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. Elliptic FitzHugh Nagumo
dVt = 1ε(Vt−Vt3−Ut−s)dt+σ1dB1t,
dUt = (γVt−Ut+β)dt+σ2dBt2, (4) where the variableVtrepresents the membrane potential of the neuron at timet, andUtrepresents the channel kinetic. Parameter sis the magnitude of the stimulus current.
(a) Write the Euler scheme for the FitzHugh Nagumo process and implement the scheme.
(b) Simlate trajectories with the following parameters : ε= 0.1,s= 0,γ= 1.5,β= 0.8,σ1= 0.3, σ2= 0.3. The time step isδ= 0.02. Simulate a trajectory of lengthn= 1000.
(c) Simlate trajectories for the hypoelliptic system with the following parameters : ε= 0.1,s= 0, γ= 1.5,β = 0.8,σ1= 0, σ2= 0.3. The time step isδ= 0.02. Simulate a trajectory of length n= 1000.
2. 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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