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

Population genetics: Moran model 2 alleles, N haploid organisms

Population genetics: Moran model 2 alleles, N haploid organisms choose 2 individual at random: 1 dies, the other reproduces

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

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 with probability x(1 – x)x = n 1 /N

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 with probability x(1 – x)x = n 1 /N n 1 – 1 with probability x(1 – x)

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 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.

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 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

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 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

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 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

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 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

continuum limit: FP equation ( N steps/generation)

continuum limit: FP equation ( N steps/generation)

continuum limit: FP equation ( N steps/generation) boundary conditions: P(0,t) = P(1,t) = 0

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)

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:

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

heterozygocity Eventually P(x,t) gets concentrated at one boundary,

heterozygocity Eventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other.

heterozygocity Eventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other. But it is random which one.

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

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:

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:

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:

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:

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

fluctuations of x

So mean-square fluctuations of x grow initially linearly in t and then saturate

with mutation: Mutation induces a drift term in the FP and sd equation

with mutation: Mutation induces a drift term in the FP and sd equation

with mutation: Mutation induces a drift term in the FP and sd equation

with mutation: Mutation induces a drift term in the FP and sd equation stationary solution:

with mutation: Mutation induces a drift term in the FP and sd equation stationary solution:

fluctuations Use Ito’s lemma on F(x) = x 2 :

fluctuations Use Ito’s lemma on F(x) = x 2 :

fluctuations Use Ito’s lemma on F(x) = x 2 : at steady state:

fluctuations Use Ito’s lemma on F(x) = x 2 : at steady state:

fluctuations Use Ito’s lemma on F(x) = x 2 : at steady state:

fluctuations Use Ito’s lemma on F(x) = x 2 : at steady state: mean square fluctuations:

heterozygocity:

small noise (large population):

heterozygocity: small noise (large population):

heterozygocity: small noise (large population): large noise (small population):

heterozygocity: small noise (large population): large noise (small population):

heterozygocity: small noise (large population): large noise (small population): usually one allele dominates, rare transitions

selection Let the alleles chosen to reproduce do so with with probabilities

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

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 with probability p 1 (1 – x)

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 with probability p 1 (1 – x) n 1 – 1 with probability p 2 x

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 with probability p 1 (1 – x) n 1 – 1 with probability p 2 x

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 with probability p 1 (1 – x) n 1 – 1 with probability p 2 x

selection: large population limit with selection but no mutations:

selection: large population limit with selection but no mutations: solution:

selection: large population limit with selection but no mutations: solution:

selection: large population limit with selection but no mutations: solution:

Neurons Neurons receive synaptic input from other neurons

Neurons Neurons receive synaptic input from other neurons ~ injected current

Neurons Neurons receive synaptic input from other neurons ~ injected current V measured from resting potential

Neurons Neurons receive synaptic input from other neurons ~ injected current V measured from resting potential with leak g = membrane conductance

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

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 )

Spikes The above is approximately true as long as V stays below a critical value V T.

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).

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.

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

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?

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

with no leak:

Assume at t = 0, x = 0

with no leak: Assume at t = 0, x = 0 boundary condition at θ : P(θ) = 0.

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:

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:

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.

solution with images: Add an extra source, of opposite sign, at x = 2θ :

solution with images: Add an extra source, of opposite sign, at x = 2θ : cumulative probability of firing by t:

solution with images: Add an extra source, of opposite sign, at x = 2θ : cumulative probability of firing by t: interspike interval density:

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

another way to get the answer: The event rate is just the (diffusive) current evaluated at x = θ.

another way to get the answer: The event rate is just the (diffusive) current evaluated at x = θ.

another way to get the answer: The event rate is just the (diffusive) current evaluated at x = θ.

a problem: The mean interspike interval is infinite:

a problem: The mean interspike interval is infinite:

a problem: The mean interspike interval is infinite: so the firing rate (= 1/ ) is zero!

adding a constant drift term:

solution with no boundary:

adding a constant drift term: solution with no boundary: need a moving image:

adding a constant drift term: solution with no boundary: need a moving image:

adding a constant drift term: solution with no boundary: need a moving image:

adding a constant drift term: solution with no boundary: need a moving image: solution:

ISI distribution: from

ISI distribution: from

ISI distribution: from

ISI distribution: from Now all moments of f are finite

ISI distribution: from Now all moments of f are finite

leaky I&F neuron ( γ = 1/τ = g/C 

leaky I&F neuron ( γ = 1/τ = g/C 

leaky I&F neuron ( γ = 1/τ = g/C  = Brownian motion with an added constant drift

leaky I&F neuron ( γ = 1/τ = g/C  = Brownian motion with an added constant drift

leaky I&F neuron ( γ = 1/τ = g/C  = Brownian motion with an added constant drift (set γ = 1 for convenience)

Looking for stationary solution

i.e.

Looking for stationary solution i.e. =>

Looking for stationary solution Boundary conditions: i.e. =>

Looking for stationary solution Boundary conditions: sink at firing threshold x  i.e. =>

Looking for stationary solution Boundary conditions: sink at firing threshold x  source at x = 0 i.e. =>

Looking for stationary solution Boundary conditions: sink at firing threshold x  source at x = 0 i.e. =>

Looking for stationary solution Boundary conditions: sink at firing threshold x  source at x = 0 i.e. =>

Looking for stationary solution Boundary conditions: sink at firing threshold x  source at x = 0 i.e. => Firing rate: current out at threshold:

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:

Stationary solution (2) Also need normalization:

Stationary solution (2) Also need normalization: Below reset level, J  :

Stationary solution (2) Also need normalization: Below reset level, J  : has solution

Stationary solution (2) Also need normalization: Below reset level, J  : has solution Between rest and threshold:

Stationary solution (2) Also need normalization: Below reset level, J  : has solution Between rest and threshold: B.C. at x  :

Stationary solution (2) Also need normalization: Below reset level, J  : has solution Between rest and threshold: B.C. at x  : =>

Stationary solution (3) Continuity at x =  =>

Stationary solution (3) Continuity at x =  => i.e.,

Stationary solution (3) Continuity at x =  => i.e., algebra … =>

Stationary solution (3) Continuity at x =  => i.e., algebra … => with refractory time τ r

A simple model of evolution: the Bak-Sneppen model N species,each with fitness x i, each uniformly distributed on (0,1)

A simple model of evolution: the Bak-Sneppen model N species,each with fitness x i, each uniformly distributed on (0,1)

A simple model of evolution: the Bak-Sneppen model N species,each with fitness x i, each uniformly distributed on (0,1) evolutionary step:

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 )

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

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

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

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)

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

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

getting a master equation P(n,t) = prob that n species have fitness values < θ at time t

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

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 :

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 :

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

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

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.

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

Traffic Nagel-Paczuski model: cars can move with speed v=+1 step/time unit or 0.

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

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

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

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:

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:

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:

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

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.