Extracting Time and Space Extracting Time and Space Scales with Scales with

Feedback and NonlinearityFeedback and Nonlinearity

André Longtin

Physics +Cellular and Molecular Medicine

CENTER FOR NEURAL DYNAMICS

UNIVERSITY OF OTTAWA

Funding by NSERC, CIHR, PREA

Processing of spatio-temporal signals Processing of spatio-temporal signals Global Feedback + common noise: Global Feedback + common noise:

oscillations oscillations Spatial scale for feedback and inputSpatial scale for feedback and input Envelope processing for narrowband in time Envelope processing for narrowband in time Information resonancesInformation resonances Coincidence transforms for synchronous Coincidence transforms for synchronous

firingfiring Short term Plasticity and information Short term Plasticity and information

processingprocessing

Overview

Research Program: Research Program:

Stochastic neural networkStochastic neural network

driven bydriven by

Stochastic input in space and timeStochastic input in space and time

Experimental <-> TheoreticalExperimental <-> Theoretical

Relevance to this group:Relevance to this group:

What nonlinearity (if any) supports What nonlinearity (if any) supports patterns or computations ? patterns or computations ?

HIGHER BRAIN AREA I

HIGHER BRAIN AREA II

THALAMUS

RECEPTORS

PHYSICAL STIMULI

Brain Diagram by Arab philosopher Avicenna (circa 1300)Brain Diagram by Arab philosopher Avicenna (circa 1300)Five ventriclesFive ventricles: common sense, imagination, judging, : common sense, imagination, judging,

second imagination (composing/combining images), memory.second imagination (composing/combining images), memory.((University University LibraryLibrary, Cambridge, Cambridge) )

From Da Vinci’s notes

Courtesy W. Ellis (1991)

Electrosensory lateralline lobe (ELL)

Courtesy N. Berman and L. Maler, J. Exp. Biol., 1999

Electrosensory Lateral Line Lobe (ELL)

Electroreceptors

Higher Brain

Electrosensory Input

Amplitude Modulation (AM)

Krahe and Gabbiani (2004) Nat. Neurosci.Rev. 5:13-23

Electric Organ Discharge (EOD)

afferents

The electric sense

Temporal Characteristics: Spatial Characteristics:

Harmonic

“local”

“global”

Broadband (noise)

Chacron, et al., Nature, 2003

- Frequency tuning is highly correlated with spatial frequency

- Tuning for harmonics or broadband signals are qualitatively the same

Weakly Electric Fish: Weakly Electric Fish: main negative feedback loopmain negative feedback loop

ELL Pyramidal Cells: the first stage of sensory processing

Prey StimuliPrey Stimuli

Prey (bug) excites a fraction of the electroreceptors:

Local stimulation

Communication StimuliCommunication Stimuli

Communication calls between fish stimulate the whole body:

Global Stimulation

Oscillation MechanismOscillation Mechanism(Doiron, Chacron, Bastian, Longtin, Maler, (Doiron, Chacron, Bastian, Longtin, Maler, Nature 2003Nature 2003))

Local Stimuli :applied heterogeneously in space: No stimulus-induced correlations.

Global Stimuli :acts homogenously in space (strong spatial correlations). Correlated activity and delay cause “waves” of inhibition

Electrosensory CircuitryElectrosensory Circuitry

Sodium channel blocker can open the feedback loop.

Network Model – Global Network Model – Global StimuliStimuli

To mimic communication stimuli we apply the external stimulus to all neurons equally.

Autocorrelation Histogram

Experimental VerificationExperimental VerificationDoiron, Chacron, Maler, Longtin and Bastian, Doiron, Chacron, Maler, Longtin and Bastian, NatureNature 42, 539 42, 539

(2003)(2003)

ISI Histogram Autocorrelation

control block recover

Correlated Stimuli in Correlated Stimuli in ExperimentsExperiments

Dipole 1

Dipole 2

Dipole 3

Dipole 4

)()(1)( tctct Gii

Each dipole emits an intrinsic noise i(t), and global source, G(t). Their relative strengths is c, i.e. the covariance between dipoles.

Integrate-and-fire dynamics Integrate-and-fire dynamics

y

Linear ResponseLinear Response

Consider the spike train from the ith neuron in our network, . Assuming weak inputs, the Fourier transform of the spike train is

)()( iji ttty

)()()()( ibgi XAYY

A( intrinsic frequency response of the noisy neuron. Xi(w): Fourier transform of input (external + feedback) to

neuron i.

(1)

POWER SPECTRUMPOWER SPECTRUM

Single Neuron Power SpectrumSingle Neuron Power Spectrumvsvs

percentage of common noise (c) percentage of common noise (c)

For an infinite network: For an infinite network:

2

~2

22

)(~

1

)(~

)(~

Re2

)()()(

AKg

AKgAKg

AcSS

d

dd

bg

)(|)(| 2 inSA

(Input-output sync) (spike-spike sync.)

Fokker-Planck analysis on noisy Leaky Integrate-and-fire Fokker-Planck analysis on noisy Leaky Integrate-and-fire Neurons + Delays+ Spatial InputNeurons + Delays+ Spatial Input

Doiron, Lindner, Longtin, Bastian and Maler, Doiron, Lindner, Longtin, Bastian and Maler, Phys. Rev. LettPhys. Rev. Lett. 93, 048101 (2004). 93, 048101 (2004)

Linear Fluctuation Theory: needs noise. Linear Fluctuation Theory: needs noise. 30

25

20

15

S (

spik

es2 /s

)

140120100806040200

frequency (Hz)

simulation c=1 theory c=1 simulations c=0 theory c=0

Input-output coherence for Input-output coherence for delayed feedback networkdelayed feedback network

(global feedback)(global feedback)

Coherence function:

CXY ( f ) SXY ( f )

2

SXX ( f ) SYY ( f )

Correlation coefficient (in the frequency domain) between two signals, X and Y

R(t) (t tii )

Response: spike trainStimulus:

S(t) - narrowband stimulus(linear)

E(t) - envelope of narrowbandstimulus (non-linear)

Network of stochastic Perfect IF’s Network of stochastic Perfect IF’s with with

+ and - global delayed feedback+ and - global delayed feedback

)2()tt(N

1)t(K

)1()t(Kg)t(Kg)t(S)t(D2VN

1k

)t(M

1jkji,ei,e

iiieeekk

k

S(t) is the stimulus

Chacron, Longtin, Maler, Phys.Rev.E (2005)

Network of Perfect IF’s with Network of Perfect IF’s with global feedback: Information global feedback: Information

theorytheory

0 20 40 60 80

0.0

0.2

0.4

0.6

0.8

1.0

coh

ere

nce

frequency (Hz)

ge=g

i=0

ge=0, g

i=-0.8

ge=0.8, g

i=0

Coherence = | H(f)|2 Pss/Pxx

INFORMATION RESONANCEINFORMATION RESONANCE(Chacron, Longtin, Maler, PRE 2005)(Chacron, Longtin, Maler, PRE 2005)

G<0

G=0

Experimental DATA !!

Introducing…Introducing…

Spatial scale for feedbackSpatial scale for feedback Spatial scale for noiseSpatial scale for noise

Two regimes with respect to Two regimes with respect to

gamma oscillationsgamma oscillations

In linear response, only the ratio of length scales matters(Hutt, Sutherland, Longtin, submitted)

GLOBAL IN SPACEGLOBAL IN SPACE

NARROWBAND IN TIME: NARROWBAND IN TIME:

2 TIME SCALES2 TIME SCALES

EOD amplitude

EOD

Hey guys

EOD amplitude

EOD

EODamplitude

EOD

From: E. W. Tan et al, Behav.Brain Res., 164:83-92 (2005)

Most probable populationsize 3-5 fish

Average f inblack (white) waters:

Day: 35.3 (54.1) HzNight: 54.6 (65.8) Hz

P-units (primary receptors)Feed forward:

- P-units respond as linear encoders

frequency (Hz)

- global stimulation: linear response to narrowband signal and its low frequency envelope

- Envelope response is absent under local stimulation

PyramidalCells

S(t)

E(t)

Middleton, Longtin, Benda, Maler, PNAS (2006)

input

output

- Generation of envelope signal is likely due to spike threshold nonlinearity

- Output spike train is phase- locked to fast oscillation and modulated at lower frequencies

Middleton, Harvey-Girard, Maler, Longtin,

Phys. Rev. E. (2006)

MECHANISM

transferfunction

input signal output signal

(rectification)

signals Spectral composition

time frequency

Network instead of single cellNetwork instead of single cell

Stochastic Envelope GatingStochastic Envelope Gating(and not SR! See Middleton et al., PRE 2006)(and not SR! See Middleton et al., PRE 2006)

Leaky Integrate-and-Fire (LIF) neuron:

dv

dt

v

2D

(t) S(t)

r() dzez2

erfc(z) vR 2 D

vR 2 D

1

(t) o a S(t)

Mean firing rate:

where

GLOBAL SPATIAL SIGNALSGLOBAL SPATIAL SIGNALS

EXTRACTING EXTRACTING

HIGH FREQUENCY CHIRPS HIGH FREQUENCY CHIRPS

FROM FROM

LOWER FREQUENCY BEATSLOWER FREQUENCY BEATS

““SYNC-DESYNC CODE”SYNC-DESYNC CODE”

Context: Context: electrocommunicationelectrocommunication

Male-male or female-female call Male-male or female-female call causes synchronization of receptorscauses synchronization of receptors

Male-female or female-male call Male-female or female-male call causes desynchronization of causes desynchronization of receptorsreceptors

(Benda, Longtin, Maler, Neuron 2006)(Benda, Longtin, Maler, Neuron 2006)

Encoding a modulatory Encoding a modulatory signalsignal

Coincidence transforms…Coincidence transforms…

Middleton, Longtin, Benda, Maler (submitted)

Short-term Plasticity Short-term Plasticity

Broadband Coding

Depression

dominates

Facilitation

dominates

-Gamma rhythms for global correlated inputs

- Gamma strength proportional to correlation

- Spatial feedback can assess spatial correlation of input

- Information resonances with delayed feedback

- Envelope generation due to spike threshold nonlinearity

- Envelope generation is dependent on mean bias and noise

Intrinsic noise can gate a signal competing with envelope

Plasticity: paradoxical effects on coding

Importance of spatiotemporal statistics of input

Brent Doiron

Maurice Chacron

Jason Middleton Carlo Laing Eric Harvey-Girard John Lewis

Jan Benda

Benjamin Lindner

Len Maler & André Longtin

Joe Bastian

Connie Sutherland

Axel Hutt

COHERENCE AND STOCHASTIC RESONANCE COHERENCE AND STOCHASTIC RESONANCE WITH DELAYED FEEDBACKWITH DELAYED FEEDBACK

Morse and Longtin, Phys. Lett. A (2006)Morse and Longtin, Phys. Lett. A (2006)

MULTIPLE RESONANCESMULTIPLE RESONANCES(fixed driving frequency)(fixed driving frequency)

The analytic signal:• The Hilbert transform: 90o phaseshift•Mapping of a time varying signal onto a 2D phase plane•Allows for the definition of phase and amplitude variables

A(t) x(t)2 xH (t)2 (t) arctanxH (t)

x(t)

H[x](t)

A(t)

(t)

z(t) x(t) ixH (t)

xH (t) H[x](t) 1

P

x( )

t

d

- Ovoid Cells are high-pass

- Ovoid spike trains are coherent with narrowband signals (blue) and their envelopes (red)

- Subthreshold voltage shows no coherence with signal envelope

Courtesy R. Krahe and F. Gabbiani,

Nat. Neurosci. Rev. (2004)

Electric Fields

Courtesy G. Hupe and J. Lewis (2005)

Apteronotus Leptorhynchus