Vol

August

doi

LETTERS

Correlation

between

neural

spike

trains

increases

with

firing

rate

Jaime

de

la

Brent

-LCB-

Eric

Kresimir

Alex

Populations

of

neurons

in

the

olfactory

and

thalamus

and

several

cortical

show

temporal

correlation

between

the

discharge

times

of

their

action

potentials

-LRB-

spike

trains

-RRB-

Correlated

firing

has

been

linked

to

stimulus

stimulus

and

motor

Nevertheless

the

mechanisms

underlying

correlated

spiking

are

poorly

and

its

coding

implications

are

still

It

is

not

clear

for

instance

whether

correlations

between

the

discharges

of

two

neurons

are

determined

solely

by

the

correlation

between

their

afferent

currents

or

whether

they

also

depend

on

the

mean

and

variance

of

the

input

We

addressed

this

question

by

computing

the

spike

train

correlation

coefficient

of

unconnected

pairs

of

in

vitro

cortical

neurons

receiving

correlated

inputs

Notably

even

when

the

input

correlation

remained

fixed

the

spike

train

output

correlation

increased

with

the

firing

rate

but

was

largely

independent

of

spike

train

variability

With

a

combination

of

analytical

techniques

and

numerical

simulations

using

integrate-and-fire

neuron

models

we

show

that

this

relationship

between

output

correlation

and

firing

rate

is

robust

to

input

heterogeneities

Finally

this

overlooked

relationship

is

replicated

by

a

standard

threshold-linear

model

demonstrating

the

universality

of

the

result

This

connection

between

the

rate

and

correlation

of

spiking

activity

links

two

fundamental

features

of

the

neural

code

How

do

cortical

cells

transform

correlation

between

their

synaptic

currents

into

correlation

between

their

output

spike

trains

We

addressed

this

question

by

studying

pairwise

spike

train

correlations

a

strategy

that

can

capture

the

full

statistical

structure

of

a

neural

Correlated

fluctuating

currents

resembling

synaptic

were

injected

into

the

somata

of

unconnected

cortical

neurons

-LRB-

see

Methods

-RRB-

The

input

current

to

cell

i

-LRB-

i

-RRB-

was

pffiffiffiffiffiffiffiffiffiffi

pffiffiffi

Ii

mi

zsi

-LCB-

c

ji

-LRB-

t

-RRB-

z

c

jc

-LRB-

t

-RRB-

where

mi

is

the

temporal

average

of

the

current

The

next

term

represents

gaussian

fluctuations

with

a

temporal

structure

consistent

with

that

in

and

was

composed

of

two

weighted

factors

ji

which

was

independent

for

each

cell

and

jc

which

was

common

to

all

cells

-LRB-

Fig.

-RRB-

The

input

correlation

coefficient

c

-LRB-

c

-RRB-

set

the

relative

weight

of

the

shared

fluctuations

whereas

si

set

the

variance

of

the

total

input

current

These

parameters

were

adjusted

so

that

the

variability

and

covariability

of

the

membrane

and

spike

were

similar

to

those

observed

in

vivo

-LRB-

Fig.

-RRB-

Typical

spike

train

cross-correlation

functions

had

positive

central

peaks

with

short

timescales

-LRB-

ms

Fig.

left

-RRB-

indicative

of

spiketime

To

quantify

spike

train

correlation

between

two

cells

we

computed

the

correlation

of

the

spike

counts

and

over

a

sliding

window

of

length

T

Cov

rT

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Var

Var

where

Cov

is

the

covariance

and

Var

the

variance

The

correlation

coefficient

rT

is

a

dimensionless

quantity

ranging

between

for

independent

and

for

fully

correlated

spike

trains

For

large

T

rT

saturates

to

a

value

-LRB-

see

Methods

-RRB-

As

expected

rT

increased

with

-LRB-

Fig.

-RRB-

however

rT

was

always

less

than

c

indicating

that

the

correlation

in

the

input

currents

bounds

the

correlation

between

the

output

spikes

Our

central

and

more

surprising

finding

was

that

rT

increased

with

the

output

firing

rate

n

when

c

was

fixed

-LRB-

Fig.

-RRB-

We

delivered

currents

with

a

fixed

c

and

various

values

of

m

and

s

thus

eliciting

spike

trains

with

a

range

of

firing

rates

and

inter-spike

interval

coefficients

of

variation

-LRB-

CV

-RRB-

Spike

trains

with

similar

n

were

paired

to

compute

rT.

Despite

rT

increasing

strongly

with

rate

it

did

not

depend

on

the

CV

-LRB-

inset

Fig.

-RRB-

The

same

increasing

trend

holds

for

all

T

values

examined

and

occurred

across

several

identified

cell

types

-LRB-

see

Supplementary

Information

-RRB-

The

correlation

rate

relationship

-LRB-

Fig.

-RRB-

might

be

expected

owing

to

an

increased

probability

of

spikes

occurring

close

in

time

at

high

rates

The

coefficient

rT

corrects

for

this

spurious

ratedependence

of

correlation

as

illustrated

by

the

following

example

of

two

cells

receiving

inputs

constructed

from

three

independent

Poisson

trains

and

xc

with

rates

-LRB-

c

-RRB-

n

-LRB-

c

-RRB-

n

and

cn

respectively

-LRB-

Fig.

-RRB-

For

a

simple

model

that

converts

every

input

spike

into

an

output

spike

the

output

trains

xc

and

xc

have

rate

n

and

a

correlation

that

depends

on

the

rate

of

the

common

train

xc

-LRB-

Fig.

red

spikes

-RRB-

The

correlation

measured

as

the

average

product

cnT

-LRB-

nT

-RRB-

depends

on

n

through

-LRB-

-RRB-

the

baseline

chance

correlation

-LRB-

nT

-RRB-

which

gives

a

nonzero

correlation

even

in

the

case

c

-LRB-

-RRB-

the

scaling

of

the

first

term

by

n

which

gives

the

absolute

number

of

synchronous

spikes

The

coefficient

rT

-LRB-

equation

-LRB-

-RRB-

-RRB-

corrects

for

these

n-dependencies

through

baseline

subtraction

and

appropriate

normalization

i

-LCB-

i

rT

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Specifically

noting

ni

nT

and

i

-LCB-

i

-LCB-

nT

-LRB-

nT

-RRB-

yields

rT

cnT/nT

c

thus

showing

how

rT

does

not

have

a

built-in

dependence

on

n.

Therefore

the

correlation

rate

relationship

exhibited

by

the

data

-LRB-

Fig.

-RRB-

is

a

priori

unexpected

and

instead

must

follow

from

an

input

output

property

of

spiking

neurons

To

analyse

further

the

correlation

rate

relationship

we

replaced

the

in

vitro

neurons

with

a

pair

of

leaky

integrate-and-fire

-LRB-

LIF

-RRB-

model

-LRB-

see

Methods

-RRB-

We

fixed

m

and

s

and

performed

simulations

of

the

LIF

neuron

pair

to

compute

r

-LRB-

c

-RRB-

The

coefficient

c

ranged

from

to

thus

yielding

r

values

that

are

comparable

to

Center

for

Neural

Science

New

York

University

New

York

USA

Institute

of

Mathematical

Sciences

New

York

University

New

York

USA

of

Mathematics

University

of

Houston

Houston

Texas

USA

-LCB-

Present

address

Department

of

Mathematics

University

of

Pittsburgh

Pittsburgh

Pennsylvania

USA

These

authors

contributed

equally

to

this

work

Nature

Publishing

Group

NATURE

Vol

August

LETTERS

those

measured

in

In

this

range

r

-LRB-

c

-RRB-

was

approximately

linear

-LRB-

Fig.

b

-RRB-

allowing

us

to

use

perturbation

to

obtain

an

expression

for

r

-LRB-

c

-RRB-

-LRB-

see

Supplementary

Information

-RRB-

dn

dm

r

S

m

s

c

c

n

We

define

the

slope

S

-LRB-

m

s

-RRB-

relating

r

to

c

as

correlation

susceptibility

Here

dn/dm

is

the

derivative

of

the

spike-rate

transfer

function

-LRB-

that

is

the

relationship

between

firing

rate

n

-LRB-

m

s

-RRB-

and

the

mean

injected

current

-RRB-

Equation

-LRB-

-RRB-

accurately

predicts

the

transformation

r

-LRB-

c

-RRB-

obtained

via

simulations

-LRB-

Fig.

b

dashed

lines

-RRB-

As

with

the

experiments

we

varied

m

or

s

while

keeping

c

fixed

to

obtain

different

values

of

n

A

plot

of

r

versus

n

when

m

-LRB-

or

s

-RRB-

was

increased

while

keeping

s

-LRB-

or

m

-RRB-

fixed

-LRB-

inset

of

Fig.

-RRB-

shows

that

r

rapidly

increased

in

the

physiological

range

of

spikes

and

asymptoted

to

c

at

high

rates

-LRB-

Fig.

-RRB-

Notably

the

correlation

rate

curves

obtained

by

increasing

m

or

s

were

nearly

identical

even

though

voltage

traces

correlation

functions

and

CVs

were

very

different

-LRB-

Fig.

-RRB-

To

show

that

these

results

did

not

depend

on

specific

m

and

s

values

we

used

equation

-LRB-

-RRB-

to

sweep

a

large

region

of

the

m

s

parameter

space

The

resultant

r

n

values

were

confined

to

a

narrow

band

around

the

stereotyped

curve

-LRB-

Fig.

shaded

areas

-RRB-

This

allowed

us

to

rewrite

equation

-LRB-

-RRB-

in

an

approximate

form

in

which

the

susceptibility

S

depends

only

on

the

output

rate

and

not

on

the

input

variables

m

and

s

r

S

n

c

The

empirically

measured

rT

also

increased

with

n

for

all

T

examined

however

the

approximation

that

S

depends

only

on

n

-LRB-

equation

-LRB-

-RRB-

-RRB-

becomes

progressively

less

accurate

as

T

decreases

-LRB-

see

Supplementary

Information

-RRB-

To

determine

whether

the

correlation

rate

relationship

emerges

in

a

less

controlled

and

more

biologically

relevant

setting

we

performed

simulations

of

a

two-layer

network

where

neurons

from

layer

were

randomly

connected

to

neurons

in

layer

-LRB-

Fig.

-RRB-

Layer

consisted

a

a

xc

b

Current

mV

ms

ms

s

c

d

Output

correlation

rT

b

s

Output

correlation

r

Time

-LRB-

ms

-RRB-

-LRB-

spikes

s

-RRB-

Input

correlation

c

e

Output

correlation

rT

Input

correlation

c

Input

correlation

c

c

rT

CV

s

-LRB-

mV

-RRB-

c

-LRB-

mV

-RRB-

Geometric

mean

output

rate

ninj

-LRB-

spikes

s

-RRB-

n

-LRB-

c

-RRB-

nc

rate

n

correlation

rT

c

f

-LRB-

n

-RRB-

Output

correlation

r

ms

f

xc

c

n

-LRB-

c

-RRB-

T

Figure

Relationship

between

output

spike

correlation

and

rate

in

in

vitro

cells

a

Twenty

neurons

from

cortical

slices

-LRB-

n

-RRB-

were

stimulated

with

gaussian

currents

-LRB-

filtered

with

a

time

constant

of

ms

-RRB-

with

both

common

and

independent

components

b

Fluctuating

current

injection

-LRB-

bottom

-RRB-

often

evoked

nearly

synchronous

spikes

-LRB-

top

arrowheads

-RRB-

Red

and

black

traces

are

membrane

potentials

from

cells

and

respectively

c

Typical

cross

-LRB-

left

-RRB-

and

auto-correlation

-LRB-

right

-RRB-

functions

from

a

pair

of

cells

firing

at

low

-LRB-

blue

-RRB-

medium

-LRB-

red

-RRB-

and

high

-LRB-

green

-RRB-

rate

d

A

typical

rT

versus

c

for

a

pair

of

cells

e

The

correlation

rT

in

a

population

-LRB-

n

pairwise

ffi

pffiffiffiffiffiffiffi

comparisons

-RRB-

increases

with

ni

nj

and

does

not

co-vary

with

CVi

CVj

-LRB-

inset

linear

regression

r

-RRB-

c

was

fixed

at

Squares

represent

mean

and

s.d.

of

the

population

and

coloured

circles

the

examples

shown

in

c.

f

A

simple

cell

model

that

generates

an

output

spike

for

every

input

spike

The

correlation

rT

is

fixed

at

c

and

is

independent

of

n.

Throughout

we

used

T

ms.

mV

ms

Output

firing

rate

n

-LRB-

spikes

s

-RRB-

ms

Figure

The

correlation

rate

relationship

in

an

integrate-and-fire

neuron

model

a

b

r

versus

c

from

simulations

-LRB-

solid

line

-RRB-

and

theory

-LRB-

dashed

line

from

equation

-LRB-

-RRB-

-RRB-

Increases

in

either

m

-LRB-

a

-RRB-

or

s

-LRB-

b

-RRB-

increase

the

slope

of

r

-LRB-

c

-RRB-

Grey

lines

are

the

diagonal

r

c.

Inset

shows

r

-LRB-

c

-RRB-

for

all

c.

c

As

m

-LRB-

red

-RRB-

or

s

-LRB-

blue

-RRB-

were

varied

following

the

paths

drawn

in

the

inset

r

and

n

covaried

as

shown

Note

that

this

relationship

holds

only

when

r

-LRB-

c

-RRB-

is

linear

-LRB-

a

and

b

-RRB-

Simulations

-LRB-

dots

-RRB-

and

theory

-LRB-

solid

line

-RRB-

show

a

dependence

of

r

and

n

along

a

stereotyped

curve

The

shaded

area

shown

in

m

s

space

-LRB-

inset

-RRB-

was

swept

using

equation

-LRB-

-RRB-

and

yielded

the

shaded

area

around

the

r

n

curves

Voltage

traces

cross

and

auto-correlation

functions

-LRB-

upper

and

lower

panels

respectively

-RRB-

are

illustrated

for

two

sets

of

-LRB-

m

s

-RRB-

values

-LRB-

see

and

in

inset

-RRB-

We

used

tm

ms.

Nature

Publishing

Group

LETTERS

NATURE

Vol

August

a

Layer

presynaptic

Layer

postsynaptic

b

ij

CV

cij

c

Output

correlation

ij

d

Correlation

susceptibility

Sij

ij

cij

Geometric

mean

rate

ij

-RRB-

-LRB-

spikes

s

Geometric

mean

rate

ij

-RRB-

-LRB-

spikes

s

Figure

Correlation

rate

relationship

in

a

simple

network

a

Two-layer

network

with

sparse

random

connectivity

-LRB-

connection

probability

was

-RRB-

where

layer

contained

excitatory

and

inhibitory

cells

firing

with

Poisson

statistics

Layer

cells

were

conductance-based

LIF

neurons

with

randomly

chosen

synaptic

weights

b

Distribution

of

n

-LRB-

top

-RRB-

CV

-LRB-

middle

-RRB-

and

cij

and

pffiffiffiffiffiffiffi

rij

-LRB-

bottom

-RRB-

across

layer

c

rij

versus

ni

nj

for

layer

-LRB-

linear

regression

pffiffiffiffiffiffiffi

r

-RRB-

d

Sij

rij/cij

versus

ni

nj

for

each

pair

shown

in

c.

Pairs

were

grouped

according

to

qij

ni/nj

-LRB-

with

ni

nj

-RRB-

qij

-LRB-

black

-RRB-

and

qij

-LRB-

grey

-RRB-

of

excitatory

or

inhibitory

neurons

that

fired

stationary

Poisson

spike

trains

with

fixed

rate

whereas

layer

neurons

were

conductancebased

LIF

model

neurons

that

received

randomly

weighted

AMPA

-LRB-

propionic

acid

-RRB-

and

GABAA

-LRB-

c-aminobutyric

acid

-RRB-

synapses

-LRB-

see

Methods

-RRB-

Each

cell

i

in

layer

received

an

input

current

with

a

different

mean

and

variance

producing

a

natural

heterogeneity

in

their

firing

rates

ni

and

variability

CVi

-LRB-

Fig.

-RRB-

In

addition

each

pair

of

layer

cells

i

and

j

shared

a

different

number

of

afferents

giving

rise

to

a

wide

distribution

of

input

correlation

coefficients

cij

and

in

turn

a

wide

distribution

of

output

correlation

coefficients

rij

-LRB-

Fig.

-RRB-

a

c

c

c

Input

Output

f

f

f

pffiffiffiffiffiffiffi

A

plot

of

rij

versus

the

geometric

mean

of

the

rates

ni

nj

shows

a

positive

correlation

-LRB-

Fig.

-RRB-

However

the

trend

is

not

nearly

as

tight

as

the

one

obtained

in

the

controlled

setting

-LRB-

compare

Fig.

with

Fig.

-RRB-

primarily

owing

to

the

heterogeneity

in

cij

-LRB-

Fig.

bottom

panel

-RRB-

After

normalization

by

the

measured

cij

values

-LRB-

obtainable

experimentally

using

intracellular

recordings

-RRB-

we

obtained

Sij

rij/cij

and

a

tight

clustering

of

points

along

the

same

stereotyped

curve

emerged

-LRB-

Fig.

-RRB-

If

we

group

the

pairs

according

to

how

different

their

rates

are

we

find

that

spike

trains

with

similar

rates

-LRB-

Fig.

black

points

-RRB-

showed

slightly

larger

Sij

and

a

tighter

correlation

rate

relationship

than

pairs

with

more

different

rates

-LRB-

Fig.

grey

points

-RRB-

The

same

qualitative

results

were

obtained

in

networks

where

synapses

had

slow

kinetics

-LRB-

NMDA

-LRB-

N-methyl-Daspartate

-RRB-

and

GABAB

-RRB-

or

where

the

total

conductance

was

highly

non-gaussian

-LRB-

see

Supplementary

Information

-RRB-

The

experiments

simulations

and

theory

although

showing

a

clear

relationship

between

r

and

n

do

not

give

a

mechanistic

understanding

of

the

relationship

To

develop

this

intuition

we

examined

phenomenological

neuron

models

defined

by

a

simple

function

f

that

transforms

an

input

random

variable

I

with

gaussian

statistics

-LRB-

m

s

-RRB-

to

an

output

random

variable

n

-LRB-

Fig.

-RRB-

The

inputs

and

to

a

pair

of

such

neuron

models

were

constructed

in

analogy

to

our

two-cell

system

so

that

their

correlation

coefficient

was

c

-LRB-

compare

Fig.

to

Fig.

-RRB-

We

varied

the

mean

input

m

while

keeping

s

and

c

fixed

-LRB-

as

in

Fig.

red

curve

-RRB-

and

computed

the

mean

output

n.

and

the

output

correlation

coefficient

r

-LRB-

equation

-LRB-

-RRB-

-RRB-

When

f

was

strictly

linear

in

I

-LRB-

Fig.

top

row

-RRB-

then

r

c

for

all

values

of

m

making

r

independent

of

n.

However

when

f

was

threshold-linear

which

captures

the

rectifying

property

of

spike

generation

r

increased

with

n.

-LRB-

Fig.

-RRB-

as

was

observed

previously

-LRB-

Figs

and

-RRB-

The

fact

that

the

simple

thresholdlinear

model

reproduced

the

correlation

rate

relationship

indicates

that

the

specific

dynamics

of

the

cell

model

are

not

vital

to

the

relationship

Modifying

the

threshold-linear

transfer

to

include

saturation

produced

a

non-monotonic

relationship

between

r

and

n.

-LRB-

Fig.

middle

row

-RRB-

Similarly

adding

a

boosting

nonlinearity

to

the

threshold-linear

model

introduces

a

small

concavity

in

the

r

n.

relationship

-LRB-

Fig.

bottom

row

-RRB-

Despite

this

diversity

of

behaviour

across

these

examples

one

general

feature

is

clear

when

the

input

distribution

P

-LRB-

I

-RRB-

-LRB-

coloured

gaussians

-RRB-

overlapped

with

the

nonlinear

region

of

f

then

r

was

significantly

less

than

c.

c

Linear

c

d

f

Threshold-linear

saturating

c

f

f

f

b

Threshold-linear

c

f

c

Threshold-linear

boosting

c

f

I

<n>

I

<n>

Figure

Nonlinearities

shape

the

correlation

rate

relationship

in

a

phenomenological

neural

model

a

The

transfer

ni

f

-LRB-

Ii

-RRB-

for

a

pair

of

models

note

the

parallels

with

Fig.

b

f

and

the

r

n

relationship

for

the

threshold-linear

model

Overlaid

are

the

input

densities

P

-LRB-

I

-RRB-

-LRB-

coloured

gaussians

-RRB-

for

three

values

of

m.

c

Same

as

in

b

using

three

different

f

functions

-LRB-

see

text

-RRB-

d

The

joint

densities

P

-LRB-

-RRB-

and

P

-LRB-

-RRB-

for

the

threshold-linear

model

for

the

three

values

of

m

in

b

The

threshold

-LRB-

white

lines

-RRB-

partitions

the

input

plane

-LRB-

-RRB-

into

four

quadrants

P

-LRB-

-RRB-

is

transformed

differently

in

each

quadrant

-LRB-

middle

column

-RRB-

In

b

d

we

set

s

and

the

results

are

valid

for

c

in

the

interval

-LRB-

-RRB-

whereas

c

in

panel

c.

Nature

Publishing

Group

NATURE

Vol

August

LETTERS

For

the

threshold

and

saturating

nonlinearities

this

effect

was

especially

prominent

as

r

limited

to

zero

The

correlation

between

and

-LRB-

or

and

-RRB-

is

graphically

represented

as

the

eccentricity

of

their

joint

probability

density

function

P

-LRB-

-RRB-

-LRB-

or

P

-LRB-

-RRB-

-RRB-

The

eccentricity

loosely

defined

as

the

elongation

of

the

probability

density

function

along

the

diagonal

increases

with

correlation

The

eccentricity

of

P

-LRB-

-RRB-

is

by

construction

uniquely

determined

by

c

so

that

for

c

the

probability

density

function

has

a

circular

distribution

whereas

as

c

approaches

the

probability

density

function

becomes

increasingly

confined

to

the

diagonal

In

contrast

the

eccentricity

of

P

-LRB-

-RRB-

depends

on

both

c

and

m

-LRB-

Fig.

bottom

row

-RRB-

as

the

following

analysis

using

the

threshold-linear

model

shows

The

threshold

associated

with

f

partitions

the

input

plane

-LRB-

-RRB-

into

four

quadrants

-LRB-

Fig.

middle

column

-RRB-

In

each

quadrant

the

transformation

of

P

-LRB-

-RRB-

into

P

-LRB-

-RRB-

via

f

is

different

When

m

is

large

most

of

the

mass

of

P

-LRB-

-RRB-

is

contained

in

quadrant

so

that

P

-LRB-

-RRB-

is

mapped

approximately

linearly

to

P

-LRB-

-RRB-

As

a

consequence

P

-LRB-

-RRB-

inherits

much

of

the

eccentricity

of

P

-LRB-

-RRB-

-LRB-

Fig.

bottom-left

panel

-RRB-

implying

r

c

As

m

decreases

the

mass

of

P

-LRB-

-RRB-

is

shifted

from

quadrant

to

the

other

three

quadrants

and

f

distorts

P

-LRB-

-RRB-

so

that

the

eccentricity

of

P

-LRB-

-RRB-

and

hence

r

decreases

-LRB-

Fig.

middle

and

right

panels

of

bottom

row

-RRB-

Thus

as

m

is

varied

both

r

and

n.

are

expected

to

co-vary

yielding

the

correlation

rate

relationship

shown

in

Figs

and

An

analogous

description

can

be

made

using

the

spiking

LIF

model

-LRB-

see

Supplementary

Information

-RRB-

The

above

analysis

shows

that

nonlinearities

in

the

transfer

function

can

cause

different

n-dependencies

of

r

and

allows

us

to

explore

the

effect

of

other

nonlinearities

present

in

neurons

Phenomena

such

as

spike

refractoriness

-LRB-

see

Fig.

of

ref

-RRB-

synaptic

depression

-LRB-

see

Fig.

of

ref

-RRB-

or

dendritic

sublinear

summation

-LRB-

see

Fig.

of

ref

-RRB-

impose

a

saturation

on

rate

that

would

cause

the

r

n

relationship

to

reach

a

maximum

and

begin

to

decrease

-LRB-

Fig.

middle

row

-RRB-

Dendritic

supra-linear

summation

-LRB-

Fig.

of

ref

and

Fig.

of

ref

-RRB-

introduces

boosting

effects

that

would

yield

small

inflections

in

the

r

n

relationship

-LRB-

Fig.

bottom

row

-RRB-

These

observations

predict

that

single

neuron/synaptic

nonlinearities

can

be

an

important

determinant

of

network

correlation

A

relationship

between

firing

rate

and

pairwise

correlation

has

until

now

been

A

consequence

of

the

correlation

rate

relationship

is

that

if

the

firing

rate

of

a

population

of

neurons

is

tuned

to

a

certain

feature

-LRB-

for

example

orientation

of

a

bar

-RRB-

the

population

correlation

inherits

the

same

tuning

Consistent

with

this

prediction

rT

measured

from

neurons

in

primary

visual

cortex

shows

a

tuning

to

stimulus

orientation

when

T

A

direct

test

of

the

correlation

rate

relationship

would

require

intracellular

recordings

where

measurements

of

both

the

afferent

current

and

spike

train

correlations

as

a

stimulus

is

varied

could

be

Nevertheless

analysis

of

extracellular

multi-unit

recordings

from

macaque

-LRB-

A.

Kohn

and

M.

Smith

personal

communication

-RRB-

and

electrosensory

receptors

-LRB-

J.

Middleton

J.

Benda

A.

Longtin

and

L.

Maler

personal

communication

-RRB-

shows

a

correlation

rate

relationship

similar

to

the

one

obtained

in

our

network

-LRB-

Fig.

-RRB-

supporting

the

generality

of

our

result

The

correlation

rate

relationship

prompts

a

re-examination

of

rate

and

correlation-based

coding

because

these

measures

are

mechanistically

intertwined

METHODS

SUMMARY

Whole-cell

somatic

recordings

were

made

from

layer

neurons

in

slices

-LRB-

mm

-RRB-

from

auditory

and

somatosensory

cortices

of

young

mice

-LRB-

postnatal

day

to

-RRB-

Surgical

slicing

and

recording

techniques

were

done

as

described

and

followed

guidelines

established

by

the

NYU

Animal

Welfare

Committee

Computer-generated

currents

-LRB-

equation

-LRB-

-RRB-

duration

L

s

-RRB-

were

delivered

to

neurons

every

s

for

a

total

of

N

trials

The

terms

ji

-LRB-

t

-RRB-

and

jc

-LRB-

t

-RRB-

were

gaussian

noises

low-pass-filtered

with

a

time

constant

of

ms.

The

spike

count

from

the

ith

cell

in

the

kth

trial

nki

-LRB-

t

-RRB-

was

the

number

of

spikes

that

occurred

in

the

temporal

window

-LRB-

t

t

T

-RRB-

Time

was

discretized

with

Dt

ms.

The

spike

count

shift-corrected

covariance

was

Cov

ni

nj

N

X

X

L

-LCB-

T

nk

-LRB-

t

-RRB-

nk

-LRB-

t

-RRB-

-LCB-

nk

-LRB-

t

-RRB-

-LRB-

t

-RRB-

i

j

i

j

N

-LRB-

L

-LCB-

T

-RRB-

k

t

where

the

interior

sum

ranges

over

time

bins

The

variance

was

Var

-LRB-

ni

-RRB-

Cov

-LRB-

ni

ni

-RRB-

and

the

correlation

coefficient

rT

was

computed

as

in

equation

-LRB-

-RRB-

In

Fig.

we

used

T

ms

-LRB-

see

Supplementary

Information

for

analysis

using

T

ms

-RRB-

We

used

a

leaky

integrate-and-fire

-LRB-

LIF

-RRB-

neural

which

obeyed

pffiffi

ffi

pffiffiffiffiffiffi

pffiffiffiffiffiffiffiffiffiffi

tm

dVi

-LCB-

Vi

zmzs

tm

-LCB-

c

ji

-LRB-

t

-RRB-

z

c

jc

-LRB-

t

-RRB-

where

Vi

is

the

membrane

dt

potential

of

the

ith

neuron

and

tm

ms

threshold

mV

and

reset

mV

Here

ji

-LRB-

t

-RRB-

and

jc

-LRB-

t

-RRB-

are

white

noise

currents

Correlation

coefficients

r

defined

as

the

limit

of

rT

as

T

increases

were

estimated

from

the

ratio

of

the

area

of

the

cross-correlation

function

normalized

by

the

areas

of

the

autocorrelation

-LRB-

see

Supplementary

Information

-RRB-

Other

parameters

that

differ

from

the

experimental

data

analysis

were

L

s

and

Dt

ms.

The

network

in

Fig.

had

NE

excitatory

and

NI

inhibitory

cells

in

layer

randomly

connected

-LRB-

probability

-RRB-

with

cells

in

layer

Layer

neurons

fired

Poisson

trains

with

a

rate

of

spikes

Layer

neurons

were

LIF

models

with

conductance-based

excitatory

and

inhibitory

synapses

with

time

constants

tE

ms

tI

ms

mean

weights

JE

nS

JI

nS

and

standard

deviation

Da

Ja

-LRB-

where

a

E

I

-RRB-

Full

Methods

and

any

associated

references

are

available

in

the

online

version

of

the

paper

at

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J.

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to

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online

version

of

the

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at

Acknowledgements

We

thank

C.

Colbert

A.

Kohn

L.

Maler

D.

Nikolic

A.-M

Oswald

and

A.

Renart

for

their

critical

reading

of

the

manuscript

and

R.

Moreno-Bote

M.

Schiff

and

J.

Rinzel

for

insightful

discussions

Funding

was

provided

by

the

Spanish

MEC

-LRB-

J.R.

-RRB-

HFSP

-LRB-

B.D.

-RRB-

a

Burroughs

Welcome

Fund

career

award

and

an

NSF

postdoctoral

fellowship

-LRB-

E.S.-B

-RRB-

NSF

-LRB-

K.J.

-RRB-

and

NIH

-LRB-

A.R.

-RRB-

Author

Information

Reprints

and

permissions

information

is

available

at

The

authors

declare

no

competing

financial

interests

Correspondence

and

requests

for

materials

should

be

addressed

to

B.D.

-LRB-

bdoiron@cns.nyu.edu

-RRB-

or

J.R

-LRB-

jrocha@cns.nyu.edu

-RRB-

Nature

Publishing

Group

doi

METHODS

Experiments

Surgical

slicing

and

recording

techniques

were

done

as

described

and

followed

guidelines

established

by

the

NYU

Animal

Welfare

Committee

Slices

were

made

from

auditory

and

somatosensory

cortices

of

young

mice

-LRB-

postnatal

day

to

-RRB-

Whole-cell

somatic

recordings

were

made

from

layer

neurons

identified

under

an

infrared

video

microscopy

During

recordings

slices

-LRB-

mm

thick

-RRB-

were

perfused

at

room

temperature

or

at

uC

with

ACSF

-LRB-

with

-RRB-

Pipettes

were

filled

with

mM

K-gluconate

mM

KCl

mM

phosphocreatine

mM

HEPES

mM

ATP-Mg

and

mM

GTP

at

pH

Filled

electrode

resistances

were

in

the

range

of

to

MV

and

recordings

were

performed

under

current-clamp

conditions

Voltage

and

current

signals

were

filtered

at

KHz

and

digitized

at

kHz

Stimulus

protocol

Computer-generated

currents

-LRB-

equation

-LRB-

-RRB-

duration

L

s

after

removing

the

initial

ms

-RRB-

were

delivered

to

neurons

every

s

The

common

jc

-LRB-

t

-RRB-

and

the

independent

terms

ji

-LRB-

t

-RRB-

were

gaussian

noises

low-pass-filtered

with

a

time

constant

of

ms.

We

presented

each

N

trials

stimulus

block

for

repetitions

-LRB-

see

Supplementary

Information

for

details

-RRB-

Data

analysis

Voltage

traces

were

differentiated

and

action

potential

times

were

detected

by

setting

a

threshold

on

the

differential

traces

The

spike

trains

emitted

by

the

ith

cell

in

the

kth

trial

were

represented

by

a

binary

time

series

yik

-LRB-

t

-RRB-

which

equalled

if

there

was

a

spike

at

time

t

and

zero

otherwise

The

spike

count

nik

-LRB-

t

-RRB-

was

simply

the

number

of

spikes

that

occurred

in

a

window

of

length

T

that

is

tzT

P

k

yi

-LRB-

t

-RRB-

Time

was

discretized

with

Dt

ms

resolution

and

the

first

nk

-LRB-

t

-RRB-

i

t

t

ms

of

the

spike

trains

were

removed

to

correct

for

spike

frequency

adaptation

The

spike

count

shift-corrected

covariance

was

computed

as

N

X

N

X

X

L

-LCB-

T

X

L

-LCB-

T

nk

-LRB-

t

-RRB-

nk

-LRB-

t

-RRB-

-LCB-

nk

-LRB-

t

-RRB-

-LRB-

t

-RRB-

Cov

-LRB-

ni

nj

-RRB-

i

j

j

N

L

-LCB-

T

k

t

N

L

-LCB-

T

k

t

i

The

auto-correlation

function

was

obtained

by

making

i

j

in

equation

-LRB-

-RRB-

Note

that

as

in

the

covariance

we

have

corrected

for

chance

correlations

in

Cij

-LRB-

t

-RRB-

by

subtracting

the

corresponding

shift-corrector

Computational

model

We

used

a

leaky

integrate-and-fire

-LRB-

LIF

-RRB-

neural

integrating

white

noise

currents

pffiffiffi

pffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi

dVi

-LCB-

Vi

zmzs

tm

-LCB-

c

ji

-LRB-

t

-RRB-

z

c

jc

-LRB-

t

-RRB-

tm

dt

where

Vi

is

the

membrane

potential

of

the

ith

neuron

and

tm

ms

threshold

mV

and

reset

mV

Correlation

coefficients

r

defined

as

the

limit

of

rT

as

T

increases

were

estimated

by

computing

the

ratio

of

the

area

of

the

cross-correlation

function

Cij

-LRB-

t

-RRB-

normalized

by

the

areas

of

the

autocorrelation

functions

Cii

-LRB-

t

-RRB-

-LRB-

see

Supplementary

Information

-RRB-

Other

parameters

in

the

simulations

that

differ

from

the

data

analysis

were

L

s

and

Dt

ms.

r

in

Fig.

was

numerically

obtained

from

a

linear

regression

of

r

-LRB-

c

-RRB-

in

the

interval

c

Parameter

values

in

Fig.

m

and

mV

s

mV

-LRB-

panel

a

-RRB-

m

mV

s

and

mV

-LRB-

panel

b

-RRB-

m

mV

and

s

mV

-LRB-

red

in

panel

c

-RRB-

m

mV

and

s

mV

-LRB-

blue

in

panel

c

-RRB-

The

theoretical

curves

in

Fig.

were

computed

from

standard

first

passage

time

formulae

appropriate

for

a

LIF

neuron

receiving

white

noise

-LRB-

see

Supplementary

Information

-RRB-

Network

simulation

The

network

in

Fig.

had

NE

excitatory

-LRB-

E

-RRB-

and

NI

inhibitory

-LRB-

I

-RRB-

cells

in

layer

and

cells

in

layer

Connections

were

randomly

established

from

layer

to

layer

with

a

probability

P

Layer

cells

fired

Poisson

spike

trains

at

rate

nE

nI

spikes

Each

neuron

in

layer

was

a

LIF

neuron

model

with

conductance-based

E

and

I

synapses

described

by

Cm

dV

gL

EL

-LCB-

V

zgE

-LRB-

t

-RRB-

EE

-LCB-

V

zgI

-LRB-

t

-RRB-

EI

-LCB-

V

dt

where

the

interior

sums

range

over

time

bins

Note

that

in

the

second

term

named

the

shift-corrector

spike

counts

from

consecutive

trials

were

multiplied

This

term

equals

the

product

ni.

nj

for

data

that

are

stationary

across

trials

-LRB-

like

in

our

LIF

model

-RRB-

and

corrected

for

both

chance

correlations

-LRB-

baseline

correlation

expected

for

independent

spike

trains

-RRB-

and

co-fluctuations

due

to

drifts

in

the

excitability

across

trials

The

variance

was

obtained

by

noting

that

Var

-LRB-

ni

-RRB-

Cov

-LRB-

ni

ni

-RRB-

and

the

correlation

coefficient

rT

was

computed

as

in

equation

-LRB-

-RRB-

The

analysis

shown

in

Fig.

used

T

ms

whereas

in

the

Supplementary

Information

we

explore

the

range

T

ms.

The

output

spike

shift-corrected

cross-correlation

function

is

defined

as

Cij

-LRB-

t

-RRB-

N

L

N

L

XX

XX

y

k

-LRB-

t

-RRB-

yjk

-LRB-

tzt

-RRB-

-LCB-

y

k

-LRB-

t

-RRB-

-LRB-

tzt

-RRB-

N

Dt

L

-LCB-

jtj

k

t

i

N

Dt

L

-LCB-

jtj

k

t

i

dga

ga

X

X

k

-LCB-

z

Ja

j

d

-LRB-

t

-LCB-

ta

j

-RRB-

dt

ta

j

k

-LRB-

a

E

I

-RRB-

where

-LCB-

tk

-RCB-

are

the

times

of

the

spikes

emitted

by

the

jth

pre-synaptic

cell

and

Ja

j

is

the

synaptic

weight

which

was

zero

if

the

jth

cell

did

not

establish

a

connection

or

was

randomly

chosen

from

a

gaussian

distribution

with

mean

JE

nS

-LRB-

JI

nS

-RRB-

and

standard

deviation

Da

Ja

-LRB-

a

E

I

-RRB-

if

there

was

one

The

synaptic

time

constants

were

tE

ms

tI

ms

resembling

AMPA

and

GABAA

kinetics

-LRB-

see

Supplementary

Information

for

a

slow

synapses

analysis

-RRB-

We

also

set

threshold

mV

reset

mV

refractory

period

ms

membrane

capacitance

Cm

nF

leak

conductance

gL

nS

excitatory

reversal

potential

EE

mV

and

inhibitory

reversal

potential

EI

mV

Nature

Publishing

Group

O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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