Download - Theoretical Neuroscience
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Theoretical NeurosciencePhysics 405, Copenhagen University
Block 4, Spring 2007
John Hertz (Nordita)Office: rm Kc10, NBI Blegdamsvej
Tel 3532 5236 (office) 2720 4184 (mobil)[email protected]
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
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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
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Lecture I: Introduction
ca 1011 neurons/human brain
104/mm3
soma 10-50 m
axon length ~ 4 cm
total axon length/mm3 ~ 400 m
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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”
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Communication: synapses
Integrating synaptic input:
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Brain anatomy: functional regions
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Visual system
General anatomy Retina
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Neural coding: firing rates depend on stimulus
Visual cortical neuron:variation with orientationof stimulus
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Neural coding: firing rates depend on stimulus
Visual cortical neuron:variation with orientationof stimulus
Motor cortical neuron:variation with directionof movement
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Neuronal firing is noisy
Motion-sensitive neuron in visual area MT: spike trains evoked by multiplepresentations of moving random-dotpatterns
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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:
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Quantifying the response of sensory neurons
spike-triggered average stimulus (“reverse correlation”)
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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
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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
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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)
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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)(
)(
/)(
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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)(
)(
/)(
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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
)()(
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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
)()(
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Homogeneous Poisson process (2)
Probability of exactly 1 spike in [0,T):
rTtTrT rtT rTrdtP eee)1( )(
0
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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
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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)(
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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
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Poisson distribution
rTn
T nrT
nP e!)(
)(
Pprobability of n spikes in interval of duration T:
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Poisson distribution
rTn
T nrT
nP e!)(
)(
Pprobability of n spikes in interval of duration T:
rTn Mean count:
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Poisson distribution
rTn
T nrT
nP e!)(
)(
Pprobability of n spikes in interval of duration T:
rTn Mean count:
variance: nrTnn 2)(
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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
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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
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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
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Poisson process (2): interspike interval distribution
rtrtP e)(Exponential distribution: (like radioactive Decay)
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Poisson process (2): interspike interval distribution
rtrtP e)(
rt
1
Exponential distribution: (like radioactive Decay)
Mean ISI:
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Poisson process (2): interspike interval distribution
rtrtP e)(
rt
1
22
2 1)( t
rtt
Exponential distribution: (like radioactive Decay)
Mean ISI:
variance:
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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:
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Poisson process (3): correlation function
)()( f
ftttS Spike train:
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Poisson process (3): correlation function
)()( f
ftttS
rtS )(
Spike train:
mean:
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Poisson process (3): correlation function
)())()()(()( rrtSrtSC
)()( f
ftttS
rtS )(
Spike train:
mean:
Correlation function:
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Stationary renewal processDefined by ISI distribution P(t)
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Stationary renewal processDefined by ISI distribution P(t)
Relation between P(t) and C(t):
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Stationary renewal process
)())((1
)( 2 trtCr
tC
Defined by ISI distribution P(t)
Relation between P(t) and C(t): define
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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
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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:
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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:
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Fano factor
1
)( 2
Fnnn
F spike count variance / mean spike count
for stationary Poisson process
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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
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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
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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)
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Nonstationary point processes
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Nonstationary point processes
Nonstationary Poisson process: time-dependent rate r(t)
Still have Poisson count distribution, F=1
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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
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Experimental results (1)
Correlation functions
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Experimental results (1)
Correlation functions Count variance vs mean
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Experimental results (2)
ISI distribution
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Experimental results (2)
ISI distribution CV’s for many neurons
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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.