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please take the three handouts,and start filling out the exciting
survey
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