SLIDE 1
The question
- What is the input-output relationship of single neurons?
Stochastic dynamics of spiking neuron models and implications for - - PowerPoint PPT Presentation
Stochastic dynamics of spiking neuron models and implications for - - PowerPoint PPT Presentation
Stochastic dynamics of spiking neuron models and implications for network dynamics Nicolas Brunel The question What is the input-output relationship of single neurons? The question What is the input-output relationship of single neurons?
SLIDE 2
SLIDE 3
The question
- 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?
Isyn(t) = µ(t) + Noise ⇒ ν(t)?
SLIDE 4
The question
- 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?
Isyn(t) = µ(t) + Noise ⇒ ν(t)?
- Simplest case: response to time-independent input
µ(t) = µ0 ⇒ ν(t) = ν0
SLIDE 5
The question
- 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?
Isyn(t) = µ(t) + Noise ⇒ ν(t)?
- Simplest case: response to time-independent input
µ(t) = µ0 ⇒ ν(t) = ν0
- Next step: response to time-dependent inputs
µ(t) = µ0 + ǫµ1(t) ⇒ ν(t) = ν0 + ǫ
t
−∞
K(t − u)µ1(u)du + O(ǫ2)
SLIDE 6
The question
- 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?
Isyn(t) = µ(t) + Noise ⇒ ν(t)?
- Simplest case: response to time-independent input
µ(t) = µ0 ⇒ ν(t) = ν0
- Next step: response to time-dependent inputs
µ(t) = µ0 + ǫµ1(t) ⇒ ν(t) = ν0 + ǫ
t
−∞
K(t − u)µ1(u)du + O(ǫ2)
- Fourier transform: response to sinusoidal inputs
µ1(ω) ⇒ ν1(ω) = ˜ K(ω)µ1(ω)
Of particular interest: high frequency limit (tells us how fast a neuron instantaneous firing rate can react to time-dependent inputs)
SLIDE 7
20 40 60
- 20
20 40 60
Noisy input current (mV)
20 40 60
Spikes
20 40 60
t (ms)
10 20 30 40
Firing rate (Hz)
A B C
SLIDE 8
How to compute the instantaneous firing rate
- Consider a LIF neuron with deterministic + white noise inputs,
τm ˙ V = −V + µ(t) + σ(t)η(t)
- P(V, t) is described by Fokker-Planck equation
τm ∂P(V, t) ∂t = σ2(t) 2 ∂2P(V, t) ∂V 2 + ∂ ∂V [(V − µ(t))P(V, t)]
- Boundary conditions ⇒ links P and instantaneous firing probability ν
– At threshold Vt: absorbing b.c. + probability flux at Vt = firing probability ν(t):
P(Vt, t) = 0, ∂P ∂V (Vt, t) = − 2ν(t)τm σ2(t)
– At reset potential Vr: what comes out at Vt must come back at Vr
P(V −
r , t) = P(V + r , t),
∂P ∂V (V −
r , t) − ∂P
∂V (V +
r , t) = − 2ν(t)τm
σ2(t)
SLIDE 9
How to compute the instantaneous firing rate
- Consider a LIF neuron with deterministic + white noise inputs,
τm ˙ V = −V + µ(t) + σ(t)η(t)
- P(V, t) is described by Fokker-Planck equation
τm ∂P(V, t) ∂t = σ2(t) 2 ∂2P(V, t) ∂V 2 + ∂ ∂V [(V − µ(t))P(V, t)]
- Boundary conditions ⇒ links P and instantaneous firing probability ν
– At threshold Vt: absorbing b.c. + probability flux at Vt = firing probability ν(t):
P(Vt, t) = 0, ∂P ∂V (Vt, t) = − 2ν(t)τm σ2(t)
– At reset potential Vr: what comes out at Vt must come back at Vr
P(V −
r , t) = P(V + r , t),
∂P ∂V (V −
r , t) − ∂P
∂V (V +
r , t) = − 2ν(t)τm
σ2(t) ⇒ Time independent solution P0(V ), ν0; ⇒ Linear response P1(ω, V ), ν1(ω).
SLIDE 10
LIF model
- ν1(ω) can be computed analytically for all ω in the
case of white noise; in low/high frequency limits in the case of colored noise with τn ≪ τm
- Resonances at
f = nν0
for high rates and low noise;
- Attenuation at high f
Gain ∼
- ν0
σ√ωτm
(white noise)
ν0 σ
τs
τm
(colored noise)
- Phase lag at high f
Lag ∼
- π
4
(white noise) (colored noise) Brunel and Hakim 1999; Brunel et al 2001; Lindner and Schimansky-Geier 2001; Fourcaud and Brunel 2002
SLIDE 11
Spike generation: exponential integrate-and-fire
- EIF: exponential integrate-and-fire neuron
C dV dt = −gL(V − VL) + ψ(V ) + Isyn(t) ψ(V ) = gL∆T exp V − VT ∆T
- Captures quantitatively very well the dynamics of a
Hodgkin-Huxley-type neuron (Wang-Buszaki).
- This is because activation curve of sodium currents
can be well fitted by an exponential close to firing threshold.
Fourcaud-Trocm´ e et al 2003
SLIDE 12
EIF vs cortical pyramidal cells - I-V curve
dV dt = F(V ) + Iin(t) C F(V ) = 1 τm
- Em − V + ∆T exp
V − VT
∆T
- Badel et al 2008
SLIDE 13
EIF - dynamical response
- ν1(ω) can be computed in low/high
frequency limits
- In the high frequency limit,
|ν1(ω)| ∼ ν0 ∆T ωτm φ(ω) ∼ π/2
- The
whole function ν1(ω) can be computed numerically using a method introduced by Richardson (2007)
10 10
1
10
2
10
3
Input frequency, f (Hz)
10
- 3
10
- 2
10
- 1
Modulation amplitude, ν1/Ι1
ν0=33Hz, σ=12.7mV
A-2
high rate, high noise
10 10
1
10
2
10
Input frequency, f (Hz)
- 90
- 45
- Phase shift, φ
ν0=33Hz, σ=12.7mV
B-2
10 10
1
10
2
10
3
Input frequency, f (Hz)
10
- 3
10
- 2
10
- 1
Modulation amplitude, ν1/Ι1
ν0=38Hz, σ=1.6mV
A-3
high rate, low noise ν0 2ν0
10 10
1
10
2
10
Input frequency, f (Hz)
- 90
- 45
- Phase shift, φ
ν0=38Hz, σ=1.6mV
B-3
Fourcaud-Trocm´ e et al 2003, Richardson 2007
SLIDE 14
Summary of high frequency behaviors
Model Exponent Phase lag
α φ(f → ∞)
LIF, colored noise LIF, white noise
0.5 45◦
EIF
1 90◦
QIF
2 180◦
SLIDE 15
Response of cortical pyramidal cells
Boucsein et al 2009
SLIDE 16
Two variable models
A second variable can be coupled to voltage to:
- Include the effects of ionic currents which are activated below threshold (possibly
leading to sub-threshold resonance): RF, GIF, aQIF , aEIF
⇒ Linear response can be computed as an expansion in ratio of time scales
(Richardson et al 2003, Brunel et al 2003)
- Include firing rate adaptation: aLIF
, aQIF , aEIF
- Include the effects of currents leading to bursting: IFB, aQIF
, aEIF
- Include a second compartment (soma + dendrite)
⇒ What are the effects of the second variable on the firing rate dynamics (linear
response)?
SLIDE 17
Two compartmental model of a Purkinje cell
- Can be fitted by two-compartment model
Cs dVs dt = −gsVs + gj(Vd − Vs) + Is Cd dVd dt = −gdVd + gj(Vs − Vd) + Id
SLIDE 18
Two compartmental model of a Purkinje cell
- or equivalently
τs dVs dt = −Vs + γW + Is τd dW dt = −W + V + Id
- Fits give τs ≪ 1ms, τd ∼ 5ms (even though Cs/gs = Cd/gd ∼ 50ms), γ ∼ 0.9
This is due to As ≪ Ad, gs ≪ gd ≪ gj
SLIDE 19
Linear firing rate response of 2C model
- 2C model with exponential spike-generating current (2C-EIF)
- Oscillatory input injected at the soma, noise at the dendrite
- Very similar results obtained with multi-compartmental model based on a reconstructed
PC with HH-type currents (Khaliq-Raman model)
SLIDE 20
Linear firing rate response: low frequency limit
- When ωτs ≪ 1, the soma is driven instantaneously by dendritic + injected currents,
V = γW + I0 + I1eiωt.
- Dynamics for the dendritic compartment becomes
τd ˙ W = −(1 − γ)W + I0 + I1eiωt + σ√τdη(t)
with spikes occurring when W = (Vt − I0 − I1eiωt)/γ.
- To recover a LIF model with constant threshold one can define
X = W − Vt − I0 − I1eiωt γ
and obtain
τd 1 − γ ˙ X = −X−VT γ + I0 γ(1 − γ)+ I1 γ(1 − γ)(1+iωτd)eiωt+ σ √1 − γ
- τd
1 − γ η(t) ⇒ ν1 ∼ ν1,LIF (1 + iωτd) ⇒ Amplitude of the response increases as a function of frequency
SLIDE 21
High frequency limit
- At high frequency, response should be dominated by the spike-generating current as in
the standard EIF . This should give
νHF
1
= ν0 ∆T iωτs
- Setting |νLF
1
| = |νHF
1
| we get ω⋆ =
- γσ
√ 2∆T 2/3 1 τ 1/3
d
τ 2/3
s
SLIDE 22
Low and high frequency asymptotics vs simulations
SLIDE 23
Response of real Purkinje cells
SLIDE 24
Summary: single cell dynamics
- High frequency behavior: controlled by spike generation dynamics
- Low frequency behavior: determined by f-I curve
- Intermediate frequencies: resonances can be due to various mechanisms:
– Low noise regime: resonances at firing rate/harmonics (all models) – Subthreshold resonance can lead to firing rate resonance if noise is strong enough (Richardson et al 2003) – Specific spatial geometry of PC: high frequency resonance in response to somatic inputs
- Linear response gives a good approximation of the dynamics as long as |ν1| < ν0
- Cortical pyramidal cell response qualitatively well described by EIF;
- Cerebellar Purkinje cell response qualitatively well described by 2-C EIF
SLIDE 25
Implications for network oscillations
- At the onset of network oscillations
S(ω)Jν1(ω) = 1
– S(ω) = AS(ω) exp(iΦS(ω)) = synaptic filtering (how PSCs respond to oscillatory pre-synaptic rate) – J = total synaptic strength – ν1(ω) = AN(ω) exp(iΦN(ω)) = neuronal filtering: (how instantaneous firing rate of single cell responds to oscillatory in- put current)
neuron neuron synapse
rI(t) II(t) sI(t) 5 10 15 20 25 time [ms] rI(t) φI,syn
- π
φI,cell
SLIDE 26
Implications for network oscillations
- At the onset of network oscillations
S(ω)Jν1(ω) = 1
– S(ω) = AS(ω) exp(iΦS(ω)) = synaptic filtering (how PSCs respond to oscillatory pre-synaptic rate) – J = total synaptic strength – ν1(ω) = AN(ω) exp(iΦN(ω)) = neuronal filtering: (how instantaneous firing rate of single cell responds to oscillatory in- put current)
neuron neuron synapse
rI(t) II(t) sI(t) 5 10 15 20 25 time [ms] rI(t) φI,syn
- π
φI,cell
- Phase ⇒ frequency(ies) ω of instability (ies):
ΦN(ω) + ΦS(ω) = 2kπ, k = 0, 1, . . .
(excitatory network)
ΦN(ω) + ΦS(ω) = (2k + 1)π k = 0, 1, . . .
(inhibitory network)
SLIDE 27
Implications for network oscillations
- At the onset of network oscillations
S(ω)Jν1(ω) = 1
– S(ω) = AS(ω) exp(iΦS(ω)) = synaptic filtering (how PSCs respond to oscillatory pre-synaptic rate) – J = total synaptic strength – ν1(ω) = AN(ω) exp(iΦN(ω)) = neuronal filtering: (how instantaneous firing rate of single cell responds to oscillatory in- put current)
neuron neuron synapse
rI(t) II(t) sI(t) 5 10 15 20 25 time [ms] rI(t) φI,syn
- π
φI,cell
- Phase ⇒ frequency(ies) ω of instability (ies):
ΦN(ω) + ΦS(ω) = 2kπ, k = 0, 1, . . .
(excitatory network)
ΦN(ω) + ΦS(ω) = (2k + 1)π k = 0, 1, . . .
(inhibitory network)
- Amplitude ⇒ associated critical total coupling strength
|J| = 1 AN(ω)AS(ω)
SLIDE 28
SLIDE 29
Example: oscillations in Purkinje cell network
- Multi-unit/LFP oscillate at about 200Hz
while single cells fire in average at about 40Hz;
SLIDE 30
Example: oscillations in Purkinje cell network
- Multi-unit/LFP oscillate at about 200Hz
while single cells fire in average at about 40Hz;
de Solages et al 2008
SLIDE 31
Reproducing quantitatively cerebellar fast oscillations
Network with ‘realistic’ parameters
- 200 two-compartmental PCs with parame-
ters fitted from data; GABAergic inputs on soma, noise (AMPA) on dendrite;
- Randomly connected, 40 connections per
cell;
- Connections with realistic conductances (∼
1 nS) and kinetics (0.5ms rise, 3ms decay,
taken from slice recordings of IPSCs)
- Resonance induced by spatial geometry
necessary to account for oscillations, given the relatively weak coupling. de Solages et al 2008
SLIDE 32