Page 1
Biol Cybern (2012) 106:155–167
DOI 10.1007/s00422-012-0487-5
ORIGINAL PAPER
Dynamical estimation of neuron and network properties II:
path integral Monte Carlo methods
Mark Kostuk · Bryan A. Toth · C. Daniel Meliza ·
Daniel Margoliash · Henry D. I. Abarbanel
Received: 16 January 2012 / Accepted: 14 March 2012 / Published online: 13 April 2012
© Springer-Verlag 2012
Abstract Hodgkin–Huxley(HH)modelsofneuronalmem-
brane dynamics consist of a set of nonlinear differential
equations that describe the time-varying conductance of var-
ious ion channels. Using observations of voltage alone we
show how to estimate the unknown parameters and unob-
served state variables of an HH model in the expected
circumstance that the measurements are noisy, the model
has errors, and the state of the neuron is not known when
observations commence. The joint probability distribution
of the observed membrane voltage and the unobserved state
variables and parameters of these models is a path integral
through the model state space. The solution to this integral
allows estimation of the parameters and thus a character-
ization of many biological properties of interest, including
channel complement and density, that give rise to a neuron’s
electrophysiologicalbehavior.Thispaperdescribesamethod
for directly evaluating the path integral using a Monte Carlo
numerical approach. This provides estimates not only of the
expected values of model parameters but alsoof their poster-
ioruncertainty.Usingtestdatasimulatedfromneuronalmod-
elscomprisingseveralcommonchannels,weshowthatshort
(<50ms) intracellular recordings from neurons stimulated
M. Kostuk · B. A. Toth
Department of Physics, University of California, 9500 Gilman
Drive, San Diego, La Jolla, CA 92093-0402, USA
C. D. Meliza (B ) · D. Margoliash
Department of Organismal Biology and Anatomy, University
of Chicago, 1027 E 57th Street, Chicago, IL 60637, USA
e-mail: dmeliza@uchicago.edu
H. D. I. Abarbanel
Marine Physical Laboratory, Department of Physics (Scripps
Institution of Oceanography), Center for Theoretical Biological
Physics, University of California, 9500 Gilman Drive,
San Diego, La Jolla, CA 92093-0374, USA
e-mail: habarbanel@ucsd.edu
with a complex time-varying current yield accurate and pre-
cise estimates of the model parameters as well as accurate
predictions of the future behavior of the neuron. We also
show that this method is robust to errors in model specifica-
tion, supporting model development for biological prepara-
tions in which the channel expression and other biophysical
properties of the neurons are not fully known.
Keywords
Ion channel properties · Markov Chain Monte Carlo
Data assimilation · Neuronal dynamics ·
1 Introduction
Thedynamicalresponsesofneuronsandnervoussystemnet-
works to time-varying inputs depend on ion channels that
are gated by membrane voltage and chemical ligands. The
complex, nonlinear dynamics of the voltage-gated channels
(Johnston and Wu 1995; Graham 2002) control the genera-
tion of action potentials and determine their shape and fre-
quency. Determining which channels are present in a class
of neuron, their biophysical properties, and how these con-
tribute to the phenomenological behavior of the neurons is
generally a painstaking process involving extensive pharma-
cological manipulation.
An alternative approach that we explore here is to record
a neuron’s response to a time-varying current and use these
datatoestimatepropertiesofabiophysicalmodeloftheneu-
ron. Accurate predictions of the neuron’s response to novel
stimuli by such a model indicates that the model may be
used to make inferences about the biological properties of
the neuron. The measurements are necessarily limited to a
small subset of the many state variables in such a model, and
there can be many unknown parameters. We have developed
a systematic approach to this problem that provides an exact
123
Page 2
156 Biol Cybern (2012) 106:155–167
statistical setting for transferring information from measure-
ments to a model, which we described in Part I of this paper
(Toth et al. 2011), along with a variational approximation to
thehigh-dimensionalpathintegralinvolvedinthisapproach.
Here, in Part II we move beyond the variational approxi-
mation, to a numerical evaluation (also approximate) of the
integraloverpathsofthemodelstatethroughandbeyondthe
temporalwindowofobservations.Bysamplingfromthedis-
tributionoflikelypathswecanobtainanestimateofthestate
and parameter values that best represent the data, as well as
statistics about the uncertainty of these estimates. This pos-
terior uncertainty indicates how much the data, as well as
the dynamics of the model, constrain the parameter and state
estimates, thereby providing additional information that can
be used to guide model selection and inference as well as
experimental design.
Small error bounds indicate that the amount of data is
sufficient to make confident statements about the underlying
biophysicalpropertiesoftheneuron;largeerrorboundsfora
parameter indicate either that the data is insufficient in quan-
tity or dynamical range, that the behavior of the neuron is
insensitive to the value of that parameter, or that the model
is in error. We can also integrate forward in time beyond
the observation interval using samples from the full poster-
ior distribution to obtain a predictive distribution, which is
useful for model validation and selection.
After a brief summary of the family of Hodgkin–Huxley
(HH) biophysical models we utilize and of the path integral
formulationofthedataassimilationproblem(formoredetail
see Toth et al. 2011), we describe a Markov Chain Monte
Carlo (MCMC) algorithm for sampling from the distribution
ofpathsconditionedontherecordeddata.Wealsopresentan
implementation of the algorithm suitable for highly parallel-
ized devices such as graphical processing units (GPUs). We
then repeat a number of the “twin experiments” from Part I,
in which data simulated from a model is used in our proce-
dure and the estimates are compared to the parameters used
to generate the data as well as the observed and unobserved
states.
The goal of these numerical experiments is to determine
under what conditions the data assimilation procedure is
accurate. We do not pursue two questions addressed in Part I
related to the frequency of the input current and the amount
of noise in the measurements, taking values that we dem-
onstrated work well with the variational method. Instead we
focusontherobustnessofthemethodtomodelerrors—when
the model contains channels that are absent in the data and
vice versa. As in Toth et al. (2011) we analyze two sim-
ple HH models: one has Na, K, and leak currents (NaKL
model), and the other has in addition an Ihcurrent (NaKLh
model). We test for model robustness by generating data
using one model and estimating parameters using the other,
and show that we can identify missing or extra channels in
the model. The numerical approach to evaluating the path
integral described here is particularly suited to these situa-
tions, because it takes into account such model errors, which
are inevitable in the study of biological neurons.
2 Methods
2.1 General framework
In studying the biological properties of neurons, we can typ-
ically measure only the membrane voltage potential, V(t),
while injecting a known stimulus current Iapp(t). We would
like to infer properties of the voltage-gated ion channels that
open and close in response to the current injection. The task
is to select a model that is consistent with the observations
and to estimate the values of the parameters that have some
connection to biologically interesting properties of the neu-
ron and the system of which it is a part. Other types of
measurements are possible, including voltage-clamp mea-
surementsofcurrentfloworopticalmeasurementsfromfluo-
rescentreporters;thesearenotwithinthescopeofthecurrent
paper.Thebasicproblem,ofestimatingunobservableparam-
etersandstatesfromalimitedsetofobservations,inprinciple
remains the same, although the kinetics of some measure-
ments (such as calcium indicators) may be sufficiently slow
so as to challenge the modeling effort.
In making such inferences, models that are explic-
itly based on biophysical entities are preferable to more
abstract phenomenological ones (e.g., integrate-and-fire).
The biophysical models typically comprise a set of ordinary
differential equations, for current conservation across the
membrane (and possibly between different compartments)
and for the kinetics of the gating variables that describe the
opened and closed configurations of various voltage-gated
channels (Sect. 2.5). In general, the system is described
by a D-dimensional state vector that includes the observ-
able voltage V(t) and a set of unobservable variables ai(t);
i = 1, 2,..., D − 1 associated with other compartments
and the permeability of each of the voltage-gated channels
included in the model. The models we examine in this paper
are relatively simple, with a single compartment and up to
five state variables associated with three channel types. The
methods developed here and in our earlier paper (Toth et al.
2011) can be applied to specific, more complex settings, but
will require richer models and possibly additional data.
The second component of the model is the process
of observation. We make measurements of the voltage at
discrete times over some interval tn={t0, t1,...,tm=T}.
We label these observations y(tn)=y(n);n =0, 1,...,m,
and they are related to the true voltage of the neuron through
somemeasurementfunctionh(x)thatincorporatesnoisearis-
ing from various sources. The discrete time nature of the
123
Page 3
Biol Cybern (2012) 106:155–167157
observations suggests that the model be stated as a rule in
discrete time taking the system at time tn= n to the state at
time n + 1. This rule may be the specific formulation of a
numerical solution algorithm for the underlying differential
equations. Both the discrete time rule and the observation
function may involve a collection of unknown parameters,
which we denote as p.
Using the observations y(n) of the voltage, we wish to
establish the full collection of state variables x(n) at times
in the observation window, in particular at the end of the
measurements t = T. With estimates of x(T), of the param-
eters p, and knowledge of the stimulus for t > T we may
use the model to predict the neuron’s behavior for t > T.
Our goal is to achieve y(t) ≈ h(x(t)) for t > T, as these
are the observed quantities. If these predictions are accu-
rate for a broad range of biologically plausible stimuli, then
the estimates of the model parameters provide a parsimoni-
ous, biophysically interpretable description of the neuron’s
behavior.
In the present paper the data are simulated from a model
which has the form of a set of ordinary differential equations
(Sect.2.5),whichwesolvebydiscretizingtime.Afterchoos-
ingarealisticsetofparametersandinitialstates,wegenerate
a solution to the equations. We select a subset of the output,
here the voltage alone, and add noise to yield the observed
quantities y(n). This transfer of information from measure-
ments to models is called data assimilation, following the
name given in the extensive and well developed geophysical
literature on this subject (Evensen 2009). The technique for
testing data assimilation methods in cases where the data is
generated by the model is known as a “twin experiment.”
2.2 Path integral formulation
As discussed in Toth et al. (2011) one can cast the general
set of questions in data assimilation when one has noisy
data, model errors, and uncertainty about the initial state
of the model x(t0) = x(0) into an integral over the states
x(tn) = x(n); {t0, t1,...,tm = T} during the measure-
ment interval [0, T] at the measurement times tn. If we
denote the path in state space through this time interval as
X = {x(0), x(1),...,x(m)}, then the path is a location in
(m + 1)D-dimensional space.
The expected value, conditional on the measurements
Y = {y(0), y(1),...,y(m)}, of any function along the path
G(X) is given by the (m + 1)D dimensional integral
?dXe−A0(X,Y)G(X)
E[G(X)|Y] =
?dXe−A0(X,Y)
.
(1)
The action A0(X, Y) is given in terms of (1) the con-
ditional mutual information of a state x(n) and a measure-
menty(n),conditionedonmeasurementsuptothetimetn−1:
Y(n − 1) = {y(0), y(1),...,y(n − 1)}, (2) the transition
probability P(x(n + 1)|x(n)) to arrive in the state x(n + 1)
at time tn+1given the state x(n) at tn, and (3) the distribution
of states at t0P(x(0)), as
−A0(X, Y)
=
n=0
m
?
m−1
?
m
?
m−1
?
log
?
P(x(n), y(n)|Y(n − 1))
P(x(n)|Y(n − 1)) P(y(n)|Y(n − 1))
?
+
n=0
log[P(x(n + 1)|x(n))] + log[P(x(0)]
?P(y(n)|x(n), Y(n − 1))
=
n=0
log
P(y(n)|Y(n − 1))
?
+
n=0
log[P(x(n + 1)|x(n))] + log[P(x(0)].
(2)
The first sum gives the information transferred from the
measurement y(n) to the state x(n), conditioned on previ-
ous measurements Y(n−1). The second term represents the
underlying dynamics of the model that moves the state for-
ward one step in time, and the last term is the uncertainty in
the state at the time t0of the initial measurement.
Approximationsto A0(X, Y)werediscussedinTothetal.
(2011):
– If the dynamical rule is written as ga(x(n + 1), x(n), p)
= 0; a = 1, 2,..., D in the case of no model errors,
then the transition probability P(x(n+1)|x(n)) is a delta
function P(x(n +1)|x(n)) = δD(g(x(n +1), x(n), p)).
Withmodelerrors,thisisbroadened,andwithaGaussian
approximation to these model errors we write
P(x(n + 1)|x(n)) ∝
⎡
×exp
⎣−1
2
m−1,D
?
n=0,a=1
Rf a(ga(x(n + 1), x(n), p))2
⎤
⎦.
(3)
When the differential equation solver is explicit g(x
(n + 1), x(n), p) = x(n + 1) − f(x(n), p).
– If the measurements are independent at different times,
and the noise in each measurement is Gaussian, then the
information transfer contribution to A0(X, Y) is propor-
tional to
Rm
2
m,L
?
n=0,l=1
(yl(n) − xl(n))2.
(4)
Thesestandardassumptionsmaybeeasilyreplacedwithin
the context of the path integral, introducing little additional
123
Page 4
158Biol Cybern (2012) 106:155–167
computational challenge using the Monte Carlo approach
utilized below.
For these standard assumptions
A0(X, Y) =Rm
2
m,L
?
?
n=0,l=1
m−1,D
(yl(n) − xl(n))2
+1
2
n=0,a=1
Rf a(ga(x(n + 1), x(n), p))2
−log[P(x(0)].
(5)
Further we take the initial distribution of states to reflect
total ignorance of the initial state: P(x(0)) is a uniform dis-
tribution. It then factors out of all calculations such as those
in Eq. (1). Note that A0(X, Y) is not Gaussian in the state
variables as the function f(x, p) is nonlinear in the x.
The expected value of a state variable comes from
G(X)=X, and the marginal distribution of xc(n); Pc,n(z)
arises from G(X) = δ(z − xc(n)). From quantities such as
this we can answer important sets of questions in the data
assimilation process.
2.3 Monte Carlo evaluation
Using our formulation of the conditional probability den-
sity P(X|Y) for a path through the state space, we would
like to evaluate approximations to quantities such as means,
covariances about those means, and marginal distributions
of parameters or state variables. These can be used to make
estimates and predictions of quantities in the model, consis-
tent with the observed data. Each of these quantities can be
written as path integrals of the form given in Eq. (1), where
G(X) is chosen to be some interesting function of the path.
Thenumericalchallengeistoevaluatethesepathintegrals.
One approach is to seek a stationary path where
∂A0(X, Y)
∂X
and we have explored this in Toth et al. (2011) when the
dynamics is deterministic, namely Rf → ∞.
In working with data from experiments, the model is
an incomplete representation of the underlying processes,
and the deterministic variational method, where Rf → ∞,
imposes an equality constraint in the optimization that is
likely to be too strong. The minimization of the action in
Eq. (6) should be better as it embodies the notion of model
errors.Inpractice,onecanconsiderusingEq.(6)toselectan
initial path from which to begin the Monte Carlo procedure
we outline here. Indeed, that is how we selected an initial
guess for the path in the iterative algorithm described below.
An alternative to the variational approach is to generate
a series of paths {X(1),...,X(Npaths)} that are distributed in
(m + 1)D dimensional path space according to P(X|Y) ∝
= 0,
(6)
exp[−A0(X, Y)].Wecanusethesepathstoapproximatethe
distribution with
P(X|Y) ≈
1
Npaths
Npaths
?
j=1
δ(X − X(j)).
An estimate of the expected value of a function ?G(X)?
on the path follows as
?G(X)? =?dXG(X)P(X|Y)
≈
j=1
These paths can be thought of as representing the many
possible time evolutions of the system state from an ensem-
bleofinitialconditions.Theyeachevolvethroughpathspace
according to the dynamics embodied in the transition prob-
abilities entering the action A0(X) (we drop the explicit
reference to the observations Y now). The parameters are
considered as constants with respect to the dynamics, i.e.,
dp(t)/dt = 0, and the distribution that is obtained reflects
their influence on the state vector x(n) through the model
equations f(x(n), p).
We require a method which will produce paths distrib-
uted according to exp[−A0(X)]. There are several path
integral Monte Carlo (PIMC) methods, such as Metropo-
lis–Hastings or Hybrid Monte Carlo, that are designed to
achieve this (Metropolis et al. 1953; Mackay 2003; Neal
1993). These methods make a biased random walk through
the path (search) space that approaches the desired distribu-
tion.
1
Npaths
Npaths
?
G(X(j)).
(7)
2.4 Metropolis–Hastings Monte Carlo
One of the earliest developed methods for sampling from
a high-dimensional distribution is the Metropolis–Hastings
MonteCarlomethod(Metropolisetal.1953;Hastings1970).
ThisgeneratesasequenceofpathsX(0), X(1),...,X(k)from
a random walk that is biased through an acceptance crite-
rion that depends upon the distribution of interest. It is an
example of a MCMC method because the sequence of paths
maythemselvesbeconsideredasstatesofaMarkovProcess.
A Markov chain consists of a set of accepted paths, the set
being indexed by the iterations k of the procedure.
The Metropolis–Hastings Monte Carlo method works by
generating a new path X(k+1)from the current path X(k)
in two steps. First, a candidate path Xproposedis proposed
by adding an unbiased random displacement to the current
path Xcurrent. The displacement may be to any subset of the
components of Xcurrent; we restrict the distribution of this
perturbation to be symmetric, assuring that the transition
Xcurrent→ Xproposedis as likely as the reverse Xproposed→
Xcurrent, to insure that the chain remains Markov.
123
Page 5
Biol Cybern (2012) 106:155–167 159
Next, the proposed path is either accepted as the next
path in the sequence X(k+1)= Xproposedor it is rejected
so X(k+1)= X(k). The probability for acceptance is
min1, exp
−(A0(Xproposed) − A0(Xcurrent))
This says that proposed changes that lower the action are
accepted, while those that increase the action are accepted
with probability exp[−?A0].
??
??
.
(8)
2.4.1 General procedure
An initial path X(0)is supplied by the user and set to be
the current path at iteration k = 0; the observed time
series Y is data that is loaded in from a file. MCMC
algorithms may take some time to converge to the cor-
rect distribution, and selecting an initial path that is close
to the true solution generally leads to much faster conver-
gence rates. In principle, the Metropolis–Hastings algorithm
will eventually sample from the entire posterior distribu-
tion, but when distributions are multimodal, in practice the
sampling chains will tend to stay in local minima corre-
sponding to the initial guess. Our approach to this prob-
lem, which is common to all MCMC algorithms, is to use
another optimization scheme (Toth et al. 2011) to perform a
coarse search over broad parameter bounds and supply the
result of this procedure as the starting point for the MCMC
chains.
The MCMC calculation proceeds in two distinct phases;
the first “burn-in” phase iterates this initial path guess while
adjusting the size of the random perturbation α so that
the acceptance rate of paths according to the Metropolis–
Hastingscriteriaequationisapproximatelyone-half.Adjust-
ing the step size between iterations violates the Markovian
requirement for symmetric transitions between the subse-
quent paths in the chain; however it also allows the chain to
evolvemoreefficiently.Duetothisviolation,nostatisticsare
gathered during the first phase.
During the second phase we uniformly sample from the
chain to collect the constituent paths of our distribution. It is
therefore important that α be held fixed during this period,
at a value determined during the first phase. Other than that,
the iterations proceed identically during both phases.
A single iteration consists of the following steps:
(1) If k = 0 evaluate A0(X(0)).
(2) If k > 0, propose a change to component(s) of X:
Xproposed
i
i
+ U(−α/2, α/2) · ri.
(3) Evaluate A0(Xproposed) and
?A0= A0(Xproposed) − A0(X(k)).
(4) If U(0, 1) < min[1, exp(−?A0)], then X(k+1)←
Xproposed(accept change).
Otherwise let X(k+1)← X(k)(reject change).
← X(k)
This loop is repeated until k = kfinal, when sufficient sta-
tistics on the paths distributed as exp[−A0(X)] have been
collected.Thesubscripti referstoanycomponentofthepath
X,includingindividualtimestepsandtheglobalparameters.
ri is a scale factor for perturbations to the ith component
of X.
The perturbation step size α is a dimensionless param-
eter between zero and one; it is set to be the same for all
dimensions of X, allowing any inter-dimensional scaling to
be done by the user via the bounds. This is a simplification,
not a requirement of the procedure. In this way, the actual
perturbation to a component of X is given by a random num-
berU(−α/2, α/2)timesthebounds-range(ri),whereU isa
uniformly distributed random value between −α/2 and α/2.
2.4.2 Implementation for parallel architectures
TheMetropolis–HastingsMonteCarlomethodissimpleand
powerful, but it requires many path updates to achieve accu-
rate statistics. One way to deal with this is to take advantage
of parallel computing technology, using a GPU. With GPU
technology it is possible to execute hundreds of threads of
execution simultaneously on a single GPU. Typically each
thread will perform the same operations, but on different
piecesofthedata.Becausethepathsareupdatedsequentially,
the iteration steps cannot be parallelized in their entirety.
However, during each iteration the path is perturbed in all
(m+1)D+K dimensionstogivethecandidatepathXproposed,
and A0(Xproposed) is calculated, both of which can be effi-
ciently parallelized across the dimensions of X.
All MCMC calculations reported here first generated 107
sample paths as a burn-in phase for the Metropolis–Hastings
iterations. These were followed by a statistics collection
phase of 108iterations during which 103accepted paths
were uniformly sampled to create the approximate distri-
bution P(X|Y). The data assimilation window comprised
m + 1 = 4,096 points. The expression of the model error
g(x(n)) is a discrete time version of deterministic equations
for the neuron. We selected a fourth order Runge–Kutta inte-
gration scheme using a time step of ?t = 0.01ms.
When the assimilation procedure was completed, a pre-
diction using a fourth order Runge–Kutta scheme was per-
formed on each of the accepted paths using state variables
x(T) and the parameters associated with that path. The pre-
dicted trajectories were averaged to determine the expected
value of x(t > T) and the standard deviation was evaluated
about this mean. This gives the predicted quantities and their
RMS errors as reported in the figures.
In order to assign values for Rf, the normalized deviation
of the noise was estimated at 1 part in 103for all dimen-
sions of the model error. This normalized deviation was then
scaled by the full range of the state variable and squared to
123
Page 6
160Biol Cybern (2012) 106:155–167
get the variance for that dimension, so for V ∈ [−200, 200],
Rf = 6.25, while for the gating variables, Rf = 106.
We considered an experimental error of ±0.5 mV giving
Rm ≈ 1. We adjusted the Monte Carlo step size using a
scaling factor α to achieve an acceptance rate near 0.5. The
time required to perform each of our reported calculations
with 108candidate paths, each of dimension 16,402 (NaKL)
to 20,498 (NaKLh), took about 10h to complete on a single
NVIDIAGTX470GPU.Inourexperience,providedthatthe
dimension of the problem is roughly constant, the amount of
timeforacalculationscalesroughlylinearlywiththenumber
of GPUs.
In practice, as the Metropolis–Hastings procedure seeks
paths distributed about the maxima of the probability dis-
tribution exp[−A0(X)], a statistical minimization of A0(X)
occurs when paths are accepted and rejected. This makes
it a natural generalization of the procedure used in Toth
et al. (2011) applicable to the situation where there is model
error as well. As emphasized in this paper, the MCMC
approach also results in expected errors for estimations and
predictions.
2.5 Model neurons
2.5.1 NaKL model
The simplest HH model describes the dynamics of the volt-
age V(t) across the membrane of a one compartment neuron
containing two voltage-gated ion channels, Na and K, a pas-
sive leak current, and an electrode through which an external
current Iapp(t)canbeapplied.Themodelconsistsofanequa-
tion for voltage (Johnston and Wu 1995)
dV(t)
dt
=1
+gKn(t)4(EK− V(t))
+gL(EL− V(t)) + IDC+ Iapp(t)?,
= FV(V(t), m(t), h(t), n(t))
C
?
gNam(t)3h(t)(ENa− V(t))
(9)
where the g terms indicate maximal conductances and the E
termsreversalpotentials,foreachoftheNa,K,andleakchan-
nels. IDCis a DC current. Equations for the voltage depen-
dent gating variables ai(t) = {m(t), h(t), n(t)} complete
the model. We refer to this as the NaKL model.
Each gating variable ai(t) = {m(t), h(t), n(t)} satisfies
a first order kinetic equation of the form
dai(t)
dt
=ai0(V(t)) − ai(t)
τi(V(t))
.
(10)
The kinetic terms ai0(V) and τi(V) are taken here in the
form
ai0(V) =1
2
?
1 + tanh(V − va)
?
+ta2
2
dva
?
τi(V) = ta0+ ta1
1 − tanh2(V − vat)
1 + tanh(V − vatt)
dvat
?
?
dvatt
?
.
(11)
Theconstantsva, dva,...areselectedtomatchthefamil-
iar forms for the kinetic coefficients usually given in terms
of sigmoidal functions (1 ± exp((V − V1)/V2))−1. As dis-
cussed in Part I, the tanh forms are numerically the same
over the dynamic range of the neuron models but have better
controlledderivativeswhenonegoesoutofthatrangeduring
the required search procedures. This is less important here
because the MCMC method does not use the derivatives, but
we retain the same form for consistency.
In terms of the formalism of Sect. 2.1, the the model state
variables are the x(t) = {V(t), m(t), h(t), n(t)}, and the
parameters are p = {C, gNa, ENa, gK, EK,...,dvatt}. In a
twin experiment, the data are generated by solving these HH
equations for some initial condition x(0) = {V(0), m(0),
h(0), n(0)} and some choice for the parameters, the DC cur-
rent and Iapp(t), the stimulating current. The measured data
y(t) = h(x(t)) is here just the voltage V(t) produced by
the model plus additive, independent noise at each time step
drawn from a Gaussian distribution with zero mean and var-
iance of 1mV.
2.5.2 NaKLh model
Most neurons express a number of voltage-gated channels
in addition to the sodium and potassium channels directly
responsible for action potential generation (Graham 2002).
These additional channels contribute to bursting, firing rate
adaptation, and other behaviors. As in Toth et al. (2011),
we are interested in whether the data assimilation procedure
described here can be applied to more complex models than
NaKL, and whether it can be used in the context of model
specificationtodeterminewhichchannelsshouldbeincluded
in the model. A model incorporating all the channels likely
tobeinatypicalneuronisbeyondthescopeofthispaper,but
as a first step we extended the NaKL model to include the Ih
current, which has moderately slower kinetics than the other
channels in the model, and is activated by hyperpolarization
(McCormickandPape1990).The Ihcurrentwasrepresented
by an additional term in the HH voltage equation (9),
Ih(t) = ghhc(t)(Eh− V(t)),
as well as an additional equation for the dynamics of the Ih
gating variable:
(12)
dhc(t)
dt
=hc0(V(t)) − hc(t)
τhc(V)
123
Page 7
Biol Cybern (2012) 106:155–167161
hc0(V) =1
2
?
1 + tanh
?(V − vhc)
?(V − vhct)
dvhc
??
τhc(V) = thc0+ thc1tanh
dvhct
?
.
(13)
3 Results
3.1 NaKL model parameters estimated from NaKL data
We begin with an examination of the PIMC data assimila-
tion procedure using the NaKL model. We selected a set of
parameters (Table 1, “Data”) using standard textbook values
for maximal conductances (gNa, gK, gL) and reversal poten-
tials (ENa, EK, EL). The parameters in the kinetic equations
for the gating variables {m(t), h(t), n(t)} came from a fit to
standard expressions for the time constants τi(V) and driv-
ing functions ai0(V) using the hyperbolic tangent functions
in Eq. (11). Choosing appropriate initial conditions, we inte-
grated the NaKL HH equations with an input current Iapp(t)
consisting of a scaled waveform taken from the solution to
a chaotic dynamical system. The amplitude of the waveform
was selected so it depolarized and hyperpolarized the model
neuron, evoking action potentials and traversing the biolog-
ically realistic regions of the neuron’s state space. Based on
our findings in Toth et al. (2011), the frequency of Iapp(t)
was slow relative to the response rate of the neuron.
Table1 ParametervaluesinsimulateddatafromNaKLHHmodeland
estimates from the PIMC algorithm
Parameter‘Data’ EstimateSDUnits
gNa
ENa
gK
EK
gL
EL
vm=vmt
dvm=dvmt
tm0
tm1
vh=vht
dvh=dvht
th0
th1
vn=vnt
dvn=dvnt
tn0
tn1
120.0
55.0
20.0
−77.0
0.3
−54.4
−34.0
34.0
0.01
0.5
−60.0
−19.0
0.2
8.5
−65.0
45.0
0.8
5.0
108.6
55.8
18.2
−78.0
0.316
−55.3mV
−36.5
35.6
0.136
0.407
−62.1
−23.6
0.157
8.3
−64.2
44.6
0.35
4.84
7.3
0.70
1.0
0.67
0.021
1.5
1.5
0.77
0.096
0.087
1.8
2.0
0.071
0.16
1.3
0.40
0.24
0.086
mS/cm2
mV
mS/cm2
mV
mS/cm2
mV
mV
mV
ms
ms
mV
mV
ms
ms
mV
mV
ms
ms
The SD column indicates the standard deviation of the posterior distri-
bution
Toconstruct A0(X, Y)wetookm+1 = 4,096datapoints
with?t = 0.01mswritingg(x(n), x(n+1), p) = x(n+1)−
?tf(x(n), p) where f(x(n), p) is represented as an explicit
4th-order Runge–Kutta integration scheme. Using the meth-
ods described in Sect. 2.3, we evaluated expected values for
the state variables ?xa(n)? and parameters through the obser-
vation period, and also evaluated second moments to yield
standard deviations about these expected values. The dimen-
sion of the integral we are approximating is 4 (4,095)+18=
16,398.
Using the model with these parameters and state vari-
ables at T as initial conditions, we predict forward from
the end of the observation/assimilation time interval, also
using 4th-order Runge–Kutta integration. In the data assim-
ilation procedure we have many accepted paths distributed
as exp[−A0(X)]. We predict into t > T using x(T) for each
path and the parameters estimated with that accepted path.
This permits us to calculate a mean and standard deviation
for each state variable as the system continues to evolve.
Figure 1a shows the estimated membrane voltage (red
dots) ± standard deviation (green band) overlaid on the true
voltage(blackline)fordatageneratedwiththeNaKLmodel.
The data assimilation window consists of the time points
before the vertical blue line. Time points after the blue line
compare the predicted response of the model to the true volt-
age after time T. Figure 1b compares the estimated and
predicted values for the Na+activation variable (m(t)), an
unobserved state variable, with the known values.
The accuracy with which the path integral estimates track
the observed and unobserved states through the observa-
tion window is clear in Fig. 1. In the prediction window
the expected voltages and m(t) give quite good values for
the times when spikes occur, indicating that the Na and K
currents are well represented when an action potential is
generated. The dynamical response in the regions of hyper-
polarized response is less accurate, though all estimates lie
within one standard deviation of the expected value. As in
Toth et al. (2011) we could take this as a signal that the stim-
ulus current did not spend enough timein thehyperpolarized
voltage region to stimulate the dynamics there very well.
Table 1 compares the parameter values used to simulate
the data from the NaKL model with the estimates (± stan-
dard deviation) made from the noisy voltage alone. For all
buttm0andtn0theexpectedvaluesarealmostidenticaltothe
true values.
3.1.1 Bias in the conditional expected values arising from
model error
Itisnoteworthythatwhiletheposteriorerrorsaresmallinthe
estimates in Table 1 (and later tables of parameter estimates)
the expected or mean value appears to be biased away from
the known value. This bias comes from our procedure and
123
Page 8
162Biol Cybern (2012) 106:155–167
Fig. 1 NaKL model. a Comparison of the membrane voltage and the
estimates and predictions. In the assimilation window (t < 40.95ms,
blueline),theexpectedvalueofthevoltageconditionedontheobserva-
tions (Vest) is shown by red dots, with the standard deviation of this dis-
tribution shown in green. For t ≥ 40.95ms, the red and green symbols
show the mean and standard deviation of the distribution of responses
predicted forward in time using the estimates of the parameters and the
state variables at T = 40.95ms. The membrane voltage is shown as
a black line. b Comparison of the known value of the Na+activation
variable (black trace) and the distribution of the estimates (red±green;
prior to blue line) and predictions from the model (red±green; after
blue line). (Color figure online)
is associated with having model error as part of the action
A0(X).
The distribution of paths exp[−A0(X)] is the solution to
a Fokker–Planck equation of the form
dX(s)
ds
= −∂A0(X(s))
∂X(s)
+
√2 N(0, 1),
(14)
where X(s) is the state space path as a function of “time” s,
and N(0, 1)isGaussianwhilenoisewithzeromeanandvari-
anceunity.AnequivalenttoourMetropolis–HastingsMonte
Carlo procedure is to solve this stochastic differential equa-
tion in (m + 1)D-dimensions, where one can show that as
s → ∞, the distribution of paths is precisely exp[−A0(X)].
The Monte Carlo algorithm is seen as a search for minima of
the action along with accounting for the fluctuations about
the minima. All of this is guided by the observations as they
enter the action.
To demonstrate the point about biases in the estimation,
suppose we had two measurements y1, y2and two model
outputs with the model taken as linear x2= Mx1. Then the
action we would associate with this, including model error,
is
A(x1, x2)=1
2
?
(y1−x1)2+(y2−x2)2+ R(x2−Mx1)2?
,
(15)
and this has its minimum at
x1
=(1+R)y1+RMy2
x2=(1+RM2)y2+RMy1
This clearly show the bias we anticipated. As R → ∞,
we see that the bias remains, but x2 = Mx1is enforced.
If R = 0, however, the minimum is at x1 = y1, x2 = y2,
1+R(1+M2)
1+R(1+M2)
.
(16)
and if the dynamics underlying the data source satisfies y2=
My1, the same holds for the model.
3.2 NaKLh model parameters estimated from NaKLh data
To increase the complexity of the model, an additional volt-
age-gated current was added (Ih; see Sect. 2.5). As in the
previous section, data were simulated from known param-
eters and initial conditions (Table 2, ‘Data’). Based on our
findings in Toth et al. (2011), we used a strong stimulus cur-
rent to ensure that the Ihcurrent was sufficiently activated.
Then, using the noisy voltage output from the simulation,
the PIMC algorithm was used to estimate the parameters and
unobserved states of the model. These estimates are as good
or better than for the simpler NaKL model (Table 2). More-
over, the additional 4,096+8 dimensions added to the inte-
gral did not increase the posterior variance or substantially
decrease the speed of the calculation.
Figure 2 compares the known values for V(t) and the
unobserved gating variable hc(t) of the newly added Ihcur-
rent against the estimates and their posterior error. As with
the NaKL model, the estimates of voltage (A) and the Ihgat-
ingvariable(B)closelyfollowthetruevaluesduringthedata
assimilation window and beyond.
3.3 NaKLh model parameters estimated from NaKL data
A critical consideration in applying this method to experi-
mentally obtained data is model selection. If the goal is to
make inferences about biologically relevant properties such
as the set of channels expressed by an individual neuron,
then we need some method of determining which chan-
nels contribute to a neuron’s electrophysiological behav-
ior and should be included in the model. One approach
123
Page 9
Biol Cybern (2012) 106:155–167163
Table 2 Parameter values in simulated data from NaKLh HH model
and estimates from the PIMC algorithm
Parameter ‘Data’Estimate SDUnits
gNa
ENa
gK
EK
gL
EL
vm=vmt
dvm=dvmt
tm0
tm1
vh=vht
dvh=dvht
th0
th1
vn=vnt
dvn=dvnt
tn0
tn1
gh
Eh
vhc
dvhc
thc0
thc1
vhct
dvhct
120.0
55.0
20.0
−77.0
0.30
−54.4
−34.0
34.0
0.01
0.5
−60.0
−19.0
0.2
8.5
−65.0
45.0
0.8
5.0
1.21
40.0
−75.0
−11.0
0.1
193.5
−80.0
21.0
116.0
55.17
19.40
−77.38
0.2987
−56.8
−33.64
36.58
0.051
0.45
−67.29
−18.9
0.34
6.25
−68.2
44.6
0.97
4.8
1.18
37.8
−77.7
−11.1
0.118
177.24
−81.3
21.4
2.8
0.19
0.20
0.27
0.0098
1.8
1.2
1.15
0.023
0.060
3.2
1.1
0.14
2.5
2.6
0.63
0.12
0.38
0.029
0.65
0.64
0.26
0.043
3.1
0.67
0.26
mS/cm2
mV
mS/cm2
mV
mS/cm2
mV
mV
mV
ms
ms
mV
mV
ms
ms
mV
mV
ms
ms
mS/cm2
mV
mV
mV
ms
ms
mV
mV
The SD column indicates the standard deviation of the posterior distri-
bution
is to start with a relatively complex model that includes
all of the currents likely to be in the neuron of inter-
est (possibly using genetic expression data to restrict the
list of candidates). All such currents would have the HH
form
Icurrent(t) = gcurrent(mc(t))q(hc(t))p(Ecurrent− V(t)), (17)
where mc(t) is a state variable associated with transitions
to the open state, hc(t) is a state variable associated with
transitions to closed states, and p and q are integers. Differ-
ential equations for the kinetics of the state variables would
need to be specified as well. If this term is included in the
model but gcurrentis not distinguishable from zero, or if the
other parameters are nonsensical, then we could infer that
the current does not contribute appreciably to the neuron’s
behavior.
Using the variational approximation introduced in Part I,
wefoundthatifweestimatedparametersofanNaKLhmodel
fromdatathatweregeneratedfromanNaKLmodel,theesti-
mated conductance for the missing Ihcurrent was extremely
close to zero, supporting this approach. We repeated this
experiment using the numerical approximation to the full
integral and obtained the same result. NaKL data were simu-
lated by using an NaKLh model in which ghwas fixed at
zero. Parameters associated with the Ih current were left
at their estimated values, though they had no effect on the
voltage behavior of the neuron. The responses of this model
wereidenticaltotheNaKLmodelinwhichno Ihcurrentwas
specified.Table3comparesvaluesfortheparametersusedto
simulate the data with the values estimated using the PIMC
method. The maximal conductance ghis small, indicating
that Ihdoes not contribute to the responses of the neuron.
We note that the estimate of ghin Toth et al. (2011) is much
smaller, but in that experiment we did not include observa-
tional noise. Reducing the observational noise in this exper-
iment leads to a corresponding decrease in the ghestimate
(not shown).
As important as the absolute value of the estimated con-
ductance is the posterior error associated with the estimate,
which we are able to estimate using the PIMC method.
Fig. 2 NaKLh model. a Comparison of the membrane voltage with
estimates during the assimilation window (t < 40.95ms) and predic-
tions from the model (t ≥ 40.95ms). As in Fig. 1, the known values
are indicated by the black trace, and the mean and standard deviation
of the posterior distributions are indicated by red dots and green bars,
respectively. b Comparison of the known values for the Ihactivation
gatingvariablehc(t)withtheestimatesandpredictionsfromthemodel.
(Color figure online)
123
Page 10
164Biol Cybern (2012) 106:155–167
Table 3 Parameter values in simulated data from NaKLh HH model
and estimates from the PIMC algorithm
Parameter‘Data’ Estimate SDUnits
gNa
ENa
gK
EK
gL
EL
vm=vmt
dvm=dvmt
tm0
tm1
vh=vht
dvh=dvht
th0
th1
vn=vnt
dvn=dvnt
tn0
tn1
gh
Eh
vhc
dvhc
thc0
thc1
vhct
dvhct
120
55.0
20.0
−77.0
0.30
−54.4
−34.0
34.0
0.01
0.5
−60.0
−19.0
0.2
8.5
−65.0
45.0
0.8
5.0
114.1
55.2
19.9
−77.1
0.292
−55.3
−31.1
34.0
0.071
0.58
−52.2
−20.1
1.1
8.8
−63.9
43.8
0.88
5.3
2.2
0.12
0.19
0.33
0.0059
0.51
1.7
1.2
0.030
0.025
2.7
0.66
0.42
2.3
1.2
1.5
0.083
0.15
mS/cm2
mV
mS/cm2
mV
mS/cm2
mV
mV
mV
ms
ms
mV
mV
ms
ms
mV
mV
ms
ms
mS/cm2
mV
mV
mV
ms
ms
mV
mV
0.00.019
−27.0
7.4
−43.5
2.8
207.6
−99.6
56.0
0.015
1.3
1.1
0.55
0.057
1.9
0.27
0.42
−40.0
−75.0
−11.0
0.1
193.5
−80
21.0
gh= 0.0 in the simulated data, indicating the absence of this channel
In contrast to the other parameters where the posterior error
is a small fraction of the expected value, the error for ghis
nearlyaslargeastheexpectedvalue.Theestimatesforthe Ih
kineticsarewrong,butcanbeignoredbecause gh≈ 0.Ifthe
current does not contribute to the data, estimates of its prop-
erties are unreliable. The idea that we could build a “large”
model of all neurons and use the data to prune off currents
that are absent is plausible and supported by this calculation.
Whetherthisoptimisticviewpointwillpersistasweconfront
laboratory data with our methods is yet to be seen.
Figure3comparesthemeasuredvoltageandK+activation
variable against the estimated values during the observation
window (0–40.95ms). The expected values follow the true
values closely, with small posterior errors, indicating that
the presence of the Ihcurrent in the estimation model does
not negatively impact the estimation procedure even though
this current is not present in the data. More importantly, the
predicted voltage obtained by integrating forward using the
estimated parameters and state variables at T = 40.95ms
closely follows the true voltage. The quality of the predic-
tion would support the inference, based on the voltage data
alone,that Ihdoesnotcontributesubstantiallytothebehavior
of this neuron.
3.4 NaKL model parameters estimated from NaKLh data
The complementary approach to the model selection prob-
lem is to build up from a simple model, adding complexity
to address aspects of the data that are not fit well. To address
the feasibility of this approach, we used data simulated from
the NaKLh model to fit parameters from the NaKL model,
which is missing key parameters and state variables. Figure
4ashowsacomparisonoftheestimatedandpredictedvoltage
from the NaKL model with the true values from the NaKLh
model. In the observation window there is no indication that
thereisanythingwrongwiththemodel,butthedramaticfail-
uretopredicttheresponseoftheneuronaftertheobservation
window leaves no doubt that something is not consistent.
Although one cannot do this easily in a laboratory experi-
ment, in a twin experiment we are able to examine how well
Fig. 3 NaKLh model parameters estimated from NaKL data. a Com-
parison of the membrane voltage with estimatesduring the assimilation
window(t < 40.95ms)andpredictionsfromthemodel(t ≥ 40.95ms).
This panel is almost identical to Fig. 2a, except that the Ihcurrent does
not contribute to the simulated data. b Comparison of the known val-
ues for the K+inactivation gating variable n(t) with the estimates and
predictions from the model
123
Page 11
Biol Cybern (2012) 106:155–167165
Fig. 4 NaKL model parameters estimated from NaKLh data. a Com-
parison of the known membrane voltage with estimates during the
assimilation window (t < 40.95ms) and predictions from the model
(t ≥ 40.95ms). This panel is almost identical to Fig. 2a, except that
model does not include the Ihcurrent that is present in the simulated
data. b Comparison of the known values for the Na+inactivation gating
variable h(t) with the estimates and predictions from the model
the estimations behave when estimating the gating variables.
Figure4bshowstheestimatedNa+inactivationvariable h(t)
and its known value from our knowledge of the full state of
the observed neuron model. It is clear that the estimates of
the unobserved variable during the observation window fails
to reflect the underlying system. One interpretation is that
because the model is too simple to represent the actual sys-
tem, the estimated parameters and state variables are driven
to unrealistic values, which is revealed when the model is
used to predict novel data. Although the failure of the NaKL
modeltocapturetheNaKLhbehaviordoesnotgiveanydirect
indicationofwhatismissing,itdoesprovideaclearbasisfor
comparing families of models to determine which one best
represents the underlying process.
4 Discussion
The equations describing the dynamics of a broad range of
voltage-gated ion channels and their effect on the membrane
voltage of neurons have been known for many years, but due
to the large number of channels expressed in nature and the
overall nonlinearity of neuronal systems it has not been pos-
sible to use recordings of membrane voltage alone to deter-
mine what channels are present and their kinetic parameters.
The problem is a general one of finding the paths through
a model state space that are consistent with observations of
some subset of the state variables (here, voltage), as well as
with the internal dynamics of the model.
When the measurements are noisy, the model has errors,
and the state of the model is uncertain when observations
commence, this is a problem in statistical physics. We pre-
sentedanexactformulationofthepathintegralthatdescribes
this problem previously (Toth et al. 2011; Abarbanel 2009),
alongwithavariationalapproximationbasedonasaddlepath
estimation. We found that this variational method provided
accurate estimates of channel densities and kinetic parame-
ters when applied to simulated data, and that the estimation
procedure was robust to additive noise as well as errors in
the model specification.
We have extended our earlier work here with effective
numerical approximations to the full path integral, using
a Metropolis–Hastings Monte Carlo technique. The chief
advantage of this approach is that it allows one to find not
only the optimal path through the state space (and the asso-
ciated parameters), but to sample from the joint probability
distribution of the paths and parameters, conditioned on the
data observations. The variance of this posterior distribution
provides valuable information about the degree to which the
observed data constrains the model, and can indicate when
additional data may be needed. It also allows generation
of posterior predictive distributions—the expected behavior
given the model and the observed data—which are useful in
model validation and selection. In other words, the numeri-
calapproximationtothefullintegralnotonlyallowstransfer
of information from the data to the model, but also indicates
how much information was transferred.
Interestingly the Metropolis–Hastings method, as with
other approaches (Andrieu et al. 2010), seeks paths near the
maxima of the distribution exp[−A0(X)] so that it repre-
sents, in effect, a statistical version of the stationary path
method discussed in Toth et al. (2011). It has the computa-
tionaladvantageofnotrequiringanyderivativesoftheaction,
so if the model contains thresholds or “switches” there is no
problem in the PIMC method.
Using this method, we repeated several of the numeri-
cal experiments with simulated data described earlier (Toth
et al. 2011) in order to show that estimates are accurate and
provide good forward predictions for simple HH-type bio-
physical models. The posterior variance of these estimates
was small, indicating that a small amount of data (41ms)
was sufficient to provide a high degree of confidence in the
123
Page 12
166Biol Cybern (2012) 106:155–167
estimates, and furthermore, that the behavior of the models
was strongly dependent on the parameter values.
We also demonstrated, by estimating model states and
parameters with the wrong complement of channels, one too
many or one too few, that we could easily identify these situ-
ations. In the case where the model contained a channel that
was not present in the data, the estimated conductance for
that additional channel was small and the posterior uncer-
tainty was large, allowing the “extra” channel to be pruned
from the model with confidence. When the model contained
too few channels the forward predictions were highly inac-
curate, indicating that the action of the missing channels
plays an important dynamical role. From these observations,
it appears that in dealing with preparations where the full
complementofchannelsisnotknown,itispreferabletostart
with a larger model that includes any channel with a reason-
able probability of being present, which might be informed
by the known biology of the system such as prior pharmaco-
logical experiments or genetic expression patterns, and then
removechannelsforwhichthemaximumconductanceisesti-
mated to be close to zero or has a high error. We recommend
this as a general strategy for selecting good models when
experimental data is used.
We noted, as exhibited by our estimates in Tables 1, 2, 3,
that because the estimation procedure is seeking the minima
of A0(X)alongwiththefluctuationsaboutthoseminima,and
since the action contains terms representing the error in the
models, the expected values of the estimates will be biased,
even when the posterior error about the expected values is
small. This bias is discernable in the twin experiments we
discuss here, but would not be known in the application of
ourmethodstolaboratorydata.Itisimportanttobecognizant
of the bias, however.
From a practical standpoint, the major disadvantage in
usingthefullpathintegralratherthanthevariationalapprox-
imation is that it can require much more computational time.
ThisislargelyamelioratedbytheparallelGPUalgorithmwe
utilized,andthelowcostofusingaGPUforthecomputation.
Furthermore, given that both methods provide similar levels
of accuracy, the variational method can be used for explor-
atoryanalysis,followedbyasubsequentin-depthanalysisof
the full distribution. We have used the variational principle,
whether implemented through IPOPT or via another optimi-
zation routine, as a source of the first guess for a path in the
PIMC method. This appears to provide a much better start-
ing path than a random guess—better, in the sense that the
PIMC method converges in fewer iterations to a distribution
with good expectation values. Other strategies are explored
by Andrieu et al. (2010).
Therehasbeensubstantialinterestinrecentyearsinusing
noisymeasurementsofneuronstoinferbiologicallyrelevant
properties, using a variety of optimization methods (Druck-
mann et al. 2007; Abarbanel 2009; Huys and Paninski 2009;
Lepora et al. 2011). The ability to make such inferences
opens possibilities of using very brief intracellular record-
ings to closely characterize individual neurons, to reveal the
distribution of various biological properties over large pop-
ulations of neurons, or to track changes in these properties
throughlearningorchangesinbehavioralstate.Themethods
we describe here and in Part I have several new advantages.
They provide estimates not only of the maximal conduc-
tances offixed channel types,butalsooftheparameters gov-
erning the gating kinetics of unknown channels. The Monte
Carlo method also provides information about the posterior
uncertaintyintheparameterestimates.Thesefeaturesmaybe
of particular value in analyzing systems where the candidate
channels are not well known.
The experiments we describe here necessarily focused on
relatively simple and generalizable model neurons. Extend-
ing the method to more complex systems will require the
incorporation of additional knowledge about the types of
channels most likely to be present or the anatomy of the neu-
rons.Dendriticdynamicsarenotveryimportantinresponses
toinjectedcurrent,butwillbecomesowhenweconsidersyn-
apticinputsandnetworkdynamics(JohnstonandNarayanan
2008).Twinexperimentsliketheoneswehavepresentedhere
willcontinue toplayanimportantroleinstudying biological
data:nomatterhowcomplexthemodelbecomes,twinexper-
iments can be used to generate simulated data to determine
what conditions are necessary for obtaining a good estimate
oftheparameters,whichinturncanbeusedtooptimizestim-
ulation protocols and other aspects of experimental design.
Acknowledgements
(Grant DE-SC0002349) and the National Science Foundation (Grants
IOS-0905076, IOS-0905030, and PHY-0961153) are gratefully ackno-
wledged. Partial support from the NSF sponsored Center for Theo-
retical Biological Physics is also appreciated. Discussions with Jack
Quinn on GPU computing were very valuable in our numerical work
reported here. He provided us with the GPU computing strategy we
have employed.
Support from the US Department of Energy
References
Abarbanel HD (2009) Effective actionsfor statisticaldataassimilation.
Phys Lett A 373(44):4044–4048
Andrieu C, Doucet A, Holenstein R (2010) Particle Markov chain
Monte Carlo methods. J R Stat Soc B 72(3): 269–342. doi:10.
1111/j.1467-9868.2009.00736.x
Druckmann S, Banitt Y, Gidon A, Schürmann F, Markram H, Segev I
(2007) A novel multiple objective optimization framework for
constraining conductance-based neuron models by experimental
data. Front Neurosci 1(1): 7–18. doi:10.3389/neuro.01.1.1.001.
2007
Evensen G (2009) Data assimilation: the ensemble Kalman filter. 2nd
edn. Springer, Berlin
Graham L (2002) In: . Arbib MA (ed) The handbook for brain theory
and neural networks. 2nd edn. MIT Press, Cambridge pp 164–170
123
Page 13
Biol Cybern (2012) 106:155–167167
Hastings WK (1970) Monte Carlo sampling methods using Markov
chains and their applications. Biometrika 57(1): 97–109. doi:10.
1093/biomet/57.1.97
Huys QJM, Paninski L (2009) Smoothing of, and parameter esti-
mation from, noisy biophysical recordings. PLoS Comput Biol
5(5):e1000379. doi:10.1371/journal.pcbi.1000379
JohnstonD,NarayananR (2008) Activedendrites:colorfulwingsofthe
mysterious butterflies. Trends Neurosci 31(6): 309–316. doi:10.
1016/j.tins.2008.03.004
Johnston D, Wu SMS (1995) Foundations of cellular neurophysiology.
MIT Press, Cambridge
Lepora NF, Overton PG, Gurney K (2011) Efficient fitting of conduc-
tance-basedmodelneuronsfromsomaticcurrentclamp.JComput
Neurosci. doi:10.1007/s10827-011-0331-2
Mackay DJC (2003) Information theory, inference and learning algo-
rithms. Cambridge University Press, Cambridge
McCormick DA, Pape HC (1990) Properties of a hyperpolarization-
activated cation current and its role in rhythmic oscillation in tha-
lamic relay neurones. J Physiol 431(1):291–318
Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E
(1953) Equationofstatecalculationsbyfastcomputingmachines.
J Chem Phys 21: 1087–1092. doi:10.1063/1.1699114
Neal RM (1993) Probabilistic inference using Markov chain Monte
Carlo methods. Tech. Rep. CRG-TR-93-1, University of Toronto
Toth BA, Kostuk M, Meliza CD, Abarbanel HDI, Margoliash D
(2011) Dynamical estimation of neuron and network properties
I: variational methods. Biol Cybern 105(3):217–237
123
Download full-text