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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
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Population genetics: Moran model 2 alleles, N haploid organisms
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Population genetics: Moran model 2 alleles, N haploid organisms choose 2 individual at random: 1 dies, the other reproduces
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Population genetics: Moran model 2 alleles, N haploid organisms choose 2 individual at random: 1 dies, the other reproduces If there are n 1 organisms of type 1 before this step, then afterwards there are
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Population genetics: Moran model 2 alleles, N haploid organisms choose 2 individual at random: 1 dies, the other reproduces If there are n 1 organisms of type 1 before this step, then afterwards there are n 1 + 1 with probability x(1 – x)x = n 1 /N
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Population genetics: Moran model 2 alleles, N haploid organisms choose 2 individual at random: 1 dies, the other reproduces If there are n 1 organisms of type 1 before this step, then afterwards there are n 1 + 1 with probability x(1 – x)x = n 1 /N n 1 – 1 with probability x(1 – x)
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Population genetics: Moran model 2 alleles, N haploid organisms choose 2 individual at random: 1 dies, the other reproduces If there are n 1 organisms of type 1 before this step, then afterwards there are n 1 + 1 with probability x(1 – x) x = n 1 /N n 1 – 1 with probability x(1 – x) n 1 with probability x 2 + (1 – x) 2.
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Population genetics: Moran model 2 alleles, N haploid organisms choose 2 individual at random: 1 dies, the other reproduces If there are n 1 organisms of type 1 before this step, then afterwards there are n 1 + 1 with probability x(1 – x) x = n 1 /N n 1 – 1 with probability x(1 – x) n 1 with probability x 2 + (1 – x) 2. So
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Population genetics: Moran model 2 alleles, N haploid organisms choose 2 individual at random: 1 dies, the other reproduces If there are n 1 organisms of type 1 before this step, then afterwards there are n 1 + 1 with probability x(1 – x) x = n 1 /N n 1 – 1 with probability x(1 – x) n 1 with probability x 2 + (1 – x) 2. So
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Population genetics: Moran model 2 alleles, N haploid organisms choose 2 individual at random: 1 dies, the other reproduces If there are n 1 organisms of type 1 before this step, then afterwards there are n 1 + 1 with probability x(1 – x)x = n 1 /N n 1 – 1 with probability x(1 – x) n 1 with probability x 2 + (1 – x) 2. So or
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Population genetics: Moran model 2 alleles, N haploid organisms choose 2 individual at random: 1 dies, the other reproduces If there are n 1 organisms of type 1 before this step, then afterwards there are n 1 + 1 with probability x(1 – x) x = n 1 /N n 1 – 1 with probability x(1 – x) n 1 with probability x 2 + (1 – x) 2. So or
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continuum limit: FP equation ( N steps/generation)
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continuum limit: FP equation ( N steps/generation)
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continuum limit: FP equation ( N steps/generation) boundary conditions: P(0,t) = P(1,t) = 0
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continuum limit: FP equation ( N steps/generation) boundary conditions: P(0,t) = P(1,t) = 0 (once an allele dies out, it can not come back)
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continuum limit: FP equation ( 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:
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continuum limit: FP equation ( 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
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heterozygocity Eventually P(x,t) gets concentrated at one boundary,
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heterozygocity Eventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other.
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heterozygocity Eventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other. But it is random which one.
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heterozygocity Eventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other. But it is random which one. Measure this by the heterozygocity
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heterozygocity Eventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other. But it is random which one. Measure this by the heterozygocity use Ito’s lemma:
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heterozygocity Eventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other. But it is random which one. Measure this by the heterozygocity use Ito’s lemma:
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heterozygocity Eventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other. But it is random which one. Measure this by the heterozygocity use Ito’s lemma:
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heterozygocity Eventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other. But it is random which one. Measure this by the heterozygocity use Ito’s lemma:
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heterozygocity Eventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other. But it is random which one. Measure this by the heterozygocity use Ito’s lemma: i.e., diversity dies out in about N generations
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fluctuations of x
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So mean-square fluctuations of x grow initially linearly in t and then saturate
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with mutation: Mutation induces a drift term in the FP and sd equation
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with mutation: Mutation induces a drift term in the FP and sd equation
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with mutation: Mutation induces a drift term in the FP and sd equation
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with mutation: Mutation induces a drift term in the FP and sd equation stationary solution:
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with mutation: Mutation induces a drift term in the FP and sd equation stationary solution:
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fluctuations Use Ito’s lemma on F(x) = x 2 :
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fluctuations Use Ito’s lemma on F(x) = x 2 :
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fluctuations Use Ito’s lemma on F(x) = x 2 : at steady state:
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fluctuations Use Ito’s lemma on F(x) = x 2 : at steady state:
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fluctuations Use Ito’s lemma on F(x) = x 2 : at steady state:
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fluctuations Use Ito’s lemma on F(x) = x 2 : at steady state: mean square fluctuations:
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heterozygocity:
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small noise (large population):
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heterozygocity: small noise (large population):
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heterozygocity: small noise (large population): large noise (small population):
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heterozygocity: small noise (large population): large noise (small population):
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heterozygocity: small noise (large population): large noise (small population): usually one allele dominates, rare transitions
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selection Let the alleles chosen to reproduce do so with with probabilities
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selection Let the alleles chosen to reproduce do so with with probabilities Now, if there are n 1 organisms of type 1 before this step, then afterwards there are
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selection Let the alleles chosen to reproduce do so with with probabilities Now, if there are n 1 organisms of type 1 before this step, then afterwards there are n 1 + 1 with probability p 1 (1 – x)
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selection Let the alleles chosen to reproduce do so with with probabilities Now, if there are n 1 organisms of type 1 before this step, then afterwards there are n 1 + 1 with probability p 1 (1 – x) n 1 – 1 with probability p 2 x
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selection Let the alleles chosen to reproduce do so with with probabilities This leads to a drift in x proportional to x(1 - x) : Now, if there are n 1 organisms of type 1 before this step, then afterwards there are n 1 + 1 with probability p 1 (1 – x) n 1 – 1 with probability p 2 x
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selection Let the alleles chosen to reproduce do so with with probabilities This leads to a drift in x proportional to x(1 - x) : Now, if there are n 1 organisms of type 1 before this step, then afterwards there are n 1 + 1 with probability p 1 (1 – x) n 1 – 1 with probability p 2 x
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selection: large population limit with selection but no mutations:
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selection: large population limit with selection but no mutations: solution:
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selection: large population limit with selection but no mutations: solution:
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selection: large population limit with selection but no mutations: solution:
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Neurons Neurons receive synaptic input from other neurons
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Neurons Neurons receive synaptic input from other neurons ~ injected current
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Neurons Neurons receive synaptic input from other neurons ~ injected current V measured from resting potential
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Neurons Neurons receive synaptic input from other neurons ~ injected current V measured from resting potential with leak g = membrane conductance
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Neurons Neurons receive synaptic input from other neurons ~ injected current V measured from resting potential with leak g = membrane conductance (experimental fact:) input current is noisy, very small τ c compared to membrane time constant τ = C/g
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Neurons Neurons receive synaptic input from other neurons ~ injected current V measured from resting potential with leak g = membrane conductance (experimental fact:) input current is noisy, very small τ c compared to membrane time constant τ = C/g V(t) is described by Wiener process ( g = 0 ) or Brownian motion ( g ≠ 0 )
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Spikes The above is approximately true as long as V stays below a critical value V T.
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Spikes The above is approximately true as long as V stays below a critical value V T. Above this threshold, active ion channels amplify incoming currents and produce an action potential (spike).
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Spikes The above is approximately true as long as V stays below a critical value V T. Above this threshold, active ion channels amplify incoming currents and produce an action potential (spike). After a few ms, V returns to its sub-threshold equilibrium level.
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Spikes The above is approximately true as long as V stays below a critical value V T. Above this threshold, active ion channels amplify incoming currents 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
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Spikes The above is approximately true as long as V stays below a critical value V T. Above this threshold, active ion channels amplify incoming currents 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 of interspike intervals?
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Spikes The above is approximately true as long as V stays below a critical value V T. Above this threshold, active ion channels amplify incoming currents 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 of interspike intervals? This is a first-passage-time problem
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with no leak:
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Assume at t = 0, x = 0
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with no leak: Assume at t = 0, x = 0 boundary condition at θ : P(θ) = 0.
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with no leak: Assume at t = 0, x = 0 boundary condition at θ : P(θ) = 0. We have solved this problem when there is no threshold:
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with no leak: Assume at t = 0, x = 0 boundary condition at θ : P(θ) = 0. We have solved this problem when there is no threshold:
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with no leak: Assume at t = 0, x = 0 boundary condition at θ : P(θ) = 0. We have solved this problem when there is no threshold: But it does not satisfy the boundary condition.
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solution with images: Add an extra source, of opposite sign, at x = 2θ :
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solution with images: Add an extra source, of opposite sign, at x = 2θ : cumulative probability of firing by t:
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solution with images: Add an extra source, of opposite sign, at x = 2θ : cumulative probability of firing by t: interspike interval density:
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solution with images: Add an extra source, of opposite sign, at x = 2θ : cumulative probability of firing by t: interspike interval density: Levy distribution (one-sided stable distribution with α = ½
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another way to get the answer: The event rate is just the (diffusive) current evaluated at x = θ.
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another way to get the answer: The event rate is just the (diffusive) current evaluated at x = θ.
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another way to get the answer: The event rate is just the (diffusive) current evaluated at x = θ.
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a problem: The mean interspike interval is infinite:
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a problem: The mean interspike interval is infinite:
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a problem: The mean interspike interval is infinite: so the firing rate (= 1/ ) is zero!
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adding a constant drift term:
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solution with no boundary:
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adding a constant drift term: solution with no boundary: need a moving image:
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adding a constant drift term: solution with no boundary: need a moving image:
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adding a constant drift term: solution with no boundary: need a moving image:
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adding a constant drift term: solution with no boundary: need a moving image: solution:
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ISI distribution: from
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ISI distribution: from
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ISI distribution: from
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ISI distribution: from Now all moments of f are finite
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ISI distribution: from Now all moments of f are finite
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leaky I&F neuron ( γ = 1/τ = g/C
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leaky I&F neuron ( γ = 1/τ = g/C
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leaky I&F neuron ( γ = 1/τ = g/C = Brownian motion with an added constant drift
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leaky I&F neuron ( γ = 1/τ = g/C = Brownian motion with an added constant drift
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leaky I&F neuron ( γ = 1/τ = g/C = Brownian motion with an added constant drift (set γ = 1 for convenience)
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Looking for stationary solution
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i.e.
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Looking for stationary solution i.e. =>
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Looking for stationary solution Boundary conditions: i.e. =>
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Looking for stationary solution Boundary conditions: sink at firing threshold x i.e. =>
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Looking for stationary solution Boundary conditions: sink at firing threshold x source at x = 0 i.e. =>
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Looking for stationary solution Boundary conditions: sink at firing threshold x source at x = 0 i.e. =>
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Looking for stationary solution Boundary conditions: sink at firing threshold x source at x = 0 i.e. =>
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Looking for stationary solution Boundary conditions: sink at firing threshold x source at x = 0 i.e. => Firing rate: current out at threshold:
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Looking for stationary solution Boundary conditions: sink at firing threshold x source at x = 0 i.e. => Firing rate: current out at threshold:= reinjection rate at reset:
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Stationary solution (2) Also need normalization:
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Stationary solution (2) Also need normalization: Below reset level, J :
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Stationary solution (2) Also need normalization: Below reset level, J : has solution
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Stationary solution (2) Also need normalization: Below reset level, J : has solution Between rest and threshold:
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Stationary solution (2) Also need normalization: Below reset level, J : has solution Between rest and threshold: B.C. at x :
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Stationary solution (2) Also need normalization: Below reset level, J : has solution Between rest and threshold: B.C. at x : =>
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Stationary solution (3) Continuity at x = =>
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Stationary solution (3) Continuity at x = => i.e.,
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Stationary solution (3) Continuity at x = => i.e., algebra … =>
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Stationary solution (3) Continuity at x = => i.e., algebra … => with refractory time τ r
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A simple model of evolution: the Bak-Sneppen model N species,each with fitness x i, each uniformly distributed on (0,1)
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A simple model of evolution: the Bak-Sneppen model N species,each with fitness x i, each uniformly distributed on (0,1)
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A simple model of evolution: the Bak-Sneppen model N species,each with fitness x i, each uniformly distributed on (0,1) evolutionary step:
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A simple model of evolution: the Bak-Sneppen model N species,each with fitness x i, each uniformly distributed on (0,1) evolutionary step: eliminate the weakest species (smallest x i )
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A simple model of evolution: the Bak-Sneppen model N species,each with fitness x i, each uniformly distributed on (0,1) evolutionary step: eliminate the weakest species (smallest x i ) replace it with another species with a random x i
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A simple model of evolution: the Bak-Sneppen model N species,each with fitness x i, each uniformly distributed on (0,1) evolutionary step: eliminate the weakest species (smallest x i ) replace it with another species with a random x i (random-neighbour version) assume another (“neighboring”) species also becomes extinct; replace it with a new one, too
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A simple model of evolution: the Bak-Sneppen model N species,each with fitness x i, each uniformly distributed on (0,1) evolutionary step: eliminate the weakest species (smallest x i ) replace it with another species with a random x i (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 θ replace them and their (randomly chosen) neighbours with new ones until all fitnesses are > θ
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A simple model of evolution: the Bak-Sneppen model N species,each with fitness x i, each uniformly distributed on (0,1) evolutionary step: eliminate the weakest species (smallest x i ) replace it with another species with a random x i (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 θ replace them and their (randomly chosen) neighbours with new ones until all fitnesses are > θ (avalanche)
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A simple model of evolution: the Bak-Sneppen model N species,each with fitness x i, each uniformly distributed on (0,1) evolutionary step: eliminate the weakest species (smallest x i ) replace it with another species with a random x i (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 θ replace them and their (randomly chosen) neighbours with new ones until all fitnesses are > θ (avalanche) start over
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A simple model of evolution: the Bak-Sneppen model N species,each with fitness x i, each uniformly distributed on (0,1) evolutionary step: eliminate the weakest species (smallest x i ) replace it with another species with a random x i (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 θ replace them and their (randomly chosen) neighbours with new ones until all fitnesses are > θ (avalanche) start over Want to know the avalanche length distribution
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getting a master equation P(n,t) = prob that n species have fitness values < θ at time t
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getting a master equation P(n,t) = prob that n species have fitness values < θ at time t At each step 2 species are reassigned random new fitnesses
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getting a master equation P(n,t) = prob that n species have fitness values < θ at time t At each step 2 species are reassigned random new fitnesses transition matrix: T mn :
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getting a master equation P(n,t) = prob that n species have fitness values < θ at time t At each step 2 species are reassigned random new fitnesses transition matrix: T mn :
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getting a master equation P(n,t) = prob that n species have fitness values < θ at time t At each step 2 species are reassigned random new fitnesses transition matrix: T mn : simple random walk in n with first step n=0 -> n=1, thereafter
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getting a master equation P(n,t) = prob that n species have fitness values < θ at time t At each step 2 species are reassigned random new fitnesses transition matrix: T mn : net drift per step: mean square change: simple random walk in n with first step n=0 -> n=1, thereafter
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getting a master equation P(n,t) = prob that n species have fitness values < θ at time t At each step 2 species are reassigned random new fitnesses transition matrix: T mn : net drift per step: mean square change: simple random walk in n with first step n=0 -> n=1, thereafter Walk (avalanche) ends when n=0 again for the first time.
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getting a master equation P(n,t) = prob that n species have fitness values < θ at time t At each step 2 species are reassigned random new fitnesses transition matrix: T mn : net drift per step: mean square change: 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
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Traffic Nagel-Paczuski model: cars can move with speed v=+1 step/time unit or 0.
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Traffic Nagel-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
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Traffic Nagel-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 first car in stau can change speed from 0 to +1 with prob p /step
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Traffic Nagel-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 first car in stau can change speed from 0 to +1 with prob p /step new cars enter the stau with prob q
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Traffic Nagel-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 first car in stau can change speed from 0 to +1 with prob p /step new cars enter the stau with prob q transition matrix for stau length:
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Traffic Nagel-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 first car in stau can change speed from 0 to +1 with prob p /step new cars enter the stau with prob q transition matrix for stau length:
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Traffic Nagel-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 first car in stau can change speed from 0 to +1 with prob p /step new cars enter the stau with prob q transition matrix for stau length: net drift per step: mean square change:
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Traffic Nagel-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 first car in stau can change speed from 0 to +1 with prob p /step new cars enter the stau with prob q transition matrix for stau length: net drift per step: mean square change: biased random walk again
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Traffic Nagel-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 first car in stau can change speed from 0 to +1 with prob p /step new cars enter the stau with prob q transition matrix for stau length: net drift per step: mean square change: biased random walk again, critical (long-tail distribution of stau lengths, lifetimes) fpr p = q.
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