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

Brownian 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 conditionalintensity 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 distributionof firing at exactly the observed spike times

the probability of not firing any other spikes in theobservation 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:

AICKSQ-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 forspike 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

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