Vorlesung Computational Neuroscience

Gene

SynaptischeKopplung

kleine Netzwerke

Neuronen

Gehirnregionen undlokale Schaltkreise

Verbindung vonGehirnarealen

Verhalten

Zelluläre Ebene

Netzwerk Ebene

Areale

Global

überlebenswichtige Proteine (Kanäle,Membran, Messenger ...)

Kanalaktivität, Signalempfang, Signalweiterleitung

Synaptische Kopplung, Neurotransmitter, Rezeptoren

Zusammenfassung von funktionellen Einheiten

Makroskopische Informationsverarbeitung

Molekulare Ebene

(biochemische Messungen)

(Elektrophysiologie, Imaging-Verfahren)

(Multielektrodensysteme)

(MRT, EEG ...)

(Beobachtung)

tischeung

etzwer

onen

ionenaltkre

ung vorealen

apppl

Ne

uro

regSch

ndrna

SynaptischeKopplung

kleine Netzwerke

Neuronen

Gehirnregionen und Schaltkreise

Verbindung vonGehirnareale

Gene

•

•

•

•

•

•

⇒

•

•

•

•

•

•

• ⇒

•

•

•

•

•

•

•

•

•→

•

•

•

•→

•

•

•

•

•

•

•

•

⇒

•

•

•

⇒

→

•

•

•

•→ →

→ →

•

•

•

•

•

•

•

•

•

•

•

+

+

2+

2+

−−3

⇒ ⇒

IL RL rLL πa2 x = 0

V1 x = L V2

IL =V2 − V1

RL

RL = rLL

πa2

⇒

Cm Q Vm

Q = CmVm

Cm A

cm

cm :=Cm

A

cm = 10nF/mm2

Rm =∆V

Ie

Ie ∆V

rmτm

τm

τm := rmcm.

Vm =Vi − Va

z · q E

E ≥ −zqV

T kb

P (E ≥ −zqV ) = (zqV /kbT )

R FVT

kbT

VT =R · TF

(=

kbT

q

)

V = Vgg

[ ] · 1 = [ ] · P (E ≥ −zqV )

⇔ [ ] = [ ] (zVgg/VT )

⇔ Vgg =VT

z

([ ]

[ ]

)( )

⇒

Vm =RT

F

∑K

PK [K+]a +∑APA[A−]i

∑K

PK [K+]i +∑APA[A−]a

•

•

jIon

(∗) j = −D

(d [ ]

dx− z F

RT· Vm

L[ ]

)

z :L :

j = −D

d [ ]

dx︸ ︷︷ ︸− z F

RT· Vm

L[ ]

︸ ︷︷ ︸

(2.11) ⇔d[I]dx

− jIDI

+ zIFRT · Vm

L [I]= 1

⇒L

0

d[I]dx

− jIDI

+ zIFRT · Vm

L [I]dx =

L

0

1 dx

⇔ RTL

zIFVm

[ (− jIDI

+zIF

RT· Vm

L[I]a

)−

(− jIDI

+zIF

RT· Vm

L[I]i

)]= L

⇔ zIFVm

RT=

(− jI

DI+ zIF

RT · VmL [I]a

− jIDI

+ zIFRT · Vm

L [I]i

)

⇔ ezIFVm

RT =− jI

DI+ zIF

RT · VmL [I]a

− jIDI

+ zIFRT · Vm

L [I]i

µ µ :=FVm

RT:

⇔ ezIµ(− jIDI

+zIF

RT· Vm

L[I]i

)= − jI

DI+

zIF

RT· Vm

L[I]a

⇔ jI = −DIzIµ

L

[I]i · ezIµ − [I]a(1− ezIµ)

PI=DIL⇔ jI = PI · zI · µ · [I]a − [I]i · ezIµ

(1− ezIµ)

JI IJI := zIF · jI

J =∑

I JI = 0 nI = ±1+ + −

K zK = +1

JK = PKµF[K]a − [K]i e

µ

1− eµ,

A zA = −1

JA = PAµF[A]a − [A]i e

−µ

1− e−µ= PAµF

[A]i − [A]a eµ

1− eµ.

∑I JI = 0

∑

K

PK [K]a +∑

A

PA[A]i

︸ ︷︷ ︸=:u

· Fµ

1− eµ=

∑

K

PK [K]i +∑

A

PA[A]a

︸ ︷︷ ︸=:v

· Fµeµ

1− eµ

⇔ eµ =u

v

⇔ µ =u

v

⇔ FVm

RT=

∑K

PK [K]a +∑APA[A]i

(∑K

PK [K]i +∑APA[A]a

⇔ Vm =RT

F

∑K

PK [K]a +∑APA[A]i

∑K

PK [K]i +∑APA[A]a

+ + −

Vm =RT

F

(PNa+ [Na+]a + PK+ [K+]a + PCl− [Cl−]iPNa+ [Na+]i + PK+ [K+]i + PCl− [Cl−]a

)

+

Vm =RT

F

(PNa+ [Na+]aPNa+ [Na+]i

)=

RT

F

([Na+]a[Na+]i

)

⇒

V − Ei i

gi

im =∑

i

gi (V − Ei)

+

ENa = +55EK = −75

+ + Vm ≈ −60

PNa+ ≈ 125 · PK+

•

• ENa+ ≈ 55

• + ⇒

• +

+ +

• +

+

+

Vm = 60− 150

+

+

+

• Vm+

•+

• Vm ↑ ⇒ + ↑ ⇒ Vm ↑↑ ⇒ ↓ +

•

+ +

⇒

CmV = Q

V

CmdV

dt=

dQ

dt.

dQdt

Im

Ie

dQ

dt= −Im + Ie

⇒ CmdV

dt= −Im + Ie

⇔ cmdV

dt= −im +

IeA

( im =∑

i

gi (V − Ei))

⇒ −

cmdV

dt= −gL (V − EL) +

IeA

⇔rm= 1

gL

τm︸︷︷︸=cmrm

dV

dt= EL − V +RmIe

V = Vth ! V = Vreset

Ie

V (t) = EL +RmIe + (V (to)− (EL +RmIe)) e− t−t0

τm

Ie tk = t0 + k ·∆t, k ∈ N[tk, tk+1] Ie

I(k)e

V (tk+1) = EL +RmIe + (V (tk)− (EL +RmI(k)e )) e−∆tτm .

t ∈ [tk, tk+1]

V (t) = EL +RmIe + (V (tk)− (EL +RmI(k)e )) e−t−tkτm .

∆t → 0

r :=1

t.

•I (t )

• rV

t V

V (t ) = V = EL +RmI + (V − EL −RmI )

(− t

τm

)

⇔V − (EL +RmI )

V − EL −RmI=

(− t

τm

)

⇔ t = τm ·(V − EL −RmI

V − EL −RmI

)

⇒ r =

(τm

(V − EL −RmI

V − EL −RmI

))−1

(1) = 0 V += V

(x) x > 0

⇒V − EL −RmI

V − EL +RmI> 0 !

V < EL VI > 0

RmI > V − EL

V

r =

(τm

(V −EL−RmIV −EL−RmI

))−1, RmI > V − E

0.

I

(V − EL −RmI

V − EL −RmI

)=

(1 +

V − V

V − EL −RmI

)≈ V − V

V − EL −RmI,

r ≈V −EL−RmIτm(V −V )

(1 + x) ≈ x x

fa

f(x) = f(a) +f ′(a)

1!(x− a) +

f ′′(a)

1!(x− a)2 +

f ′′′(a)

1!(x− a)3 + . . . .

f(x) = (1 + x) 0

(1 + x) = (1 + 0)︸ ︷︷ ︸=0

+ln′(1 + 0)

1!︸ ︷︷ ︸=1

(x− 0) + · · · ≈ x.

⇒ K+

K+

K+

τmdV

dt= EL − V − rmg (V − EK)︸ ︷︷ ︸+RmIe

g τ dgdt = −g

g −→ g +∆g

Si Ti = [ti, ti+1]

Si :=( [ti, ti+1])

ti+1 − ti

Si (i = 1 . . . n)T ki Sk

i k

Si

Ski

⇒

a

dgdt

↔

2+

⇒

≈

Pi :=( i)

( i)

gi : igi := gi · Pi :

PK(Vm)

kk n

PK = nk n ∈ [0, 1]

k k = 4 +

αn(V ) βn(V )

n

dn

dt= αn(V )(1− n)− βn(V )n

⇔ τn(V )dn

dt= n∞(V )− n

τn(V ) =1

αn(V ) + βn(V )

n∞ =αn(V )

αn(V ) + βn(V )

αn βn qBαV

Bα

(−qBα/kBT )

⇒ αn(V ) := Aα (−qBα/kBT )(≡ Aα (−BαV /VT )

Aα

βn(V ) := Aβ (−qBβV /kBT )

⇒ n∞(V ) =1

1 + βnαn

(V )=

1

1 +

(Aβ

Aα·

((Bα−Bβ)·V

VT

))

•

•+

mk ( k = 3) →hi ( i = 1)

→ PNa+ = m3h

m h

dm

dt= αm(V )(1−m)− βm(V ) ·m

dh

dt= αh(V )(1− h)− βh(V ) · h

αm,αh; βm,βh αn βm

τm(V )dm

dt= m∞(V )−m

τm(V ) =1

αm(V ) + βm(V ))

m∞(V ) =αm(V )

αm(V ) + βm(V )

+ +

im = gi(V − EL)︸ ︷︷ ︸+ gKn4(V − EK)︸ ︷︷ ︸+ gNam3h(V − ENa)︸ ︷︷ ︸

gi = . − EL = − .

gK = . − EK = −gNa = . − ENa = +

CmdV

dt= −im +

IeA

τm(V )dm

dt= m∞(V )−m

τn(V )dn

dt= n∞(V )− n

τh(V )dh

dt= h∞(V )− h

αn(V ) =0.01(V + 55)

1− (−0.1 (V + 55))

βn(V ) = 0.125 (−0.0125 (V + 65))

αm(V ) =0.1(V + 40)

1− (−0.1 (V + 40))

βm(V ) = 4 (−0.0556 (V + 65))

αh(V ) = 0.07 (−0.05 (V + 65))

βh(V ) =1

1 + (−0.1 (V + 35))

−

• + +

•

•

•

+ +

→ 2+

→ + +

→

Si (i = 1 . . . n)Si ! Sj , (i, j) ∈ 1, . . . , n2

P (Si, t)t Si

dP (Si, t)

dt=

n∑

j=1

P (Sj , t)P (Sj → Si)−n∑

j=1

P (Si, t)P (Si → Sj)

si Si

(i, j) ∈ 1 . . . n2 rij rji Si

Sj Si

rij !! Sjrji""

i

dsidt

=n∑

j=1

sjrji −n∑

j=1

sirij .

rijV

Si

rij(V )!! Sj .

rji(V )""

Si Sj Uij

Si (−Uij/kbT )

rij(V ) = Rij (−Uij(V )/kbT )

kbRij

Uij(V )

Uij(V ) ≈ c0 + c1V

⇒ rij(V ) = Rij (−Uij(V )/kbT )

= Rij (−(c0 + c1V )/kbT ) = Rij

−c0kbT ·

−c1VkbT

aij := Rij

−c0kbT , bij :=

kbT

c1.

⇒ rij(V ) = aij ·(− V

bij

)

aij bij

Cr1(V )

!! Or2(V )""

mαm(V )

!! m∗,βm(V )""

hαh(V )

!! h∗.βh(V )""

o = m3h

Cr1 !!

r6

##!!!

!!!!

! Or2

""

r3$$""""""""

I

r4%%""""""""r5

&&!!!!!!!!

r1, . . . , r6

Cr6

##!!!

!!!!

! Or2

""

I

r4%%""""""""r5

&&!!!!!!!!

r1 = 0 r3 = 0 r5 = . −

r2 = . − r4 = − r6 = − −

Cr1 !!

r6

##!!!

!!!!

! Or2

""

r3$$""""""""

I

r4%%""""""""r5

&&!!!!!!!!

r1 = − − r3 = − r5 = −

r2 = − r4 = − r6 = − −

Cr5 !! C1r6

""r5 !! C2r6

""

r5''

Or4 !!

r9''

C4r3""

r2 !! C3r1""

r6

((

r10''

I

r7

((

r1 !! I4r2""

r3 !! I3r4""

r8

((

Cr1 !! Or2

""

dC

dt= r2 ·O − r1 · C

dO

dt= r1 · C − r2 ·O

C(1−O)

dO

dt= r1 · (1−O)− r2 ·O

O(t0) = O∗

O(t) = O∞ +K1 (−(t− t0)/τ1)

K1 = O∗ −O∞

O∞ =r1

r1 + r2

τ1 =1

r1 + r2

dO

dt= K1

(− 1

τ1

)(−t/τ1)−

1

τ1O∞ +

1

τ1O∞

=

(− 1

τ1

)O(t) +

1

τ1O∞

= (r1 + r2) ·O∞ − (r1 + r2)O(t)

= r1 − (r1 + r2)O(t).

O(t0) = O∞ +K1 · 1= O∞ +O∗ −O∞

= O∗.

Cr1 !!

r6

##!!!

!!!!

! Or2

""

r3$$""""""""

I

r4%%""""""""r5

&&!!!!!!!!

O I

dO

dt= r1 (1−O − I)− (r2 + r3)O + r4 I

dI

dt= r6 (1−O − I)− (r4 + r5) I + r3O

O(t0) = O∗ I(t0) = I∗

O(t− t0) = O∞ +K1 (−(t− t0)/τ1) +K2 (−(t− t0)/τ2)

I(t− t0) = I∞ +K3 (−(t− t0)/τ1) +K4 (−(t− t0)/τ2

K1 =(O∗ −O∞)(a+ 1/τ2) + b(I∗ − I∞)

1τ2

− 1τ1

K2 = (O∗ −O∞)−K1

K3 = K1−a− 1/τ1

b

K4 = K2−a− 1/τ1

b

O∞ =br6 − dr1ad− bc

I∞ =cr1 − ar6ad− bc

a = −(r1 + r2 + r3) , b = −r1 + r4,

c = r3 − r6 , d = −(r4 + r5 + r6)

τ1/2 = −a+ d

2± 1

2

√(a− b)2 + 4bc.

• a

• x

→ V (x, t) x t

•

∆xQ

Cm∂V

∂t=∂Q

∂t= IL(x)− IL(x+∆x)− Im + Ie,

Cm

VILImIe

IL RL Φdx

Φ(x+ dx)− Φ(x) = −RL(x) · IL(x),

RL

RL(x) = rLdx

πa2(x)

rL

Φ(x+ dx)− Φ(x) = −rLdx

πa2(x)· IL(x).

dx dx → 0

∂Φ

∂x= − rL

πa2· IL.

Φa ≡ 0 ∂Φ∂x

∂V∂x

(= ∂(Φi−Φa)

∂x

)

IL = −πa2

rL

∂V

∂x.

Cm

V = E · d

d E = −∇Φ

ρi Ω

−∆Φ =ρiε0.

ε0

ˆ

∂Ω

−∇Φ · n dν =

ˆ

Ω

ρiε0

dµ ⇔ 2πa∆xE =Q

ε0⇔ E =

Q

2πa∆xε0,

E

V =d

ε0

1

2πa∆x︸ ︷︷ ︸=C−1

m

Q ⇒ Cm =ε0d︸︷︷︸

=:cm

2πa∆x.

cm2πa∆x︸ ︷︷ ︸Cm

∂V

∂t= −πa

2(x)

rL

∂V

∂x(x)

︸ ︷︷ ︸IL(x)

− (−1) · πa2(x+∆x)

rL

∂V

∂x(x+∆x)

︸ ︷︷ ︸IL(x+∆x)

− Im + Ie.

2πa(x)∆xIm Ie im ie

∆x → 0

cm∂V

∂t=

1

2arL

∂

∂x

(a2∂V

∂x

)− im + ie

d ! a

dV

dX= 0.

V = 0.

VL

V = V .

V (·, t0) ≡ V .

x∗1 . . . n x∗

V1(x∗) = V2(x∗) = · · · = Vn(x∗).

n∑

i=1

Ii(x∗) =n∑

i=1

πa2

rL

∂Vi

∂x x∗

= 0.

a x

im

im =V − V

rm.

v := V − V

cm∂v

∂t=

a

2rL

∂2v

∂x2︸ ︷︷ ︸− v

rm+ ie

︸ ︷︷ ︸.

τm := rmcm λ :=√

arm2rL

τm∂v

∂t= λ2

∂2v

∂x2− v + rmie.

∂v∂t = 0

• v → 0 |x| → ∞

• Ie x = 0

2ε |x| < εie = Ie

2πa·2ε ε → 0

λ2d2v

dx2= v − rmie.

ie ≡ 0 x < −ε x > ε

λ2d2v

dx2= v,

v(x) = B1 (−x

λ) +B2 (

x

λ)

x < ε v(x) → 0 (x → −∞)

v(x) → B1

(−x

λ

)= 0 (x → −∞) ⇒ B1 = 0,

x > ε v(x) → 0 (x → ∞)

v(x) → B2

(xλ

)= 0 (x → ∞) ⇒ B2 = 0.

B1 = B2 =: Bx /∈ [−ε, ε]

v(x) = B

(− |x|λ

).

[−ε, ε]

λ2d2v

dx2= v − rmie.

εˆ−ε

λ2d2v

dx2dx =

εˆ−ε

(v − rmie) dx

⇔ λ2(dv

dx(ε)− dv

dx(−ε)

)=

εˆ−ε

v dx− rmie · 2ε =

εˆ−ε

v dx− rmIe2πa

dvdx(−ε)

dvdx(ε)

t↑−ε

dvdx(t) t↓ε

dvdx(t)

dv

dx(t) =

Bλ

(tλ

), t < −ε

−Bλ

(− t

λ

), t > ε

−2λB(− ελ

)= λ2

(−B

λ

(− ελ

)− B

λ

(−ελ

))=

εˆ−ε

v dx− rmIe2πa

.

vε→ 0 v

−2λB · 1 = 0− rmIe2πa

B =rmIe4πaλ

.

x ∈ R Rλ := rm2πaλ

v(x) =RλIe2

(− |x|λ

).

Lλ(

λ :=√

arm2rL

).

2πaL := SD

λ a a = SD2πL

Lλ SD

→

•

•

• V µ

µ

µ

Cm∂Vµ

∂t= IL

(xµ − 1

2Lµ

)− IL

(xµ +

1

2Lµ

)− Im + Ie

IL

Vµ Vµ+1

IL

(xµ − 1

2Lµ

)=

Φµ − Φµ−1

rL12Lµ−1

πa2µ−1+ rL

12Lµ

πa2µ

, IL

(xµ +

1

2Lµ

)=

Φµ+1 − Φµ

rLLµ

2πa2µ+ rL

Lµ+1

2πa2µ+1

.

Φ V

Cm =ε0d︸︷︷︸

=:cm

2πaµLµ

cm∂Vµ

∂t= −iµm + iµe + gµ−1,µ (Vµ − Vµ−1)− gµ,µ+1 (Vµ+1 − Vµ)

gµ−1,µ =

(rL

Lµ−1

2πa2µ−1

+ rLLµ

2πa2µ

)−1

(2πaµLµ)−1 =

aµa2µ−1

rLLµ

(Lµ−1a2µ + Lµa2µ−1

) ,

gµ,µ+1 =

(rL

Lµ

2πa2µ+ rL

Lµ+1

2πa2µ+1

)−1

(2πaµLµ)−1 =

aµa2µ+1

rLLµ

(Lµa2µ+1 + Lµ+1a2µ

)

gµ,µ+1

µ µ+1

•

•

j

FB(Gj) =1

k

k∑

i=1

Gi.

G k

j

FM (Gj) = Gj1 , . . . , Gjk .

j

j

FG(Gj) =k∑

i=1

g(j, i) ·Gi

g(j, i) =

1

(2π)d2 σ

·(−1

2‖i−j‖2

σ2

)

k∑l=1

1

(2π)d2 σ

·(−1

2‖i−l‖2

σ2

) .

d σ‖i−j‖ i j

g(j, i)

u

,j = −D,∇u,

= ,∇ =

∂/∂x∂/∂y∂/∂z

,

D

V uV V

ˆ

V

∂u

∂t(,x) d,x.

V uV

−ˆ

∂V

,j · ,n ds =

ˆ

V

∂u

∂t(,x) d,x.

,j · ,n

V ,Fˆ

V

,F (,x) d,x =

ˆ

∂V

,F (,x) · ,n ds

u = ,∇ · u

ˆ

V

∂u

∂t(,x) d,x = −

ˆ

∂V

,j · ,n ds = −ˆ

V

,j d,x,

V ,j

∂u

∂t=

(D,∇u

).

D

∂u

∂t= D ,∇u = ∆u

∆ := =n∑

i=1

∂2

∂x2i

n

D

D =

1 0 00 1 00 0 1

⇒

D =

5 0 00 1 00 0 1

⇒

D

: M :=∑

i

mi

: R :=1

M

∑mi,ri ,ri : xi

TR =1

2

3∑

l,m=1

Jlm ωl ωm

J :

ω :

Jlm =∑

i

mi(r2i · δlm − ril rim

),

δlm =

1, l = m0,

ril rim l m

i

J

v1, v2, v3 0 < λ1 ≤ λ2 ≤ λ3

J =(v1 v2 v3

)

λ1

λ2λ3

(v1 v2 v3

)T.

λ1 λ3

λ1 λ2,λ3 →λ1,λ2 λ3 →λ1 ≈ λ2 ≈ λ3 →

λ1λ2

2 1,λ2λ3

≈ 1 : D = DL :=(v1 v2 v3

)

1εε

(v1 v2 v3

)T.

λ1λ3

2 1,λ1λ2

≈ 1 : D = DP :=(v1 v2 v3

)

1

1ε

(v1 v2 v3

)T.

•

•

G∗

G∗

G∗ = 90

⇒→

Ω ⊂ Rd

∂u

∂t= D∆u.

dh

h

∂huh∂t

= −D∆huh Ωh.

∂u∂t

u′(t) = f(t, u(t))f

u′(t) ≈ u(t+ ht)− u(t)

ht

uht(t+ ht)− uht(t)

ht= f(t, uht(t))

⇔ uht(t+ ht) = uht + ht · f(t, uht(t))

uht(t+ ht) u(t+ ht)

uht → u ht → 0.

u′(t) = f(t, u(t))

t+ k · ht, (k = 1 . . . n)

uh(t+ h) = uh(t) + h · Φh(t, uh(t), uh(t+ h))

Φ f

Φh → f

h → 0

uh → u

h → 0

u(t+ h) = u(t) + h · u′(t)︸ ︷︷ ︸+h2

2u′′(t)

︸ ︷︷ ︸

+ . . .+hp

p!u(p)(t) +

∆u

∆u = u′′

−∆u = f′

u′(x) =h→0

u(x+ h)− u(x)

h︸ ︷︷ ︸

u′(x) ≈ u(x+ h)− u(x)

h

u′(x) ≈ u(x)− u(x− h)

h

u′(x) ≈ u(x+ h)− u(x− h)

2h

ξ1 ∈ (x − h, x), ξ2 ∈ (x, x + h)

u(x± h) = u(x)± hu′(x) +h2

2u′′(ξ2/1)

⇒ u(x+ h)− u(x)

h= u′(x) +

h

2u′′(ξ2)

u(x)− u(x− h)

h= u′(x)− h

2u′′(ξ1)

u(x± h) = u(x)± hu′(x) +h2

2u′′(x)± h3

6u′′′(ξ2/1)

⇒ u(x+ h)− u(x− h)

2h= u′(x) +

h2

6

(u′′′(ξ1) + u′′′(ξ2)

)

∂+ ∂−

′′( )

(∂−∂+u)(x) :=u(x+h)−u(x)

h − u(x)−u(x−h)h

h

=u(x+ h)− 2u(x) + u(x− h)

h2

u(x± h) = u(x)± hu′(x) +h2

2u′′(x)± h3

6u′′′(x) +

h4

4!u(4)(ξ2/1)

⇒ u(x+ h) + u(x− h) = 2u(x) + h2u′′(x) +h4

4!

(u(4)(ξ1) + u(4)(ξ2)

)

⇒ (∂−∂+u)(x) = u′′(x) +h2

24

(u(4)(ξ1) + u(4)(ξ2)

)

u′′

u ∈ C4(Ω)

−∆u = f ( Ω)

− ∂−∂+uh(x) = f(x) ( Ωh)

O(h2)

Ωh

n+ 1n − 1

Ωh (0, 1) h = n−1

uh =

uh(h)uh(2h)

uh(1− h)

.

Lhuh = qh

Lh =1

h2

2 −1−1 2 −1

−1 2

−1−1 2

qh =

f(h) + h−2ϕ0f(2h)

f(1− h) + h−2ϕ1

,

ϕ0 ϕ1

∂u

∂t︸︷︷︸= D · (∆u)︸ ︷︷ ︸

Lhuh = qh

u′(t) ≈ u(t+ ht)− u(t)

ht

x∂−∂+

uh(t+ ht, x) = uh(t, x) +htD

h2(uh(t+ ht, x− h)− 2uh(t+ ht, x) + uh(t+ ht, x+ h)) ,

Ωh = (0, 1) h = n−1

htD

h2

2 + h2

htD−1

−1 2 + h2

htD

−1

−1 2 + h2

htD

uh(t+ ht, h)uh(t+ ht, 2h)

uh(t+ ht, 1− h)

=

uh(t, h) +htDh2 ϕ0

uh(t, 2h)

uh(t, 1− h) + htDh2 ϕ1

.

Ω = (0, 1)× (0, 1) = (x, y) : 0 < x < 1, 0 < y < 1 .

Ω Ωh (n−1)× (n−1)∂Ω Γh 4n h

Ωh = (x, y) ∈ Ω : x/h, y/h ∈ Z h =1

n,

Γh = (x, y) ∈ ∂Ω : x/h, y/h ∈ Z .

−∆u = −uxx − uyy = f Ω,

u = ϕ Γ = ∂Ω.

(−∆hu)(x, y) :=(−∂−x ∂+x − ∂−y ∂+y

)u(x, y)

= −h−2(u(x− h, y) + u(x+ h, y)

+u(x, y − h) + u(x, y + h)− 4u(x, y))

−∆h = h−2

−1

−1 4 −1−1

.

(h, h), (2h, h), . . . , (1− h, h); (h, 2h), . . . , (1− h, 2h); . . . ; (h, 1− h), . . . , (1− h, 1− h).

Lhuh = qh

Lh = h−2

T −I

−I T

−I−I T

, T =

4 −1

−1 4

−1−1 4

.

T I (n− 1)× (n− 1) I

(n − 1)

•

•u ∈ C4(Ω)

A ∈ Rn×n AT ∈ Rn×n

u, v ∈ Rn

(Au, v) = (u,AT v)

(·, ·) Rn

Rn

u, v(u, v)

(u, v) :=

1ˆ0

u(x)v(x) dx.

A AT AAT A∗

u, v

(Au, v) = (u,A∗v).

A = ddx

u, v [0, 1]

(Au, v) =

(d

dxu, v

)=

1ˆ0

d

dxu(x)v(x) dx =

1ˆ0

u(x) ·(− d

dxv(x)

)dx + u v

∣∣∣1

0︸︷︷︸= 0

⇒(

d

dx

)∗= − d

dx.

L2(Ω) Ω = [0, 1]

u

− d

dx

(D(x)

d

dxu

)= f.

v(x)

(− d

dx

(D(x)

d

dxu

), v

)= (f, v)

⇔1ˆ

0

− d

dx

(D(x)

d

dxu

)v(x) dx =

1ˆ

0

f(x)v(x) dx

D = 1

1ˆ

0

− d

dx

(du

dx(x)

)· v(x) dx =

1ˆ

0

du

dx(x)

dv

dx(x) dx − du

dx(x)v(x)

∣∣∣1

0

u(x) = v(x) = 0 Γ

−dudx(x)v(x)

∣∣∣1

0= 0

v

vU

uU

Φ1(x), . . .Φn(x) u

u(x) ≈ U(x) = U1Φ1(x) + . . .+ UnΦn(x).

U1 . . . Un

V1 . . . Vn U1 . . . Un

Vi

Vi

Vi = Φi i = 1 . . . n

KU = F

K F

i

1ˆ0

dU

dx(x)

dVi

dx(x) dx =

1ˆ0

f(x)Vi(x) dx

⇔1ˆ

0

n∑

j=1

dUj

dx

dΦjdx

(x)

Vi(x) dx =

1ˆ0

f(x)Vi(x) dx.

i K(Ui)i=1...n i F (i, j)

Kij =

1ˆ

0

dΦidx

(x)dVj

dx(x) dx.

K

K = h−1

2 −1

−1 2

−1−1 2

.