noise and delays in neurophysics
DESCRIPTION
NOISE and DELAYS in NEUROPHYSICS. Andre Longtin Center for Neural Dynamics and Computation Department of Physics Department of Cellular and Molecular Medicine UNIVERSITY OF OTTAWA, Canada. OUTLINE. Modeling Single Neuron noise leaky integrate and fire - PowerPoint PPT PresentationTRANSCRIPT
NOISE and DELAYS in NOISE and DELAYS in NEUROPHYSICSNEUROPHYSICS
Andre LongtinAndre Longtin
Center for Neural Dynamics and ComputationCenter for Neural Dynamics and ComputationDepartment of PhysicsDepartment of Physics
Department of Cellular and Molecular MedicineDepartment of Cellular and Molecular Medicine
UNIVERSITY OF OTTAWA, CanadaUNIVERSITY OF OTTAWA, Canada
OUTLINEOUTLINE
Modeling Single Neuron noiseModeling Single Neuron noise
leaky integrate and fireleaky integrate and fire
quadratic integrate and firequadratic integrate and fire
“ “transfer function” approachtransfer function” approach Modeling response to signalsModeling response to signals Information theoryInformation theory Delayed dynamicsDelayed dynamics
MOTIVATION for STUDYING NOISE
““Noise” in the neuroscience Noise” in the neuroscience literatureliterature
•As an input with many frequency components over a particular band, of similar amplitudes, and scattered phases
•As the resulting current from the integration of many independent, excitatory and inhibitory synaptic events at the soma
•As the maintained discharge of some neurons
•As « cross-talk » responses from indirectly stimulated neurons
•As « internal », resulting from the probabilistic gating of voltage-dependent ion channels
•As « synaptic », resulting from the stochastic nature of vesicle release at the synaptic cleft
Segundo et al., Origins and Self Organization, 1994
Leaky Integrate-and-fire with Leaky Integrate-and-fire with + and - Feedback+ and - Feedback
f = firing rate function
Firing Rate FunctionsFiring Rate Functions
Or stochastic:
Noise free:
Noise induced Stochastic Noise induced Stochastic Gain Control ResonanceGain Control Resonance
For Poisson input (Campbell’s theorem): For Poisson input (Campbell’s theorem):
mean conductance mean conductance ~ ~ mean input ratemean input ratestandard deviation standard deviation σσ ~ sqrt(mean rate) ~ sqrt(mean rate)
NOISE smoothes out f-I curvesNOISE smoothes out f-I curves
WHAT QUADRATIC WHAT QUADRATIC INTEGRATE-AND-FIRE MODEL?INTEGRATE-AND-FIRE MODEL? Technically more difficultTechnically more difficult Which variable to use? Which variable to use?
On the real line? On the real line?
On a circle?On a circle?
Information-theoretic approachesInformation-theoretic approaches Linear encoding versus nonlinear processingLinear encoding versus nonlinear processing Rate code, long time constant , integratorRate code, long time constant , integrator Time code, small time constant, coincidence Time code, small time constant, coincidence
detector (reliability)detector (reliability) Interspike interval code (ISI reconstruction)Interspike interval code (ISI reconstruction) Linear correlation coefficientLinear correlation coefficient CoherenceCoherence Coding fractionCoding fraction Mutual informationMutual information Challenge: Biophysics of coding Challenge: Biophysics of coding Forget the biophysics? Use better Forget the biophysics? Use better
(mesoscopic ?) variables? (mesoscopic ?) variables?
dI )log(
ii
s(t) (t t )
Neuroscience101 (Continued):
Number of spikesIn time interval T:
T
0
N(T) s(t)dt
Spike train:
60 70 80420
450
480
T
rial N
um
ber
time (msec)
Raster Plot:
Random variables
Interspike Intervals (ISI):
i i 1 i i 1{I t t }
Information Theoretic Calculations:Information Theoretic Calculations:
Gaussian Noise Stimulus S Spike Train XNeuron
???
S~
S~
X~
X~
S~
X~
)f(C**
2*
Coherence Function: Mutual Information Rate:
c
c
f
f
2 )]f(C1[logdf2
1MI
Stimulus Protocol:
f
fc
Study effect of (stimulus contrast) and fc (stimulus bandwidth) on coding.
Information Theory:
)S/R(H)R(H)S,R(I
60 70 80 90 100
time (EOD cycles)
60 70 80 90 100
time (EOD cycles)
Linear Response Calculation for Linear Response Calculation for Fourier transform of spike train:Fourier transform of spike train:
)(~
)()(~
)(~
0 fSffXfX st
susceptibilityunperturbed spike train
Spike Train Spec = Background Spec + (transfer function*Signal Spec)
20 )(
0 100 200 30010-1
100
101
102
103
104
Pow
er (
spk2 /s
ec)
f (Hz)
LIFDT Nelson
1ii
2
21I
CV)0f(P
CV == INTERVAL mean / INTERVAL Standard deviation
Wiener KhintchineWiener Khintchine
diSC
diCS
)exp()()(
)exp()()( Power spectrum
Autocorrelation
Integral of S over all frequencies = C(0) = signal variance
Integral of C over all time lags = S(0) = signal intensity
P
(n,T
)
P0(n,T)
P1(n,T)
n (spikes)
Signal Detection Theory:
1 0
2 21 0
d '
d ' 3 5 0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
typical
chance
perfect
PD
PFA
D 1n k
FA 0n k
P P (n,T)
P P (n,T)
ROC curve:
Information Theory Information Theory
Actual signal
Reconstructed signal
The stimulus can be well characterized (electric field). This allows for detailed signal processing analysis.
Gabbiani et al., Nature (1996) 384:564-567.Bastian et al., J. Neurosci. (2002) 22:4577-4590. Krahe et al., (2002) J. Neurosci. 22:2374-2382.
Linear Stimulus ReconstructionLinear Stimulus Reconstruction Estimate filter which, when convolved with the Estimate filter which, when convolved with the
spike train, yields an estimated stimulated “closest” spike train, yields an estimated stimulated “closest” to real stimulusto real stimulus
)(
)()]([)(
)]()([1
)'()'(')(
)_()(
)(
0
22
0
0
fS
fSthfH
tstsdtT
txtthdtts
xitttx
stimulusts
xx
sx
T
est
T
est
i
Spike train (zero mean)
Estimated stimulus
Mean square error
Optimal Wiener filter
NOISE smoothes out f-I curvesNOISE smoothes out f-I curves
““stochastic resonance above stochastic resonance above threshold”threshold”
Coding fraction versus noise intensity:
Modeling Electroreceptors:Modeling Electroreceptors: The Nelson Model (1996) The Nelson Model (1996)
High-PassFilterInput
Stochastic Spike Generator
Spike generator assigns 0 or 1 spike per EOD cycle: multimodal histograms
Modeling Electroreceptors: Modeling Electroreceptors: The Extended LIFDT ModelThe Extended LIFDT Model
High-PassFilterInput LIFDT Spike Train
Parameters: without noise, receptor fires periodically
(suprathreshold dynamics – no stochastic resonance)
Signal Detection: Count Spikes During Interval TSignal Detection: Count Spikes During Interval T
2 4 6 8 10 120.0
0.2
0.4
P(n
)
P0(n,T) (no stimulus)
P1(n,T) (with stimulus)
n (spikes)
20
21
01SNR
46 48 50 52 54 56 58
0.0
0.1
0.2
0.3
0.4 (b) baseline LIFDT stimulus LIFDT baseline Nelson stimulus Nelson
P(n
)
n
T=255 msec
10-1 100 101 102 103 104 105 106
10-2
10-1
100
n=5
CV2
LIFDT shuffled LIFDT Nelson
Fan
o fa
ctor
F(T
)
counting time T (msec)
)T(
)T()T(F
2
Fano Factor:
1ii
2 21CV)(F
0 5 10 15-0.5
0.0
0.5
1.0
j
j
Asymptotic Limit(Cox and Lewis, 1966)
Regularisation:Regularisation:
Sensory NeuronsSensory Neurons
ELL Pyramidal Cell
Sensory Input
Higher Brain
Feedback: Open vs Closed Loop ArchitectureFeedback: Open vs Closed Loop Architecture
Higher Brain
Higher Brain
Loop time d
Delayed Feedback Neural NetworksDelayed Feedback Neural Networks
Afferent Input
Higher Brain Areas
The ELL; first stage of sensory processing
Jelte Bos’ data
Andre’s data
Longtin et al., Phys. Rev. A 41, 6992 (1990)
If one defines:
corresponding to the stochastic diff. eq. :
one gets a Fokker-Planck equation:
One can apply Ito or Stratonovich calculus, as for SDE’s.
However, applicability is limited if there are complex eigenvalues or system is strongly nonlinear
TWO-STATE DESCRIPTION: TWO-STATE DESCRIPTION: S=S=±1±1
2 transition probabilities:
For example, using Kramers approach:
DETERMINISTIC DELAYED DETERMINISTIC DELAYED BISTABILITYBISTABILITY
Stochastic approach does not yet Stochastic approach does not yet get the whole picture!get the whole picture!
ConclusionsConclusions
NOISE: many sources, many approaches, NOISE: many sources, many approaches, exercise caution (Ito vs Strato)exercise caution (Ito vs Strato)
INFORMATION THEORY: usually makes INFORMATION THEORY: usually makes assumptions, and even when it doesn’t, ask the assumptions, and even when it doesn’t, ask the question whether next cell cares. question whether next cell cares.
DELAYS: SDDE’s have no Fokker-Planck DELAYS: SDDE’s have no Fokker-Planck equivalentequivalent
tomorrow: linear response-like theory tomorrow: linear response-like theory
OUTLOOKOUTLOOK
Second order field theory for stochastic neural Second order field theory for stochastic neural dynamics with delaysdynamics with delays
Figuring out how intrinsic neuron dynamics Figuring out how intrinsic neuron dynamics (bursting, coincidence detection, etc…) (bursting, coincidence detection, etc…) interact with correlated inputinteract with correlated input
Figuring out interaction of noise and burstingFiguring out interaction of noise and bursting Forget about steady state! Forget about steady state! Whatever you do, think of the neural Whatever you do, think of the neural
decoder…decoder…