寺前順之介 - Osaka University...observed6,8,9. Firing rate distributions are well fitted by...

脳の自発揺らぎの数理科学 -‐その起源と神経情報処理における役割-‐

寺前順之介大阪大学大学院情報科学研究科

[email protected]‐u.ac.jp

自己紹介出身は群馬

大学から京都，物理「非線形物理学」自然の秩序形成，自己組織化

理化学研究所「理論神経科学」脳の情報処理メカニズム

2012.12-‐

大阪大学情報科学研究科

キーワードは『ゆらぎと確率』

脳・大脳皮質

膨大な数の神経細胞からなるネットワーク

大脳皮質だけで数百億個，それぞれ数千の入出力を持つ

“Brainbow” pyramidal neuron

スパイク発火による情報伝達

時間時間

つながりの強さ：興奮性シナプス後電位（EPSP）

-‐70mV

大脳皮質の自発活動

大脳皮質では入力がなくても活動が持続自発的持続発火活動（spontaneous ongoing acDvity）

6 sec

神経細胞

時間

時間

膜電位

Destexhe et al. 2003 Nat. Rev. Neurosci.

Takekawa et al. 非同期、不規則、低頻度（1-‐2Hz）

膜電位も乱雑に大きく変動

自発発火活動の特徴

744 | SEPTEMBER 2003 | VOLUME 4 www.nature.com/reviews/neuro

R E V I EW S

POWER SPECTRUM

After analysing a waveform witha Fourier transform, itsamplitude spectrum is thecollection of amplitudes of thesinusoidal components thatresult from the analysis. Thepower spectrum is the square ofthe amplitude spectrum.

COLOURED NOISE

White noise is a signal thatcovers the entire range ofcomponent sound frequencieswith equal intensity. In colourednoise, the signal covers a narrowband of frequencies.

Box 1 | Synaptic noise

The term ‘synaptic noise’ is commonly used to describe the irregular subthreshold dynamics of the membrane potentials ofneurons in vivo, which are caused by the discharge activity of a large number of presynaptic neurons.Despite carryingneuronal information, this activity seems to be nearly random, resulting in stochastic dynamics of the membranepotential, with statistical properties and a broadband POWER SPECTRUM that resemble those of COLOURED NOISE.Panel a showssynaptic ‘noise’ in neocortical neurons in vivoduring activated periods with a desynchronized electroencephalogram(EEG).Panel b illustrates a detailed biophysical model of synaptic noise in a reconstructed layer VI pyramidal neuron, withNa+ and K+ channels in dendrites and soma.Randomly releasing excitatory (n≈ 16,000) and inhibitory (n≈ 4,000) synapseswere modelled using AMPA (α-amino-3-hydroxy-5-methyl-4-isoxazole propionic acid) and GABAA (γ-aminobutyric acidsubtype A)-receptor kinetics17. Their distribution in soma and dendrites was based on ultrastructural measurements1.Panel c shows a ‘point conductance’model of synaptic noise; a single-compartment model with two global excitatory (ge)and inhibitory (gi) synaptic conductances,modelled by stochastic processes69.Panel d shows the results of dynamic-clampinduction of synaptic noise in neocortical neurons in vitro. In each case,an example of the membrane potential time course(left), its amplitude distribution (middle) and its power spectral density (right; logarithmic scale) are shown. The powerspectral densities were computed in the absence of spikes (hyperpolarized,or using passive models). In all cases, thedistributions were approximately symmetric,and power spectral densities were broadband and behaved as a negativepower of frequency (1/fk, k≈ 2.6; green lines) at high frequencies (as expected for low-pass filtered noise). The data used forthe analysis in dwere kindly provided by M. Badoual and T. Bal.

20 mV

20 mV

20 mV

20 mV

–60 mV

–60 mV

–60 mV

–60 mV

500 ms

500 ms

500 ms

500 ms

AMPA

GABAA

c Point-conductance models

0.11

10–3

10–6

0.06

0.02

0.15

0.1

0.05

–80 –70 –60

–80 –70 –60

0.15

0.1

0.05

–80 –70 –60

0.15

0.1

0.05

–80 –70 –60

–50 0.1 1 10 100 1,000

1

10–3

10–6

0.1 1 10 100 1,000

1

10–3

10–6

0.1 1 10 100 1,000

1

10–3

10–6

0.1 1 10 100 1,000

Vm (mV) Frequency (Hz)




Am

plitu

de d

istr

ibut

ion

Pow

er s

pect

ral d

ensi

ty

Am

plitu

de d

istr

ibut

ion

Pow

er s

pect

ral d

ensi

ty

Am

plitu

de d

istr

ibut

ion

Pow

er s

pect

ral d

ensi

ty

Am

plitu

de d

istr

ibut

ion

Pow

er s

pect

ral d

ensi

ty

ge(t)

ge(t)

gi(t)

gi(t)

d Dynamic-clamp experiments

a In vivo experiments

b Detailed biophysical models

744 | SEPTEMBER 2003 | VOLUME 4 www.nature.com/reviews/neuro

R E V I EW S

POWER SPECTRUM

After analysing a waveform witha Fourier transform, itsamplitude spectrum is thecollection of amplitudes of thesinusoidal components thatresult from the analysis. Thepower spectrum is the square ofthe amplitude spectrum.

COLOURED NOISE

White noise is a signal thatcovers the entire range ofcomponent sound frequencieswith equal intensity. In colourednoise, the signal covers a narrowband of frequencies.

Box 1 | Synaptic noise

The term ‘synaptic noise’ is commonly used to describe the irregular subthreshold dynamics of the membrane potentials ofneurons in vivo, which are caused by the discharge activity of a large number of presynaptic neurons.Despite carryingneuronal information, this activity seems to be nearly random, resulting in stochastic dynamics of the membranepotential, with statistical properties and a broadband POWER SPECTRUM that resemble those of COLOURED NOISE.Panel a showssynaptic ‘noise’ in neocortical neurons in vivoduring activated periods with a desynchronized electroencephalogram(EEG).Panel b illustrates a detailed biophysical model of synaptic noise in a reconstructed layer VI pyramidal neuron, withNa+ and K+ channels in dendrites and soma.Randomly releasing excitatory (n≈ 16,000) and inhibitory (n≈ 4,000) synapseswere modelled using AMPA (α-amino-3-hydroxy-5-methyl-4-isoxazole propionic acid) and GABAA (γ-aminobutyric acidsubtype A)-receptor kinetics17. Their distribution in soma and dendrites was based on ultrastructural measurements1.Panel c shows a ‘point conductance’model of synaptic noise; a single-compartment model with two global excitatory (ge)and inhibitory (gi) synaptic conductances,modelled by stochastic processes69.Panel d shows the results of dynamic-clampinduction of synaptic noise in neocortical neurons in vitro. In each case,an example of the membrane potential time course(left), its amplitude distribution (middle) and its power spectral density (right; logarithmic scale) are shown. The powerspectral densities were computed in the absence of spikes (hyperpolarized,or using passive models). In all cases, thedistributions were approximately symmetric,and power spectral densities were broadband and behaved as a negativepower of frequency (1/fk, k≈ 2.6; green lines) at high frequencies (as expected for low-pass filtered noise). The data used forthe analysis in dwere kindly provided by M. Badoual and T. Bal.

20 mV

20 mV

20 mV

20 mV

–60 mV

–60 mV

–60 mV

–60 mV

500 ms

500 ms

500 ms

500 ms

AMPA

GABAA

c Point-conductance models

0.11

10–3

10–6

0.06

0.02

0.15

0.1

0.05

–80 –70 –60

–80 –70 –60

0.15

0.1

0.05

–80 –70 –60

0.15

0.1

0.05

–80 –70 –60

–50 0.1 1 10 100 1,000

1

10–3

10–6

0.1 1 10 100 1,000

1

10–3

10–6

0.1 1 10 100 1,000

1

10–3

10–6

0.1 1 10 100 1,000





Am

plitu

de d

istr

ibut

ion

Pow

er s

pect

ral d

ensi

ty

Am

plitu

de d

istr

ibut

ion

Pow

er s

pect

ral d

ensi

ty

Am

plitu

de d

istr

ibut

ion

Pow

er s

pect

ral d

ensi

ty

Am

plitu

de d

istr

ibut

ion

Pow

er s

pect

ral d

ensi

ty

ge(t)

ge(t)

gi(t)

gi(t)

d Dynamic-clamp experiments

a In vivo experiments

b Detailed biophysical modelsDestexhe et al. 2003 Nat. Rev. Neurosci.

静止膜電位発火しきい値

膜電位が持ち上がり、大きく揺らぐ

自発活動と神経応答

(1996 Science)

神経応答　＝　刺激入力　＋　自発活動

自発活動と神経応答

神経応答の空間構造　〜　自発活動の空間構造

(1999 Science)

QuesDon

揺らぎの起源は何か？

その機能は何か？

神経ネットワークの数理的な記述

C dvdt

= −gL v − EL( )− gNam3h v − ENa( )− gKn4 v − EK( ) + Iext

Hodgikn – Huxley equaDon

単一の神経細胞の記述

ところが，神経ネットワークのモデルは自発揺らぎを説明できなかった

活動が持続しない　か　爆発してしまう

神経ダイナミクスの大問題

•  No noise source in the brain.

•  Single neurons are silent.

なぜか? ニューロンは多数の弱入力を積算する多数決素子

EPSP ~ 1mV Vthr = -‐50 mv

Vrest = -‐70 mv

v

Dme

20 mv

強い同期発火 or

高発火率

仮説：多数の弱い結合と少数の極めて強い結合の共存が鍵ではないか

多数の弱い結合と少数の極めて強い結合の共存

S. Song, P. J. Sjoestroem, M.Reigl, S. Nelson, D. B. Chklovskii

PLoS Biology, 2005, 3(3) 0507-‐0519

対数正規分布 Lognormal distribuDon

興奮性細胞 10000個

抑制性細胞 2000個

Random net, P = 0.1 for exc. 0.5 for inh.

Gmax ~ 10 mv

Lognormal

極めて不均一なネットワーク結合強度

( ) ( )22

log1 exp22x

P xx

µσπσ

⎡ ⎤−= −⎢ ⎥

⎢ ⎥⎣ ⎦

神経細胞モデルは単純に

Vthr = -‐50 mv

Vrest = -‐70 mv

v

Dme

20 mv

( ) ( ) ( )

( )

1

j

rest E I I

j

Em

j jj ss

dv v V g gdtdg

v V v V

t sGdt

gτ

δ ττ

− −

−

⎧ = − − − −⎪⎪⎨⎪ = − +⎪

−⎩

∑ ∑

Leaky integrate-‐and-‐fire neuron with conductance synapses

外部ノイズや、背景ノイズは用いなくて良い

Poisson spike trains to all neurons during initial 100 [ms] to trigger aspontaneous firing. In the absence of external input, the model sus-tains a stable asynchronous firing initiated by a brief external stimu-lus (Fig. 2a). The spontaneous network activity emerges purely fromreverberating synaptic input, is stable in a very low-frequency regime(Fig. 2b) and is highly irregular (Fig. 2c) as experimentallyobserved6,8,9. Firing rate distributions are well fitted by lognormaldistributions7,46,47. Each neuron exhibits large membrane potentialfluctuations, on top of which spikes are generated occasionally(Fig. 2d), owing to the dynamic balance between excitatory andinhibitory activities (Fig. 2a and 2e)18,20,24,48. All these properties areconsistent with the spontaneous activity observed in cortical neu-rons20. Importantly, the average values of the membrane potentials

are around –60 mV in excitatory neurons (Fig. 2f)20,49, at which spiketransmission at strong-sparse synapses becomes most reliable(Fig. 1a, shaded area). Inputs to weak-dense synapses maintain theaverage membrane potential of each neuron (Fig. 2g), whereas inputsto strong-sparse synapses govern sparse spiking. Therefore, weak-dense and strong-sparse synapses have different roles in stochasticneural dynamics, although they distribute continuously.

Long-tailed distributions of coupling strengths offer a much widerregion of the parameter space to stable spontaneous activity thanGaussian-distributed coupling strengths (Supplementary Fig. 1).Furthermore, a linear stability analysis reveals the homeostasis ofthe ongoing state of the SSWD network (Methods).

What is the underlying mechanism and functional implicationsof the spontaneous noise generation? Strong-sparse synapses formmultiple synaptic pathways in the recurrent neural network(Fig. 3a). Owing to the stochastic resonance effect at thesesynapses, spike sequences are routed reliably along these pathways(Fig. 3b: Supplementary Methods) that may branch and converge(Fig. 3c). Since strong synapses are rare, spike propagation along apathway is essentially unidirectional, as indicated by the cross-correlograms for presynaptic and postsynaptic neuron pairs(Fig. 3d). If, therefore, external stimuli elicit spikes from the initialneurons of some strong pathways, the spikes can stably travelalong these pathways without much interference (Fig. 3e). Thenumber of spikes received at the end of a pathway is proportionalto that of spikes evoked at the start, although fluctuations in thespike number increase with the distance of travel (Fig. 3f). Theseresults imply that spikes can carry rate information along themultiple synaptic pathways embedded by strong-sparse synapses.The presence of precise spike sequences has been reported in thebrain of behaving animals50–52. We note that the same spikes aresensed as noise if they are input to weak synapses.

DiscussionIn this study, we have explored a coordinating principle in neuralcircuit function based on a long-tailed distribution of connectionweights in a model neural network. The network properties con-ferred by the long-tailed EPSP distribution account for a role of noisein information routing and present a novel hypothesis for neuralnetwork information processing. Namely, we have demonstratedthat a single neuron shows spike-based aperiodic stochastic res-onance; the cross-correlation coefficient between output spikes of asingle neuron and inputs to the strongest synapses are maximizedwhen the neuron receives a certain amount of background noise.Stochastic resonance has been studied in neuronal systems in variouscontexts. The presence of sensory noise improved behavioral per-formance in humans38,41 and other animals39. Synaptic bombard-ment enhanced the responsiveness of neurons to periodic sub-threshold stimuli20,40,42. Asynchronous neurotransmitter release cangive a noise source for stochastic resonance in local circuits of modelneurons with short-term synaptic plasticity43,44. A surprising resulthere is that the networks may internally generate optimal noise with-out external noise sources for the spike-based stochastic resonanceon sparse-strong connections. Weak-dense connections redistributeexcitatory activity routed reliably on strong connections over thenetwork as optimal noise sources to sustain spontaneous firing ofrecurrent networks.

Internal noise or asynchronous irregular firing may provide theneural substrate for probabilistic computations by the brain, and howsuch activity emerges in cortical circuits has been a fundamentalproblem in cortical neurobiology. Such neuronal firing has beenreplicated by sparsely connected networks of binary or spikingneurons18,19,21–23, and the importance of excitation-inhibition balancehas been repeatedly emphasized. However, the mechanism togenerate extremely low-rate spontaneous asynchronous firing(=10 Hz) remained unclear, and our model gives a possible solution

Firing rate (Hz)

Pro

ba

bili

ty d

en

sity

00

0.2

0.4

20 40

Excitatory pool (Hz)1.0

0

20

40

2.0 3.0

Inh

ibito

ry p

oo

l (H

z)

Coefficient of variation0.0

0

3

6

1.0 2.0Pro

ba

bili

ty d

en

sity

c

d

e

fTime (ms)

2,000-70-50

0

3,000

Me

mb

ran

e p

ote

ntia

l (m

V)

-70-50

0

2,000 3,000

a

Membrane potential (mV)

Pro

ba

bili

ty d

en

sity

-700

0.1

-60 -50

b

Ne

uro

ns

10,00012,000

0

Time (ms)

Exc

itato

ry p

ool (

Hz)

Inhi

bito

ry p

ool (

Hz)

2,4000

1

2

0

20

40

2,500 2,600

2,400 2,500 2,600

Me

an

me

nb

ran

ep

ote

ntia

l (m

V)

SD

of m

em

bra

ne p

ote

ntia

ls (

mV

)

minimum EPSP (mV)

00

-74

-68

-62

2

2

4

4

g

Figure 2 | Spontaneous noise in the SSWD recurrent network. Thenetwork receives neither external input nor background noise, and henceactivity is spontaneous. (a) Upper, Spike raster of excitatory (red) andinhibitory (blue) neurons in the noisy spontaneous firing state. Lower, Thepopulation firing rates of excitatory (red) and inhibitory (blue) neurons.(b) Firing rate distributions of excitatory (red) and inhibitory (blue)neurons can be fitted by lognormal distributions (black lines). Mean firingrates are 1.6 and 14 [Hz] for excitatory and inhibitory neurons respectively.(c) CVs of inter-spike intervals are distributed around unity in excitatory(red) and inhibitory (blue) neurons. (d) Time courses of the membranepotentials of excitatory (red) and inhibitory (blue) neurons exhibit largeamplitude fluctuations. (e) Scatter plot of the instantaneous populationactivities of excitatory and inhibitory neurons. The solid line representslinear regression. (f) Distribution functions of the fluctuating membranepotentials show the depolarized states of excitatory (red) and inhibitory(blue) neurons. (g) The mean (solid) and standard deviation (dashed) ofthe membrane potential fluctuations of an excitatory neuron when allEPSPs smaller than the minimum value given in the abscissa areeliminated. Here, we remove a portion of excitatory synapses on a neuronfrom the weakest ones.

www.nature.com/scientificreports

SCIENTIFIC REPORTS | 2 : 485 | DOI: 10.1038/srep00485 3

Poisson spike trains to all neurons during initial 100 [ms] to trigger aspontaneous firing. In the absence of external input, the model sus-tains a stable asynchronous firing initiated by a brief external stimu-lus (Fig. 2a). The spontaneous network activity emerges purely fromreverberating synaptic input, is stable in a very low-frequency regime(Fig. 2b) and is highly irregular (Fig. 2c) as experimentallyobserved6,8,9. Firing rate distributions are well fitted by lognormaldistributions7,46,47. Each neuron exhibits large membrane potentialfluctuations, on top of which spikes are generated occasionally(Fig. 2d), owing to the dynamic balance between excitatory andinhibitory activities (Fig. 2a and 2e)18,20,24,48. All these properties areconsistent with the spontaneous activity observed in cortical neu-rons20. Importantly, the average values of the membrane potentials

are around –60 mV in excitatory neurons (Fig. 2f)20,49, at which spiketransmission at strong-sparse synapses becomes most reliable(Fig. 1a, shaded area). Inputs to weak-dense synapses maintain theaverage membrane potential of each neuron (Fig. 2g), whereas inputsto strong-sparse synapses govern sparse spiking. Therefore, weak-dense and strong-sparse synapses have different roles in stochasticneural dynamics, although they distribute continuously.

Long-tailed distributions of coupling strengths offer a much widerregion of the parameter space to stable spontaneous activity thanGaussian-distributed coupling strengths (Supplementary Fig. 1).Furthermore, a linear stability analysis reveals the homeostasis ofthe ongoing state of the SSWD network (Methods).

What is the underlying mechanism and functional implicationsof the spontaneous noise generation? Strong-sparse synapses formmultiple synaptic pathways in the recurrent neural network(Fig. 3a). Owing to the stochastic resonance effect at thesesynapses, spike sequences are routed reliably along these pathways(Fig. 3b: Supplementary Methods) that may branch and converge(Fig. 3c). Since strong synapses are rare, spike propagation along apathway is essentially unidirectional, as indicated by the cross-correlograms for presynaptic and postsynaptic neuron pairs(Fig. 3d). If, therefore, external stimuli elicit spikes from the initialneurons of some strong pathways, the spikes can stably travelalong these pathways without much interference (Fig. 3e). Thenumber of spikes received at the end of a pathway is proportionalto that of spikes evoked at the start, although fluctuations in thespike number increase with the distance of travel (Fig. 3f). Theseresults imply that spikes can carry rate information along themultiple synaptic pathways embedded by strong-sparse synapses.The presence of precise spike sequences has been reported in thebrain of behaving animals50–52. We note that the same spikes aresensed as noise if they are input to weak synapses.

DiscussionIn this study, we have explored a coordinating principle in neuralcircuit function based on a long-tailed distribution of connectionweights in a model neural network. The network properties con-ferred by the long-tailed EPSP distribution account for a role of noisein information routing and present a novel hypothesis for neuralnetwork information processing. Namely, we have demonstratedthat a single neuron shows spike-based aperiodic stochastic res-onance; the cross-correlation coefficient between output spikes of asingle neuron and inputs to the strongest synapses are maximizedwhen the neuron receives a certain amount of background noise.Stochastic resonance has been studied in neuronal systems in variouscontexts. The presence of sensory noise improved behavioral per-formance in humans38,41 and other animals39. Synaptic bombard-ment enhanced the responsiveness of neurons to periodic sub-threshold stimuli20,40,42. Asynchronous neurotransmitter release cangive a noise source for stochastic resonance in local circuits of modelneurons with short-term synaptic plasticity43,44. A surprising resulthere is that the networks may internally generate optimal noise with-out external noise sources for the spike-based stochastic resonanceon sparse-strong connections. Weak-dense connections redistributeexcitatory activity routed reliably on strong connections over thenetwork as optimal noise sources to sustain spontaneous firing ofrecurrent networks.

Internal noise or asynchronous irregular firing may provide theneural substrate for probabilistic computations by the brain, and howsuch activity emerges in cortical circuits has been a fundamentalproblem in cortical neurobiology. Such neuronal firing has beenreplicated by sparsely connected networks of binary or spikingneurons18,19,21–23, and the importance of excitation-inhibition balancehas been repeatedly emphasized. However, the mechanism togenerate extremely low-rate spontaneous asynchronous firing(=10 Hz) remained unclear, and our model gives a possible solution

Firing rate (Hz)

Proba

bility d

ensity

00

0.2

0.4

20 40

Excitatory pool (Hz)1.0

0

20

40

2.0 3.0

Inhibit

ory po

ol (Hz

)

Coefficient of variation0.0

0

3

6

1.0 2.0Proba

bility d

ensity

c

d

e

fTime (ms)

2,000-70-50

0

3,000

Memb

rane p

otentia

l (mV)

-70-50

0

2,000 3,000

a

Membrane potential (mV)

Proba

bility d

ensity

-700

0.1

-60 -50

b

Neuro

ns

10,00012,000

0

Time (ms)

Excitato

ry pool

(Hz)

Inhibito

ry pool

(Hz)

2,4000

1

2

0

20

40

2,500 2,600

2,400 2,500 2,600

Mean

men

brane

poten

tial (m

V)

SD of

memb

rane p

otentia

ls (mV

)

minimum EPSP (mV)

00

-74

-68

-62

2

2

4

4

g

Figure 2 | Spontaneous noise in the SSWD recurrent network. Thenetwork receives neither external input nor background noise, and henceactivity is spontaneous. (a) Upper, Spike raster of excitatory (red) andinhibitory (blue) neurons in the noisy spontaneous firing state. Lower, Thepopulation firing rates of excitatory (red) and inhibitory (blue) neurons.(b) Firing rate distributions of excitatory (red) and inhibitory (blue)neurons can be fitted by lognormal distributions (black lines). Mean firingrates are 1.6 and 14 [Hz] for excitatory and inhibitory neurons respectively.(c) CVs of inter-spike intervals are distributed around unity in excitatory(red) and inhibitory (blue) neurons. (d) Time courses of the membranepotentials of excitatory (red) and inhibitory (blue) neurons exhibit largeamplitude fluctuations. (e) Scatter plot of the instantaneous populationactivities of excitatory and inhibitory neurons. The solid line representslinear regression. (f) Distribution functions of the fluctuating membranepotentials show the depolarized states of excitatory (red) and inhibitory(blue) neurons. (g) The mean (solid) and standard deviation (dashed) ofthe membrane potential fluctuations of an excitatory neuron when allEPSPs smaller than the minimum value given in the abscissa areeliminated. Here, we remove a portion of excitatory synapses on a neuronfrom the weakest ones.



自発発火活動が再現されるノイズ源は要らない！

神経細胞

時間

時間

膜電位

非同期、不規則、低頻度（1-‐2Hz）

膜電位も乱雑に大きく変動

膜電位の強い揺らぎ

興奮性神経細胞抑制性神経細胞

静止膜電位発火閾値

膜電位のUP状態

ゆらぎの機能は何か？

VL5270 [mV], VE50 [mV], VI5280 [mV], respectively. The excit-atory and inhibitory synaptic conductances gE and gI [ms21] normal-ized by the membrane capacitance obey

dgX

dt~{

gX

tszX

j

GX,j

X

sj

d t{sj{dj! "

, X~E,I !2"

where d(t) is the delta function, Gj, dj, sj are the weight, delay andspike timing of synaptic input from the j-th neuron, respectively.The decay constant ts is 2 [ms] and synaptic delays are chosenrandomly between d021 to d011 [ms], where d0 5 2 for excit-atory-to-excitatory connections and d0 5 1 for other connectiontypes. The values are determined from the stability of spontaneous

activity (Methods). Spike threshold is Vthr5 250 [mV] and v is resetto Vr 5 260 mV after spiking. The refractory period is 1 [ms].

The values of Gi for excitatory-to-excitatory connections are dis-tributed such that the amplitude of EPSPs x measured from theresting potential obey a lognormal distribution

p x! "~exp { log x{m! "2

#2s2

$ %&&&&&2pp

sx!3"

on each neuron (Fig. 1a), where the values s51.0 and m-s2 5 log(0.2)well replicate the experimentally observed long-tailed distributionsof EPSP amplitudes33,34. We declined any unrealistic value of Gi thatgives an amplitude larger than 20 [mV] by drawing a new valuefrom the distribution. The resultant amplitude of strongest EPSPwas about 10 [mV] on each neuron. For simplicity, excitatory-to-inhibitory, inhibitory-to-excitatory and inhibitory-to-inhibitorysynapses have uniform values of Gi50.018, 0.002 and 0.0025,respectively. Excitatory-to-excitatory synaptic transmissions fail atan EPSP amplitude-dependent rate of pE 5 a/(a1EPSP), wherea50.1 [mV]34.

We first demonstrate numerically that the long-tailed distributionof EPSP amplitudes achieves aperiodic stochastic resonance for spikesequence on a single neuron receiving random synaptic inputs(Fig. 1b). Stochastic resonance refers to a phenomenon wherein aspecific level of noise enhances the response of a nonlinear system toa weak periodic or aperiodic stimulus35–37, and has been observed inmany physical and biological systems38–45. We vary the average mem-brane potential of the neuron by changing the rate of presynapticspikes at a portion of the weakest excitatory synapses (EPSP ampli-tudes , 3 mV). Interestingly, the cross-correlation coefficients(C.C.) between output spikes and inputs to the strongest synapsesare maximized at a subthreshold membrane potential value about 10[mV] above the resting potential and 10 [mV] below firing threshold(Fig. 1c). At more hyperpolarized levels of the average membranepotential, even an extremely strong EPSP (,10 mV) cannot evoke apostsynaptic spike, and the fidelity of spike transmission is reduced.On the contrary at more depolarized average membrane potentials,the neuron can fire without strong inputs, also degrading the fidelity.

We can express the C.C.s in terms of the conditional probability ofspiking by strong-sparse input, which we can analytically obtainfrom the stochastic differential equations for weak-dense synapses(Methods). The analytic results well explain the optimal neuronalresponse obtained numerically (Fig. 1c). The phenomena can beregarded as stochastic resonance for aperiodic spike inputs36,37. Wefind that the stochastic enhancement of spike transmission is muchweaker in a neuron (Fig. 1c, dashed curve) having Gaussian-distrib-uted EPSP amplitude, which give the same mean and variance ofsynaptic conductances as the lognormal distribution but no tails ofstrong synapses (Supplementary Methods). The results prove theadvantage of long-tailed distributions of EPSP amplitude.

We confirmed the above model’s prediction by performingdynamic clamp recordings from cortical neurons (n514). To mimicsynaptic bombardment with long-tailed distributed EPSP ampli-tudes, we injected the synaptic current given in equation (2) by usingthe same values of excitatory and inhibitory conductances as used inFig. 1c (Supplementary Methods). The rate of random synapticinputs was varied in a low-frequency regime. The physiological resultalso demonstrated the maximization of the fidelity of synaptic trans-mission (Fig. 1d, e).

Now, we ask whether the above stochastic resonance is achievableby the noise generated internally by SSWD recurrent neural net-works. To see this, we conduct numerical simulations of equations(1) and (2) for a network model of 10000 excitatory and 2000inhibitory neurons that are randomly connected with couplingprobabilities of excitatory and inhibitory connections being 0.1 and0.5, respectively. Since the network has a trivial stable state in whichall neurons are in the resting potentials, we briefly apply external

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Figure 1 | Maximizing the fidelity of spike transmission with long-tailedsparse connectivity. (a) Each excitatory neuron has a lognormal amplitudedistribution of EPSPs. The resultant mean and variance of the model are0.89 [mV] and 1.12 [mV2], respectively, whereas those shown in a previousexperiment [1] were 0.77 [mV] and 0.92 [mV2]. Inset is a normal plot of thesame distribution. (b) Schematic illustration of the neuron model withstrong-sparse and weak-dense synaptic inputs. Colors (red, green and blue)indicate inputs to the top three strongest weights. (c) C.C.s between theoutput spike train and input spike trains at the 1st (red), 2nd (green) and3rd (blue) strongest synapses on a neuron are plotted against the meanmembrane potential and the corresponding input firing rate at each synapse.The dashed line and shaded area show the mean and SD of the membranepotential distribution of excitatory neurons shown in Fig. 2f for the SSWDnetwork. Vertical bars represent SEM over different realizations of randominput. The dashed line indicates an analytical curve for the strongest synapseof the long-tailed distribution, while the dot-dashed line is the C.C.s for thestrongest synapse when EPSP amplitudes obey Gaussian distribution. (d)Similar C.C.s obtained by dynamic clamp recordings from a cortical neuron.The color code and vertical bars are the same as in C. (e) The trial-averagedC.C.s for the strongest synapses on n514 neurons.




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多数の弱結合への入力背景ゆらぎ

少数の強結合

膜電位の UP state

理論数値計算

揺らぎがスパイク伝達効率を最適化！

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by Yasuhiro Tsubo

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多数決素子 ...

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大脳皮質自発揺らぎの起源と機能

•  ネットワークの非常に強い不均一性（多数の弱結合と少数の非常に強い結合の共存）が脳の自発揺らぎを生む

•  自発揺らぎがスパイク情報伝達を最適化　　自己組織的確率共鳴

•  文脈依存でノイズ強度をコントロールできる　→　内的なノイズ制御に

　　　　　よる情報処理

稀な強結合

ゆらぎ不規則性

多数の弱結合

正確な情報の伝播

Collaborators Tomoki Fukai (RIKEN)

Yasuhiro Tsubo (Ritsumeikan Univ.) Naoki Hiratani (Univ. Tokyo)

寺前順之介 - Osaka University...observed6,8,9. Firing rate distributions are well fitted by...

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Transcript of 寺前順之介 - Osaka University...observed6,8,9. Firing rate distributions are well fitted by...