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sdwfrost/morris-lecar.ipynb

Created November 7, 2017 19:36
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Simulation of a stochastic Morris-Lecar model
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morris-lecar.ipynb
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"# Simulating the Morris-Lecar model\n",
"\n",
"I consider the [Morris-Lecar model](https://en.wikipedia.org/wiki/Morris%E2%80%93Lecar_model), originally derived to model calcium and potassium conductance in the muscle fibre of the giant barnacle. A simple version of this model consists of two variables, the membrane potential, $v$, and the recovery variable, $n$, the probability that the potassium channel is conducting.\n",
"\n",
"$$\n",
"\\begin{array}{lcl}\n",
"C\\frac{dv}{dt} &= &I(t)-g_{Ca}m_\\infty(v)(v-v_{Ca}) -g_L(v-v_L)-g_Kn(v-v_K)\\\\\n",
"\\frac{dn}{dt} &=& \\frac{n_\\infty(v)-n}{\\tau(v)}\\\\\n",
"&=&\\alpha(v)(1-n)-\\beta(v)n\n",
"\\end{array}\n",
"$$\n",
"\n",
"where\n",
"\n",
"$$\n",
"\\begin{array}{lcl}\n",
"m_\\infty(v) &=& \\frac{1}{2}\\cdot\\left(1+\\rm{tanh}\\left[\\frac{v-v_1}{v_2}\\right]\\right)\\\\\n",
"n_\\infty(v) &=& \\frac{1}{2}\\cdot\\left(1+\\rm{tanh}\\left[\\frac{v-v_3}{v_4}\\right]\\right)\\\\\n",
"\\tau(v) &=& 1/\\left(\\phi \\hspace{1mm}\\rm{cosh}\\left[\\frac{v-v_3}{2v_4}\\right]\\right)\n",
"\\end{array}\n",
"$$\n",
"\n",
"The parameters of the system are as follows.\n",
"\n",
"- $I$: applied current;\n",
"- $C$: membrane capacitance;\n",
"- $g_L,g_{Ca},g_K$: leak, calcium and potassium conductances through membranes channel;\n",
"- $v_L,v_{Ca},v_{K}$: equilibrium potentials of ion channels;\n",
"- $v_1,v_2,v_3,v_4$: tuning parameters for steady state and time constant; and\n",
"- $\\phi$: reference frequency.\n",
"\n",
"To simulate the deterministic Morris-Lecar model in Julia, we first load the appropriate libraries."
]
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"using DifferentialEquations\n",
"using ParameterizedFunctions\n",
"using LSODA\n",
"using Plots"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Next, we define some convenience functions."
]
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"function n∞(v,v₃,v₄)\n",
" ξ=(v-v₃)/v₄\n",
" (1+tanh(ξ))/2\n",
"end\n",
"\n",
"function τ(v,v₃,v₄,ϕ)\n",
" ξ=(v-v₃)/v₄\n",
" 1/(ϕ*cosh(ξ/2))\n",
"end;\n",
"\n",
"function m∞(v,v₁,v₂)\n",
" (1+tanh((v-v₁)/v₂))/2\n",
"end\n",
"\n",
"function α(v,v₃,v₄,ϕ)\n",
" ξ=(v-v₃)/v₄\n",
" ϕ*cosh(ξ/2)/(1+exp(-2*ξ))\n",
"end\n",
"\n",
"function β(v,v₃,v₄,ϕ)\n",
" ξ=(v-v₃)/v₄\n",
" ϕ*cosh(ξ/2)/(1+exp(2*ξ)) \n",
"end;"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Next, we define the Morris-Lecar model using the ParameterisedFunctions interface."
]
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"text": [
"\u001b[1m\u001b[33mWARNING: \u001b[39m\u001b[22m\u001b[33mUndefined derivative found. If you are using a non-elementary function, you must define the derivative in order to calculate Jacobians et. al. Please refer to the documentation.\u001b[39m\n",
"\u001b[1m\u001b[33mWARNING: \u001b[39m\u001b[22m"
]
}
],
"source": [
"ml = @ode_def MorrisLecar begin\n",
" dv=(I-gL*(v-vL)-gCa*m∞(v,v₁,v₂)*(v-vCa)-gK*n*(v-vK))/C\n",
" dn=(α(v,v₃,v₄,ϕ)*(1-n)-β(v,v₃,v₄,ϕ)*n)\n",
"end I=>100.0 C=>20.0 gL=>2.0 gCa=>4.4 gK=>8.0 vL=>-60.0 vCa=>120.0 vK=>-84.0 v₁=>-1.2 v₂=>18.0 v₃=>2.0 v₄=>30 ϕ=>0.04"
]
},
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"source": [
"The following provide the initial conditions and time steps"
]
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"u0 = [-50;0.5]\n",
"tf = 1000.0\n",
"tstep = 1;\n",
"tspan = (0.0,tf)\n",
"tp=0.0:tstep:tf;"
]
},
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"metadata": {},
"source": [
"The differential equation can be solved as follows."
]
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"p1 = MorrisLecar()\n",
"prob1 = ODEProblem(p1,u0,tspan);\n",
"sol1=solve(prob1);"
]
},
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"source": [
"The following plots the probability of a channel being open."
]
},
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"source": [
"p_ode = plot(tp,[sol1(t)[2] for t in tp],xlab=\"Time\",ylab=\"Probability of open channel\",title=\"Deterministic\",legend=false)"
]
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"## Stochastic version\n",
"\n",
"In the stochastic analogue of $n(t)$, the potassium channel kinetics is modelled as a population of independent two-state (open/closed) Markov chains. For a total number $N_{tot}$ of potassium channels, the number of open channels, $N(t)$, follows a birth-death process with transitions rates $\\alpha(v),\\beta(v)$.\n",
"\n",
"[Anderson et al. (2015)](https://link.springer.com/article/10.1007/s10827-014-0528-2) described two algorithms for simulating a stochastic Morris-Lecar model, both of which require simulating an ordinary differential equation until an equality condition is reached. Below, [uniformisation](https://en.wikipedia.org/wiki/Uniformization_(probability_theory)) is used instead.\n",
"\n"
]
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"source": [
"function f_ml_rej!(tt,x,xdot,data)\n",
" (I,C,gL,gCa,gK,vL,vCa,vK,v₁,v₂,v₃,v₄,ϕ,ntot,λ,n) = data\n",
" (v,) = x\n",
" xdot[1] = (I-gL*(v-vL)-gCa*m∞(v,v₁,v₂)*(v-vCa)-gK*(n/ntot)*(v-vK))/C\n",
" nothing\n",
"end;"
]
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"ml_rej (generic function with 1 method)"
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"function ml_rej(x0,parms,tf)\n",
" (v,N) = x0\n",
" (I,C,gL,gCa,gK,vL,vCa,vK,v₁,v₂,v₃,v₄,ϕ,ntot,λ) = parms\n",
" eparms = [parms;N]\n",
" t=0.0\n",
" ta = Vector{Float64}(0)\n",
" va = Vector{Float64}(0)\n",
" Na = Vector{Float64}(0)\n",
" push!(ta,t)\n",
" push!(va,v)\n",
" push!(Na,N)\n",
" while t<tf\n",
" s = -log(rand())/λ\n",
" res = LSODA.lsoda((tt,x,xdot,data)->f_ml_rej!(tt,x,xdot,eparms),\n",
" [v],\n",
" [t,t+s],\n",
" abstol=1.e-9,\n",
" reltol=1e-7,\n",
" nbsteps=10000)\n",
" v = res[end,1]\n",
" t = t+s\n",
" # Update N\n",
" opn = α(v,v₃,v₄,ϕ)*(ntot-N)/λ\n",
" cls = β(v,v₃,v₄,ϕ)*N/λ\n",
" if opn+cls>1.0\n",
" @printf(\"Rate=%s\\tλ=%s\",λ*(opn+cls),λ)\n",
" error(\"Error: the rate parameter λ has been exceeded.\")\n",
" end\n",
" u = rand()\n",
" if u<opn\n",
" N=N+1\n",
" eparms[end]=N\n",
" push!(ta,t)\n",
" push!(va,v)\n",
" push!(Na,N)\n",
" elseif u<(opn+cls)\n",
" N=N-1\n",
" eparms[end]=N\n",
" push!(ta,t)\n",
" push!(va,v)\n",
" push!(Na,N)\n",
" end\n",
" end\n",
" return(ta,va,Na)\n",
"end"
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"parms_rej=[100.0,20.0,2.0,4.4,8.0,-60.0,120.0,-84.0,-1.2,18.0,2.0,30,0.04,40,3.0] # λ is last\n",
"x0_rej=[-50.0;20.0]\n",
"tf_rej=1000.0;"
]
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"source": [
"srand(1)\n",
"sol_rej=ml_rej(x0_rej,parms_rej,tf_rej);"
]
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"Below is a single realisation of the model."
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"source": [
"plot(sol_rej[1],sol_rej[3],xlab=\"Time\",ylab=\"Number of open channels\",title=\"Stochastic, uniformisation\",legend=false)"
]
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"A crude benchmark for simulating 100 realisations is as follows."
]
},
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"text": [
" 3.620145 seconds (27.86 M allocations: 1.446 GiB, 10.21% gc time)\n"
]
}
],
"source": [
"@time begin\n",
" for i in 1:100\n",
" out=ml_rej(x0_rej,parms_rej,tf_rej)\n",
" end\n",
"end"
]
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"source": [
"### Using common interface\n",
"\n",
"The above uses the bespoke interface from [LSODA.jl](https://github.com/rveltz/LSODA.jl). To use the common interface from DifferentialEquations.jl, we first define the Morris-Lecar model using the `@ode_def` macro, inlining the functions so that derivatives are calculated automatically."
]
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{
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"text": [
"\u001b[1m\u001b[33mWARNING: \u001b[39m\u001b[22m\u001b[33mFailed to build the parameter derivatives.\u001b[39m\n"
]
},
{
"data": {
"text/plain": [
"(::MorrisLecar2) (generic function with 26 methods)"
]
},
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"source": [
"ml2 = @ode_def MorrisLecar2 begin\n",
" dv=(I-gL*(v-vL)-gCa*((1+tanh((v-v₁)/v₂))/2)*(v-vCa)-gK*(n/ntot)*(v-vK))/C\n",
"end I=>100.0 C=>20.0 gL=>2.0 gCa=>4.4 gK=>8.0 vL=>-60.0 vCa=>120.0 vK=>-84.0 v₁=>-1.2 v₂=>18.0 v₃=>2.0 v₄=>30 ϕ=>0.04 ntot=>40 n=>40"
]
},
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{
"data": {
"text/plain": [
"ml_rej_ci (generic function with 1 method)"
]
},
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"source": [
"function ml_rej_ci(x0,parms,tf)\n",
" (v,N) = x0\n",
" (I,C,gL,gCa,gK,vL,vCa,vK,v₁,v₂,v₃,v₄,ϕ,ntot,λ) = parms\n",
" t=0.0\n",
" ta = Vector{Float64}(0)\n",
" va = Vector{Float64}(0)\n",
" Na = Vector{Float64}(0)\n",
" push!(ta,t)\n",
" push!(va,v)\n",
" push!(Na,N)\n",
" ml = MorrisLecar2(I=I,C=C,gL=gL,gCa=gCa,gK=gK,vL=vL,vCa=vCa,vK=vK,v₁=v₁,v₂=v₂,v₃=v₃,v₄=v₄,ϕ=ϕ,ntot=ntot,n=N)\n",
" while t<tf\n",
" s = -log(rand())/λ\n",
" ml.n=N\n",
" p = ODEProblem(ml,[v],(t,t+s))\n",
" sol = solve(p,lsoda(),reltol=1e-7,abstol=1e-9)\n",
" t = t+s\n",
" v = sol(t)[1]\n",
" # Update N\n",
" opn = α(v,v₃,v₄,ϕ)*(ntot-N)/λ\n",
" cls = β(v,v₃,v₄,ϕ)*N/λ\n",
" if opn+cls>1.0\n",
" @printf(\"Rate=%s\\tλ=%s\",λ*(opn+cls),λ)\n",
" error(\"Error: the rate parameter λ has been exceeded.\")\n",
" end\n",
" u = rand()\n",
" if u<opn\n",
" N=N+1\n",
" push!(ta,t)\n",
" push!(va,v)\n",
" push!(Na,N)\n",
" elseif u<(opn+cls)\n",
" N=N-1\n",
" push!(ta,t)\n",
" push!(va,v)\n",
" push!(Na,N)\n",
" end\n",
" end\n",
" return(ta,va,Na)\n",
"end"
]
},
{
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"source": [
"srand(1)\n",
"sol_rej_ci=ml_rej_ci(x0_rej,parms_rej,tf_rej);"
]
},
{
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},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(sol_rej_ci[1],sol_rej_ci[3],xlab=\"Time\",ylab=\"Number of open channels\",title=\"Stochastic, uniformisation, common interface\",legend=false)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The benchmark using the common interface is below."
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {
"collapsed": false,
"deletable": true,
"editable": true
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" 25.146869 seconds (118.46 M allocations: 5.613 GiB, 7.39% gc time)\n"
]
}
],
"source": [
"@time begin\n",
" for i in 1:100\n",
" out=ml_rej_ci(x0_rej,parms_rej,tf_rej)\n",
" end\n",
"end"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Julia 0.6.0",
"language": "julia",
"name": "julia-0.6"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "0.6.0"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
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