An Introductory to Statistical Models of Neural Data - IPM

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20 Οκτ 2013 (πριν από 3 χρόνια και 7 μήνες)

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An Introductory
to


Statistical Models
of
Neural Data

SCS
-
IPM

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Spike trains carry information in their temporal patterning, yet they are often


highly irregular across time and across experimental replications.



all
-
or
-

nothing nature
of a sequence
of neuronal action potentials together


with
their stochastic
structure suggests
that a neuronal spike train can
be
viewed
as a point process.



A
point process is
a stochastic


process
composed of a sequence of


binary
events that occur in


continuous
time.



Three
primary ways to characterize a


point process: probability model of


Spike time
,
interspike

intervals
(ISI)


and
counting processes
.

Statistical model for spike trains:


A probabilistic description of the sequence of spikes.

Modeling of analyses of neural system


Deterministic models:



Hodgkin and Huxley



Integrate
-
and
-
fire(IF)



Neural network


-

For
actual neurons,
the deterministic
representation is never completely true as many



factors
which these models
assume are
rarely known with certainty, even in


controlled experiments.


-
In
general, the deterministic models
cannot suggest strategies
or methods
to analyze


the
non
-
deterministic properties of neural spike trains
.



Stochastic models: ???

Stochastic Integrate
-
and
-
fire models (IF)

-

Non
-
leaky
integrator with excitatory Poisson inputs

Poisson process with
constant rate parameter

Magnitude
of
each
excitatory
input

Neuron
discharges
an action
potential when

-

Non
-
leaky integrator with excitatory and inhibitory Poisson inputs

Stochastic integrate
-
and
-
fire (IF)

While the stochastic IF model has a long history (1960’s), it has provided much
insight into the behavior of single neurons and neural populations.

3
approach of the IF model:

-
As a diffusion process

-
As a state
-
space model

-
Via simpler point
-
process models

Leaky stochastic IF model

Brownia
n motion

After each threshold crossing,

is reset to

Membrane
conductance

Input current

“Spike
-
response” model
(Gerstner and
Kistler
,
2002
;
Paninski

et al.,
2004
)

-
By changing the shape and magnitude of
h(.)
, we can model a variety of
interspike

interval behavior
,
refractory period
,
firing rate saturation
or
spike
-
rate adaptation
,

burst effects
in the spike train.


-
It is also natural to consider similar models for the conductance
g(t)

following a
spike time
(Stevens and
Zador
, 1998;
Jolivet

et al., 2004).

-
We would like to model the effects of an external stimulus on the observed
spike train.

: Stimulus

Fitting models to data:

The IF model as a state
-
space model (hidden Markov” models)

This model consists of two processes: an unobserved (“hidden”)
Markovian

process,

and an observed process which is related to the hidden process in a simple instantaneous
manner
(Brown et al., 1998; Smith and Brown, 2003; Czanner et al., 2008;, Salimpour et al., 2011; Shimazaki et al., 2012
)


V (t) is a hidden
Markovian

process which we observe only indirectly, through the
spike times , which may be considered a simple function of V (t): the observed
spike variable at time t is zero if V (t) is below threshold, and one if

Conditional Intensity Function

Since most neural systems have a
history
-
dependent structure
that makes Poisson
models inappropriate, it is necessary to define probability models that account for
history dependence.

Any point process can be completely characterized by its conditional

intensity function

(Daley and
Vere
-
Jones, 2003)

history of the spiking
activity up to time
t

Above equation states that the conditional intensity function multiplied by
gives
the probability of a spike
event in a small time interval

By using this Bernoulli approximation and

joint probability density of observing a spike train

N(T):total number of spikes
observed in the interval

characterizes the distribution

of firing at exactly the observed
spike times

the probability of not firing any
other spikes in the

observation interval

The functional relation between a neuron’s spiking activity and these biological and
behavioral signals is often called the
neuron’s
receptive field
.

Log conditional intensity = stimulus + stimulus history + spiking history + trial + LFP

Ex:

Conditional intensity model for a
hippocampal

place cell

The covariates for this model are
x(t) and y(t), the
animal’s
x
-

and y position.

Generalized Linear Models



Generalized linear models (GLMs) provide a simple, flexible approach to modeling


relationships between spiking data and a set of covariates to which they are associated


(an extension of the Linear regression model)




Generalized Linear Models Are a Flexible Class of Spiking Models That Are Easy to


Fit by Maximum Likelihood


is a collection of covariates that are related to the spiking activity

is a collection of functions of those covariates.

is a parameter set

The goal of a single neuron GLM is to predict the current number of spikes using the
recent spiking history and the preceding stimulus.

represent the vector of preceding stimuli up to
but not including time t.

be a vector of preceding spike counts up to but
not including time t.

distributed according to a Poisson distribution whose conditional intensity

Stimulus
filter

of the neuron
(i.e. receptive
field
),


post
-
spike
filter

to account for
spike history dynamics (e.g.
refractoriness, bursting, etc.)

Link function

Parameter

How can we efficiently fit the model to spike train data?

GLM Network Models

Autoregressive Model (AR)

Criterion Autoregressive Transfer (Minimum Description Length)

p:Model order, N: length of the signal, variance of the
error sequence

Evaluating a statistical Models:

AIC

KS

Q
-
Q Plot

Determining the uncertainty of the parameter estimates

Confidence intervals around the parameter estimates based on the
Fisher information

Examine the confidence intervals computed
foreach

parameter of model based on Fisher
information

Cramér

Rao bound:

Goodness
-

of
-
fit of neural spiking models

Measuring quantitatively the agreement between a proposed model for

spike train data

Time
-
rescaling theorem:



To transform point processes data into continuous measures and then


assess goodness
-
of
-
fit.

Given a point process with conditional intensity function
and occurrence
times

Then these

are independent, exponential random
variables with rate parameter 1.

-
If
is constant
and equal to 1 everywhere, then this is a simple Poisson


process with independent, exponential ISIs, and time does not need to be rescaled.

Define

Because the transformation is
oneto
-

one, any statistical
assessment that measures the agreement between the


values and an exponential
distribution directly evaluates
how well the original model agrees with the spike train
data.

A
Kolmogorov


Smirnov (KS) plot is a
plot of the empirical cumulative
distribution function (CDF) of the
rescaled
zj’s

against an exponential CDF

Another approach to measuring agreement between the model and data is to
construct a
quantile

quantile

(Q
-
Q) plot, which plots the
quantiles

of the
rescaled ISIs against those of exponential distribution

(Q
-
Q) plot

http://www.sfn.org/index.aspx?pagename=ShortCourse3_2008

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