Andre Longtin

Physics Department

University of Ottawa

Ottawa, Canada

Effects of Non-Renewal Firing on Information Transfer in

Neurons

-Weakly Electric Fish

- Electroreceptor data

- Modeling

- Effects of ISI correlations

- Linear response models

Biology

Computation

Theory

Overview

Collaborators

Benjamin Lindner, postdoc, Physics, U. Ottawa

Maurice Chacron, postdoc, Physics, U. Ottawa

Leonard Maler, Cell. Molec. Med, U. Ottawa

Khashayar Pakdaman, INSERM, Paris

Martin St-Hilaire, M.Sc. Student, U. Ottawa

90 100 110 120-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

mV

time (EOD cycles)

transepidermal voltage amplitude

Weakly Electric Fish: Electrolocation

90 95 100 105 110 115 120

0.7

0.8

0.9

1.0

1.1

1.2

1.3

mV

time (EOD cycles)

Electroreceptor Neurons: Anatomy

Pore

SensoryEpithelium

Axon(To Higher Brain)

Electroreceptor Neurons: Electrophysiology

0 10 20-0.5

0.0

0.5

1.0

0 4 80

4

8

0 5000 100000

4

8

(c) i

i 0 4 80

500

1000

1500(d)

Cou

nts

ISI

(b)

ISI i+

1

ISIi

(a)

ISI

ISI Number

data courtesy of Mark Nelson, U. Illinois

2i

2i

2ijii

j II

III

Modeling Electroreceptors: The Nelson Model (1996)

High-PassFilterInput

Stochastic Spike Generator

Fit of Nelson Model to Data:

0 5 10 15 20

0.0

0.4

0.8

0 2 4 6 8 100

2

4

6

8

10

0 5000 10000

2

4

6

8

10

(d)(c)

(b)(a)

i

lag i0 2 4 6 8 10

0

500

1000

1500

2000

# C

oun

tsISI

ISI i+

1

ISIi

ISI

ISI numberRenewal Process(No ISI correlations)

1015 1020 1025 1030 1035 1040

0.00

0.04

0.08

0.12

0.16

membrane potential threshold

time (ms)

Ii Ii+1w

Leaky Integrate-and-fire Model with Dynamic Threshold Chacron, Longtin, St-Hilaire, Maler, Phys.Rev.Lett. 85, 1576 (2000)

)t(w)t(vifw)t(w)t(w

)t(w)t(vif0)t(v

ww)Ttt(Hw

)t()]ft2[sin(H)ft2sin()]t(a[H)t(av

v

firefirefirefire

firefirefire

w

0rfire

v

Modeling Electroreceptors: The Extended LIFDT Model

High-PassFilterInput LIFDT Spike Train

Fitting the Experimental Data (Part 2):

0 5000 10000

2

4

6

8

10

0 5 10 15 20

-0.4

0.0

0.4

0.8

0 2 4 6 8 100

2

4

6

8

10

(d)(c)

(b)(a)

ISI

ISI number

i

lag i0 2 4 6 8 10

0

500

1000

1500

2000

# co

unts

ISIIS

I i+1

ISIi

Non-renewalProcess

Summary of Fitting:

0 5 10 15 20-0.5

0.0

0.5

1.0

0 5 10 15 20-0.5

0.0

0.5

1.0

0 5 10 15 20-0.5

0.0

0.5

1.0

0 2 4 6 8 100

500

1000

1500

2000

SC

C j

lag j0 2 4 6 8 10

0

500

1000

1500

2000

# c

ount

s

ISI

data

0 2 4 6 8 100

500

1000

1500

2000

Experimental Data:

LIFDT Model:

Nelson Model:

What Else We Know about LIFDT

• 1D map for consecutive threshold values

• Negative correlation appear when fixed point of map is perturbed by noise: it is a deterministic property.

• Strength of correlation depends on system parameters

• With sinusoidal forcing, 2D annulus map: simple and complex phase locking, chaos

See: Chacron, Pakdaman, Longtin, Neural Comput. (2003).

Chacron, Longtin, Pakdaman, Physica D (2004).

Comparison Approach to Assess Effects of ISI Correlations:

Nelson Model

(renewal process)

LIFDT Model

(non-renewal process)

vs.

Weak Signal Detection:

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:

Stimulation Protocol:

f

fc

Gaussian white noise

Low-pass filter Stimulus

Stimuli are Gaussian with standard deviation and cutoff frequency fc

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

0.00 0.02 0.04

0

30

60

90

120

Nelson LIFDT

MI (

bits

/s)

stimulus contrast (mV)0 50 100 150 200

0

10

20

30

0 50 100 150 200

0

50

100

150

(b)

MI (

bits

/s)

fc (Hz)

(a)

LIFDT NelsonM

I (bi

ts/s

)

fc (Hz)

Comparison using Info Theory

An Important Clue: Reduction of Power at Low Frequencies:

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

Theory for why certain correlations are useful: Need simpler models !!

• Simple Intrinsic Dynamics only, no extra filtering

perfect integrator neuron instead of leaky: dv/dt = μ + signal(t)

• Noise on threshold and reset only

• Assume simple noise distribution and action (uniform distribution, piecewise constant in time)

Two identical models, except for correlationsChacron, Lindner, Longtin, Phys.Rev.Lett. (in press 2004)

Model A: Model B:

2VU

VUI

01jj

jjj

Successive intervals are thus correlated

Successive intervals are not correlated

Statistics and Spectra

ISI Statistics: Power Spectra:

)f(sin)fI2cos()f(sin)f(2)f(

I/)f(sin)f()f(S

I

nf

I

11

)f(

)f(sin1

I

1)f(S

4224

44

0B

n2

2

0A

Noise Shaping

where β=2πD/µ

Linear Response Calculation for Fourier transform of spike train:

)(~

)()(~

)(~

0 fSffXfX st

susceptibilityunperturbed spike train

0)()( ff BAIt turns out:

Spike Train Spectrum= Background Spectrum + Signal Spectrum20 )(

Linear Response Calculation (Part 2):

1

st20

2

00B,A

2

B,A )f(S

S

1)f(C

Coherence Function

Linear Response Calculation (Part 3):

)]f(C1[logdf2

1)f(MI

c

c

f

f

2B,A

Mutual Information Rate

Conclusions- Weakly electric fish must detect prey (low freq. stimuli, less than 0.1 V)

- Negative ISI Correlations Can Regularize a Spike Train through spike count variance reduction and noise reduction at low frequencies.

- This is achieved through noise shaping in the power spectrum and this is greatest for weak low frequency stimuli.

- Outlook:

1) Experimentally prove that the negative correlations are really being used for computations.

2) Deal with mixtures of positive and negative correlations at lags >= 1

3) Extend to more realistic models of excitability with memory

4) Use the ideas presented here in devices to improve SNR and detectability

References:- Chacron, Longtin, St-Hilaire, Maler, PRL 85, 1576 (2000).

- Chacron, Longtin, Maler, J. Neurosci. 21, 5328 (2001).

- Chacron, Lindner, Longtin, (submitted).

- Cover, Thomas, Elements of Information Theory (1991).

- Cox, Lewis, The Statistical Analysis of Series of Events (1966).

- Nelson, Xu, Payne, J. Comp. Physiol. A 181, 532 (1997).

- Ratnam, Nelson, J. Neurosci. 20, 6672 (2000).

“Why should we explore exotic sensory systems such as electrosensation in fish or echolocation in bats?...

More highly evolved organisms derive their superior qualities not so much from novel mechanisms at the cellular level but rather from a richer complexity in the orchestration of basic designs that they share with simpler organisms. Fundamental mechanisms of perception and neuronal processing of sensory information are shared by animals as diverse as flies and primates, but a larger number of neuronal structures and interconnecting pathways bestow more powerful computational abilities and memory capacities upon the brains of primates.”

--Walter Heiligenberg

Food for Thought: