Theoretical Neuroscience
Physics 405, Copenhagen University
Block 4, Spring 2007
John Hertz (Nordita)
Office: rm Kc10, NBI Blegdamsvej
Tel 3532 5236 (office) 2720 4184 (mobil)
hertz@nordita.dk
www.nordita.dk/~hertz/course.html
Texts:
P Dayan and L F Abbott, Theoretical Neuroscience (MIT Press)
W Gerstner and W Kistler, Spiking Neuron Models (Cambridge U
Press) http://diwww.epfl.ch/~gerstner/SPNM/SPNM.html
Outline
• Introduction: biological background, spike trains
• Biophysics of neurons: ion channels, spike generation
• Synapses: kinetics, medium- and long-term synaptic
modification
• Mathematical analysis using simplified models
• Network models:
–
–
–
–
noisy cortical networks
primary visual cortex
associative memory
oscillations in olfactory circuits
Lecture I: Introduction
ca 1011 neurons/human brain
104/mm3
soma 10-50 mm
axon length ~ 4 cm
total axon length/mm3 ~ 400 m
Cell membrane, ion channels, action potentials
Na in: V rises,
more channels open
 “spike”
Membrane potential: rest at ca -70 mv
Na-K pump maintains excess K inside,
Na outside
Communication: synapses
Integrating synaptic input:
Brain anatomy: functional regions
Visual system
General anatomy
Retina
Neural coding: firing rates depend on stimulus
Visual cortical neuron:
variation with orientation
of stimulus
Neural coding: firing rates depend on stimulus
Visual cortical neuron:
variation with orientation
of stimulus
Motor cortical neuron:
variation with direction
of movement
Neuronal firing is noisy
Motion-sensitive neuron in visual
area MT: spike trains evoked by multiple
presentations of moving random-dot
patterns
Neuronal firing is noisy
Motion-sensitive neuron in visual
area MT: spike trains evoked by multiple
presentations of moving random-dot
patterns
Intracellular recordings of
membrane potential:
Isolated neurons fire regularly;
neurons in vivo do not:
Quantifying the response of sensory neurons
spike-triggered average stimulus (“reverse correlation”)
Examples of reverse correlation
Electric sensory neuron in
electric fish:
s(t) = electric field
Motion-sensitive neuron in blowfly
Visual system:
s(t) = velocity of moving pattern in
visual field
Note: non-additive effect for spikes
very close in time (Dt < 5 ms)
Spike trains: Poisson process model
Homogeneous Poisson process: r = rate = prob of firing per unit time,
(Dt  0)
i.e., rDt  prob of spike in interval [t , t  Dt )
Spike trains: Poisson process model
Homogeneous Poisson process: r = rate = prob of firing per unit time,
(Dt  0)
i.e., rDt  prob of spike in interval [t , t  Dt )
Survivor function: probability of not firing in [0,t):
S(t)
Spike trains: Poisson process model
Homogeneous Poisson process: r = rate = prob of firing per unit time,
(Dt  0)
i.e., rDt  prob of spike in interval [t , t  Dt )
Survivor function: probability of not firing in [0,t):
r
 dS (t ) / dt
S (t )
 S (t )  e rt
S(t)
Spike trains: Poisson process model
Homogeneous Poisson process: r = rate = prob of firing per unit time,
(Dt  0)
i.e., rDt  prob of spike in interval [t , t  Dt )
Survivor function: probability of not firing in [0,t):
r
 dS (t ) / dt
S (t )
 S (t )  e rt
Probability of firing for the first time in [t, t + Dt)/ Dt :
S(t)
Spike trains: Poisson process model
Homogeneous Poisson process: r = rate = prob of firing per unit time,
(Dt  0)
i.e., rDt  prob of spike in interval [t , t  Dt )
Survivor function: probability of not firing in [0,t):
r
 dS (t ) / dt
S (t )
 S (t )  e rt
Probability of firing for the first time in [t, t + Dt)/ Dt :
P (t )  
dS (t )
 re  rt
dt
S(t)
Spike trains: Poisson process model
Homogeneous Poisson process: r = rate = prob of firing per unit time,
(Dt  0)
i.e., rDt  prob of spike in interval [t , t  Dt )
Survivor function: probability of not firing in [0,t):
r
 dS (t ) / dt
S (t )
S(t)
 S (t )  e rt
Probability of firing for the first time in [t, t + Dt)/ Dt :
P (t )  
dS (t )
 re  rt
dt
(interspike interval distribution)
Homogeneous Poisson process (2)
Probability of exactly 1 spike in [0,T):
T
PT (1)   dt rert  er (T t )  rTerT
0
Homogeneous Poisson process (2)
Probability of exactly 1 spike in [0,T):
T
PT (1)   dt rert  er (T t )  rTerT
0
Probability of exactly 2 spikes in [0,T):
T
t2
0
0
PT (2)   dt2  dt1 rert1  rer ( t2 t1 )  er (T t2 )  12 (rT )2 erT
Homogeneous Poisson process (2)
Probability of exactly 1 spike in [0,T):
T
PT (1)   dt rert  er (T t )  rTerT
0
Probability of exactly 2 spikes in [0,T):
T
t2
0
0
PT (2)   dt2  dt1 rert1  rer ( t2 t1 )  er (T t2 )  12 (rT )2 erT
… Probability of exactly n spikes in [0,T):
PT ( n ) 
1
( rT ) n e  rT
n!
Homogeneous Poisson process (2)
Probability of exactly 1 spike in [0,T):
T
PT (1)   dt rert  er (T t )  rTerT
0
Probability of exactly 2 spikes in [0,T):
T
t2
0
0
PT (2)   dt2  dt1 rert1  rer ( t2 t1 )  er (T t2 )  12 (rT )2 erT
… Probability of exactly n spikes in [0,T):
PT ( n ) 
1
( rT ) n e  rT
n!
Poisson distribution
Poisson distribution
Pprobability of n spikes in interval of duration T:
(rT ) n rT
PT (n) 
e
n!
Poisson distribution
Pprobability of n spikes in interval of duration T:
(rT ) n rT
PT (n) 
e
n!
Mean count:
n  rT
Poisson distribution
Pprobability of n spikes in interval of duration T:
(rT ) n rT
PT (n) 
e
n!
Mean count:
variance:
n  rT
(n  n ) 2  rT  n
Poisson distribution
Pprobability of n spikes in interval of duration T:
(rT ) n rT
PT (n) 
e
n!
Mean count:
variance:
n  rT
(n  n ) 2  rT  n
i.e., n
 n spikes
Poisson distribution
Pprobability of n spikes in interval of duration T:
(rT ) n rT
PT (n) 
e
n!
Mean count:
variance:
n  rT
(n  n ) 2  rT  n
large rT :  Gaussian
i.e., n
 n spikes
Poisson distribution
Pprobability of n spikes in interval of duration T:
(rT ) n rT
PT (n) 
e
n!
Mean count:
variance:
n  rT
(n  n ) 2  rT  n
large rT :  Gaussian
i.e., n
 n spikes
Poisson process (2): interspike interval distribution
Exponential distribution:
P(t )  re  rt
(like radioactive Decay)
Poisson process (2): interspike interval distribution
Exponential distribution:
Mean ISI:
P(t )  re  rt
t
1
r
(like radioactive Decay)
Poisson process (2): interspike interval distribution
Exponential distribution:
Mean ISI:
variance:
P(t )  re  rt
t
(like radioactive Decay)
1
r
1
(t  t )  2  t 2
r
2
Poisson process (2): interspike interval distribution
Exponential distribution:
Mean ISI:
variance:
Coefficient of variation:
P(t )  re  rt
t
(like radioactive Decay)
1
r
1
(t  t )  2  t 2
r
2
std dev
CV 
1
mean
Poisson process (3): correlation function
Spike train:
S (t )    (t  t f )
f
Poisson process (3): correlation function
Spike train:
S (t )    (t  t f )
f
mean:
S (t )  r
Poisson process (3): correlation function
Spike train:
S (t )    (t  t f )
f
mean:
S (t )  r
Correlation function:
C ( )  (S (t )  r )( S (t   )  r )  r ( )
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)
1
2
C
(
t
)

(
C
(
t
)

r
)(t )
Relation between P(t) and C(t): define 
r
Stationary renewal process
Defined by ISI distribution P(t)
1
2
C
(
t
)

(
C
(
t
)

r
)(t )
Relation between P(t) and C(t): define 
r
t
C (t )  P (t )   dt 'P (t ' )P (t  t ' )  
0
t
 P (t )   dt 'P (t ' )C (t  t ' )
0
Stationary renewal process
Defined by ISI distribution P(t)
1
2
C
(
t
)

(
C
(
t
)

r
)(t )
Relation between P(t) and C(t): define 
r
t
C (t )  P (t )   dt 'P (t ' )P (t  t ' )  
0
t
 P (t )   dt 'P (t ' )C (t  t ' )
0
Laplace transform:
C ( )  P( )  P( )C ( )
Stationary renewal process
Defined by ISI distribution P(t)
1
2
C
(
t
)

(
C
(
t
)

r
)(t )
Relation between P(t) and C(t): define 
r
t
C (t )  P (t )   dt 'P (t ' )P (t  t ' )  
0
t
 P (t )   dt 'P (t ' )C (t  t ' )
0
Laplace transform:
Solve:
C ( )  P( )  P( )C ( )
C  ( ) 
P ( )
1  P ( )
Fano factor
(n  n ) 2
spike count variance / mean spike count
F
n
F  1 for stationary Poisson process
Fano factor
(n  n ) 2
spike count variance / mean spike count
F
n
F  1 for stationary Poisson process
T
n   S (t ) dt rT
0
T
T

0
0

(n)   dt  dt ' S (t )S (t ' )  T  C ( )d
2
Fano factor
(n  n ) 2
spike count variance / mean spike count
F
n
F  1 for stationary Poisson process
T
n   S (t ) dt rT

0
T
T

0
0

(n)   dt  dt ' S (t )S (t ' )  T  C ( )d
2

F
 C ( )d

r
Fano factor
(n  n ) 2
spike count variance / mean spike count
F
n
F  1 for stationary Poisson process
T
n   S (t ) dt rT

0
T
T

0
0

(n)   dt  dt ' S (t )S (t ' )  T  C ( )d
2
F  CV 2

F
 C ( )d

for stationary renewal process (prove this)
r
Nonstationary point processes
Nonstationary point processes
Nonstationary Poisson process: time-dependent rate r(t)
Still have Poisson count distribution, F=1
Nonstationary point processes
Nonstationary Poisson process: time-dependent rate r(t)
Still have Poisson count distribution,
F=1
Nonstationary renewal process: time-dependent ISI distribution
P (t )  Pt0 (t )
= ISI probability starting at t0
Experimental results (1)
Correlation functions
Experimental results (1)
Correlation functions
Count variance vs mean
Experimental results (2)
ISI distribution
Experimental results (2)
ISI distribution
CV’s for many neurons
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.