Post on 05-Jan-2016
Theoretical NeurosciencePhysics 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 m
axon length ~ 4 cm
total axon length/mm3 ~ 400 m
Cell membrane, ion channels, action potentials
Membrane potential: rest at ca -70 mvNa-K pump maintains excess K inside,Na outside
Na in: V rises,more channels open “spike”
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 orientationof stimulus
Neural coding: firing rates depend on stimulus
Visual cortical neuron:variation with orientationof stimulus
Motor cortical neuron:variation with directionof movement
Neuronal firing is noisy
Motion-sensitive neuron in visual area MT: spike trains evoked by multiplepresentations of moving random-dotpatterns
Neuronal firing is noisy
Motion-sensitive neuron in visual area MT: spike trains evoked by multiplepresentations of moving random-dotpatterns
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 inelectric fish:s(t) = electric field
Motion-sensitive neuron in blowflyVisual system:s(t) = velocity of moving pattern invisual field
Note: non-additive effect for spikesvery close in time (t < 5 ms
Spike trains: Poisson process model
Homogeneous Poisson process: = rate = prob of firing per unit time,
i.e., prob of spike in interval
rtr )0(),[ tttt
Spike trains: Poisson process model
Homogeneous Poisson process: = rate = prob of firing per unit time,
i.e., prob of spike in interval
rtr )0(),[ tttt
Survivor function: probability of not firing in [0,t): S(t)
Spike trains: Poisson process model
Homogeneous Poisson process: = rate = prob of firing per unit time,
i.e., prob of spike in interval
rtr )0(),[ tttt
Survivor function: probability of not firing in [0,t): S(t)
rttStS
dttdSr
e)(
)(
/)(
Spike trains: Poisson process model
Homogeneous Poisson process: = rate = prob of firing per unit time,
i.e., prob of spike in interval
rtr )0(),[ tttt
Survivor function: probability of not firing in [0,t): S(t)
Probability of firing for the first time in [t, t +t)/ t :
rttStS
dttdSr
e)(
)(
/)(
Spike trains: Poisson process model
Homogeneous Poisson process: = rate = prob of firing per unit time,
i.e., prob of spike in interval
rtr )0(),[ tttt
Survivor function: probability of not firing in [0,t): S(t)
Probability of firing for the first time in [t, t +t)/ t :
rttStS
dttdSr
e)(
)(
/)(
rtrdt
tdStP e
)()(
Spike trains: Poisson process model
Homogeneous Poisson process: = rate = prob of firing per unit time,
i.e., prob of spike in interval
rtr )0(),[ tttt
Survivor function: probability of not firing in [0,t): S(t)
Probability of firing for the first time in [t, t +t)/ t :
(interspike interval distribution)
rttStS
dttdSr
e)(
)(
/)(
rtrdt
tdStP e
)()(
Homogeneous Poisson process (2)
Probability of exactly 1 spike in [0,T):
rTtTrT rtT rTrdtP eee)1( )(
0
Homogeneous Poisson process (2)
Probability of exactly 1 spike in [0,T):
rTtTrT rtT rTrdtP eee)1( )(
0
Probability of exactly 2 spikes in [0,T):
rTtTrttrT rtt
T rTrrdtdtP e)(eee)2( 221)()(
0 0 122121
2
Homogeneous Poisson process (2)
Probability of exactly 1 spike in [0,T):
rTtTrT rtT rTrdtP eee)1( )(
0
Probability of exactly 2 spikes in [0,T):
rTtTrttrT rtt
T rTrrdtdtP e)(eee)2( 221)()(
0 0 122121
2
… Probability of exactly n spikes in [0,T):
rTnT rT
nnP e)(
!
1)(
Homogeneous Poisson process (2)
Probability of exactly 1 spike in [0,T):
rTtTrT rtT rTrdtP eee)1( )(
0
Probability of exactly 2 spikes in [0,T):
rTtTrttrT rtt
T rTrrdtdtP e)(eee)2( 221)()(
0 0 122121
2
… Probability of exactly n spikes in [0,T):
rTnT rT
nnP e)(
!
1)( Poisson distribution
Poisson distribution
rTn
T nrT
nP e!)(
)(
Pprobability of n spikes in interval of duration T:
Poisson distribution
rTn
T nrT
nP e!)(
)(
Pprobability of n spikes in interval of duration T:
rTn Mean count:
Poisson distribution
rTn
T nrT
nP e!)(
)(
Pprobability of n spikes in interval of duration T:
rTn Mean count:
variance: nrTnn 2)(
Poisson distribution
rTn
T nrT
nP e!)(
)(
Pprobability of n spikes in interval of duration T:
rTn Mean count:
variance: nrTnn 2)( i.e., spikesnn
Poisson distribution
rTn
T nrT
nP e!)(
)(
Pprobability of n spikes in interval of duration T:
rTn Mean count:
variance: nrTnn 2)( i.e., spikesnn
large : GaussianrT
Poisson distribution
rTn
T nrT
nP e!)(
)(
Pprobability of n spikes in interval of duration T:
rTn Mean count:
variance: nrTnn 2)( i.e., spikesnn
large : GaussianrT
Poisson process (2): interspike interval distribution
rtrtP e)(Exponential distribution: (like radioactive Decay)
Poisson process (2): interspike interval distribution
rtrtP e)(
rt
1
Exponential distribution: (like radioactive Decay)
Mean ISI:
Poisson process (2): interspike interval distribution
rtrtP e)(
rt
1
22
2 1)( t
rtt
Exponential distribution: (like radioactive Decay)
Mean ISI:
variance:
Poisson process (2): interspike interval distribution
rtrtP e)(
rt
1
22
2 1)( t
rtt
1mean
devstd CV
Exponential distribution: (like radioactive Decay)
Mean ISI:
variance:
Coefficient of variation:
Poisson process (3): correlation function
)()( f
ftttS Spike train:
Poisson process (3): correlation function
)()( f
ftttS
rtS )(
Spike train:
mean:
Poisson process (3): correlation function
)())()()(()( rrtSrtSC
)()( f
ftttS
rtS )(
Spike train:
mean:
Correlation function:
Stationary renewal processDefined by ISI distribution P(t)
Stationary renewal processDefined by ISI distribution P(t)
Relation between P(t) and C(t):
Stationary renewal process
)())((1
)( 2 trtCr
tC
Defined by ISI distribution P(t)
Relation between P(t) and C(t): define
Stationary renewal process
)())((1
)( 2 trtCr
tC
)'()'(')(
)'()'(')()(
0
0
ttCtPdttP
ttPtPdttPtC
t
t
Defined by ISI distribution P(t)
Relation between P(t) and C(t): define
Stationary renewal process
)())((1
)( 2 trtCr
tC
)'()'(')(
)'()'(')()(
0
0
ttCtPdttP
ttPtPdttPtC
t
t
)()()()( CPPC
Defined by ISI distribution P(t)
Relation between P(t) and C(t): define
Laplace transform:
Stationary renewal process
)())((1
)( 2 trtCr
tC
)'()'(')(
)'()'(')()(
0
0
ttCtPdttP
ttPtPdttPtC
t
t
)()()()( CPPC
)(1)(
)(
P
PC
Defined by ISI distribution P(t)
Relation between P(t) and C(t): define
Laplace transform:
Solve:
Fano factor
1
)( 2
Fnnn
F spike count variance / mean spike count
for stationary Poisson process
Fano factor
1
)( 2
Fnnn
F
dCTtStSdtdtn
rTdttSn
T T
T
)()'()(')(
)(
0 0
2
0
spike count variance / mean spike count
for stationary Poisson process
Fano factor
1
)( 2
Fnnn
F
dCTtStSdtdtn
rTdttSn
T T
T
)()'()(')(
)(
0 0
2
0
r
dC
F
)(
spike count variance / mean spike count
for stationary Poisson process
Fano factor
1
)( 2
Fnnn
F
dCTtStSdtdtn
rTdttSn
T T
T
)()'()(')(
)(
0 0
2
0
r
dC
F
)(
2CVF
spike count variance / mean spike count
for stationary Poisson process
for stationary renewal process (prove this)
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
)()(0
tPtP 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.