 # Stochastic dynamics of spiking neuron models

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32 Pages

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Stochastic dynamics of spiking neuron models and implications for network dynamics Nicolas Brunel

• stochastic dynamics

• neuron models

• neuron

• independent input

• input- output relationship

• dependent

• syn noise

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##### Dependency

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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

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