university of copenhagen
Neurons (nerve cells)
Faculty of Science
The Morris Lecar neuron model gives
rise to the Ornstein-Uhlenbeck leaky
integrate-and-fire model
=⇒
Susanne Ditlevsen
Cindy Greenwood
Stochastic Models in Neuroscience
Marseille 2009
January 18, 2010
Slide 1/32
university of copenhagen
department of mathematical sciences
The model
Xt : membrane potential at time t after a spike
x0 : initial voltage (the reset value following a spike)
An action potential (a spike) is produced when the
membrane voltage Xt exceeds a firing threshold
X(t)
dXt = µ(Xt ) dt + σ(Xt ) dW (t) ; X0 = x0
S
S(t) = S > X (0) = x0
After firing the process is reset to x0 . The interspike
interval T is identified with the first-passage time of the
threshold,
T = inf{t > 0 : Xt ≥ S}.
Slide 3/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010
x0
T
T
time
•
university of copenhagen
department of mathematical sciences
university of copenhagen
department of mathematical sciences
Two commonly used Leaky
Integrate-and-Fire neuron models (I)
Two commonly used Leaky
Integrate-and-Fire neuron models (II)
The Ornstein-Uhlenbeck process:
Xt
dXt =
− + µ dt + σ dWt ; X0 = x0 .
τ
The Feller process (also CIR or square root process):
p
Xt − VI
d(Xt − VI ) =
−
+ µ dt + σ Xt − VI dWt ;
τ
X0 = x0 ≥ VI .
where
Xt : membrane potential at time t after a spike
τ : membrane time constant, reflects spontaneous
voltage decay (> 0)
µ: characterizes constant neuronal input
σ: characterizes erratic neuronal input
x0 : initial voltage (the reset value following a spike)
where
VI : inhibitory reversal potential
and
1
Slide 5/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010
university of copenhagen
2µ ≥ σ 2
Slide 6/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010
department of mathematical sciences
Tem
OU and square-root process
S2
S1
µτ
VI
time
Slide 7/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010
From Berg, Ditlevsen and Hounsgaard (2008)
university of copenhagen
department of mathematical sciences
1.0
The Hodgkin-Huxley model
0.6
0.4
Monoexponential: τ = 43.1 ms
0.2
autocorrelation
0.8
Hodgkin and Huxley (1952). Explains the ionic
mechanisms underlying the initiation and propagation of
action potentials in the squid giant axon.
Nobel Prize in Medicine in 1963.
0.0
Biexponential: τ1 = 12.1 ms; τ2 = 53.8 ms
0
10
20
30
40
50
time in ms
university of copenhagen
Slide 10/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010
department of mathematical sciences
with the auxiliary functions given by
v − V1
1
1 + tanh
m∞ (v ) =
2
V2
1
v − V3
v
α(v ) =
φ cosh
1 + tanh
2
2V4
1
v − V3
v
β(v ) =
φ cosh
1 − tanh
2
2V4
Slide 11/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010
− V3
V4
− V3
V4
0
−20
dWt = (α(Vt )(1 − Wt ) − β(Vt )Wt ) dt
−40
1
(−gCa m∞ (Vt )(Vt − VCa ) − gK Wt (Vt − VK )
C
−gL (Vt − VL ) + I )dt
membrane voltage, V(t)
dVt =
20
The Morris Lecar model
0
200
400
600
time
800
1000
university of copenhagen
Chaos, Vol. 14, No. 3, 2004
department of mathematical sciences
T. Tateno and K. Pakdaman
Bifurcation diagram, Morris Lecar model
0.4
0.3
0.2
normalized conductance, W(t)
0.5
514
0.1
●
From Tateno and Pakdaman (2004)
−40
−20
0
20
40
membrane voltage, V(t)
Slide 14/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010
Chaos, Vol. 14, No. 3, 2004
university of copenhagen
Random dynamics of the ML neural model
519
department of mathematical sciences
The stochastic Morris Lecar model
Where to put the noise?
dVt =
1
(−gCa m∞ (Vt )(Vt − VCa ) − gK Wt (Vt − VK )
C
−gL (Vt − VL ) + I )dt + σ1 (Vt , Wt )dBt
dWt
FIG. 2. 共A兲 Bifurcation diagram of the class I ML model. The thick curves
stand for stable solutions and the thin curve for unstable ones. Repetitive
= (α(Vfiring
t )(1 − Wt ) − β(Vt )Wt ) dt
occurs for the critical current I c ⯝40 ␮ A/cm2 , where the stable rest
state and saddle coalesce. Branches labeled ‘‘osc’’ respectively represent
+σ2 (V
maximum
and
t , Wt )dB
t minimum values of v in each periodic solution. Abscissa:
stimulus current intensity I ( ␮ A/cm2 ), ordinate: membrane voltage v 共mV兲.
共B兲 Frequency of stable periodic solutions versus I. With increasing the
current intensity, the frequency is monotonically increasing from zero frequency at the critical current.
FIG. 3. 共A兲 Bifurcation diagram of the class II ML model. The system
possesses a unique equilibrium point for all values of I in the parameter
region shown here. The thick curve stands for stable equilibrium points for
I⬍I H⯝93.86 ( ␮ A/cm2 ) and the thin curve for unstable ones. Amplitude of
stable periodic solutions 共labeled ‘‘osc’’兲 is indicated by maximum and
minimum values of v over one period for I⬎I DC⯝88.29 ( ␮ A/cm2 ). Stability of the periodic solutions is also shown as filled 共stable兲 and unfilled
共unstable兲 circles. Abscissa: stimulus current intensity I ( ␮ A/cm2 ), ordinate: membrane voltage v 共mV兲. 共B兲 Frequency of stable periodic solutions
versus I. Frequency is monotone over the I parameter range of periodic
solutions and the minimum firing frequency has a nonzero value.
FIG. 6. Stationary distributions of the class II ML model in the bistable regime. The initial conditions were on the equilibrium point 共column panels A兲 and
the stable limit cycle 共column panels B兲 for three different values of noise intensity in row panels 1, 2, and 3. In order to make small changes visible the
logarithmic scale (log(1⫹z)) on the vertical axis was used. Parameters: 共A兲 and 共B兲 I⫽88.3 ( ␮ A/cm2 ). 共1兲 ␴ 0 ⫽0.0001, 共2兲 ␴ 0 ⫽0.3, and 共3兲 ␴ 0 ⫽0.8.
Voltage noise
From Tateno and Pakdaman (2004)
Slide 15/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010
to display distinct maxima, one associated with the peak at
spent by the system in tight vicinities of the equilibrium and
university of copenhagen
department of mathematical sciences
university of copenhagen
department of mathematical sciences
The stochastic Morris Lecar model
The stochastic Morris Lecar model
Channel noise
Wt can be interpreted as a probability and should stay
between 0 and 1. For one-dimensional diffusions
dVt
1
=
(−gCa m∞ (Vt )(Vt − VCa ) − gK Wt (Vt − VK )
C
−gL (Vt − VL ) + I )dt
dWt = (α(Vt )(1 − Wt ) − β(Vt )Wt ) dt + σ(Vt , Wt )dBt
dXt = b(Xt )dt + σ(Xt )dWt
it is easy to find conditions such that boundaries are not
hit by use of the scale measure. Density:
Z x
2b(y )
s(x) = exp −
dy , x ∈ (l, r )
2
x ∗ σ (y )
for some x ∗ ∈ (l, r ). Density of speed measure:
How should σ(Vt , Wt ) look?
m(x) =
Slide 17/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010
university of copenhagen
department of mathematical sciences
1
,
σ 2 (x)s(x)
x ∈ (l, r )
Slide 18/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010
university of copenhagen
department of mathematical sciences
The stochastic Morris Lecar model
The stochastic Morris Lecar model
If
Back to business... We look at
Z
x∗
Z
r
s(y )dy = ∞
s(y )dy =
x∗
l
then the boundaries l and r are non-attracting. If
moreover
Z r
M=
m(y )dy < ∞
l
then X is ergodic with invariant measure
µ(x) = m(x)/M. In particular,
D
Xt → µ as t → ∞.
If X0 ∼ µ, then X is stationary and Xt ∼ µ for all t ≥ 0.
Slide 19/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010
dWt = (α(Vt )(1 − Wt ) − β(Vt )Wt ) dt + σ2 (Vt , Wt )dBt
Consider Vt fixed, then for Wt to stay between 0 and 1,
first of all we need the noise to go to zero when W
approaches the boundaries. Natural choice is a Jacobi
diffusion
p
dWt = −θ (Wt − µ) dt + σ 2θWt (1 − Wt )dBt
where σ 2 ≤ µ and σ 2 ≤ 1 − µ.
The invariant distribution is a Beta-distribution with
parameters σ 2 /µ and σ 2 /(1 − µ).
Slide 20/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010
university of copenhagen
department of mathematical sciences
university of copenhagen
The stochastic Morris Lecar model
department of mathematical sciences
The stochastic Morris Lecar model
In our case we have
θ = α(Vt ) + β(Vt ),
µ=
We end up with the model
α(Vt )
α(Vt ) + β(Vt )
Requirements translates to
dVt =
α(Vt )
α(Vt ) + β(Vt )
β(Vt )
≤
α(Vt ) + β(Vt )
σ2 ≤
σ2
1
(−gCa m∞ (Vt )(Vt − VCa ) − gK Wt (Vt − VK )
C
−gL (Vt − VL ) + I )dt
dWt = (α(Vt )(1 − Wt ) − β(Vt )Wt ) dt
Fulfilled if
2α(Vt )β(Vt )Wt (1 − Wt )dBt
where σ ≤ 1. Now Vt is not fixed, but still okay...
when 0 < α(Vt ), β(Vt ) < 1.
0.4
0.3
normalized conductance, W(t)
0.5
0.4
0.3
0.2
normalized conductance, W(t)
0.4
0.3
0.5
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0.5
Slide 21/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010
0.2
0.1
●
0.1
●
−40
−20
0
membrane voltage, V(t)
●
0.1
normalized conductance, W(t)
p
0.2
σ2
+σ
α(Vt )β(Vt )
≤
α(Vt ) + β(Vt )
−40
−20
0
membrane voltage, V(t)
20
40
20
40
−40
−20
0
membrane voltage, V(t)
20
40
normalized conductance, W(t)
0.4
0.2
0.2
0.15
membrane voltage, V(t)
membrane voltage, V(t)
membrane voltage, V(t)
membrane voltage, V(t)
−50
−50
0
2000
4000
0
2000
time
department of mathematical sciences
Linearization around the equilibrium
point
Vt − Veq
Wt − Weq
, Yt =
Veq
Weq
Small noise: the dynamics concentrate around the
equilibrium point (x, y ) = (0, 0). Linear approximation:
Xt
Xt
d
≈ M
dt + GdBt
Yt
Yt
where
M =
0
2000
∂f ∗
∂y
∂g ∗
∂y
!
=
0.026
0.112
−0.069 −0.045
4000
0
2000
time
σ = 0.2
university of copenhagen
∂f ∗
∂x
∂g ∗
∂x
4000
time
σ = 0.1
Xt =
−50
−30
0
0
0
−20
normalized conductance, W(t)
0.5
normalized conductance, W(t)
0.4
normalized conductance, W(t)
σ = 0.5
university of copenhagen
σ=1
department of mathematical sciences
Deterministic approximation
Xt
b cos ωt
b sin ωt −λt
−λt
≈ C1
e
+ C2
e
Yt
− sin ωt
cos ωt
where
−λ ± ωi = −0.0094 ± 0.09i
are (nearly) the eigenvalues of M.
Note typical time scales of the system:
λt and ωt and λ ω.
(x,y )=(0,0)
Slide 27/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010
4000
time
Slide 28/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010
university of copenhagen
department of mathematical sciences
Deterministic approximation
Xt
b cos ωt
b sin ωt
≈ A(T )
+ B(T )
Yt
− sin ωt
cos ωt
exact
approx
where A(T ) and B(T ) incorporate the slow decay on the
time scale T = λt and the stochastic component:
dA
f1 (A, B)
dξ1 (T )
=
dT + Σ
dB
f2 (A, B)
dξ2 (T )
0.12
0.14
department of mathematical sciences
Stochastic approximation
normalized conductance, W(t)
−25
membrane voltage, V(t)
After some calculations:
σ
dA
−A
1 0
dξ1 (T )
.
=
dT + √
dB
−B
dξ2 (T )
2 λ 0 1
−30
0
university of copenhagen
500
1000
time
Slide 29/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010
Slide 30/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010
university of copenhagen
department of mathematical sciences
Conclusions
0.6
0.4
Y(t)
0.2
z
0.0
0.0
0.2
Y(t)
0.4
0.6
• One-dimensional diffusion models have limitations
●
●
−0.4
−0.2
0.0
X(t) b
0.2
0.4
−0.4
−0.2
0.0
X(t) b
0.2
0.4
when describing neuron membrane potential
dynamics
• Data clearly show two time scales in the system
• Biophysical models are difficult to fit to data
because of limited experimental data
• Biophysical models can be related to
two-dimensional diffusion models with linear drift in
a meaningful way.
Firing then corresponds to the first-exit time from
an ellipse.
Slide 32/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010