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An Introductory to
Statistical Models of
Neural Data
SCS-IPM
 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.
Statistical model for spike trains:
A probabilistic description of the sequence of spikes.
 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.
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
Magnitude of each
excitatory input
Poisson process with
constant rate parameter
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
Membrane
conductance
After each threshold crossing,
Input current
is reset to
Brownia
n motion
“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
history of the spiking
activity up to time t
Above equation states that the conditional intensity function multiplied by
the probability of a spike event in a small time interval
gives
(Daley and Vere-Jones, 2003)
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 parameter set
is a collection of covariates that are related to the spiking activity
is a collection of functions of those covariates.
The goal of a single neuron GLM is to predict the current number of spikes
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.
using the
distributed according to a Poisson distribution whose conditional intensity
Stimulus filter of the neuron
(i.e. receptive field),
Link function
Parameter
post-spike filter to account for
spike history dynamics (e.g.
refractoriness, bursting, etc.)
How can we efficiently fit the model to spike train data?
GLM Network Models
Evaluating a statistical Models:
AIC
KS
Q-Q Plot
Autoregressive Model (AR)
Criterion Autoregressive Transfer (Minimum Description Length)
p:Model order, N: length of the signal, variance of the
error sequence
Determining the uncertainty of the parameter estimates
Confidence intervals around the parameter estimates based on the Fisher information
Cramér–Rao bound:
Examine the confidence intervals computed
foreach parameter of model based on Fisher
information
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
times
and occurrence
Define
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.
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
(Q-Q) plot
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
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