Molekulare Ebene (biochemische Messungen) Zelluläre Ebene (Elektrophysiologie, Imaging-Verfahren) Netzwerk Ebene (Multielektrodensysteme) Areale (MRT, EEG...) Gene Neuronen Synaptische Kopplung kleine Netzwerke Gehirnregionen eg ionen und lokale Schaltkreise Verbindung von Gehirnarealen realen überlebenswichtige Proteine (Kanäle, Membran, Messenger...) Kanalaktivität, Signalempfang, Signalweiterleitung Synaptische Kopplung, Neurotransmitter, Rezeptoren Zusammenfassung von funktionellen Einheiten Makroskopische Informationsverarbeitung Global (Beobachtung) Verhalten

+ + 2+ 2+ 3

I L R L r L L πa 2 x =0 V 1 x = L V 2 I L = V 2 V 1 R L L R L = r L πa 2 C m Q V m Q = C m V m C m A

c m c m := C m A c m = 10nF /mm 2 R m = V I e I e V r m τ m τ m τ m := r m c m. V m = V i V a z q E E zqv

T k b P (E zqv )=(zqv /k b T ) R F V T k b T V T = R T ( = k ) bt F q V = V gg [] 1=[] P (E zqv ) [] =[] (zv gg /V T ) V gg = V ( ) T [] z () [] V m = RT P K [K + ] a + P A [A ] i F K A P K [K + ] i + P A [A ] a K A j Ion ( ) z : L : j = D ( d [] dx z F RT ) Vm L []

j = D d [] dx }{{} z F RT L [] Vm } {{ } (2.11) j I D I d[i] dx + z IF RT Vm L [I] =1 ˆL 0 j I D I RT L z I FV m z IFV m RT e z I FVm RT d[i] dx + z IF [ = RT Vm L ( j I = j I D I ( j I D I j I D I ˆL dx = 1 dx [I] 0 + z ) IF D I RT Vm L [I] a + z IF RT ) Vm L [I] a j I D I + z IF RT Vm L [I] i + z IF RT Vm L [I] a + z IF RT Vm L [I] i µ µ := FV m RT : ( e z Iµ j I + z ) IF D I RT Vm L [I] i j I = D Iz I µ L [I] i e z Iµ [I] a (1 e z Iµ ) P I = D I L j I = P I z I µ [I] a [I] i e z Iµ (1 e z Iµ ) ( j I + z )] IF D I RT Vm L [I] i = j I D I + z IF RT Vm L [I] a J I I J I := z I F j I = L

J = I J I =0 n I = ±1 + + K z K =+1 A z A = 1 J K = P K µf [K] a [K] i eµ 1 e µ, J A = P A µf [A] a [A] i e µ 1 e µ = P A µf [A] i [A] a eµ 1 e µ. I J I =0 P K [K] a + P A [A] i K A Fµ 1 e µ = }{{} =:u e µ = u v P K [K] i + K A }{{} =:v P A [A] a Fµeµ 1 e µ µ = u v FV P K [K] a + P A [A] i m RT = K A ( P K [K] i + P A [A] a K A V m = RT P K [K] a + P A [A] i F K A P K [K] i + P A [A] a K A + + V m = RT F ( PNa +[Na + ] a + P K +[K + ] a + P Cl [Cl ) ] i P Na +[Na + ] i + P K +[K + ] i + P Cl [Cl ] a + V m = RT F ( PNa +[Na + ) ] a P Na +[Na + ] i = RT F ( [Na + ) ] a [Na + ] i

i m = i V E i g i g i (V E i ) i

+ E Na = +55 E K = 75

+ + V m 60 P Na + 1 25 P K + E Na + 55

+ + + + + + + V m = 60 150 + + + + V m + V m + V m +

+ +

C m V = Q V C m dv dt = dq dt. dq dt I m I e dq dt = I m + I e C m dv dt = I m + I e dv c m dt = i m + I e A ( i m = i g i (V E i ))

r m = 1 g L dv c m dt dv dt τ m }{{} =c m r m = g L (V E L )+ I e A = E L V + R m I e V = V th V = V reset I e V (t) = E L + R m I e +(V (t o ) (E L + R m I e )) e t t 0 τm I e t k = t 0 + k t, k N [t k,t k+1 ] I e I (k) e V (t k+1 ) = E L + R m I e +(V (t k ) (E L + R m I e (k) )) e t τm. t [t k,t k+1 ] V (t) = E L + R m I e +(V (t k ) (E L + R m I e (k) )) e t t k τm. t 0 r := 1 t. I (t )

r V t V ( V (t )=V = E L + R m I +(V E L R m I ) V (E L + R m I ) t = τm V E L R m I ( ) V E L R m I t = τ m V E L R m I ( ( V E L R m I r = τ m V E L R m I (1) = 0 V V (x) x>0 ( ) )) 1 V E L R m I V E L + R m I > 0! t τm ) V <E L V I > 0 R m I >V E L V { ( ( )) V E τ r = m L R 1 mi V E L R m I, Rm I >V E. 0 I ( ) ( ) V E L R m I V V V V = 1+, V E L R m I V E L R m I V E L R m I r V E L R m I τ m (V V ) (1 + x) x x

f a f(x) =f(a)+ f (a) 1! (x a)+ f (a) 1! f(x) =(1 + x) 0 (1 + x) =(1 + 0) }{{} =0 + ln (1 + 0) 1! (x a) 2 + f (a) (x a) 3 +... 1! } {{ } =1 (x 0) + x. K + K + K + dv τ m dt = E L V r m g (V E K ) +R }{{} m I e dg g τ dt = g g g + g S i T i =[t i,t i+1 ] S i := ( [t i,t i+1 ]) t i+1 t i S i (i =1...n) Ti k Si k k S i S k i a

dg dt 2+ ( i) ( i) P i := g i : i g i := g i P i :

P K (V m ) k k n P K = n k n [0, 1] k k =4 + α n (V ) β n (V ) n dn dt = α n(v )(1 n) β n (V )n τ n (V ) dn dt = n (V ) n 1 τ n (V )= α n (V )+β n (V ) α n (V ) n = α n (V )+β n (V ) α n β n qb α V B α ( qb α /k B T ) α n (V ):=A α ( qb α /k B T )( A α ( B α V /V T ) A α β n (V ):=A β ( qb β V /k B T ) n (V )= 1 1+ β n α n (V ) = 1 ( 1+ A β A α ( (Bα B β) V V T ))

+ m k ( k = 3) h i ( i = 1) P Na + = m 3 h m h dm dt = α m(v )(1 m) β m (V ) m dh dt = α h(v )(1 h) β h (V ) h α m,α h ; β m,β h α n β m τ m (V ) dm dt τ m (V ) = m (V ) = = m (V ) m 1 α m (V )+β m (V )) α m (V ) α m (V )+β m (V ) + + i m = g i (V E L ) + g }{{} K n 4 (V E K ) + g }{{} Na m 3 h(v E Na ) }{{} g i =. E L =. g K =. E K = g Na =. E Na =+

dv C m dt = i m + I e A τ m (V ) dm dt τ n (V ) dn dt τ h (V ) dh dt = m (V ) m = n (V ) n = h (V ) h α n (V ) = 0.01(V + 55) 1 ( 0.1(V + 55)) β n (V ) = 0.125 ( 0.0125 (V + 65)) α m (V ) = 0.1(V + 40) 1 ( 0.1(V + 40)) β m (V ) = 4( 0.0556 (V + 65)) α h (V ) = 0.07 ( 0.05 (V + 65)) β h (V ) = 1 1+( 0.1(V + 35)) + + + + 2+

+ + S i (i =1...n) S i S j, (i, j) {1,...,n} 2 P (S i,t) t S i dp (S i,t) dt n = P (S j,t)p(s j S i ) j=1 j=1 n P (S i,t)p(s i S j ) s i S i (i, j) {1...n} 2 r ij r ji S i S j S i r ij S j r ji i ds i dt = n s j r ji j=1 n s i r ij. j=1 r ij V r ij (V ) S i S j. r ji (V )

S i S j U ij S i ( U ij /k b T ) r ij (V )=R ij ( U ij (V )/k b T ) k b R ij U ij (V ) U ij (V ) c 0 + c 1 V r ij (V ) = R ij ( U ij (V )/k b T ) = R ij ( (c 0 + c 1 V )/k b T )=R ij c 0 k bt c 1 V k b T a ij := R ij c 0 k bt, b ij := k bt. c 1 ) r ij (V )=a ij ( Vbij a ij b ij C r 1(V ) O r 2 (V ) m α m(v ) m, β m(v ) h α h(v ) h. β h (V ) o = m 3 h

C r 5 r 1 r r 2 6 r 4 8 r 3 8 8888888 I O r 1,...,r 6 C r 5 r r 2 6 r 4 8 8888888 I r 1 =0 r 3 =0 r 5 =. r 2 =. r 4 = r 6 = C r 5 r 1 r r 2 6 r 4 8 r 3 8 8888888 I r 1 = r 3 = r 5 = r 2 = r 4 = r 6 = O O r 7 C r 5 C 1 r 6 O r 4 I r 3 r 9 r 1 r 2 C 4 I 4 r 5 r 6 r 2 r 1 r 3 r 4 C 2 r 6 r 5 C 3 r 8 r 10 I 3

C r 1 O r 2 dc dt = r 2 O r 1 C do dt = r 1 C r 2 O C (1 O) do dt = r 1 (1 O) r 2 O O(t 0 )=O O(t) =O + K 1 ( (t t 0 )/τ 1 ) K 1 = O O O = τ 1 = r 1 r 1 + r 2 1 r 1 + r 2 do dt ( = K 1 1 ) ( t/τ 1 ) 1 O + 1 O τ 1 τ 1 τ 1 ) = ( 1τ1 O(t)+ 1 O τ 1 = (r 1 + r 2 ) O (r 1 + r 2 )O(t) = r 1 (r 1 + r 2 )O(t). O(t 0 ) = O + K 1 1 = O + O O = O.

C r 5 r 1 r r 2 6 r 4 8 r 3 8 8888888 I O I O do dt di dt = r 1 (1 O I) (r 2 + r 3 ) O + r 4 I = r 6 (1 O I) (r 4 + r 5 ) I + r 3 O O(t 0 )=O I(t 0 )=I O(t t 0 ) = O + K 1 ( (t t 0 )/τ 1 )+K 2 ( (t t 0 )/τ 2 ) I(t t 0 ) = I + K 3 ( (t t 0 )/τ 1 )+K 4 ( (t t 0 )/τ 2 K 1 = (O O )(a + 1/τ 2 )+b(i I ) 1 τ 2 1 τ 1 K 2 = (O O ) K 1 K 3 = K 1 a 1/τ 1 b K 4 = K 2 a 1/τ 1 b O = br 6 dr 1 ad bc I = cr 1 ar 6 ad bc a = (r 1 + r 2 + r 3 ), b = r 1 + r 4, c = r 3 r 6, d = (r 4 + r 5 + r 6 ) τ 1/2 = a + d ± 1 (a b) 2 2 2 +4bc.

a x V (x, t) x t

x Q C m V t = Q t = I L(x) I L (x + x) I m + I e, C m V I L I m I e I L R L Φ dx Φ(x + dx) Φ(x) = R L (x) I L (x), R L dx R L (x) =r L πa 2 (x) r L dx Φ(x + dx) Φ(x) = r L πa 2 (x) I L(x). dx dx 0 Φ x = r L πa 2 I L. ( ) Φ a 0 Φ x V x = (Φ i Φ a ) x I L = πa2 r L V x. C m V = E d

d E = Φ ρ i Ω Ω Ω Φ = ρ i ɛ 0. ɛ 0 ˆ ˆ ρ i Φ ndν= dµ 2πa xe = Q Q E =, ɛ 0 ɛ 0 2πa xɛ 0 E V = d 1 Q C m = ɛ 0 2πa x. ɛ 0 2πa x d }{{}}{{} =:c m =C 1 m V c m 2πa x }{{} t C m = πa2 (x) V r L x (x) ( 1) πa2 (x + x) V (x + x) r }{{} L x }{{} I L (x) I L (x+ x) I m + I e. 2πa(x) x I m I e i m i e x 0 c m V t = 1 ( a 2 V ) i m + i e 2ar L x x d a

dv =0. dx V =0. V L V = V. V (,t 0 ) V. x 1...n x V 1 (x )=V 2 (x )= = V n (x ). n n πa 2 V i I i (x )= =0. r L x x i=1 i=1

a x i m i m = V V r m. v := V V c m v t = a 2 v 2r L x }{{ 2 v + i e. r }} m {{} τ m := r m c m λ := arm 2r L τ m v t = λ2 2 v x 2 v + r mi e. v t =0 v 0 x I e x =0 2ε x <ε i e = Ie 2πa 2ε ε 0 λ 2 d2 v dx 2 = v r mi e. i e 0 x< ε x>ε λ 2 d2 v dx 2 = v,

v(x) =B 1 ( x λ )+B 2 ( x λ ) x <ε v(x) 0(x ) ( v(x) B1 x ) =0(x ) B1 =0, λ x >ε v(x) 0(x ) ( x v(x) B2 =0(x ) B2 λ) =0. B1 = B2 =: B x / [ ε, ε] ( v(x) =B x ). λ [ ε, ε] λ 2 d2 v dx 2 = v r mi e. ˆε ε λ 2 d2 v dx 2 dx = ˆε ε ( ) dv λ 2 dv (ε) dx dx ( ε) (v r m i e ) dx = ˆε ε vdx r m i e 2ε = ˆε ε I e vdx r m 2πa dv dv dx ( ε) dx (ε) dv dv t ε dx (t) t ε dx (t) dv dx (t) = ( 2λB ε ) λ { B λ ( t λ), t < ε B λ ( t λ), t > ε ( = λ 2 B ( λ ε ) B ( )) ε λ λ λ = ˆε ε vdx r mi e 2πa.

v ε 0 v 2λB 1=0 r mi e 2πa B = r mi e 4πaλ. x R R λ := v(x) = R λi e 2 ( x λ ). r m 2πaλ ( ) L λ λ := arm 2r L. 2πaL := S D λ a a = S D 2πL L λ S D V µ µ µ C m V µ t = I L (x µ 1 ) ( 2 L µ I L x µ + 1 ) 2 L µ I m + I e

I L V µ V µ+1 ) ) Φ µ Φ µ 1 Φ µ+1 Φ µ I L (x µ 1 2 L µ = r L 1 2 L µ 1 πa 2 µ 1 + r L 1 2 Lµ πa 2 µ, I L (x µ + 1 2 L µ = L r µ L L + r µ+1 2πa 2 L µ 2πa 2 µ+1 Φ V C m = ɛ 0 2πa µ L µ }{{} d =:c m c m V µ t g µ 1,µ = g µ,µ+1 = = i µ m + i µ e + g µ 1,µ (V µ V µ 1 ) g µ,µ+1 (V µ+1 V µ ) ( ( L µ 1 L µ r L 2πa 2 + r L µ 1 2πa 2 µ L µ L µ+1 r L 2πa 2 + r L µ 2πa 2 µ+1. ) 1 (2πa µ L µ ) 1 a µ a 2 µ 1 = ), r L L µ (L µ 1 a 2 µ + L µ a 2 µ 1 ) 1 (2πa µ L µ ) 1 a µ a 2 µ+1 = ) r L L µ (L µ a 2 µ+1 + L µ+1a 2 µ g µ,µ+1 µ µ+1

j F B (G j )= 1 k G i. k G k j F M (G j )= {G j1,...,g jk }. i=1

j j F G (G j ) = g(j, i) = k g(j, i) G i i=1 1 (2π) d 2 σ k 1 l=1 (2π) d 2 σ ( ) 1 i j 2 2 σ 2 ( ). 1 i l 2 2 σ 2 d σ i j i j g(j, i)

u j = D u, = = / x / y / z, D V u V V ˆ u ( x ) d x. t V V u V ˆ ˆ u j n d s = ( x ) d x. t V V j n V F ˆ ˆ F ( x ) d x = F ( x ) n d s V u = u V ˆ ˆ ˆ u ( x ) d x = j n d s = t V V V j d x,

V j u ( t = D u ). D u t = D u = u := = n i=1 2 x 2 i n D D = D = 1 0 0 0 1 0 0 0 1 5 0 0 0 1 0 0 0 1 D : M := i m i : R := 1 M mi r i r i : x i T R = 1 2 3 J lm ω l ω m l,m=1 J : ω :

J lm = i m i ( r 2 i δ lm r il r im ), { 1, l = m δ lm = r 0, il r im l m i J v 1,v 2,v 3 0 <λ 1 λ 2 λ 3 J = ( ) λ 1 v 1 v 2 v 3 λ 2 ( ) T v 1 v 2 v 3. λ 1 λ 3 λ 1 λ 2,λ 3 λ 1,λ 2 λ 3 λ 1 λ 2 λ 3 λ 3 λ 1 λ 2 1, 1: D = D L := ( ) 1 v 1 v 2 v 3 ε ( ) T v 1 v 2 v 3. λ 2 λ 3 ε λ 1 λ 1 1, 1: D = D P := ( ) 1 v 1 v 2 v 3 1 ( ) T v 1 v 2 v 3. λ 3 λ 2 ε

G G G = 90

Ω R d u t = D u. d h

h h u h t = D h u h Ω h. u t u (t) =f(t, u(t)) f u (t) u(t + h t) u(t) h t u ht (t + h t ) u ht (t) h t = f(t, u ht (t)) u ht (t + h t ) = u ht + h t f(t, u ht (t)) u ht (t + h t ) u(t + h t ) u ht u h t 0. u (t) =f(t, u(t)) t + k h t, (k =1...n) u h (t + h) =u h (t)+h Φ h (t, u h (t),u h (t + h)) Φ f

Φ h f h 0 u h u h 0 u(t + h) =u(t)+h u (t) }{{} + h2 2 u (t) } {{ } +...+ hp p! u(p) (t)+ u u = u u = f u u(x + h) u(x) (x) = h 0 }{{ h } u (x) u (x) u (x) u(x + h) u(x) h u(x) u(x h) h u(x + h) u(x h) 2h

ξ 1 (x h, x), ξ 2 (x, x + h) u(x ± h) =u(x) ± hu (x)+ h2 2 u (ξ 2/1 ) u(x + h) u(x) = u (x)+ h h 2 u (ξ 2 ) u(x) u(x h) = u (x) h h 2 u (ξ 1 ) u(x ± h) =u(x) ± hu (x)+ h2 2 u (x) ± h3 6 u (ξ 2/1 ) u(x + h) u(x h) = u (x)+ h2 ( u (ξ 1 )+u (ξ 2 ) ) 2h 6 + () ( + u)(x) := = u(x+h) u(x) h u(x) u(x h) h h u(x + h) 2u(x)+u(x h) h 2 u(x ± h) =u(x) ± hu (x)+ h2 2 u (x) ± h3 6 u (x)+ h4 u(x + h)+u(x h) =2u(x)+h 2 u (x)+ h4 4! ( + u)(x) =u (x)+ h2 ( u (4) (ξ 1 )+u (4) (ξ 2 ) 24 u 4! u(4) (ξ 2/1 ) ) ( u (4) (ξ 1 )+u (4) (ξ 2 ) )

u C 4 ( Ω) u = f ( Ω) + u h (x) =f(x) ( Ω h ) O(h 2 ) Ω h n +1 n 1 Ω h (0, 1) h = n 1 u h (h) u h (2h) u h =. u h (1 h) L h u h = q h L h = 1 h 2 2 1 1 2 1 1 2 1 1 2 q h = f(h)+h 2 ϕ 0 f(2h) f(1 h)+h 2 ϕ 1, ϕ 0 ϕ 1

u }{{} t u (t) u(t + h t) u(t) h t = D ( u) }{{} L h u h = q h x + u h (t + h t,x)=u h (t, x)+ h td h 2 (u h(t + h t,x h) 2u h (t + h t,x)+u h (t + h t,x+ h)), Ω h =(0, 1) h = n 1 2+ h2 h td 1 h t D u h (t + h t,h) u h (t, h)+ htd ϕ h 1 2+ h2 h 2 h t D u h (t + h t, 2h) 2 0 1 = u h (t, 2h). 1 u h (t + h t, 1 h) u h (t, 1 h)+ h td ϕ h 2 1 2+ h2 h td Ω=(0, 1) (0, 1) = {(x, y) : 0<x<1, 0 <y<1}. Ω Ω h (n 1) (n 1) Ω Γ h 4n h Ω h = {(x, y) Ω:x/h, y/h Z} h = 1 n, Γ h = {(x, y) Ω :x/h, y/h Z}. u = u xx u yy = f Ω, u = ϕ Γ = Ω.

( h u)(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)) 1 h = h 2 1 4 1. 1 (h, h), (2h, h),...,(1 h, h); (h, 2h),...,(1 h, 2h);...;(h, 1 h),...,(1 h, 1 h). L h u h = q h T I 4 1 L h = h 2 I T I, T = 1 4 1. I T 1 4 T I (n 1) (n 1) I (n 1) u C 4 (Ω)

A R n n A T R n n u, v R n (Au, v) =(u, A T v) (, ) R n R n u, v (u, v) ˆ1 (u, v) := u(x)v(x) dx. 0 A A T A A T A u, v (Au, v) =(u, A v). A = d dx u, v [0, 1] (Au, v) = ( ) d = d dx dx. ( ) d dx u, v = ˆ1 0 d u(x)v(x) dx = dx ˆ1 0 ( u(x) d ) dx v(x) dx + uv 1 0 }{{} =0 L 2 (Ω) Ω=[0, 1]

u d ( D(x) d ) dx dx u = f. v(x) ( d ( D(x) d ) ) dx dx u,v = (f,v) ˆ1 d ( D(x) d ) ˆ1 dx dx u v(x) dx = f(x)v(x) dx 0 D =1 0 ˆ1 0 d ( ) du dx dx (x) v(x) dx = ˆ1 0 du dv du (x) (x) dx dx dx dx (x)v(x) 1 0 u(x) =v(x) =0 Γ du dx (x)v(x) 1 =0 0 v v U

u U Φ 1 (x),...φ n (x) u u(x) U(x) =U 1 Φ 1 (x)+...+ U n Φ n (x). U 1...U n V 1...V n U 1...U n V i V i V i = Φ i i =1...n KU = F K F i ˆ1 0 ˆ1 0 ˆ1 du dx (x)dv i (x) dx = f(x)v i (x) dx dx 0 ˆ1 n du j dφ j dx dx (x) V i (x) dx = f(x)v i (x) dx. j=1 i K (U i ) i=1...n i F (i, j) 0 K ij = ˆ1 0 dφ i dx (x)dv j (x) dx. dx

K 2 1 K = h 1 1 2 1. 1 2