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please take the three handouts, and start filling out the exciting survey. MathTools for Neuroscience. Greg. Ilana. Today:. Introduction Equations are your friends <break> Remember Calculus?. The two questions:. What am I going to learn?. Math. a foreign language:. Why should I care?.

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today

Today:

  • Introduction
  • Equations are your friends
  • <break>
  • Remember Calculus?
the two questions

The two questions:

What am I going to learn?

Math

a foreign language:

Why should I care?

Because you will be tested on it.

why is important for a neuroscientist to learn math

Why Is Important for a Neuroscientist to Learn Math?

To calculate stuff

To prove stuff

To understand

to express, to describe, to communicate

just like a language

math is a language

cat, truth,

transcendence

her, him,

somestuff

Mathis a language

3, π, ∞

nouns:

pronouns:

x, y, 

verbs:

run, conjure

+, ∫ dx

clauses:

the gray cat

3x2

sentences:

God is dead.

E = mc2

but i m an american

a foreign language?

…why can’t you all just speak English?

But I’m an American…

Mathis aspeciallanguage

Universal and Stable

Precise, Unambiguous Expression

Truth-Preserving Manipulation

why do we care

Why do we care?

Analysis

Description

why do we care1

Why do we care?

Statistics

Description

why do we care2

Why do we care?

Statistics

Modeling

what are the goals

What are the Goals?

Broad Introduction

Intuition & Comfort

Solid Statistics

Read Any Paper

Propose Novel Analyses

what are the goals1

What are the Goals?

I. The Basics

II. Probability and Statistics

III. Advanced Topics

but i m not a systems neuroscientist

But I’m Not A Systems Neuroscientist

Cognitive Neuroscience

Molecular Genetics

technical stuff1

Technical Stuff

mathtools.stanford.edu

technical stuff2

Technical Stuff

mathtools.stanford.edu

Problem Sets

technical stuff3

Technical Stuff

mathtools.stanford.edu

Problem Sets

Survey & Sign Up Sheet

Lecture Notes

Feedback

equations are your friends

Equations are Your Friends

How to Speak Math

Your Pet Equation

why speak math

Why Speak Math?

Universal and Stable

Precise Expression

Truth-Preserving Manipulation

what s in an equation really1

What’s in an Equation, Really?

It’s an ‘is’ statement.

17 - 3  5 = 2

what s in an equation really2

What’s in an Equation, Really?

It’s an ‘is’ statement.

17 - 3  5 ‘is’ 2

three types of equations

Three Types of Equations

Evaluation

Equivalence

Description

three types of equations1

Three Types of Equations

Evaluation

Equivalence

Description

evaluation

Evaluation

‘is the numerical value’

three types of equations2

Three Types of Equations

Evaluation - ‘is the numerical value’

Equivalence

Description

three types of equations3

Three Types of Equations

Evaluation - ‘is the numerical value’

Equivalence

Description

equivalence

Equivalence

‘is equivalent to’ ‘can be rewritten as’

three types of equations4

Three Types of Equations

Evaluation - ‘has the numerical value’

Equivalence - ‘can be rewritten’

Description

three types of equations5

Three Types of Equations

Evaluation - ‘has the numerical value’

Equivalence - ‘can be rewritten’

Description

description

Description

‘is defined as’ ‘has the form’

three types of equations6

Three Types of Equations

Evaluation - ‘has the numerical value’

MATLAB

Arithmetic(get a calculator)

Equivalence - ‘can be rewritten’

Manipulation(get a geek)

Mathematica

Description - ‘has the form’

Science(get a clue)

three types of equations7

Three Types of Equations

Evaluation - ‘has the numerical value’

MATLAB

Arithmetic(get a calculator)

Equivalence - ‘can be rewritten’

Manipulation(get a geek)

Mathematica

Description - ‘has the form’

Science(get a clue)

more on descriptive equations

More on Descriptive Equations

Functions and Relations

Metrics and Statistics

what are functions

What are Functions?

Mappings from Input to Outputs

what are functions1

response

firing rate

frustration

dose

contrast

time

What are Functions?

Mappings from Input to Outputs

slide43

f(x)

Still a function

y

x

slide44

f(x)

Not a function

y

12

f(50) = ?

63.081

46

x

types of equations

Types of Equations

Evaluation - Arithmetic (a la MATLAB)

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

Equivalence - Manipulation (a la Mathematica)

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

Description -Science

Functions - relating input to output

(e.g. sinusoidal oscillation)

Metrics

types of equations1

Types of Equations

Evaluation - Arithmetic (a la MATLAB)

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

Equivalence - Manipulation (a la Mathematica)

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

Description -Science

Functions - relating input to output

(e.g. sinusoidal oscillation)

Metrics

what are metrics

What are Metrics?

Measures of a Quantity of Interest

Often formulaic ‘something-ness’

‘fat-ness’

types of equations2

Types of Equations

Evaluation - Arithmetic (a la MATLAB)

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

Equivalence - Manipulation (a la Mathematica)

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

Description -Science

Functions - relating input to output

(e.g. sinusoidal oscillation)

Metrics - formulaic ‘something-ness’

(e.g. variance and mean)

how to read an equation

How To Read an Equation

I. Consider the Context

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 equation1

How To Read an Equation

I. Consider the Context

II. Identify the Variables

identify the variables

Identify the Variables

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

other content based names

Parameters: , a, b, m

Indices: i, j, k, m, n

Special Numbers: e, i, 

how to read an equation2

How To Read an Equation

I. Consider the Context

II. Identify the Variables

III. Chunk It

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 equation3

How To Read an Equation

I. Consider the Context

II. Identify the Variables

III. Chunk It

IV. Consider the Form(s)

forms
Forms
  • Functions: sin, cos, log, e
  • Operations: ∫ dx, d/dx
  • Compact Sums and Products: ∑, ∏
how to read an equation4

How To Read an Equation

I. Consider the Context

II. Identify the Variables

III. Chunk It

IV. Consider the Form(s)

V. Imagine the Effect of Change

let s do an example

Let’s Do An Example

Fruit Salad!!!

calculus review

Calculus Review

Differentiation

Integration

calculus concepts

Calculus Concepts

Limits

Fundamental Theorem

differentiation1
Differentiation

Meaning:

rate of change

local slope

instantaneous rate

differentiation2
Differentiation

Neuroscience Examples:

calculus review1

Calculus Review

Differentiation

Integration

integration
Integration

Notation:

integration1
Integration

Meaning:

infinite sum

area under the curve

cumulative

integration2
Integration

Neuroscience Examples?

calculus review2

Calculus Review

Differentiation

Integration

the fundamental theorem of calculus3
The FundamentalTheorem of Calculus

differentiation & integration

are inverses