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Vorlesung Computational Neuroscience

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Page 1: Vorlesung Computational Neuroscience
Page 2: Vorlesung Computational Neuroscience
Page 3: Vorlesung Computational Neuroscience
Page 4: Vorlesung Computational Neuroscience
Page 5: Vorlesung Computational Neuroscience
Page 6: 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

Page 7: Vorlesung Computational Neuroscience

Page 8: Vorlesung Computational Neuroscience

• ⇒

•→

•→

Page 9: Vorlesung Computational Neuroscience

Page 10: Vorlesung Computational Neuroscience

•→ →

→ →

Page 11: Vorlesung Computational Neuroscience

+

+

2+

2+

−−3

Page 12: Vorlesung Computational Neuroscience

⇒ ⇒

IL RL rLL πa2 x = 0

V1 x = L V2

IL =V2 − V1

RL

RL = rLL

πa2

Cm Q Vm

Q = CmVm

Cm A

Page 13: Vorlesung Computational Neuroscience

cm

cm :=Cm

A

cm = 10nF/mm2

Rm =∆V

Ie

Ie ∆V

rmτm

τm

τm := rmcm.

Vm =Vi − Va

z · q E

E ≥ −zqV

Page 14: Vorlesung Computational Neuroscience

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 :

Page 15: Vorlesung Computational Neuroscience

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

Page 16: Vorlesung Computational Neuroscience

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

)

Page 17: Vorlesung Computational Neuroscience

V − Ei i

gi

im =∑

i

gi (V − Ei)

Page 18: Vorlesung Computational Neuroscience

+

ENa = +55EK = −75

Page 19: Vorlesung Computational Neuroscience

+ + Vm ≈ −60

PNa+ ≈ 125 · PK+

• ENa+ ≈ 55

Page 20: Vorlesung Computational Neuroscience

• + ⇒

• +

+ +

• +

+

+

Vm = 60− 150

+

+

+

• Vm+

•+

• Vm ↑ ⇒ + ↑ ⇒ Vm ↑↑ ⇒ ↓ +

Page 21: Vorlesung Computational Neuroscience

+ +

Page 22: Vorlesung Computational Neuroscience

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))

⇒ −

Page 23: Vorlesung Computational Neuroscience

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 )

Page 24: Vorlesung Computational Neuroscience

• 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

Page 25: Vorlesung Computational Neuroscience

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

Page 26: Vorlesung Computational Neuroscience

dgdt

2+

Pi :=( i)

( i)

gi : igi := gi · Pi :

Page 27: Vorlesung Computational Neuroscience

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

(−qBα/kBT )

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

βn(V ) := Aβ (−qBβV /kBT )

⇒ n∞(V ) =1

1 + βnαn

(V )=

1

1 +

(Aβ

Aα·

((Bα−Bβ)·V

VT

))

Page 28: Vorlesung Computational Neuroscience

•+

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 = +

Page 29: Vorlesung Computational Neuroscience

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+

Page 30: Vorlesung Computational Neuroscience

→ + +

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 )""

Page 31: Vorlesung Computational Neuroscience

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

Page 32: Vorlesung Computational Neuroscience

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

((

Page 33: Vorlesung Computational Neuroscience

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∗.

Page 34: Vorlesung Computational Neuroscience

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.

Page 35: Vorlesung Computational Neuroscience

• a

• x

→ V (x, t) x t

Page 36: Vorlesung Computational Neuroscience

∆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

Page 37: Vorlesung Computational Neuroscience

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

Page 38: Vorlesung Computational Neuroscience

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.

Page 39: Vorlesung Computational Neuroscience

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,

Page 40: Vorlesung Computational Neuroscience

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) =

(tλ

), t < −ε

−Bλ

(− t

λ

), t > ε

−2λB(− ελ

)= λ2

(−B

λ

(− ελ

)− B

λ

(−ελ

))=

εˆ−ε

v dx− rmIe2πa

.

Page 41: Vorlesung Computational Neuroscience

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

Page 42: Vorlesung Computational Neuroscience

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

2πa2µ+ rL

Lµ+1

2πa2µ+1

)−1

(2πaµLµ)−1 =

aµa2µ+1

rLLµ

(Lµa2µ+1 + Lµ+1a2µ

)

gµ,µ+1

µ µ+1

Page 43: Vorlesung Computational Neuroscience

j

FB(Gj) =1

k

k∑

i=1

Gi.

G k

j

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

Page 44: Vorlesung Computational Neuroscience

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)

Page 45: Vorlesung Computational Neuroscience

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,

Page 46: Vorlesung Computational Neuroscience

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 :

Page 47: Vorlesung Computational Neuroscience

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

(v1 v2 v3

)T.

Page 48: Vorlesung Computational Neuroscience

G∗

G∗

G∗ = 90

⇒→

Page 49: Vorlesung Computational Neuroscience

Ω ⊂ Rd

∂u

∂t= D∆u.

dh

Page 50: Vorlesung Computational Neuroscience

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

Page 51: Vorlesung Computational Neuroscience

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

Page 52: Vorlesung Computational Neuroscience

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′′

Page 53: Vorlesung Computational Neuroscience

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

Page 54: Vorlesung Computational Neuroscience

∂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.

Page 55: Vorlesung Computational Neuroscience

(−∆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)

Page 56: Vorlesung Computational Neuroscience

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]

Page 57: Vorlesung Computational Neuroscience

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 =

0

f(x)v(x) dx

D = 1

0

− d

dx

(du

dx(x)

)· v(x) dx =

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

Page 58: Vorlesung Computational Neuroscience

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 =

0

dΦidx

(x)dVj

dx(x) dx.

Page 59: Vorlesung Computational Neuroscience

K

K = h−1

2 −1

−1 2

−1−1 2

.

Page 60: Vorlesung Computational Neuroscience