Equation

II. Identify the Variables

I. Consider the Context

Identify the Variables

Variables: x, y, z, t, v, u

Parameters: a, b, m

Indices: i, j, k, m, n

other content based names

Special Numbers: e, i,

How To Read an Equation

II. Identify the Variables

I. Consider the Context

III. Chunk It

Chunk It

• Break It Down Into Digestible Parts �• Look for Terms you recognize • Let Parentheses Guide You ( �) ( �)• Look for separate Additive Terms � + �• Look at Multiplicative Terms � �

How To Read an Equation

II. Identify the Variables

I. Consider the Context

III. Chunk It

IV. Consider the Form(s)

Forms

• Functions: sin �, cos �, log �, e �

• Operations: ∫ �dx, d �/dx

• Compact Sums and Products: ∑ �, ∏ �

How To Read an Equation

II. Identify the Variables

I. Consider the Context

III. Chunk It

IV. Consider the Form(s)

V. Imagine the Effect of Change

Let’s Do An Example

Fruit Salad!!!

<break>

Calculus Review

Differentiation

Integration

Calculus Concepts

Limits

Fundamental Theorem

Differentiation

€

d

dxf (x)

€

∂∂x

f (x)

€

′ f (x)

€

˙ f (x)

Notation:

Differentiation

Meaning:

local slope

rate of change

instantaneous rate

Differentiation

Neuroscience Examples:

€

y = f x( )

x

y

f(x)

x

y

€

g(x) =d

dxf (x)

g(x)

y

x

f(x)->g(x)

Differential Edge Detection

Differential Edge Detection

Differential Edge Detection

Center-Surround = Spatial Differentiation

Calculus Review

Differentiation

Integration

Integration

g(x)∫ dx

Notation:

€

g(x)4

17

∫ dx

g(x)—∫

Integration

Meaning:

cumulative

area under the curve

infinite sum

Integration

Neuroscience Examples?

x

y

g(x)

€

y = g x( )

x

y

g(x)

€

y = g x( )

x

y

g(x)

€

y = g x( )

€

f (x) = g(x)dx∫x

y

f(x)

f (x) = g(x)dx∫x

y

f(x)

y

x

g(x)->f(x)

The Famous Neural Integrator

Calculus Review

Differentiation

Integration

The FundamentalTheorem of Calculus

g(x) <-> f(x)

The FundamentalTheorem of Calculus

€

g(x) =d

dxf (x)

€

f (x) = g(x)dx∫

The FundamentalTheorem of Calculus

€

f (x) =d

dxf (x)

⎡ ⎣ ⎢

⎤ ⎦ ⎥dx∫

€

f (x) =d

dxf (x)dx∫[ ]

The FundamentalTheorem of Calculus

differentiation & integrationare inverses