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?
• 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)?
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
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
Z
µ(t) = µ0 + µ1 (t)
⇒
t
ν(t) = ν0 + −∞
K(t − u)µ1 (u)du + O(2 )
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
Z
µ(t) = µ0 + µ1 (t)
⇒
t
ν(t) = ν0 + K(t − u)µ1 (u)du + O(2 )
−∞
• Fourier transform: response to sinusoidal inputs
µ1 (ω)
⇒
ν1 (ω) = K̃(ω)µ1 (ω)
Of particular interest: high frequency limit (tells us how fast a neuron instantaneous firing rate can react to
time-dependent inputs)
Noisy input current (mV)
A
60
40
20
0
-20
0
20
40
60
0
20
40
60
0
20
40
60
Spikes
B
40
Firing rate (Hz)
C
30
20
10
0
t (ms)
How to compute the instantaneous firing rate
• Consider a LIF neuron with deterministic + white noise inputs,
τm V̇ = −V + µ(t) + σ(t)η(t)
• P (V, t) is described by Fokker-Planck equation
σ 2 (t) ∂ 2 P (V, t)
∂
∂P (V, t)
=
+
[(V − µ(t))P (V, t)]
τm
2
∂t
2
∂V
∂V
• 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
2ν(t)τm
(Vt , t) = − 2
∂V
σ (t)
– At reset potential Vr : what comes out at Vt must come back at Vr
P (Vr− , t) = P (Vr+ , t),
∂P
∂P
2ν(t)τm
(Vr− , t) −
(Vr+ , t) = − 2
∂V
∂V
σ (t)
How to compute the instantaneous firing rate
• Consider a LIF neuron with deterministic + white noise inputs,
τm V̇ = −V + µ(t) + σ(t)η(t)
• P (V, t) is described by Fokker-Planck equation
σ 2 (t) ∂ 2 P (V, t)
∂
∂P (V, t)
=
+
[(V − µ(t))P (V, t)]
τm
2
∂t
2
∂V
∂V
• 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
2ν(t)τm
(Vt , t) = − 2
∂V
σ (t)
– At reset potential Vr : what comes out at Vt must come back at Vr
P (Vr− , t) = P (Vr+ , t),
⇒ Time independent solution P0 (V ), ν0 ;
⇒ Linear response P1 (ω, V ), ν1 (ω).
∂P
∂P
2ν(t)τm
(Vr− , t) −
(Vr+ , t) = − 2
∂V
∂V
σ (t)
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
∼
ν
√0
σ ωτm
ν0
τs
σ
τm
p
(white noise)
(colored noise)
• Phase lag at high f
(
Lag
∼
π
4
(white noise)
0
(colored noise)
Brunel and Hakim 1999; Brunel et al 2001; Lindner and Schimansky-Geier 2001; Fourcaud and Brunel 2002
Spike generation: exponential integrate-and-fire
• EIF: exponential integrate-and-fire neuron
dV
C
dt
= −gL (V − VL ) + ψ(V ) + Isyn (t)
V − VT
ψ(V ) = gL ∆T exp
∆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-Trocmé et al 2003
EIF vs cortical pyramidal cells - I-V curve
dV
dt
F (V )
=
F (V ) +
=
1
τm
Iin (t)
C
Em − V + ∆T exp
V − VT
∆T
Badel et al 2008
EIF - dynamical response
ν0
∆T ωτm
φ(ω) ∼ π/2
|ν1 (ω)|
∼
• The whole function ν1 (ω) can
be computed numerically using a
method introduced by Richardson
10
Fourcaud-Trocmé et al 2003, Richardson 2007
0
-2
10
-45
o
-90
o
ν0=33Hz, σ=12.7mV
ν0=33Hz, σ=12.7mV
-3
10
0
10
1
2
10
10
3
10
0
1
2
10
10
B-3
A-3
10
10
10
Input frequency, f (Hz)
Input frequency, f (Hz)
high rate,
low noise
-1
0
-2
10
(2007)
high rate, high noise
-1
Phase shift, φ
10
Phase shift, φ
• In the high frequency limit,
Modulation amplitude, ν1/Ι1
frequency limits
B-2
A-2
Modulation amplitude, ν1/Ι1
• ν1 (ω) can be computed in low/high
ν0=38Hz, σ=1.6mV
-3
10
0
10
1
ν0
-45
o
-90
o
ν0=38Hz, σ=1.6mV
2ν0
2
10
Input frequency, f (Hz)
10
3
10
0
10
1
2
10
Input frequency, f (Hz)
10
Summary of high frequency behaviors
Model
Exponent
Phase lag
α
φ(f → ∞)
LIF, colored noise
0
0
LIF, white noise
0.5
45◦
EIF
1
90◦
QIF
2
180◦
Response of cortical pyramidal cells
Boucsein et al 2009
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)?
Two compartmental model of a Purkinje cell
• Can be fitted by two-compartment model
dVs
dt
dVd
Cd
dt
Cs
=
−gs Vs + gj (Vd − Vs ) + Is
=
−gd Vd + gj (Vs − Vd ) + Id
Two compartmental model of a Purkinje cell
• or equivalently
dVs
dt
dW
τd
dt
τs
=
−Vs + γW + Is
=
−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
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)
Linear firing rate response: low frequency limit
• When ωτs 1, the soma is driven instantaneously by dendritic + injected currents,
V = γW + I0 + I1 eiωt .
• Dynamics for the dendritic compartment becomes
√
τd Ẇ = −(1 − γ)W + I0 + I1 eiωt + σ τd η(t)
with spikes occurring when W
= (Vt − I0 − I1 eiωt )/γ .
• To recover a LIF model with constant threshold one can define
Vt − I0 − I1 eiωt
X=W −
γ
and obtain
τd
VT
I0
I1
σ
iωt
Ẋ = −X− +
+
(1+iωτd )e + √
1−γ
γ γ(1 − γ) γ(1 − γ)
1−γ
⇒ ν1 ∼ ν1,LIF (1 + iωτd )
⇒ Amplitude of the response increases as a function of frequency
r
τd
η(t)
1−γ
High frequency limit
• At high frequency, response should be dominated by the spike-generating current as in
the standard EIF. This should give
ν1HF =
ν0
∆T iωτs
2/3
• Setting |ν1LF | = |ν1HF | we get
ω? =
γσ
√
2∆T
1
1/3 2/3
τd τs
Low and high frequency asymptotics vs simulations
Response of real Purkinje cells
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
Implications for network oscillations
φI,cell
• At the onset of network oscillations
neuron rI(t)
S(ω)Jν1 (ω) = 1
–
-π
II(t)
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 input current)
φI,syn
synapse
sI(t)
neuron rI(t)
0
5
10
15
time [ms]
20
25
Implications for network oscillations
φI,cell
• At the onset of network oscillations
neuron rI(t)
S(ω)Jν1 (ω) = 1
–
-π
II(t)
S(ω) = AS (ω) exp(iΦS (ω)) = synaptic filtering
(how PSCs respond to oscillatory pre-synaptic rate)
φI,syn
synapse
sI(t)
–
J = total synaptic strength
–
ν1 (ω) = AN (ω) exp(iΦN (ω)) = neuronal filtering:
(how instantaneous firing rate of single cell responds to oscillatory in-
neuron rI(t)
0
put current)
• 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)
5
10
15
time [ms]
20
25
Implications for network oscillations
φI,cell
• At the onset of network oscillations
neuron rI(t)
S(ω)Jν1 (ω) = 1
–
-π
II(t)
S(ω) = AS (ω) exp(iΦS (ω)) = synaptic filtering
(how PSCs respond to oscillatory pre-synaptic rate)
φI,syn
synapse
sI(t)
–
J = total synaptic strength
–
ν1 (ω) = AN (ω) exp(iΦN (ω)) = neuronal filtering:
(how instantaneous firing rate of single cell responds to oscillatory in-
neuron rI(t)
0
put current)
• 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)
5
10
15
time [ms]
20
25
Example: oscillations in Purkinje cell network
• Multi-unit/LFP oscillate at about 200Hz
while single cells fire in average at
about 40Hz;
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
Reproducing quantitatively cerebellar fast oscillations
Network with ‘realistic’ parameters
• 200 two-compartmental PCs with parameters 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
Acknowledgements
LIFs : Vincent Hakim; Nicolas Fourcaud-Trocmé; Frances Chance; Larry Abbott
EIFs : Nicolas Fourcaud-Trocmé; Carl van Vreeswijk; David Hansel
GIFs : Magnus Richardson; Vincent Hakim
Two compartment models : Srdjan Ostojic; Vincent Hakim; German Szapiro; Boris
Barbour