Universit´e Joseph Fourier and ENSIMAG Master SIAM
Stochastic modelling for neurosciences
Lab 1: simulation of neuronal models
Objectives: Being able to simulate any neuronal model, with exact or approximate numerical schemes.
1 Exact simulation of one-dimensional neuronal model
1. Wiener process with drift
dV(t) =Idt+σdB(t), V(0) =V0 (1)
(a) Compute the exact distribution of (V(t)).
(b) Simulate a trajectory ofV(t) at discrete timesti=i∆,i= 1, . . . , n, with ∆ = 0.0001, n= 1000 andI=−10,V0=−65,σ= 1.
(c) Change the value of the diffusion coefficient: σ= 10 orσ= 20. Comment.
2. Ornstein-Uhlenbeck process
dV(t) =
−V(t)−α τ
dt+σ dB(t), V(0) =V0, withα=V0+τ I (2) (a) Compute the exact distribution of (V(t)).
(b) Simulate a trajectory ofV(t) at discrete timesti=i∆,i= 1, . . . , n, with ∆ = 0.001, n= 5000.
Parameters areτ= 0.5,V0=−65,I= 50,σ= 10.
(c) Compute the spiking times with the thresholdS=−45.
(d) Plot the distribution of the spiking times. Plot the membrane potential evolution.
(e) Change the value of the inputI= 20. Comment.
2 Approximate simulation of one-dimensional neuronal model
1. Ornstein-Uhlenbeck process
(a) Write the Euler scheme for the OU process (2) and implement the scheme.
(b) Compare the trajectories obtained with the exact scheme, depending on the value of ∆.
2. Feller process
dV(t) =
−V(t)−α τ
dt+σp
V(t)−VIdB(t), V(0) =V0 (3) (a) Write the the Euler scheme for the Feller process (3) and implement the scheme.
(b) Simulate a trajectory ofV(t) at discrete timesti=i∆,i= 1, . . . , n, with ∆ = 0.001, n= 5000.
Parameters areτ= 0.5,V0=−65,I= 50,VI =−70,σ= 10.
(c) Compute the spiking times with the thresholdS=−45.
(d) Plot the distribution of the spiking times.
(e) Change the value of VI. Comment.
3 Approximate simulation of multi-dimensional neuronal model
1. Morris-Lecar process
dV(t) = −(gf astm∞(t) (V(t)−Vf ast) +gslowU(t)(V(t)−Vslow) +gL(V(t)−VL) +I(t))dt+ +γdB(t)˜
dU(t) = − 1
τu(V(t))(U(t)−u0(V(t))) = (α(V(t))(1−U(t))−β(V(t))U(t))dt+σ(V(t), U(t))dB(t)
with
m∞(v) = 1 2
1 + tanh(v−V1 V2
)
u0(v) = 1 2
1 + tanh(v−V3
V4 )
τu(v) = τu
cosh
v−V3
V4
α(v) = 1 2φcosh
v−V3
2V4
1 + tanh
v−V3
V4
, β(v) = 1
2φcosh
v−V3
2V4
1−tanh
v−V3
V4
σ(v, u) = σ s
2 α(v)β(v)
α(v) +β(v)u(1−u).
(a) Write the Euler scheme for the Morris-Lecar process and implement the scheme.
(b) Simlate trajectories with the following parameters Vf ast=−84;VL =−60;Vslow = 120;I = 88/20;gL = 2/20;gslow = 4.4/20;gf ast = 8/20;V1 = −1.2;V2 = 18;V3 = 2;V4 = 30;φ = 0.04;σ= 0.03;γ= 1. The initial conditions areV0=−26;U0= 0.2. The time step isδ= 0.1.
Simulate a trajectory of lengthn= 5000.
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