Transcript Theoretical Neuroscience

Lecture 2: Everything you need to know
to know about point processes
Outline:
• basic ideas
• homogeneous (stationary) Poisson processes
• Poisson distribution
• inter-event interval distribution
• coefficient of variance (CV)
• correlation function
• stationary renewal process
• relation between IEI distribution and correlation function
• Fano factor F
• relation between F and CV
• nonstationary (inhomogeneous) Poisson process
• time rescaling
Point Processes
Point process: discrete set of points (events) on the real numbers
(or some interval on the reals)
  t1  t 2 L tN  

Point Processes
Point process: discrete set of points (events) on the real numbers
(or some interval on the reals)
  t1  t 2 L tN  
Usually we are thinking of times of otherwise identical events.
(but sometimes space or space-time)

Point Processes
Point process: discrete set of points (events) on the real numbers
(or some interval on the reals)
  t1  t 2 L tN  
Usually we are thinking of times of otherwise identical events.
(but sometimes space or space-time)

Examples: radioactive decay, arrival times, earthquakes,
neuronal spike trains, …
Point Processes
Point process: discrete set of points (events) on the real numbers
(or some interval on the reals)
  t1  t 2 L tN  
Usually we are thinking of times of otherwise identical events.
(but sometimes space or space-time)

Examples: radioactive decay, arrival times, earthquakes,
neuronal spike trains, …
Stochastic: characterized by the probability (density) of every set
{t1, t2, … tN}
Neuronal spike trains
Action potential:
Neuronal spike trains
Action potential:
spike trains evoked by many
presentations of the same stimulus:
Neuronal spike trains
Action potential:
spike trains evoked by many
presentations of the same stimulus:
(apparently) stochastic
Homogeneous Poisson process
Homogeneous Poisson process
Homogeneous Poisson process: r = rate = prob of event per unit time,
i.e., rΔt = prob of event in interval [t, t + Δt)
(Δt 0)
Homogeneous Poisson process
Homogeneous Poisson process: r = rate = prob of event per unit time,
i.e., rΔt = prob of event in interval [t, t + Δt)
(Δt 0)
Survivor function: probability of no event in [0,t):
S(t)
Homogeneous Poisson process
Homogeneous Poisson process: r = rate = prob of event per unit time,
i.e., rΔt = prob of event in interval [t, t + Δt)
(Δt 0)
Survivor function: probability of no event in [0,t):
dS
 rS
dt

 S(t)  ert
S(t)
Homogeneous Poisson process
Homogeneous Poisson process: r = rate = prob of event per unit time,
i.e., rΔt = prob of event in interval [t, t + Δt)
(Δt 0)
Survivor function: probability of no event in [0,t):
dS
 rS
dt
 S(t)  ert
Probability /unit time of first event in [t, t + Dt)) :

S(t)
Homogeneous Poisson process
Homogeneous Poisson process: r = rate = prob of event per unit time,
i.e., rΔt = prob of event in interval [t, t + Δt)
(Δt 0)
Survivor function: probability of no event in [0,t):
dS
 rS
dt
 S(t)  ert
Probability /unit time of first event in [t, t + Dt)) :

P(t)  

dS(t)
 rert
dt
S(t)
Homogeneous Poisson process
Homogeneous Poisson process: r = rate = prob of event per unit time,
i.e., rΔt = prob of event in interval [t, t + Δt)
(Δt 0)
Survivor function: probability of no event in [0,t):
dS
 rS
dt
S(t)
 S(t)  ert
Probability /unit time of first event in [t, t + Dt)) :

P(t)  

dS(t)
 rert
dt
(inter-event interval distribution)
Homogeneous Poisson process (2)
Probability of exactly 1 event in [0,T):
PT (1) 


T
0
dt rert  er(T t )  rTerT
Homogeneous Poisson process (2)
Probability of exactly 1 event in [0,T):
PT (1) 

T
0
dt rert  er(T t )  rTerT
Probability of exactly 2 events in [0,T):


PT (2) 

T
0
dt 2  dt1 rert1  rer(t 2 t1 )  er(T t 2 )  12 (rT) 2 erT
t2
0
Homogeneous Poisson process (2)
Probability of exactly 1 event in [0,T):
PT (1) 

T
0
dt rert  er(T t )  rTerT
Probability of exactly 2 events in [0,T):

PT (2) 

T
0
dt 2  dt1 rert1  rer(t 2 t1 )  er(T t 2 )  12 (rT) 2 erT
t2
0
… Probability of exactly n events in [0,T):

1
PT (n) 
n!
(rT)n erT
Homogeneous Poisson process (2)
Probability of exactly 1 event in [0,T):
PT (1) 

T
0
dt rert  er(T t )  rTerT
Probability of exactly 2 events in [0,T):

PT (2) 

T
0
dt 2  dt1 rert1  rer(t 2 t1 )  er(T t 2 )  12 (rT) 2 erT
t2
0
… Probability of exactly n events in [0,T):

1
PT (n)  (rT)n erT
Poisson distribution
n!
Poisson distribution
Probability of n events in interval of duration T:
Poisson distribution
Probability of n events in interval of duration T:
mean count:
<n> = rT
Poisson distribution
Probability of n events in interval of duration T:
mean count:
<n> = rT
variance: <(n -<n>)2> = rT,
i.e. <n> ± <n>1/2
Poisson distribution
Probability of n events in interval of duration T:
mean count:
<n> = rT
variance: <(n -<n>)2> = rT,
large rT:  Gaussian
i.e. <n> ± <n>1/2
Poisson distribution
Probability of n events in interval of duration T:
mean count:
<n> = rT
variance: <(n -<n>)2> = rT,
large rT:  Gaussian
i.e. <n> ± <n>1/2
Characteristic function
Poisson distribution with mean a:

P(n)  e
a
an
n!
Characteristic function
Poisson distribution with mean a:
P(n)  e
a
Characteristic function G(k)  e ikn  ea  e ikn
n


an
n!
n
a
 ea 
n!
n
e ika
n
n!
Characteristic function
Poisson distribution with mean a:
P(n)  e
a
an
n!
Characteristic function G(k)  e ikn  ea  e ikn
n
a
 ea 
n!
n


 ea exp(e ik a)  exp (e ik 1)a

eika
n
n
n!
Characteristic function
Poisson distribution with mean a:
P(n)  e
a
an
n!
Characteristic function G(k)  e ikn  ea  e ikn
n
a
 ea 
n!
n


 ea exp(e ik a)  exp (e ik 1)a
Cumulant generating function

eika
n
n
n!
Characteristic function
Poisson distribution with mean a:
P(n)  e
a
an
n!
Characteristic function G(k)  e ikn  ea  e ikn
n
a
 ea 
n!
n

eika
n
n
n!

 ea exp(e ik a)  exp (e ik 1)a
Cumulant generating function logG(k)  e ik 1a  ika  12 k 2 a  L


Characteristic function
Poisson distribution with mean a:
P(n)  e
a
an
n!
Characteristic function G(k)  e ikn  ea  e ikn
n
a
 ea 
n!
n

eika
n
n
n!

 ea exp(e ik a)  exp (e ik 1)a
Cumulant generating function logG(k)  e ik 1a  ika  12 k 2 a  L

2
 n  a, n  n   a, L

Characteristic function
Poisson distribution with mean a:
P(n)  e
a
an
n!
Characteristic function G(k)  e ikn  ea  e ikn
n
a
 ea 
n!
n

eika
n
n
n!

 ea exp(e ik a)  exp (e ik 1)a
Cumulant generating function logG(k)  e ik 1a  ika  12 k 2 a  L

2
 n  a, n  n   a, L
All cumulants = a

Homogeneous Poisson process
(3): inter-event interval distribution

rt (like radioactive Decay)
Exponential distribution: P
(
t
)
r
e
Homogeneous Poisson process
(3): inter-event interval distribution

rt (like radioactive Decay)
Exponential distribution: P
(
t
)
r
e
t
mean IEI:

1
r
Homogeneous Poisson process
(3): inter-event interval distribution

rt (like radioactive Decay)
Exponential distribution: P
(
t
)
r
e
mean IEI:
variance:

t
1
r
1
2
2
(t

t
)


t

r2
Homogeneous Poisson process
(3): inter-event interval distribution

rt (like radioactive Decay)
Exponential distribution: P
(
t
)
r
e
t
mean IEI:
1
r
1
2
2
(t

t
)


t

r2
variance:
Coefficient of variation:


CV 
stddev
1
mean
Homogeneous Poisson process
(4): correlation function
notation:
S
(
t
)

(
t
tf)

f
Homogeneous Poisson process
(4): correlation function
notation:
S
(
t
)

(
t
tf)

f
Homogeneous Poisson process
(4): correlation function
notation:
S
(
t
)

(
t
tf)

f
mean:

S(t)  r
Homogeneous Poisson process
(4): correlation function
S
(
t
)

(
t
tf)

notation:
f
mean:
S(t)  r

correlation function:



C
(
)

(
S
(
t
)

r
)(
S
(
t

)

r
)

r
(
)
Proof:
(1):

S(t)S(t ) , t  t :
Proof:
(1):
S(t)S(t ) , t  t :
S(t) and S(t’) are independent, so S(t)S(t )  S(t) S(t )  r 2


Proof:
(1):
S(t)S(t ) , t  t :
S(t) and S(t’) are independent, so S(t)S(t )  S(t) S(t )  r 2
 (2):
t  t: Use finite-width, finite-height delta functions:
1


 (t)  ,   /2  t   /2


Proof:
(1):
S(t)S(t ) , t  t :
S(t) and S(t’) are independent, so S(t)S(t )  S(t) S(t )  r 2
 (2):
t  t: Use finite-width, finite-height delta functions:
1


 (t)  ,   /2  t   /2


Proof:
(1):
S(t)S(t ) , t  t :
S(t) and S(t’) are independent, so S(t)S(t )  S(t) S(t )  r 2
 (2):
t  t: Use finite-width, finite-height delta functions:
1

 (t)  ,   /2  t   /2

12 r
S (t)  r      r (0)
  
2



Proof:
(1):
S(t)S(t ) , t  t :
S(t) and S(t’) are independent, so S(t)S(t )  S(t) S(t )  r 2
 (2):
t  t: Use finite-width, finite-height delta functions:
1

 (t)  ,   /2  t   /2


12 r
S (t)  r      r (0)
  
2

S(t)S( t )  r (t  t )  r 2 1  (t  t )
so


Proof:
(1):
S(t)S(t ) , t  t :
S(t) and S(t’) are independent, so S(t)S(t )  S(t) S(t )  r 2
 (2):
t  t: Use finite-width, finite-height delta functions:
1

 (t)  ,   /2  t   /2


12 r
S (t)  r      r (0)
  
2

S(t)S( t )  r (t  t )  r 2 1  (t  t )
so
 r(1 r) (t  t )  r 2



Proof:
(1):
S(t)S(t ) , t  t :
S(t) and S(t’) are independent, so S(t)S(t )  S(t) S(t )  r 2
 (2):
t  t: Use finite-width, finite-height delta functions:
1

 (t)  ,   /2  t   /2


12 r
S (t)  r      r (0)
  
2


S(t)S(t )  r (t  t )  r 2 1  (t  t )
so
 r(1 r) (t  t )  r 2
S(t) 
S(t)
S(t ) 
S(t )

S(t)S( t )  r(1 r) (t  t )

Proof:
(1):
S(t)S(t ) , t  t :
S(t) and S(t’) are independent, so S(t)S(t )  S(t) S(t )  r 2
 (2):
t  t: Use finite-width, finite-height delta functions:
1

 (t)  ,   /2  t   /2


12 r
S (t)  r      r (0)
  
2


S(t)S(t )  r (t  t )  r 2 1  (t  t )
so
 r(1 r) (t  t )  r 2
S(t) 
S(t)
S(t ) 
S(t )

S(t)S( t )  r(1 r) (t  t ) 
r (t  t )
0
General stationary point process:
For any point process,

C(t)  r (t)  A(t)
General stationary point process:
For any point process,
with A(t) continuous,


C(t)  r (t)  A(t)
A(t) t
0

General stationary point process:
For any point process,
with A(t) continuous,
C(t)  r (t)  A(t)
A(t) t
0


(because S(t) is composed of delta-functions)

Stationary renewal process
Defined by ISI distribution P(t)
Stationary renewal process
Defined by ISI distribution P(t)
Relation between P(t) and C(t):
Stationary renewal process
Defined by ISI distribution P(t)
Relation between P(t) and C(t):
1
define C (t)  (C(t)  r 2 )(t)
r

Stationary renewal process
Defined by ISI distribution P(t)
Relation between P(t) and C(t):
define
Stationary renewal process
Defined by ISI distribution P(t)
Relation between P(t) and C(t):
C (t)  P(t) 
1
define C (t)  (C(t)  r 2 )(t)
r
t
 dt P(t )P(t  t )  L
0

 P(t)   dt P( t )C (t  t )
t
0

Stationary renewal process
Defined by ISI distribution P(t)
Relation between P(t) and C(t):
C (t)  P(t) 
1
define C (t)  (C(t)  r 2 )(t)
r
t
 dt P(t )P(t  t )  L
0

 P(t)   dt P( t )C (t  t )
t
0
Laplace transform:


C ( )  P()  P( )C ( )
Stationary renewal process
Defined by ISI distribution P(t)
Relation between P(t) and C(t):
C (t)  P(t) 
1
define C (t)  (C(t)  r 2 )(t)
r
t
 dt P(t )P(t  t )  L
0

 P(t)   dt P( t )C (t  t )
t
0
Laplace transform:
Solve:
C ( )  P()  P( )C ( )
C ( ) 


P(  )
1 P(  )
Fano factor
F

(n  n ) 2
n
spike count variance / mean spike count
Fano factor
F
F 1

(n  n ) 2
n
spike count variance / mean spike count
for stationary Poisson process
Fano factor
(n  n ) 2
F
n
F 1
n 
T

for stationary Poisson process
S(t) dt rT
0
n 
2
T
T
 dt  dt
1
0
2
0
spike count variance / mean spike count
S(t1 )S(t 2 )
Fano factor
(n  n ) 2
F
n
F 1
n 
T

spike count variance / mean spike count
for stationary Poisson process
S(t) dt rT
0
n 
2
T
T
 dt  dt
1
0
2
S(t1 )S(t 2 )
0

t2
t1 
Fano factor
(n  n ) 2
F
n
F 1
n 
T

spike count variance / mean spike count
for stationary Poisson process
S(t) dt rT
0
n 
2
T
T
 dt  dt
1
0
2
0

S(t1 )S(t 2 )  T  C( )d


τ
t
Fano factor
(n  n ) 2
F
n
F 1
n 
T

spike count variance / mean spike count
for stationary Poisson process
S(t) dt rT
0
n 
2
T
T
 dt  dt
1
0
2

S(t1 )S(t 2 )  T  C( )d

0

 F
 C( )d

r

τ
t
F=CV2 for a stationary renewal process
F=CV2 for a stationary renewal process
F depends on integral of C(t), CV depends on moments of P(t).
F=CV2 for a stationary renewal process
F depends on integral of C(t), CV depends on moments of P(t).
Use relation between C(λ) and P(λ) to relate F and CV.
F=CV2 for a stationary renewal process
F depends on integral of C(t), CV depends on moments of P(t).
Use relation between C(λ) and P(λ) to relate F and CV.
The algebra:
1
F
r

1
 dtC(t)  r




dt r (t)  A(t)
F=CV2 for a stationary renewal process
F depends on integral of C(t), CV depends on moments of P(t).
Use relation between C(λ) and P(λ) to relate F and CV.
The algebra:
1
F
r

1
 dtC(t)  r

2
 dt r (t)  A(t)  1 r

 rC

0
2
(t)

r
dt

F=CV2 for a stationary renewal process
F depends on integral of C(t), CV depends on moments of P(t).
Use relation between C(λ) and P(λ) to relate F and CV.
The algebra:
2 
 dt r (t)  A(t)  1 r  0 rC (t)  r 2 dt


r 
 t
 1 2lim  dte C (t)  r  1  2limC ( )  
 0 0
 0
 
1
F
r

1
 dtC(t)  r


F=CV2 for a stationary renewal process
F depends on integral of C(t), CV depends on moments of P(t).
Use relation between C(λ) and P(λ) to relate F and CV.
The algebra:
2 
 dt r (t)  A(t)  1 r  0 rC (t)  r 2 dt


r 
 1 2lim  0 dte t C (t)  r  1  2limC ( )  
 0
 0
 
 P( )
r 
 1 2lim
 
 01 P(  )
 
1
F
r

1
dtC(t)


r


F=CV2 for a stationary renewal process
F depends on integral of C(t), CV depends on moments of P(t).
Use relation between C(λ) and P(λ) to relate F and CV.
The algebra:
2 
 dt r (t)  A(t)  1 r  0 rC (t)  r 2 dt


r 
 1 2lim  0 dte t C (t)  r  1  2limC ( )  
 0
 0
 
 P( )
r 
 1 2lim
 
 01 P(  )
 
1
F
r
Now use


1
dtC(t)


r


P() 1  t  12 2 t 2 L
F=CV2 for a stationary renewal process
F depends on integral of C(t), CV depends on moments of P(t).
Use relation between C(λ) and P(λ) to relate F and CV.
The algebra:
2 
 dt r (t)  A(t)  1 r  0 rC (t)  r 2 dt


r 
 1 2lim  0 dte t C (t)  r  1  2limC ( )  
 0
 0
 
 P( )
r 
 1 2lim
 
 01 P(  )
 
1
F
r
Now use


1
dtC(t)


r


P() 1  t   t L
1
2
2
2
and r 
1
t
F=CV2 (continued)
1  t  1 2 t 2
1
2
F  1 2lim

1 2 2
 0
t

t

 t

2






F=CV2 (continued)
1  t  1 2 t 2

1
2

F  1 2lim

1 2 2
 0
 t 

  t  2  t

1 2 2

1 1  t  2  t

 1 2lim
1
2
1
 0  t


 1 2  t / t


F=CV2 (continued)
1  t  1 2 t 2

1
2

F  1 2lim

1 2 2
 0
 t 

  t  2  t

1 2 2


1 1  t  2  t

 1 2lim
1
2
1
 0  t


 1 2  t / t

1 2 2
1 12  t 2 / t 
1 1  t  2  t


 1 2lim

2
2
1
1
 0  t
1 2  t / t 

 1 2  t / t


F=CV2 (continued)
1  t  1 2 t 2

1
2

F  1 2lim

1 2 2
 0
 t 

  t  2  t

1 2 2


1 1  t  2  t

 1 2lim
1
2
1
 0  t


 1 2  t / t

1 2 2

1 12  t 2 / t 
1 1  t  2  t


 1 2lim

2
2
1
1
 0  t
1 2  t / t 

 1 2  t / t

2
1
1 2 2 

1  t  2  t / t  2  t


 1 2lim
2
1
 0  t
1 2  t / t





F=CV2 (continued)
1  t  1 2 t 2

1
2

F  1 2lim

1 2 2
 0
 t 

  t  2  t

1 2 2


1 1  t  2  t

 1 2lim
1
2
1
 0  t


 1 2  t / t

1 2 2

1 12  t 2 / t 
1 1  t  2  t


 1 2lim

2
2
1
1
 0  t
1 2  t / t 

 1 2  t / t

2
1
1 2 2
1   t  2  t / t  2  t 


 1 2lim
2
1
 0  t
1 2  t / t





 1 2 1

1
2
t2 / t
2

F=CV2 (continued)
1  t  1 2 t 2

1
2

F  1 2lim

1 2 2
 0
 t 

  t  2  t

1 2 2

1 1  t  2  t

 1 2lim
1
2
1
 0  t


 1 2  t / t

1 2 2
1 12  t 2 / t 
1 1  t  2  t


 1 2lim

2
2
1
1
 0  t
1 2  t / t 

 1 2  t / t

2
1
1 2 2
1   t  2  t / t  2  t 


 1 2lim
2
1
 0  t
1 2  t / t





 1 2 1
t2 / t
2
t  t  t

1
2
2
 1
t
2

2
F=CV2 (continued)
1  t  1 2 t 2

1
2

F  1 2lim

1 2 2
 0
 t 

  t  2  t

1 2 2

1 1  t  2  t

 1 2lim
1
2
1
 0  t


 1 2  t / t

1 2 2
1 12  t 2 / t 
1 1  t  2  t


 1 2lim

2
2
1
1
 0  t
1 2  t / t 

 1 2  t / t

2
1
1 2 2
1   t  2  t / t  2  t 


 1 2lim
2
1
 0  t
1 2  t / t





 1 2 1
t2 / t
2
t  t  t

1
2
2
 1
t
2

2

t  t
t
2

2
F=CV2 (continued)
1  t  1 2 t 2

1
2

F  1 2lim

1 2 2
 0
 t 

  t  2  t

1 2 2

1 1  t  2  t

 1 2lim
1
2
1
 0  t


 1 2  t / t

1 2 2
1 12  t 2 / t 
1 1  t  2  t


 1 2lim

2
2
1
1
 0  t
1 2  t / t 

 1 2  t / t

2
1
1 2 2
1   t  2  t / t  2  t 


 1 2lim
2
1
 0  t
1 2  t / t





 1 2 1
t2 / t
2
t  t  t

1
2
2
 1
t
2

2

t  t
t
2

2
 CV 2
Nonstationary Poisson processes
Nonstationary Poisson process: time-dependent rate r(t)
Nonstationary Poisson processes
Nonstationary Poisson process: time-dependent rate r(t)
Time rescaling: instead of t, use

s

t
r( t )dt 
Nonstationary Poisson processes
Nonstationary Poisson process: time-dependent rate r(t)
Time rescaling: instead of t, use
s

t
r( t )dt 
Then the event rate per unit s is 1, i.e. the process is stationary
when viewed as a function of s.

Nonstationary Poisson processes
Nonstationary Poisson process: time-dependent rate r(t)
Time rescaling: instead of t, use
s

t
r( t )dt 
Then the event rate per unit s is 1, i.e. the process is stationary
when viewed as a function of s. In particular, still have
 in any interval
• Poisson count distribution
Nonstationary Poisson processes
Nonstationary Poisson process: time-dependent rate r(t)
Time rescaling: instead of t, use
s

t
r( t )dt 
Then the event rate per unit s is 1, i.e. the process is stationary
when viewed as a function of s. In particular, still have
 in any interval
• Poisson count distribution
• F =1
Nonstationary Poisson processes
Nonstationary Poisson process: time-dependent rate r(t)
Time rescaling: instead of t, use
s

t
r( t )dt 
Then the event rate per unit s is 1, i.e. the process is stationary
when viewed as a function of s. In particular, still have
 in any interval
• Poisson count distribution
• F =1
Nonstationary renewal process: time-dependent inter-event
distribution
Nonstationary Poisson processes
Nonstationary Poisson process: time-dependent rate r(t)
Time rescaling: instead of t, use
s

t
r( t )dt 
Then the event rate per unit s is 1, i.e. the process is stationary
when viewed as a function of s. In particular, still have
 in any interval
• Poisson count distribution
• F =1
Nonstationary renewal process: time-dependent inter-event
distribution
P
(
t
)

P
(
t
)= inter-event probability starting at t0
t
0
homework
• Prove that the ISI distribution is exponential for a stationary Poisson
process.
• Prove that the CV is 1 for a stationary Poisson process.
• Show that the Poisson distribution approaches a Gaussian one for
large mean spike count.
• Prove that F = CV2 for a stationary renewal process.
• Show why the spike count distribution for an inhomogeneous
Poisson process is the same as that for a homogeneous Poisson
process with the same mean spike count.