Lecture 9: Population genetics, first-passage problems
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Transcript of Lecture 9: Population genetics, first-passage problems
![Page 1: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/1.jpg)
Lecture 9: Population genetics, first-passage problems
Outline:• population genetics
• Moran model• fluctuations ~ 1/N but not ignorable• effect of mutations• effect of selection
• neurons: integrate-and-fire models• interspike interval distribution
•no leak• with leaky cell membrane
• evolution• traffic
![Page 2: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/2.jpg)
Population genetics: Moran model
2 alleles, N haploid organisms
![Page 3: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/3.jpg)
Population genetics: Moran model
2 alleles, N haploid organismschoose 2 individual at random: 1 dies, the other reproduces
![Page 4: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/4.jpg)
Population genetics: Moran model
2 alleles, N haploid organismschoose 2 individual at random: 1 dies, the other reproducesIf there are n1 organisms of type 1 before this step, then afterwards there are
![Page 5: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/5.jpg)
Population genetics: Moran model
2 alleles, N haploid organismschoose 2 individual at random: 1 dies, the other reproducesIf there are n1 organisms of type 1 before this step, then afterwards there are
n1 + 1 with probability x(1 – x) x = n1/N
![Page 6: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/6.jpg)
Population genetics: Moran model
2 alleles, N haploid organismschoose 2 individual at random: 1 dies, the other reproducesIf there are n1 organisms of type 1 before this step, then afterwards there are
n1 + 1 with probability x(1 – x) x = n1/Nn1 – 1 with probability x(1 – x)
![Page 7: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/7.jpg)
Population genetics: Moran model
2 alleles, N haploid organismschoose 2 individual at random: 1 dies, the other reproducesIf there are n1 organisms of type 1 before this step, then afterwards there are
n1 + 1 with probability x(1 – x) x = n1/Nn1 – 1 with probability x(1 – x)n1 with probability x2 + (1 – x)2.
![Page 8: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/8.jpg)
Population genetics: Moran model
2 alleles, N haploid organismschoose 2 individual at random: 1 dies, the other reproducesIf there are n1 organisms of type 1 before this step, then afterwards there are
n1 + 1 with probability x(1 – x) x = n1/Nn1 – 1 with probability x(1 – x)n1 with probability x2 + (1 – x)2.
So
€
n1(t + δt) = n1(t)
![Page 9: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/9.jpg)
Population genetics: Moran model
2 alleles, N haploid organismschoose 2 individual at random: 1 dies, the other reproducesIf there are n1 organisms of type 1 before this step, then afterwards there are
n1 + 1 with probability x(1 – x) x = n1/Nn1 – 1 with probability x(1 – x)n1 with probability x2 + (1 – x)2.
So
€
n1(t + δt) = n1(t)
n1(t + δt) − n1(t + δt)( )2
= 2n1(t)n2(t) = 2n1(t) 1− n1(t)( )
![Page 10: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/10.jpg)
Population genetics: Moran model
2 alleles, N haploid organismschoose 2 individual at random: 1 dies, the other reproducesIf there are n1 organisms of type 1 before this step, then afterwards there are
n1 + 1 with probability x(1 – x) x = n1/Nn1 – 1 with probability x(1 – x)n1 with probability x2 + (1 – x)2.
So
€
n1(t + δt) = n1(t)
n1(t + δt) − n1(t + δt)( )2
= 2n1(t)n2(t) = 2n1(t) 1− n1(t)( )
or
€
x(t + δt) = x(t)
![Page 11: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/11.jpg)
Population genetics: Moran model
2 alleles, N haploid organismschoose 2 individual at random: 1 dies, the other reproducesIf there are n1 organisms of type 1 before this step, then afterwards there are
n1 + 1 with probability x(1 – x) x = n1/Nn1 – 1 with probability x(1 – x)n1 with probability x2 + (1 – x)2.
So
€
n1(t + δt) = n1(t)
n1(t + δt) − n1(t + δt)( )2
= 2n1(t)n2(t) = 2n1(t) 1− n1(t)( )
or
€
x(t + δt) = x(t)
x(t + δt) − x(t + δt)( )2
=2x(t) 1− x(t)( )
N 2
![Page 12: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/12.jpg)
continuum limit: FP equation
€
δt = 1N (N steps/generation)
![Page 13: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/13.jpg)
continuum limit: FP equation
€
δt = 1N
⇒∂P(x, t)
∂t= 1
2 σ 2 ∂ 2
∂x 2x(1− x)P(x, t)[ ], σ 2 =
2
N
(N steps/generation)
![Page 14: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/14.jpg)
continuum limit: FP equation
€
δt = 1N
⇒∂P(x, t)
∂t= 1
2 σ 2 ∂ 2
∂x 2x(1− x)P(x, t)[ ], σ 2 =
2
N
(N steps/generation)
boundary conditions: P(0,t) = P(1,t) = 0
![Page 15: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/15.jpg)
continuum limit: FP equation
€
δt = 1N
⇒∂P(x, t)
∂t= 1
2 σ 2 ∂ 2
∂x 2x(1− x)P(x, t)[ ], σ 2 =
2
N
(N steps/generation)
boundary conditions: P(0,t) = P(1,t) = 0(once an allele dies out, it can not come back)
![Page 16: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/16.jpg)
continuum limit: FP equation
€
δt = 1N
⇒∂P(x, t)
∂t= 1
2 σ 2 ∂ 2
∂x 2x(1− x)P(x, t)[ ], σ 2 =
2
N
(N steps/generation)
boundary conditions: P(0,t) = P(1,t) = 0(once an allele dies out, it can not come back)
stochastic differential equation:
€
dx = σ x(1− x)dW (t)
![Page 17: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/17.jpg)
continuum limit: FP equation
€
δt = 1N
⇒∂P(x, t)
∂t= 1
2 σ 2 ∂ 2
∂x 2x(1− x)P(x, t)[ ], σ 2 =
2
N
(N steps/generation)
boundary conditions: P(0,t) = P(1,t) = 0(once an allele dies out, it can not come back)
stochastic differential equation:
notice
€
dx = σ x(1− x)dW (t)
dx = 0
![Page 18: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/18.jpg)
heterozygocityEventually P(x,t) gets concentrated at one boundary,
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heterozygocityEventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other.
![Page 20: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/20.jpg)
heterozygocityEventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other. But it is randomwhich one.
![Page 21: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/21.jpg)
heterozygocity
€
H(t) = 2x(t) (1− x(t)( )
Eventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other. But it is randomwhich one. Measure this by the heterozygocity
![Page 22: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/22.jpg)
heterozygocity
€
H(t) = 2x(t) (1− x(t)( )
Eventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other. But it is randomwhich one. Measure this by the heterozygocityuse Ito’s lemma:
€
dF =∂F
∂xu(x) +
∂F
∂t+ 1
2 σ 2 ∂ 2F
∂x 2G2(x)
⎛
⎝ ⎜
⎞
⎠ ⎟dt + σ
∂F
∂xG(x)dW
![Page 23: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/23.jpg)
heterozygocity
€
H(t) = 2x(t) (1− x(t)( )
Eventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other. But it is randomwhich one. Measure this by the heterozygocityuse Ito’s lemma:
€
dF =∂F
∂xu(x) +
∂F
∂t+ 1
2 σ 2 ∂ 2F
∂x 2G2(x)
⎛
⎝ ⎜
⎞
⎠ ⎟dt + σ
∂F
∂xG(x)dW
G(x) = x(1− x), u(x) = 0
![Page 24: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/24.jpg)
heterozygocity
€
H(t) = 2x(t) (1− x(t)( )
Eventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other. But it is randomwhich one. Measure this by the heterozygocityuse Ito’s lemma:
€
dF =∂F
∂xu(x) +
∂F
∂t+ 1
2 σ 2 ∂ 2F
∂x 2G2(x)
⎛
⎝ ⎜
⎞
⎠ ⎟dt + σ
∂F
∂xG(x)dW
G(x) = x(1− x), u(x) = 0
d x(1− x)[ ] = 12 σ 2(−2)x(1− x)dt + σ (1− 2x) x(1− x)dW ⇒
![Page 25: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/25.jpg)
heterozygocity
€
H(t) = 2x(t) (1− x(t)( )
Eventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other. But it is randomwhich one. Measure this by the heterozygocityuse Ito’s lemma:
€
dF =∂F
∂xu(x) +
∂F
∂t+ 1
2 σ 2 ∂ 2F
∂x 2G2(x)
⎛
⎝ ⎜
⎞
⎠ ⎟dt + σ
∂F
∂xG(x)dW
G(x) = x(1− x), u(x) = 0
d x(1− x)[ ] = 12 σ 2(−2)x(1− x)dt + σ (1− 2x) x(1− x)dW ⇒
dH
dt= −σ 2H
![Page 26: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/26.jpg)
heterozygocity
€
H(t) = 2x(t) (1− x(t)( )
Eventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other. But it is randomwhich one. Measure this by the heterozygocityuse Ito’s lemma:
€
dF =∂F
∂xu(x) +
∂F
∂t+ 1
2 σ 2 ∂ 2F
∂x 2G2(x)
⎛
⎝ ⎜
⎞
⎠ ⎟dt + σ
∂F
∂xG(x)dW
G(x) = x(1− x), u(x) = 0
d x(1− x)[ ] = 12 σ 2(−2)x(1− x)dt + σ (1− 2x) x(1− x)dW ⇒
dH
dt= −σ 2H i.e., diversity dies out in about N generations
![Page 27: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/27.jpg)
fluctuations of x
€
x(t) − x(t)( )2
= x 2(t) − x(t)2
=
![Page 28: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/28.jpg)
fluctuations of x
€
x(t) − x(t)( )2
= x 2(t) − x(t)2
=
= x 2(t) − x(t) + x(t) − x(t)2
![Page 29: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/29.jpg)
fluctuations of x
€
x(t) − x(t)( )2
= x 2(t) − x(t)2
=
= x 2(t) − x(t) + x(t) − x(t)2
= − 12 H(t) + x(t) 1− x(t)( )
![Page 30: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/30.jpg)
fluctuations of x
€
x(t) − x(t)( )2
= x 2(t) − x(t)2
=
= x 2(t) − x(t) + x(t) − x(t)2
= − 12 H(t) + x(t) 1− x(t)( )
= − 12 H(t) + x(0) 1− x(0)( )
![Page 31: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/31.jpg)
fluctuations of x
€
x(t) − x(t)( )2
= x 2(t) − x(t)2
=
= x 2(t) − x(t) + x(t) − x(t)2
= − 12 H(t) + x(t) 1− x(t)( )
= − 12 H(t) + x(0) 1− x(0)( ) t →∞
⏐ → ⏐ ⏐ x(0) 1− x(0)( )
![Page 32: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/32.jpg)
fluctuations of x
€
x(t) − x(t)( )2
= x 2(t) − x(t)2
=
= x 2(t) − x(t) + x(t) − x(t)2
= − 12 H(t) + x(t) 1− x(t)( )
= − 12 H(t) + x(0) 1− x(0)( ) t →∞
⏐ → ⏐ ⏐ x(0) 1− x(0)( )
So mean-square fluctuations of x grow initially linearly in t and then saturate
![Page 33: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/33.jpg)
with mutation:Mutation induces a drift term in the FP and sd equation
€
n1(t + δt) = n1(t) + 1N μ12 n2 − μ21 n1[ ] = n1(t) + 1
N μ12 N − n1( ) − μ21 n1⇒[ ]
![Page 34: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/34.jpg)
with mutation:Mutation induces a drift term in the FP and sd equation
€
n1(t + δt) = n1(t) + 1N μ12 n2 − μ21 n1[ ] = n1(t) + 1
N μ12 N − n1( ) − μ21 n1⇒[ ]
⇒
dx = μ12 − μ12 + μ21( )x[ ]dt + σ x(1− x)dW (t)
![Page 35: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/35.jpg)
with mutation:Mutation induces a drift term in the FP and sd equation
€
n1(t + δt) = n1(t) + 1N μ12 n2 − μ21 n1[ ] = n1(t) + 1
N μ12 N − n1( ) − μ21 n1⇒[ ]
⇒
dx = μ12 − μ12 + μ21( )x[ ]dt + σ x(1− x)dW (t)
∂P
∂t= −
∂
∂xμ12 − μ12 + μ21( )x( )P[ ] + 1
2 σ 2 ∂ 2
∂x 2x(1− x)P( )
![Page 36: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/36.jpg)
with mutation:Mutation induces a drift term in the FP and sd equation
€
n1(t + δt) = n1(t) + 1N μ12 n2 − μ21 n1[ ] = n1(t) + 1
N μ12 N − n1( ) − μ21 n1⇒[ ]
⇒
dx = μ12 − μ12 + μ21( )x[ ]dt + σ x(1− x)dW (t)
∂P
∂t= −
∂
∂xμ12 − μ12 + μ21( )x( )P[ ] + 1
2 σ 2 ∂ 2
∂x 2x(1− x)P( )
stationary solution:
![Page 37: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/37.jpg)
with mutation:Mutation induces a drift term in the FP and sd equation
€
n1(t + δt) = n1(t) + 1N μ12 n2 − μ21 n1[ ] = n1(t) + 1
N μ12 N − n1( ) − μ21 n1⇒[ ]
⇒
dx = μ12 − μ12 + μ21( )x[ ]dt + σ x(1− x)dW (t)
∂P
∂t= −
∂
∂xμ12 − μ12 + μ21( )x( )P[ ] + 1
2 σ 2 ∂ 2
∂x 2x(1− x)P( )
stationary solution:
€
dx = 0 ⇒ x =μ12
μ12 + μ21
![Page 38: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/38.jpg)
fluctuationsUse Ito’s lemma on F(x) = x2:
![Page 39: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/39.jpg)
fluctuationsUse Ito’s lemma on F(x) = x2:
€
d x 2 = 2 μ12 − μ12 + μ21( )x[ ]xdt + σ 2x(1− x)dt
![Page 40: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/40.jpg)
fluctuationsUse Ito’s lemma on F(x) = x2:
at steady state:
€
d x 2 = 2 μ12 − μ12 + μ21( )x[ ]xdt + σ 2x(1− x)dt
€
μ12 + 12 σ 2
( ) x = μ12 + μ21 + 12 σ 2
( ) x 2
![Page 41: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/41.jpg)
fluctuationsUse Ito’s lemma on F(x) = x2:
at steady state:
€
d x 2 = 2 μ12 − μ12 + μ21( )x[ ]xdt + σ 2x(1− x)dt
€
μ12 + 12 σ 2
( ) x = μ12 + μ21 + 12 σ 2
( ) x 2
x 2 =μ12 + 1
2 σ 2( ) x
μ12 + μ21 + 12 σ 2
( )
![Page 42: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/42.jpg)
fluctuationsUse Ito’s lemma on F(x) = x2:
at steady state:
€
d x 2 = 2 μ12 − μ12 + μ21( )x[ ]xdt + σ 2x(1− x)dt
€
μ12 + 12 σ 2
( ) x = μ12 + μ21 + 12 σ 2
( ) x 2
x 2 =μ12 + 1
2 σ 2( ) x
μ12 + μ21 + 12 σ 2
( )=
μ12 μ12 + 12 σ 2
( )
μ12 + μ21( ) μ12 + μ21 + 12 σ 2
( )
![Page 43: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/43.jpg)
fluctuationsUse Ito’s lemma on F(x) = x2:
at steady state:
€
d x 2 = 2 μ12 − μ12 + μ21( )x[ ]xdt + σ 2x(1− x)dt
€
μ12 + 12 σ 2
( ) x = μ12 + μ21 + 12 σ 2
( ) x 2
x 2 =μ12 + 1
2 σ 2( ) x
μ12 + μ21 + 12 σ 2
( )=
μ12 μ12 + 12 σ 2
( )
μ12 + μ21( ) μ12 + μ21 + 12 σ 2
( )
€
x 2 − x2
=12 σ 2μ12μ21
μ12 + μ21( )2
μ12 + μ21 + 12 σ 2
( )
mean square fluctuations:
![Page 44: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/44.jpg)
heterozygocity:
€
H = 2 x − x 2( ) = 2 x 1−
μ12 + 12 σ 2
( )
μ12 + μ21 + 12 σ 2
( )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
![Page 45: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/45.jpg)
heterozygocity:
€
H = 2 x − x 2( ) = 2 x 1−
μ12 + 12 σ 2
( )
μ12 + μ21 + 12 σ 2
( )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=2μ12μ21
μ12 + μ21( ) μ12 + μ21 + 12 σ 2
( )
![Page 46: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/46.jpg)
heterozygocity:
€
H = 2 x − x 2( ) = 2 x 1−
μ12 + 12 σ 2
( )
μ12 + μ21 + 12 σ 2
( )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=2μ12μ21
μ12 + μ21( ) μ12 + μ21 + 12 σ 2
( )
small noise (large population):
![Page 47: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/47.jpg)
heterozygocity:
€
H = 2 x − x 2( ) = 2 x 1−
μ12 + 12 σ 2
( )
μ12 + μ21 + 12 σ 2
( )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=2μ12μ21
μ12 + μ21( ) μ12 + μ21 + 12 σ 2
( )
small noise (large population):
€
H →2μ12μ21
μ12 + μ21( )2 = x 1− x ; x 2 − x
2∝σ 2 → 0
![Page 48: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/48.jpg)
heterozygocity:
€
H = 2 x − x 2( ) = 2 x 1−
μ12 + 12 σ 2
( )
μ12 + μ21 + 12 σ 2
( )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=2μ12μ21
μ12 + μ21( ) μ12 + μ21 + 12 σ 2
( )
small noise (large population):
€
H →2μ12μ21
μ12 + μ21( )2 = x 1− x ; x 2 − x
2∝σ 2 → 0
large noise (small population):
![Page 49: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/49.jpg)
heterozygocity:
€
H = 2 x − x 2( ) = 2 x 1−
μ12 + 12 σ 2
( )
μ12 + μ21 + 12 σ 2
( )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=2μ12μ21
μ12 + μ21( ) μ12 + μ21 + 12 σ 2
( )
small noise (large population):
€
H →2μ12μ21
μ12 + μ21( )2 = x 1− x ; x 2 − x
2∝σ 2 → 0
large noise (small population):
€
H →4μ12μ21
μ12 + μ21( )σ 2⇒
![Page 50: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/50.jpg)
heterozygocity:
€
H = 2 x − x 2( ) = 2 x 1−
μ12 + 12 σ 2
( )
μ12 + μ21 + 12 σ 2
( )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=2μ12μ21
μ12 + μ21( ) μ12 + μ21 + 12 σ 2
( )
small noise (large population):
€
H →2μ12μ21
μ12 + μ21( )2 = x 1− x ; x 2 − x
2∝σ 2 → 0
large noise (small population):
€
H →4μ12μ21
μ12 + μ21( )σ 2⇒ usually one allele dominates, rare transitions
![Page 51: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/51.jpg)
selectionLet the alleles chosen to reproduce do so with with probabilities
€
p1 =w1x
w1x + w2(1− x), p2 =
w2(1− x)
w1x + w2(1− x)
![Page 52: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/52.jpg)
selectionLet the alleles chosen to reproduce do so with with probabilities
€
p1 =w1x
w1x + w2(1− x), p2 =
w2(1− x)
w1x + w2(1− x)
Now, if there are n1 organisms of type 1 before this step, then afterwards there are
![Page 53: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/53.jpg)
selectionLet the alleles chosen to reproduce do so with with probabilities
€
p1 =w1x
w1x + w2(1− x), p2 =
w2(1− x)
w1x + w2(1− x)
Now, if there are n1 organisms of type 1 before this step, then afterwards there are
n1 + 1 with probability p1(1 – x)
![Page 54: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/54.jpg)
selectionLet the alleles chosen to reproduce do so with with probabilities
€
p1 =w1x
w1x + w2(1− x), p2 =
w2(1− x)
w1x + w2(1− x)
Now, if there are n1 organisms of type 1 before this step, then afterwards there are
n1 + 1 with probability p1(1 – x)n1 – 1 with probability p2x
![Page 55: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/55.jpg)
selectionLet the alleles chosen to reproduce do so with with probabilities
€
p1 =w1x
w1x + w2(1− x), p2 =
w2(1− x)
w1x + w2(1− x)
This leads to a drift in x proportional to x(1 - x):
Now, if there are n1 organisms of type 1 before this step, then afterwards there are
n1 + 1 with probability p1(1 – x)n1 – 1 with probability p2x
![Page 56: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/56.jpg)
selectionLet the alleles chosen to reproduce do so with with probabilities
€
p1 =w1x
w1x + w2(1− x), p2 =
w2(1− x)
w1x + w2(1− x)
This leads to a drift in x proportional to x(1 - x):
Now, if there are n1 organisms of type 1 before this step, then afterwards there are
n1 + 1 with probability p1(1 – x)n1 – 1 with probability p2x
€
dx = μ12 − μ12 + μ21( )x[ ]dt + sx(1− x)dt + σ x(1− x)dW (t);
s =2(w1 − w2)
(w1 + w2)(s <<1)
![Page 57: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/57.jpg)
selection: large population limit
with selection but no mutations:
€
dx
dt= sx(1− x)
![Page 58: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/58.jpg)
selection: large population limit
with selection but no mutations:
€
dx
dt= sx(1− x)
logx
1− x⋅1− x0
x0
⎛
⎝ ⎜
⎞
⎠ ⎟= stsolution:
![Page 59: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/59.jpg)
selection: large population limit
with selection but no mutations:
€
dx
dt= sx(1− x)
logx
1− x⋅1− x0
x0
⎛
⎝ ⎜
⎞
⎠ ⎟= st
x =1
1+1− x0
x0
exp(−st)
solution:
![Page 60: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/60.jpg)
selection: large population limit
with selection but no mutations:
€
dx
dt= sx(1− x)
logx
1− x⋅1− x0
x0
⎛
⎝ ⎜
⎞
⎠ ⎟= st
x =1
1+1− x0
x0
exp(−st)
t >>1
slog
1− x0
x0
⎛
⎝ ⎜
⎞
⎠ ⎟: x →1
solution:
![Page 61: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/61.jpg)
Neurons
Neurons receive synaptic input from other neurons
![Page 62: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/62.jpg)
Neurons
Neurons receive synaptic input from other neurons~ injected current
![Page 63: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/63.jpg)
Neurons
Neurons receive synaptic input from other neurons~ injected current
€
CdV
dt= I(t) V measured from resting potential
![Page 64: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/64.jpg)
Neurons
Neurons receive synaptic input from other neurons~ injected current
€
CdV
dt= I(t)
CdV
dt= −gV + I(t)
V measured from resting potential
with leak g = membrane conductance
![Page 65: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/65.jpg)
Neurons
Neurons receive synaptic input from other neurons~ injected current
€
CdV
dt= I(t)
CdV
dt= −gV + I(t)
V measured from resting potential
with leak g = membrane conductance
(experimental fact:) input current is noisy, very small τc compared tomembrane time constant τ = C/g
![Page 66: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/66.jpg)
Neurons
Neurons receive synaptic input from other neurons~ injected current
€
CdV
dt= I(t)
CdV
dt= −gV + I(t)
V measured from resting potential
with leak g = membrane conductance
(experimental fact:) input current is noisy, very small τc compared tomembrane time constant τ = C/g
V(t) is described by Wiener process (g = 0) or Brownian motion (g ≠ 0)
![Page 67: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/67.jpg)
SpikesThe above is approximately true as long as V stays below a criticalvalue VT.
![Page 68: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/68.jpg)
SpikesThe above is approximately true as long as V stays below a criticalvalue VT. Above this threshold, active ion channels amplify incomingcurrents and produce an action potential (spike).
![Page 69: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/69.jpg)
SpikesThe above is approximately true as long as V stays below a criticalvalue VT. Above this threshold, active ion channels amplify incomingcurrents and produce an action potential (spike). After a few ms, V returns to its sub-threshold equilibrium level.
![Page 70: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/70.jpg)
SpikesThe above is approximately true as long as V stays below a criticalvalue VT. Above this threshold, active ion channels amplify incomingcurrents and produce an action potential (spike). After a few ms, V returns to its sub-threshold equilibrium level.
“integrate-and-fire” or “leaky integrate-and-fire” model of a neuron
![Page 71: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/71.jpg)
SpikesThe above is approximately true as long as V stays below a criticalvalue VT. Above this threshold, active ion channels amplify incomingcurrents and produce an action potential (spike). After a few ms, V returns to its sub-threshold equilibrium level.
“integrate-and-fire” or “leaky integrate-and-fire” model of a neuron
our question here: if I(t) is white noise, what is the distribution ofinterspike intervals?
![Page 72: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/72.jpg)
SpikesThe above is approximately true as long as V stays below a criticalvalue VT. Above this threshold, active ion channels amplify incomingcurrents and produce an action potential (spike). After a few ms, V returns to its sub-threshold equilibrium level.
“integrate-and-fire” or “leaky integrate-and-fire” model of a neuron
our question here: if I(t) is white noise, what is the distribution ofinterspike intervals?
This is a first-passage-time problem
![Page 73: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/73.jpg)
with no leak:
€
CV ≡ x
dx
dt= I(t), x < θ = CVT ; I(t)I( ′ t ) = σ 2δ(t − ′ t )
![Page 74: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/74.jpg)
with no leak:
Assume at t = 0, x = 0
€
CV ≡ x
dx
dt= I(t), x < θ = CVT ; I(t)I( ′ t ) = σ 2δ(t − ′ t )
![Page 75: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/75.jpg)
with no leak:
Assume at t = 0, x = 0boundary condition at θ: P(θ) = 0.
€
CV ≡ x
dx
dt= I(t), x < θ = CVT ; I(t)I( ′ t ) = σ 2δ(t − ′ t )
![Page 76: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/76.jpg)
with no leak:
Assume at t = 0, x = 0boundary condition at θ: P(θ) = 0.
€
CV ≡ x
dx
dt= I(t), x < θ = CVT ; I(t)I( ′ t ) = σ 2δ(t − ′ t )
We have solved this problem when there is no threshold:
![Page 77: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/77.jpg)
with no leak:
Assume at t = 0, x = 0boundary condition at θ: P(θ) = 0.
€
CV ≡ x
dx
dt= I(t), x < θ = CVT ; I(t)I( ′ t ) = σ 2δ(t − ′ t )
We have solved this problem when there is no threshold:
€
P(x, t) =1
2πσ 2texp −
x 2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
![Page 78: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/78.jpg)
with no leak:
Assume at t = 0, x = 0boundary condition at θ: P(θ) = 0.
€
CV ≡ x
dx
dt= I(t), x < θ = CVT ; I(t)I( ′ t ) = σ 2δ(t − ′ t )
We have solved this problem when there is no threshold:
€
P(x, t) =1
2πσ 2texp −
x 2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
But it does not satisfy the boundary condition.
![Page 79: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/79.jpg)
solution with images:
Add an extra source, of opposite sign, at x = 2θ:
€
P(x, t) =1
2πσ 2texp −
x 2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟− exp −
(x − 2θ)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥
![Page 80: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/80.jpg)
solution with images:
Add an extra source, of opposite sign, at x = 2θ:
€
P(x, t) =1
2πσ 2texp −
x 2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟− exp −
(x − 2θ)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥
cumulative probability of firing by t:
€
F(t) = 2dx
2πσ 2texp −
x 2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
θ
∞
∫
![Page 81: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/81.jpg)
solution with images:
Add an extra source, of opposite sign, at x = 2θ:
€
P(x, t) =1
2πσ 2texp −
x 2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟− exp −
(x − 2θ)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥
cumulative probability of firing by t:
€
F(t) = 2dx
2πσ 2texp −
x 2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
θ
∞
∫
€
f (t) =dF(t)
dt= 2
d
dt
du
2πe− 1
2 u2
θ
σ t
∞
∫ =θ
2πσ 2t 3exp −
θ 2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
interspike interval density:
![Page 82: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/82.jpg)
solution with images:
Add an extra source, of opposite sign, at x = 2θ:
€
P(x, t) =1
2πσ 2texp −
x 2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟− exp −
(x − 2θ)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥
cumulative probability of firing by t:
€
F(t) = 2dx
2πσ 2texp −
x 2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
θ
∞
∫
€
f (t) =dF(t)
dt= 2
d
dt
du
2πe− 1
2 u2
θ
σ t
∞
∫ =θ
2πσ 2t 3exp −
θ 2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
interspike interval density:
Levy distribution (one-sided stable distribution with α = ½
![Page 83: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/83.jpg)
another way to get the answer:
The event rate is just the (diffusive) current
€
J(x) = −D∂P
∂x= − 1
2 σ 2 ∂P
∂x
evaluated at x = θ.
![Page 84: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/84.jpg)
another way to get the answer:
The event rate is just the (diffusive) current
€
J(x) = −D∂P
∂x= − 1
2 σ 2 ∂P
∂x
evaluated at x = θ.
€
f (t) =d
dx
1
2πσ 2texp −
x 2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟−
1
2πσ 2texp −
(x − 2θ)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥x=θ
![Page 85: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/85.jpg)
another way to get the answer:
The event rate is just the (diffusive) current
€
J(x) = −D∂P
∂x= − 1
2 σ 2 ∂P
∂x
evaluated at x = θ.
€
f (t) =d
dx
1
2πσ 2texp −
x 2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟−
1
2πσ 2texp −
(x − 2θ)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥x=θ
=2
2πσ 2t
d
dxexp −
x 2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥x=θ
=θ
2πσ 2t 3exp −
θ 2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
![Page 86: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/86.jpg)
a problem:The mean interspike interval is infinite:
![Page 87: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/87.jpg)
a problem:The mean interspike interval is infinite:
€
t = tf (t)dt0
∞
∫ ~tdt
t 3 / 2∫ = ∞
![Page 88: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/88.jpg)
a problem:The mean interspike interval is infinite:
€
t = tf (t)dt0
∞
∫ ~tdt
t 3 / 2∫ = ∞
so the firing rate (= 1/<t>) is zero!
![Page 89: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/89.jpg)
adding a constant drift term:
€
dx
dt= μ + I(t), x < θ = CVT ; I(t)I( ′ t ) = σ 2δ(t − ′ t )
![Page 90: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/90.jpg)
adding a constant drift term:
€
dx
dt= μ + I(t), x < θ = CVT ; I(t)I( ′ t ) = σ 2δ(t − ′ t )
solution with no boundary:
€
P(x, t) =1
2πσ 2texp −
(x − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
![Page 91: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/91.jpg)
adding a constant drift term:
€
dx
dt= μ + I(t), x < θ = CVT ; I(t)I( ′ t ) = σ 2δ(t − ′ t )
solution with no boundary:
€
P(x, t) =1
2πσ 2texp −
(x − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
need a moving image:
€
P(x, t) =1
2πσ 2texp −
(x − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟−
C
2πσ 2texp −
(x − 2θ − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
![Page 92: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/92.jpg)
adding a constant drift term:
€
dx
dt= μ + I(t), x < θ = CVT ; I(t)I( ′ t ) = σ 2δ(t − ′ t )
solution with no boundary:
€
P(x, t) =1
2πσ 2texp −
(x − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
need a moving image:
€
P(x, t) =1
2πσ 2texp −
(x − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟−
C
2πσ 2texp −
(x − 2θ − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
€
P(θ, t) = 0 ⇒ exp −(θ − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟= C exp −
(θ + μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
![Page 93: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/93.jpg)
adding a constant drift term:
€
dx
dt= μ + I(t), x < θ = CVT ; I(t)I( ′ t ) = σ 2δ(t − ′ t )
solution with no boundary:
€
P(x, t) =1
2πσ 2texp −
(x − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
need a moving image:
€
P(x, t) =1
2πσ 2texp −
(x − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟−
C
2πσ 2texp −
(x − 2θ − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
€
P(θ, t) = 0 ⇒ exp −(θ − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟= C exp −
(θ + μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟ ⇒ C = exp
2μθ
σ 2
⎛
⎝ ⎜
⎞
⎠ ⎟
![Page 94: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/94.jpg)
adding a constant drift term:
€
dx
dt= μ + I(t), x < θ = CVT ; I(t)I( ′ t ) = σ 2δ(t − ′ t )
solution with no boundary:
€
P(x, t) =1
2πσ 2texp −
(x − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
need a moving image:
€
P(x, t) =1
2πσ 2texp −
(x − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟−
C
2πσ 2texp −
(x − 2θ − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
€
P(θ, t) = 0 ⇒ exp −(θ − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟= C exp −
(θ + μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟ ⇒ C = exp
2μθ
σ 2
⎛
⎝ ⎜
⎞
⎠ ⎟
€
P(x, t) =1
2πσ 2texp −
(x − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟−
1
2πσ 2texp
2μθ
σ 2
⎛
⎝ ⎜
⎞
⎠ ⎟exp −
(x − 2θ − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
solution:
![Page 95: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/95.jpg)
ISI distribution:
€
J(x) = −D∂P
∂x= − 1
2 σ 2 ∂P
∂xfrom
![Page 96: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/96.jpg)
ISI distribution:
€
J(x) = −D∂P
∂x= − 1
2 σ 2 ∂P
∂xfrom
€
f (t) = −12 σ 2
2πDσ 2t
d
dxexp −
(x − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟− C exp −
(x − μt − 2θ)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥x=θ
![Page 97: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/97.jpg)
ISI distribution:
€
J(x) = −D∂P
∂x= − 1
2 σ 2 ∂P
∂xfrom
€
f (t) = −12 σ 2
2πDσ 2t
d
dxexp −
(x − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟− C exp −
(x − μt − 2θ)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥x=θ
=θ
2πσ 2 t 3 / 2exp −
(θ − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
![Page 98: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/98.jpg)
ISI distribution:
€
J(x) = −D∂P
∂x= − 1
2 σ 2 ∂P
∂xfrom
€
f (t) = −12 σ 2
2πDσ 2t
d
dxexp −
(x − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟− C exp −
(x − μt − 2θ)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥x=θ
=θ
2πσ 2 t 3 / 2exp −
(θ − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
Now all moments of f are finite
![Page 99: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/99.jpg)
ISI distribution:
€
J(x) = −D∂P
∂x= − 1
2 σ 2 ∂P
∂xfrom
€
f (t) = −12 σ 2
2πDσ 2t
d
dxexp −
(x − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟− C exp −
(x − μt − 2θ)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥x=θ
=θ
2πσ 2 t 3 / 2exp −
(θ − μt)2
2σ 2t
⎛
⎝ ⎜
⎞
⎠ ⎟
Now all moments of f are finite
![Page 100: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/100.jpg)
leaky I&F neuron
€
dx
dt= −γx + I0 + δI(t) (γ = 1/τ = g/C
![Page 101: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/101.jpg)
leaky I&F neuron
€
dx
dt= −γx + I0 + δI(t) (γ = 1/τ = g/C
€
δI(t) = 0
€
δI(t)δI( ′ t ) = σ 2δ(t − ′ t ) = 2Dδ(t − ′ t )
![Page 102: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/102.jpg)
leaky I&F neuron
€
dx
dt= −γx + I0 + δI(t) (γ = 1/τ = g/C
€
δI(t) = 0
€
δI(t)δI( ′ t ) = σ 2δ(t − ′ t ) = 2Dδ(t − ′ t )
= Brownian motion with an added constant drift
![Page 103: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/103.jpg)
leaky I&F neuron
€
dx
dt= −γx + I0 + δI(t) (γ = 1/τ = g/C
€
δI(t) = 0
€
δI(t)δI( ′ t ) = σ 2δ(t − ′ t ) = 2Dδ(t − ′ t )
= Brownian motion with an added constant drift
€
∂P(x, t)
∂t= −
∂J
∂x= −
∂
∂xI0 − γx( )P(x, t) − D
∂P(x, t)
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥
![Page 104: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/104.jpg)
leaky I&F neuron
€
dx
dt= −γx + I0 + δI(t) (γ = 1/τ = g/C
€
δI(t) = 0
€
δI(t)δI( ′ t ) = σ 2δ(t − ′ t ) = 2Dδ(t − ′ t )
= Brownian motion with an added constant drift
€
∂P(x, t)
∂t= −
∂J
∂x= −
∂
∂xI0 − γx( )P(x, t) − D
∂P(x, t)
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥
(set γ = 1 for convenience)
![Page 105: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/105.jpg)
Looking for stationary solution
€
∂P
∂t= 0
![Page 106: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/106.jpg)
Looking for stationary solution
€
∂P
∂t= 0
€
dJ
dx=
d
dxI0 − x( )P(x) − D
∂P(x)
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥= 0i.e.
![Page 107: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/107.jpg)
Looking for stationary solution
€
∂P
∂t= 0
€
dJ
dx=
d
dxI0 − x( )P(x) − D
∂P(x)
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥= 0
€
J(x) = I0 − x( )P(x) − D∂P(x)
∂x= const
i.e.
=>
![Page 108: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/108.jpg)
Looking for stationary solution
€
∂P
∂t= 0
€
dJ
dx=
d
dxI0 − x( )P(x) − D
∂P(x)
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥= 0
€
J(x) = I0 − x( )P(x) − D∂P(x)
∂x= const
Boundary conditions:
i.e.
=>
![Page 109: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/109.jpg)
Looking for stationary solution
€
∂P
∂t= 0
€
dJ
dx=
d
dxI0 − x( )P(x) − D
∂P(x)
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥= 0
€
J(x) = I0 − x( )P(x) − D∂P(x)
∂x= const
Boundary conditions: sink at firing threshold x
i.e.
=>
![Page 110: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/110.jpg)
Looking for stationary solution
€
∂P
∂t= 0
€
dJ
dx=
d
dxI0 − x( )P(x) − D
∂P(x)
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥= 0
€
J(x) = I0 − x( )P(x) − D∂P(x)
∂x= const
Boundary conditions: sink at firing threshold x source at x = 0
i.e.
=>
![Page 111: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/111.jpg)
Looking for stationary solution
€
∂P
∂t= 0
€
dJ
dx=
d
dxI0 − x( )P(x) − D
∂P(x)
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥= 0
€
J(x) = I0 − x( )P(x) − D∂P(x)
∂x= const
Boundary conditions: sink at firing threshold x source at x = 0
€
P(x) = 0, x ≥ θ
i.e.
=>
![Page 112: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/112.jpg)
Looking for stationary solution
€
∂P
∂t= 0
€
dJ
dx=
d
dxI0 − x( )P(x) − D
∂P(x)
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥= 0
€
J(x) = I0 − x( )P(x) − D∂P(x)
∂x= const
Boundary conditions: sink at firing threshold x source at x = 0
€
P(x) = 0, x ≥ θ
€
J(x) = 0, x > θ and x < 0
i.e.
=>
![Page 113: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/113.jpg)
Looking for stationary solution
€
∂P
∂t= 0
€
dJ
dx=
d
dxI0 − x( )P(x) − D
∂P(x)
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥= 0
€
J(x) = I0 − x( )P(x) − D∂P(x)
∂x= const
Boundary conditions: sink at firing threshold x source at x = 0
€
P(x) = 0, x ≥ θ
€
J(x) = 0, x > θ and x < 0
€
r = J(θ−) = −DdP
dx
⎛
⎝ ⎜
⎞
⎠ ⎟x=θ
i.e.
=>
Firing rate: current out at threshold:
![Page 114: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/114.jpg)
Looking for stationary solution
€
∂P
∂t= 0
€
dJ
dx=
d
dxI0 − x( )P(x) − D
∂P(x)
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥= 0
€
J(x) = I0 − x( )P(x) − D∂P(x)
∂x= const
Boundary conditions: sink at firing threshold x source at x = 0
€
P(x) = 0, x ≥ θ
€
J(x) = 0, x > θ and x < 0
€
r = J(θ−) = −DdP
dx
⎛
⎝ ⎜
⎞
⎠ ⎟x=θ
€
r = J(0+) = I0P(0) − DdP
dx
⎛
⎝ ⎜
⎞
⎠ ⎟x= 0
i.e.
=>
Firing rate: current out at threshold: = reinjection rate at reset:
![Page 115: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/115.jpg)
Stationary solution (2)
Also need normalization:
€
dxP(x) =1−∞
∞
∫
![Page 116: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/116.jpg)
Stationary solution (2)
Also need normalization:
€
dxP(x) =1−∞
∞
∫
Below reset level, J :
€
I0 − x( )P(x) − D∂P(x)
∂x= 0
![Page 117: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/117.jpg)
Stationary solution (2)
Also need normalization:
€
dxP(x) =1−∞
∞
∫
Below reset level, J :
€
I0 − x( )P(x) − D∂P(x)
∂x= 0
has solution
€
P(x) = c1 exp −(x − I0)2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥
![Page 118: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/118.jpg)
Stationary solution (2)
Also need normalization:
€
dxP(x) =1−∞
∞
∫
Below reset level, J :
€
I0 − x( )P(x) − D∂P(x)
∂x= 0
has solution
€
P(x) = c1 exp −(x − I0)2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥
€
P(x) = c2 exp −(x − I0)2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥ dy exp
(y − I0)2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥
x
θ
∫Between rest and threshold:
![Page 119: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/119.jpg)
Stationary solution (2)
Also need normalization:
€
dxP(x) =1−∞
∞
∫
Below reset level, J :
€
I0 − x( )P(x) − D∂P(x)
∂x= 0
has solution
€
P(x) = c1 exp −(x − I0)2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥
€
P(x) = c2 exp −(x − I0)2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥ dy exp
(y − I0)2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥
x
θ
∫Between rest and threshold:
B.C. at x :
€
r = −DdP
dx
⎛
⎝ ⎜
⎞
⎠ ⎟x=θ
= −Dc2 exp −(θ − I0)2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥ −exp
(θ − I0)2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥
⎛
⎝ ⎜
⎞
⎠ ⎟= Dc2
![Page 120: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/120.jpg)
Stationary solution (2)
Also need normalization:
€
dxP(x) =1−∞
∞
∫
Below reset level, J :
€
I0 − x( )P(x) − D∂P(x)
∂x= 0
has solution
€
P(x) = c1 exp −(x − I0)2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥
€
P(x) = c2 exp −(x − I0)2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥ dy exp
(y − I0)2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥
x
θ
∫Between rest and threshold:
B.C. at x :
€
r = −DdP
dx
⎛
⎝ ⎜
⎞
⎠ ⎟x=θ
= −Dc2 exp −(θ − I0)2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥ −exp
(θ − I0)2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥
⎛
⎝ ⎜
⎞
⎠ ⎟= Dc2
=>
€
c2 =r
D
![Page 121: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/121.jpg)
Stationary solution (3)
Continuity at x = =>
€
c1 exp −I0
2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥= c2 exp −
I02
2D
⎡
⎣ ⎢
⎤
⎦ ⎥ dy exp
(y − I0)2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥
0
θ
∫
![Page 122: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/122.jpg)
Stationary solution (3)
Continuity at x = =>
€
c1 exp −I0
2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥= c2 exp −
I02
2D
⎡
⎣ ⎢
⎤
⎦ ⎥ dy exp
(y − I0)2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥
0
θ
∫
i.e.,
€
c1 = c2 dy exp(y − I0)2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥
0
θ
∫
![Page 123: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/123.jpg)
Stationary solution (3)
Continuity at x = =>
€
c1 exp −I0
2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥= c2 exp −
I02
2D
⎡
⎣ ⎢
⎤
⎦ ⎥ dy exp
(y − I0)2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥
0
θ
∫
i.e.,
€
c1 = c2 dy exp(y − I0)2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥
0
θ
∫
algebra … =>
€
r =1
π dx−I 0 / 2D
(θ −I 0 ) / 2D
∫ exp(x 2)(1+ erf x)=
1
π dx−I 0 /σ
(θ −I 0 ) /σ
∫ exp(x 2)(1+ erf x)
![Page 124: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/124.jpg)
Stationary solution (3)
Continuity at x = =>
€
c1 exp −I0
2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥= c2 exp −
I02
2D
⎡
⎣ ⎢
⎤
⎦ ⎥ dy exp
(y − I0)2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥
0
θ
∫
i.e.,
€
c1 = c2 dy exp(y − I0)2
2D
⎡
⎣ ⎢
⎤
⎦ ⎥
0
θ
∫
algebra … =>
€
r =1
π dx−I 0 / 2D
(θ −I 0 ) / 2D
∫ exp(x 2)(1+ erf x)=
1
π dx−I 0 /σ
(θ −I 0 ) /σ
∫ exp(x 2)(1+ erf x)
with refractory time τr
€
r =1
τ r + π dx−I 0 /σ
(θ −I 0 ) /σ
∫ exp(x 2)(1+ erf x)
![Page 125: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/125.jpg)
A simple model of evolution: the Bak-Sneppen model
N species,each with fitness xi, each uniformly distributed on (0,1)
![Page 126: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/126.jpg)
A simple model of evolution: the Bak-Sneppen model
N species,each with fitness xi, each uniformly distributed on (0,1)
![Page 127: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/127.jpg)
A simple model of evolution: the Bak-Sneppen model
N species,each with fitness xi, each uniformly distributed on (0,1)
evolutionary step:
![Page 128: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/128.jpg)
A simple model of evolution: the Bak-Sneppen model
N species,each with fitness xi, each uniformly distributed on (0,1)
evolutionary step: eliminate the weakest species (smallest xi)
![Page 129: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/129.jpg)
A simple model of evolution: the Bak-Sneppen model
N species,each with fitness xi, each uniformly distributed on (0,1)
evolutionary step: eliminate the weakest species (smallest xi)replace it with another species with a random xi
![Page 130: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/130.jpg)
A simple model of evolution: the Bak-Sneppen model
N species,each with fitness xi, each uniformly distributed on (0,1)
evolutionary step: eliminate the weakest species (smallest xi)replace it with another species with a random xi
(random-neighbour version) assume another (“neighboring”)species also becomes extinct; replace it with a new one, too
![Page 131: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/131.jpg)
A simple model of evolution: the Bak-Sneppen model
N species,each with fitness xi, each uniformly distributed on (0,1)
evolutionary step: eliminate the weakest species (smallest xi)replace it with another species with a random xi
(random-neighbour version) assume another (“neighboring”)species also becomes extinct; replace it with a new one, too
Now one or more of these new ones may get fitnesses below θreplace them and their (randomly chosen) neighbours with new ones until all fitnesses are > θ
![Page 132: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/132.jpg)
A simple model of evolution: the Bak-Sneppen model
N species,each with fitness xi, each uniformly distributed on (0,1)
evolutionary step: eliminate the weakest species (smallest xi)replace it with another species with a random xi
(random-neighbour version) assume another (“neighboring”)species also becomes extinct; replace it with a new one, too
Now one or more of these new ones may get fitnesses below θreplace them and their (randomly chosen) neighbours with new ones until all fitnesses are > θ (avalanche)
![Page 133: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/133.jpg)
A simple model of evolution: the Bak-Sneppen model
N species,each with fitness xi, each uniformly distributed on (0,1)
evolutionary step: eliminate the weakest species (smallest xi)replace it with another species with a random xi
(random-neighbour version) assume another (“neighboring”)species also becomes extinct; replace it with a new one, too
Now one or more of these new ones may get fitnesses below θreplace them and their (randomly chosen) neighbours with new ones until all fitnesses are > θ (avalanche)
start over
![Page 134: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/134.jpg)
A simple model of evolution: the Bak-Sneppen model
N species,each with fitness xi, each uniformly distributed on (0,1)
evolutionary step: eliminate the weakest species (smallest xi)replace it with another species with a random xi
(random-neighbour version) assume another (“neighboring”)species also becomes extinct; replace it with a new one, too
Now one or more of these new ones may get fitnesses below θreplace them and their (randomly chosen) neighbours with new ones until all fitnesses are > θ (avalanche)
start over
Want to know the avalanche length distribution
![Page 135: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/135.jpg)
getting a master equationP(n,t) = prob that n species have fitness values < θ at time t
![Page 136: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/136.jpg)
getting a master equationP(n,t) = prob that n species have fitness values < θ at time tAt each step 2 species are reassigned random new fitnesses
![Page 137: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/137.jpg)
getting a master equationP(n,t) = prob that n species have fitness values < θ at time tAt each step 2 species are reassigned random new fitnessestransition matrix: Tmn:
![Page 138: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/138.jpg)
getting a master equationP(n,t) = prob that n species have fitness values < θ at time tAt each step 2 species are reassigned random new fitnessestransition matrix: Tmn:
€
Tn−1,n = (1−θ)2
Tnn = 2θ(1−θ)
Tn +1,n = θ 2
![Page 139: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/139.jpg)
getting a master equationP(n,t) = prob that n species have fitness values < θ at time tAt each step 2 species are reassigned random new fitnessestransition matrix: Tmn:
€
Tn−1,n = (1−θ)2
Tnn = 2θ(1−θ)
Tn +1,n = θ 2
simple random walk in n with first step n=0 -> n=1, thereafter
![Page 140: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/140.jpg)
getting a master equationP(n,t) = prob that n species have fitness values < θ at time tAt each step 2 species are reassigned random new fitnessestransition matrix: Tmn:
€
Tn−1,n = (1−θ)2
Tnn = 2θ(1−θ)
Tn +1,n = θ 2
net drift per step:
mean square change:
€
Tn +1,n − Tn−1,n = θ 2 − (1−θ)2 = 2 θ − 12( )
Tn +1,n + Tn−1,n =1− 2θ(1−θ)
simple random walk in n with first step n=0 -> n=1, thereafter
![Page 141: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/141.jpg)
getting a master equationP(n,t) = prob that n species have fitness values < θ at time tAt each step 2 species are reassigned random new fitnessestransition matrix: Tmn:
€
Tn−1,n = (1−θ)2
Tnn = 2θ(1−θ)
Tn +1,n = θ 2
net drift per step:
mean square change:
€
Tn +1,n − Tn−1,n = θ 2 − (1−θ)2 = 2 θ − 12( )
Tn +1,n + Tn−1,n =1− 2θ(1−θ)
simple random walk in n with first step n=0 -> n=1, thereafter
Walk (avalanche) ends when n=0 again for the first time.
![Page 142: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/142.jpg)
getting a master equationP(n,t) = prob that n species have fitness values < θ at time tAt each step 2 species are reassigned random new fitnessestransition matrix: Tmn:
€
Tn−1,n = (1−θ)2
Tnn = 2θ(1−θ)
Tn +1,n = θ 2
net drift per step:
mean square change:
€
Tn +1,n − Tn−1,n = θ 2 − (1−θ)2 = 2 θ − 12( )
Tn +1,n + Tn−1,n =1− 2θ(1−θ)
simple random walk in n with first step n=0 -> n=1, thereafter
Walk (avalanche) ends when n=0 again for the first time.
critical case (no drift): θ =½
![Page 143: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/143.jpg)
TrafficNagel-Paczuski model:cars can move with speed v=+1 step/time unit or 0.
![Page 144: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/144.jpg)
TrafficNagel-Paczuski model:cars can move with speed v=+1 step/time unit or 0.jam/kø/queue/file/stau = n cars in a row not moving
![Page 145: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/145.jpg)
TrafficNagel-Paczuski model:cars can move with speed v=+1 step/time unit or 0.jam/kø/queue/file/stau = n cars in a row not movingfirst car in stau can change speed from 0 to +1 with prob p/step
![Page 146: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/146.jpg)
TrafficNagel-Paczuski model:cars can move with speed v=+1 step/time unit or 0.jam/kø/queue/file/stau = n cars in a row not movingfirst car in stau can change speed from 0 to +1 with prob p/stepnew cars enter the stau with prob q
![Page 147: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/147.jpg)
TrafficNagel-Paczuski model:cars can move with speed v=+1 step/time unit or 0.jam/kø/queue/file/stau = n cars in a row not movingfirst car in stau can change speed from 0 to +1 with prob p/stepnew cars enter the stau with prob q transition matrix for stau length:
![Page 148: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/148.jpg)
TrafficNagel-Paczuski model:cars can move with speed v=+1 step/time unit or 0.jam/kø/queue/file/stau = n cars in a row not movingfirst car in stau can change speed from 0 to +1 with prob p/stepnew cars enter the stau with prob q transition matrix for stau length:
€
Tn−1,n = q(1− p)
Tnn = (1− p)(1− q) + pq
Tn +1,n = p(1− q)
![Page 149: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/149.jpg)
TrafficNagel-Paczuski model:cars can move with speed v=+1 step/time unit or 0.jam/kø/queue/file/stau = n cars in a row not movingfirst car in stau can change speed from 0 to +1 with prob p/stepnew cars enter the stau with prob q transition matrix for stau length:
€
Tn−1,n = q(1− p)
Tnn = (1− p)(1− q) + pq
Tn +1,n = p(1− q)
€
Tn +1,n − Tn−1,n = p(1− q) − q(1− p) = p − q
Tn +1,n + Tn−1,n = p(1− q) + q(1− p) = p + q − 2pq
net drift per step:
mean square change:
![Page 150: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/150.jpg)
TrafficNagel-Paczuski model:cars can move with speed v=+1 step/time unit or 0.jam/kø/queue/file/stau = n cars in a row not movingfirst car in stau can change speed from 0 to +1 with prob p/stepnew cars enter the stau with prob q transition matrix for stau length:
€
Tn−1,n = q(1− p)
Tnn = (1− p)(1− q) + pq
Tn +1,n = p(1− q)
€
Tn +1,n − Tn−1,n = p(1− q) − q(1− p) = p − q
Tn +1,n + Tn−1,n = p(1− q) + q(1− p) = p + q − 2pq
net drift per step:
mean square change:
biased random walk again
![Page 151: Lecture 9: Population genetics, first-passage problems](https://reader035.fdocuments.in/reader035/viewer/2022062321/56813191550346895d980390/html5/thumbnails/151.jpg)
TrafficNagel-Paczuski model:cars can move with speed v=+1 step/time unit or 0.jam/kø/queue/file/stau = n cars in a row not movingfirst car in stau can change speed from 0 to +1 with prob p/stepnew cars enter the stau with prob q transition matrix for stau length:
€
Tn−1,n = q(1− p)
Tnn = (1− p)(1− q) + pq
Tn +1,n = p(1− q)
€
Tn +1,n − Tn−1,n = p(1− q) − q(1− p) = p − q
Tn +1,n + Tn−1,n = p(1− q) + q(1− p) = p + q − 2pq
net drift per step:
mean square change:
biased random walk again, critical (long-tail distribution of staulengths, lifetimes) fpr p = q.