Basic Models in Theoretical Neuroscience

Oren Shriki

2010

Differential Equations

Two Types of Dynamical Systems

• Differential equations:Describe the evolution of systems in continuous time.

• Difference equations / Iterated maps:Describe the evolution of systems in discrete time.

What is a Differential Equation?• Any equation of the form:

• For example:

0,,,2

2

dx

yd

dx

dyyF

0644

4

dx

ydy

dx

dy

Order of a Differential Equation

• The order of a differential equation is the order of the highest derivative in the equation.

• A differential equation of order n has the form:

0,,,,2

2

n

n

dx

yd

dx

yd

dx

dyyF

1st Order Differential Equations

• A 1st order differential equation has the form:

• For example:

yxfdx

dy,

yxdx

dy 2

Separable Differential Equations

• Separable equations have the form:

• For example:

yhxgdx

dy

yxdx

dy 2

Separable Differential Equations

• How to solve separable equations?

• If h(y)≠0 we can write:

• Integrating both sides with respect to x we obtain:

xgyh

xy

'

dxxgdxxyyh

'1

Separable Differential Equations

• By substituting:

• We obtain:

dxxydy

xyx

'

dxxgyh

dy

Example 1

xeydx

dy 21

Cey

Cey

dxey

dy

x

x

x

tan

tan

1

1

2

Example 2

xyydx

dye x sinln

xdxeyy

dy x sinln

1lnlnln

Cyyy

dy

Integrating the left side:

Example 2 (cont.)

2cossin2

1sin

sincossincossinsin

Cxxexdxe

xdxexexexdxexexdxe

xx

xxxxxx

Cxxey

Cxxey

x

x

cossin2

1expexp

cossin2

1lnln

Integrating the right side:

Thus:

Linear Differential Equations

• The standard form of a 1st order linear differential equation is:

• For example:

xQyxPdx

dy

xyxdx

dysin

Linear Differential Equations

General solution:

• Suppose we know a function v(x) such that:

• Multiplying the equation by v(x) we obtain:

yxvxPdx

dyvyxv

dx

d

dxxQxvxv

y

xQxvyxvdx

d

xQxvyxvxPdx

dyxv

1

)(

Linear Differential Equations

• The condition on v(x) is:

• This leads to:

yxvxPdx

dyvyxv

dx

d

yxvxPdx

dvy

yxvxPdx

dyv

dx

dvy

dx

dyv

Linear Differential Equations

• The last equation will be satisfied if:

• This is a separable equation:

xvxPdx

dv

dxxPev

dxxPv

dxxPv

dv

ln

Linear Differential Equations

• To sum up:

• Where:

dxxQxvxv

y

xQyxPdx

dy

1

dxxPexv

Example

• Solution:

xeyxdx

dyx sinhcosh

xeeexv xdxxdxxPcosh)cosh(ln)tanh()(

x

eyx

dx

dy x

coshtanh

)cosh(x

exQ

x

Example (cont.)

dxx

ex

x

dxxQxvxv

y

x

cosh)cosh(

)cosh(

1

1

Cex

xy x

cosh

1)(

Derivative with respect to time

• We denote (after Newton):

dt

dxx