Stochasticdynamicsofspikingneuronmodels

andimplicationsfornetworkdynamics

Nicolas BrunelThequestion

• What is the input output relationship of single neurons?Thequestion

• What is the input output relationship of single neurons?

• Given a (time dependent) input, and a given statistics of the noise, what is the instantaneous ﬁring rate of a

neuron?

I (t) = μ(t)+ Noise ⇒ ν(t)?synThequestion

• What is the input output relationship of single neurons?

• Given a (time dependent) input, and a given statistics of the noise, what is the instantaneous ﬁring rate of a

neuron?

I (t) = μ(t)+ Noise ⇒ ν(t)?syn

• Simplest case: response to time independent input

μ(t) = μ ⇒ ν(t) = ν0 0Thequestion

• What is the input output relationship of single neurons?

• Given a (time dependent) input, and a given statistics of the noise, what is the instantaneous ﬁring rate of a

neuron?

I (t) = μ(t)+ Noise ⇒ ν(t)?syn

• Simplest case: response to time independent input

μ(t) = μ ⇒ ν(t) = ν0 0

• Next step: response to time dependent inputs

Z t

2μ(t) = μ + μ (t) ⇒ ν(t) = ν + K(t−u)μ (u)du+O( )0 1 0 1

−∞Thequestion

• What is the input output relationship of single neurons?

• Given a (time dependent) input, and a given statistics of the noise, what is the instantaneous ﬁring rate of a

neuron?

I (t) = μ(t)+ Noise ⇒ ν(t)?syn

• Simplest case: response to time independent input

μ(t) = μ ⇒ ν(t) = ν0 0

• Next step: response to time dependent inputs

Z t

2μ(t) = μ + μ (t) ⇒ ν(t) = ν + K(t−u)μ (u)du+O( )0 1 0 1

−∞

• Fourier transform: response to sinusoidal inputs

˜μ (ω) ⇒ ν (ω) = K(ω)μ (ω)1 1 1

Of particular interest: high frequency limit (tells us how fast a neuron instantaneous ﬁring rate can react to

time dependent inputs)60A

40

20

0

-20

0 20 40 60

B

0 20 40 60

40

C

30

20

10

0

0 20 40 60

t (ms)

Firing rate (Hz) Spikes Noisy input current (mV)Howtocomputetheinstantaneousﬁringrate

• Consider a LIF neuron with deterministic+ white noise inputs,

˙τ V =−V +μ(t)+σ(t)η(t)m

• P(V,t) is described by Fokker Planck equation

2 2∂P(V,t) σ (t) ∂ P(V,t) ∂

τ = + [(V −μ(t))P(V,t)]m 2∂t 2 ∂V ∂V

• Boundary conditions⇒ linksP and instantaneous ﬁring probabilityν

– At thresholdV : absorbing b.c. + probability ﬂux atV = ﬁring probabilityν(t):t t

∂P 2ν(t)τm

P(V ,t) = 0, (V ,t) =−t t 2∂V σ (t)

– At reset potentialV : what comes out atV must come back atVr t r

∂P ∂P 2ν(t)τm− + − +P(V ,t) = P(V ,t), (V ,t)− (V ,t) =−r r r r 2∂V ∂V σ (t)Howtocomputetheinstantaneousﬁringrate

• Consider a LIF neuron with deterministic+ white noise inputs,

˙τ V =−V +μ(t)+σ(t)η(t)m

• P(V,t) is described by Fokker Planck equation

2 2∂P(V,t) σ (t) ∂ P(V,t) ∂

τ = + [(V −μ(t))P(V,t)]m 2∂t 2 ∂V ∂V

• Boundary conditions⇒ linksP and instantaneous ﬁring probabilityν

– At thresholdV : absorbing b.c. + probability ﬂux atV = ﬁring probabilityν(t):t t

∂P 2ν(t)τm

P(V ,t) = 0, (V ,t) =−t t 2∂V σ (t)

– At reset potentialV : what comes out atV must come back atVr t r

∂P ∂P 2ν(t)τm− + − +P(V ,t) = P(V ,t), (V ,t)− (V ,t) =−r r r r 2∂V ∂V σ (t)

⇒ Time independent solutionP (V),ν ;0 0

⇒ Linear responseP (ω,V),ν (ω).1 1LIFmodel

• ν (ω) can be computed analytically for all ω in the1

case of white noise; in low/high frequency limits in the

case of colored noise withτ τn m

• Resonances at

f = nν0

for high rates and low noise;

• Attenuation at highf

(

ν0√ (white noise)σ ωτmGain∼ p

ν τ0 s (colored noise)

σ τm

• Phase lag at highf

(

π

(white noise)4Lag∼

0 (colored noise)

Brunel and Hakim 1999; Brunel et al 2001; Lindner and Schimansky Geier 2001; Fourcaud and Brunel 2002