Neurons (nerve cells) Faculty of Science The Morris Lecar neuron - - PowerPoint PPT Presentation

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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

Neurons (nerve cells)

= ⇒

• u n i v e r s i t y o f c o p e n h a g e n

d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s

The model

dXt = µ(Xt) dt + σ(Xt) dW (t) ; X0 = x0 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 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}.

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time X(t) T T S x0

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Two commonly used Leaky Integrate-and-Fire neuron models (I)

The Ornstein-Uhlenbeck process: dXt =

• −Xt

τ + µ

• dt + σ dWt ; X0 = x0.

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)

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Two commonly used Leaky Integrate-and-Fire neuron models (II)

The Feller process (also CIR or square root process): d(Xt − VI) =

• −Xt − VI

τ + µ

• dt + σ
• Xt − VI dWt;

X0 = x0 ≥ VI. where VI: inhibitory reversal potential and 2µ ≥ σ2

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OU and square-root process

S1 S2 µτ VI

time

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From Berg, Ditlevsen and Hounsgaard (2008)

time in ms autocorrelation 10 20 30 40 50 0.0 0.2 0.4 0.6 0.8 1.0 Monoexponential: τ τ = = 43.1 ms Biexponential: τ1 = 12.1 ms; τ τ2 = 53.8 ms

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s

The Hodgkin-Huxley model

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.

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The Morris Lecar model

dVt = 1 C (−gCam∞(Vt)(Vt − VCa) − gKWt(Vt − VK) −gL(Vt − VL) + I)dt dWt = (α(Vt)(1 − Wt) − β(Vt)Wt) dt with the auxiliary functions given by m∞(v) = 1 2

• 1 + tanh

v − V1 V2

• α(v)

= 1 2φ cosh v − V3 2V4 1 + tanh v − V3 V4

• β(v)

= 1 2φ cosh v − V3 2V4 1 − tanh v − V3 V4

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200 400 600 800 1000 −40 −20 20 time membrane voltage, V(t)

−40 −20 20 40 0.1 0.2 0.3 0.4 0.5 membrane voltage, V(t) normalized conductance, W(t)

• u n i v e r s i t y o f c o p e n h a g e n

d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s

Bifurcation diagram, Morris Lecar model

From Tateno and Pakdaman (2004)

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The stochastic Morris Lecar model

Where to put the noise? dVt = 1 C (−gCam∞(Vt)(Vt − VCa) − gKWt(Vt − VK) −gL(Vt − VL) + I)dt + σ1(Vt, Wt)dBt dWt = (α(Vt)(1 − Wt) − β(Vt)Wt) dt +σ2(Vt, Wt)dBt

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Voltage noise From Tateno and Pakdaman (2004)

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The stochastic Morris Lecar model

Channel noise dVt = 1 C (−gCam∞(Vt)(Vt − VCa) − gKWt(Vt − VK) −gL(Vt − VL) + I)dt dWt = (α(Vt)(1 − Wt) − β(Vt)Wt) dt + σ(Vt, Wt)dBt How should σ(Vt, Wt) look?

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u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s

The stochastic Morris Lecar model

Wt can be interpreted as a probability and should stay between 0 and 1. For one-dimensional diffusions 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: s(x) = exp

• −

x

x∗

2b(y) σ2(y)dy

• ,

x ∈ (l, r) for some x∗ ∈ (l, r). Density of speed measure: m(x) = 1 σ2(x)s(x), x ∈ (l, r)

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The stochastic Morris Lecar model

If x∗

l

s(y)dy = r

x∗ s(y)dy = ∞

then the boundaries l and r are non-attracting. If moreover M = r

l

m(y)dy < ∞ then X is ergodic with invariant measure µ(x) = m(x)/M. In particular, Xt

D

→ µ as t → ∞. If X0 ∼ µ, then X is stationary and Xt ∼ µ for all t ≥ 0.

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u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s

The stochastic Morris Lecar model

Back to business... We look at 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 dWt = −θ (Wt − µ) dt + σ

• 2θWt(1 − Wt)dBt

where σ2 ≤ µ and σ2 ≤ 1 − µ. The invariant distribution is a Beta-distribution with parameters σ2/µ and σ2/(1 − µ).

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u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s

The stochastic Morris Lecar model

In our case we have θ = α(Vt) + β(Vt), µ = α(Vt) α(Vt) + β(Vt) Requirements translates to σ2 ≤ α(Vt) α(Vt) + β(Vt) σ2 ≤ β(Vt) α(Vt) + β(Vt) Fulfilled if σ2 ≤ α(Vt)β(Vt) α(Vt) + β(Vt) when 0 < α(Vt), β(Vt) < 1.

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u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s

The stochastic Morris Lecar model

We end up with the model dVt = 1 C (−gCam∞(Vt)(Vt − VCa) − gKWt(Vt − VK) −gL(Vt − VL) + I)dt dWt = (α(Vt)(1 − Wt) − β(Vt)Wt) dt +σ

• 2α(Vt)β(Vt)Wt(1 − Wt)dBt

where σ ≤ 1. Now Vt is not fixed, but still okay...

Slide 22/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010

−40 −20 20 40 0.1 0.2 0.3 0.4 0.5 membrane voltage, V(t) normalized conductance, W(t)

• −40

−20 20 40 0.1 0.2 0.3 0.4 0.5 membrane voltage, V(t) normalized conductance, W(t)

• −40

−20 20 40 0.1 0.2 0.3 0.4 0.5 membrane voltage, V(t) normalized conductance, W(t)

time

−30 −20 2000 4000

membrane voltage, V(t)

0.15

normalized conductance, W(t) time

−50 2000 4000

membrane voltage, V(t)

0.2 0.4

normalized conductance, W(t)

σ = 0.1 σ = 0.2

time

−50 2000 4000

membrane voltage, V(t)

0.2 0.4

normalized conductance, W(t) time

−50 2000 4000

membrane voltage, V(t)

0.5

normalized conductance, W(t)

σ = 0.5 σ = 1

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s

Linearization around the equilibrium point

Xt = Vt − Veq Veq , Yt = Wt − Weq Weq Small noise: the dynamics concentrate around the equilibrium point (x, y) = (0, 0). Linear approximation: d Xt Yt

• ≈

M Xt Yt

• dt + GdBt

where M =

• ∂f ∗

∂x ∂f ∗ ∂y ∂g∗ ∂x ∂g∗ ∂y

• (x,y)=(0,0)

= 0.026 0.112 −0.069 −0.045

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Deterministic approximation

Xt Yt

• ≈

C1 b cos ωt − sin ωt

• e−λt + C2

b sin ωt cos ωt

• e−λt

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.

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Deterministic approximation

time

−30 −25 500 1000

membrane voltage, V(t)

0.12 0.14

normalized conductance, W(t) exact approx

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Stochastic approximation

Xt Yt

• ≈

A(T) b cos ωt − sin ωt

• + B(T)

b sin ωt cos ωt

• where A(T) and B(T) incorporate the slow decay on the

time scale T = λt and the stochastic component: dA dB

• =

f1(A, B) f2(A, B)

• dT + Σ

dξ1(T) dξ2(T)

• After some calculations:

dA dB

• =

−A −B

• dT +

σ 2 √ λ 1 1 dξ1(T) dξ2(T)

• .

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−0.4 −0.2 0.0 0.2 0.4 0.0 0.2 0.4 0.6 X(t) b Y(t) z

• −0.4

−0.2 0.0 0.2 0.4 0.0 0.2 0.4 0.6 X(t) b Y(t)

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Conclusions

• One-dimensional diffusion models have limitations

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.

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