Rewriting a pymc script for parameter estimation i

Question

I'd like to use pymc3 to estimate unknown parameters and states in a Hodgkin Huxley neuron model. My code in pymc is based off of http://healthyalgorithms.com/2010/10/19/mcmc-in-python-how-to-stick-a-statistical-model-on-a-system-dynamics-model-in-pymc/ and executes reasonably well.

#parameter priors
@deterministic
def HH(priors in here)
    #model equations
    #return numpy arrays that somehow contain the probability distributions as elements
    return V,n,m,h

#Make V deterministic in one line. Seems to be the magic that makes this work.
V = Lambda('V', lambda HH=HH: HH[0])

#set up the likelihood
A = Normal('A',mu=V,tau=2.0,value=voltage_data,observed = True)
#run the sampling...

In pymc3, the Lambda trick is not available to me. Here is one of my attempts:

import numpy as np
import theano.tensor as tt
from pymc3 import Model, Normal, Uniform, Deterministic, sample, NUTS, Metropolis, find_MAP
import scipy

#observed data
T = 10
dt = 0.02

voltage_data_file = 'noise_measured.dat'
voltage_data = np.loadtxt(voltage_data_file)
voltage_data = voltage_data[0:T]

current_data_file = 'current.dat'
current_data = np.loadtxt(current_data_file)
current_data = current_data[0:T]

#functions needed later
def x_inf(V,vx,dvx):
    return 0.5*(1 + np.tanh((V - vx)/dvx))
def tau(V,vx_t,dvx_t,tx_0,tx_1):
    return tx_0 + 0.5*tx_1*(1 + np.tanh((V- vx_t)/dvx_t))

#Model
NaKL_Model = Model()
with NaKL_Model:
    #priors
    g_Naa = Uniform('g_Naa',0.0,150.0)
    E_Na = Uniform('E_Na',30.0,70.0)
    g_K = Uniform('g_K',0.0,150.0)
    E_K = Uniform('E_K',-100.0,-50.0)
    g_L = Uniform('g_L',0.1,1.0)
    E_L = Uniform('E_L',-90.0,-50.0)
    vm = Uniform('vm',-60.0,-30.0)
    vm_t = Uniform('vm_t',-60.0,-30.0)
    dvm = Uniform('dvm',10.0,20.0)
    dvm_t = Uniform('dvm_t',10.0,20.0)
    tmm_0 = Uniform('tmm_0',0.05,0.25)
    tm_1 = Uniform('tm_1',0.1,1.0)
    vh = Uniform('vh',-80.0,-40.0)
    vh_t = Uniform('vh_t',-80.0,-40.0)
    dvh = Uniform('dvh',-30.0,-10.0)
    dvh_t = Uniform('dvh_t',-30.0,-10.0)
    th_0 = Uniform('th_0',0.5,1.5)
    th_1 = Uniform('th_1',5.0,9.0)
    vn = Uniform('vn',-70.0,-40.0)
    vn_t = Uniform('vn_t',-70.0,-40.0)
    dvn = Uniform('dvn',10.0,40.0)
    dvn_t = Uniform('dvn_t',10.0,40.0)
    tn_0 = Uniform('tn_0',0.5,1.5)
    tn_1 = Uniform('tn_1',3.0,7.0)
    #Initial Conditions
    V_0 = Uniform('V_0',-100.0,50.0)
    n_0 = Uniform('n_0',0.0,1.0)
    m_0 = Uniform('m_0',0.0,1.0)
    h_0 = Uniform('h_0',0.0,1.0)

    def HH(i,V_current,n_current,m_current,h_current,g_Naa=g_Naa,E_Na=E_Na,g_K=g_K,E_K=E_K,g_L=g_L,E_L=E_L,vm=vm,vm_t=vm_t,dvm=dvm,dvm_t=dvm_t,tmm_0=tmm_0,tm_1=tm_1,vh=vh,vh_t=vh_t,dvh=dvh,dvh_t=dvh_t,th_0=th_0,th_1=th_1,vn=vn,vn_t=vn_t,dvn=dvn,dvn_t=dvn_t,tn_0=tn_0,tn_1=tn_1):

        V_new = V_current + dt*(g_L*(E_L - V_current) + g_K*n_current**4*(E_K - V_current) + g_Naa*m_current**3*h_current*(E_Na - V_current) + current_data[i])

        n_new = n_current + dt*(x_inf(V_current,vn,dvn)-n_current)/tau(V_current,vn_t,dvn_t,tn_0,tn_1)
        m_new = m_current + dt*(x_inf(V_current,vm,dvm)-m_current)/tau(V_current,vm_t,dvm_t,tmm_0,tm_1)
        h_new = h_current + dt*(x_inf(V_current,vh,dvh)-h_current)/tau(V_current,vh_t,dvh_t,th_0,th_1)

        return V_new,n_new,m_new,h_new

    V_state = [V_0]
    n_state = [n_0]
    m_state = [m_0]
    h_state = [h_0]

    #A = [Normal('A0',mu=V_0,tau=2.0,observed = voltage_data[0])]
    for i in range(1,T):
        V_state.append(Deterministic('V' + str(i), HH(i-1,V_state[i-1],n_state[i-1],m_state[i-1],h_state[i-1])[0]))
        n_state.append(Deterministic('V' + str(i), HH(i-1,V_state[i-1],n_state[i-1],m_state[i-1],h_state[i-1])[1]))
        m_state.append(Deterministic('V' + str(i), HH(i-1,V_state[i-1],n_state[i-1],m_state[i-1],h_state[i-1])[2]))
        h_state.append(Deterministic('V' + str(i), HH(i-1,V_state[i-1],n_state[i-1],m_state[i-1],h_state[i-1])[3]))
        #A.append(Normal('A' + str(i),mu=V_state[i],sd=2.0,observed = voltage_data[0]))
    A = Normal('A',mu=V_state,sd=2.0,observed = voltage_data)
    start = find_MAP()
    #step = NUTS(scaling=start)
    step = Metropolis(start=start)
    trace = sample(100,step)

The only other example that I've seen asked on these forums is here: Simple Dynamical Model in PyMC3

However, that doesn't help answer my question because neither proposed solution there works. My solution doesn't work either - I don't get an error, but when I run the script the sampling appears never to start. In any case, my solution seems inefficient because I have to run a python for loop to define all of the Deterministic distributions. I'm not sure if pymc3 even recognizes the way I've put them all in a pure python list. If my function HH() could return a numpy array or some kind of theano object, maybe that would help. But I'm lost as to how to implement it.

Answer 1

Your approach works for me, after a tiny change in the last two lines to read:

step = Metropolis()
trace = sample(100, step, start=start)

It takes a while, though, so you can make T smaller if you want to see results sooner.

Here is a notebook that shows it in action.